THERMOELECTRIC COOLING DEVICES: THERMODYNAMIC … · Appendix E Programming Flow Chart of the...

THERMOELECTRIC COOLING DEVICES: THERMODYNAMIC

MODELLING AND THEIR APPLICATION IN ADSORPTION

COOLING CYCLES

ANUTOSH CHAKRABORTY

(B.Sc Eng. (BUET), M.Eng. (NUS))

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2005

i

Acknowledgements

I am deeply grateful to my supervisor, Professor Ng Kim Choon, for giving me the

guidance, insight, encouragement, and independence to pursue a challenging project.

His contributions to this work were so integral that they cannot be described in words

here.

I would like to thank Associate Professor Bidyut Baran Saha of Kyushu University,

Japan, for the encouragement and helpful technical advice.

I am deeply grateful to Mr. Sai Maung Aye for his assistance in the electro-adsorption

chiller experimentation program and Mr. R Sacadeven for kindly assisting in the

procurement of equipment, and construction of the constant-volume-variable-pressure

(CVVP) experimental test facility.

I would like to extend my deepest gratitude to my parents for their complete moral

support. Finally, I wish to thank my wife, Dr. Antara Chakraborty and my son Amitosh

Chakraborty, for being a constant source of mental support.

Last but not least, I wish to express my gratitude for the honor to be co-author with my

supervisor in six international peer-reviewed journal papers, three international peer-

reviewed conference papers and one patent (US Patent no 6434955). I also thank A*

STAR for providing financial assistance to a patent application on the electro-

adsorption chiller: a miniaturized cooling cycle design, fabrication and testing results. I

extend my appreciation to the National University of Singapore for the research

scholarship during the course of candidature, to the Micro-system technology initiative

(MSTI) laboratory for giving me full support in the setting up of the test facility.

ii

Table of Contents

Acknowledgements i

Table of Contents ii

Summary vi

List of Tables viii

List of Figures x

List of Symbols xv

Chapter 1 Introduction 1

Chapter 2 Thermodynamic Framework for Mass, Momentum,

Energy and Entropy Balances in Micro to Macro Control

Volumes 9

2.1 Introduction 9

2.2 General form of balance equations 13

2.2.1 Derivation of the Thermodynamic Framework 13

2.2.2 Mass Balance Equation 16

2.2.3 Momentum Balance Equation 18

2.2.4 Energy Balance Equation 20

2.2.5 Summary of section 2.2 26

2.3 Conservation of entropy 26

2.4 Summary of Chapter 2 30

Chapter 3 Thermodynamic modelling of macro and micro

thermoelectric coolers 32

3.1 Introduction 32

3.2 Literature review 34

3.3 Thermoelectric cooling 37

iii

3.3.1 Energy Balance Analysis 38

3.3.2 Entropy Balance Analysis 41

3.3.3 Temperature-entropy plots of bulk

thermoelectric cooling device 42

3.4 Transient behaviour of thermoelectric cooler 51

3.4.1 Derivation of the T-s relation 54

3.4.2 Results and discussions 55

3.5 Microscopic Analysis: Super-lattice type devices 60

3.5.1 Thermodynamic modelling for thin-film

thermoelectrics 63

3.5.2 Results and discussions 67


Chapter 4 Adsorption Characteristics of Silica gel + water 74

4.1 Characterization of Silica gels 75

4.2 Isotherms of the silica gel + water system 80

4.2.1 Experiment 80

4.2.2 Results and analysis for adsorption isotherms 85


Chapter 5 An electro-adsorption chiller: thermodynamic modelling

and its performance 94

5.1 Introduction 94

5.2 Thermodynamic property fields of

adsorbate-adsorbent system 95

5.2.1 Mass balance 96

5.2.2 Enthalpy energy and entropy balances 96

iv

5.2.3 Specific heat capacity 103

5.2.4 Results and discussion 104

5.3 Thermodynamic modelling of an

electro-adsorption chiller 106

5.3.1 Mathematical model 109

5.3.2 COP of the electro-adsorption chiller 116

5.3.3 Results and discussion 120


Chapter 6 Experimental investigation of an electro-adsorption chiller 128

6.1 Design development and fabrication 128

6.2 Experiments 138

6.3 Results and discussions 143

6.4 Comparison with theoretical modelling 152


Chapter 7 Conclusions and Recommendations 156

References 160

Appendices

Appendix A Gauss Theorem approach 171

Appendix B The Thomson effect in equation (3.7) 180

Appendix C Energy balance of a thermoelectric element 185

Appendix D Programming Flow Chart of the thin film thermoelectric

cooler (Superlattice thermoelectric element) 190

Appendix E Programming Flow Chart of the electro-adsorption chiller

and water properties equations 192

v

Appendix F The energy flow of major components of an electro-

adsorption chiller 201

Appendix G Design of an electro-adsorption chiller (EAC) 207

Appendix H The Transmission band of the fused silica or quartz 215

Appendix I Calibration certificates 216

vi

Summary

This thesis presents a thermodynamic framework, which is developed from the basic

Boltzmann Transport Equation (BTE), for the mass, momentum and energy balances

that are applicable to solid state cooling devices and electro-adsorption chiller.

Combining with the concept of Gibbs law, the thermodynamic approach has been

extended to give the entropy flux and the entropy generation analyses which are crucial

to quantify the impacts of various dissipative mechanisms or “bottlenecks” on the solid

state cooler’s efficiency.

The thesis examines the temperature-entropy (T-s) formulation, which successfully

depicts the energy input and energy dissipation within a thermoelectric cooler and a

pulsed thermoelectric cooler by distinguishing the areas under process paths. The

Thomson heat effect, which has been omitted in literature, is now incorporated in the

present analysis. The simulation shows that the total energy dissipation from the

Thomson effect is about 5-6% at the cold junction. On a micro-scale superlattice

thermoelement level, the BTE approach to thermodynamic framework enables the

temperature-entropy flux formulation to be developed. As the physical scale

diminishes, the collision effects of electrons, holes and phonons become significant

and such effects are accounted as entropy generation sources and the corresponding

energy dissipation due to collision effects is mapped using the T-s diagram.

Extending the BTE to a miniaturized adsorption chiller such as the electro-adsorption

chiller (EAC), the adsorbent (silica gel) properties and the isotherm characteristics of

silica gel-water systems are first investigated experimentally before a full-scale

simulation could be performed. The thermodynamic property fields of adsorbate-

adsorbent systems such as the internal energy, enthalpy and entropy as a function of

pressure (P), temperature (T) and the amount of adsorbate (q) have been developed and

vii

the formulation of the specific heat capacity of adsorbate-adsorbent system is proposed

and verified with available experimental data in the literature. Assuming local

thermodynamic equilibrium, the proposed thermodynamic framework is applied

successfully to model the electro-adsorption chiller. A parametric study of the EAC is

performed to locate its optimal operating conditions.

Based on the electro-adsorption chiller modelling, an experimental investigation is

performed to verify its performances at the optimal and rating conditions. The bench-

scale prototype is the first-ever experimentally built EAC where the dimensions are

based on the earlier simulations. To provide uniform heat flux and the necessary power

level to emulate heat input similar to heat generation of computer processors or CPUs,

infra-red heaters are designed to operate through a four-sided and tapered

kaleidoscope. The optimum COP of the EAC has been measured to be 0.78 at a heat

flux of about 5 W/cm2, and the load surface temperature is maintained below or just

above the ambient temperature, a region that could never be achieved by forced-

convective fan cooling. The experiments and the predictions from mathematical model

agree well.

The performance investigation of EAC’s evaporator provided a successful study of the

pool boiling heat flux. Such pool boiling data, typically at 1.8-2.2 kPa, are not

available in the literature. Using the water properties at the working pressure and the

measured boiling data, a novel and yet accurate boiling correlation for the copper-foam

cladded evaporator has been achieved.

viii

List of Tables Chapter 2 Table 2.1 Examples of the source term, Φ, for two common applications. 20 Table 2.2 Explanation of energy source in different fields. 26

Table 2.3 Example of processes leading to the irreversible production of entropy. 29 Table 2.4 Summary of the conservation equations. 31 Chapter 3 Table 3.1 Physical parameters of a single thermoelectric couple. 43 Table 3.2 Description of close loop a-b-c-d-a (Figure 3.3). 47 Table 3.3 Physical parameters of a pulsed thermoelectric cooler. 60 Table 3.4 the balance equations. 64 Table 3.5 the energy, entropy flux and entropy generation equations of the well and the

barrier. 65 Table 3.6 the thermo-physical properties of bulk SiGe and superlattice Si.8Ge.2/Si

thermoelectric element. 67 Table 3.7 Performance analysis of the superlattice thermoelectric cooler at various

electric field. 71 Chapter 4 Table 4.1 Characterization data for the type ‘RD’ and ‘A’ silica gel. 77 Table 4.2. Thermophysical properties of silica gels. 79 Table 4.3. The measured isotherm data for type ‘A’ silica gel. 86

Table 4.4. The measured isotherm data for type ‘RD’ silica gel. 87

Table 4.5. Correlation coefficients for the two grades of Fuji Davison silica gel + water systems (The error quoted refers to the 95% confidence interval of the least square regression of the experimental data). 92

ix

Chapter 5 Table5.1 Specifications of component and material properties used in the simulation

code. 121 Table 5.2 Energy balance schedule of the electro-adsorption chiller (EAC). 127 Chapter 6 Table 6.1 Energy utilization schedule of an Electro-Adsorption Chiller (Refer to Figure

6.6). 142 Table 6.2 Heat flux calibration table (refer to Figure 6.5). 144 Table 6.3 The heat flux density results from measurements and ray-tracing simulation

(the source have a total power of 3.8 KW, refer to Figure 6.7). 145 Table 6.4 Evaporator temperature and cycle average COP (Experimental and simulated

values) 154

x

List of Figures Chapter 1 Figure 1.1 The dependence of the chip surface temperature with the power intensity of

CPU. 2 Chapter 2 Figure 2.1 Schematic diagram of a thermal transport model showing the applicability

of Boltzmann Transport Equation (BTE) in the thin films. 10 Figure 2.2: The motion of a particle specified by the particle position r and its velocity

v''. 14 Chapter 3 Figure 3.1 The schematic view of a thermoelectric cooler. 37 Figure 3.2 T-α diagram of a thermoelectric pair. 45 Figure 3.3 Temperature-entropy flux diagrams for the thermoelectric cooler at

maximum coefficient of performance identifies the principal energy flows. 46 Figure 3.4 Characteristics performance curve of the thermoelectric cooler. COP is

plotted against entropy flux for the both N and P legs. 49 Figure 3.5 Temperature-entropy flux diagram for the thermoelectric pair at the

optimum current flow by the present study and the other researcher (Chua et al., 2002(b)). 49

Figure 3.6 Temperature-entropy generation diagram of a thermoelectric cooler at

optimum cooling power. Area (abcd) is the total heat dissipation along the thermoelectric element, where area (aefd) is the irreversibility due to heat conduction and area (aghd) indicates the entropy generation due to joule effect. 50

Figure 3.7 Schematic diagram of the first transient thermoelectric cooler. (a) indicates

the normal operation and cold substrate is out of contact, (b) defines the thermoelectric pulse operation and cold substrate is in contact with the cold junction, (c) shows the current profile during normal thermoelectric operation, (d) indicates that the pulse current is applied to the device and (e) the cycle of (tno + tp) is repeated such that the device is operated in an ac mode. 53

Figure 3.8 the cold reservoir temperature of a super-cooling thermoelectric cooler as a

function of time. The small loop indicates the super-cooling during current pulse operation. Temperature drop between the cold and hot junctions

xi

subjected to a step current pulse of magnitude 3 times the DC current and of width 4 s. A indicates the beginning of pulse period, B defines the maximum temperature drop and C represents the end of pulse period, C’ is the beginning of non-contact period, B’ indicates maximum temperature rise due to residual heat. 56

Figure 3.9 T-s diagram of a pulsed thermoelectric cooler during non-pulse operation;

C′ defines the beginning of non-contact, B′ indicates the maximum temperature rise of cold junction during non-pulse period and A′ represents the maximum temperature drop of cold junction with no cooling power. 58

Figure 3.10 Temperature-entropy diagram of a pulsed thermoelectric cooler during

pulsed current operation; A defines the beginning of cooling, B represents the maximum temperature drop and C indicates the maximum cooling power. 59

Figure 3.11 A schematic of the superlattice thermoelectric element (Elsner et al., 1996) 61 Figure 3.12 Schematic diagram of a superlattice thermoelement with electric

conducting wells and electric insulating barriers (Antonyuk et al., 2001). 66 Figure 3.13 the variations of electron temperature in a thin film thermoelectric element

at different electric field; (---) indicates the phonon temperature, (—) defines the electron temperature at 12.5 Kamp/cm2, (---) shows the electron temperature at 25 Kamp/cm2 and () represents the electron temperature at 125 Kamp/cm2. 68

Figure 3.14 The temperature-entropy diagram of a superlattice thermoelectric couple

(both the p and n-type thermoelectric element are shown here). 69 Figure 3.15 the temperature-entropy diagrams of p and n-type thin film thermoelectric

elements. The close loop A-B-C-D shows the T-s diagram when the collision terms are taken into account and the close loop A′-B′-C′-D′ indicates the T-s diagram when collisions are not included. Energy dissipation due to collisions is shown here. 72

Chapter 4 Figure 4.1. Pore size distribution (DFT-Slit model) for two types of silica gel. 78 Figure 4.2. Schematic diagram of the constant volume variable pressure test facility:

T1, T2, T3 = resistance temperature detectors; Ps, Pd, Pe = capacitance manometers. 81

Figure 4.3 The experimental set-up of the constant volume variable pressure (C.V.V.P) test facility. 83

xii

Figure 4.4: Isotherm data for water vapor onto the type ‘RD’ silica gel; for

experimental data points with: () T = 303 K; () T = 308 K; () T = 313 K; () T = 323 K; (+) T = 338 K, for computed data points with solid lines for the Tóth’s equation, for manufacturer’s data points with: (--------) T = 323 K of NACC; 2 (⎯ − − ⎯) T = 338 K of NACC,2 and for one experimental data point with: () T = 298 K for the estimation of monolayer saturation uptake. 89

Figure 4.5: Isotherm data for water vapor onto the type ‘A’ silica gel; for experimental

data points with: () T = 304 K; () T = 310 K; () T = 316 K; () T = 323 K; (+) T = 338 K, for computed data points with solid lines for the Tóth’s equation, and for one experimental data point with: () T =298 K for the estimation of monolayer saturation uptake. 90

Figure 4.6. Isotherm data for water vapor with the silica gel by the present study and

other researchers with: (⎯⎯) type ‘A’ by the present study; (⎯⎯) type ‘A’ by Chihara and Suzuki; (⎯∆⎯) type ‘RD’ by the present study; () type ‘RD’ by Cho and Kim; (-------) type RD by NACC. 92

Chapter 5

Figure 5.1: Different specific heat capacities of a single component type 125 silica gel + water system at 298 K: a data point (---) indicates refers to the liquid adsorbate specific heat capacity of the adsorption system, (----) defines the gas adsorbate specific heat capacity of adsorbent + adsorbate system, () refers to the newly interpreted specific heat capacity with average isosteric heat of adsorption 2560 KJ kg-1, () the specific heat capacity of type 125 silica gel + water system as obtained through DSC and () the specific heat capacity of the adsorption system with the THS technique. 105

Figure 5.2 Schematic showing the principal components and energy flow of the electro-adsorption chiller. 108

Figure 5.3 A block diagram to highlight the sensible and latent heat flow and the

energy balance of a thermoelectrically driven adsorption chiller. 119 Figure 5.4. Temporal history of major components of the Electro-Adsorption Chiller. 123 Figure 5.5 Effects of cycle time on chiller cycle average cooling load temperature,

evaporator temperature and COP. The chosen 500 s cycle time, shown by the solid dotted vertical line, is close to, the minimum cooling load temperature. 124

Figure 5.6 Dűhring diagram for 400 s and 700 s cycles. 125

xiii

Figure 5.7 COP, the cycle average load temperature and the evaporator temperature as functions of the cycle average input current to the thermoelectric modules.

126 Chapter 6 Figure 6.1 The test facility of a bench-scale electro-adsorption chiller. 129 Figure 6.2 Assembly of the reactor; (a) the bottom plate (outside), (b) the bottom plate

(inside view), (c) the heat exchanger, (d) the heat exchanger with bottom plate, (e) the PTFE enclosure and (f) two reactors. 131

Figure 6.3 the condenser unit. 132 Figure 6.4 The evaporator unit; (a) shows the body of the evaporator, (b) indicates the

quartz bottom plate through which infra red heat passes, (c) represents the top part of the evaporator, (d) the IR heater system and (e) is the evaporator. 135

Figure 6.5: Drawing of the kaleidoscope/cone concentrator developed for radiative

heat transfer between the heat source (point 1, four rods of 4 kW total power), and the bottom plate (quartz point 3) of the evaporator (exit at the top). 136

Figure 6.6 Schematic diagram of a prototyped Electro-Adsorption chiller (EAC) to

explain Energy Utilization. 141 Figure 6.7 Tracing 20 rays from the lambertian sources through the kaleidoscope

systems. The heat flux after the source is measured in point 1, then at the exit of the kaleidoscope in point 2, and finally after having passed the fused silica window in point 3. 145

Figure 6.8 The experimentally measured temporal history of the electro-adsorption

chiller at a fixed cooling power of 120 W: () defines the temperature of reactor 1; () indicates the silica gel temperature of reactor 2; () shows the condenser temperature; () represents the load surface (quartz) temperature and () is the evaporator temperature. Hence the switching and cycle time intervals are 100s and 600s, respectively. 146

Figure 6.9 The DC current profile of the thermo-electric cooler for the first half-cycle. 147 Figure 6.10 Effects of cycle time on average load surface (quartz) temperature,

evaporator temperature and cycle average net COP. The constants used in this experiment are the heat flux qevap = 4.7 W/cm2 and the terminal voltage of thermoelectric modules V = 24 volts. 149

Figure 6.11 Experimentally measured load and evaporator temperatures as a function

of heat flux. 150

xiv

Figure 6.12. Heat flux (q flux) versus temperature difference (DT) for water at 1.8 kPa (author’s experiment), 4 kPa (McGillis et al., 1990). and 9 kPa (McGillis et al., 1990). The bottom most data (of 1.8 kPa) is observed to have insignificant boiling, i.e., mainly convective heat transfer by water.

152 Figure 6.13. Experimentally measured temporal history versus the simulated

temperatures of major components of electro-adsorption chiller at a fixed cooling power of 80 W. +++ indicates the simulated load surface temperature using the proposed correlation (Csf = 0.0136, n = 0.33 and m = 0.37) and ×××× shows the simulated load surface temperature using Rohsenow pool boiling correlation (Csf = 0.0132 and n = 0.33).

153

xv

List of Symbols

A area m2

Abed,ads adsorber bed heat transfer area m2

Abed,des desorber bed heat transfer area m2

Acond condenser heat transfer area m2

Aevap Evaporator heat transfer area m2

Ate cross sectional area of a thermoelectric element m2

Atube,ads tube heat transfer area m2

COP the Coefficient of Performance --

COP eac Coefficient of performance of electro-adsorption chiller --

COP net net Coefficient of Performance --

c the mass fraction kg kg-1

cas the proposed specific heat capacity J kg-1 K-1

cgs specific heat capacity at gaseous phase J kg-1 K-1

cls specific heat capacity at liquid phase J kg-1 K-1

cp specific heat capacity J kg-1 K-1

cp,CE specific heat capacity of ceramic J kg-1 K-1

cp,mf specific heat capacity of copper plate J kg-1 K-1

cp,sg specific heat of silica gel J kg-1 K-1

cp,te specific heat capacity of thermoelectric material J kg-1 K-1

cv specific heat capacity at constant volume J kg-1 K-1

DT temperature difference between Quartz surface

and the water vapour temperature at saturation K

Dso a kinetic constant for the silica gel water system

d3 3-D operator

xvi

E electric field volt

Ea activation energy of surface diffusion J kg-1

e the total energy per unit mass J kg-1

F the Faraday constant Coulomb mol-1

F the external force N

f the statistical distribution function

GF the geometric factor m

H the total enthalpy of the system J kg-1

h enthalpy per unit mass J kg-1 K-1

htuin convective heat transfer coefficient (equation 5.33) Wm-2 K-1

I the current amp

J heat flux Wm-2

Js entropy flux W m-2 K-1

JE electric field W m-3

J(=I/A) the current density amp m-2

Jk diffusion flow of component k

Jp the momentum flux N m-3

Js,pn the entropy flux both the p and n leg W m-2 K-1

k the thermal conductivity W m-1 K-1

K thermal conductance W K-1

Ko pre-exponential coefficient kPa-1

kte thermal conductivity of thermoelectric element

(equations 5.28 and 5.31) W m-1 K-1

L length m

Lte characteristics length of thermoelectric element m

xvii

M current pulse magnitude --

Mcond mass of condenser including fins kg

MCE mass of ceramic plate kg

Mevap mass of evaporator including foams kg

Mref,cond mass of condensate in the condenser kg

Mref,evap mass of refrigerant in the evaporator kg

MHX tube weight + fin weight + bottom plate weight kg

Mmf mass of thin copper plate kg

Msg silica gel mass kg

m* the mass of a single molecular particle kg

m mass kg

ma* mass of adsorbate under equilibrium condition kg

mc mass flow rate at the condenser (Figure 5.3) kg s-1

me mass flow rate at the evaporator (Figure 5.3) kg s-1

mw mass flow rate of water (= 0.01 kg s-1) kg s-1

N number

Nte no of thermoelectric couples

n carrier density m-3

nm number of thermoelectric modules

p the momentum vector

P pressure Pa

P1 saturated pressure Pa

ncolliinP , power input with non-collision effect mW

colliinP , power input with collision effect mW

lpr Prandtl number (equation 5.35)

xviii

Q the total heat or energy W or J

Qo the heat transport of electrons/ holes J mol-1

ncolliHQ , heating power at hot side with non-collision effect mW

colliHQ , heating power at hot side with collision effect mW

ncolliLQ , cooling power at cold side with non-collision effect mW

colliLQ , cooling power at hot side with collision effect mW

colliQ power dissipation due to collisions mW

qm the monolayer capacity kg kg-1

q* the adsorbed quantity of adsorbate by the adsorbent

under equilibrium conditions kg kg-1

q fraction of refrigerant adsorbed by the adsorbent kg kg-1

fluxq ′′ Heat flux at the evapotrator Wcm-2

R total resistance ohm

Rp average radius of silica gel particle m

Rg Universal gas constant J kg-1 K-1

r the particle position vector

r′ the new position vector at an infinitesimally time dt

S total entropy J kg-1 K-1

Ste the total Seebeck coefficient V K-1

s entropy per unit mass J kg-1 K-1

T temperature K

Tquartz quartz surface temperature K

Tevap evaporator temperature K

t time s

xix

t1 the töth constant -

tcycle cycle time s

U a peculiar velocity m s-1

U total internal energy of the system J.kg-1

Uads overall heat transfer coefficient of adsorber W m-2 K-1

Ucond condenser heat transfer coefficient W m-2 K-1

Udes Overall heat transfer coefficient of desorber W m-2 K-1

u internal energy per unit mass J kg-1 K-1

V volume of the system m3

V Input voltage to the TE modules volt

v ′′ the particle velocity vector m s-1

v′ the particle velocity vector at an infinitesimally time dt m s-1

v velocity m s-1

υ (= 1/ρ) specific volume m3 kg-1

w energy flow (the combination of thermal and drift energy) J

X any sufficiently smooth vector field

Xtu distance between fins (equation 5.33) m

Y energy dissipation during collision (= ρ e) J m-3

Z thermoelectric Figure of merit K-1

z the total electric charge coulomb

T∆ temperature difference between the hot and

cold junctions K

adsH∆ enthalpy of adsorption kJ kg-1

Φ the source of momentum

Π tensor part of a pressure tensor

xx

π Peltier coefficient V

α the Seebeck coefficient VK-1

Tα thermal expansion factor K-1

β isothermal compressibility factor of the system Pa-1

γ flag which governs condenser transients

δ flag which governs adsorber transients

χ the substantive derive of any variable

ρ the effective density kg m-3

ρte thermoelectric material density kg m-3

τ average collision time fs

Γ the Thomson coefficient VK-1

λte the thermal conductivity Wm-1K-1

λ the phonon wavelength nm

ψ potential energy per unit mass J kg-1

µ the thermodynamic or chemical potential

µl liquid viscosity (equation 5.35) kg/ms

σ the entropy source strength W m-3 K-1

σs Surface tension (equation 5.35) N m-1

σ ′ the electrical conductivity ohm-1 m-1

κ the mass transfer coefficient s-1

θ flag which governs desorber transients

φ the fin efficiency

φ∇ the electric potential gradient V m-1

Subscripts

a adsorbate

xxi

ads adsorption

ads,T adsorption by external cooling and thermoelectrics (Table 5.2)

ads,TL latent heat in the adsorber bed (Table 5.2)

ads,w adsorption by external cooling (Figure 5.3)

ave average temperature

abe adsorbent

b barrier

b,e electrons in the barrier of superlattice

b,h holes in the barrier of superlattice

b,g phonons in the barrier of superlattice

bed,ads adsorption bed

beds, ads, TE bed adsorption by thermoelectric modules

bed,des desorption bed

CE ceramic plate

colli the collision term

colli e-h collision between electrons and holes

colli e-g collision between electrons and phonons

colli h-e collision between holes and electrons

colli h-g collision between holes and phonons

colli g-e collision between phonons and electrons

colli g-h collision between phonons and holes

cond condenser

condL latent heat in the condenser (Table 5.2)

des desorption

des,L latent heat in the desorber bed (Table 5.2)

xxii

dissip dissipation

e electron

evap evaporator

evapL latent heat at the evaporator (Table 5.2)

f liquid phase

fin fin

g phonons or gas

gas gaseous phase

H or hj hot junction

HX heat exchanger

h hole

IR Infrared heater

in inlet

irr irreversible

L or cj cold junction

Load Load surface

l liquid phase

max maximum

mf metal film

min minimum

np non-pulsed period

o outlet

p the pulse period

pr Prediction

q Joulean heat flux

xxiii

quartz quartz (fused silica) surface

r& rate of change of position vector

ref,cond refrigeration (here water) of condenser

ref,evap refrigerant (water vapour) at the evaporator

rev reversible

sys system

TE thermoelectric modules

Tube,ads tube of the adsorption bed

t the total amount

te thermoelectric device

th theoretical

tu tube

v ′′& rate of change of velocity vector

w,e electrons in the well of superlattice

w,h holes in the well of superlattice

w,g phonons in the well of superlattice

w well or water refrigerant

w,i water inlet

w,o water outlet

1

Chapter 1

Introduction

One of the major problems facing the electronics industry is the thermal management

problem where the heat dissipation by conventional fan cooling from a single CPU

(Central Processor Unit) chip has reached a bottleneck situation. With increasing heat

rejection from higher designed clock speeds*, temperatures on the chip surfaces have

reached the thermal design point (TDP) of fan-fins cooling devices, about 73o C. A

single CPU chip containing both power and logic circuits, can no longer sustain the

designed clock speed* because of high thermal dissipation. Consequently, major chip

manufacturers have embarked on two or more processors designed on a single

footprint, distributing the CPU generated heat to a wider area of its casing so as to have

a capability for over-clocking**. A plot of the chip surface temperature with the power

intensity of CPUs is shown in Figure 1.1 (Tomshardware guide, 2005).

A central challenge to the thermal management problem of CPUs is the development

of cooling systems which can handle not only the level of heat dissipation of

computer’s CPU (around 120 W at 3.8 GHz) but they should also have the potential of

being scaled down or miniaturized without being severely bounded by the thermal

bottlenecks of the convective air cooling or boiling.

_____________________________________________________________________

* The clock speed of a CPU is defined as the frequency that a processor executes instructions or that data is processed. This clock speed is measured in millions of cycles per second or megahertz (MHz) and gigahertz (GHz). ** Overclocking is the act of increasing the speed of certain components in a computer other than that specified by the manufacturer. It mainly refers to making the CPU run at a faster rate although it could also refer to making graphics card or other peripherals run faster.

2

A survey of the literature (Phelan et al., 2002) shows that there are two categories of

cooling devices: (i) The passive-type cooling devices in which the load-surface

temperatures are operated well above the ambient temperature and these devices

include forced-convective air cooling, liquid immersion cooling, heat pipe, and

thermo-syphon cooling, and (ii) the active-type cooling device where electricity or

power is consumed. For example, a scale-down vapour-compression refrigeration

system has the capability of lowering the load surface temperature below the ambient.

However, the efficiency of such a vapour-compression type cooler decrease rapidly

when its physical dimensions are down scaled or miniaturized because the dissipative

losses of the refrigeration cycle could not be scaled accordingly. The surface and fluid

friction of the refrigeration cycle increase relatively with reduced volume of devices.

In addition, mechanical devices have high maintenance cost.

Figure 1.1. The dependence of the chip surface temperature with the power intensity of CPU.

50

55

60

65

70

75

20 30 40 50 60 70 80 90 100 110 120

power intensity of CPU (W)

chip

sur

face

tem

pera

ture

(o C)

Heat sink design point

Fan

Heat sink

3

The simplest is forced air convection (Chu, 2004) with the option of an extended heat

sink that effectively increases the heat source surface area for heat exchange and/or the

possibility of introducing ribs or barriers on the surfaces to be cooled to increase air

turbulence so as to realize better heat dissipation. This method is adequate for many

types of current microelectronic cooling applications, but might cease to satisfy the

constraint of compactness for future generations that will require at least an order of

magnitude higher cooling density.

Passive thermo-syphons (Haider et al., 2002) have been proposed. These devices

involve virtually no moving parts except with the possibility of one or more cooling

fans at the condenser. Such a device, however, is highly orientation-dependent as it

relies on gravity to feed condensate from a condenser located at a higher elevation so

as to provide liquid flush back to the evaporator, which is located at a lower elevation.

Thermo-syphons equipped with one or more mini pumps have also been proposed

(Kevin et al., 1999). Instead of relying on gravity, condensate is pumped from the

condenser back to the evaporator. This scheme is orientation-independent and also

allows for the possibilities of forced convective boiling, spraying of condensate or jet-

impingement of condensate at the evaporator, which will effectively enhance boiling

characteristics and therefore cooling performance.

Laid-out heat pipes (Yeh, 1995 and Kim et al., 2003) have found applications

especially in laptop and desktop computers. The evaporating ends of the heat pipes are

judiciously arranged over the CPU while the condensing ends of the same are laid out

so as to effectively increase the surface area of the heat sink.

4

Recently, interest in jet impingement cooling (Lee et al., 1999) of electronics cooling

has been significantly increased. The working fluids can be liquids or gases. In

addition, jet impingement is produced by single or two-phase jets. The advantages are;

direct contact with the hot spot, resulting in a higher heat transfer coefficient, and the

simple structure. In impinging jet cooling, the heat transfer coefficient is found to be

high at the stagnation point only and decreases rapidly away from the stagnation point

with significant temperature gradients over the heating surface.

Mini vapour compression chillers (Roger, 2000) have also found applications. In one

design, the evaporator is arranged over the heat source surface while the mini

condensing unit is positioned away from the heat source. The advantage of such a

system lies in its higher COP. However, many moving parts are involved in the

compressor and they have to be highly reliable. Further scaling down of the

compressor for miniaturized cooling applications may also be a technical challenge,

and this may lead to a sizable loss of compressor efficiency due to high flow leakages

and in turn the low chiller COP.

Thermoelectric chillers (Goldsmid and Douglas, 1954 and Rowe, 1995) are also in use,

but suffer from inherently low COP (typically in the range of 0.1-0.5 for the

temperature ranges characteristic of many microelectronic applications) and high cost.

The low COP means that major increases in cooling density will require unacceptably

high levels of electrical power input and rates of heat rejection to the environment that

will be difficult to satisfy in a compact package, all at increased cost. Thermoelectric

chillers satisfy the requirement of compactness, the absence of moving parts except for

the possibility of one or more cooling fans, and an insensitivity to scale (since energy

5

transfers derive from electron flows). Typically, commercial thermoelectric devices

comprise semiconductors, most commonly Bismuth Telluride. The semiconductor is

doped to produce an excess of electrons in one element (n-type), and a dearth of

electrons in the other element (p-type). Electrical power input drives electrons through

the device. At the cold end, electrons absorb heat as they move from a low energy

level in the p-type semiconductor to a higher energy level in the n-type element. At the

hot side, electrons pass from a high energy level in the n-type element to a lower

energy level in the p-type material, and heat is rejected to a reservoir.

Adsorption chillers (Saha et al., 1995(a), 1995(b), Amar et al., 1996) are also capable

of being miniaturized (Viswanathan et al., 2000) since adsorption of refrigerant into

and desorption of refrigerant from the solid adsorbent are primarily surface, rather than

bulk, processes. A micro-scale reactor consists of micro-channel cavities to house the

adsorbents. The reactor includes a combination of heat exchanger media or adsorbate,

porous adsorbents, and a thin micro-machined contractor media.

The technology of coupling a thermoelectric device (often referred to as a Peltier

device), to an adsorber and a desorber is not new (Edward, 1970 and Bonnissel et al.,

2001). It is typically applied to humidification, dehumidification, gas purification, and

gas detection. In the electro-adsorption chiller (Ng et al., 2002), the two junctions of a

thermoelectric device are separately attached in a thermally conductive but electrically

non-conductive manner to two beds (hot and cold) or reactors. When direct current is

applied to the thermoelectric device, the reactor attached to the cold junction acts as an

adsorber, while the second reactor attached to the hot junction acts as a desorber.

When the direction of flow of direct current through the thermoelectric device is

6

reversed, the original cold junction is switched into a hot junction which in turn also

switches the reactor from an adsorber or absorber to a desorber; concomitantly, the

original hot junction is switched into a cold junction which in turn also switches the

reactor from a desorber to an adsorber or absorber.

This thesis aspires to develop the thermodynamic modelling of solid state cooling

devices namely thermoelectric refrigeration (Rowe, 1995), the pulsed thermoelectric

cooler (Snyder et al., 2002 and Miner et al., 1999), micro thermoelectric refrigeration

using superlattice quantum well structure (Elsner et al., 1996), and electro-adsorption

chiller (Ng et al., 2002) such that their performances can be analyzed and understood.

These modeling equations are derived from the basic Boltzmann Transport Equation

(BTE) to explain both the macroscopic and microscopic thermodynamic concepts.

Based on the Gibbs relationship and the conservation equations, the theoretical

formulations of the temperature-entropy (T-s) for macro and micro scale solid state

semiconductor cooling devices have been developed. These T-s formulations produce

the net energy flow and categorise all types of dissipative mechanisms, which include

(i) the transmission losses, which occur due to input energy sources to the system such

as electrical energy, thermal energy sources, (ii) external losses due to interact with the

environment (iii) the internal losses including friction, mass transfer, internal

regeneration, finite temperature difference heat transfer and (iv) the dissipation due to

collisions of electrons, holes and phonons in solids. This thesis also investigates

experimentally the adsorption isotherms and kinetics to model the electro-adsorption

chiller. The experimental investigation of a bench-scale electro-adsorption chiller is

performed such that the key parameters in evaluating its performance and the energy

7

flow can be understood. The simulations and the experimentally measured results are

compared for verification.

A chapter-wise overview of this report is given below.

Chapter 2 develops the general form of energy balance equations, which are derived

from the Boltzmann Transport Equation (BTE). The balance equations have two terms

“drift” and “collision” and these terms are discussed. This chapter also describes the

conservation of entropy using Gibbs relation and the energy balance equation for

macro and micro-scales and found that the entropy generation in a solid state device

occurs due to irreversible transport processes, collision processes (electrons-holes and

phonons) and free energy changes associated with recombination and generation of

electron-hole pairs are also provided in this chapter.

Based on the conservation laws and the entropy balance equations, Chapter 3 develops

the temperature-entropy formulations of a solid state thermoelectric cooling device, a

pulsed thermoelectric cooler and a micro-scale thermoelectric cooler. The

thermodynamic performances of these solid state coolers are discussed graphically

with T-s diagrams. The mathematical modelling of the thin film superlattice

thermoelement is developed from the basic Boltzmann Transport Equation where the

collision terms are included.

Prior to studying the proposed electro-adsorption chiller, the adsorption characteristics

and isotherms are described briefly in Chapter 4. This chapter starts off with the

experimental measurements of the characterization of two types of silica gel (Fuji

8

Davison type “A” and type “RD”). This chapter also investigates experimentally the

isotherm characteristics of two types of silica gel. The experimental results are

compared with similar experimental data in the literature for verification.

Chapter 5 consists of two subsections. The first sub-section starts with the

development of the full thermodynamic property maps for the adsorbent-adsorbate

system. The property fields express the extensive thermodynamic quantities such as

enthalpy, internal energy and entropy as a function of pressure, temperature and the

amount of adsorbate. In this section, the newly interpreted specific heat capacity of the

adsorbate-adsorbent system has been proposed. Based on the theory of

thermoelectricity, the adsorption characteristics, isotherms, kinetics and the fully

thermodynamic maps of adsorption the second section models the electro-adsorption

chiller. For simplicity, the lumped modelling approach is applied.

Having gone through the analyses of three major chapters (i.e., Chapters 3, 4 and 5),

Chapter 6 investigates a bench-scale electro-adsorption chiller (EAC) experimentally,

where the experimental quantifications are described. The formulations of the EAC

developed in chapter 5 are verified against experimental data. This thesis ends with

conclusions (Chapter 7), where the originality and contributions of the author are

discussed.

9

Chapter 2

Thermodynamic Framework for Mass, Momentum, Energy and

Entropy Balances in Micro to Macro Control Volumes

2.1 Introduction

In the past few decades, significant progress has been made in development of semi-

conductor materials for cooling and heating applications, for example, the

thermoelectric chillers using the Bismuth-Telluride (Bi2Te3), Antimony-Telluride

(Sb2Te3), Silicon-Germanium (SiGe) (Rowe, 1995). With greater needs for

miniaturization of micro or mini-chillers, these semi-conductors are fabricated to

within a few microns thick using the chemical-vapor deposition (CVD) techniques for

making the thin-film materials in layers of the positive-negative or P-N junctions. The

law of equilibrium thermodynamics as well as the conservation laws, which are well

established for analyzing energy and mass transfers of macro control volumes or

systems, may not be sufficient to handle the systems of micro dimensions (Chen, 2001

and 2002), in particular, their inability to handle the dissipation phenomena associated

with high density electrons, holes and phonons collisions or flows and these

dissipations are known to have set real limits to the performance of micro systems.

In this chapter, the Boltzmann Transport Equation (BTE) (Tomizawa, 1993 and

Parrott, 1996) is used to formulate the transport laws for equilibrium and irreversible

thermodynamics. The BTE equations are deemed to be suitable for analyzing micro-

scaled systems because they account for the collisions terms associated with the high

density fluxes of electrons and phonons in the thin films. In Figure 2.1, the relative

merits of applying the classical Gauss theorem (control volume approach), BTE and

10

the molecular dynamics model is elaborated with respect to the length and time scales.

For example, at wavelengths ranging from a phonon (1~2 nm) to its mean free path

(300 nm), the molecular dynamic model is usually employed (Lee, 2005). In regions

higher the mean free path of phonons, the Boltzmann Transport Equation (BTE)

models could be utilized to capture the collision contributions of particles and the BTE

model can be extended into the realm of the Gauss theorem or control volume

approach.

Figure 2.1. Schematic diagram of a thermal transport model showing the applicability of Boltzmann Transport Equation (BTE) in the thin films, where τc is the wave interaction time (100 fs), τ defines the average time between collisions, τr is the relaxation time, λ indicates the phonon wavelength (1~2 nm), Λ is the mean free path (~300 nm) and lr is the distance corresponding to relaxation time. (Lee, 2005). Hence when L ~ λ, wave phenomena exhibit, when L ~ Λ and t >> τ, τr, ballistic transport and there is no local thermal equilibrium, when L >>lr and t ~ τ, τr, statistical transport equations are used, when L >> lr and t >> τ, τr, local thermal equilibrium can be applied over space and time, leading to macroscopic transport laws.

τr

Gauss Theorem Approach

(Control volume)

Boltzmann Transport Equation (BTE)

Molecular Dynamics

λ Λ lr >Λ L >> lr Length Scale

t >>τr

τ

τc

Tim

e Sc

ale

11

In this regard, particular attention is paid to the energy, momentum and mass

conservation properties of the collision operator in sub-micron semiconductor devices.

It is noted that the dissipative effects from the collision terms are translated into unique

expressions of entropy generation where one could analysis systems performance

using simply the classical temperature-entropy (T-s) diagrams that capture the rate of

energy input, the dissipation and the useful effects. Another advantage of using the

BTE approach is that when the systems to be analyzed enters into a macro-scale

domain, the collision terms could simply be omitted (as the effects are known to be

additive) and the conservation laws revert back to that of the classical

thermodynamics, a method similar to the direction taken by the works of de Groot and

Mazur. (1962).

Prior to writing down the BTE formulation below, some aspects of classical

thermodynamic development in the recent years are first reviewed. One such topic that

has been greatly discussed in the literature is the finite time thermodynamics (FTT)

(Novikov II, 1958) which has been applied to problems of isothermal transfers to

chiller cycles called the reversible cycle. In the FTT analysis, the processes of heat

transfer (from and to the heat reservoir) are time dependent but all other processes (not

involving heat transfers) are assumed reversible. Take for example, an endo-reversible

chiller that has been commonly modelled (Mey and Vos, 1994). Invariably, they

assumed reversibility for the flow processes of its working fluid but irreversible

processes are arbitrary applied and restricted to the heat interactions with the

environment or heat reservoirs. Owing to the arbitrary assumption of reversibility for

the flow processes of working fluid within the chiller cycle, the finite time

12

thermodynamics (FTT) faces an “uncertain treatment” (Moron, 1998) and hence, it

renders itself totally impractical in the real world.

On the other hand, there has been an excellent development in the general argument of

equilibrium and irreversible that applies to macro systems. It is built upon an approach

following the method of de Groot and Mazur (1962). They proposed the theory of

fluctuations that describes non-equilibrium (irreversible) phenomena, based on the

basic reciprocity relations (Onsager, 1931), and relied on the microscopic as well as

drawing phenomenon about macroscopic behavior. Most non-equilibrium

thermodynamics assumes linear processes occurring close to an equilibrium

thermodynamic state and assumes that the phenomenological coefficients are constant

(Denton, 2002).

For the microscopic and sub-microscopic domains, the transport equations are

developed from the rudiments of Boltzmann Transport Equation (BTE) but

conforming to the First and Second Laws of Thermodynamics. In the sections to

follow, the main objective is to discuss and formulate the basic conservation laws

(mass, momentum and energy balances) for the thin films and based on the specific

dissipative losses encountered in these layers, the entropy generation with respect to

the electrons, holes and phonons fluxes will be formulated.

The organization of this chapter is as follows: Section 2.2 describes the general form of

conservation equations. The equations to be discussed are the mass, momentum and

energy conservation and the properties of the collision operator in sub-micron

semiconductor devices. In section 2.3, the mathematical rigor of the BTE with respect

13

to irreversible production of entropy is presented. Following the Gibbs expression on

entropy, the entropy balance equation is established in relation with the conservation

equations, which comprise a source term or entropy source strength term. The

concluding section of this chapter tabulates a list of the common entropy generation

mechanisms and demonstrates the manner the terms are deployed in the BTE

formulation. An application example of this demonstration is given in Chapter 3.

2.2 General form of balance equations

In this section, the transport processes involving the mass, momentum, and energy are

described from the basic conservation laws, developed together with the transport of

heat or energy by molecular fluxes such as electrons, holes and phonons within the

micro layers in a system.

2.2.1 Derivation of the Thermodynamic Framework

Figure 2.2 shows a schematic of an ensemble or a group of molecular particles, such as

an electrons, holes or phonons, represented initially at a time t where their positions

and velocities are expressed in the range ( ) ( )vr ′′33 , dd near r and v ′′ , respectively.

The operator, 3d indicates the three dimensional spaces of the variables r and v ′′ .

At an infinitesimally small time dt later, the molecular particles move, under the

presence and influence of an external force ) ,( vrF ′′ , to a new position dtrrr &+=′ and

attaining a velocity dtvvv ′′+′′=′ & . Owing to collisions of molecular particles, the

number of molecules in the range vr ′′33 dd could be also changed where particles or

molecules originally outside the range ( ) ( )vr ′′33 dd can be scattered into the domain

under consideration. Conversely, molecules originally inside the domain range could

14

be scattered out. Following the framework of Reif, F. (1965), the generic form of the

conservation laws is additive and is given by,

4444 84444 764444 84444 764444 84444 76"&)_(_

33

&)(),_(_

"33"

&)(),_(_

33''

''''

)"()(),()()(),,()()(),,(

vratMolecules

colli

vrtatMoleculesvrtatMolecules

ddfddtfddtf vrvrvrvrvrvr '"''' +=

(2.1)

where t’= t + dt, the range ( )dtrrr &+=′ and ( )dtvvv ′′+′′=′ & . ( )tf ,, vr is the statistical

distribution function of an ensemble particle, which varies with time t, particle position

vector r, and velocity vector v.

For convenience, we drop the implied 3-D operator, d3, the equation reduces to a more

readable form, i.e.,

Position

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14 16 18 20

v'

v''

t

t + dt

r'r

Vel

ocity

Figure 2.2. The motion of a particle specified by the particle position r and its velocity v''.

15

( ) ( ) ( )

( ) ( ) ( )tfdttdtdtfdttfistermcollisiontheor

dttftfdttdtdtf

colli

colli

,,,,,,

,,,,,,

vrvvrrvr

vrvrvvrr

′′−+′′+′′+=′′

′′+′′=+′′+′′+

&&

&&

By invoking the partial derivatives, the basic Boltzmann transport equation is now

expressed for a situation that handles the molecular flux over a thin layer of materials,

i.e.,

tf

tf

tf

grearrangin

tf

tv

vf

tr

rf

tf

colli

drift

colli

∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

−=∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

=∂

∂ "

"

and the subscript “drift” refers to the flow of flux through space (r) and convective

velocity (v) and the second term is the effect of collisions with time. It is noted that the

term “drift” is a non-equilibrium transport mechanism that associates with the external

force ) ,( vrF ′′ , i.e.,

dtdf

dtdf

tf

Drift

vv

rr

′′′′∂

∂+

∂∂

=⎟⎠⎞

⎜⎝⎛∂∂ , (2.2)

where r is the position vector, v ′′ is the velocity vector, i.e., dtdr

=′′v , and the rate of

change of v ′′ is *mdt

d Fv=′′

, where m* is the mass of a single molecular particle.

The physical meanings of velocity and electric field are described in the drift term of

equation (2.2) is now fully elaborated as,

pF

rv

vF

rv

∂∂

−∂∂′′=

′′∂∂

−∂∂′′=⎟

⎠⎞

⎜⎝⎛∂∂ fff

mf

tf

Drift *,

16

where the momentum *m vp ′′= , is that associated with a molecular particle.

Substituting back into equation (2.2), the general form of the transport equation with

respect to r, p and t becomes

tfff

tf colli

∂∂

+∂∂

+∂∂′′−=

∂∂

pF

rv . (2.3)

Having formulated the basic form of the Boltzmann transport equation or BTE in

short, and two other functions will be used along the BTE and they are: Firstly, the

carrier density of the molecular flux is now incorporated by defining the carrier density

as the integral of the product between the molecular particles and the change in their

momentum (Tomizawa, 1993), i.e.,

∫= pfdn . (2.4)

Secondly, the substantive derive of any variable, χ, that varies both in time and in

space can be expressed in vector notation as χ∇+∂∂

=tdt

d .v = +∂∂t

(v.∇ ) χ, or simply

can be written as ∇⋅+∂∂

= vtdt

d . The following conservation equations can now be

formulated with the BTE format and they are elaborated here below.

2.2.2 Mass Balance Equation:

For a given space and time domain, the basic Boltzman transport equation (2.3) is now

invoked to give

tfff

tf colli

∂∂

+∂∂

+∂∂′′−=

∂∂

pF

rv

where the partials of f to the momentum and space in 3-dimensions can be represented

by applying the “Del” operator, i.e., rp ∂∂

=∇∂∂

=∇ffffp , . Thus, integrating on both

17

sides of equation (2.3) with respect to the momentum space, and noting the definition

of the carrier density n (equation 2.4), yields:

( )t

ndfntn colli

p ∂∂

+∇+′′−∇=∂∂

∫ pFv (2.5)

Should dp be small in the space, the second term of right hand side of equation (2.5)

can be omitted and equation (2.5) reduces to a familiar expression of the form;

tnn

tn colli

∂∂

+′′−∇=∂∂ )(v , (2.6)

where the collision effects from molecular particles are still retained. Defining the

effective density as nm *=ρ , where m* is the mass of single molecular particle. For

simplicity, assuming the particles move with an average velocity, then vv ′′= , and

rearranging gives the mass conservation as

collitt ∂∂

+−∇=∂∂ ρρρ )(v . (2.7)

The advantage of equation (2.7) is that it is perfectly valid for describing electrons or

phonons flow within submicron semiconductor devices and the collision term takes

them into account. In the case of a macro-scale domain, the mass balance omits the

collision term, i.e.,

).( vρρ−∇=

∂∂

t. (2.8)

The form of equation 2.8 can also be derived using the Gauss theorem within a control

volume method (see Appendix A). For example, the rectangular coordinates

nomenclature for the component velocities for kjiv ˆˆˆ wvu ++= would yield the

continuity equation as

( ) ( ) ( ) 0=∂

∂+

∂∂

+∂

∂+

∂∂

zw

yv

xu

tρρρρ . (2.9)

18

Only when steady state is considered, i.e., 0=∂∂

tρ ; and a case of constant-fluid

density, the following simplified continuity equation reduces to

0=∂∂

+∂∂

+∂∂

zw

yv

xu (2.10)

In vectorial notation, the mass conservation for a macro domain and constant density

situation is simply expressed as

( )

v

v

⋅∇−=

⋅−∇=∂∂

ρρ

ρρ

dtd

t . (2.11)

2.2.3 Momentum Balance Equation

Starting from the basic transport equation, t

ffftf colli

∂∂

+∂∂

+∂∂′′−=

∂∂

pF

rv , and multiply

the momentum, p, on both sides before integrating over the momentum space yields;

( ) ( ) pppFppvppp dt

fdfdfdft

collip ∂

∂+∇+∇′′−=

∂∂

∫∫∫∫ .. (2.12)

Invoking the definition of the carrier density, n. and the left hand term of equation

(2.12) can be expanded as, ( ) ( ) ( )tt

nmt

ndft ∂

∂=

∂∂

=∂

∂=

∂∂∫

vvppp ρ* whilst the first term

on the right hand side of (2.12) is reduced to ( ) ( ) pdfdf Jppvpvp ... ∇=′′∇=∇′′ ∫∫ ,

where pJ is a tensor. This flux density of momentum can further be transformed to

( ) ( ) ( ),.... PvvvvppvJ +∇=′′′′∇=′′∇=∇ ∫ ρρdfp

where according to Reif, F. (1965), the velocity vector ( )Uvv +=′′ consists of a mean

velocity v and a peculiar velocity U, and the pressure tensor is given by,

UUUUP ρρ γα == .

19

The second term on the right hand side of equation (2.12) is given by,

( ) ( ) ΦppFppFpFp =∇+∇=∇ ∫∫∫ dfdfdf ppp ... , known as the source of

momentum, and the last term on the right hand side refers to as the collision integral,

( )collicolli t

dtf

∂∂

=⎟⎠⎞

⎜⎝⎛∂∂

∫vpp ρ .Combining these expressions from above, equation (2.7) is

now summarized in terms of the physical variables ( ),, pvρ as

( ) ( ) ( )collitt ∂

∂+++−∇=

∂∂ vΦvvPv ρρρ . . (2.13)

The vector product vv, is also known as the Dyadic product which is given by a set of

matrix functions with respect to the component velocities. Detailed manipulation of the

Dyadic product can be found in many mathematics texts. In general, two kinds of

external forces may act on a fluid: (i) the body forces which are proportional to the

volume such as the gravitational, centrifugal, magnetic, and/or electric fields, and (ii)

the surface forces which are proportional to area, for example, the static pressure drop

due to viscous stresses. From a microscopic viewpoint, the pressure tensor, P, is a

related to short-range interaction between the particles of system, whereas the body

forces, Φ, pertains to the external forces in the system. Equation (2.13) describes a

balance for the momentum density ( vρ ), the momentum flow, ( )vvP ρ+ with a

convective portion ( )vvρ , and Φ, is a source of momentum. Some examples of the

type of source terms employed in two different application scenarios are outlined in

table 2.1.

For a macro scale flow situation, the collision term is omitted and equation (2.13)

reduces to the conventional form as

20

( ) ( )

ΦPv

ΦvvPv

+⋅−∇=

++−∇=∂

∂

dtd

t

ρ

ρρ .. (2.14)

It is noted that the total pressure tensor can be further divided into a scalar hydrostatic

part p, and a tensor part, Π.

Table 2.1 Examples of the source term, Φ, for two common applications

2.2.4 Energy Balance Equation

Associated with the flow of a carrier density, n, there is a corresponding energy flow,

w, such as the thermal energy and the drift energy term. By multiplying both sides of

the basic BTE, i.e., equation (2.3), by w and integrating over the momentum space

(product of mass and velocity) yields

( ) ( ) ∫∫∫∫ ∂∂

+∇+∇′′−=∂∂ pp.p.p d

tfdfdfdf

tcolli

p wFwvww (2.15)

Expanding the left hand side of equation (2.15), eeww ρ===∂∂∫ *p nmndf

t, where

e is the total energy per unit mass and ue +ψ+= 2

21 v . On the right hand side of

equation (2.15), the spatial gradient (i.e., the first term) can be placed outside the

Field Φ

Fluid mechanics (Bird, 1960)

gρ+∇− τ. , the first term indicates the viscous force per unit volume where τ is the stress tensor. The second term shows the gravitational force.

Electric flow (de Groot, and Mazur, 1962)

Ezρ , where z is the total electric charge per unit mass in the control volume and E is the electric field acting on the conductor.

21

integral to give, ( ) wdfdf Jwvvw .p.p. ∇=′′∇=∇′′ ∫∫ , where wJ represents the energy

flux, defined here as

q

n

kkkw df JJvPvwvJ +++=′′= ∑∫

=1

.ep ψρ , and veρ is a convective term, ( vP. ) is

the mechanical work in the system, ∑=

n

kkk J

1ψ equals to a potential energy flux due to

diffusion, qJ is the heat flow. The second term on the right hand side of equation

(2.15) can be expanded as:

( ) vΦvFpwFw .*

.p. ==∇=∇ ∫∫ mdfdfF pp

ρ

The collision term can also be expressed in terms of the energy dissipated Y, i.e.,

( )collicolli

colli

ttnd

tf

∂∂

=∂

∂=

∂∂

∫Yww p .

Combining the above expressions, the energy balance equation is now summarized

that includes the collision terms as;

( )colli

w tt ∂∂

++−∇=∂

∂ YvΦJe ..ρ (2.16)

To further understand the usage of the energy balance equation (i.e., equation 2.16), a

minor digression is needed to demonstrate the presence of kinetic and potential energy

terms. Multiplying the equation of motion [equation (2.13)] by v, a balance equation

for the kinetic energy can be obtained, i.e.,

( ) ( )[ ] vvPvPvvv

vvPvvvv2

.:.21.2

1

.... 2 Φ+∇+⎟⎠⎞

⎜⎝⎛ +∇−=

∂

⎟⎠⎞

⎜⎝⎛∂

⇒Φ++∇−=∂

∂ ρρ

ρρtt

(2.17)

22

which is commonly known as the mechanical energy term. Note that the vectorial

reduction could be performed on equation 2.14, as given by de Groot, (1962),

( )[ ] ⎟⎠⎞

⎜⎝⎛⋅∇=

∂∂

==⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=⋅⋅∇ ∑∑==

vvvvv 2

1

23

1 21

21,,1, ρρρρ

n

iji

ij

iji

i

vvx

njivvvx

L

and

( )

( ) v

vv

⋅⋅∇−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=∇+⋅⋅∇−

∑ ∑∑∑∑∑

∑∑∑ ∑

= == == =

= == =

PPx

vvx

Pvx

PPx

v

njivx

PvPx

PP

n

j

n

iij

ij

n

i

n

jj

iij

n

i

n

jj

iijij

ij

n

i

n

jj

iij

n

i

n

jjij

i

1 11 11 1

1 11 1,,3,2,1,: L

Similar simplifications can also be made on the rate of change of potential energy

density, as shown below;

( )

( ) ( ) JFv

vv

−+−∇=∂

∂

+−⎟⎠

⎞⎜⎝

⎛+−∇=

∂∂ ∑∑∑∑

= ===

Jt

JFJJt

n

k

r

jjkjk

n

kkk

n

kkk

ψρψρψ

ψψρψρψ

.

.1 111 (2.18)

The above results indicate that should ( )rjn

kkjk L,10

1==∑

=

vψ , this implies that the

potential energy is conserved into the chemical reaction situation. JF=∑=

n

kkk FJ

1 which

is referred to the source term where the particles interact with a force field such as the

gravitational field or the electric charge experienced within an electric field. Adding

equations (2.17) and (2.18), the combined effects become

( ) ( )

vJFvPvvPvvv

JFvvvvPvvv

2

2

.:.21.2

1

..:.21.2

1

2

2

Φ+−∇+⎟⎠⎞

⎜⎝⎛ +++−∇=

∂

⎟⎠⎞

⎜⎝⎛ +∂

−+∇−Φ+∇+⎟⎠⎞

⎜⎝⎛ +−∇=

∂∂

+∂

⎟⎠⎞

⎜⎝⎛∂

Jt

JPtt

ψρψρψρ

ψρψρρψρ

23

From the energy balance equation (2.16) and substituting for these derived functions,

one obtains working form of the energy equation which gives in terms of the internal

energy, u, heat flux, Jq, and the associated source and collision terms;

( ) ( )

( ) ( )colli

q

colliw

tu

tu

tt

∂∂

++∇−+−∇=∂

∂∴

∂∂

+Φ+−∇=∂

∂

YJFvPJv

evJe

:.

..

ρρ

ρρ

(2.19)

It is noted that ρuv+Jq is the energy flow due to convection and thermal energy

transfers which contributes to the internal energy change, while the vectors P:∇v + JF

is deemed as the source term associated with the internal energy.

Further simplification of the energy balance Equation (2.19) leads to its concise form

of having only 4 terms and it is applicable to micro or thin layer flow of electrons or

phonons;

( ) ( )

colliq

colliq

tdtdu

tuu

t

∂∂

++⋅−∇=ρ

∂∂

++∇−+ρ⋅−∇=ρ∂∂

YΨJ

YJFvPJv : (2.20)

It is noted that the last term of the right hand side indicates the rate of change of energy

as a result of collisions of the molecular particles such as electrons or phonons.

Collision occurs by four processes in the micro-sublayer, i.e., (i) the increase in carrier

kinetic energy due to interactions between electrons and holes or phonons, (ii) the heat

dissipation arising from electrons or phonons interaction with the lattice vibrations,

(iii) the heat dissipation by the electrons to the holes and vice versa, and (iv) the heat

generation arising from the recombination and generation of the molecular particles.

Equation 2.20 could be extended to macro analysis by simply dropping the collision

term which has its form known conventionally as the energy balance equation;

24

ΨJ +⋅−∇= qdtduρ (2.21)

Another name for equation (2.20) is the equation of internal energy, as it is known in

some texts (Bird, 1960, de Groot and Mazur, 1962).

At this juncture, it is perhaps appropriate to elaborate these terms in terms of some

common application examples. For example, the source term Ψ, in equilibrium and

irreversible thermodynamics has two parts: (i) One aspect is the reversible rate of the

internal energy attributed by effects of compression and (ii) the other is the irreversible

rate of internal energy contributed by any dissipative effect. For most engineering

applications involving real gasses, the internal energy is expressed in terms of the fluid

temperature and the heat capacity. In such cases, the internal energy u may be

considered as a function of υ and T, and assuming constant specific heats, i.e.,

dTcdTpTpdT

Tududu v

T

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

+−=⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

= υυυ υυ

and the substantial derivative term becomes

dtdTc

TpTp

dtdTc

dtd

TpTp

dtdu

vv ρρυρρυυ

+∇⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

+−=+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

+−= v.

Invoking the energy balance equation (2.21), it becomes

( ) JFvJ +∇⎟⎠⎞

⎜⎝⎛∂∂

−⋅−∇= .V

qv TpT

dtdTcρ (2.22)

Should the gas be assumed ideal, then Tp

Tp

V

=⎟⎠⎞

⎜⎝⎛∂∂

ˆ and the second term of right hand

side equation (2.22) becomes zero, i.e.,

25

JFJ +⋅−∇= qv dtdTcρ . (2.23)

In another example, the internal energy u is considered to be a function of p and T, as

in a real gas. Hence,

dtdp

dtdp

dtdu

dtdhpuh υυυ ++=⇒+= .

Invoking the constant pressure assumption, it can be shown that dtdTc

dtdh

pρρ = .

dtdTc

dtdh

dtdp

dtdu

dtdh

pρρυ=⇒+= (2.24)

From equation (2.24) and substituting for the energy balance equation for dtduρ , it can

be demonstrated that the substantial derivative for the enthalpy, dtdTcpρ , comprise

only two terms, i.e.,

dtdpp

dtdTc

dtdp

dtdu

dtdTc

qp

p

υρρ

υρρρ

++⋅∇−⋅−∇=

+=

JFvJ (2.25)

From the continuity equation (2.11) we show that

v⋅∇= pdtdp υρ

Equation (2.25) is written as,

JFJ +⋅−∇= qp dtdTcρ (2.26)

The right hand side of equation 2.26 comprises (i) the heat flux term, qJ and (ii) the

energy source term JF. Table 2.2 tabulates three scenarios where the source energy

could be elaborated. For example, in the study of adsorption, where many chapters of

the thesis are devoted, the source energy is derived by considering adsorbate mass and

the isosteric heat of adsorption, i.e., adsHq∆& . Other examples in fluid mechanics and

electrically powered semi-conductors are also explained.

26

Table 2.2 Explanation of energy source in different fields.

Field Source of energy JF

Fluid mechanics v∇:τ ; This term indicates the irreversible internal energy increase due to viscous dissipation.

Electrical ρelectJ2; this is the joulean heat.

Chemical / Adsorption adsHq∆& , where q& is the adsorption or desorption rate and adsH∆ is the isosteric enthalpy of adsorption.

It should be emphasized that equations (2.21-2.26) are valid for macro-scale devices

simulation. However, for the analysis of sub-micron devices, the collision terms, which

are provided in equation (2.20), must be included.

2.2.5 Summary of section 2.2

In section 2.2, the conservation laws, i.e., mass, momentum and energy are derived

from the transportation of molecules using the Boltzmann Transport equation. The

thermodynamic framework for describing the conservation laws is applicable to both

the macro-scale and micro-scale systems. This approach differs from the control

volume approach of de Groot and Mazur (1962), Bird, R. B. (1965), etc., where the

Gauss theorem was employed. The latter approach could not embrace the collision

terms, as what has been shown here using the BTE approach. For thin films or micro-

scale systems, the collisions of electrons with electrons, electrons with holes, holes

with holes are deemed to have significant dissipative effects, a source of irreversibility

as quantified in the next section.

2.3 Conservation of entropy

The change of entropy of a system relates not only to the entropy (s) crossing the

boundary between the system and its surroundings, but also to the entropy produced or

generated by processes taking place within the system. Processes inside the system

27

may be either reversible or irreversible (Richard and Fine, 1948). The reversible

process may occur during the transfer of entropy from one portion to another of the

interior without entropy generation. On the other hand the irreversible process

invariably leads to entropy generation inside the system.

The entropy per unit mass is a function of the internal energy u, the specific volume υ

and the mass fractions ck i.e. ),,( kcuss υ= . In equilibrium the total differential of s is

given by the Gibbs relation (de Groot, 1961)

∑=

−+=n

kkk dcpdduTds

1

µυ , where p is the equilibrium pressure and kµ is the

thermodynamic or chemical potential of component k.

∑=

−+=n

k

kk dt

dcdtdp

dtdu

dtdsT

1

µυ (2.27)

From mass balance we get

collitTP

TP

dtd

TP

∂∂

+⋅∇=ρ

ρυρ v

The mass balance equation, equation (2.7), is written as,

colli

kkk

k

tdtd

∂∂

+⋅∇−⋅∇−=ρρρ Jv

where Jk ( )( )vv −= kkρ is the diffusion flow of component k defined with respect to

the ‘barycentric motion’. Defining the component mass fractions ck as

11

== ∑=

n

kk

kk cthatsoc

ρρ ,

and equation (2.7) can be simplified as

28

kk

colli

kkk

k

dtdc

tdtd

J

Jv

⋅−∇=

∂∂

+⋅∇−⋅∇−=

ρ

ρρρ

From equation (2.27) we have,

collicolli

n

k

kkq

n

kkkq

n

k

kk

n

k

kk

tTP

tT

TTT

TT

TT

dtdc

Tdtd

Tp

dtdu

Tdtds

dtdc

dtdp

dtdu

dtdsT

∂∂

+∂∂

+

+⎟⎠⎞

⎜⎝⎛ ∇⋅−∇⋅−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛−

⋅−∇=

−+=

−+=

∑∑

∑

∑

=

=

=

=

ρρ

µµ

ρµυρρρ

µυ

Y

JFJJJJ

1

1111

21

1

1

(2.28)

where Jk ( )( )vv −ρ= kk is the diffusion flow of component k defined with respect to

the ‘barycentric motion’.

From equation (2.28), one observes that the entropy flow is given by two terms:

⎟⎠

⎞⎜⎝

⎛−= ∑

=

n

kkkqs T 1

1 JJJ µ , (2.29)

and the entropy source strength by

collisionleirreversib

collicolli

n

k

kkq tT

PtTTT

TT

TT

σσσ

ρρ

µσ

+=⇒∂∂

+∂∂

++⎟⎠⎞

⎜⎝⎛ ∇⋅−∇⋅−= ∑

=

YJFJJ 11111

2 (2.30)

The expression of entropy generation (equation 2.30) provides the key to irreversible

thermodynamics. The first two terms indicate entropy generation as a result of

irreversible transport processes whilst the third term represents entropy increase due to

source applied to the system. The last two terms on the right hand side entropy

29

increase due to (1) heat transfers from the other systems via collision processes, (2)

free energy changes associated with recombination and generation of electron-hole

pairs.

From (2.28) the general form of entropy balance equation is written as,

σ+∇= sJdtdS . (2.31)

It is noted here that equation (2.31) has a similar form to the expression for total

entropy, i.e.,

irrrev dtdS

dtdS

dtdS

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= , (2.32)

Comparing the first term of the right hand side of both equations, the reversible

entropy flux is ⎥⎦

⎤⎢⎣

⎡∇=⎟

⎠⎞

⎜⎝⎛

srev

JdtdS . . This is the net sum of quantities carried into the

system by transfers of matter, plus the net sum of quantities of entropy carried in by

the transfer of heat. The second term yields ⎥⎦

⎤⎢⎣

⎡=⎟

⎠⎞

⎜⎝⎛ σ

irrdtdS , which is the entropy

generation rate produced by irreversible processes taking place inside the system.

Table 2.3 shows some common examples of the irreversible processes (hence the

internal entropy generation) of common engineering processes.

Table 2.3 Example of processes leading to the irreversible production of entropy.

Entropy generation irrdt

dS⎟⎠⎞

⎜⎝⎛=σ Equations

Dissipation Mechanical energy (due to friction or viscosity)

⎟⎠⎞

⎜⎝⎛

dtdF

T1 (Richard et. al., 1948), where

dtdF is the rate at which work is done

against force or viscous force.

30

Electrical energy (by the passage of electric current through a resistance)

TR2I , where I is the flow of current

through a resistance R in conductor or semiconductor at temperature T.

Irreversible heat flow in a conducting medium

⎟⎠⎞

⎜⎝⎛∇

T1.q . Here ( )TK∇=q is the vector

which expresses the rate of heat flow. The

final expression is 2TTTk

dtdS

irr

∇∇=⎟

⎠⎞

⎜⎝⎛ .

Free expansion of gas without absorbing heat (Richard et. al., 1948)

The approximate expression for the irreversible increase in entropy accompanying the transfer of any particle element of the gas , consisting of dN moles, from pressure P1 to P2 is

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1logPPRdNdSirr

Chemical reaction (Richard et. al., 1948)

dtdx

fBfAfDfcKR

dtdS

ba

dc

pirr

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛

...

...loglog ,

where Kp is equilibrium constant at temperature T, R is the gas constant, fAfB…fcfd…etc., are the actual instantaneous values of the fugacities, a. b, …, c, d, … are the number of moles and

dtdx is a factor.

Collisions (electron and hole transport in submicron semiconductor or thin film devices)

collicollicolli tTP

tTdtdS

∂ρ∂

ρ+

∂∂

=⎟⎠⎞

⎜⎝⎛ Y1 , where

the first term of right hand side indicates the collision due to change of energy between electrons-electrons, electrons-holes, holes-holes interactions, and the second term defines the collision because of change of carrier density.

2.4 Summary of Chapter 2

The details of the Boltzmann Transport equation for analyzing the mass, momentum,

energy and entropy conservation equations in macro and micro scales have been

discussed in this chapter. This chapter concludes with the summary of all conservation

31

equations as shown in Table 2.4 using Gauss and BTE approaches so that these two

methods could be compared.

Table 2.4 Summary of the conservation equations

Gauss theorem approach (Appendix A) Boltzmann Transport Equation approach

Ω=Ω=∇ ∫∫∫

ΩΩ

ddndVV

... XXX r t

ftf

tf colli

drift ∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

−=∂∂

Mass ).( vρρ−∇=

∂∂

t

collitt ∂∂

+−∇=∂∂ ρρρ )(v

Momentum ( ) Φ++ρ⋅−∇=∂ρ∂ Pvvvt

( ) ( ) ( )

collitt ∂∂

+Φ++−∇=∂

∂ vvvPv ρρρ .

Energy ΨJ +⋅−∇=ρ qdtdu

colliq tdt

du∂∂

++⋅−∇=ρYΨJ

Entropy

JF

JJ

JJ

T

TT

TT

T

Tdtds

n

k

kkq

n

kkkq

1

111

2

1

+

⎟⎠⎞

⎜⎝⎛ µ∇⋅−∇⋅−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛µ−

⋅−∇=ρ

∑

∑

=

=

collicolli

n

k

kk

q

n

kkkq

tTP

tTT

TT

T

TTTdt

ds

∂ρ∂

ρ+

∂∂

++

⎟⎠⎞

⎜⎝⎛ µ∇⋅−

∇⋅−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛µ−

⋅−∇=ρ

∑

∑

=

=

YJF

J

JJJ

11

1

1

1

21

32

Chapter 3

Thermodynamic modeling of macro and micro thermoelectric coolers

3.1 Introduction

The reversible thermoelectric effects are the Seebeck, Peltier and Thomson effects

(Burshteyn, 1964). These are related to the transformation of thermal into electrical

energy, and vice versa. The physical nature of the thermoelectric effects is not well

explained in the literature, as well as the dependence of Seebeck coefficient on

temperature and materials. From the electron theory, the absorption of Thomson heat

in the interior of a thermally non-uniform conductor has been reported to be the

additive superposition of two effects: Firstly, a part of Thomson effect is the internal

Peltier effect which is caused by the non-equilibrium electron distribution functions in

a thermally non-uniform conductor. Secondly, the other part is heat absorbed due to

current flow against the drift potential difference. These effects are assumed to be

reversible and hence, the sign of the Thomson heat changes with the reversal of current

direction. The thermoelectric effects in semi-conductors are stronger than those found

in metals. This is attributed to the fact that electron gas in semi-conductors is far from

being degenerated and obeys the classical Boltzmann statistics. The non-equilibrium

changes in the distribution function are more noticeable than in a degenerated gas and

this affects the magnitude of the thermoelectric effects. The magnitude of non-

equilibrium change in the distribution function (and consequently the magnitudes of

the thermoelectric effects) depends on the relative importance of the factors (such as

electric field, thermal non-uniformity) that are responsible for the departure from

equilibrium, as well as the mechanism for re-establishing the equilibrium.

33

When an electric field is applied to a thermoelectric device, the following irreversible

processes are deemed to occur in each elemental volume:

(i) Electric conduction (electric current due to an electric potential gradient).

(ii) Heat conduction (heat flow due to a temperature gradient).

(iii) A cross effect (electric current due to a temperature gradient)

(iv) The appropriate reciprocal effect (heat flow due to an electric potential

gradient).

The last two belongs to a new class of irreversible process (Onsager, 1931).

The organization of this chapter is as follows:

Section 3.2 describes the literature review of the thermoelectric cooling systems. The

thermodynamic modelling of the bulk thermoelectric cooler is discussed in section 3.3.

Although other researchers excluded the Thomson effect, this is included in the thesis.

The explanation of Thomson effect is discussed in Appendix B and in addition, the

entropy analysis of a transient thermoelectric cooler is also discussed. The

thermodynamic behaviour of a thin-film thermoelectric cooler is investigated in

section 3.4 and the analysis is compared with the work of Arenas et al., (2001). The

governing equations employed in this chapter are derived from the Boltzmann

Transport Equation (BTE) that was reported in the previous chapter.

34

3.2 Literature review

Richard and Fine (1948) first generalized the thermodynamic treatment of

thermoelectrics involving the irreversible processes at steady state operation. Applying

the generalized laws of thermodynamics, Lord Kelvin (Thomson) proved the validity

of the relationships dTd

Tππ

−=Γ and Tαπ = where π is the Peltier Coefficient and α

is the Seebeck Coefficient. In 1931, the Onsager symmetry relationships were

developed. These complemented the theory of thermoelectricity and using these

relationships, the thermodynamic foundations for analyzing the irreversible steady

state processes have been laid. A detailed account and justification of the

thermodynamic framework in establishing the connections between all the

thermoelectric effects could be found in the monographs of de Groot, (1962) and

Haase, (1990). The use of semiconductors in thermoelectric refrigeration was proposed

(Goldsmid and Douglas, 1954) as the materials have high mean atomic weights and

high thermoelectric powers. The relationship between cooling capacity and the

Coefficient of Performance (COP) were examined for a single stage unit (Parrott,

1960) with respect to a given temperature difference between the hot and cold

junctions, and the design equations and performance graphs were presented. Recently,

a new approach to design thermoelectric cooling systems optimally has been proposed

by Yamanashi in 1996. He succinctly introduced the dimensionless entropy flow

equations to analyze the thermoelectric cooling performance.

Interface effects in thermoelectric micro-refrigerators were studied by Sungtaek and

Ghoshal (2000). They employed a phenomenological model to examine the behavior

of thermoelectric refrigerators as a function of thermal and electrical contact resistance,

the Seebeck coefficient and heat sink conductance. Ghoshal et al. (2002) described a

35

structured point-contact thermoelectric device, which defines the thermal gradient and

electric fields at the cold junction, and exploited the reduction of thermal conduction,

tunneling properties of point contacts and poor electron-phonon coupling at the

junctions. They have proposed a thermoelectric figure of merit, called the ZT

parameter, which has a range of 1.4-1.7 at room temperature.

In 1961, Landecker and Findlay studied extensively the transient behavior of Peltier

junctions by devising a method for measuring the thermo-junction temperature after

the passage of transient current pulse. They have theoretically demonstrated that the

cold junction temporal profile is a function of both pulse current and duration. Snyder

et al., (2002) reported the possibility of having a super cooler that employs a Peltier

device and exploiting the short time-scale behaviour of the current pulse at a

magnitude several folds higher than that of the non-pulsing period. The Peltier cold

junction breaks contact momentarily with the surface to be cooled prior to the arrival

of the Joulean heat, where the latter is a heat transfer phenomenon with a slower time-

scale. Such a transient operation of the thermoelectrics has enable these cooling

devices to reach an unprecedented low temperature level, typically about 220 - 240K

which widens the application potential of pulsed thermoelectrics, for example, a cryo-

cooler for surgery applications or the cooling of infrared detectors. In 1999, Miner et

al. modified a traditional thermoelectric cooler by using the intermittent contact of a

mechanical element, which was synchronized with an applied pulsed current for a

finite time. The proposed technique effectively increases the thermoelectric figure of

merit by a factor of 1.8 but there is no experimental result to justify the proposed

concept.

36

Temperature entropy (T-s) formulation of thermoelectric cycles has been developed

(Chua et al., 2002(b)) within the framework of irreversible thermodynamics and they

have presented the steady state results. Similar work on the T-s diagram of the

thermoelectric module was also performed by Arenas et al. (2000) but they found a

close link between the Seebeck coefficient and the entropy per unit electric current.

Also in 2000 and using a finite time method, Nuwayhid et al. presented a procedure of

minimizing the entropy generation so as to maximize the cooling power of

thermoelectric device.

Solid state superlattice structures could also be used to enhance the thermoelectric

device performance by reducing the thermal conductivity between the hot and cold

junctions and provide selective emission of hot carriers through the thermionic

emission (Mahan, 1994). With much work performed in the study of superlattice

thermoelectric properties, (Antonyuk et. al., 2001), the idea of increasing ZT parameter

could be derived by enhancing the density of electron flow with reduced dimensions.

Recent study shows that superlattice thermal conductivity in the cross-plane direction

is lower than that of in-plane direction and such hetrostructures could be used for

thermionic emission to increase the cooling. The superlattice thermoelectric module

consists of n-type and p-type super-lattices (Shakuri and Bowers, 1997 and Elsner et

al., 1996).

37

3.3 Thermoelectric cooling:

The refrigeration capability of a semiconductor material depends on parameters such

as the Peltier effect, Joulean heat, Thomson heat and material’s physical properties

over the operational temperatures between the hot and cold ends. A thermoelectric

device is composed of P-type and N-type thermoelements such as Bi2Te3 and they are

connected electrically in series and thermally in parallel. These thermoelectric

elements are sandwiched between two ceramic substrates. As shown in Figure 3.1,

thermoelectric cooling is generated by passing a direct current through one or more

pairs of n and p-type thermoelements, the temperature of cold reservoir decreases

because the electrons and holes pass from the low energy level in p-type material

through the interconnecting conductor to the higher energy level in the n-type material.

Similarly, the arrival of electrons and holes in the opposite end results in an increase in

the local temperature forming the hot junction.

P N

J

J

J J

Hot reservoir

Cold reservoir

TH

TL

+ ++ + + +

- -- -- -

J J P-arm

N-arm

Hot reservoir

Cold reservoir

(a) (b)

Figure 3.1. The schematic view of a thermoelectric cooler

38

When a temperature differential is established between the hot and cold junctions, a

Seebeck voltage is generated and the voltage is directly proportional to the temperature

differential. The amount of heat absorbed at the cold-end and dissipated at the hot end

depends on the product of Peltier coefficient (π ), Fourier effect (λte) and the current (I)

flowing through the semiconductor material, whilst the heat generation due to the

Thomson effect occurs along the thermoelectric element. Thermoelectric effects are

caused by coupling between charge transport and heat transport and from which the

basic mass, energy and entropy equations can be formulated to form a thermodynamic

framework.

3.3.1 Energy Balance Analysis:

The Peltier and the Thomson effects enhance the cooling effect in thermoelectric

materials. The Thomson effect is a function of the gradients from the Seebeck

coefficient and the temperature in an operating thermoelectric pallet or element. The

energy balance equation of the thermoelectric cooler, as shown in figure 3.1, follows

the methodology of the Boltzmann transport equation which has been discussed in the

previous chapter, namely:

( ) ( ) ( )colli

q tuEu

tu

∂∂

++∇−+−∇=∂

∂ ρρρ JvPJv :. ,

where u is the internal energy per unit mass, ρ is the density, v defines velocity, Jq

indicates the heat flow, P represents the pressure and JE is the energy field term.

For the bulk thermoelectric material analysis, the contributions from the collision

terms are deemed negligible and can be neglected;

( ) ( ) Eutu

qtete JvPJv +∇−+−∇=∂

∂ :. ρρ .

39

Hence the subscript te indicates the thermoelectric cooler. As there is no velocity and

pressure fields in a thermoelectric device,

Etu

qte JJ +−∇=∂∂ .ρ (3.1)

Invoking the definition of enthalpy;

( )tu

tpV

tu

thpVuh

∂∂

=∂

∂+

∂∂

=∂∂

⇒+=

( )t

Tctp

ph

tT

Th

thpThh te

tepT

te

pte te∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=∂∂

⇒= ,,

where the pressure gradient term is zero.

The first term of right hand side of equation (3.1) is the heat current density, qJ (in

W/m2), and the second term is the electric current field, ( )φ∇= JEJ (in W/m3). Using

the thermodynamic-phenomenological treatment (Haase, 1990), the heat current

density

TJFQ

teo

q ∇−−= λJ , (3.2)

where J (in amp/m2) denotes the electric current density, Qo indicates the heat

transport of electrons/ holes, teλ defines the thermal conductivity of the bulk

thermoelectric element and F is the Faraday constant and the electric current density

can be expanded as

tete

otete T

FTQJ ∇′

+∇′=σφσ , (3.3)

where teσ ′ defines the electrical conductance and φ is the electrical potential.

Equation (3.3) is re-arranged as,

tete

o

te

TFTQJ

∇−′

=∇σ

φ (3.4)

40

Hence, ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∇+

′+∇∇=+∇−

tete

teteteq T

QTFJJTE

σλ

2

.. JJ

where ⎟⎟⎠

⎞⎜⎜⎝

⎛

teTQ is equivalent to the entropy term.

Now, substitute all these relations into equation (3.1), the expression becomes

( ) tepte

tete

tetete

tetepte T

Ts

FJTJT

tTc ∇⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+′

+∇∇=∂∂

σλρ

2

, . , (3.5)

where ste introduces the transported entropy of the bulk thermoelectric arms. Defining

the Thomson coefficient ( )Γ as (Haase, 1990),

pte

tete

Ts

FT

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

≡Γ− ,

the governing thermal transport in the bulk semiconductor arms is given by

( ) tete

tetete

tepte TJJTt

Tc ∇Γ−′

+∇∇=∂∂

σλρ

2

, . (3.6)

Using the Kelvin relations (de Groot, 1961),

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−−=Γtete TTππ , where ( )Tαπ = is the Peltier coefficient.

Equation (3.6) can now be written as (derivations are shown in Appendix B),

teTte

tetete

tepte TT

JTJTJTt

Tcte

∇⎟⎠⎞

⎜⎝⎛∂∂

−∇+′

+∇=∂∂ αα

σλρ

22

, (3.7)

41

3.3.2 Entropy Balance Analysis

The basic entropy balance equation has been introduced in the previous chapter, as

shown below:

( ) ( ) σρρ

σρ

++−∇=∂

∂

+⋅−∇=

svts

dtds

s

s

J

J

. (3.8)

Introducing the phenomenological relations:

JT

TT

vste

tete

tetetetes

πλρ +∇−=+,J , (3.9)

and invoking the Kelvin laws, one could observe

πα =T . (3.10)

Assuming only the bulk thermoelectric materials, the collision terms are usually

neglected and the entropy source strength is given by (as discussed in the previous

Chapter);

EJTT

TT

TT

n

k

kkq ⋅+⎟

⎠⎞

⎜⎝⎛ ∇⋅−∇⋅−= ∑

=

1111

2

µσ JJ

Hence, the entropy generation of thermoelectric arm can now be modified into a form

that is inclusive of the Peltier coefficient and the local temperature as (de Groot, 1961)

( )tetete

tete

tetete

JTJT

TJT

TTT

σαπλσ ′+∇⋅+∇⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−−= /11

Finally, the entropy balance equation (3.8) becomes;

( ) ( )tetetete

tetetete

TJ

TJ

TT

ts

σπλρ

′+⋅∇−

∇⋅∇=

∂∂ 2

(3.11)

42

The first term on the right-hand side of equation (3.11) represents the change of

entropy due to heat conduction; the second term indicates the entropy flux due to

Seebeck effect and the last term is the entropy generation due to Joulean heat.

A more elegant method of writing the entropy fluxes of thermoelectric arm is to

express them in terms of (i) the heat and carrier fluxes, (ii) the internal dissipation, and

(iii) the entropy generation that comes from conduction and Joulean heat transfer:

( )

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

′+⎟⎟

⎠

⎞⎜⎜⎝

⎛∇∇−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

∇⋅∇=

∂∂

87644 844 76444 8444 76 heatJouleantoduestrengthsourceentropy

tete

ndissipatioernal

tetete

fluxentropy

tete

tetetete

TJ

TT

TJ

TT

ts

σλπλρ 2

int

1.

(3.12)

where the units of entropy flux is in W/m2 K and it is given by,

tete

teteflux T

JT

TS πλ−

∇= ,

(3.13)

and entropy generation per unit volume is in W/m3 K which is expressed as:

tetetetetegen T

JT

TSσ

λ′

+⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−=

21. .

(3.14)

Equation 3.14 is always a positive term.

3.3.3 Temperature-entropy plots of bulk thermoelectric cooling device

From the work of Carnot and Clasius, Kelvin deduced that the reversible heat flow

discovered by Peltier must have entropy associated with it. It has been shown that the

coefficient discovered by Seebeck was a measure of entropy associated with electric

current (Nolas et. al., 2001). A temperature-entropy (T-s) has been proven to be a

pedagogical tool in analyzing the performance of thermoelectrics undergoing both

43

reversible and irreversible processes. The parameters used in the computation are listed

in Table 3.1.

For an ideal thermoelectric device, the Seebeck (α) coefficient is constant and the heat

flux at the hot and cold junctions are given by αJTi where the subscript i refers either

to the hot or cold junction. Thus, the temperature Seebeck coefficient (T-α) plot is

simply a simple rectangle. However, the temperature effect on the Seebeck (α)

coefficient induces the dissipative losses along the p and n legs of thermoelectrics and

an indication of these losses are shown by the shaded area on the T-α plot, as shown in

Figure 3.2, in which the physical properties of the Bismuth Telluride thermoelectric

pairs are used. The two isotherms are the hot and cold junctions whilst the inclined

lines indicate the effect of temperature on the α values when the current (I) travels

along the p- and n-legs.

Table 3.1 Physical parameters of a single thermoelectric couple

Property value

Hot junction temperature, TH 300 K

Cold junction temperature, TL 270 K

Thermoelectric element length, L 1.15 × 10 -3 m *

Cross-sectional area, A 1.96 × 10-6 m2 *

Electrical conductivity, σ' 97087.38 ohm-1.m-1 *

Seebeck coefficient α (V/K)

α (T) = αo + α1 ln (T/To)

αo = 210 × 10-6 V/K, α1 = 120 × 10-6 V/K

To = 300 K (Seifert et al., 2002)

Thermal conductivity, λ 1.70 W/m K *

* Remarks Melcor thermoelectric catalogue, MELCOR CORPORATION, 1040 Spruce Street, Trenton, NJ 08648, USA. Web site: www.melcor.com

For thermodynamic consistency, the temperature-entropy (or temperature-entropy

flux) diagram is plotted instead using the entropy flux expression derived previously.

44

In an optimum current operation of the thermoelectric cooler, the presence of entropy

loss (due to conduction heat transfer and Joule heat) along the p-n legs is given by

TK∇ /Ti and they are manifested by enclosed areas “k-i-u-r-d-c-k” and “j-i-n-q-a-b-j”

of Figure 3.3. Both mentioned dissipative losses have the consequence of reducing the

cooling effect, represented by area “a-d-r-q”. Within the internal framework (adiabatic

chiller), the cycle “a-b-c-d” operates in an anti-clockwise manner with process with

two isotherms b-c and d-a, as well as two non-adiabats a-b and c-d. Details of these

processes are described in Table 3.2. Thus, the T-s diagram provides an effective

means of analyzing how the real chiller cycle of thermoelectrics is inferior to the ideal

Carnot cycle by capturing the key losses and useful effects in the cycle.

45

265

270

275

280

285

290

295

300

305 -0.0

0025

-0.0

002

-0.0

0015

-0.0

001

-0.0

0005

00.

0000

50.

0001

0.00

015

0.00

020.

0002

5

Seeb

eck

coef

ficie

nt (V

/K)

Temperature (K)

A

BC

D

hot j

unct

ion

cold

junc

tion

P-ar

mN

-arm

J

Figu

re 3

.2. T

-α d

iagr

am o

f a th

erm

oele

ctri

c pa

ir.

46

Figure 3.3. Temperature-entropy flux diagram for the thermoelectric cooler at the maximum coefficient of performance identifies the principal energy flows.

0

50

100

150

200

250

300

350

-500 -400 -300 -200 -100 0 100 200 300 400 500

Entropy flux density (W/m2 K)

Tem

pera

ture

(K)

P-armN-arm

a

bc

d eh i

jk

lJ

mnoqrtuv

TH

TL

Q dissip Q dissip

12

Net cooling power

47

Table 3.2 Description of close loop a-b-c-d-a (Figure 3.3)

Process Description

a--b

This process is developed along P-arm of the thermoelectric pair. Points a and b are the states at the ends of the thermoelement of which the temperatures are TL and TH respectively (as current J flows from P-arm to N-arm). The process a-b can not be adiabatic because it absorbs heat due to the Thomson effect which depends on the Seebeck coefficient of the thermoelectric element. The areas beneath the a-b process represent dissipation due to heat conduction, and heat absorbed by the Thomson effect. Area 2-i-j represents the energy dissipation due to Seebeck coefficient, which varies along the thermoelectric p-arm.

b--c This process represents the heat exchange between system and medium (environment) which occurs at the union between P-arm and N-arm at temperature TH.

c--d

The process c-d is developed along N-arm, where points c and d are at the ends of the thermoelement whose temperature are TH and TL. The areas beneath c-d process indicate heat losses due to dissipation. If this process is adiabatic, the Thomson effect would be null. Area 1-l-k indicates the energy losses due to Seebeck effect along the n-arm.

d--a

This process represents the heat exchange between the system and the heat load at isothermal condition; however it occurs in the physic union between thermoelectric elements at temperature TL.

From temperature-entropy flux diagram, the thermodynamic performance variables

can be easily tallied with the enclosed rectangles. The heat fluxes at the cold (process

d-a) and hot (process b-c) reservoirs are as follows (for details please see Appendix C)

( )

876876

876heatsonT

heatconductionheatJoulean

coolingPeltier

L

cjLL

cjLpnSLL

TITKRIT

TATT

TATJTadrqQ

_hom_

_

2_

,

21

21

∆−∆−−=

∇−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∇−===

τα

λαλ

α

I

IIArea

, (3.15)

48

( )

876876

876heatsonT

lossConductionheatJoulea

heatPeltier

H

hjHH

hjHhjpnSHH

TITKRIT

TATT

TATJTbctoQ

_hom_

_

2_

,

21

21

∆+∆−+=

∇−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∇−===

τα

λαλ

α

I

IIArea , (3.16)

where T∆ is the temperature difference across the hot and cold junctions. The first

term of right hand side of both equations (3.15) and (3.16) indicates Peltier cooling and

heating, the second term is the Joulean heat, the third term represents heat conduction

between hot and cold junctions and the heat absorbed by the Thomson effect. The

Thomson heat is transmitted by thermal conduction across the thermoelement and this

effect is not reversible. From the first law of thermodynamics, the required electrical

power input is given by,

( ) ( )

( ) TIRITTP

TITKRITTITKRIT

JTJTadrqbctoP

LH

Q

L

Q

H

coldpnSLhotpnSH

LH

∆++−=

∆+∆++−∆+∆−+=

−=−=

τα

τατα

2

22

,,

21

21

21

21

I

II

AreaArea

44444 844444 7644444 844444 76

(3.17)

The coefficient of performance is defined as PQCOP L= , could be non-

dimensionalized with its maximum COP and plotted against entropy flux of cold

reservoir for both the P and N arms, as shown in Figure 3.4. There are two curves for

the p and n arms of thermoelectrics, and the four points are indicated on each curve

are; (i) Point “A” refers to the low COP at low current limit, (ii) Point “B” is the

maximum COP at optimum current, (iii) Point “C” indicates the maximum entropy

flux or output power and (iv) Point “D” represents the vanishing COP at higher current

limit. It is instructive to compare the present T-s plot with those of the published

reference (Chua et al., 2002 (b)) for the same thermoelectric module.

49

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Sflux,L/Sflux,L max

CO

P/C

OP m

ax

P-armN-arm

JJ

A

B

C

DA

B

C

D

Figure 3.4. Characteristics performance curve of the thermoelectric cooler. COP is plotted against entropy flux for the both N and P legs.

Figure 3.5. Temperature-entropy flux diagram for the thermoelectric pair at the optimum current flow by the present study and the other researcher (Chua et al., 2002 (b)).

265

270

275

280

285

290

295

300

305

-400 -300 -200 -100 0 100 200 300 400

Entropy flux (W/m2 K)

Tem

pera

ture

(K)

P-armN-arm

a

bc

d a'd'

CO

P/C

OP m

ax

Js/Js,max

Present study Area abcd

Chua et al., 2002 (b) Area a′b′c′d′ ----

50

The recent work of Chua et al. (2002 (b)) assumes the adiabatic process for the T-

entropy flux diagram, as shown in Figure 3.5 where the adiabats are represented by

paths “a-b” and “c-d”. The shaded area (in Figure 3.5) is the heat dissipation due to

Thomson effect. In the Carnot cycle, internal and external dissipations (thermal and

electrical) are zero and the processes a-b and c-d becomes adiabatic. The equivalent

cycle at the same TH and TL is denoted by a rectangle 1-2-j-k (see Figure 3.3), for

which ( ) ( ) ( )[ ]2121 vmAreakvmjAreavmAreaCOPcarnot −= or

( ) ( ) TTjkAreavmAreaCOP Lcarnot ∆== 1221 .

The irreversibility is the heat dissipation, which is shown in Figure 3.3, occurs due to

the Joulean heat and to the heat conduction along the thermoelectric arm. For a better

Figure 3.6. Temperature-entropy generation diagram of a thermoelectric cooler at optimum cooling power. Area (abcd) is the total heat dissipation along the thermoelectric element, where area (aefd) is the irreversibility due to heat conduction and area (aghd) indicates the entropy generation due to joule effect.

265

270

275

280

285

290

295

300

305

0 40000 80000 120000 160000 200000

Entropy generation (W/m3 K)

Tem

pera

ture

(K)

P or N-arm

J J

a b

cd

e

f

g

h

Cold reservoir

Hot reservoirPN

51

understanding of the dissipative effects in the thermoelectric element, the temperature-

entropy generation (T- σ ′ ) plot is used and this is shown in Figure 3.6. Two major

dissipative mechanisms in the thermoelectrics can be observed: Firstly, Joule heat

dissipation occurs (enclosed by area a-g-h-d) evenly in the two sides of thermoelectric

arms and secondly, a smaller loss from the heat conduction (a-e-f-d) affects along the

thermoelectric element. The combined losses are given by the area (a-b-c-d) of Figure

3.6.

3.4 Transient behavior of thermoelectric cooler

The performance of a Peltier or thermoelectric cooler can be enhanced by utilizing the

transient response of a current pulse (Snyder et al., 2002). At fast transient, a high but

short-period (4 seconds) current pulse is injected so that additional Petlier cooling is

developed instantly at the cold junction: The joule heating developed in the

thermoelectric element is a slow phenomenon as heat travels slowly over the materials

and could not reached the cold junction in the short transient. Hence, the cold junction

temperature is at the lowest possible value and heat flows only from the source to the

cold junction. An example of such an application of the short transient cooling is an

infrared detector operating in a ‘winking mode, which needs to be super cold only

during the wink.

Figure 3.7 shows the schematic diagram of a current “pulse train” thermoelectric

cooler with the p- and n- doped semiconductor elements are electrically connected in

series and are thermally in parallel. The hot side of the device is mounted directly onto

a substrate to reject heat to the environment and the thermal contact between the cold

junction of thermoelectric device and the cold substrate (load) is made periodically.

52

There are two periods of operation: Firstly, an “open contact” period with tno and

current flow Jnp to maintain the lowest temperature Tmin at the cold junction but the

cold substrate (load) is not connected with the cold junction. Secondly, a large current

pulse Jp (where Jp=MJnp, M > 1) is applied for a duration tp and concomitantly, the

cold substrate makes contact with the cold junction. Intense Peltier cooling is achieved

during tp, which reduces the cold junction temperature below Tmin. However, Joule and

Thomson heating are developed in the bulk thermoelectric module and diffuse towards

the hot and cold ends of the device. Thermal contact of the cold substrate with the cold

end of the thermoelectric module is maintained over the short current pulse period but

these surface break contact before the cooling could be degraded by the conduction

heat. In the open contact manner, the cold substrate temperature could reach below that

of the cold junction temperature under the normal thermoelectric operation. The total

operation consists of two periods; one is the normal thermoelectric operation and the

other is the pulse or effective operation. The cycle of (normal operation + pulse

operation) is repeated such that the device is continuously produced cooling.

53

Figure 3.7. Schematic diagram of the first transient thermoelectric cooler. (a) indicates the normal operation and cold substrate is out of contact, (b) defines the thermoelectric pulse operation and cold substrate is in contact with the cold junction, (c) shows the current profile during normal thermoelectric operation, (d) indicates that the pulse current is applied to the device and (e) the cycle of (tno + tp) is repeated such that the device is operated in an ac mode.

J np

time t

J p

P N P N P N P N P N P N

Hot substrate H ot junction

Cold junction

Cold substrate (a)

P N P N P N P N P N P N

Hot substrate H ot junction

Cold junction

Cold substrate (b)

(e)

J

t ime t

J np t no

(c)

J

time t

Jp

tp

(d)

J np

J p

54

This section formulates a non-equilibrium thermodynamic model to demonstrate the

energy and entropy balances of a transient thermoelectric cooler. In analyzing the fast

transient cycle in the classical T-s plane, one distinguishes the entropy fluxes of the

working fluid (which is the current flux J) from the external entropy fluxes in the hot

and cold reservoirs and at the interface with the reservoirs. The T-s relation is

formulated that identifies the key heat and work flows, the heat conduction and

electrical resistive dissipative losses.

3.4.1. Derivation of the T-s relation

The methodology of energy and entropy balances of the commercial available

thermoelectric device has been discussed briefly in the previous sections. Starting from

the general entropy balance equation, it can be expressed in terms of the key

parameters such as Peltier effect ( )π , electrical conductivity ( )σ ′ , current density (J)

and Fourier effect (λ), as shown below:

( )4444 84444 76444 8444 76

generationentropy

pp

p

ppp

fluxentropy

tep

p

p

pppp

TJ

TT

TJ

TT

ts

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

′+⎟

⎟⎠

⎞⎜⎜⎝

⎛∇∇−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

∇⋅∇=

∂

∂

σλ

πλρ 2

,

1. , (3.18)

hence the subscript p defines the pulsed thermoelectric cooler, the first term of the

right hand side indicates the entropy flux in a given control volume and the second

terms represent the total entropy generation. The total entropy flux density (in W/m2

K) is expressed by

pp

pp

ptots JT

Tα

λ−

∇=,J , (3.19)

and the entropy generation (in W/m3 K) is written as

pp

p

ppptotp T

JT

Tσ

λσ′

+⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−=

2

,1. . (3.20)

55

The energy conservation equation is as follows

pp

ppTteppp

ppp

ppp T

TTJTJ

JT

tT

cp

∇⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∇+′

+∇=∂

∂ αασ

λρ ,

22 . (3.21)

To solve the problem, two boundary conditions are defined: Firstly, during the normal

(open-contact) operation, there is only Peltier cooling at the cold junction

p

cj

cjat

p JTx

Tλ

α=

∂

∂, (3.22)

and at hot junction, a hot side heat sink is represented by, hjhjatp TT = . But during the

pulse (close-contact) operation, at the cold junction the boundary condition becomes

p

subs

p

cj

cjat

p qJTx

Tλλ

α .

+=∂

∂, (3.23)

where .subsq is the cooling load of the cold substrate.

3.4.2 Results and discussions

The transient behavior is demonstrated here using the typical physical and mechanical

properties of a thermoelectric cooler and the simulated results for the hot and cold

junctions are shown in Figure 3.8. A pulse-type cooler is useful not only for testing the

theoretical models but also is in designing the device. Firstly, the thermoelectric cooler

is supplied a steady current until no cooling is produced at the cold end but the

junction temperature of the cold junction reaches a TA’ (at point A’) whilst the hot-

junction is at a constant temperature of TH, as shown in Figure 3.8. A pulse current, at

a magnitude of M (=Ip/Inp) folds from the steady current, is then supplied to the

thermoelectric device which increase the cooling momentarily and depressing the cold

junction temperature further from A’ to B. At which time, the effect of Joulean heat,

56

conducted across the hot to cold junctions, is being felt causing a rise in the cold

junction temperature from B to C. Although the current pulse is terminated at point C,

breaking physical contact to the cold substrate, temperature within the cold junction

continue to rise to point B’ due to the delayed Joulean heat transfer in the legs of the

thermoelectrics.

The parameters used in this modeling are shown in Table 3.3. The energy flows in the

thermoelectric device can be captured by the use of a temperature-entropy diagram, as

shown in Figures 3.9 and 3.10. For the non-pulsing or contact interval, Fig 3.9 shows

220

230

240

250

260

270

280

290

300

310

320

0 20 40 60 80 100 120time (s)

tem

pera

ture

(K)

C

C'

B

B'

A

A'

hot junction

cold junction

Figure 3.8. the cold reservoir temperature of a super-cooling thermoelectric cooler as a function of time. The small loop indicates the super-cooling during current pulse operation. Temperature drop between the cold and hot junctions subjected to a step current pulse of magnitude 3 times the DC current and of width 4 s. A indicates the beginning of pulse period, B defines the maximum temperature drop and C represents the end of pulse period, C’ is the beginning of non-contact period, B’ indicates maximum temperature rise due to residual heat.

Time (s)

Tem

pera

ture

(K)

57

the typical anti-clockwise loop (A’) for the entropy fluxes in the p and n-legs, as

indicated by the process states “a-g-h-a”. The area under the isothermal path “g-h”

indicates the total energy supplied to the thermoelectrics. However, it is noted that the

cold isotherm (T_a) converges to a point at “a” where no area is enclosed under the

isotherm. Thus, no cooling power is generated at the cold junction, i.e., the cooling

generated at the cold junction is totally overwhelmed by the flow of Joulean heat from

the hot junction. Similar T-s plots are also found for points C’ and B’ during the open-

contact operation of the device where the cold isotherms become a single point at “e”

and “f”. It is also noted that the maximum temperature attained within the p and n-legs

might exceed that of the hot junction temperature.

During closed contact operation, as shown in Fig.3.10, point A indicates the start of

contact interval where the imposed current pulse generates substantial cooling,

depressing the cold junction and the cold template temperatures from “f” to “a” (see

the smaller insert diagram of Fig. 3.10). The area below the isotherm b-b' indicates the

amount of cooling on the T-s plot at point A. The isotherms c-c' and d-d' correspond to

the points B and C. It is noted that although the enclosed area for point C is larger than

that of point B, but the temperature of point C has risen due to the conduction of

Joulean heat from the hot junction.

58

220

240

260

280

300

320

340 -3

00-2

00-1

000

100

200

300

entr

opy

flux

(W/m

2 K)

temperature (K)

p-ar

mn-

arm

gh

aef

I np

C'

B'

A'

Figu

re 3

.9.

T-s

dia

gram

of

a pu

lsed

the

rmoe

lect

ric

cool

er d

urin

g no

n-pu

lse

oper

atio

n; C′

defin

es t

he b

egin

ning

of

non-

cont

act,

B′

indi

cate

s th

e m

axim

um t

empe

ratu

re r

ise

of c

old

junc

tion

duri

ng n

on-p

ulse

per

iod

and

A′ r

epre

sent

s th

e m

axim

um te

mpe

ratu

re d

rop

of c

old

junc

tion

with

no

cool

ing

pow

er.

59

220

240

260

280

300

320

340 -3

00-2

00-1

000

100

200

300

entr

opy

flux

(W/m

2 K)

temperature (K)

p-ar

mn-

arm

I p

km

cc'

bb'

dd'

ef

AB

C

225

230

235

240

245

250

255

260

265

-6-4

-20

24

6

cc'

bb'

dd'

e

A

B

C

B

aef

Figu

re 3

.10.

Tem

pera

ture

-ent

ropy

dia

gram

of

a pu

lsed

the

rmoe

lect

ric

cool

er d

urin

g pu

lsed

cu

rren

t op

erat

ion;

A

de

fines

th

e be

ginn

ing

of

cool

ing,

B

re

pres

ents

th

e m

axim

um

tem

pera

ture

dro

p an

d C

indi

cate

s the

max

imum

coo

ling

pow

er.

60

Table 3.3 Physical parameters of a pulsed thermoelectric cooler

Property value Hot reservoir temperature, Thj (K) 308

Thermoelectric element length, Lte (mm) 5.8 (Snyder et al., 2002)

Cross-sectional area, Ate (mm2) 1 (Snyder et al., 2002)

Geometric factor, GFte (mm) 0.1724 (Snyder et al., 2002)

Current, I (amp) 0.675 (Snyder et al., 2002)

Electrical restivity (ohm-cm)

82210 10)( −×++= aveave TT ρρρρ

ρ0 = 5112.0, ρ1 = 163.4, ρ2 = 0.6279

(Melcor)

Seebeck coefficient (V/K)

( ) 92210 10−×++= aveave TT αααα

α0=22224.0, α1= 930.6, α2 = -0.9905

(Melcor)

Thermal conductivity (W/cm K)

( ) 92210 10−×++= aveave TT λλλλ

λ0= 62605.0, λ1= -277.7, λ2 = 0.4131

(Melcor)

Tave = (Thj + Tcj)/2 Melcor thermoelectric catalogue, MELCOR CORPORATION, 1040 Spruce Street, Trenton, NJ 08648, USA. Web site: www.melcor.com

3.5 Microscopic Analysis: Super-lattice type devices

As dimensions of thermoelectric elements (both p and n arms) are reduced to super-

lattice or quantum well scale, they form a category of thin-film thermoelectric cooler

where the conventional macro-scale modeling may not be adequate. Such studies are

of importance because of their potential for achieving high cooling and power

densities. At the micro dimensions, the crystalline structures form the quantum wells

which increase the electron mobility as well as reducing the electrical conductivity. An

example of a super-lattice that is made of silicon as a barrier material and Si0.8Ge0.2 as

conducting material is shown in Figure 3.11.

61

As the silicon (Si) and germanium (Ge) both have the similar crystalline structure, the

Si and solid solution SiGe are fused such that the atoms fit together to form a well-

ordered lattice. The super-lattice quantum well materials are grown by depositing

semiconductors in layers, whose thickness is in the range of a few to up to about 100 Ǻ

(Angstroms), comprising only a few atoms thick. The superlattice structures enhance

the thermoelectric arm performance by reducing the thermal conductivity between the

hot and cold junctions (Elsner et al., 1996) whilst the p-type and n-type superlattice

thermoelectric elements are formed into an “egg-crate” configuration (Elsner et al.,

1996) with metal contacts.

When an electric field is applied to the thin-film thermoelectric cooler, a fraction of

electrons possessing a sufficiently high value of kinetic energy absorbs heat from the

cold reservoir, seeps through the thin film and releases their kinetic energy to the hot

reservoir. Meanwhile, heat is conducted back to the cold reservoir by phonons from the

hot side, which is also heated up the energetic electrons by collisions. Electrons are

…..…..…..

………………

Hot reservoir

Cold reservoir

J

J

Figure 3.11 A schematic of the superlattice thermoelectric element (Elsner et al., 1996)

62

assumed to pass ballistically through the thin film between the hot and cold reservoirs

such that the kinetic energy of electrons is converted to phonons at a minimum level

and a high cooling efficiency is achieved. This would be possible if the thin film

thickness is less than the mean free path of the electrons in the thin film or quantum

well superlattice (Mahan and Woods, 1998). Since the length of the practical device is

typically larger than the electron mean free path, electrons collisions in the thin film

have to be considered in the modeling process.

The efficiency of a thermoelectric module is fundamentally limited by the material

properties of the n- and p-type semiconductors which is determined by a dimensionless

parameter ZT, where T is the temperature and Z characterizes the material’s electrical

and thermal transport properties. Effective thermoelectric materials have a low thermal

conductivity but a high electrical conductivity. The best thermoelectric materials

commercially available today have a ZT ≈ 0.9. For a thermoelectric device to be

competitive to the performance of refrigerators, it would require a ZT ≈ 3 at room

temperature. Recently, several materials with ZT > 1 for both refrigeration and power

generation applications have been reported, in particular the work of

Venkatasubramian et al. (2001), who reported that a high ZT = 2.4 for p-type

superlattices of Bi2Te3/Sb2Te3 at room temperature and ZT = 1.2 for n type

superlattices.

The use of quantum-well structures to enhance the thermoelectric figure of merit was

proposed (Hicks and Dresselhaus, 1993). Since then much work has been done in the

study of superlattice thermoelectric properties, mostly for the in-plane direction where

the enhanced density of electrons (thus, increase in ZT) was achieved by reducing

63

dimensionality. Recent study shows that large ZT improvement is possible for the

cross plane transport as thermal conductivity of cross-plane direction is lower than that

of in-plane direction

This section describes the modeling framework of the superlattice thermoelectric

device to understand the energy transport in semiconductors. The mass, momentum,

energy and entropy balances of the superlattice thermoelectric device are based on the

general arguments from equilibrium and irreversible thermodynamics (de Groot and

Mazur, 1962). These equations contain collision terms that are related to the variations

of the distribution function due to collisions. The Monte Carlo method (Jacoboni and

Reggiani, 1983), which is based on the successive and simultaneous calculation of the

motions of many particles during a small time increment and suitable for the analysis

of transient carrier motion, is used to solve collision terms. Monte Carlo method relies

on the generation of a sequence of random numbers with given distribution

probabilities.

3.5.1 Thermodynamic modeling for thin-film thermoelectrics

The interactions of electrical and thermal phenomena are important to simulate the

superlattice thermoelectric element. The most important thermal effect is the thermal

runaway, in which the electrical dissipation of energy causes an increase in

temperature which in turn causes a greater dissipation of energy. Without repeating the

basic derivation (as derived in Chapter 2), a complete model of electron, hole and

phonon conservation equations for a thin film thermoelectric element is summarized in

Table 3.4

64

Table 3.4 the balance equations

As seen in Table 3.4, the Boltzmann Transport Equation is employed together with (as

discussed in Chapter 2) the phenomenological relationship (Onsager, 1931). The

energy balance equations for electrons, holes and phonon are computed using Gibbs

law. Based on these formulation, the entropy flux and entropy generation for electrons,

holes and phonons in the well and barrier of the superlattice thermoelement are

developed and they are summarized in Table 3.5. The Joule heating in the wells

produces a temperature variation along the x axis; however, the temperature remains

constant along the z axis as shown in Figure 3.12.

Particles Equations

mass ( )colli

eee

e

tnnv

tn

∂∂

+−∇=∂∂ (3.24)

energy colli

eee

e

tUQ

tU

∂∂

++−∇=∂∂ JF. (3.25)Electrons

entropy gecollihecollieirrsee Jt

S−− +++∇=

∂∂

,,,. σσσ (3.26)

mass ( )colli

hhe

h

tnnv

tn

∂∂

+−∇=∂∂ (3.27)

energy colli

hhh

h

tUQ

tU

∂∂

++−∇=∂∂ JF. (3.28)Holes

entropy ghcolliehcollihirrshh Jt

S−− +++∇=

∂∂

,,,. σσσ (3.29)

energy colli

hg

g

tUQ

tU

∂∂

+−∇=∂

∂. (3.30)

Phonons entropy hgcolliegcolligirrgs

g Jt

S−− +++∇=

∂

∂,,,,. σσσ (3.31)

energy collit

UQt

U∂∂

++−∇=∂∂ JF. (3.32)Combined

(Electron + Hole + Phonon) entropy σ+∇= sJ

dtdS . (3.33)

65

Table 3.5 the energy, entropy flux and entropy generation equations of the well and the barrier.

well

Particles Energy balance Entropy flux W/m2 K

Entropy generation W/m3K

Electron ( )

colli

eww

ew

ew

ewewew

w

tT

cJ

Tt

Tc

∂∂

+′

+

∇∇=∂

∂

,

,

2,

,,, .

σ

λ

ew

ewe

ew

ewew

TJ

TT

,

,

,

,, πλ+

∇−

colli

we

we

w

wewe

we

wewewe

tT

Tc

TJ

TT

∂∂

+

′+⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

σλ

21.

Hole ( )

colli

hww

hw

hw

hwhwhw

w

tT

cJ

Tt

Tc

∂∂

+′

+

∇∇=∂

∂

,

,

2,

,,, .

σ

λ

hw

hwh

hw

hwhw

TJ

TT

,

,

,

,, πλ+

∇−

colli

wh

wh

w

whwh

wh

whwhwh

tT

Tc

TJ

TT

∂∂

+

′+⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

σλ

21.

Phonon ( )

colli

gwwgwgw

gww t

TcT

tT

c∂

∂+∇∇=

∂

∂ ,,,

, . λ

gw

gwgw

TT

,

,, ∇−λ

colli

wg

wg

w

wgwgwg

tT

Tc

TT

∂

∂+

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

1.λ

Combined ( )

colli

ww

w

www

ww

tTc

JTt

Tc

∂∂

+

′+∇∇=

∂∂

σλ

2

.

w

w

w

ww

TJ

TT πλ

+∇

−

colli

w

w

w

ww

w

www

tT

Tc

TJ

TT

∂∂

+

′+⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

σλ

21.

barrier Energy balance Entropy flux Entropy generation

Electron ( )colli

ebbebeb

ebb t

TcT

tT

c∂∂

+∇∇=∂∂ ,

,,, . λ

eb

ebeb

TT

,

,, ∇λ−

colli

be

be

b

bebebe

tT

Tc

TT

∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

1.λ

Hole ( )colli

hbbhbhb

hbb t

TcT

tT

c∂

∂+∇λ∇=

∂∂ ,

,,, .

hb

hbhb

TT

,

,, ∇−λ

colli

bh

bh

b

bhbhbh

tT

Tc

TT

∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

1.λ

Phonon ( )colli

gbbgbgb

gbb t

TcT

tT

c∂

∂+∇∇=

∂

∂ ,,,

, . λ gb

gbgb

TT

,

,, ∇−λ

colli

bg

bg

b

bgbgbg

tT

Tc

TT

∂

∂+

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

1.λ

66

Combined ( )colli

bbbb

bb t

TcTt

Tc∂∂

+∇∇=∂∂ λ.

b

bb

TT∇

−λ

colli

b

b

b

bbb

tT

Tc

TT

∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇−

1.λ

The wells are electric conducting and the barriers are electric non-conducting. At y = 0

the sample is in contact with a cold thermal reservoir of temperature TL, and at y = L

with a hot reservoir of temperature TH. An electric field J is applied parallel to the

thermal gradient. The sample is bulk in size in the xz plane.

. . . . . . … . . . . . . …

Hot reservoir, TH

Cold reservoir, TL

b b

a

barrier barrier well

J

x

y z

Figure 3.12. Schematic diagram of a superlattice thermoelement with electric conducting wells and electric insulating barriers (Antonyuk et al., 2001).

67

3.5.2 Results and discussions

Due to current flow into the superlattice thermoelectric element, the Joulean heat,

which is generated in a well, flows eventually along the thermoelectric element into

the cold and hot reservoirs either via the well or via the barrier. The amount of heat

transported through the well and barrier depends on their cross-sectional areas and

thermal conductivity. The bulk and superlattice thermoelement properties and their

dimensions used in this simulation are taken from the experimental analysis (Elsner et

al., 1996 and Antonyuk et al., 2001), are shown in Table 3.6. The Flow Chart of the

simulation is provided in Appendix D.

Table 3.6 the thermo-physical properties of bulk SiGe and superlattice Si.8Ge.2/Si thermoelectric element.

Sample Electrical resistivity mΩ-cm

Seebeck coefficient

µV/oC

Thermal conductivity

W/m K

Carrier concentration

Figure of merit

Si.8Ge.2/Si 2N 4 -1250 7.66 1019 3101.5 −×

Si.8Ge.2/Si 6P 1.4 850 12.9 20105× 3104 −×

Bulk SiGe P 1 130 5.63 1020 3103.0 −×

Bulk SiGe N 2 -200 6.06 1020 31033.0 −×

Length, m Width, m Thickness, m

Well 6103 −× 6102 −× 101020 −×

Barrier 6103 −× 6102 −× 1010100 −×

The electrons flowing in the well are taken as electron gas. When the hot electron gas

flows through the thin film, it loses energy to holes and phonons, however, it also

gains energy from the external applied voltage. At high electric field the thin film

68

thermoelectric semiconductor produces high energetic electrons and the equilibrium

between electrons and phonons decreases because of high rate of heat generation. At

higher electric field, the electron temperature is generally higher than the phonon

temperature, as shown in Figure 3.13. In addition, the electron and phonon interaction

rate are assumed high enough to saturate the electron drift velocity where the electrons

become so energetic that they can not dissipate energy to the phonons through

collisions. Therefore, they operate far out of equilibrium from the phonons or lattice

vibrations.

260

270

280

290

300

310

320

330

0 0.5 1 1.5 2 2.5 3co-ordinate (micro m)

tem

pera

ture

(K)

260

270

280

290

300

310

320

330

tem

pera

ture

(K)

electron temperature

phonon temperature

Figure 3.13. the variations of electron temperature in a thin film thermoelectric element at different electric field; (---) indicates the phonon temperature, (—) defines the electron temperature at 12.5 Kamp/cm2, (---) shows the electron temperature at 25 Kamp/cm2 and () represents the electron temperature at 125 Kamp/cm2.

69

The micro-scale heat transport phenomena include electron transport, phonon transport

and more importantly, heat generation during electron-phonon interactions. From T-s

diagram, which is shown in Figure 3.14, it is observed that the electron flow in n-type

thermoelectric element is opposite to the current flow whilst the hole flow in p-type

thermoelectric element follows the same direction of current flow. The direction of

phonons is the opposite of electrons and holes. The phonons and the collisions of

electrons and phonons create the inherent dissipation and hence, decrease the device

performance.

The close loop, given by g-h-i-j-k-l-g, establishes the T-s diagram based on electron

temperature and electron current. In this cycle, cooling capacity is equal to (TL x g-l)

Figure 3.14. The temperature-entropy diagram of a superlattice thermoelectric couple (both the p and n-type thermoelectric element are shown here)

250

260

270

280

290

300

310

-800000 -600000 -400000 -200000 0 200000 400000 600000 800000

entropy flux (W/m2 K)

tem

pera

ture

(K)

a

b

cd

e

f a'

b'c'

d' g

h

ij

k

lTL

TH

p-armn-arm

cold reservoir (Isotherm)

hot reservoir (Isotherm)

J

phonon flowphonon flow

electron flow hole flow

70

and the heating power is (TH x i-j). The coefficient of performance due to electron and

hole flow is given by (TL x g-l)/(TH x i-j- TL x g-l), which is large and quite close to the

Carnot COP. This is because the electron thermal conductivity is small as compared

with the phonon thermal conductivity and the total thermal conductivity is the sum of

electron and phonon thermal conductivities.

The energy dissipation due to the transfer of phonons from hot reservoir to cold

reservoir is shown by points a′-b′-c′-d′-a′, as shown in T-s plot of Figure 3.14, and it is

noted that more heat is dissipated at the cold end (a′d′ > b′c′). The performance of the

device is given by state points “a-b-c-d-e-f-a. For example, the net cooling power is

defined by, QL, net = TL(gl - a′d′) = TL (af), where “gl” is the entropy flux at the cold end

due to electrons and holes flowing from cold side to the hot end and “a′d′” is the heat

dissipation due to phonons or lattice vibrations and phonon-electron collisions. The net

heating power is QH, net = TH(ij - b′c′) = TH (cd), where “ij” is the entropy flux at the hot

side due to electron and hole and “b′c′” is the phonon flow at the hot end. So the net

performance is calculated as TL (af)/ TH (cd) - TL (af).

It is informative to show the effect of collisions between electrons and phonons. Figure

3.15 shows the temperature-entropy diagrams for collisions and non collision effects.

Due to collisions, the decrease of cooling power is ( )DDAATSST LcjfluxcjfluxL ′+′=− 1,2, ,

and the increase of heating power is ( )CCBBTSST HhjfluxhjfluxH ′+′=− 1,2, . So the net

amount of input power is increased by ( ) ( )[ ]DDAATCCBBT LH ′+′−′+′ as shown in

Figure 3.15. The net coefficient of performance is dropped by approximately 7.4 %

due to collisions of electrons and phonons, and holes and phonons when the electric

field is 75 KA/cm2. Here it is important to note that the cooling power density of the

71

commercial thermoelectric cooler is limited to several watts per square centimeters,

and the thin film thermoelectric coolers can achieve cooling power densities as high as

several thousands of watts per square centimeters (Vashaee, and Shakouri, 2004).The

performances of the superlattice thermoelectric cooler are furnished in table 3.7.

Table 3.7 Performance analysis of the superlattice thermoelectric cooler at various electric field.

With out collision effects With collision effects J

kA/cm2 Entropy flux at hj W/cm2K

Entropy flux at cj W/cm2K

COP Entropy flux at hj W/cm2K

Entropy flux at cj W/cm2K

COP

Input energy loss

due to collisions

(%) 25 95.8 37.9 0.55 96.6 37.1 0.52 4.8 75 185.9 116.9 1.30 189.7 115.8 1.21 7.4 125 414.9 301.4 1.88 427.2 299.1 1.7 11.8 250 658.2 467.1 1.77 684.3 462.8 1.55 15.3 375 911.8 616.1 1.55 958.1 607.2 1.32 19.4 500 2524.6 1102.6 0.65 2830.7 1063.3 0.51 32.7

For the case of optimum operation 125 kA/cm2

QH, ncolli 0.744 mW QH, colli 0.768 mW

QL, ncolli 0.489 mW QL,colli 0.484 mW Pin, ncolli 0.255 mW

Pin, colli 0.302 mW Loss due to collisions Qcolli = 0.027 mW

72

Figu

re 3

.15.

the

tem

pera

ture

-ent

ropy

dia

gram

s of

p a

nd n

-typ

e th

in fi

lm th

erm

oele

ctri

c el

emen

ts. T

he c

lose

loop

A-B

-C-D

sho

ws

the

T-s

dia

gram

whe

n th

e co

llisi

on t

erm

s ar

e ta

ken

into

acc

ount

and

the

clo

se l

oop

A′-B

′-C′-D

′ in

dica

tes

the

T-s

dia

gram

whe

n co

llisi

ons a

re n

ot in

clud

ed. E

nerg

y di

ssip

atio

n du

e to

col

lisio

ns is

show

n he

re.

240

250

260

270

280

290

300

310

320 -500

000

-400

000

-300

000

-200

000

-100

000

010

0000

2000

0030

0000

4000

0050

0000

entr

opy

flux

(W/m

2 K)

temperature (K)

hot r

eser

voir

(Isot

herm

al)

cold

rese

rvoi

r (I

soth

erm

al)

J

p-ar

mn-

arm

ener

gy d

issi

patio

n du

e to

col

lisio

n

S flu

x, c

j1

S flu

x, c

j2

S flu

x, h

j1

S flu

x, h

j2

AA′

B

B′

C C

′

D

D′

TH

TL

73


From the macro viewpoints, this chapter presents the pedagogical value of a

temperature-entropy flux diagram, where it shows clearly how the energy input in a

thermoelectric cooling device has been consumed by both reversible (Peltier, Seebeck

and Thomson) and irreversible (Joule and Fourier) phenomena. A similarity has been

found between the Seebeck coefficient and entropy flux. T-α diagram indicates the

reversible effect of a thermoelectric pair. T-α is same as the Carnot cycle if the

thermoelectric arms are considered adiabatic. But the temperature dependent seebeck

coefficient indicates the real T-α cycle. The effect of Thomson heat has been

incorporated in the performance estimation of a thermoelectric cooler. This effect,

albeit 5% or so, increases irreversibility at the cold junction. Using the macroscopic

BTE equation approach, the transient behavior of a pulsed-operated thermoelectric

cooling device is accurately presented. The distribution of input power in such a device

for both the generation of useful effects as well as in overcoming the dissipative losses

could be accurately captured by a T-s diagram. The present model predicts accurately

the maximum temperature difference between the hot and cold junctions with respect

to the functions of the pulse current. From the micro viewpoints, this Chapter also

presents the T-s diagram of a superlattice thermoelectric element where the collision

effects of electrons, holes and phonons become significant.

The next Chapter starts with the experimental realization of adsorption isotherms and

characteristics to model an electro-adsorption chiller (EAC).

74

Chapter 4

Adsorption Characteristics of Silica gel + Water

Prior to electro-adsorption chiller (EAC) cycle simulation, it is essential to determine

the adsorption characteristics of water vapor on silica gel, namely the isotherms,

monolayer vapour-uptake capacity and the enthalpy of adsorption. Such data are

needed to determine the energetic performance of adsorption cooling cycles. This

chapter presents (1) the measurement of surface characteristics such as surface areas,

pore size distributions, pore volume, and skeletal densities of silica gels through

nitrogen gas adsorption at liquid nitrogen temperature, (2) the adsorption isotherm

characteristics of water vapor onto two types of silica gel (namely the Fuji Davison

type ‘A’, and ‘RD’) by using a volumetric technique at controlled temperatures

ranging from 303 K to 338 K and at pressures between 500 and 7000 Pa. The

experimental isotherms are then fitted with the Tóth equation which could be used in

the simulation code and it has been found that the predicted isotherms are within the

experimental errors. Extending the Tóth isotherm equation, the monolayer uptake

capacity and the enthalpy of adsorption are also calculated. For verification, these

experimental isotherms and the corresponding computed isosteric enthalpies of

adsorption are compared with published data available from the literature.

75

4.1 Characterization of silica gels

The common types of silica gel found in the commercial chillers are the Fuji Davison

type ‘A’, and ‘RD’. It is well known that the thermophysical properties of silica gels

can directly affect adsorption performance via parameters such as the equilibrium

adsorption capacity and the molecular and surface pore sizes, etc. For an adsorbent to

work well, it should possess a large specific surface area and a relatively good thermal

conductivity. The thermophysical properties of silica gel to be measured are (i) the

surface area, (ii) pore size, (iii) pore volume, (iv) porosity and (v) skeletal density.

Pore size and surface area measurements are performed on the Micrometrics Gas

Adsorption analyzer (model ASAP-2010 with analysis software Version 4) and it

employs the liquid Nitrogen at 77K as the filing fluid. Prior to the experiment, samples

of silica gel are degassed at 413 K for 48 h, to achieve a residual vacuum within the

sample tube of less than 10 mPa. To compute the surface areas, the Brunauer Emmet

Teller (BET) plots of nitrogen adsorption data are selected over the range Pr = 0.03 –

0.3 (following the procedure outlined by IUPAC, 1985) to get the best linear fit - Pr

denotes the relative pressure or the ratio of the prevailing vapor pressure to the

corresponding saturated pressure. A sample of the characterization data are tabulated

in Table 4.1. Using the software provided by the analyzer (Density Functional Theory

DFT- Version 2), the pore volumes and size distributions of the silica gel samples are

computed. The pore size distributions for the two types of silica gel (type “RD” and

type “A”), are depicted in Figure 4.1. It is noted that the Barrett-Joyner-Halenda (BJH)

method has also been investigated for calculating the pore size distributions of the

silica gels but the results are found to be less satisfactory as the method is usually used

to calculate the mesopore structures.

76

The skeletal densities of dry silica gels are determined by the automated Micromeritics

AccuPyc 1330 pycnometer. Helium gas is employed in the test because it behaves

ideally. In the experiments, each silica gel sample is regenerated at 413 K for 24 h and

a dry mass sample is then introduced to the pycnometer. Each sample is tested three

times and the average value is taken. Using the measured skeletal densities and pore

volumes (DFT method), the particle bulk densities of silica gels could then be

calculated. The experimentally determined thermo-physical properties are summarized

in Table 4.2 and one observes that the thermophysical properties of these two types of

silica gel are similar. Although the type 'RD' silica gel has a slightly higher thermal

conductivity than type ‘A’ silica gel, the most significant difference between these two

types of silica gels lies in their water vapor uptake characteristics.

77

Table 4.1 Characterization data for the type ‘RD’ and ‘A’ silica gel

Type RD Type A

Pr 0.048688 0.055774 0.063324 0.071464 0.080186 0.089197 0.098848 0.108946 0.119269 0.130128 0.141184 0.152436 0.164243 0.176378 0.188519 0.201084 0.213885 0.226838

q* (cm3/g STP)

184.2954 188.3066 192.1586 195.9647 199.7403 203.3650 207.0079 210.5666 213.9935 217.4258 220.7799 224.0624 227.3611 230.6294 233.7987 236.9616 240.0596 243.1273

Pr

0.050404 0.059275 0.069060 0.079633 0.091123 0.103225 0.115798 0.129227 0.143242 0.157589 0.172680 0.188005

q* (cm3/g STP)

160.577 164.4663 168.2734 171.9793 175.4915 179.0015 182.3687 185.7076 188.9687 192.0915 195.2137 198.2016

78

Figu

re 4

.1. P

ore

size

dis

trib

utio

n (D

FT-S

lit m

odel

) for

two

type

s of s

ilica

gel

.

Incremental Pore Volume/ cm3 g-1

Pore

wid

th/ n

m

79

Table 4.2. Thermophysical properties of silica gels*

Property Value

Type ‘A’ Type ‘RD’

BET/N2 surface area c

BET constant c

BET Volume STP c

Range of Pr c

(716 ± 3.3) m2⋅g-1

293.8

164.5 cm3⋅g-1

0.05 ~ 0.19

(838 ± 3.8) m2⋅g-1

258.6

192.5 cm3⋅g-1

0.05 ~ 0.23

Pore size c (0.8 ~ 5) nm (0.8 ~ 7.5) nm

Porous volume c

Micropore volume c

Mesopore volume c

0.28 cm3⋅g-1

57%

43%

0.37 cm3⋅g-1

49%

51%

Skeletal density d 2060 kg⋅m-3 2027 kg⋅m-3

Particle bulk density e 1306 kg⋅m-3 1158 kg⋅m-3

Surface area f 650 m2⋅g-1 720 m2⋅g-1

Average pore diameter f 2.2 nm 2.2 nm

Porous volume d 0.36 cm3⋅g-1 0.4 cm3⋅g-1

Apparent density f** 730 kg⋅m-3 700 kg⋅m-3

Mesh Size f 10 - 40 10 – 20

pH f 5.0 4.0

Water content f < 2.0 mass %

Specific heat capacity f 0.921 kJ⋅kg-1⋅K-1 0.921 kJ⋅kg-1⋅K-1

Thermal conductivity f 0.174 W⋅m-1⋅K-1 0.198 W⋅m-1⋅K-1

* Manufacturer's representative chemical composition (dry mass basis): SiO2 (99.7%), Fe2O3 (0.008%), Al2O3 (0.025%), CaO (0.01%), Na2O (0.05%). c Remarks ASAP 2010 d Remarks AccuPyc 1330 e Remarks computed f Remarks Manufacturer ** This value is inclusive of bed porosity.

80

4.2 Isotherms of the silica gel + water

The next step is to determine the isotherm characteristics of silica gel + water, the

isosteric heat of adsorption, and the monolayer uptake capacity which are useful in the

simulation of the adsorption cooling cycle. In the literature, two methods are available

for the above-mentioned measurements, namely (i) the thermal-gravimetric analyzer

(TGA) and (ii) the constant-volume-variable pressure (CVVP) apparatus. Owing to

budget limitation, the CVVP apparatus has been designed and employed for the

experiment. This section investigates experimentally the isotherm adsorption of water

vapor onto Fuji Davison type ‘RD’ and type ‘A’ silica gel.

4.2.1 Experiment

Figure 4.2 shows the schematic diagram of the CVVP which comprises (i) a SS 304

charging tank with a volume of 572.64 ± 27 cm3, inclusive of related piping and

valves, (ii) a SS 304 dosing tank of volume 698.47 ± 32 cm3, inclusive of related

piping and valves, (iii) an evaporator flask with a volume of 2000 cm3 and (iv) the

electro-pneumatic gate valve. The charging tank is designed to have a high aspect

ratio, so that the adsorbent could be spread on the large flat base. Its external wall is

coiled with copper pipes to enable in-situ regeneration of the silica gel using a hot

silicone oil bath supplied at 140 °C.

81

`

The volumes of the charging and dosing tanks is calibrated by using calibrated

standard volume (210.9 ± 0.2 cm3) with a helium gas where the helium is used to fill

the mentioned tanks, sequentially. By measuring the pressures of the combined

volumes at isothermal conditions, the volumes of both the charging and the dosing

tanks are calculated using ideal gas law. Pressures in the two tanks are measured with

two calibrated capacitance manometers (Edwards barocel type 622; 10 kPa span; total

error in Pa: if 90 < P/Pa < 900, total error in Pa = 100⋅[0.0025 + 2.25⋅10-6 + 1.764⋅10-

3⋅P2]1/2, if 900 < P/Pa < 9000, total error in Pa = 100⋅[0.0025 + 2.25⋅10-6 + 7.84⋅10-

4⋅P2]1/2; maximum operating temperature: T = 338 K). Similarly, the temperatures are

measured with three Pt 100 Ω Class A resistance temperature detectors (RTD). The

ARGON GAS CYLINDER

OVERFLOW

DRAIN TEMPERATURE CONTROLLED BATH

TO VACUUM PUMP

PNEUMATIC VALVE TEMPERATURE LINING

+ HEATING TAPE

+ INSULATION

HEATING COILS

SILICA GEL

TANK DOSING TANK

MAGNETIC STIRRER ROD

TEMPERATURE CONTROLLED BATH

DISTILLED WATER IN EVAPORATOR

Pc Pd Pe T1 T2 T3

Figure 4.2. Schematic diagram of the constant volume variable pressure test facility: T1, T2, T3 = resistance temperature detectors; Ps, Pd, Pe = capacitance manometers.

82

RTD for the charging tank, T1 in Figure 4.2, is in direct contact with the silica gel to

give representative adsorbent temperature. Except for the 3.175 mm vacuum adaptors

used to seal the RTD probes, all other inter-connecting piping (SS 316), stainless steel

vacuum fittings, and vacuum-rated stainless steel diaphragm valves (Nupro, Swagelok)

are chosen to be 12.7 mm to ensure good conductance during evacuation. Isothermal

condition in the two tanks is obtained by immersing them in a temperature-controlled

bath (± 0.01 K). Measurement errors caused by a temperature gradient along the

piping, fittings and valves, are mitigated with thermal jacket, whose temperature is

maintained by the same bath. All other exposed piping and fittings are also well

insulated.

A pictorial view of the CVVP apparatus is shown in Figure 4.3. Vacuum environment

is maintained by a two-stage rotary vane vacuum pump (Edwards bubbler pump) with

a water vapor pumping rate of 315 ×10-6 m3 s-1. To prevent back migration of oil mist

into the apparatus, an alumina-packed foreline trap is installed upstream of the vacuum

pump. Argon, with a purity of 99.9995 per cent, is sent through a column of packed

calcium sulphate before being used to purge the vacuum system. Upon evacuation, a

residual argon pressure of 100 Pa could be achieved in the apparatus.

The evaporator flask is immersed in a different temperature-controlled bath (precision

of control: ± 0.01 K). It is first purged with argon with a purity of 99.9995 per cent and

evacuated by the vacuum pump. Distilled water is then charged into the evaporator. Its

temperature could be regulated to supply water vapor at the desired pressure. An

immersion magnetic stirrer is used to ensure uniformity of temperature inside the

evaporator flask.

83

All pressure and temperature readings are continuously monitored by a calibrated 18-

bit ADDA data logger (Fluke, Netdaq 2640A). The calibration of all pressure

transmitters, temperature sensors, and the data logger are traceable to national

standards. In the present test facility, the major measurement bottleneck lies in the

measurement of vapor pressure.

The dry mass of silica gel is determined by the calibrated moisture balance (Satorious

MA40 moisture analyzer, uncertainty: 0.05 percent traceable to DKD standard) at 413

Figure 4.3: The experimental set-up of the constant volume variable pressure (C.V.V.P) test facility

Water bath

Pressure sensor

Evaporator

Charging and dosing tanks

84

K. The mass of silica gel is continuously monitored and recorded when there is no

change in dry silica gel mass and silica gels (range from 0.1 g to 1.25 g) are introduced

into the charging tank.

The dosing tank, the charging tank and all related piping systems are purged by

purified and dried argon, and evacuated. The silica gel is again regenerated in situ at

413 K for 24 h. At the end of regeneration process, the test system is purged with

argon and evacuated. The silica gel is further regenerated at 413 K for another 8 hours,

and the test system is evacuated again. Based on measurements, there is no measurable

interaction between the inert gas and the adsorbent. The effect of the partial pressure of

argon in the tanks is found to be small. Nevertheless, the partial pressure of water

vapor has been adjusted for the presence of argon so as to avoid additional systematic

error.

Water vapor is firstly charged into the dosing tank. Prior to charging, the tank

temperature is initially maintained at about 5 K higher than the evaporator temperature

to eliminate the possibility of condensation. The effect of condensation would cause

significant error in pressure reading. This would lead to significant error in calculating

actual mass of water vapor, which is charged into the dosing tank. The evaporator is

isolated from the dosing tank soon after the equilibrium pressure is achieved inside the

dosing tank. Taking the effect of residual argon, the mass of water vapor is determined

via pressure and temperature. The temperatures of the charging tank and the dosing

tank are subsequently raised to the typical adsorber and desorber temperature of an

adsorption cooling cycle. Once the test system reaches thermodynamic equilibrium at

the desired temperature, the electro-pneumatic gate valve between the dosing and

85

charging tank is opened, and both tanks are allowed to approach equilibrium. The

temperatures of both tanks are adjusted to the desired temperature via water baths

before pressure and temperature measurements are taken for both tanks.

With the known initial mass of dry silica gel adsorbent, the temperature of the test

system is varied to measure the uptake of water vapour at a different temperature and

pressure. The test system is subsequently evacuated. Silica gel is regenerated, and the

apparatus is purged with purified and dried argon before water vapor at different initial

pressure is charged into the dosing tank to enable measurement along the isotherms. A

number of test conditions are repeated at least twice with a fresh batch of silica gel to

ensure that repeatability is achieved to within the reported experimental uncertainty.

4.2.2 Results and analysis for adsorption isotherms

Adsorption equilibrium data of water vapor on these two different types of silica gels

are obtained. The measured isotherm data for type ‘A’ and type ‘RD’ silica gels are

furnished in table 4.3 and table 4.4 respectively. Figures 4.4 and 4.5 depict the

experimental equilibrium data for the type ‘RD’ and ‘A’ silica gel with water vapor,

respectively. The uncertainty of pressure ranges from ( 10± to 200 ) Pa.

86

Table 4.3. The measured isotherm data for type ‘A’ silica gel.

T P1 q* K

kPa

kg kg-1

298.15 5.1054 0.4014 304.15 0.6675 0.0753 0.9523 0.1084 1.3888 0.1593 2.3351 0.2682 3.3496 0.3535 4.1878 0.3651 5.2226 0.3948 310.15 0.9473 0.0717 1.3302 0.1019 1.9090 0.1469 3.0696 0.2376 3.9451 0.3159 4.7446 0.3465 5.6442 0.3860 316.15 1.2730 0.0673 1.7980 0.0946 2.5116 0.1334 3.7915 0.2087 4.7202 0.2705 5.3888 0.2986 323.15 1.7948 0.0624 2.4281 0.0855 3.1658 0.1192 4.7487 0.1725 5.7795 0.2121 338.15 3.2807 0.0489 4.0972 0.0637 5.4151 0.0814

87

Table 4.4. The measured isotherm data for type ‘RD’ silica gel.

T P1 q* K

kPa

kg kg-1

298.15 2.9512 0.4535 304.15 0.5653 0.0916 0.7774 0.1246 1.4679 0.2331 2.2804 0.3507 2.8829 0.4018 3.4849 0.4409 4.4327 0.4465 310.15 0.7591 0.0897 1.0747 0.1206 2.0115 0.2182 2.9984 0.3180 3.5595 0.3640 4.1266 0.4089 4.6402 0.4270 316.15 1.0948 0.0853 1.5080 0.1132 2.6504 0.2012 3.8004 0.2876 4.3298 0.3235 4.7023 0.3454 4.7865 0.3572 5.3335 0.3827 323.15 1.5294 0.0794 2.0491 0.1037 3.4923 0.1744 4.7954 0.2366 5.3003 0.2630 338.15 2.8814 0.0638 3.7069 0.0786 5.3634 0.1191

The performance of type ‘RD’ silica gel water system employed by the adsorption

chiller manufacturer (NACC. PTX data, 1992) available in the form of the Henry's

isotherm (Suzuki, 1990) is also superimposed for comparison. The data of the

88

manufacturer have been obtained for temperatures from 328 to 358 K, and for

pressures from 1000 to 5000 Pa. However, the uncertainties of measurements are not

available from the manufacturer. Apparently the manufacturer is focusing more on the

desorber conditions. Comparing the manufacturer’s data of type ‘RD’ silica gel with

the experimental data for the same type of silica gel, the performance of type ‘RD’

silica gel-water system is found to be consistent in trend with the manufacturer’s data,

as shown in Figure 4.6. From Figures 4.4 and 4.5, the isotherms of T = 304 K and T =

310 K for both type ‘RD’ and ‘A’ silica gel-water systems exhibit signs of onset of

monolayer saturation at higher pressures. The Tóth’s equation (Tóth, 1971 and Suzuki,

1990) is found to fit the experimental data of both type ‘RD’ and type ‘A’ silica gel-

water systems well and is therefore used to find the isotherm parameters and isosteric

heats of adsorption The form of the Tóth’s equation is given as:

( ) ( ) [ ] 11 1

10

1

)(exp1exp

*tt

gadsm

gadso

PTRHqKPTRHK

q⋅⋅∆⋅+

⋅⋅∆⋅= , (4.1)

where q* is the adsorbed quantity of adsorbate by the adsorbent under equilibrium

conditions, qm denotes the monolayer capacity, P1 the equilibrium pressure of the

adsorbate in gas phase, T the equilibrium temperature of gas phase adsorbate, Rg the

universal gas constant, adsH∆ the isosteric enthalpies of adsorption, Ko the pre-

exponential constant, and 1t is the dimensionless Tóth’s constant. Tóth (Tóth, 1971)

observed that the empirical constant 1t , was between the interval of zero to one insofar

as his experiments concerned but 1t could actually be greater than one (Valenzuela and

Myers., 1989).

89

0

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

01

23

45

67

P 1(k

Pa)

q (kg kg-1

) Figu

re 4

.4.

Isot

herm

dat

a fo

r w

ater

vap

or o

nto

the

type

‘R

D’

silic

a ge

l; fo

r ex

peri

men

tal d

ata

poin

ts w

ith: ()

T =

303

K; ()

T =

308

K; (

) T =

31

3 K

; (

) T

= 32

3 K

; (+

) T

= 33

8 K

, for

com

pute

d da

ta p

oint

s w

ith

solid

lin

es f

or t

he T

óth’

s eq

uatio

n, f

or m

anuf

actu

rer’

s da

ta p

oint

s w

ith: (

----

----

) T =

323

K o

f NA

CC

; 2 (⎯ −

− ⎯

) T =

338

K o

f NA

CC

,2 an

d fo

r on

e ex

peri

men

tal

data

poi

nt w

ith:

()

T =

298

K f

or t

he

estim

atio

n of

mon

olay

er sa

tura

tion

upta

ke.

90

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

01

23

45

67

P 1 (k

Pa)

q (kg kg-1

)

Figu

re 4

.5. I

soth

erm

dat

a fo

r w

ater

vap

or o

nto

the

type

‘A

’ si

lica

gel;

for

expe

rim

enta

l dat

a po

ints

with

: ()

T =

304

K; ()

T =

310

K; (

) T =

316

K; ()

T =

323

K; (+)

T =

338

K

, for

com

pute

d da

ta p

oint

s w

ith s

olid

line

s fo

r th

e T

óth’

s eq

uatio

n, a

nd f

or o

ne

expe

rim

enta

l da

ta p

oint

with

: (

) T

=298

K f

or t

he e

stim

atio

n of

mon

olay

er

satu

ratio

n up

take

.

91

The adsorption isotherm parameters and the isosteric enthalpies of adsorption for the

two grades of silica gel (type ‘RD’ and type ‘A’) + water system are summarized in

Table 4.5. Cremer and Davis (1958) first reported a value of 2.98⋅103 kJ⋅kg-1 for

adsH∆ , though neither the experimental uncertainty nor the grade of silica gel was

reported. This value has since been widely used. While Chihara and Suzuki (1983)

reported a adsH∆ that ranged from (2.8⋅103 to 3.5⋅103) kJ⋅kg-1 for the type ‘A’ silica gel,

Sakoda and Suzuki (1984) later adopted the value of 2.8⋅103 kJ⋅kg-1. Referring to Table

4.5, the derived monolayer capacity of the type ‘A’ silica gel is estimated to be 0.4

kg⋅kg-1, which is higher than the value, 0.346 kg⋅kg-1, reported by Chihara and Suzuki

(1983). The derived monolayer capacity of the type ‘RD’ silica gel is similarly

estimated to be 0.45 kg kg-1.

Referring to Figures 4.4 and 4.5, the estimation of the monolayer capacity for both

types of silica gel is based on experimental measurement of water vapor uptake for

both types of silica gel at 298 K near the saturation pressure while maintaining (8 to

10) K of superheat. It is observed that there is a significant difference between the

present measurements and those of references (Chihara et al., 1983 and Sakoda et al.,

1984) for the type ‘A’ silica gel. Moreover, while the Tóth’s equation is employed to

correlate the present experimental data, the Freundlich’s equation is used to correlate

the data in the two said references. The present data for the type ‘RD’ silica gel with

water vapor shows good consistency with the chiller manufacturer’s correlation

(Manufacturer’s Proprietary Data, Nishiyodo Air Conditioning Co Ltd., Tokyo, Japan,

1992).

92

Table 4.5. Correlation coefficients for the two grades of Fuji Davison silica gel -

water systems (The error refers to the 95% confidence interval of the

experimental data)

Ko ∆Hads qm Type (kg⋅kg-1⋅kPa-1)

(kJ⋅kg-1)

(kg⋅kg-1)

t1 Remarks

‘A’ (4.65 ± 0.9)⋅10-10 (2.71 ± 0.1)⋅103 0.4 10 By Tóth

‘RD’ (7.30 ± 2)⋅10-10 (2.693 ± 0.1)⋅103 0.45 12 By Tóth

‘RD’ (Manufacturer) 2.0⋅10-9 2.51⋅103

By Henry

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 2 3 4 5 6 7

q* (k

g kg

-1)

P1 (kPa)

T = 303 K

T = 338 K

Figure 4.6. Isotherm data for water vapor with the silica gel by the present study and other researchers with: (⎯⎯) type ‘A’ by the present study; (⎯⎯) type ‘A’ by Chihara and Suzuki; (⎯∆⎯) type ‘RD’ by the present study; () type ‘RD’ by Cho and Kim; (-------) type RD by NACC.

93

It is evident that the water vapor uptake capacity of the type ‘RD’ silica gel, as

presented by Cho and Kim (1992) is much higher than those obtained by other

researchers. However, details of the experimental method and procedures are not

available.


Equilibrium studies of water vapor on the two Fuji Davison type silica gels have been

investigated experimentally. The ranges of pressure and temperature covered in these

experiments are within the operating conditions expected of an adsorption chiller. The

derived monolayer capacities (q*) for water vapor of the type ‘A’ and type ‘RD’ silica

gels have been calculated to be about 0.4 kg⋅kg-1 and 0.45 kg⋅kg-1 respectively. The

Tóth’s equation is found to be able to sufficiently describe the performance of the type

‘A’ and type ‘RD’ silica gels with water vapor. The computed values of the isosteric

heats of adsorption are lower than those found in the literatures (Chihara and Suzuki,

1983, Sakoda and Suzuki, 1984 and Cremer and Davis, 1958).

Based on the measured adsorption isotherms (fitted Tóth correlation) of water vapor on

silica gel at different temperatures and pressures, isosteric enthalpy of adsorption

( adsH∆ ), the mathematical model for the electro-adsorption chiller (EAC) is now fully

equipped with the real behaviour of the silica gel + water pair and details of the

simulation will be described in the chapters to follow.

94

Chapter 5

An electro-adsorption chiller: thermodynamic modelling and its

performance

5.1 Introduction

This chapter presents (1) a general thermodynamic approach for formulating the mass,

energy and entropy balance equations of an adsorbate-adsorbent system and (2) the

theoretical modeling of an electro-adsorption chiller (EAC). Such a modeling approach

enables one to perform detailed parametric study of the key variables that affect the

performance of an EAC and hence, optimizing the EAC with respect to a selected

cooling capacity. The model incorporates the physical characteristics of adsorbent,

adsorption isotherms and adsorption kinetics, expressing them in terms of pressure (P),

the amount of adsorbate(q), and temperature (T) of silica gel + water systems. In

addition, the model also accounts for the behaviour of thermoelectrics and the overall

heat transfer conductance of the reactor beds.

For the theoretical development, this chapter employs the thermodynamic property

fields of adsorbate-adsorbent system that is based on the works of Feuerecker et al.

(1994) where it was originally developed for absorption (liquid-vapour) system. This

method yields the energy and entropy balances, from which a specific heat capacity

expression for silica gel - water as a function of P, T and q, are derived.

95

5.2 Thermodynamic property fields of adsorbate-adsorbent system

The complete thermodynamic property fields for a single-component adsorbent +

adsorbate system is proposed and developed in the sections below. The equations for

computing the actual specific heat capacity, partial enthalpy and entropy are essential

for the analyses of single-component adsorption processes. Using the framework of

adsorption thermodynamics (Lopatkin, 2001), the general equations for the differential

and the integral heat capacities of an adsorbed substance on the surface of solid

adsorbent at constant pressure could be described. It is assumed that the differential

and integral heat capacities are taken to be equal for the case of ideal gases, but

additional terms are needed for them to take into account of temperature increment

dT , i.e., non-equilibrium conditions, within the control volume, and this additional

terms are added only to the heat capacity of gaseous phase.

Recently, the thermodynamic functions such as Gibbs free energy, enthalpy and

entropy on the basis of isothermal immersion of adsorbent in the gas phase for

describing the adsorbate-adsorbent systems have been derived in details by Myers

(Myers, 2002), and followed by defining the heat capacity of the adsorbate-adsorbent

system under condensed (liquid) phase and gaseous phase in terms of the constituents’

properties such as the solid adsorbent and the adsobate, where the latter is assumed to

be perfect gas.

Using the proposed formulation (developed in the thesis) and the experimentally

measured adsorption isotherm data from the literature (Bjurström et al., 1984), the

specific heat capacity of silica gel + water system is analysed and compared to

highlight the major differences between the conventional uncorrected form of the

96

specific heat capacity (Ruthven, 1984, Suzuki, 1990, Saha et. al, 1997) and the

improved simplified expression is also presented.

5.2.1 Mass balance

The mass balance of the adsorbent + adsorbate system can be represented by

gasgastabea Vmmm ρ−=.* , (5.1)

where abem is the mass of adsorbent, tm the total amount of gas in the system, gasV the

gas phase volume, and gasρ the gas phase density. *am is the adsorbed quantity of

adsorbate under equilibrium conditions

5.2.2 Enthalpy Energy and Entropy Balances

The thermodynamic state of the adsorbent-adsorbate control volume can be accurately

expressed in terms of its three measurable properties, namely, the P, T, and ma, where

ma denotes the mass of adsorbate (or q Mabe in kg). Tracking the change of a

thermodynamic variable from an initial to a final state or process, the expressions of an

extensive thermodynamic quantity, such as enthalpy, internal energy or entropy

involving a single component adsorbate + adsorbent system at a given mass of

adsorbent, Mabe, are shown as follows:

( )

( ) ( )∫∫

∫∫

∫ ∫∫

⋅⋅−⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⋅−⋅+

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⋅⋅=

∂∂

+∂∂

+∂∂

=

P

PT

sys

sysa

m

TaT

sys

sys

m

aTmTPa

T

Tabepabe

aa

a

dPTM

dmmPT

M

dmmHdTcM

dPPHdm

mHdT

THmTPH

a

a

ao

1

1

11

),,(

0

1

0 ,,,

αρ

αρ

,

(5.2)

97

where sysM is the mass, sysρ the density of the adsorbent + adsorbate, and abepc , the

specific heat capacity of the adsorbent.

Using the definition (U = H-PV), the internal energy for the adsorbent + adsorbate

system is given by

( ) ( ) ( )

( )

( )

( )∫∫

∫

∫∫

∫∫∫∫

⋅⋅−⋅⋅−⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅−⋅−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⋅−⋅⋅−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⋅⋅=

∂∂

−∂∂

−∂

∂−=

P

PT

sys

sysm

oa

TPa

sys

sys

sys

sys

a

m

TaT

sys

sys

m

aTmTPa

T

Tabepabe

aa

a

dPPTM

dmm

MP

dmmPPT

M

dmmHdTcM

dPP

PVdmmPVdT

TPVdHmTPU

a

a

a

ao

11

1

,21

0

11

0 ,,,

1

),,(

βαρ

ρ

ρρ

βαρ

, (5.3)

and similarly, the entropy is given by

( )

∫

∫∫∫

∫∫∫

⋅⋅−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⋅−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⋅⋅=

∂∂

+∂∂

+∂∂

=

P

PT

sys

sys

a

m

TaT

sys

sysm

aTmTPa

T

T

abepabe

aa

a

dPM

dmmPM

dmmSdT

Tc

M

dPPSdm

mSdT

TSmTPS

aa

ao

1

1 0

1

0 ,,

,

),,(

αρ

αρ

(5.4)

where β⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅=

amT

sys

sys P,

1 ρρ

is the isothermal compressibility factor and

Tα ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅−=

amP

sys

sys T,

1 ρρ

is the thermal expansion factor of the system.

Expressing ( ) TmTPa

a

mH

,,1

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ ,

( ) TmTPaa

mS

,,1

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ in terms of the P, T, and ma, they

can be further expanded in the following forms:

98

( )

( )

( ) ( )

( )aa

a

m

a

TmTPa

sysag

agTmTPa

TmTP

mV

TmTPT

TmTPhmH

⎟⎠⎞

⎜⎝⎛

∂∂

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−⋅

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

,,,

,,

1

,,1

1,,

1

1

ν

, (5.5)

( )

( )

( ) ( )

( )aa

a

m

a

TmTPa

sysag

agTmTPa

TmTP

mV

TmTP

TmTPsmS

⎟⎠

⎞⎜⎝

⎛∂

∂⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−ν

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

,,,

,,

1

,,1

1,,

1

1

, (5.6)

It is noted ( )TamP ∂∂ 1 , as found in equations 5.2, 5.3 and 5.4, can be obtained directly

from the isotherm characteristics of Chapter 4.

It is noted in all above equations, integration is performed from an initial reference

temperature, To, to temperature, T, initially at zero adsorbate state, then follows by

isothermal tracking along zero adsorbate state to adsorbate mass, ma, and finally from a

saturated pressure P1(T, ma), to a non-equilibrium pressure, P at constant T and

constant ma. Owing to the cumulative tracking of variables along the mentioned paths,

the equations define accurately the respective instantaneous states of an adsorbent-

adsorbate system insofar as local thermodynamic equilibrium could be assumed.

In equations 5.5 and 5.6, ( )amTP ∂∂ 1 relates to the Clapeyron relation as determined by

the local pressure and temperature, and they could be further expressed respectively as,

( )

( ) adsagTmTPa

HTmTPhmH

a

∆−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ ,,1

,,1

, (5.7)

and

99

( )

( ) THTmTPs

mS ads

agTmTPa

a

∆−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂ ,,1

,,1

. (5.8)

Based on the above equations, the total differential of the extensive enthalpy, internal

energy, and entropy for a single component adsorbate + adsorbent system can therefore

be respectively given by:

( )

( ) ( )

( )( ) ( )

444444444444 8444444444444 76444 8444 76

4444444444444 84444444444444 76

44444444 844444444 76

44444444444444444 844444444444444444 76

44 844 76

pressureduechange

Tsys

sysa

P

P TP

Tsys

sys

a

adsorbateofmasstoduechange

aTmTpa

TPtoduechange

P

Tsys

sysP

P mP

Tsys

sys

amsionthermaltoduechange

m

a

mPTaT

sys

sys

uptakemassalongchange

m

amP

ads

P

g

a

sysg

ads

a

sysg

gadsgp

adsorbent

abepabea

dPTM

dmdPTM

mdm

mH

dTTPT

MdPT

MT

dTdmmPT

MT

dTdmT

HT

mV

H

mVT

HTPc

dTcMmTPdH

a

a

a

a

a

a

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅+⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅

∂∂

+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂∂⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅

∂∂

+

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⋅−⋅∂∂

+

⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛

∂∆∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∆−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

⋅∆

++

⋅⋅=

∫

∫

∫

∫

αρ

αρ

αρ

αρ

αρ

ν

νν

ν

11

11

1

),(

),,(

1

11

1

,,,

&

1

,

exp

0 ,

1

0 ,1,

,

, (5.9)

Similarly, the change in the internal energy is given by

100

( )

( ) ( )

( )

( ) ( ) dPPTM

dmdPPTM

m

dmm

MPPdmmH

dTTPPT

MdPPT

MT

dTdmm

MPPT

dTdmmPPT

MT

dTdmTH

TmV

H

mVT

HTPc

dTcMmTPdU

Tsys

sysa

P

P TP

Tsys

sys

a

aTPa

sys

sys

sys

sysa

TmTPa

P

Tsys

sysP

P mP

Tsys

sys

m

a

mPTPa

sys

sys

sys

sys

m

a

mPTaT

sys

sys

m

amP

ads

P

g

a

sysg

ads

a

sysg

gadsgp

abepabea

a

a

a

a

a

a

a

a

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅⋅−⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅⋅

∂∂

−

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

⋅−−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂∂⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅⋅−⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅⋅

∂∂

−

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

⋅−

∂∂

−

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⋅−⋅⋅∂∂

−

⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛

∂∆∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∆−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

⋅∆

++

⋅⋅=

∫

∫

∫

∫

∫

βαρ

βαρ

ρ

ρρ

βαρ

βαρ

ρ

ρρ

βαρ

ν

νν

ν

1

1

11

1

1

,

,2

11

,,

1

,

0 ,,2

11

0 ,

11

0 ,1,

,

),(

),,(

(5.10)

and the change in the entropy is

101

( )

dPM

dmdPM

mdm

mS

dTTPM

dPM

T

dTdmmPM

T

TdTdm

TH

TmV

H

mVT

HTPc

dTT

cMmTPdS

Tsys

sysa

P

P TP

Tsys

sys

aa

TmTPa

p

Tsys

sysP

P mP

Tsys

sys

a

m

mPTaT

sys

sys

a

m

mP

ads

P

g

a

sysg

ads

a

sysg

gadsgp

abePabea

a

a

a

a

a

a

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂∂⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

−

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⋅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⋅∂∂

−

⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛

∂∆∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∆−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

⋅∆

++

⋅⎥⎦

⎤⎢⎣

⎡ ⋅=

∫

∫

∫

∫

αρ

αρ

αρ

αρ

αρ

ν

νν

ν

1

11

1

,,,

1

,

0 ,

1

0 ,1,

,

),(

),,(

. (5.11)

From equations (5.9-5.11), it is noted that they contain the partial terms of enthalpy,

internal energy and entropy of adsorbate only. The details of these partial terms can be

further expanded in terms of the P and T and these are given as follows:

( ) ( ) dPTm

TTm

MT

mmTPH

mmTPH

P

P a

sysT

TPa

sys

sys

sysT

sys

TPa

a

TPa

a

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂⋅∂

∂⋅−⋅⋅−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂⋅−⋅−⋅+

⎭⎬⎫

⎩⎨⎧

∂∂

=⎭⎬⎫

⎩⎨⎧

∂∂

∫1

1

2

,2

,

1

,

2111

),,(),,(

ρα

ρ

ρα

ρ,

(5.12)

where P1 is the pressure at equilibrium (maximum) uptake of the adsorbate. For

simplicity, the contribution from the second term of equation 5.12 (from P1 to P) to the

partial enthalpy with adsorbate uptake is assumed negligible and thus,

102

abeabe MTPa

abea

MTPa

abea

mMmTPH

mMmTPH

,,

1

,, 1

),,,(),,,(

⎭⎬⎫

⎩⎨⎧

∂∂

≈⎭⎬⎫

⎩⎨⎧

∂∂

, (5.13)

and similarly,

( )

dP

PmP

TmT

M

mMPT

mM

P

mmTPH

mmTPU

P

P

a

sys

a

sys

sys

sys

TPa

sys

sys

sys

sys

T

TPa

sys

sys

sys

sys

TPa

a

TPa

a

⋅

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂⋅∂

∂⋅+

∂⋅∂

∂⋅⋅−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅−

⋅−⋅

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅−⋅−

⎭⎬⎫

⎩⎨⎧

∂∂

=⎭⎬⎫

⎩⎨⎧

∂∂

∫11

1

22

2

,

,21

,

1

,

21

1

),,(),,(

ρρ

ρ

ρρρ

βα

ρ

ρρ

(5.14)

TPa

a

TPa

a

mmTPH

mmTPU

,

1

, 1

),,(),,(

⎭⎬⎫

⎩⎨⎧

∂∂

≈⎭⎬⎫

⎩⎨⎧

∂∂

, (5.15)

and also

∫ ⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂⋅∂

∂⋅+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅⋅+

⎭⎬⎫

⎩⎨⎧

∂∂

=⎭⎬⎫

⎩⎨⎧

∂∂

P

P a

sys

sys

sys

TPa

sys

syssys

T

MTPa

a

MTPa

a

dPTm

Mm

mmTPS

mmTPS

abeabe

1

1

2

2,

,,

1

,,

12

),,(),,(

ρ

ρ

ρρρ

α. (5.16)

abeabe MTPa

abea

MTPa

abea

mMmTPS

mMmTPS

,,

1

,, 1

),,,(),,,(

⎭⎬⎫

⎩⎨⎧

∂∂

≈⎭⎬⎫

⎩⎨⎧

∂∂

(5.17)

103

5.2.3 Specific heat capacity

The specific heat capacity of adsorbent-adsorbate system has two parts; Firstly, the

specific heat capacity derives from the solid adsorbent and secondly, the specific heat

capacity contributes from adsorbate that emanates from different processes associated

with the isothermal conditions. The specific heat capacity of the adsorbate has,

hitherto, assumed to be equal to that of the liquid-phase (Ruthven, 1984 and Chi Tien,

1994 and Saha et al., 1997) that is expressed as;

( )TPcMmcc lp

abe

aabepls ,1,, += (5.18)

where abepc , is the specific heat capacity of solid adsorbent, am is the amount of

adsorbate, abeM is the mass of adsorbent, P1 is the saturated pressure, T is the

temperature and lpc , indicates the specific heat capacity of adsorbate at liquid phase.

Recently some researchers view the specific heat capacity of the adsorbate to be equal

to the gas phase specific heat capacity (Chua et al., 1999, Saha et al., 2003 and Sami et

al., 1996) in the design and simulation of adsorption chiller, which is defined as,

( )TPcMmcc gp

abe

aabepgs ,1,, += , (5.19)

where gpc , is the specific heat capacity at gaseous phase.

In this thesis, the specific heat capacity of a single component adsorbent + adsorbate

system is proposed to be dependent on the isosteric heat of adsorption T

H ads

∂∆∂

as a

function of T, system volume Vsys, specific heat capacity of solid adsorbent and a small

value of the gaseous phase ( ) ( )Tgg ∂ν∂⋅ν1 , where vg is the specific volume at gaseous

phase, and the full expression is given by,

104

( ) ∫ ⋅

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛

∂∆∂

−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∆−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

⋅∆

++=a

a

m

a

mP

ads

P

g

a

sysg

ads

a

sysg

gads

abeap

abe

aabepas dm

TH

TmV

H

mVT

H

MTPc

Mmcc

0

,

1,,

11,

ν

νν

ν

(5.20)

Assuming that amP

ads

TH

,

⎟⎠

⎞⎜⎝

⎛∂∆∂

is negligible and a constant skeletal system volume sysV ,

the specific heat capacity of adsorbent + adsorbate system is simplified to

( ) ∫ ⋅⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

ν∂⋅

ν∆

−∆

++=am

aP

g

g

adsads

abegp

abe

aabepas dm

TH

TH

MTPc

Mm

cc0

1,,

1

1,

(5.21)

The correction term, namely ∫ ⋅⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

∆−

∆am

aP

g

g

adsads dmT

HTH

0 1

νν

, is found to be

significant, because of the large variation of isosteric heat of adsorption at low

adsorbate uptake and the non-ideality of the gas phase. The effect of the adsorbent in

the adsorbed phase is discussed below.

5.2.4 Results and discussion

Using the proposed specific heat capacity expression, the specific heat capacity of

adsorbate is analyzed with respect to a single-component silica gel + water system,

using the published adsorption isotherms and the isosteric heat of adsorption data of

Bjurström (Bjurström et al., 1984). The specific heat capacity of silica gel + water

system is plotted against the amount of adsorbate uptake (type silica gel - Grace type

125 + water) , as shown in Figure 5.1. The measurement methods employed in their

experiments were the transient hot strip (THS – denoted in square symbols) and the

105

DSC (denoted by filled circles) and these values are superimposed in Figure 5.1 for

comparison. It can be observed that the proposed specific heat capacity agrees better

with the experimentally measured data, whilst the liquid-phase based cls and the

gaseous phase based cgs forms the upper and lower bounds of the experimental data.

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Amount adsorbed (kg kg-1)

Spec

ific

heat

cap

acity

(kJ

kg-1

K-1

)

Figure 5.1: Different specific heat capacities of a single component type 125 silica gel + water system at 298 K: a data point (---) indicates refers to the liquid adsorbate specific heat capacity of the adsorption system, (----) defines the gas adsorbate specific heat capacity of adsorbent + adsorbate system, () refers to the newly interpreted specific heat capacity with average isosteric heat of adsorption 2560 KJ kg-1, () the specific heat capacity of type 125 silica gel + water system as obtained through DSC and () the specific heat capacity of the adsorption system with the THS technique.

106

5.3 Thermodynamic modelling of an electro-adsorption chiller

This section focuses on the thermodynamic modelling of an electro-adsorption chiller

(EAC) (Ng et al., 2002) that symbiotically combines the transient behaviours of

thermoelectric modules and the batch-operated adsorption cooling cycle. The main

processes in an EAC are (i) the electric input to the thermoelectric device that produces

the hot and cold junctions where the heat from the former desorbes the vapor from the

adsorbent (silica gel) into the condenser and the latter adsorbes part of the heat

released by the adsorption of vapour into the silica gel, (ii) useful cooling effect is

achieved at the evaporator, (3) partial heat recovery process is effected by the

thermoelectric devices that occurs mainly during the switching period.

This section consists of three sub-sections; Section one describes the electro-

adsorption chiller, the second section explains the lumped-parameter modeling of the

EAC, usually comprising the energy balance, mass balance and adsorption equilibrium

equations, and the final section discusses the operating conditions on the performance

of electro-adsorption chiller.

Figure 5.2 shows the schematic layout of an electro-adsorption chiller. It comprises the

evaporator, the condenser, the reactors or beds and the thermoelectric modules and

water is used as the refrigerant. For continuous cooling operations, the batch-operated

cycle of an EAC consists of two time intervals, namely the switching and the operation

time intervals. During the operation, the DC power is supplied to thermoelectrics,

producing the hot and cold junctions in relation to the designated desorber and

adsorber. As the thermoelectric device is found to have poorer cooling performance as

107

compared to its heating capacity, external (water) cooling system is incorporated in the

EAC in order to remove the heat during the adsorption process. The air-cooled

condenser condenses the desorbed vapor from the designated desorber and the

condensate (refrigerant) is refluxed back to the evaporator via a pressure reducing

valve. Pool boiling is affected in the evaporator by the vapor uptake at the adsorber,

and thus completing the refrigeration close loop or cycle.

The roles of the reactors (containing the adsorbent) are refreshed by switching which is

performed by reversing the direction of the DC power supply (current flow direction is

reversed). With reversed polarity, what was formerly the cold junction becomes the hot

junction, and vice versa. Concomitantly, the external cooling is also switched to the

designated adsorber and similarly, the evaporator and condenser are also switched to

the respective adsorber and desober. It is noted that during switching interval, no mass

transfers occur between the hot bed and the condenser or the cold bed and the

evaporator.

108

p

n

p

n

p

n

p

n

Heat absorbed from the load surface

Heat rejected to the environment

Control box

Power supply

Electro-magnetic valve

Evaporator

Thermoelectric modules

Copper foam

Reactor or bed Silica-gel + copper fins

Electro-pneumatic gate valve

Condenser

Metering valve

Cooling water loop for external cooling

Figure 5.2. Schematic showing the principal components and energy flow of the electro-adsorption chiller (EAC).

109

5.3.1 Mathematical model

To simplify the mathematical modeling of an electro-adsorption chiller, the following

assumptions are generally made: (i) the temperature is uniform in the adsorbent (silica

gel) layer; (ii) the water vapor is adsorbed uniformly in the reactor; and, (3) both solid

(silica gel) and gas phases exist at a thermodynamic equilibrium. The rate of

adsorption or desorption is calculated by the linear driving force equation and this is

given by (Ruthven, 1984 and Suzuki, 1990),

( )q(t) -*q)( κ=dt

tdq , (5.22)

where κ is the mass-transfer coefficient, expressed in terms of other parameters as

follows (Chihara and Suzuki, 1983)

2

15

p

TRaE

so

ReD g−

=κ , (5.23)

where Rp denotes the average radius of a silica gel particle; R= universal gas constant;

E a = activation energy of surface diffusion; and D so = a kinetic constant for the silica

gel water system. Hence t indicates time in equation (5.22).

It is noted from Chapter 4 that the water vapor uptake is linearly proportional to the

vapor pressure and the Henry’s isotherm equation can be applied (Suzuki, M., 1990).

At high pressure and low temperature, the vapor uptake shows a saturation

phenomenon, and the isotherms are appropriately captured by the Tóth’s equation

(Tóth, 1971), which is given as follows,

( ) ( ) [ ] 11 1

10

1

)(exp1exp

*tt

gadsm

gadso

PTRHqKPTRHK

q⋅⋅∆⋅+

⋅⋅∆⋅= , (5.24)

110

where q* is the adsorbed quantity of adsorbate by the adsorbent under equilibrium

conditions, qm denotes the monolayer capacity, P1 the equilibrium pressure of the

adsorbate in gas phase, T the equilibrium temperature of gas phase adsorbate, Rg the

universal gas constant, adsH∆ the isosteric enthalpies of adsorption, Ko the pre-

exponential constant, and 1t is the dimensionless Tóth’s constant.

At the beginning of the each cycle, the adsorber bed and the desorber bed are isolated

from the evaporator and the condenser by valves connected between them, i.e., the

isosteric phase. The designated valves are opened only when the pressures in the beds

are equalized to those at Pcond and Pevap, and the mass transfer across the bed and

condenser or evaporator begins. The mass balance of refrigerant (here water) is

expressed by neglecting the gas phase as,

( )( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−γ−δ−=dt

dqdt

dqM

dtdM adsbed

returncondensatetodue

desbedabe

evapref ,,, 11

444 8444 76

(5.25)

where δ and γ are the switching functions to designate the role of desorber where the

vapour are condensed at the condenser.

The governing energy balance equation of thermoelectric element, which is discussed

in Chapter 3, is given by

xtxT

cJ

cJ

xtxT

cttxT te

teptetetepte

te

tepte

tete

∂∂Γ

−′

+∂

∂=

∂∂ )),(),(),( 2

2

2

ρσρρλ

. (5.26)

111

The adsorption reactors comprise the silica gel, sandwiched between the heat

exchanger fins. The energy balance for the sorption bed, ignoring heat losses, during

its interaction with the evaporator may be written as follows:

( )

( )

( ) ( ) ( ) ( )[ ]( )( )inwowwwpw

adsadsadsbedevapgevapgadsbed

abeLadsbed

adsbedTEadsbedsadsbed

m

aP

g

g

adsadsgpa

adsbedmfpmfCEpCEHXpHXabepabe

TTTcm

HHTPhThdt

dqMTT

AUdt

dTdm

TH

THTPcm

dtdT

cMcMcMcM

a

,,,

,,

,

,,,,

01,

,,,,,

1,

,1

−−

∆−+∆+−+−

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

∆−

∆+

++++

∫

&

δδ

νν (5.27)

Here the value of the switching function (δ) depends on the processes in the bed or

reactor. δ would be 1 for adsorption and 0 for desorption. The first term on the left-

hand side represents the amount of sensible heat required to cool the silica gel, and the

copper fins, whereas the second term defines the specific heat capacity of adsorbate at

different isotherms, which has been discussed in the previous section. On the other

hand, the first term on the right hand side denotes the partial absorption of heat from

adsorption reactor to the cold junction of thermoelectric modules, the second term

provides the heat generation by adsorption and the third term denotes the rejection of

heat from the adsorber by external water cooling. As the electro-adsorption chiller is

well insulated, the heat losses to the bed are assumed to be negligible.

The energy balance for the cold end plate (comprising the thermoelectrics and bed) and

is given by,

( ) ( )0

,,,,,, )(=∂

∂+−−=+

x

teteteLteLadsbedadsbedTEadsbeds

LHXpHXLpL x

TAkTtISTTAUdt

dTcMcM .

(5.28)

112

The first term of the left hand side indicates the energy input to the thermoelectric

modules, the second term is the Seebeck effect due to thermoelectric operation and

finally the third term shows the heat conduction at the cold end boundary of the

thermoelectric modules. The outlet temperature of the external cooling source (water)

is calculated by simply log mean temperature difference method and it is given by

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+=

wpw

adstubeadstubeadsbedinwadsbedow cm

AUTTTT

,

,,,,,, exp

&. (5.29)

For the regeneration step (when the bed interacts with the condenser) the energy

balance is described as follows:

( )

( )

( ) ( ) ( ) ( ) [ ]adsadsdesbedcondgcondgdesbed

abedesbedH

desbeddesbedsdesbed

m

aP

g

g

adsadsgpa

desbedmfpmfCEpCEHXpHXabepabe

HHTPhThdt

dqMTT

AUdt

dTdm

TH

THTPcm

dtdT

cMcMcMcM

a

∆θ+∆+−θ−+−

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

ν∂⋅

ν∆

−∆

+

++++

∫

,,

,

,,,

01,

,,,,,

,1

,1

(5.30)

The value (θ ) depends on adsorption or desorption. During desorption θ equals to 1

and the value of θ is 0 during adsorption. The first term on left-hand side provides the

amount of sensible heat required to heat the silica gel, water, copper fins and the

bottom surface of the chamber, and the second term shows the specific heat capacity of

adsorbate (here water vapor) for different isotherms and isosteric heat of adsorption as

a function of the amount of adsorbate. On the other hand, the first term of the right

hand side represents the heat input due to Seebeck, Joulean and conduction effects of

the thermoelectric modules and finally the second term indicates the transfer of latent

heat from the desorber bed to the condenser.

113

The energy balance for the hot end plate is given by,

( ) ( )Lx

teteteHteDESbedHdesbedTEdesbeds

HHXpHXHpH x

TAkTtISTTAUdt

dTcMcM=∂

∂−+−−=+ )(,,,,,,

(5.31)

The condenser liquefies the refrigerant (water vapor) from the desorber and delivers

the condensate to the evaporator. The influence of condenser is considered at two

levels: (a) how thermal dynamics within the sorption beds depend on condenser

performance; and (b) the energy balance with in the condenser itself. In the current

design, the condenser is of air finned type and has cross airflow. The design has been

performed in such a way that the condenser and the desorber are always maintained at

the refrigerant saturated vapour pressure. The energy balance for the condenser can be

written as

( )[ ] ( )

( ) ( ) ( ) ( ) ( )[ ] ( )acondcondconddesbed

abecondfcondgdesbedcondg

desbedabecondf

condcondcondpcondrefcondfp

TTAUdt

dqMThThTPh

dtdq

MThdt

dTMcMTc

−−−−−+−

=−+

,,

,,,,

11, γθθθ

γ

(5.32)

The first term on the left-hand side represents the sensible heat of condensate from the

condenser and that of condenser material. On the other hand the first term on the right

hand side is the heat generation by condensation, the second term provides the sensible

heat of adsorbate desorbed due to regeneration and finally the third term denotes the

heat transferred to the cooling air. In the current formalism, the micro-fin condenser is

able to contain a certain maximum amount of condensate (M maxrefcond ). Beyond this, the

condensate would flow into the evaporator. Hence, if M refcond < M maxrefcond , γ=1. If

M refcond = M maxrefcond , and (dq desbed , /dt) ≤ 0, γ= 0. But if M refcond = M max

refcond , and (dq desbed , /dt)

> 0, γ=1 (Chua et. al, 1999).

114

The overall heat transfer coefficient of the air-cooled condenser can be described by

the following equation (Sami and Tribes, 1996)

))1(1(

11

ϕ−−++=

cond

finfin

tutuavg

tucond

tuintuin

cond

cond

AA

hKAXA

hAA

U (5.33)

where (A cond ) is the total external area in contact with the air and ϕ represents the fin

efficiency. It is important to note that the fin efficiency is calculated in terms of the

heat exchanger geometry. The air cooled condenser heat transfer coefficient as well as

fin efficiency has been calculated (Briggs and Young, 1962).

The energy balance analysis for the evaporator, and its influence on the sorption bed

dynamics, are quantitatively identical to that of the condenser. The energy balance

equation reads

( )[ ] ( ) ( ) ( )

( ) ( ) ( )[ ] evaploadadsbed

abeadsbedevapgevapgdesbed

abe

condfevapref

evapfevap

evapevappevaprefevapfp

Aqdt

dqMTPhTh

dtdq

M

Thdt

dMTh

dtdT

McMTc

′′+−+−=

−+++

,,

,

,,,,

,1

1

δδ

γθ,(5.34)

where loadq ′′ is the constant heat flux of the cooling components (delivered by IR

device). The left-hand side of this equation represents the total amount of sensible heat

given away by the evaporated liquid and the evaporator material. The second term

shows the enthalpy of evaporated liquid during pool boiling. On the other hand, the

first term on the right hand side calculates the sensible heat required to cool down the

incoming condensate, the second term defines the heat of evaporation, and finally the

third term denotes the required heat generated by the load.

115

The evaporator design is based on the nucleate pool boiling technique where it occurs

at low pressure and this differs greatly from the conventional pool boiling at

atmospheric pressure. In the absence of design data (during the simulation phase of the

research), the well known Rohsenow’s boiling correlation (Rohsenow, 1952) for water

is used but extrapolated where the water properties and the constants Csf= 0.013 and

the index of 0.33 of the correlation are assumed to be valid at the vacuum conditions.

The correlation is given by,

33.0

)(Pr

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−′′

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

gl

s

fgl

load

pf

slfgsf

evapload ghq

chC

TTρρ

σµ

. (5.35)

Rohsenow correlated boiling data shows excellent agreement for pressures ranging

from 101 kPa to 16936 kPa (McGillis et al., 1991). Deviations are expected to be large

at low pressures and heat flux levels. At low heat flux levels, the boiling is not

continuous. The commonly used Rohsenow nucleate boiling correlation agrees well

with pressure variation of the continuous boiling data. In the isolated-bubble regime

(low heat flux and low pressure), the experimental data deviates from the Rohsenow

correlation (McGillis et al., 1991). This is due to the fact that the heat transfer

mechanism comprised of conduction into the fluid during intermittent boiling is not as

effective as continuous boiling heat transfer, resulting in a reduction in heat transfer

performance.

During the switching operation, the roles of reactors or beds are changed (adsorption

bed converts into desorption mode and desorption bed changes in to adsorption mode)

and no latent heat is transferred from the evaporator to the cold bed and from hot bed to

the condenser. When the bed switches to adsorption mode, the governing equation

becomes,

116

( )

( )

( ) ( )( )inwwowwpwLadsbed

adsbedTEadsbedsadsbed

m

aP

g

g

adsadsgpa

adsbedmfpmfCEpCEHXpHXabepabe

TTTcmTT

AUdt

dTdm

TH

THTPcm

dtdT

cMcMcMcM

a

,,,

,,,,

01,

,,,,,

1

,

−−−

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⋅

∆−

∆+

++++

∫

&

νν

(5.36)

When the bed switches to desorption mode, the equation now becomes,

( )

( )

( )desbedH

desbeddesbedsdesbed

m

aP

g

g

adsadsgpa

desbedmfpmfCEpCEHXpHXabepabe

TT

AUdt

dTdm

TH

THTPcm

dtdT

cMcMcMcM

a

,

,,,

01,

,,,,,

1

,

−

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

ν∂⋅

ν∆

−∆

+

++++

∫ (5.37)

During switching the behavior of condenser is described by

( )[ ] ( )

( )acondcondcond

desbedabecondf

condcondcondpcondrefcondfp

TTAUdt

dqMTh

dtdTMcMTc

−−

=+ ,,,, γ (5.38)

And finally the energy balance of evaporator is noted during switching operation.

( )[ ] evaploadevap

evapevappevaprefevapfp Aqdt

dTMcMTc ′′=+ ,,, (5.39)

These energy balance equations are based on the fact that pressure and temperature

change in tandem, the thermal swing of the beds is assumed to be isosteric. t defines

time (equations 5.25-5.39).

5.3.2 COP of the Electro-Adsorption Chiller

The cycle-average Coefficient of Performance (COPnet) of the electro-adsorption

chiller, can be expressed in terms of the cycle average COPs of the individual

adsorption (subscript ads) and thermoelectric (subscript TE) chillers by considering the

overall energy flows, however, COPads is a complicated function of COPTE via the

117

dynamics of the combined cycle. Referring to Figure 5.3, the COPs of the individual

components are defined as the nominal cycle-average cooling rate produced relative to

the particular cycle-average power input.

in

TEadsTE P

QCOP ,= (5.40)

TEdes

evapads Q

QCOP

,= (5.41)

where Qevap denotes the cycle-average cooling rate of the overall device, i.e. the rate of

heat extraction at the evaporator. The cycle-average values of the thermal power

absorbed at the cold junction Qads,TE that drives refrigerant adsorption, as well as the

thermal power rejected at the hot junction Qdes,TE that drives refrigerant desorption for

adsorption cycle. But it is noted that Qads,TE is less than Qdes,TE and the total heat

rejected during adsorption is the sum of Qads,TE plus the amount given by externally

assisted cooling. COPTE will fall below commonly observed thermoelectric COP

values due primarily to losses associated with the large thermal lift (TH- TC).

From the energy balance of the thermoelectric device (first law analysis applies

irrespective of external cooling to adsorber)

TEadsTEdesin QQP ,, −= (5.42)

The overall or net COP is

( )TEadsin

evapEAC COPCOP

PQ

COP +== 1 (5.43)

The power input from the thermoelectric device is represented by

cycle

t

inputin t

dttIvP

cycle

∫= 0

)( (5.44)

118

where vinput is the fixed terminal voltage per unit thermoelectric elements. The

designed value of vinput is taken from Melcor web site. The time response of control

signal variation I (t) for a fixed terminal voltage can be expressed by

( )[ ]LHtetem

x

tetete

Lx

tetete

Lte

teptete

TTvNn

xtxTA

xtxTAdx

ttxTAcw

tI te

te

−−⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−∂

∂−

∂∂

===

∫α

λλ00

),(),(),(

)( (5.45)

The maximum COPads of a heat driven chiller is 0.85 (COPads = 85.0≅∆ ads

fg

Hh

), where

hfg is the latent heat of water and adsH∆ is the isosteric heat of adsorption at similar

pressure and temperature conditions. At high thermal-lift, the COPTE of thermo-

electrics is generally low, typically less than 0.3. Based on the experience of operating

the EAC at high thermal lift (TH – TC) of 30 to 35 degrees, the typical COPTE is about

0.2 to 0.4. Thus, at these mentioned operating conditions, the theoretical COP of the

electro-adsorption chiller is estimated to be about

1.1 0.3)(1 0.85 ) 1 ( ≅+=+= TEads COPCOPOPmaxEACC .

119

Figure 5.3 A block diagram to highlight the sensible and latent heat flow and the energy balance of a thermoelectrically driven adsorption chiller

Thermoelectricmodule

Adsorption reactor

Desorption reactor

Condenser

Evaporator

Qevap

Qcond

Pin

Qdes

Qads,TE

Qads,w

)( evapfge Thm&

adsadsbed H

dtdq

∆,

adsdesbed H

dtdq

∆,

( )condfgc Thm&

Latent heat flow

Sensible heat flow

External cooling

[Heat recovery by thermoelectric device]

( )

( ) adsadsbed

condfgc

adsdesbed

evapfge

Hdt

dqThm

Hdt

dqThm

∆+

≅∆+

,

,

&

&

evapdes

condwadsTEads

QQQQQ

+

≅++ ,,

Sensible heat balance

Late

nt e

nerg

y ba

lanc

e

Continuous regeneration

desadsdesbed

evapfge

QHdt

dq

Qhm

<∆

>

,

&

prdes

evapads

thads

fgads

COPQQ

COP

COPHh

COP

==

=∆

=max

prth COPCOP >

120

5.3.3 Results and discussion

The performance predictions of an electro-adsorption chiller are presented here. Using

one dimensional mass and energy conservation equations and the spatially discretized

dependent variables along a control volume, the equation sets are solved by the finite

difference formulations. The updated values of the key system’s variables are obtained

by numerical integration in time. The calculation procedure commenced with the

initialization of variables, the system geometries and the adsorption characteristics of

adsorbent-adsorbate systems. A series of subroutines are used to calculate the

adsorption isotherms and kinetics. To speed up the computation, the current formalism

makes use of polynomial curve fits of the thermodynamic property, data of Wagner

and Kruse (1998). The original thermodynamic property routines of Wagner, W. et al.

(1998) have also been applied to the present formalism. The coupled equations are

solved by the fifth order Gear’s differentiation formulae method in the DIVPAG

subroutine of the IMSL FORTRAN Developer Studio software. Double precision has

been used, and the tolerance is set to 1× 10-6. Here only the initial conditions are

specified, and the total system is allowed to operate from transient to cyclic steady

state conditions. The Flow chart of the EAC modeling is shown in Appendix E.

The key parameters analyzed in the simulation code are (a) the temperature profiles of

the adsorber, desorber, condenser and evaporator, (b) the corresponding heat transfer

conductance and the input voltage of the thermoelectric modules (c) the amount of

adsorbent per reactor of the chiller and (d) the net Coefficient of Performance and the

load surface temperature of the electro-adsorption chiller for a fixed cooling capacity.

The values used in the present simulation are shown in Table 5.1.

121

Table5.1 Specifications of component and material properties used in the simulation code

Parameter or material property Value or descriptive equation, with units

Thermoelectric UT8-12-40-F1 (melcor) units Reference

No of thermoelectric couples/elements in a thermoelectric module 127 / 354 --- Melcor

Specific heat capacity of thermoelectric material 544 J/kg K Rowe, 1995

Thermoelectric material density 3102.7 × kg/m3 Rowe, 1995 Cross sectional area of a thermoelectric element

610619.3 −× m2 Melcor

Characteristics length of thermoelectric element

3103.3 −× m Melcor

Geometric factor 210171.0 −× m Melcor Ceramic plate and very thin copper substrate

Mass of ceramic plate 20 310−× kg Melcor Specific heat capacity of ceramic 775 J/kg K Melcor Mass of thin copper plate 31010 −× kg Melcor Specific heat capacity of copper plate 386 J/kg K Melcor

Sorption thermodynamic properties and sorption beds Kinetic constant 2.54 410−× m2/s Chihara and

Suzuki, 1983 Activation Energy 4.2 410−× J/mole Chihara and

Suzuki, 1983 Average radius of silica gel particle 4107.1 −× m manufacturer Isosteric heat of adsorption 6108.2 × J/kg Chua et al., 2002

Specific heat of silica gel 921 J/kg K Chihara and Suzuki, 1983

Adsorber/desorber bed heat transfer area

0.0169 m2

Adsorber/desorber tube heat transfer area

3101.5 −× m2

Finned surface area 0.10664 m2 Tube weight 0.2 kg Volume of tube 6102.3 −× m3 Bottom plate weight 1.8 kg Fin weight 1.2 Kg Silica gel mass 0.3 kg Overall heat transfer coefficient of adsorber

1090 W/m2K

Author’s experimental

data

Overall heat transfer coefficient of desorber

1850 W/m2K

122

Condenser Condenser heat transfer area 0.5 m2

Mass of condenser including fins 2 kg Author’s

experimental data

Condenser heat transfer coefficient 700 W/m2K Mass of condensate in the condenser 41084.12 −× kg

Evaporator Evaporator heat transfer area 0.0026 m Mass of evaporator including foams 2 kg Mass of refrigerant in the evaporator 0.3 kg

Figure 5.4 shows the prediction of temporal histories for the reactor (adsorber or

desorber), the condenser, the load and the evaporator by using the mathematical model

presented herein. At the very short cycle times, say a few tens of seconds, the COP of

EAC is expected to be low due to incomplete vapour uptake onto the silica gel and the

large amount of sensible heat of the thermoelectric modules as compared to latent heat

transfer. In contrast, a very large cycle time (a few hours) for the cycle would saturate

the adsorbed vapour onto the silica gel and cooling rate would decrease. Thus, there

exist in between these extreme cycle intervals that an optimal COP would be located

for the EAC. Figure 5.5 represents a narrow region of the EAC operation (cycle time

intervals) where the COP shows a monotonic increase (i.e., before the optima is

reached) and yet the load and evaporator temperatures have demonstrated their

minima. These simulations have a fixed input heat flux of 5 watt per cm2, a rate

expected of the IR (Infra-red radiant) heaters in the experiments (described in the

following chapter). From the results, the lowest load temperature is found to be about

24oC and it lies between 400 and 500 seconds, whilst the COP of EAC is computed to

be about 0. 8.

123

1020304050607080

025

050

075

010

0012

5015

0017

5020

0022

5025

0027

5030

00Ti

me

(s)

Temperature (oC)

Des

orpt

ion

bed

Ads

orpt

ion

bed

Con

dens

er

Loa

d

Eva

pora

tor

Figu

re 5

.4.

Tem

pora

l hi

stor

y of

maj

or c

ompo

nent

s of

the

ele

ctro

-ads

orpt

ion

chill

er

obta

ined

by

sim

ulat

ion.

124

Figure 5.6 shows the Dűhring or P-T diagram of the electro-adsorption chiller with two

cycle times (400 s and 600 s), each having the same fixed cooling capacity of 120 W.

At a 700s (denoted by bold line), the difference in the vapor uptake and off-take is

about 15 % and the pressure of the adsorption reactor goes down 1600 Pa, but for a

400s (denoted by dotted line), the difference between the vapour adsorption and

desorption is 10 %. From the P-T profile, one observes the cold-to-hot thermal swing

of the reactor, causing momentary desorption at the beginning of operation and

maintaining the pressure in the hot reactor and the condenser. On the hand, during hot-

to-cold thermal swing of the bed, there is a sudden rise in the bed pressure at the end of

the switching phase, causing momentary adsorption in the reactor.

Figure 5.5. Effects of cycle time on chiller cycle average cooling load temperature, evaporator temperature and COP. The chosen 500 s cycle time, shown by the solid dotted vertical line, is close to, the minimum cooling load temperature.

10

15

20

25

30

35

40

45

50

55

60

200 300 400 500 600 700 800 900Cycle time (s)

Tem

pera

ture

(o C)

0.60

0.65

0.70

0.75

0.80

0.85

Coe

ffici

ent o

f Per

form

ance

(CO

P)

Load

Evaporator

125

Figure 5.7 shows the simulated results of COP and the variations of load

surface/evaporator temperature with different cycle average input currents

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫

cyclet

cycle

dttIt

I0

)(1~ of the thermoelectric modules. These simulations are conducted

for a fixed input cooling power 5 Watt per cm2 and optimal cycle period 500 seconds.

As expected, the load surface temperature and COP decrease with increasing input

current.

Figure 5.6. Dűhring diagram for 400 s and 700 s cycles

1000

10000

30 35 40 45 50 55 60 65 70 75 80Temperature (oC)

Pres

sure

(Pa)

2%

5%

8%

10%

15%20%25%35%40%45%50%60%70%80%90%100%

j

k

30 %

cycle time 400 s

Cycle time 700 s

qads – qdes = 0.17

0.1

126

For the verification of the EAC modeling, a comparison of the predictions to the

experimental results will be presented in the next chapter. For the moment, the analysis

of energy (both the sensible and latent heat) balances of major components of EAC is

analyzed. The cyclic average energy values for the internal and external balances are

found to be within 0.2% and 2.25%, respectively, as shown in Table 5.2. A full listing

of the energy flows in the major components can be found in Appendix F.

Figure 5.7. COP, the cycle average load temperature and the evaporator temperature as functions of the cycle average input current to the thermoelectric modules.

10

15

20

25

30

35

40

45

50

5 5.5 6 6.5 7 7.5 8 Cycle average current (amp)

Tem

pera

ture

(o C)

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

Coe

ffici

ent o

f Per

form

ance

(CO

P)

Load

Evaporator

of E

AC

127

Table 5.2 Energy balance schedule of the electro-adsorption chiller (EAC)

Sensible heat balance (- indicates heat rejection and + indicates heat addition)

Qevap, (W) Qcond, W Qads,T, W Qdes, W Qevap + Qdes Qcond + Qads,T Error

%

129.05 -183.77 -190.58 237.05 366.10 -374.35 2.25

Latent heat balance

QevapL, W QcondL, W Qads,TL, W QdesL, W QevapL + QdesL QcondL +

Qads,TL Error

%

-136.28 183.95 146.39 -194.74 -331.02 330.35 0.2


The thermodynamic formulations of specific heat capacity, internal energy, enthalpy

and entropy of a single component adsorbent + adsorbate system are essential in the

development of adsorption thermodynamics and the modeling of the electro-adsorption

chiller (EAC).

A theoretical model has been successfully developed incorporating transient modelling

equations with realistic adsorption isotherms and adsorption kinetics, overall heat and

mass transfer correlations, and thermoelectric performances (data as provided by

manufacturer MELCOR). With appropriate initial and boundary conditions and for a

range of cycle time between 400s to 700s, the COP is found to be increasing with cycle

time but there is a corresponding minima for the evaporator and load surface

temperatures. The validity of the simulation will have to shown by comparison with

experimental results, which is described in the following chapter.

128

Chapter 6

Experimental investigation of an electro-adsorption chiller

This chapter describes the experimental investigation of the proposed electro-

adsorption chiller (Ng et al., 2002 and 2005(a)) where its performances at different

operating conditions have been evaluated and analyzed. Experimental procedures and

energy utilization schedules of this chiller are also described. For the verification of

lumped model as presented in chapter 5, the experimentally measured results are

compared with simulation.

Figure 6.1 shows the fully assembled bench scale electro-adsorption chiller comprising

reactors or beds, evaporator and condenser. Based on the simulation results of chapter

5, (i) the expected cooling capacity is about 120-130 W, (ii) each reactor is sized to

hold 0.3 kg of silica gel (Fuji Davison type ‘RD’), (iii) the reactor, evaporator and

condenser surface areas are 0.0169 m2, 0.05 m2 and 0.0026 m2 respectively. The

reactors are connected to an evaporator and a condenser via off on control valves

(electro-pneumatic gate valve). The thermoelectric modules (Melcor, UT8-12-40-F1, 9

pieces, 3 series and 3 parallel ways) are placed between the two reactors.

6.1 Design development and fabrication

The major components of a reactor or bed are (1) two circular copper plates, (2) a heat

exchanger with fins and tubes, (3) the PTFE (Poly-Tetra-Fluoro-Ethylene, thermal

conductivity = 0.25 W/m.K) enclosures (tensile strength 40 MPa). The copper plate as

shown in Figure 6.2 (a and b), has 3 x 3 arrangement of studs and slots (width: 40 mm,

length: 40 mm and depth: 1 mm) on the outer surface and one slot (width: 135 mm,

129

length: 135 mm and depth: 1 mm) at the inner side. The detailed design and drawings

of the electro-adsorption chiller (EAC) are furnished in Appendix G.

The studs and slots are for the positioning of heat exchanger block (packed with silica-

gel) and thermoelectric modules. The heat exchanger block as shown in Figure 6.2 (c

and d), which holds silica gels, composes of 34 slots (slot width is 3 mm and the

thickness of copper wall between two consecutive slots or fin thickness is 1 mm). The

Figure 6.1. The test facility of a bench-scale electro-adsorption chiller

Reactor


IR heater system

Pressure sensor

Gate valve

Condenser

Cooling loop

U-tube

Evaporator

Vacuum pump

Control system

130

slots are to be filled with silica gel and the silica gels are covered by copper mesh (40

meshes per inch). To measure the silica gel temperature at different points of the bed,

four RTD probes (± 0.15 °C) are placed inside the slot and RTD wires are connected to

the electrical lead through ( 8 pins, TL8K25, Edwards). The heat exchanger is attached

to the inner side of the copper plate with screw tight to maintain the contact resistance

as low as possible. The enclosure is machined from a solid PTFE block. Five short

pipe sockets (DN 10, stainless steel) are placed with viton O’ ring at the outer side of

the chamber (Figure 6.2(e)). Two ports (6 mm diameter) for copper tubes and one port

(25 mm diameter) for electrical lead-through are developed on the top surface of the

enclosure. A circular groove (diameter 250 mm, 3 mm wide and 4 mm deep) for the

location of a centering ring (DN 200) is machined at one surface of the chamber’s

flange. Thermoelectric modules (3 series and 3 parallel connections) are placed at the

outer surface of reactor. The two reactors are attached with thermoelectric devices

where compressive force is applied from four sets of stud and nut at quadrants of the

two reactors [as shown in Figure 6.2 (f)]. The fabrication of two reactors is fully

completed when electro-pneumatic gate valves (normally close, 230 volts, operating

pressure 4.5 to 7 bar), manual valves, pressure transducers (active strain gauge,

accuracy 0.2% full scale, temperature range 30 °C to 130 °C, EDWARDS) and

temperature sensors (Thermister YSI 400 series, ± 0.1 °C) are placed at their positions

on the two reactors.

131

(a) (b)

(e) (f)

Figure 6.2. Assembly of the reactor; (a) the bottom plate (outside), (b) the bottom plate (inside view), (c) the heat exchanger, (d) the heat exchanger with bottom plate, (e) the PTFE enclosure and (f) two reactors.

Thermoelectric modules location

Heat exchanger

Bottom plate

Copper fin

Copper tube

Copper mesh

Short pipe socket

Port for copper pipe

Port for electrical lead-through Gate valve

Pressure sensor

Valve

electrical lead-through PTFE enclosure

40 mm

40

(c) (d)

Silica gel RTD

132

The condenser as shown in Figure 6.3 is an air-cooled type and has a cross air flow

arrangement. It composes of two tube-centered fins bundles, a vapor collector and a

condensate collector tubes. Since all tube-centered fins bundle is machined from a

copper block to achieve 42 parallel fins centered by a 160 mm length tube (inner

diameter 10 mm and outer diameter 14 mm thus the wall thickness is 2 mm), there are

no contact resistances between fins and copper tube. The distance between two fins

(width: 50 mm, length: 50 mm thickness: 1 mm) is 3 mm. The condenser is connected

to the inlet and outlet squared collector tubes (Each collector has outside dimension 20

mm (width), 20 mm (length), 76 mm (height) and inside dimension 16 mm (width), 16

mm (length), 72 mm (height) thus, the overall thickness of the wall is 2 mm) and the

condenser might provide sufficient heat transfer area with the help of a fan (30 Watt,

140 CFM, 230 V).

`

Figure 6.3. The condenser unit.

Inlet port

Water-vapour collector port Pressure

transducer port

Condenser fins

Temperature sensor port

Outlet port

Condensate collector tube

133

Condenser pressure, inlet and outlet temperatures can be monitored through three

copper ports (DN 10), which are machined and brazed to inlet and outlet collector

tubes. For the refrigerant inflow and outflow, two copper ports are machined and

brazed to the top and bottom of two collector tubes. Two YSI (400 series, accuracy

±0.1°C) thermisters are used to measure the temperature of refrigerant at the inlet and

outlet of the condenser. The pressure of the condenser is continuously monitored by a

BOC Edwards pressure transducer (Active strain gauge ± 2% full scale, temperature

range from 30 °C to 130 °C).

The evaporator as shown in Figure 6.4 (a-f) consists of a NW 100 stainless steel tube

body, two NW 100 flanges (one is stainless steel blanking flange and the other is

quartz view port) and standard vacuum fittings. One stainless steel flange is put on top

and the quartz view port is put on the bottom of the tube to form a vacuum enclosure

with the centering O’ rings and claw clamps. Two glass view ports are made on the

tube body of the evaporator to observe the pool boiling and the level of refrigerant.

Four ports (14 mm diameter) and one big port (25 mm diameter) are provided at the

top plate (NW 100, St. Steel, 12 mm thickness) where four short pipe sockets (size DN

10) are welded. The electrical lead through (TL8K25, 8 pins EDWARDS) is placed in

the big port with the viton O’ ring and screw. Two short pipe sockets are connected to

the reactors via electro-pneumatic gate valves (DN 16, pressure range 7101 −× mbar to 2

bar) and flexible hoses. The third one is connected with a temperature sensor

(Thermister, accuracy: ± 0.1°C; YSI 400 Series) and the fourth is connected to a

pressure transducer (Active strain gauge, accuracy ± 0.2 % full scale, BOC Edwards)

with compression fittings. Water refrigerant is charged (or removed) into (or from) the

evaporator chamber by a diaphragm vacuum valve, which is connected to a 4 mm

134

diameter stainless steel tube. The tube is welded at one side of the bottom quartz plate

and connected with another 4 mm diameter, 90 mm long stainless steel tube to allow

warm condensate to flow back to the evaporator. A metering valve with U-bend is used

between the evaporator and the condenser to create a pressure difference during

operation.

The copper foam (5% density, 50 PPI, width 52 mm, length 52 mm and thickness 32

mm) is used as pool boiling enhancement material. The foam structure consists of

ligaments forming a network of inter-connected dodecahedral-like cells and the cells

are randomly oriented and mostly homogeneous in size and shape (provair, 2004).

Metal foam can be produced at various pore size varied from 0.4 mm to 3 mm and net

density from 3% to 15% of a solid of the same material. Copper metal foams have high

surface area to volume ratio and high thermal conductivity. These are potentially

excellent candidates for high heat dissipation applications. Copper foam also has

excellent capillary effect which behaves like a natural pump and has the ability to

generate refrigerant flow far greater than the usual gravity effect. As a result, the foam

is able to draw the surrounding liquid and makes all foam surface areas wet. Foam

material, owing its capillary effect, is used as a liquid transport material. In addition,

open cells of the foam also behave as the fluid re-entrance cavities which play the most

important role in pool boiling applications. Therefore high thermal conductivity copper

foam is the kind of the material that can substitute pool boiling enhancement

structures.

135

(a) (b)

(c) (d)

(e)

Short pipe sockets

View window

Bottom plate (quartz view port)

Quartz plate

Port for electrical leadthrough IR

heater

Power supply

View window

Flexible bellow

Manual valve

Electrical leadthrough

View port

Quartz bottom plate

RTD sensor

Flexible hose (to be connected to reactor)

Pressure transducer

Figure 6.4. The evaporator unit; (a) shows the body of the evaporator, (b) indicates the quartz bottom plate through which infra red heat passes, (c) represents the top part of the evaporator, (d) the IR heater system and (e) is the evaporator.

St Steel body

Blanking flange Claw

clamp

136

An infra-red radiant heater with a tapered homogenizer (kaleidoscope) as shown in

Figure 6.5 is incorporated. The kaleidoscope is used for radiation heat transfer between

a heat source and the evaporator of a bench-scale electro-adsorption chiller. For

constructive reason, the distance between the radiation heat source and the evaporator

is 1 m. The Kaleidoscope is filled with air, its inside surface has a reflectivity of 0.94.

A window made of fused silica (quartz) is the entry aperture of the evaporator. Fused

silica is highly transmissive (τ › 0.94) for radiation up to a wave length of 2500 nm.

The maximum temperature of the source is approximately 1500 K.

Wein’s displacement law yields the wavelength of the peak of the radiative energy

distribution, 2000≈=Tpaekσλ nm, where Boltzmann constant is σ = 2.898 × 10-3 mK.

The peak wavelength is well within the transmission band of the fused silica or quartz

Figure 6.5. Drawing of the kaleidoscope/cone concentrator developed for radiative heat transfer between the heat source (point 1, four rods of 4 kW total power), and the bottom plate (quartz point 3) of the evaporator (exit at the top).

IR-heater

Kaleidoscope 1

1

2 3

137

(Appendix H). The rod sources are Lambertian radiatiors i.e. radiation is equally

distributed in the sphere, the rod looks equally bright from all directions.

The connections between the evaporator and the reactors, and those between the

condenser and the reactors, are performed by the flexible hose (DN 10, stainless steel)

and electro-pneumatic gate valves (normally closed, 240 volt). The flexible hoses are

joined to the reactors (or evaporator/condenser) with standard vacuum fittings such as

centering ring with O’ rings and clamping rings. Two flexible hoses at the outlet of the

reactors are combined and led to condenser through a tee. To vacuum the system, an

Edwards rotary vane pump is used and argon gas cylinder is connected to the test

facility with a manually valve and a special DN 10 stainless steel tee. The outlet port of

the condenser is led to the evaporator via a ¼ mm metering valve and a 4 mm stainless

steel tube. The water vapor quality at the outlet of the condenser can be observed

through a DN 10, 70 mm quartz tube, which is located between the condenser outlet

port and the metering valve, and the metering valve is used to control the flow rate of

liquid refrigerant from the condenser and create a pressure differential between the

condenser and the evaporator.

In this experiment, the data acquisition unit includes an Agilent (34970 A) Data

Acquisition System, BOC Edwards TIC (Turbo Instrument Controller) pressure

acquisition system and a personal computer. The Agilent Data Acquisition unit is used

to accurately capture the required temperatures at the evaporator, reactors and

condenser sections through thermisters and RTD sensors. TIC controller unit is used to

monitor the pressure of evaporator, reactors and condenser. Agilent and TIC controller

138

softwares are installed in a personal computer. The calibration of pressure transmitters,

temperature sensors, and data logger are traceable to national standards.

Power input to the thermoelectric modules is provided by a Agilent 6032A (0-60 Vdc,

0-50A, 1200 W, GBIP auto ranging, programming accuracy: voltage 0.035% or +40

mV and current 0.2% or +85 mA, Readback accuracy: voltage 0.08% and current

0.36%) DC power supplies. As the electro-adsorption chiller is fully automatic, batch

operating machine; a control system is necessary to control the operation sequences of

the machine. A computer program using HP Visual Engineering Environment

(HPVEE) pro version 6.1 is used in conjunction with the electro-adsorption chiller to

(i) control the opening and closing of electro-pneumatic valves and electromagnetic

valves at different time intervals in a batch cycle, (ii) reverse the polarity of the voltage

supplied to the thermoelectric so that the role of the adsorber and desorber reversed

after each batch cycle, (iii) record the power supplied to the thermoelectric modules so

that the performance of the electro-adsorption chiller can be computed. The design and

fabrication of a bench scale electro-adsorption chiller has been successfully completed.

This chiller is relatively compact, fully automatic and free of moving parts (except

fans).

6.2 Experiments

A full scale test plant as shown in Figure 6.1 has been built to test-run an electro-

adsorption chiller (EAC). Prior to experiment, the test facility (evaporator, condenser,

reactors and piping systems) is first evacuated by a two-stage rotary vane vacuum

pump (BOC Edwards pump) with a water vapor pumping rate of 315 ×10-6 m3 s-1. To

prevent back migration of oil mist into the test apparatus, an alumina-packed foreline

139

trap is installed immediately upstream of the vacuum pump. During evacuation, power

is applied to the infra-red radiant heater and thermoelectric modules to heat-up the

copper foam and silica gel to remove moisture and air trapped from the reactors and

the evaporator. Systems pressures are continuously monitored by pressure transducers.

After removing moisture from the reactors, the system is cooled to nearly ambient

temperature and the required vacuum (about 2 mbar) is obtained. The vacuum pump is

switch off and argon, with a purity of 99.9995 per cent, is sent through a column of

packed calcium sulphate into the system to remove any trace of residual air or water

vapor. The argon charging and removing on the system are repeated until the

satisfactory vacuum condition is achieved. Based on measurements involving only

argon and silica gel, it is concluded that there is no measurable interaction between the

inert gas and the adsorbent. The effect of the partial pressure of argon in the reactors is

found to be small. Nevertheless, the partial pressure of water vapor is adjusted for the

presence of argon so as to avoid additional systematic error. The cooling water inlet

temperature is controlled by the room temperature and the electro-pneumatic valves

control the flow of water vapor to the reactors. Air flows to the condenser is controlled

by the power of fan.

The required amount of refrigerant water (about 300 gm) is firstly charged into the

evaporator via a refrigerant charging unit. Prior to charging, the test facility is initially

evacuated and maintained at room temperature. The test facility is fully automatic,

power on is performed to run the test facility. Three time intervals in one cycle of the

electro-adsorption chiller are (i) delay time (time requirement for the control system to

change the polarity of DC power. DC power supply itself also needs to be off before

switching the polarity to prevent electrical shock), (ii) switching time (time allowances

140

for the thermoelectric modules for the purpose of pre-heating and pre-cooling the beds

before the vapor refrigerant entering or leaving the reactors or beds. All electro-

pneumatic gate valves are closed during the switching period) and (iii) operation time

(adsorption/desorption time requirements for the reactors, only the necessary electro-

pneumatic valves and adsorption coolant loop for external sensible cooling, are opened

except the condenser’s fan which is always on to continuously reject heat to the

environment).

Water vapor is generated in the evaporator chamber as the heating power (Qflux, IR)

from the Infra-red radiant heater, is applied to the evaporator by the homogenizer

(kaleidoscope). The control system activates the DC power supply and the electro-

pneumatic valves (off/on control) to start pre-cooling and pre-heating the reactors at

the beginning of switching period. During operation period, one control valve “A” as

shown in Figure 6.6 allows vaporizes refrigerant to enter into pre-cooled reactor

(adsorber bed) from the evaporator, and valve “C” is also opened to release desorbed

refrigerant to the condenser, where the desorbed refrigerant is condensed and heat is

rejected to the environment. Finally, the condensed refrigerant flows back to the

evaporator via the metering valve “E”. At the end of operation period (or the delay

period for 1 second), the control system closes all electro-pneumatic gate valves and

changes the polarity of DC power supply system. The reactor, which is cooled in the

previous cycle, is heated sufficiently, while the other reactor that is acted as a desorber

in the previous cycle is cooled down sufficiently by external cooling and they take the

role of adsorption and desorption again. The control system activates valve “B” and

“C” at the beginning of operation time and the other two valves “A” and “D” are not

activated. After getting cooling effect at the evaporator, the evaporated water vapor is

141

adsorbed onto silica gel and the desorbed water vapor is released to the condenser for

condensation. Heat is rejected from the condenser via a fan and finned heat exchanger.

The warm condensate flows back to the evaporator via the metering valve E. The

energy utilizing schedule of the electro-adsorption chiller is furnished in Table 6.1.

p

n

p

n

p

n

p

n

R-1

R-2

Evaporator

Condenser

A B

C D

V1

V2

V3

V4

PS

Qflux, IR

Qcond

E

Figure 6.6. Schematic diagram of an electro-adsorption chiller (EAC) to explain the energy utilization.

142

All pressures are measured with BOC Edwards TIC (Turbo Instrument Controller)

pressure acquision systems and temperature readings are continuously monitored by a

calibrated data logger (Agilent 34970 Acquisition system). The calibration of pressure

transmitters, temperature sensors, and the data logger are traceable to national

standards. In the present test facility, the major measurement bottleneck lies in the

measurement of vapor pressure and reactor temperatures.

Table 6.1 Energy utilization schedule of an Electro-Adsorption Chiller (Refer to Figure 6.6)

Cycle 1

R-1 SW DES→ADS V1, V2 → ON

R-2 SW ADS→DES V3, V4 → OFF

A, B, C, D CLOSED

PS ON VPS 1

R-1 OP ADS V1, V2 → ON

R-2 OP DES V3, V4 → OFF

A,C → OPEN B,D → CLOSED

PS ON VPS 1

DELAY 1 Second, A, B, C, D Closed PS OFF Cycle 2

R-1 SW ADS→DES V1, V2 → OFF

R-2 SW DES→ADS V3, V4 → ON

A, B, C, D CLOSED

PS ON VPS 2

R-1 OP DES V1, V2 → OFF

R-2 OP ADS V3, V4 → ON

A,C → CLOSE B,D → OPEN

PS ON VPS 2

. . . . .

143

ADS: Adsorption stage, DES: Desorption Stage SW: Switching period, OP: Operation period Cycle = Switching + Operation R-1: Reactor 1 or bed 1, R-2: Reactor 2 or bed 2 PS : Power Supply, VPS: Voltage Polarity Switching A, B, C, D: Electro-pneumatic gate valve (OFF/ON control) V1, V2, V3, V4: Electro-magnetic valve (OFF/ON control)

6.3 Results and discussions

The infra-red radiation heater provides uniform heat flux to (5 cm x 5 cm) window

aperture of evaporator up to 5W/cm2. The heat flux delivery after and before the quartz

window is accurately calibrated using a water cooled heat flux meter (Transducer type:

Circular foil heat flux transducer, model: 1000-0 with amp 11, Sensor Emissivity: 0.94

at 2 microns, Vatell Corporation). The calibrated results are shown in Table 6.2.

In a ray-tracing simulation with the light toolsTM package, the lambertian sources are

set to emit 4000 W at two wavelengths 1800 nm and 2200 nm to represent Wien’s law.

Convection and conduction are neglected, only radiation is considered. The heat flux

after the source, at the exit of the kaleidoscope and after passing through the fused

silica window (points 1, 2 and 3 as shown in Figure 6.7) are predicted and simulated.

The ray tracing results of the tapered kaleidoscope are provided in Table 6.3 and

pictured in Figure 6.7. The correlation between simulated and measured values is good

however the optical efficiency of the system is low, but the heat flux density on the

evaporator meets the target values.

144

Table 6.2 Heat flux calibration table (refer to Figure 6.5)

Sensor Output (mv) *

Power output (W/cm2) Input from

variable transformer

(Volts)

near IR heater point 1

before quartz

window point 2

after quartz

window point 3

near IR heater point 1

before quartz

window point 2

after quartz

window point 3

164 3.6 1.01 0.274 4.49 1.22 0.33

175 4.85 1.27 0.35 6.97 1.95 0.53

185 5.3 1.45 0.49 8.86 2.35 0.8

196 5.72 1.65 0.66 9.68 2.78 1.11

207 6.3 1.85 1.01 11.07 3.25 1.78

218 7.2 2.03 1.22 13.3 3.75 2.26

229 8.1 2.35 1.7 15.55 4.51 3.26

240 8.9 2.52 1.9 17.54 4.97 3.75

Conversion factor x mv = 2.4438 × x -1.2189 W/ cm2

145

The net Infra-Red radient flux is absorbed by the copper foam in the evaporator and

boiling occurs instantly when sufficient superheat is achieved.

Table 6.3 The heat flux density results from measurements and ray-tracing

simulation (the source have a total power of 3.8 KW, refer to Figure 6.7)

Heat flux in W per square cm

Input voltage from variable transformer is 220 Volt

near source (point 1)

exit of kaleidoscope

(point 2)

after fused silica (quartz)

point 3

simulation 8.33 3.78 3.27

measured 13.3 3.75 2.26

Figure 6.7. Tracing 20 rays from the lambertian sources through the kaleidoscope systems. The heat flux after the source is measured in point 1, then at the exit of the kaleidoscope in point 2, and finally after having passed the fused silica window in point 3, the image view of infra-red heater has been pictured from point 3.

Infra-red radiant heater image

Image view from point 3

146

Figure 6.8 shows the temporal histories of the reactors (the adsorber and the desorber),

condenser, evaporator and the load for the cyclic steady state operation at the standard

operating condition (net input voltage 24 volts and average current 6 amp). These are

measured experimentally.

It is observed that the evaporator (vapour) temperature plunges rapidly during the first

half-cycle to 18.7 °C whilst the inner surface temperature of quartz (where IR heat flux

is being directed at) is 23 °C. At the full rating conditions of 4.7 W/cm2, steady state

performances are reached within two full cycles with load surface temperature

stabilizes at 22.5 °C and the evaporator is at 18.2°C. From experimental observation,

Figure 6.8. The experimentally measured temporal history of the electro-adsorption chiller at a fixed cooling power of 120 W: () defines the temperature of reactor 1; () indicates the silica gel temperature of reactor 2; () shows the condenser temperature; () represents the load surface (quartz) temperature and () is the evaporator temperature. Hence the switching and cycle time intervals are 100s and 600s, respectively.

10

20

30

40

50

60

70

80

90

0 450 900 1350 1800 2250 2700 3150 3600

Time (s)

Tem

pera

ture

(o C)

First half cycleswitching

Second half cycle

147

during switching the temperature of the desorber reactor is increased sharply because

of heat recovery from the other bed and continuous regeneration. Figure 6.9 shows the

typical current profile during a half-cycle operation of the electro-adsorption chiller.

A deviation in the control signal (current I (t)) at the beginning is probably due to the

thermal inertia of the cold end heat exchanger and the desorber bed attached to the hot

end. The thermal mass absorbs a relatively large amount of heat at the transient period

and causes an additional heat pumping effect. The current is thus slightly lowered.

After a long operating period the current becomes constant. The current decreases

abruptly at the end of switching. The value of I (t) increases during the switching

Figure 6.9. The DC current profile of the thermo-electric cooler for the first half-cycle.

0

1

2

3

4

5

6

7

8

9

10

1 10 100 1000

Time (s)

Cur

rent

(am

p)

I(t)

Switching period Operation period

Current increase due to regenerative heat transfer

Current due to high thermal lift between hot and cold junctions

148

operation, because of the presence of regenerative heat transfer occurring across the

junctions of thermo-electrics.

For too short a cycle time, the adsorption beds get insufficient time to respond or

saturate. In other words, the bed consumes excessive thermal power due to frequent

cycling. On the other hand, the evaporator as well as the load (quartz) surface

temperature goes higher at very long cycle time, because the adsorbent is saturated in

less than the allotted time. Figure 6.10 represents COPnet (net Coefficient of

Performance) and the load surface temperature as functions of cycle time. For a given

heat flux qevap (4.7 W per cm2) at the evaporator, the pronounced variation of Tload with

time constrains input voltage. COPeac reaches a limiting value at long cycle time,

because in this limit the impact of thermoelectric transient grows negligible. COPeac

can be expressed as dtVtI

tAqcyclet

cycleevapevap

∫0

)(

.. For a long cycle time, I becomes constant (i.e.,

roughly time-independent).

149

A plot of the load and the evaporator temperatures at different input radiant heat flux

(Watt per cm2) is shown in Figure 6.11. These measurements are conducted at

optimum cycle time operation.

Figure 6.10. Effects of cycle time on average load surface (quartz) temperature, evaporator temperature and cycle average net COP. The constants used in this experiment are the heat flux qevap = 4.7 W/cm2 and the terminal voltage of thermoelectric modules V = 24 volts.

17

20

23

26

29

32

300 350 400 450 500 550 600 650 700 750 800

Cycle time (s)

Tem

pera

ture

(o C)

0.60

0.65

0.70

0.75

0.80

0.85

Net

Coe

ffici

ent o

f Per

form

ance

Evaporator

Load

150

The present electro-adsorption chiller test facility is used here to measure the boiling

heat flux as a function of temperature difference between the quartz surface and the

evaporator at low pressure (1.8 kPa) and the measured values are plotted in Figure

6.12. In the experiments, the range of measured heat fluxes varies from 1 to 5 W/cm2

whilst the DT (Tquartz – Tevap = temperature difference between the quartz surface and

the evaporature at saturation) varies up to 5.4 K. It is noted that on the same plot, the

nucleate pool boiling experimental data of water, as performed by McGillis et al.

(1990) at two pressure ranges of 4 and 9 kPa, are also superimposed. However, these

measured heat fluxes of McGillis et al. (1990) were conducted at heat fluxes from 5 to

10 W/cm2. Using the water properties at the working pressures as well as measured

boiling fluxes and DT, a multi-variable regression is conducted to give the following

modified Rohsenow correlation, as shown below, i.e.,

Figure 6.11. Experimentally measured load and evaporator temperatures as a function of heat flux.

10

12

14

16

18

20

22

24

26

0 1 2 3 4 5 6

Heat flux (W/cm2)

Tem

pera

ture

(o C)

Tload

Tevap

151

48476444444 8444444 76 pressureduefactorCorrection

m

atm

nsCorrelatioRohsenow

n

glfgl

load

pf

slfgsf

PP

ghq

chC

DT ⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ρ−ρσ

µ′′

⎟⎟⎠

⎞⎜⎜⎝

⎛=

)(Pr

,

where the coefficients Csf and n remain unchanged as in the Rohsenow correlation, i.e.,

0.0136 and 0.33, respectively, but the exponent m for the low pressure correction is

best fitted at 0.37. This heat flux correlation is applicable at low absolute pressures

ranging from 1.8 kPa to 10 kPa and extrapolation beyond these pressures may not be

justifiable as higher system pressures tend to delay the on-set of pool boiling. It should

be highlighted that the lowest data point in Figure 6.12, which falls far from the

modified correction, is observed to have no boiling at low heat flux, i.e., the rise in is

attributed to natural convective heat transfer. Owing to the high Awetted / Abase of copper

foam, the capillary effect is found to be excellent and these effects enhance the boiling

heat transfer rates at higher heat fluxes. Hence, Awetted is the total wetted area of copper

foam and Abase is the base area of foam.

152

6.4 Comparison with theoretical modelling

The current formulations of the electro-adsorption chiller (EAC) modeling as

described in chapter 5 makes use of the present experimental data for comparison.

Using the above proposed correlation (with pressure correction), the mathematical

modeling and simulation of EAC are performed. Figure 6.13 compares the predicted

and experimentally measured temperatures of major components of the electro-

adsorption chiller.

Figure 6.12. Heat flux (q flux) versus temperature difference (DT) for water at 1.8 kPa (author’s experiment), 4 kPa (McGillis et al., 1990). and 9 kPa (McGillis et al., 1990). The bottom most data (of 1.8 kPa) is observed to have insignificant boiling, i.e., mainly convective heat transfer by water.

1

10

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8DT (K)

q flu

x (W

/cm

2 )

Pool boiling region

Natural convective

heat transfer region

1.8 kPa

4 kPa 9 kPa

∆ Author’s data at 1.8 kPa with copper-foam, Awetted/Abase = 60.

McGillis et al. (1990) data at 4 kPa, with copper fin Awetted / Abase = 10

McGillis et al. (1990) data at 9 kPa, with copper fin Awetted

/ Abase = 10

153

01020304050607080

010

020

030

040

050

060

070

0

Tim

e (s

)

Temperature (oC)

Hot

reac

tor o

r des

orbe

r

Col

d re

acto

r or a

dsor

ber

Con

dens

er

Eva

pora

tor

Load

sur

face

E

xper

imen

tally

mea

sure

d --

- Sim

ulat

ion

Figu

re

6.13

. E

xper

imen

tally

m

easu

red

tem

pora

l hi

stor

y ve

rsus

th

e si

mul

ated

te

mpe

ratu

res

of m

ajor

com

pone

nts

of e

lect

ro-a

dsor

ptio

n ch

iller

at

a fix

ed

cool

ing

pow

er

of

80

W.

+++

indi

cate

s th

e si

mul

ated

lo

ad

surf

ace

tem

pera

ture

usi

ng th

e pr

opos

ed c

orre

latio

n (C

sf =

0.0

136,

n =

0.3

3 an

d m

=

0.37

) an

d ××

×× s

how

s th

e si

mul

ated

loa

d su

rfac

e te

mpe

ratu

re u

sing

R

ohse

now

poo

l boi

ling

corr

elat

ion

(Csf

= 0

.013

2 an

d n

= 0.

33).

Switc

hing

154

The present electro-adsorption chiller modeling is based on some assumptions, which

are defined in the previous Chapter. Upon these limitations on modeling, one may

observe that the current prediction agrees well with the experimental measurements at

the reactors, condenser and evaporator. Table 6.4 shows the average cooling load

temperature and cycle average COP values obtained from for various cycle times (for a

fixed cooling load 4.7 W per cm2 and 100 s switching time) from experiments and

simulations. The experimental values of evaporator temperature are higher than the

simulated values. The experimentally measured COP is lower than its theoretical

values but both show that COP increases with cycle time.

Table 6.4 Evaporator temperature and cycle average COP (Experimental and simulated values)

Simulated values Experimental

Cycle time (s) Tevap

W Tload W

COP

Tevap ± 0.2 °C

Tload ± 0.2 °C

COP ±0.03

300

400

500

600

700

21.95

17.50

16.73

17.13

17.90

30.27

26.50

25.80

26.60

27.30

0.72

0.75

0.78

0.80

0.81

--

20.2

18.3

18.9

19.7

---

24.0

22.2

22.8

23.5

---

0.69

0.74

0.78

0.8

Heat losses from the electro-adsorption chiller are largely due to the sensible heat load

resulting from alternating heating and cooling of silica gel adsorbents, heat exchanger

and the adsorbed or desorbed water vapour. These sensible loads are included in the

system modeling but other sources of heat losses, which are not accounted in the

155

simulation, are interaction with the external environment and the sensible heating and

cooling of reactors.


The experiments show that the bench-scale electro-adsorption chiller (EAC) has been

successfully designed and commissioned. The bench-sized chiller is found to have a

COP of 0.78 with a maximum cooling capacity of 120 W, an improvement of more

than two times when the system is cooled with thermoelectric cooler only. This

cooling capacity is found in many CPUs of today’s digital processing units. For

comparison, the load surface temperature is not maintained below ambient temperature

by a commercially available thermoelectric cooling (TEC) device as TEC is not

suitable for the application at high heat flux. A new boiling correlation is proposed,

which agrees well with the present experimental data. For verification, the new

correlation has been compared with the published data.

156

Chapter 7

Conclusions and Recommendations

In this thesis, a universal thermodynamic framework that is suitable for both the macro

and micro scale modeling of thermoelectric cooler has been derived. It is based on the

Boltzmann Transport Equation (BTE) approach, incorporating the classical Gibbs law

and the energy conservation equation where their amalgamation yields the form of the

temperature-entropy flux (T-s) formulation. The modeling is applied to the energy

flows and performance investigation of the solid state cooling devices such as the

thermoelectric coolers, the pulsed-mode thermoelectric coolers and the thin-film

superlattice thermoelectric device. It demonstrates the regimes of entropy generation

that are associated to irreversible transport processes arising from the spatial thermal

gradients, as well as the contributions from the collisions of electrons, phonons and

holes in a micro- scaled type solid state cooler.

The pedagogical value of the general temperature-entropy flux diagram has been

successfully demonstrated. The area under the process paths of a thermoelectric

element indicates how energy input has been consumed by the presence of both the

reversible (Peltier, Seebeck and Thomson) and the irreversible (Joule and Fourier)

effects. The similarity between the Seebeck coefficient and the entropy flux for these

elements at adiabatic states has been shown using the T-α diagram. For the first time,

the temperature dependency of Seebeck coefficient on the semi-conductor cooler is

revealed thermodynamically. The incorporation of Thomson heat in the performance

estimation of a thermoelectric cooler, as opposed to previous studies, is estimated to be

about 5% (Chakraborty and Ng, 2005(a)).

157

The thesis demonstrates successfully the transient operation of a pulsed thermoelectric

cooler using the same thermodynamic approach and it provides the necessary

expression for the entropy flux, expressed in terms of the basic measurable variables

(Chakraborty and Ng, 2005 (b)). Using the “time-frozen” T-s diagrams, the cyclic

processes of the pulsed and non-pulsed operations of thermoelectric cooler show how

the input energy has been distributed - depicting the useful energy flows and

“bottlenecks” of cycle operation. The thesis presents also the T-s diagrams of a thin

film thermoelectric, where the effects of electrons flow in n-leg and holes flow in p-leg

generate the total cooling effects at the cold junction. However, the phonons flow in

both legs of the semi-conductors contributes to dissipative losses at the cold junction

and the energy dissipation due to collisions is about 5-10%.

Extending the thermodynamic approach to an adsorbent-adsorbate system, the totally

novel properties for describing accurately the specific heat capacity, the partial

enthalpy and the entropy are presented. These equations are essential for the modeling

and analyses of a novel electro-adsorption chiller. Contrary to the assumption of either

a liquid or gaseous phase for the specific heat capacity, the thesis shows the

theoretically defined specific heat capacity of the adsorbate-adsorbate at all

thermodynamic states (Langmuir, 19, 2254-2259, 2003). The adsorption characteristics

(monolayer, vapor uptake capacity, pre-exponential constant and isosteric heat of

adsorption) of water vapor on silica gel are determined experimentally. The

experimental quantifications of these adsorption characteristics are the author’s novel

contributions. Using the improved thermodynamic model as well as the isotherm

characteristics of the silica gel-water pair (performed prior to EAC experiments –

158

JCED, 47, 1177-1181, 2002), the cyclic operation of an EAC could now be accurately

modeled. The predictions have been shown to compare well with the EAC

experiments. For the given dimensions of a bench-scale EAC unit, the model predicted

the exact mass of adsorbent (300 gm), the overall dimensions of reactor beds and the

amount of adsorbate to be used (US Patent no 6434955). One key design of the EAC

experiments is the design of an infra-red radiant heater and a tapered homogenizer or

kaleidoscope (surface reflectivity = 0.94) which enables uniform heat flux supplied to

the quartz-evaporator. It illuminates the foam-metal cladded evaporator up to 5 W/cm2,

corresponding to a total heat load of 120 W, and this heat load is sufficient to simulate

the heat release from a computer CPU.

The design, developments and fabrication of a bench scale electro-adsorption (EAC)

chiller are the author’s original novel contributions. The design of EAC is developed

on the basis of the adsorption isotherms, adsorption property fields and the dynamic

behavior of thermoelectric modules. In the EAC experiments, tests are conducted

across assorted heat input, cycle and switching time intervals for the adsorption cycle.

The measured pressures and temperatures data agree well with the predictions obtained

from the above-mentioned thermodynamic model and they provide a better

understanding of the EAC cooling cycle. One pleasant surprise from the tests of the

evaporator (during the cyclic steady pool boiling) is excellent plot of heat flux versus

the temperature difference (DT= Twall-Tsat) (Ng and Chakraborty, 2005(b)). The boiling

heat flux data at vacuum pressures as low as 1.8 kPa is not available in the literature.

The nearest similar data were those reported by McGillis et al (1990), however, the

heat fluxes and pressures from the latter were conducted at significantly higher.

Improvising the classical Rohsenow correlation, the thesis proposes a new set of

159

coefficients for pool boiling, namely, a power index for pressure ratio correction at low

vacuum pressures. It predicts accurately the boiling q versus DT at pressures ranging

from 1.8 kPa to 10 kPa. The validity of the new correlation is demonstrated by the

good agreement between the predictions and measured load surface temperatures of

evaporator.

Based on the present work, the incremental improvements of the research are

summarized below:

(1) To conduct the experimental investigation of a thin film superlattice thermoelectric

cooler so as to verify the present model inclusive of the collisions terms.

(2) To improve the lumped modelling of EAC, the distributed modelling could be used

and the performances of the miniaturized EAC could be captured.

(3) To conduct the pressurized electro-adsorption chiller on a bench scale where the

problems associated with vacuum operation could be eliminated.

160

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171

Appendix A Gauss Theorem approach

V is the volume of an element (solids, liquids, vicoelastic materials and rigid bodies)

which is bounded by a closed surface Ω. The differential element of the volume and

surface area is written as dV and dΩ = n dΩ respectively, where n is the unit normal to

Ω and is defined as pointing out of V. with this notation the Gauss’ Theorem is written

as,

Ω=Ω=∇ ∫∫∫ΩΩ

ddndVV

... XXX r (A1)

where X is any sufficiently smooth vector field and is defined on the volume V and the

boundary surfaceΩ.

Balance equations represent the fundamental physical laws and these include the

conservation of mass, momentum, energy and some form of the second law of

thermodynamics, entropy. At any instant, the mass is taken to occupy the volume V

and has a bounding surface given by Ω. The general form of the conservation laws is

given by,

The time rate of change of a quantity = actions of the surroundings on the surface

of V + the actions of the surroundings on the volume itself.

The word “actions” means the change of quantity. Logically, the general form of the

general form of the right hand side of the conservation principle always include all

possible ways to influence the time rate of change of the quantity of interest.

VΩ

172

Conservation of mass

The rate of change of mass of component k within a given volume V is

∫ ∫ ∂∂

=V V

kk dV

tdV

dtd ρ

ρ

dVJddVt

r

j VjkjkkV

k ∑∫∫∫=

Ω+Ω−=

∂∂

1

. vvρρ (A2)

From gauss theorem

∫∫∫∫ΩΩ

Ω=∇⇒Ω=∇ ddVddV kkV

kkkkV

kk .... vvvv ρρρρ

Equation (2) is written as,

( ) ∑

∑∫∫∫

=

=

+−∇=∂∂

⇒

+−∇=∂∂

r

jjkjkk

k

r

j Vjkj

VkkV

k

Jt

dVJdVdVt

1

1

.

.

vv

vv

ρρ

ρρ

(A3)

Total density ∑=

=n

kk

1

ρρ and barycentric velocity ∑=

=n

k

kk

1 ρρ vv

Equation (3) now becomes (neglecting chemical potential)

( )vρρ .−∇=∂∂

t (A4)

Conservation of momentum:

The equation of motion is given by,

( ) ( )

( ) ( ) ZEPt

ZEPt

ZEPdtd

ρρρ

ρρρ

ρρ

++−∇=∂

∂⇒

+−∇=∇+∂

∂⇒

+−∇=

vvv

vvv

v

.

..

.

173

Here vρ = momentum density, P+vvρ = momentum flow and ZEρ = the source of

momentum.

vv = Dyadic product = ji vv (i, j = 1, 2, 3……………..)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=232313

322212

312121

vvvvvvvvvvvvvvv

vv

Conservation of energy

In Tensor notation,

( ) ( )

( ) ( )[ ]

vvvvvv

vvvvvv

vvv

2

.:.21.2

1

....

.

2 ZEPPt

ZEPt

ZEPt

ρρρ

ρρρ

ρρρ

+∇+⎟⎠⎞

⎜⎝⎛ +−∇=

∂

⎟⎠⎞

⎜⎝⎛∂

⇒

++∇−=∂

∂⇒

++−∇=∂

∂

(A5)

Note that

( )[ ]

⎟⎠⎞

⎜⎝⎛⋅∇=

∂∂

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=⋅⋅∇

∑

∑

=

=

vv

vvv

2

1

2

3

1

21

21

,,1,

ρ

ρ

ρρ

n

iji

i

ji

jii

vvx

njivvvx

L

and

174

( )

( ) v

vv

⋅⋅∇−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=∇+⋅⋅∇−

∑ ∑

∑∑∑∑

∑∑∑ ∑

= =

= == =

= == =

P

Px

v

vx

Pvx

PPx

v

njivx

PvPx

PP

n

j

n

iij

ij

n

i

n

jj

iij

n

i

n

jj

iijij

ij

n

i

n

jj

iij

n

i

n

jjij

i

1 1

1 11 1

1 11 1,,3,2,1,: L

The rate of change of potential energy density

( )

( )

( ) ( ) JFZEJt

FJFJt

JFJFJt

n

kkk

n

kkk

n

kkk

n

k

r

jjkjk

n

kkk

n

kkk

n

kkk

−−+−∇=∂

∂∴

−−⎟⎠

⎞⎜⎝

⎛+−∇=

∂∂

⇒

+−−⎟⎠

⎞⎜⎝

⎛+−∇=

∂∂

∑∑∑

∑∑∑∑∑

===

= ====

vv

vv

vvv

..

..

..

111

1 1111

ρψρψρψ

ρψρψρψ

ψρψρψρψ

(A6)

Hence, ( )rjn

kkjk L,10

1==∑

=

vψ as potential energy is conserved into chemical

reaction.

∑=

n

kkk FJ

1

= the source term.

Adding equations (A5) and (A6)

( ) ( )

JFPJPt

JFZEJZEPPtt

−∇+⎟⎠⎞

⎜⎝⎛ +++−∇=

∂

⎟⎠⎞

⎜⎝⎛ +∂

−−+∇−+∇+⎟⎠⎞

⎜⎝⎛ +−∇=

∂∂

+∂

⎟⎠⎞

⎜⎝⎛∂

vvvvvv

vvvvvvvv

2

2

:.21.2

1

...:.21.2

1

2

2

ψρψρψρ

ρψρψρρρψρ

175

For total energy

( )eJ

te .−∇=

∂∂ ρ (A7)

where total energy ue ++= ψ2

21 v ,

total energy flux ∑=

+++=n

kqkke JPeJ

1. Jvv ψρ

veρ = a convective term, v.P = Mechanical work in the system, ∑=

n

kkk J

1ψ = a potential

energy flux due to diffusion, qJ = the heat flow.

From equation (A7) we get,

( )

( )t

JPetu

JPetu

t

JPet

u

n

kqkk

n

kqkk

n

kqkk

∂

⎟⎠⎞

⎜⎝⎛ +∂

−⎟⎠

⎞⎜⎝

⎛+++−∇=

∂∂

⇒

⎟⎠

⎞⎜⎝

⎛+++−∇=

∂∂

+∂

⎟⎠⎞

⎜⎝⎛ +∂

⇒

⎟⎠

⎞⎜⎝

⎛+++−∇=

∂

⎟⎠⎞

⎜⎝⎛ ++∂

∑

∑

∑

=

=

=

ψρψρρ

ψρρψρ

ψρψρ

2

1

1

2

1

2

21

..

..21

..21

vJvv

Jvvv

Jvvv

( ) ( )

( )

JFPJP

JPutu

tJPe

tu

n

kqkk

n

kqkk

+∇−⎟⎠⎞

⎜⎝⎛ +++∇+

⎟⎠

⎞⎜⎝

⎛++∇−⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧ ++−∇=

∂∂

⇒

∂

⎟⎠⎞

⎜⎝⎛ +∂

−⎟⎠

⎞⎜⎝

⎛++∇−−∇=

∂∂

⇒

∑

∑

=

=

vvvvv

Jvvv

vJvv

:.21.

..21.

21

...

2

1

2

2

1

ψρψρ

ψψρρ

ψρψρρ

( )

JFP

JPJJPutu n

kqkk

+∇−

⎟⎠

⎞⎜⎝

⎛−−−−+++++−∇=

∂∂

⇒ ∑=

v

vvvvvvvvv

:

.21.

21. 2

1

2 ψρψρψρρψρρ

176

( ) ( ) JFPutu

q +∇−+−∇=∂

∂⇒ vJv :. ρρ (A8)

ρuv+Jq is the energy flow due to convection and thermal energy transfer that gives rise

to an internal energy change, while -P:∇v + JF is an internal energy source term.

Equation (A8) is simplified as

( )

( ) ( )

JFpdtdu

JFPuudtdu

JFPuut

q

q

q

+⋅∇−⋅−∇=

+∇−+⋅−∇=⋅∇−

+∇−+⋅−∇=∂∂

vJ

vJvv

vJv

ρ

ρρρ

ρρ

:

:

(A9)

Conservation of entropy

The entropy per unit mass is a function of the internal energy u, the specific volume υ

and the mass fractions ck:

),,( kcuss υ= .

In equilibrium the total differential of s is given by the Gibbs relation:

∑=

−+=n

kkk dcpdduTds

1µυ , where p is the equilibrium pressure and kµ is the

thermodynamic or chemical potential of component k.

∑=

−+=n

k

kk dt

dcdtdp

dtdu

dtdsT

1µυ (A10)

From mass balance we get

177

( )

( )

v

v

v

v

v

v

vvv

vvv

v

⋅∇=

⋅∇=

⋅∇−=−

⋅∇−=⎟⎠⎞

⎜⎝⎛

⋅∇−=⎟⎠⎞

⎜⎝⎛

⋅∇−=

∇⋅−⋅∇−=∇⋅−

⋅−∇=∇⋅−∇⋅+∂∂

⋅−∇=∂∂

TP

dtd

TPdtd

dtd

dtd

d

d

dt

d

dtddtd

t

t

υρ

υρ

ρυυ

ρυυυ

ρυ

ρρ

ρρρρ

ρρρρ

ρρ

2

1

1

1

Equation (A4) is written as

( )

( )

( ) ( )

( ) ( )

( )[ ]

kkk

kkkk

kkkkkk

kkkkk

kkkkkk

kkkkk

kkk

dtddt

ddt

ddt

ddt

dt

t

Jv

vvv

vvvvv

vvvvv

vvv

vvv

v

⋅∇−⋅∇−=

−⋅∇−⋅∇−=

∇⋅−−−⋅∇−⋅∇−=

∇⋅−−+−⋅∇−=

∇⋅−⋅∇−=∇⋅−

⋅−∇=∇⋅−∇⋅+∂∂

⋅−∇=∂∂

ρρ

ρρρ

ρρρρ

ρρρ

ρρρρ

ρρρρ

ρρ

where Jk ( )( )vv −= kkρ is the diffusion flow of component k defined with respect to

the barycentric motion.

178

Defining the component mass fractions ck as

1cthatsoc2

1kk

kk =

ρρ

= ∑=

and making use of equation (A6), equation (A7) can be simplified as

( )

kk

kkkk

kkkk

kk

k

kkk

dtdcdt

dcdtd

dtdc

dt

d

dtd

J

Jvv

Jv

Jv

Jv

⋅−∇=

⋅∇−⋅∇−=⋅∇−+

⋅∇−⋅∇−=+

⋅∇−⋅∇−=⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅∇−⋅∇−=

ρ

ρρρρ

ρ

ρρρρ

ρ

ρρ

ρρ

ρρ

From equation (10) we have,

( )

∑∑

∑∑

∑

∑

∑

∑

=

=

=

=

=

=

=

=

⎭⎬⎫

⎩⎨⎧ −∇⋅−∇⋅−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛−

⋅−∇=

∇⋅−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛−

⋅∇−⋅+∇⋅+⋅−∇=

⋅∇−−+⋅+−⋅∇−=

−+=

−+=

n

kk

kkq

kkkq

n

k

kk

n

kkkn

kkkq

q

n

kk

kq

n

k

kk

n

k

kk

FT

TT

TTT

TTF

TTT

Tdtd

TpFJ

Tdtd

Tp

T

dtdc

Tdtd

Tp

dtdu

Tdtds

dtdc

dtdp

dtdu

dtdsT

12

2

1

1

1

1

1

1

1

11

11

11

µµ

µµ

µυρυρ

ρµυρρρ

µυ

JJJJ

JJ

JJJ

JJ

(A11)

where Jk ( )( )vv −ρ= kk is the diffusion flow of component k defined with respect to

the barycentric motion.

179

From equation (A11), one can observe that the entropy flow is given by

⎟⎠

⎞⎜⎝

⎛−= ∑

=

n

kkkqs T 1

1 JJJ µ (A12)

and the entropy source strength by

∑= ⎭

⎬⎫

⎩⎨⎧ −∇⋅−∇⋅−=

n

kk

kkq F

TT

TT

T 12

11 µσ JJ (A13)

180

Appendix B

The Thomson effect in equation (3.7)

The electrical and thermal currents are coupled in a thermoelectric device. The Peltier

coefficient of the junction is a property depends on both the materials and is the ratio

of power evolved at the junction to the current flowing through it. When current is

applied to a thermoelectric element, thermal energy is generated or absorbed at the

junction due to Peltier effect. The Seebeck coefficient depends on temperature and this

is different at different places along the TE material. So the thermoelectric element is

thought of as a series of many small Peltier junctions and each of which is generating

or absorbing heat. This is called Thomson power evolved per unit volume ( )TJ ∇Γ .

In a Thomson effect, heat is absorbed or evolved when current is flow in a

thermoelectric element with a temperature gradient. The heat is proportional to both

the electric current and the temperature gradient. The proportionality constant is

known as Thomson coefficient.

DC power source+ -

J

Heat absorbed at the cold end

p-arm

J

Heat rejected from the hot end

n-arm

J

J

Tcj

Thj

p-arm or

n-arm

Heat absorbed at cold side ( )Jcjπ

Heat involved at the hot side ( )Jhjπ

TJ ∇Γ.Thomson heat

Figure B1 A thermoelectric element showing the transportation of heat and current.

181

If the heat current density of a thermoelectric element is only a result of temperature

gradient and the electric current density occurs due to electric potential gradient. Then,

Tq ∇−= .λJ and φσ ∇′=J . Using equation (3.1) the energy balance equation of a

thermoelectric element is obtained.

( )σ

λρ′

+∇∇=∂∂ 2

. JTtTcp (B1)

The first term of right hand side of equation (B1) is Fourier conduction term and the

second term indicates the Joule heat. But in a thermoelectric cooling device, the

junctions are maintained at different temperatures and an electromotive force will

appear in the circuit. This flow of heat has a tendency to carry the electricity along the

thermoelectric arm.

When two or more irreversible processes take place in a thermodynamic system, they

may interfere with each other. A completely consistent theory of thermoelectricity has

been developed using the Onsager symmetry relationship (Onsager, 1931). When a

current is applied to a thermoelectric device, the heat current of TE arm is developed

as a result of electric current and temperature gradient (cross flow) and the electric

current is generated due to electric potential and the temperature gradient.

Using the thermodynamic-phenomenological treatment, the heat current density is

TJFQo

q ∇−−= λJ , and the electric current density in amp per m2 becomes

TFTQJ

TFT

QJ

o

o

∇−′

=∇

∇′

+∇′=

.σ

φ

σφσ

Equation (3.1) now becomes

182

( )

( ) ⎟⎠⎞

⎜⎝⎛∇+

′+∇∇=

∇−′

+∇+∇∇=

⎟⎠⎞

⎜⎝⎛ ∇−

′+⎟

⎠⎞

⎜⎝⎛ −∇−−∇=

∂∂

TQ

FJTJT

TFTJQJQ

FJT

TFTQJJJ

FQT

tTc

o

oo

oop

σλ

σλ

σλρ

2

2

.

..

..

( ) ⎟⎠⎞

⎜⎝⎛∇+

′+∇∇=

∂∂

TQ

FJTJT

tTc o

p σλρ

2

. (B2)

Hence the third term ⎟⎠⎞

⎜⎝⎛∇

TQ

FJT o occurs due to current flow for temperature gradient

(cross flow) and heat flow for the electric potential gradient (reciprocal effect). This

term indicates the least dissipation of energy (Onsager, 1931)

( )sFJT

TQ

FJT o ∇=⎟

⎠⎞

⎜⎝⎛∇ , where ⎟

⎠⎞

⎜⎝⎛=

TQs o is the entropy (in J/mol.K) along the

thermoelectric arm.

Hence

( )

TTs

FTJ

TTs

FJT

sFJT

TQ

FJT o

∇⎟⎠⎞

⎜⎝⎛

∂∂

=

∇∂∂

=

∇=⎟⎠⎞

⎜⎝⎛∇

.

.

The Thomson heat is defined as

( ) TJQ

TTs

FTJQ

T

T

∇Γ−=

∇⎟⎠⎞

⎜⎝⎛

∂∂

= .,

where Thomson coefficient Ts

FT∂∂

=Γ− .

The temperature gradient is bound to cause a certain degradation of energy by

conduction of heat. The electric potential gradient must cause an additional

183

degradation of energy, making the total rate of increase of entropy along the

thermoelectric arm.

So the electric current due to temperature gradient (cross flow) and the heat current

due to electric potential gradient (reciprocal effect) indicate the Thomson effect, which

will be discussed in the following section. Without Thomson effect, the physics of

thermoelectricity is not completed.

Derivation of equation (3.7)

The energy balance of the thermoelectric arm is

( ) TJJTtTcp ∇Γ−

′+∇∇=

∂∂

σλρ

2

. (B3)

Expression of the third term

Using Kelvin relations ⎥⎦⎤

⎢⎣⎡ =

∂∂

−=Γ TandTT

απππ ,

TTT

JTJ ∇⎟⎠⎞

⎜⎝⎛

∂∂

−=∇Γ .ππ (B4)

TTT∇

∂∂

+∇=∇ .πππ (de Groot and Mazur, 1962)

From equation (B4) we have

184

( )

( )

T

T

T

T

T

JTTT

JT

JTTT

JTTJTJ

JTTJTJ

JJTT

J

JTT

J

TT

JTT

JTJ

αα

αααα

ααα

πππ

πππ

ππ

∇+∇∂∂

−=

∇+∇∂∂

−∇−∇=

∇+∇−∇=

∇+∇−∇=

∇−∇−∇=

∇∂∂

−∇=∇Γ

.

.

.

.

..

Equation (B3) is now written as

( )Tp JTT

TJTJT

tTc αα

σλρ ∇+∇

∂∂

−′

+∇∇=∂∂ ..

2

(B5)

Equation (B5) is same as equation (3.7).

185

Appendix C

Energy balance of a thermoelectric element

Energy balance equation at steady state becomes

022

2

=+− JdxdTJ

dxTd ρτλ , (C1)

where τ indicates Thomson coefficient, λ defines thermal conductivity and ρ is the

electrical resistivity.

The solution becomes

BeJ

AxJT

AeJdxdT

xJ

xJ

++=

+=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

λτ

λτ

τλ

τρ

τρ

and

Putting boundary conditions

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

⎟⎠⎞

⎜⎝⎛ −

=⇒=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

⎟⎠⎞

⎜⎝⎛ −

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+==−

+=

++=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

11

1

_________________________________

LJLJ

LJ

LH

L

LJ

H

e

JJLDTAA

eJ

JLDT

eJ

ALJDTTT

BJ

AT

BeJ

ALJT

λτ

λτ

λτ

λτ

λτ

τρ

τλ

τρ

τλ

τρ

τλ

τλ

τρ

Now,

186

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛−−=−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛+=

1expexp

1expexp

LJ

JJLDTxJAJA

dxdTA

LJ

JJLDTxJJ

dxdT

λτ

λτ

τρ

λτλ

τρλλ

λτ

λτ

τρ

λτ

τρ

( )

( )

( ) ⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

⎟⎠

⎞⎜⎝

⎛−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ +−

⎟⎠⎞

⎜⎝⎛+−=−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠

⎞⎜⎝

⎛−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛−−=−

xAIxL

AI

AIL

LAILA

LADT

AIA

dxdTA

xJxLJ

JLLJLA

LADT

JAdxdTA

xJxLJ

JJLDTAJA

dxdTA

LJ

JJLDTAxJJA

dxdTA

LJ

JJLDTxJAJA

dxdTA

λτ

λτ

λτ

τρλλ

τρλλ

λτ

λτ

λτ

τρλλ

τρλλ

λτ

λτ

λτ

τρλ

τρλλ

λτ

λτ

τρλ

λτ

τρλλ

λτ

λτ

τρ

λτλ

τρλλ

expexp

expexp

expexp

1expexp

1expexp

187

( )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

Lx

AIL

Lx

AIL

AILI

LADT

IdxdTA

xAIxL

AI

AILI

LADT

IdxdTA

xAIxL

AI

AILI

LADT

IdxdTA

λτ

λτ

λτ

τλρλ

τλρλ

λτ

λτ

λτ

τλρλ

τλρλ

λτ

λτ

λτ

τλρλ

τλρλ

exp1exp

expexp

expexp

(C2)

Defining Fourier heat L

ADTQFλ

= , Joule heat A

LIQJ

2ρ= and Thomson heat

DTIQT τ=

Here, ALI

LADTDTI

QQ

F

T

λτ

λτ

==

Equation (2) is written as,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=−

Lx

QQ

Lx

QQ

QQIQ

IdxdTA

F

T

F

T

F

TF

exp1exp

τλρ

τλρλ (C3)

At cold junction (x = 0)

τλρ

τλρτ

λρ

τλρλ IIQ

QQQQ

QQ

QQIQ

IdxdTA F

F

T

F

T

F

T

F

TF

x

−⎟⎠⎞

⎜⎝⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛ +−

+−=−= 1exp1exp0

As we know,

.......................................!642

1!430

1!26

12

11)exp(

642

−+−+−=−

xxxxxx

188

..................21

3601

21

61

211

1exp

42

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ F

T

F

T

F

T

F

T

F

T

QQ

QQ

QQ

QQQQ

..................21

3601

21

61

211

1exp

42

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

×⎟⎠⎞

⎜⎝⎛ +−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +−

F

T

F

T

F

T

F

F

T

F

T

F

QQ

QQ

QQ

IQ

QQQQ

IQτλρ

τλρ

After rearranging,

....................................................................................................................21

21

3601

21

21

61

21

21

3

3

3

42

0

+⎥⎦

⎤⎢⎣

⎡+−⎥

⎦

⎤⎢⎣

⎡++⎥⎦

⎤⎢⎣⎡ −−−=−

= F

TJ

F

T

F

TJ

F

TJTF

x QQQ

QQ

QQQ

QQQQQ

dxdTAλ

⎥⎦⎤

⎢⎣⎡ −−−=−

=JTF

x

QQQdxdTA

21

21

0

λ

Similarly at hot junction we can prove that (removing all higher order terms)

⎥⎦⎤

⎢⎣⎡ ++−=−

=JTF

Lx

QQQdxdTA

21

21λ (C4)

Energy analysis

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=

Lx

QQ

Lx

QQ

QQIQ

IxITxQ

F

T

F

T

F

TF

exp1exp)()( τ

λρ

τλρα (C5)

Similarly at hot junction

⎥⎦⎤

⎢⎣⎡ ++−+= JTFHH QQQITQ

21

21α (C6)

From (C6) we have,

189

DTIKDTRITQ HH τα21

21 2 +−+= I , where

LAK λ

= , ALR ρ

= .

This is same as (3.16)

At cold junction

⎥⎦⎤

⎢⎣⎡ −−−+= JTFLL QQQITQ

21

21α (C7)

(C7) is written as,

DTIKDTRIITQ LL τα21

21 2 −−−= , which is same as (3.15)

Net power input

( ) TJLH

JTFLJTFHLH

QQTTI

QQQITQQQITQQ

++−=

⎥⎦⎤

⎢⎣⎡ −−−+−⎥⎦

⎤⎢⎣⎡ ++−+=−

α

αα21

21

21

21

(C8)

Entropy flux

)(1

exp1exp)()(

)()(xT

Lx

QQ

Lx

QQ

QQIQ

xTI

xTxITxJ

F

T

F

T

F

TF

S

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +−

+−=τλρ

τλρα (C9)

190

Appendix D

Programming Flow Chart of the thin film thermoelectric cooler (Superlattice thermoelectric element (SLTE))

Figure D1 Ensemble Particle Motion. Here each horizontal solid line shows the trace of each particle. The vertical broken line shows time. The on the solid line shows the scattering or collision time.

τ τ + ∆ τ τ - ∆ τ

Equation Solver

191

Flow chart of SLTE programming and Monte Carlo method to solve collision time.

START

CONFIGURATION

Geometry of Device

DATA INPUT

Initial Conditions For mass, energy,

temperature and entropy

EMC (Ensemble Monte-Carlo) For calculating collision time

Using IMSL solver Balance equation of mass (ρ) Balance equation of momentum (v) Balance equation of energy (u and T) Balance equation of entropy (s)

t < tmax

No

END

Start

τ1 = τi τ2 = τs = scattering time

τ2 > τ + ∆ τ

Yes

τcolli = τ + ∆ τ – τ1

Cal. τcolli

τcolli = τ2 – τ1

Drift (τ)

τ1 = τ2 τ2 = new scattering time

192

Appendix E

Programming Flow Chart of the electro-adsorption chiller (EAC) and water properties equations

Initial Conditions for

EAC (t = 0)

Open Files

START

Data Storage

Read Data

Adsorption Bed Tads = Troom qads = 0 Qads = 0

Desorption Bed Tdes = Troom qdes = 0 Qdes = 0


Tte = Troom Pin = 0

Condenser and evaporator Qcond = 0 Qevap = 0

N

Y

Data Output By

WRITE

Calculation EAC Operation

by Call Function 1 By

DIVPAG in IMSL

Calculation of Adsorption isotherm Adsorption isobar Adsorption kinetics

Calculation of water saturation and superheated properties (Wagner routines) Equations are given in section below

Close

If t ? (Operation time)

Ope

ratio

n pe

riod

193

N

Y

Data Output By

WRITE

START Switching

Initial value of switching

operation (Data from

previous operation period)

ADS - DES DES - ADS TE COND & EVAP

Calculation EAC switching by

Call Function 2

By DIVPAG in IMSL

If ׀Tj – Tj-1 ׀ ? 1× 10-2 N

Y

END

If t ? (switching time)

194

Properties of water and steam

All equations for calculating water and steam thermodynamic properties are covered by the new industrial standard “IAPWS Industrial formulation 1997 for the Thermodynamic Properties of Water and Steam” [Wagner, 1997]. The basic equation for the liquid water is a fundamental equation for the specific Gibbs

free energy g. This equation is expressed in dimensionless form, )/(RTg=γ , and

reads

( ) ( ) ( ) ( )∑=

−−==34

1222.11.7,,

i

JIi

iinRT

Tpg τπτπγ ,

where =π p/p* and =τ T*/T with p*=16.53Mpa and T*=1386K;R=0.461526kJkg-1K-1.

The coefficients ni and exponents Ii and Ji are listed in Table 1.

All thermodynamic properties can be derived from the basic equation by using the

appropriate combinations of the dimensionless Gibbs free energy γ and its derivatives.

Relations between the relevant thermodynamic properties and γ and its derivatives are

summarized in Table 2.

Table E1 Coefficients and exponents of the basic equation i Ii Ji ni

1 0 -2 0.14632971213167

2 0 -2 -0.84548187169114

3 0 0 -.37563603672040×101

4 0 1 0.33855169168385×101

5 0 2 -0.95791963387872

6 0 3 0.15772038513228

7 0 4 -0.16616417199501×10-1

8 0 5 0.81214629983568×10-3

9 1 -9 0.28319080123804×10-3

10 1 -7 -0.60706301565874×10-3

11 1 -1 -0.18990068218419×10-1

12 1 0 -0.32529748770505×10-1

195

13 1 1 -0.21841717175414×10-1

14 1 3 -0.52838357969930×10-4

15 2 -3 -0.47184321073267×10-3

16 2 0 -0.30001780793026×10-3

17 2 1 0.47661393906987×10-4

18 2 3 -0.44141845330846×10-5

19 2 17 -0.72694996297594×10-15

20 3 -4 -0.31679644845054×10-4

21 3 0 -0.28270797985312×10-5

22 3 6 -0.85205128120103×10-9

23 4 -5 -0.22425281908000×10-5

24 4 -2 -0.65171222895601×10-6

25 4 10 -0.14341729937924×10-12

26 5 -8 -0.40516996860117×10-6

27 8 -11 -0.12734301741641×10-8

28 8 -6 -0.17424871230634×10-9

29 21 -29 -0.68762131295531×10-18

30 23 -31 0.14478307828521×10-19

31 29 -38 0.26335781662795×10-22

32 30 -39 -0.11947622640071×10-22

33 31 -40 0.18228094581404×10-23

34 32 -41 -0.93537087292458×10-25

196

Table E2 Relations of thermodynamic properties to the dimensionless Gibbs free

energy γ and its derivatives.

Property Relation

Specific volume

( )Tpgv ∂∂= / ( ) ππγτπ =

RTpv ,

Specific internal energy

( ) ( )Tp pgpTgTgu ∂∂−∂∂−= //

( )πτ πγτγτπ

−=RT

u ,

Specific entropy

( )pTgs ∂∂−= /

( ) γτγτπτ −=

Rs ,

Specific enthalpy

( )pTgTgh ∂∂−= /

( )ττγτπ

=RT

h ,

Specific isobaric heat capacity

( )pp Thc ∂∂= /

( )ττγτ

τπ 2,−=

Rc p

τπ π

γγ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= ,π

τ τγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

= ,τ

ππ πγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

= 2

2

,τ

πτ τπγγ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂=

2

,π

ττ τγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

= 2

2

A sample set of thermodynamic property values are shown below where the basic equations for selected temperatures and pressures are employed to determine these parameters.

Property T=300K, p=0.0035Mpa

v[m3kg-1] 0.100215168×10-2

h[kJkg-1] 0.115331273×103

u[kJkg-1] 0.112324818×103

s[kJkg-1K-1] 0.392294792

cp[kJkg-1K-1] 0.417301218×101

The basic equation for steam is a fundamental equation for the specific Gibbs free

energy g. This equation is expressed in dimensionless form, ( )RTg /=γ , and is

separated into 2 parts, an ideal-gas part 0γ and a residual part rγ , so that

( ) ( ) ( )τπγτπγτπγ ,,),(, 0 r

RTTpg

+== ,

197

The equation for the ideal-gas part 0γ of the dimensionless Gibbs free energy reads

∑=

+=9

1

00 0

lni

Ji

in τπγ ,

The form of the residual part rγ of the dimensionless Gibbs free energy is as follows:

( )∑=

−=43

1

5.0i

JIi

r iin τπγ ,

where */ pp=π and TT /*=τ with *p =1 MPa and *T =540K.The coefficients 0in and

exponents 0iJ are listed in Table 3.The coefficients in and exponents Ii and Ji are listed

in Table 4.

All thermodynamic properties can be derived from the basic equation by using the

appropriate combinations of the ideal-gas part 0γ and the residual part rγ of the

dimensionless Gibbs free energy and their derivatives.

Property Relation

Specific volume

( )Tpgv ∂∂= / ( ) ( )r

RTpv ππ γγπτπ += 0,

Specific internal energy

( ) ( )Tp pgpTgTgu ∂∂−∂∂−= // ( ) ( ) ( )rr

RTu

ππττ γγπγγττπ+−+= 00,

Specific entropy

( )pTgs ∂∂−= / ( ) ( ) ( )rr

Rs γγγγττπ

ττ +−+= 00,

Specific enthalpy

( )pTgTgh ∂∂−= / ( ) ( )r

RTh

ττ γγττπ+= 0,

Specific isobaric heat capacity

198

( )pp Thc ∂∂= / ( ) ( )rp

Rc

ττττ γγττπ

+−= 02,

τπ π

γγ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=r

r ,π

τ τγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=r

r ,π

ττ τγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

= 2

2 rr ,

τπ π

γγ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=0

0 ,π

τ τγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=0

0 ,π

ττ τγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

= 2

020

Table E3 Coefficients 0in and exponents

0iJ

i 0iJ 0

in

1 0 -0.96927686500217×101

2 1 0.10086655968018×102

3 -5 -0.56087911283020×10-2

4 -4 0.71452738081455×10-1

5 -3 -0.40710498223928

6 -2 0.14240819171444×101

7 -1 -0.43839511319450×101

8 2 -0.28408632460772

9 3 0.21268463753307×10-1

Table E4 Coefficients ni and exponents Ii ,Ji i Ii Ji ni

1 1 0 -0.17731742473213×10-2

2 1 1 -0.17834862292358×10-1

3 1 2 -0.45996013696365×10-1

4 1 3 -0.57581259083432×10-1

5 1 6 -0.50325278727930×10-1

6 2 1 -0.33032641670203×10-4

7 2 2 -0.18948987516315×10-3

8 2 4 -0.39392777243355×10-2

9 2 7 -0.43797295650573×10-1

10 2 36 0.26674547914087×10-4

199

11 3 0 0.20481737692309×10-7

12 3 1 0.43870667284435×10-6

13 3 3 -0.32277677238570×10-4

14 3 6 -0.15033924542148×10-2

15 3 35 -0.40668253562649×10-1

16 4 1 -0.78847309559367×10-9

17 4 2 0.12790717852285×10-7

18 4 3 0.48225372718507×10-6

19 5 7 0.22922076337661×10-5

20 6 3 -0.16714766451061×10-10

21 6 16 -0.21171472321355×10-2

22 6 35 -0.23895741934104×102

23 7 0 -0.59059564324270×10-17

24 7 11 -0.12621808899101×10-5

25 7 25 -0.38946842435739×10-1

26 8 8 0.11256211360459×10-10

27 8 36 -0.82311340897998×101

28 9 13 0.19809712802088×10-7

29 10 4 0.10406965210174×10-18

30 10 10 -0.10234747095929×10-12

31 10 14 -0.10018179379511×10-8

32 16 29 -0.80882908646985×10-10

33 16 50 0.10693031879409

34 18 57 -0.33662250574171

35 20 20 0.89185845355421×10-24

36 20 35 0.30629316876232×10-12

37 20 48 -0.42002467698208×10-5

38 21 21 -0.59056029685639×10-25

39 22 53 0.37826947613457×10-5

40 23 39 -0.12768608934681×10-4

41 24 26 0.73087610595061×10-28

42 24 40 0.55414715350778×10-16

200

43 24 58 -0.94369707241210×10-6

A sample set of thermodynamic property values are shown below where the basic

equations for selected temperatures and pressures are employed to determine these

parameters.

Property T=300K, p=0.0035Mpa

v[m3kg-1] 39.4913866

h[kJkg-1] 2549.91145

u[kJkg-1] 2411.69160

s[kJkg-1K-1] 8.52238967

cp[kJkg-1K-1] 1.91300162

201

Appendix F Energy flow of the major components of an electro-adsorption chiller

Latent heat balance of an electro-adsorption chiller (last half cycle) time Qe,L Qd,L Qa,L Qc,L lhs rhs Error%1.162791 -135.565 -195.282 145.6575 184.4292 -330.847 330.0867 0.229907 5.813953 -135.656 -195.293 145.7517 184.4396 -330.949 330.1914 0.229015 10.46512 -135.734 -195.279 145.8313 184.4265 -331.013 330.2578 0.228068 17.44186 -135.834 -195.255 145.9341 184.4038 -331.089 330.3379 0.226778 23.25581 -135.908 -195.238 146.0095 184.3888 -331.146 330.3983 0.22583 46.51163 -136.149 -195.213 146.2581 184.3663 -331.362 330.6243 0.222726 69.76744 -136.344 -195.227 146.4602 184.3811 -331.571 330.8413 0.220141 93.02326 -136.509 -195.259 146.6328 184.4129 -331.768 331.0457 0.217766 98.83721 -136.546 -195.269 146.672 184.422 -331.815 331.094 0.217191 127.907 -136.709 -195.316 146.8443 184.4688 -332.025 331.3131 0.214415

156.9767 -136.826 -195.36 146.9683 184.5131 -332.185 331.4814 0.211928 180.2326 -136.886 -195.39 147.0333 184.5441 -332.276 331.5774 0.210155 203.4884 -136.921 -195.414 147.072 184.57 -332.335 331.642 0.208538 226.7442 -136.934 -195.432 147.0883 184.5899 -332.366 331.6783 0.207043

250 -136.929 -195.443 147.0854 184.6036 -332.372 331.689 0.205638 279.0698 -136.901 -195.447 147.0589 184.6111 -332.348 331.67 0.203972 308.1395 -136.853 -195.437 147.0112 184.6066 -332.29 331.6178 0.202359 331.3953 -136.804 -195.404 146.9603 184.5796 -332.208 331.54 0.201037 360.4651 -136.73 -195.32 146.8833 184.5042 -332.049 331.3875 0.199305 389.5349 -136.645 -195.187 146.7938 184.3828 -331.832 331.1766 0.197492 418.6047 -136.552 -195.01 146.6939 184.2197 -331.562 330.9137 0.19561 447.6744 -136.452 -194.795 146.5856 184.02 -331.247 330.6056 0.193671 476.7442 -136.346 -194.547 146.4702 183.7887 -330.893 330.2589 0.191689 494.186 -136.28 -194.385 146.3981 183.6369 -330.665 330.035 0.190482

500 -136.258 -194.328 146.3736 183.5844 -330.586 329.958 0.190077 502 -136.238 -194.3 146.3523 183.5576 -330.538 329.9099 0.190077 510 -136.159 -194.187 146.2669 183.4505 -330.345 329.7174 0.190077 520 -136.059 -194.045 146.1603 183.3168 -330.105 329.4771 0.190077 530 -135.96 -193.904 146.0538 183.1833 -329.864 329.2371 0.190077 540 -135.861 -193.763 145.9475 183.0499 -329.624 328.9975 0.190077 550 -135.762 -193.622 145.8414 182.9168 -329.384 328.7582 0.190077 560 -135.664 -193.481 145.7354 182.7839 -329.145 328.5193 0.190077 570 -135.565 -193.341 145.6295 182.6511 -328.906 328.2807 0.190077 580 -135.467 -193.2 145.5239 182.5186 -328.667 328.0425 0.190077 590 -135.369 -193.06 145.4183 182.3862 -328.429 327.8046 0.190077

597.5 -135.295 -192.955 145.3393 182.2871 -328.25 327.6264 0.190077 600 -135.271 -192.92 145.313 182.2541 -328.191 327.567 0.190077

202

Sensible energy balance (Last half chcle) time Qevap Qcond Qadsc Qh Qads,Ex Qads,Te 1.162791 129.05 -184.368 191.0289 236.8934 114.6174 76.41157 10.46512 129.05 -184.265 191.0832 236.9165 114.6499 76.43328 23.25581 129.05 -184.177 191.1482 236.9419 114.6889 76.45929 46.51163 129.05 -184.099 191.2298 236.9799 114.7379 76.49191 63.95349 129.05 -184.085 191.2611 237.0022 114.7567 76.50444

87.2093 129.05 -184.097 191.2676 237.0244 114.7606 76.50705 110.4651 129.05 -184.127 191.2393 237.0392 114.7436 76.4957 168.6047 129.05 -184.217 191.0553 237.0603 114.6332 76.42212 197.6744 129.05 -184.256 190.9215 237.0666 114.5529 76.36858 203.4884 129.05 -184.263 190.8922 237.0675 114.5353 76.35686 209.3023 129.05 -184.27 190.8621 237.0684 114.5173 76.34484 215.1163 129.05 -184.276 190.8313 237.0692 114.4988 76.33253 220.9302 129.05 -184.282 190.7999 237.0699 114.4799 76.31994 226.7442 129.05 -184.287 190.7677 237.0705 114.4606 76.3071 232.5581 129.05 -184.292 190.735 237.0711 114.441 76.294 238.3721 129.05 -184.297 190.7016 237.0716 114.421 76.28066

244.186 129.05 -184.301 190.6677 237.072 114.4006 76.26709 250 129.05 -184.305 190.6333 237.0723 114.38 76.25331

255.814 129.05 -184.309 190.5983 237.0726 114.359 76.23931 261.6279 129.05 -184.312 190.5628 237.0728 114.3377 76.22512 267.4419 129.05 -184.315 190.5268 237.073 114.3161 76.21073 273.2558 129.05 -184.317 190.4904 237.073 114.2942 76.19616 279.0698 129.05 -184.319 190.4536 237.0731 114.2721 76.18142 284.8837 129.05 -184.32 190.4163 237.073 114.2498 76.16651 290.6977 129.05 -184.321 190.3786 237.0729 114.2272 76.15144 296.5116 129.05 -184.322 190.3405 237.0728 114.2043 76.13622 302.3256 129.05 -184.322 190.3021 237.0727 114.1813 76.12085 308.1395 129.05 -184.322 190.2633 237.0729 114.158 76.10534 313.9535 129.05 -184.32 190.2242 237.0731 114.1345 76.08969 319.7674 129.05 -184.318 190.1848 237.0735 114.1109 76.07392 325.5814 129.05 -184.314 190.1451 237.074 114.087 76.05802 331.3953 129.05 -184.309 190.105 237.0745 114.063 76.04201 337.2093 129.05 -184.302 190.0647 237.0752 114.0388 76.02588 343.0233 129.05 -184.293 190.0241 237.0759 114.0145 76.00965 348.8372 129.05 -184.283 189.9833 237.0768 113.99 75.99331 354.6512 129.05 -184.271 189.9422 237.0777 113.9653 75.97688 360.4651 129.05 -184.257 189.9009 237.0786 113.9405 75.96035 366.2791 129.05 -184.241 189.8593 237.0797 113.9156 75.94373

372.093 129.05 -184.224 189.8176 237.0809 113.8905 75.92702 377.907 129.05 -184.204 189.7756 237.0821 113.8653 75.91023

383.7209 129.05 -184.183 189.7334 237.0833 113.84 75.89336 389.5349 129.05 -184.16 189.6911 237.0847 113.8146 75.87642 395.3488 129.05 -184.135 189.6485 237.0861 113.7891 75.8594 401.1628 129.05 -184.109 189.6058 237.0876 113.7635 75.84232 406.9767 129.05 -184.08 189.5629 237.0891 113.7377 75.82516 412.7907 129.05 -184.051 189.5199 237.0908 113.7119 75.80795 418.6047 129.05 -184.019 189.4767 237.0924 113.686 75.79067 424.4186 129.05 -183.986 189.4333 237.0942 113.66 75.77333 430.2326 129.05 -183.952 189.3899 237.096 113.6339 75.75594 436.0465 129.05 -183.916 189.3462 237.0978 113.6077 75.7385 441.8605 129.05 -183.878 189.3025 237.0997 113.5815 75.721

203

447.6744 129.05 -183.839 189.2587 237.1017 113.5552 75.70346 453.4884 129.05 -183.799 189.2147 237.1037 113.5288 75.68587 459.3023 129.05 -183.758 189.1706 237.1058 113.5024 75.66824 465.1163 129.05 -183.715 189.1264 237.1079 113.4759 75.65057 470.9302 129.05 -183.671 189.0822 237.11 113.4493 75.63286 476.7442 129.05 -183.625 189.0378 237.1121 113.4227 75.61512 482.5581 129.05 -183.579 188.9933 237.1142 113.396 75.59734 488.3721 129.05 -183.531 188.9488 237.1166 113.3693 75.57952

494.186 129.05 -183.483 188.9042 237.1189 113.3425 75.56168 500 129.05 -183.433 188.8595 237.1214 113.3157 75.5438

500.5 129.05 -183.43 188.8877 237.1394 113.3326 75.55509 501 129.05 -183.428 188.9158 237.1463 113.3495 75.56633

501.5 129.05 -183.425 188.9438 237.1509 113.3663 75.57751 502 129.05 -183.422 188.9716 237.1542 113.383 75.58863

502.5 129.05 -183.419 188.9993 237.1568 113.3996 75.5997 505 129.05 -183.401 189.1355 237.1617 113.4813 75.65421

507.5 129.05 -183.382 189.2684 237.162 113.561 75.70736 510 129.05 -183.36 189.3979 237.1604 113.6388 75.75917

512.5 129.05 -183.336 189.5242 237.1579 113.7145 75.80967 515 129.05 -183.311 189.6472 237.1547 113.7883 75.85888

517.5 129.05 -183.284 189.7671 237.151 113.8603 75.90684 520 129.05 -183.257 189.8839 237.1469 113.9304 75.95357

522.5 129.05 -183.229 189.9977 237.1427 113.9986 75.99908 525 129.05 -183.2 190.1085 237.1382 114.0651 76.04342

527.5 129.05 -183.17 190.2165 237.1336 114.1299 76.08659 530 129.05 -183.14 190.3216 237.1288 114.1929 76.12863

532.5 129.05 -183.109 190.4239 237.1238 114.2543 76.16955 535 129.05 -183.078 190.5235 237.1185 114.3141 76.20938

537.5 129.05 -183.046 190.6204 237.1131 114.3722 76.24815 540 129.05 -183.015 190.7147 237.1076 114.4288 76.28586

542.5 129.05 -182.983 190.8064 237.102 114.4838 76.32255 545 129.05 -182.95 190.8956 237.0961 114.5374 76.35824

547.5 129.05 -182.918 190.9823 237.09 114.5894 76.39293 550 129.05 -182.886 191.0667 237.0838 114.64 76.42667

552.5 129.05 -182.853 191.1486 237.0775 114.6892 76.45945 555 129.05 -182.82 191.2283 237.071 114.737 76.49131

557.5 129.05 -182.788 191.3056 237.0643 114.7834 76.52225 560 129.05 -182.755 191.3808 237.0575 114.8285 76.55231

562.5 129.05 -182.722 191.4537 237.0505 114.8722 76.5815 565 129.05 -182.689 191.5246 237.0434 114.9147 76.60982

567.5 129.05 -182.656 191.5933 237.0361 114.956 76.63731 570 129.05 -182.623 191.66 237.0287 114.996 76.66398

572.5 129.05 -182.59 191.7246 237.0211 115.0348 76.68985 575 129.05 -182.557 191.7873 237.0134 115.0724 76.71492

577.5 129.05 -182.524 191.8481 237.0055 115.1088 76.73922 580 129.05 -182.491 191.9069 236.9975 115.1441 76.76276

582.5 129.05 -182.458 191.9639 236.9894 115.1783 76.78556 585 129.05 -182.425 192.0191 236.9811 115.2115 76.80764

587.5 129.05 -182.392 192.0725 236.9726 115.2435 76.829 590 129.05 -182.359 192.1241 236.9641 115.2745 76.84966

592.5 129.05 -182.326 192.1741 236.9553 115.3044 76.86963 595 129.05 -182.293 192.2223 236.9465 115.3334 76.88894

597.5 129.05 -182.26 192.269 236.9375 115.3614 76.90758 600 129.05 -182.227 192.314 236.9284 115.3884 76.92558

204

Pin COPte COPT COPnet Qc+Qaes Qe +Qdes %Error COPads 160.4818 0.476139 0.804141 0.804141 365.9434 375.3973 2.583435 0.54476160.4812 0.476157 0.804144 0.804144 365.9455 375.3872 2.580078 0.544755160.4817 0.476173 0.804142 0.804142 365.9487 375.3789 2.576904 0.544748160.4823 0.476189 0.804139 0.804139 365.9521 375.3718 2.574036 0.54474160.4827 0.476204 0.804136 0.804136 365.9553 375.3657 2.571473 0.544732

160.483 0.47622 0.804135 0.804135 365.9583 375.3604 2.569172 0.544725160.4832 0.476237 0.804134 0.804134 365.9612 375.3557 2.567088 0.544719160.4832 0.476253 0.804134 0.804134 365.9639 375.3515 2.565181 0.544713160.4832 0.47627 0.804134 0.804134 365.9665 375.3477 2.563423 0.544707160.4831 0.476286 0.804134 0.804134 365.969 375.3443 2.561791 0.544701

160.483 0.476302 0.804135 0.804135 365.9714 375.3413 2.560267 0.544695160.4829 0.476318 0.804135 0.804135 365.9738 375.3385 2.55884 0.54469160.4828 0.476334 0.804136 0.804136 365.9762 375.336 2.5575 0.544684160.4827 0.476349 0.804136 0.804136 365.9785 375.3338 2.556239 0.544679160.4826 0.476364 0.804137 0.804137 365.9808 375.3318 2.555051 0.544674160.4826 0.476434 0.804137 0.804137 365.9919 375.3248 2.550057 0.544648

160.483 0.476494 0.804135 0.804135 366.0023 375.322 2.546361 0.544624160.4841 0.476545 0.804129 0.804129 366.0121 375.3223 2.543684 0.544602160.4858 0.476588 0.804121 0.804121 366.0213 375.3248 2.541803 0.544581

160.488 0.476621 0.80411 0.80411 366.0299 375.329 2.540529 0.544561160.4907 0.476645 0.804096 0.804096 366.0379 375.3342 2.539703 0.544543

160.494 0.476662 0.80408 0.80408 366.0453 375.3399 2.53919 0.544526160.4977 0.47667 0.804061 0.804061 366.0522 375.3458 2.538871 0.54451

160.502 0.47667 0.80404 0.80404 366.0585 375.3514 2.538649 0.544495160.5067 0.476663 0.804016 0.804016 366.0643 375.3566 2.538438 0.544482160.5118 0.476649 0.803991 0.803991 366.0696 375.3611 2.538168 0.54447160.5174 0.476628 0.803963 0.803963 366.0744 375.3646 2.537777 0.544459160.5233 0.4766 0.803933 0.803933 366.0788 375.367 2.537215 0.544449160.5297 0.476566 0.803901 0.803901 366.0827 375.3681 2.53644 0.54444160.5364 0.476526 0.803868 0.803868 366.0861 375.3679 2.535415 0.544432160.5435 0.47648 0.803832 0.803832 366.0892 375.3663 2.534112 0.544425

160.551 0.476428 0.803795 0.803795 366.0919 375.3631 2.53248 0.544418160.559 0.47637 0.803754 0.803754 366.0945 375.3585 2.530498 0.544413

160.5675 0.476307 0.803712 0.803712 366.097 375.3525 2.528166 0.544407160.5765 0.476239 0.803667 0.803667 366.0993 375.3451 2.525492 0.544402160.5859 0.476166 0.80362 0.80362 366.1014 375.3362 2.52248 0.544397160.5956 0.476089 0.803571 0.803571 366.1034 375.3261 2.519135 0.544392160.6058 0.476007 0.80352 0.80352 366.1053 375.3146 2.515461 0.544388160.6163 0.475922 0.803468 0.803468 366.1071 375.3018 2.511465 0.544384160.6271 0.475833 0.803414 0.803414 366.1088 375.2877 2.50715 0.54438160.6382 0.475741 0.803358 0.803358 366.1103 375.2723 2.502523 0.544376160.6496 0.475645 0.803301 0.803301 366.1118 375.2558 2.497588 0.544373160.6613 0.475546 0.803242 0.803242 366.1131 375.238 2.492352 0.54437160.6733 0.475443 0.803183 0.803183 366.1144 375.219 2.486821 0.544367160.6855 0.475338 0.803121 0.803121 366.1155 375.1988 2.480999 0.544364

160.698 0.47523 0.803059 0.803059 366.1166 375.1776 2.474892 0.544362160.7107 0.47512 0.802996 0.802996 366.1175 375.1552 2.468507 0.54436160.7236 0.475007 0.802931 0.802931 366.1184 375.1317 2.461848 0.544358160.7367 0.474892 0.802866 0.802866 366.1192 375.1071 2.454921 0.544356

160.75 0.474774 0.8028 0.8028 366.1199 375.0815 2.447731 0.544354160.7634 0.474655 0.802732 0.802732 366.1205 375.0549 2.440284 0.544353160.7771 0.474533 0.802664 0.802664 366.1211 375.0273 2.432584 0.544351160.7909 0.474409 0.802595 0.802595 366.1216 374.9987 2.424636 0.54435

205

160.8049 0.474283 0.802525 0.802525 366.122 374.9691 2.416446 0.544349160.819 0.474156 0.802455 0.802455 366.1223 374.9386 2.408018 0.544349

160.8333 0.474027 0.802384 0.802384 366.1226 374.9072 2.399356 0.544348160.8477 0.473896 0.802312 0.802312 366.1228 374.8748 2.390465 0.544348160.8622 0.473764 0.802239 0.802239 366.123 374.8416 2.381347 0.544347160.8769 0.47363 0.802166 0.802166 366.123 374.8075 2.372008 0.544347160.8916 0.473495 0.802093 0.802093 366.1231 374.7725 2.362451 0.544347160.9065 0.473359 0.802019 0.802019 366.123 374.7367 2.352679 0.544347160.9215 0.473221 0.801944 0.801944 366.1229 374.7001 2.342695 0.544347160.9365 0.473082 0.801869 0.801869 366.1228 374.6626 2.332502 0.544348160.9519 0.472942 0.801793 0.801793 366.1227 374.6243 2.322054 0.544348160.9675 0.472799 0.801715 0.801715 366.1229 374.585 2.311292 0.544347160.9834 0.472655 0.801635 0.801635 366.1231 374.5446 2.300167 0.544347160.9996 0.47251 0.801555 0.801555 366.1235 374.5027 2.288619 0.544346

161.016 0.472363 0.801473 0.801473 366.124 374.4591 2.276594 0.544345161.0325 0.472215 0.801391 0.801391 366.1245 374.4138 2.264057 0.544344161.0493 0.472066 0.801307 0.801307 366.1252 374.3666 2.250982 0.544342161.0663 0.471915 0.801223 0.801223 366.1259 374.3174 2.237345 0.54434161.0834 0.471764 0.801138 0.801138 366.1268 374.2663 2.223139 0.544338161.1008 0.471611 0.801051 0.801051 366.1277 374.213 2.20835 0.544336161.1183 0.471457 0.800964 0.800964 366.1286 374.1578 2.19298 0.544334

161.136 0.471302 0.800876 0.800876 366.1297 374.1004 2.177023 0.544332161.1538 0.471146 0.800788 0.800788 366.1309 374.0411 2.160487 0.544329161.1718 0.470989 0.800698 0.800698 366.1321 373.9797 2.143379 0.544326

161.19 0.470832 0.800608 0.800608 366.1333 373.9163 2.125704 0.544323161.2083 0.470673 0.800517 0.800517 366.1347 373.8509 2.107474 0.54432161.2267 0.470514 0.800426 0.800426 366.1361 373.7836 2.088696 0.544317161.2453 0.470354 0.800333 0.800333 366.1376 373.7144 2.069377 0.544314

161.264 0.470193 0.800241 0.800241 366.1391 373.6433 2.049546 0.54431161.2828 0.470031 0.800147 0.800147 366.1408 373.5705 2.029196 0.544306161.3018 0.469869 0.800053 0.800053 366.1424 373.4959 2.008349 0.544302161.3208 0.469706 0.799959 0.799959 366.1442 373.4195 1.987025 0.544299

161.34 0.469542 0.799864 0.799864 366.146 373.3415 1.965222 0.544294161.3593 0.469378 0.799768 0.799768 366.1478 373.2619 1.942963 0.54429161.3787 0.469213 0.799672 0.799672 366.1497 373.1807 1.920259 0.544286161.3982 0.469048 0.799575 0.799575 366.1517 373.098 1.897118 0.544281161.4178 0.468882 0.799478 0.799478 366.1537 373.0139 1.873568 0.544277161.4376 0.468715 0.79938 0.79938 366.1558 372.9282 1.849602 0.544272161.4573 0.468548 0.799282 0.799282 366.1579 372.8412 1.825266 0.544267161.4771 0.468381 0.799185 0.799185 366.16 372.7529 1.800562 0.544262161.4969 0.468214 0.799086 0.799086 366.1621 372.6632 1.77549 0.544257161.5169 0.468046 0.798988 0.798988 366.1642 372.5723 1.750062 0.544252

161.537 0.467877 0.798888 0.798888 366.1666 372.4802 1.724258 0.544247161.5573 0.467708 0.798788 0.798788 366.1689 372.3869 1.698125 0.544242161.5776 0.467539 0.798688 0.798688 366.1714 372.2926 1.671679 0.544236161.5843 0.467589 0.798654 0.798654 366.1894 372.3182 1.673673 0.544195

161.58 0.467671 0.798676 0.798676 366.1963 372.3435 1.678656 0.544179161.5734 0.46776 0.798708 0.798708 366.2009 372.3686 1.684229 0.544168161.5656 0.467851 0.798747 0.798747 366.2042 372.3934 1.690072 0.544161161.5571 0.467944 0.798789 0.798789 366.2068 372.4179 1.69607 0.544155161.5074 0.468426 0.799034 0.799034 366.2117 372.5368 1.727192 0.544144161.4547 0.468908 0.799296 0.799296 366.212 372.6499 1.75797 0.544143161.4013 0.469384 0.79956 0.79956 366.2104 372.7576 1.787801 0.544146161.3482 0.469851 0.799823 0.799823 366.2079 372.8602 1.816525 0.544152

206

161.2958 0.470309 0.800083 0.800083 366.2047 372.9581 1.844157 0.54416161.2441 0.470757 0.800339 0.800339 366.201 373.0516 1.870725 0.544168161.1934 0.471195 0.800591 0.800591 366.1969 373.141 1.896258 0.544177160.8218 0.47435 0.802441 0.802441 366.1576 373.7292 2.067834 0.544268160.7794 0.474704 0.802653 0.802653 366.152 373.7889 2.085741 0.544281160.7378 0.475048 0.80286 0.80286 366.1461 373.846 2.102961 0.544294160.6971 0.475385 0.803064 0.803064 366.14 373.9004 2.11952 0.544308160.6572 0.475713 0.803263 0.803263 366.1338 373.9523 2.135415 0.544322

160.618 0.476033 0.803459 0.803459 366.1275 374.0017 2.15068 0.544337160.5797 0.476345 0.803651 0.803651 366.121 374.0487 2.16533 0.544352160.5421 0.476649 0.803839 0.803839 366.1143 374.0933 2.179382 0.544367160.5052 0.476946 0.804024 0.804024 366.1075 374.1357 2.19285 0.544383

160.469 0.477235 0.804205 0.804205 366.1005 374.1758 2.205753 0.544399160.4336 0.477517 0.804383 0.804383 366.0934 374.2137 2.218113 0.544415160.3988 0.477792 0.804557 0.804557 366.0861 374.2495 2.229923 0.544432160.3647 0.47806 0.804728 0.804728 366.0787 374.2833 2.241204 0.544449160.3313 0.478321 0.804896 0.804896 366.0711 374.3149 2.251973 0.544466160.2985 0.478575 0.805061 0.805061 366.0634 374.3446 2.262238 0.544484160.2663 0.478823 0.805222 0.805222 366.0555 374.3724 2.272018 0.544502160.2347 0.479064 0.805381 0.805381 366.0475 374.3982 2.281322 0.544521160.2038 0.479299 0.805537 0.805537 366.0394 374.4222 2.290154 0.544539160.1734 0.479528 0.805689 0.805689 366.0311 374.4444 2.298528 0.544558160.1436 0.479751 0.805839 0.805839 366.0226 374.4648 2.306459 0.544578160.1144 0.479967 0.805986 0.805986 366.0141 374.4834 2.313938 0.544597160.0857 0.480178 0.806131 0.806131 366.0053 374.5004 2.321006 0.544617160.0576 0.480383 0.806272 0.806272 365.9965 374.5156 2.327653 0.544638

160.03 0.480582 0.806412 0.806412 365.9875 374.5292 2.333878 0.544658160.0029 0.480776 0.806548 0.806548 365.9784 374.5413 2.339704 0.544679

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Appendix G

Design of an electro-adsorption chiller (EAC)

(1) Amount of silica gel:

When the amount of adsorbent (here silica gel) is very small, heat transfer in the

reactor or bed becomes so rapid that the temperature difference between the hot and

cold beds reaches its maximum value within a very short period and the amount of

refrigerant circulated through the system is small. So, the evaporator temperature

increases, which is shown in the region (1) of Figure G1. Region (3) of Figure G1

corresponds to relatively heavy heat exchangers, where increasing the adsorbent mass

results in reducing the cooling capacity and increasing the evaporator temperature.

Region (2) is effective as the cooling capacity increases or the load temperature

decreases. From the simulation, the optimum amount of silica gel is selected as 300

gm. This amount of silica gel is valid for 120 W at the evaporator. The amount of

Silica gel depends on the applied cooling load. On the basis of the amount of

adsorbent, the size of heat exchanger is designed.

208

(2) Bed Design:

The size of the bed is designed according to the amount of silica gel and cooling load

of the evaporator.

Figure G1: The load surface temperature and the evaporator temperature as a function of silica gel mass for the same input power to the thermoelectric modules (voltage 24 volts and the cycle average current 6.8 amp, cycle time 500 seconds, input power to the evaporator is 5 watt per cm2)

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Mass of silica gel (gm)

Tem

pera

ture

(C

)

Coe

ffic

ient

of p

erfo

rman

ce

Optimum

(1)

(2)(3)

Copper plate

40 mm × 40 mm × 1 mm depth

Dia 220 mm

Dia 180 mm

135

209

Heat exchanger inside the bed

Heat exchanger (top vies)

Copper tube (1/4” dia)

Fin (thickness 1 mm)

Flexible hose (3/8” dia)

135 mm

Heat exchanger (Isometric view)

32 mm

130 mm

Flexible hose (DN 10)

3 mm

DN 10 centering ring with O ring

¼” cu-tube

Heat exchanger (Front view)

3/8” comp fitting

210

Cover of the bed

75 mm

DN 10 short pipe socket

20 mm

PTFE Chamber (Front view)

φ 220 mm

¼”

φ 25 mm

3 × DN 10 short pipe 2 × DN 16

pipe socket

PTFE Chamber (Top view)

φ 180 mm

211

O’ ring (viton) and PTFE Bed (3-D view)

DN 200 viton O’ ring

Bed

Dia 200 mm Dia 180 mm

Bed for adsorption or desorption

212

(2) Evaporator Design (mat S. steel):

The size of the evaporator depends on the cooling load and the amount of refrigerant

for pool boiling. (the amount of refrigerant hence water must be higher than the total

amount of adsorbate during adsorption)

Size DN 100 Dia 25 mm

4 × DN 10 short pipe socket

Blanking flange (DN 100) for the top cover of the evaporator

DN 100 centering ring

Dia 120 mm

Dia 70 mm Quartz view port

Bottom part of the Evaporator

70 100 mm

DN 25 for window

Body of the Evaporator

213

DN 10 Tee

DN 16 clamping ring

DN 100 centering ring

2 × DN 25 window view port

DN 16 flexible bellow to hold the

copper foam

DN 10 short pipe socket li id

52 mm × 52 mm × 30 mm

Evaporator Design

3-D view of the Evaporator

214

(4) Design of a Kaleidoscope for uniform cooling load:

Kaleidoscope is designed on the basis of cooling load at the evaporator. From the

infra-red radiant heater (4 kW), radiation heat is developed and transferred to the

evaporator through the kaleidoscope and the quartz.

500 mm

500 mm

52 mm × 52 mm

IR heater (4 kW and 1500 K)

Mat: S. steel

Kaleidoscope

215

Appendix H The Transmission band of the fused silica or quartz

216

Appendix I Calibration certificates

THERMOELECTRIC COOLING DEVICES: THERMODYNAMIC … · Appendix E Programming Flow Chart of the...

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