Appendix AUncertainty Analysis for Experimental Data

To compute the uncertainty in the experimental data of this work, error analyseshave been conducted according to the principles proposed by Taylor [1]. The erroranalysis procedures are summarized below:

Uncertainty in Sums and Differences

Suppose that x,…, w are measured with uncertainties dx, …, dw, and themeasured values used to compute

f ¼ xþ � � � þ z� ðuþ � � � þ wÞ

If the uncertainties in x, …, w are known to be independent and random, thenthe uncertainty in f is the quadratic sum of the original uncertainties.

df ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðdxÞ2 þ � � � þ ðdzÞ2 þ ðduÞ2 þ � � � þ ðdwÞ2q

In any case, df is never larger than their ordinary sum,

df � dxþ � � � þ dzþ duþ � � � þ dw

Uncertainties in Products and Quotients

Suppose that x,…, w are measured with uncertainties dx,…, dw, and themeasured values used to compute

f ¼ x� � � � � z

u� � � � � w

If the uncertainties in x, ..., w are independent and random, then the fractionaluncertainty in f is the sum in quadrature of the original fractional uncertainties,

T. Alam et al., Flow Boiling in Microgap Channels,SpringerBriefs in Thermal Engineering and Applied Science,DOI: 10.1007/978-1-4614-7190-5, � The Author(s) 2014

75

df

fj j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dx

xj j

� �2

þ � � � þ dz

zj j

� �2

þ du

uj j

� �2

þ � � � þ dw

wj j

� �2s

In any case, it is never larger than their ordinary sum,

df

fj j �dx

xj j þ � � � þdz

zj j þdu

uj j þ � � � þdw

wj j

Uncertainty in Any Function of One Variable

If x is measured with uncertainty dx and is used to calculate the function f(x),then the uncertainty df is

df ¼ df

dx

�

�

�

�

�

�

�

�

dx

Uncertainty in a Power

If x is measured with uncertainty dx and is used to calculate the power f = xn

(where n is a fixed, known number), then the fractional uncertainty in f is nj j timesthat in x,

df

fj j ¼ nj j dx

xj j

Uncertainty in a Function of Several Variables

Suppose that x,…, z are measured with uncertainties dx,…, dz, and themeasured values used to compute the function f(x, …, z). If the uncertainties inx,…, z are independent and random, then the uncertainty in f is

df ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

of

oxdx

� �2

þ � � � þ of

ozdz

� �2s

In any case, it is never larger than their ordinary sum,

76 Appendix A: Uncertainty Analysis for Experimental Data

df � of

ox

�

�

�

�

�

�

�

�

dxþ � � � þ of

oz

�

�

�

�

�

�

�

�

dz

Table A.1 shows the measurement accuracies and experimental uncertaintiesassociated with sensors and parameters.

Table A.2 shows a set of uncertainty values in different parameters calculatedbased on the above equations for 300 lm depth microgap at mass flux, G = 390kg/m2s and heat flux, q00eff ¼ 52:5 W=cm2:

Table A.1 The measurementaccuracies and experimentaluncertainties associated withsensors and parameters

Sensors and parameters Accuracies and uncertainties

T-type thermocouples ±0.5 �CDiode temperature sensors ±0.3 �CFlow meter ±5 ml/minPressure transducer ±1.8 mbarDifferential pressure transducer ±1 mbarVoltage measurement ±0.06 VCurrent measurement ±0.15 ADimension measurement ±10 lmHeat flux 2–8 %Pressure drop 4–18 %Heat transfer coefficient 4–10 %

Appendix A: Uncertainty Analysis for Experimental Data 77

Tab

leA

.2S

ampl

eun

cert

aint

yca

lcul

atio

nfo

r30

0lm

dept

hm

icro

gap

atm

ass

flux

,G

=39

0kg

/m2s

and

heat

flux

,q0 ef

f¼

52:5

W=

cm2

Wid

th(W

)L

engt

h(L

)T

hick

ness

(t)

Vol

tage

(V)

Cur

rent

(I)

Tf

Td

Ks

1.27

±0.

001

cm1.

27±

0.00

1cm

0.06

75±

0.00

1cm

10.7

±0.

06V

8.5

±0.

15A

101.

76±

0.5

�C11

9.6

�C±

0.3

�C1.

21W

/cm

�C

Cal

cula

tion

q00 ef

f¼

q eff A¼

fV;

I;W;

Lð

Þ¼52:5

W/c

m2

dq00 ef

f

q00 ef

f

� �

� �

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dV Vjj

��

2

þdI Ijj

��

2

þdW Wjj

��

2

þdL Ljj

��

2s

¼0:

0185

5¼

1:85

5%�

2%

Tf¼

Tsa

t

dTf¼

dTsa

t¼�

0:5� C

Tw¼

Td�

q00 ef

ft

Ks

¼11

6:77� C

dTw¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

oT

w

oT

d

:dT

d

��

2

þo

Tw

oq00 ef

f

:dq00 ef

f

��

2

þoT

w

ot:d

t

��

2

þo

Tw

oK

s

:dK

s

��

2s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1:dT

dð

Þ2þ�

t Ks

:dq00 ef

f

��

2

þ�

q00 ef

f

Ks

:dt

��

2

þ0

s

¼�

0:3�

C

DT¼

Tw�

Tf¼

15:0

05� C

dT

w�

Tf

ðÞ¼

dDT¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dTw

ðÞ2þ

dTf

ðÞ2

q

¼�

0:6� C h z¼

q00 ef

f

DT

dhz

h zjj¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dq00 ef

f

q00 ef

f

� �

� �

!

2

þdD

T

DTjj

��

2

¼0:

038¼

3:8

%�

4%

v u u t

78 Appendix A: Uncertainty Analysis for Experimental Data

Appendix BNomenclature

A Footprint area (cm2)Ac Wetted area of microchannel (cm2)Agap Microgap cross-sectional area (cm2)Aman Manifold cross-sectional area (cm2)Bl Boiling numberBo Bond numberCo Confinement numbercp Specific heat, (J/kg �C)d Depth of microchannel (lm)D Microgap depth (lm)g Gravitational accelerationG Mass flux (kg/m2s)h Heat transfer coefficient (W/m2K)hfg Heat of vaporization (J/kg)ks Thermal conductivity, W/cm �CKc Loss coefficientL Length of the substrate (cm)_m Mass flow rate (kg/s)N Number of microchannelsP Pressure (bar)DP Pressure drop (bar)q Total heat dissipation (W)qeff Effective heat dissipation (W)q00eff Effective heat flux (W/cm2)qloss Heat loss (W)Ra Roughness parameter (arithmetic mean value)Re Reynolds numberRt Roughness parameter (maximum peak to valley height)t Substrate thickness (cm)T Temperature (�C)

T. Alam et al., Flow Boiling in Microgap Channels,SpringerBriefs in Thermal Engineering and Applied Science,DOI: 10.1007/978-1-4614-7190-5, � The Author(s) 2014

79

Vd Voltage drop across diode (V)w Channel width (lm)W Width of the substrate (cm)x Vapor qualityz z-Coordinate (axial distance) (cm)

Greek Symbols

q Density (kg/m3)l Dynamic viscosity (Ns/m2)r Surface tension (N/m)g Fin efficiency

Subscripts

c Contractiond Diodee Expansionf Liquidg Vaporgap Microgapi Manifold inleto Manifold outletman Manifolds Substratesat Saturatedsp Single-phasew Wallz Local

80 Appendix B: Nomenclature

Appendix CData Reduction

Heat Transfer Data Reduction

The effective heat supplied, qeff to the fluid in each test piece and the effective heatflux q00eff is calculated as given.

The effective heat transfer rate, qeff to the fluid in microgap channel is obtainedby:

qeff ¼ q� qloss ðC:1Þ

Where q is input power and qloss is heat loss.The effective heat flux q00eff that the heat sink can dissipate is calculated from:

q00eff ¼qeff

AðC:2Þ

where A is the base area of heat sink, A = W 9 L.For microchannel, the total wetted area of the microchannels is:

Ac ¼ Nðwþ 2gdÞL ðC:3Þ

where N is total number of channels; w, d and L are the width, depth, and length ofthe channel respectively and g is the efficiency of a fin with adiabatic tip which iscorrelated by:

g ¼ tanhðmdÞmd

ðC:4Þ

and

m ¼ffiffiffiffiffiffiffiffiffiffiffi

2h

Ksww

r

ðC:5Þ

where Ks is the thermal conductivity of the substrate and ww is the width of thechannel wall.

So, the wall heat flux for microchannel is defined as

q00w ¼qeff

Ac

ðC:6Þ

T. Alam et al., Flow Boiling in Microgap Channels,SpringerBriefs in Thermal Engineering and Applied Science,DOI: 10.1007/978-1-4614-7190-5, � The Author(s) 2014

81

The test section is divided into two regions: an upstream subcooled inlet regionand a downstream saturated region as subcooled (Tf,i \ Tsat) water is supplied intothe heat sink for all test conditions; the location of zero thermodynamicequilibrium quality (x = 0) serves as a dividing point between the two regions.

The local heat transfer coefficient in microgap is calculated from,

hz ¼q00eff

Tw�Tf

ðC:7Þ

The local heat transfer coefficient in microchannel is calculated from,

hz ¼qeff

AcðTw�TfÞðC:8Þ

in which Tf is the fluid temperature as defined by

Tf ¼ Tf;i þq00effWz

_mcp

single� phase regionð Þ ðC:9Þ

where z, _m and cp are the axial distance, mass flow rate, and specific heatrespectively.

Tf ¼ Tsat saturated regionð Þ ðC:10Þ

Tw, is the local wall temperature. This temperature is corrected assuming onedimensional heat conduction through the substrate

Tw ¼ Td �q00eff t

Ks

for microgapð Þ ðC:11Þ

Tw ¼ Td �q00effðt � dÞ

Ks

for microchannelð Þ ðC:12Þ

where t and Ks are the substrate thickness and thermal conductivity respectively.Td is the measured temperature by an integrated diode.

Bond number is defined as the ratio of buoyancy force to surface tension force.

Bo ¼gðqf � qgÞ

r

� �

D2 ðC:13Þ

where r is the surface tension, g is the gravitational acceleration, qf and qg areliquid and vapor densities of fluid respectively. D is the gap depth. Some othernon-dimensional parameter like Boiling number, Bl which is non-dimensional heatflux and Reynolds number, Re are defined as follows:

Bl ¼ q00eff

Ghfg

ðC:14Þ

Re ¼ GD

lf

ðC:15Þ

82 Appendix C: Data Reduction

where hfg and lf are the heat of vaporization and dynamic viscosity of fluidrespectively.

Pressure Drop Data Reduction

Pressure taps are located across the microgap and microchannel inlet and outletplenum. These taps are positioned as close as possible to the test die. Pressurelosses by the sudden contraction (DPc) and the sudden enlargement (DPe) werevery small compared with the frictional pressure drop. Though these values arevery small of total pressure changes, the pressure drop and the pressure recovery atthe sudden contraction and the sudden enlargement were considered forcalculation of the total pressure drop.

Pressure losses are calculated based on the methods described in Blevins [2],Chislom and Sutherland [3] and Collier and Thome [4]. As mentioned earlier,subcooled water (Tf, i \ Tsat) is supplied into the heat sink for all test conditions.The pressure drop associated with the liquid flow at the sudden contraction inmicrogap channel is calculated as

DPc ¼G2

2qf

1� Agap

Aman

� �2

þKc

" #

ðC:16Þ

where G is mass flux in the microgap, qf is liquid density and Kc is the non-recoverable loss coefficient for laminar flow given by

Kc ¼ 19lf

GD

�

þ 0:64 ðC:17Þ

The pressure recovery at the sudden enlargement at the exit is calculated as

DPe ¼G2

qf

Agap

Aman

� �

1� Agap

Aman

� �� �

1þ qf

qg

� 1

!

x

" #

ðC:18Þ

The microchannel pressure drops (DP) are calculated as follows. The pressuredrop associated with the liquid flow at the sudden contraction is calculated as

DPc ¼G2

2qf

1� NAch

Aman

� �2

þKc

" #

ðC:19Þ

where G is mass flux in the microgap, qf is liquid density and Kc is the non-recoverable loss coefficient for laminar flow given by

Kc ¼ 0:0088d

w

� �2

� 0:1785d

w

� �

þ 1:6027 ðC:20Þ

Appendix C: Data Reduction 83

The pressure recovery at the sudden enlargement at the exit is calculated as

DPe ¼G2

qf

NAch

Aman

� �

1� NAch

Aman

� �� �

1þ qf

qg

� 1

!

x

" #

ðC:21Þ

Therefore, the pressure drops (DP) reported below are

DP ¼ Pi � DPcð Þ � Po þ DPeð Þ½ ðC:22Þ

References

1. Taylor JR (1997) An introduction to error analysis: The study of uncertainties in physicalmeasurements, University Science Books, 2nd edn. US

2. Blevins RD (1991) Applied fluid dynamics handbook. Krieger Publishing Co., Berlin,pp 77–78

3. Chislom D, Sutherland LA (1969) Prediction of pressure gradients in pipeline systems duringtwo-phase flow. Symposium in two-phase flow systems. University of Leeds

4. Collier JG, Thome JR (1994) Convective boiling and condensation. Clarendon Press, Oxford

84 Appendix C: Data Reduction