Simplified Correlation Equations of Heat Transfer

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Journal of Energy and Power Engineering 6 (2012) 1543-1552

Simplified Correlation Equations of Heat Transfer

Coefficient during Phase Change for Flow inside Tubes

Mahmoud A Hammad1

and Abed Alrzaq S Alshqirate2

1. Mechanical Engineering Department, Faculty of Engineering and Technology, University of Jordan, Amman 11942, Jordan

2. Department of Basic and Applied Sciences, Al-Shoubak University College, Al-Balqa Applied University, Al-Salt 19117, Jordan

Received: November 24, 2011 / Accepted: Feburary 13, 2012 / Published: October 31, 2012.

Abstract: In this work, the easy to use, simple and direct equations were formulated and tested. These equations can be used to

calculate the mean values of the heat transfer coefficients of inside tube flow during phase change. Analytical and experimental

methods were used to correlate these equations. Two different forms were used, one for evaporation case and the other for condensation

case. Carbon dioxide, CO2, was used as case study. Correlated values of the mean heat transfer coefficients (hcorr) were compared with

the experimental results (hexp) and with other published result, a good agreement was noticed. The resulted correlations can be used to

simplify the design and performance studies of both condensers and evaporators.

Key words: Heat transfer, two phase flow, change of phase, refrigeration, carbon dioxide, correlations.

Nomenclature

Bo Boiling number (q/Ghfg)

Bon Bond number (g D2/)

Br Brinkman number (U2/kT)

C Constant

Cp Specific heat (J/kgK)

D Tube inner diameter (m)

Eu Euler number (P/U2)

Fn Function

G Mass velocity (kg/m2s)

Ga Galelio number (gD3/2)

H Convection heat transfer coefficient (W/m2K)

hfg Latent heat for evaporation (J/kg)

T Temperature difference (K)

Ja Jacobs number (CpT/hfg)

K Thermal conductivity (W/mK)Nu Nusselt number (hD/k)

n Constant

m Constant

Pr Prandtl number (Cp/k)

Re Reynolds number (UD/)

U,v Velocity (m/s)

We Weber number (U2D/)

Wecp Modified Weber No. (U2D/)

Corresponding author: Mahmoud A Hammad, professor,research fields: energy, solar refrigeration, new & alternativerefrigerants, PC heat transfer. E-mail: [email protected].

X Mass quality, vapour/liquid

x Distance in direction of flow

r Radius (m)

Latins

Density (kg/m3

) Surface tension (N/m)

Viscosity (Ns/m2)

Subscripts

l , f Liquid, saturated liquid

m Mean value

v, g Vapor, saturated vapor

exp, corr Experimental, correlated values

Superscripts

/ Small change_ Mean value

1. Introduction

Heat transfer and fluid flow inside tubes have many

applications such as: heat exchangers, condensers,

evaporators and boilers are examples of these

applications.

Fluid flow can be categorized as:

(1) Single phase flow

Literature shows many recent studies of heat transfer

for single phase flow inside tubes. Equations and

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Simplified Correlation Equations of Heat Transfer Coefficient duringPhase Change for Flow inside Tubes

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correlations are formulated and validated by

experimental works. Different correlations are now in

use to calculate the heat transfer coefficients. These

correlations can be found in all books and papers of

heat transfer, such as Refs. [1, 2].

(2) Two and multy phase flow

Many correlations can be found in literature

concerning the case of two phase flow, as shown in

Refs. [3, 4]. In the case of constant mass ratio of two

phase flow, the correlations used are extracted by

modifying the single phase correlation or formulated

using experimental results.

The case of having two phase flow with phase

change inside tubes shows some more complicatedprocesses. These processes can be found in most power

cycles and refrigeration cycles in evaporation (boiling)

and condensation. The mass ratio between the two

phases of the flow changes continuously between 0.0%

and 100% for both phases. Describing the process

physically is not an easy or simple matter. Calculating

heat transfer coefficients for these kinds of flows is also

complicated. Many works predicted different

correlations in literature, some of which are mentioned

as references for this work, such as Refs. [5-10].

Litterature shows different studies of CO2, heat

transfer coefficient during phase change inside tubes

that covered a domain of saturation temperature range

of -6 C up to 20 C with tube inner diameter of 0.8

mm up to 7.53 mm [11-13].

Table 1 shows correlations for evaporation and

condensation during flow inside tubes that are just

examples of the published literature in this field, with

all variables involved presented. These studies show an

acceptable agreement between experimental values of

the heat transfer coefficient and those calculated using

the shown correlations.

In the current research, understanding of the physical

processes and mechanisms of the flow of any gas inside

tubes was attempted. Change phase process and

convection heat transfer took place. Calculations of

heat flux and heat transfer coefficients were carried

out.

This work was carried out and validated against

experimental results of CO2 flow inside horizontaltubes.

2. Experiment

2.1 The Apparatus and Results

Fig. 1 shows a schematic diagram of the used test set.

The experiments are conducted for complete

evaporation and condensation for CO2 flowing inside

horizontal copper tubes. Condensation occurred inside

a chest freezer with lowest possible air temperature of

-28 C. Evaporation occurred in room air temperature

which ranged from 20 C to 25 C. The DAS (data

acquisition system) of model SCX14, made by national

instruments, is used. LAB VIEW software for data

processing is used. Visual and printed reports are

the outputs of this system. Thirty temperature readings

Table 1 Coefficient of heat transfer correlations.

No. Reference Refrigerant Correlation Case

1Jokar et al., (2006),[3]

R-134aNu = 3.37Re .55 (G2/l2 CpfT)1.39 (hfg/G2)1.05(l/lG)0.05lNu= 0.603Rel.5Pr0.1x-2 (G2/l

2CplT)-0.1 (l

2hfg /G2)0.5 (l/lG)

1.1 (l/(lv))

2

CondensationEvaporation

2 Dhaim, (2006), [5] CO2Nu = 0.25 10-16[Ja-0.0046 Ga2.196Pr1.3Bon-0.169 (U2/Dg)0.105]

Nu= 0.33 10-10 [Ja0.0192 Ga1.2379Pr-1.1357Bon-0.2715 (U2/Dg)1.3]CondensationEvaporation

3 Shqairat, (2007), [6] CO2Nu = 2.56 10-5 [Re1.2755 Ja-1.236 Ga-0.105 Pr4.367(L/D)-0.7196 Eu0.2108]

Nu = 34.92 [Re-0.3099Ja-0.7051 Ga0.1135Pr-0.3853Bon-0.7196Eu0.2109 (L/D)-0.8182

We0.8097]

CondensationEvaporation

4Choi et al., (2007),[11]

CO2h = 55Pr0.12 (-0.4343lnPr)-0.55M-0.5q0.67

M= molecular weighth = 0.023(kf/D)[G(1 x)D/ ]

-0.8 [Cpf /kf]0.4

Boiling

5Kandlikar andSteinke, (2003), [12]

CO2 hcp/hl=D1C0D2 (1 x)0.8Fn(frin) +D3Bo

D4(1 x)0.8Ffl, Fn(frin) = 1 Boiling


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Fig. 1 Schematic diagram of the test apparatus.

The main numbering components of the test apparatus are: 1Carbon dioxide gas cylinder; 2Regulating valve; 3Chest

freezer; 4Condenser; 5Cut off valve (4 units); 6Pressure gauges (3 units); 7Sight glass (2 units); 8Evaporator;

9DAS (data acquisition system); 10Volume flow meter.

were sensed by K-type thermocouples and fed to the

DAS simultaneously. The pressure was read in three

points at steady state conditions and theywere: before

condensation, after condensation and after evaporation.

Volumetric rate of flow in m3/s was read at the end

outlet of the flow by a gas flow meter calibrated for

CO2 at room temperature and local pressure conditions.

The mean experimental heat transfer coefficients

(hexp) were calculated as follows:

The tube outside surface temperatures along the

whole test sections (about 25 m) were measured by

means of the thermocouples which were fixed on the

outer surface at longitudinal locations. These

temperatures were tabulated along with the test section

length for the different tests changing the following

independent variables: the pipe diameter (three different

values), the test section inlet pressure (four different

values), and the rate of flow (four different values).

Fig. 2 shows the pipe longitudinal distribution of the

outer surface temperatures of a typical condensation

experiment. The figure shows a gas cooling part, a

liquid sub cooling part and in the condensation part.

The three lines are with different slopes. This study is

concerned only with the middle line which shows the

process of condensation.

For only the condensation part of the pipe, the

calculation of the mean heat transfer coefficient begins

by the following equation of heat balance. Latent heat

released by the gas as it condenses is equal to heat

transferred by convection and conduction radialy

through the tube:

QCO2 =.

m CO2 hfg, eff = 2 L (Tlm)/(Co+ C1 Ri) (1)

where the logarithmic mean temperature difference

equals:

Tlm= (Ti To) / ln (Ti / To) (2)

where: To = Tsat, outTsurand Ti = Tsat, in Tsur.

CO1

2

6

5

3

6

7

Flow direction ofCO2

5

Computer DASdata ac uisition s stem

Chest freezer

Printer

CondenserEvaporator

5

Thermocouples

4

8

10


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barometric pressure (Pb) and the volume flow rate (.

V).

The uncertainty can be expressed as:

(6)

Table 2 shows the values of the quantities of

different measured values of uncertainties and

sensitivities required for Eq. (6), for a typical

condensation test.

Calculations gave the following value:

W hexp = 1.28 W/ m

2K.

This is less than 1% of the original value.

In the same way, the uncertainty in the experimentalh

expfor the evaporation processes can be calculated.

3. Analytical Study

The variables affecting the convection heat flux in

the processes under consideration were gathered in one

equation. The mean values for complete evaporation

and complete condensation processes were used. Heat

flux in both the sub-cooled and the superheated regions

was not considered in this work.

The variables totaled 11. The following equation

represents the relation between all variables, whether

dependent or independent:

( h , m,km, hfg, Cpm, Tm, m, , Um, (lv),D) = 0 (7)

Table 2 Values required for Eq. (6).

No. Variable Value

1 1.28

2 (surfT

W ) 1.0 oC

3 0.23

4 (gP

W ) 0.5 bar

5 0.0061

6 (bP

W ) 0.25 mbar

7 0.0244

8 ( .V

W ) 0.5 liter/min

The basic number of dimensions contained by the

variables is four: mass in kg; length in m; time in s and

temperature in K. This implied that the suitable number

of non dimensional quantities required to represent Eq.

(7) is seven. Each quantity may be calledPi () [14, 15].

The following non-dimensional Pies were

formulated using the exponential method for

dimensional analysis:

1Reynolds number, 1=ReD = m Um D / m;

2Prandtl number, 2 =Pr= Cpmm / km;

33 = Umm / D hfgm;

4Weber number,4 = We =m U2

m D / ;

55= (lv) / m;

6Jacobs number, 6=Ja = CpmTm/ hfg;

77= h Um / m hfgCpm.

The main variable is h , this will give the followingrelation:

7= (1, 2, 3, 4, 5and 6) (8)

To simplify this equation:

1 *2 *3/ 6= U2/ Tm km =Br

5 *4 = (lv) Um2D / m = modified We

Thus: 7= (Br, We) (9)

This7is a very important non-dimensional number,

and it was not mentioned in the literature and not used

before, up to the knowledge of the authors. 7will be

given the symbol, , and thus:

= ( h Um / m Cpm hfg) (10)

The importance of this number emanates from that it

contains the main dependent variable which is the

mean convection heat transfer coefficient ( h ), and

relates the convection heat flux to the stored quantities

of heat as sensible heat presented by the mean specific

heat (Cpm) and latent heat (hfg). This number can be

compared to the diffusivity of the conduction heat flux

( = k/Cp) which relates conduction heat flux (k) to

stored heat (Cp). This number may represent the

diffusivity of the convection heat flux in the case of

phase change flow, and can be put in the form (conv =

h Um/ m Cpm hfg).

Eq. (9) can be written in the form:

= C(Brm)m

(Wem)n

(11)


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3.1 Physical Thermodynamics Discussion

The importance of the new number () can be

explained by physical analysis of the thermal processes.

Considering a control volume inside the tubethrough which fluid flows in combination with heat

transfer across tube surface. This causes continuous

phase change. Fig. 3 represents that control volume.

The following analysis will describe the process.

The differences in temperature radial wise and

longitudinal wise due to heat transfer create differences

in density and differences in local pressures. This

produces differences in local velocities. All this will be

combined with formation of vapor or liquid bubbles orfilms which will either be super heated or crushed back

to liquid in case of evaporation and sub cooled or

evaporated back to gas in case of condensation

depending on local changes in temperature, density and

pressure.

Differences in compressibility and density between

the liquid phase and vapor phase will create differences

in local pressures and differences of velocities in both

directions. Also the collapse of bubbles in the forming

phase creates collisions of pressure fronts creating

local pressure waves that travel locally and contribute

to the mechanisms of pressure differences in the

control volume.

Due to all of these factors, different processes may

take place at the same time in different points of the

control volume as follows:

In case of evaporation:

Processes taking place at different points of the

control volume may be:

(1) Sensible heating of liquid at temperatures lower

than saturation;

(2) Sensible heating of vapor at saturation temperature;

(3) Latent heat absorbed by evaporating particles;

(4) Latent heat re-released from vapor particles

collapsing to liquid due to pressure being not highenough to sustain vapor phase;

(5) Changes of energy forms and quantities in both

directions (positive and negative) between different

control volume parts may occur simultaneously with

changing in: conduction heat flux, friction and shear

energy due to viscosity, kinetic energy, energy created

by surface tension, buoyancy effect created by

difference in density.

Energy balances of the first law of thermodynamics

will domain all these processes.

In case of condensation:

Fig. 3 Representation of various effects of an arbitrary control volume.

dx

rD

dr

h h'

V'rV'r

P P'

U U'U U'

PPP'

T T'

( k, , )(CpT', hfg)


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Processes taking place at different parts of the

control volume may be:

(1) Sensible cooling of vapor at temperatures higher

than saturation;(2) Sensible cooling of liquid at saturation temperature;

(3) Latent heat released from condensing particles;

(4) Latent heat re-stored in particles which gained

some pressure and evaporated again;

(5) Changes of energy forms and quantities in both

directions (positive and negative) between different

control volume parts may occur simultaneously with

changing in: conduction heat flux, friction and shear

energy due to viscosity, kinetic energy, energy created

by surface tension, buoyancy effect created by

difference in density.

All above mentioned heat processes either store heat

or reject it at different particles in different parts of the

control volume (Fig. 3). The temperature differences

mentioned will be very minor differences. And it can

be presented as (T T/), and pressures can also be

presented as (p p/), velocities (U u

/) and v

/and

density of ( /). Minor changes in the local heat

transfer coefficient will result and can be presented as(h

h

/). Respective changes in the other thermal

properties must occur, such as minor changes in

viscosity, surface tension and buoyancy.

Energy balances of the first law will keep

dominating these changes in both cases as mentioned

before. Forms of energy involved are: kinetic energy

(U2/2), internal heat (CpT), flow work (P/), surface

tension effect (), viscosity effect (), latent heat (hfg),

buoyancy effect (lv), conduction heat transfer (k

and T) and convection heat transfer (h).

The heat storing or loosing of the control volume

(Q1) will be in two forms: latent heat (hfg), and sensible

heat (CpT). The following form can represent that

relation:

Q1 = (, Cp, T, hfg, r, dx) (12a)

While the control surface heat flux that related to the

convection heat transfer can be presented by Q2, as

follows:

Q2= (h, U, T, r, dx) (12b)

The ratio ofQ2/Q1 can be presented by convection

heat diffusivity (pcd), which is called in this work:

= hU/ Cp hfg (13)It is clear that the number relates the heat energy

parameters of: convection heat flux with stored heat,

sensible and latent. The number resembles convection

heat flux diffusivity with the case of phase change.

The difference between Q1 and Q2 represents the

energy changes occurring due to convection diffusivity

of heat. The difference will be balanced by the other

forms of energy which were mentioned before to keep

the first law of thermodynamics valid.

3.2 The Empirical Correlations

The aim of this work is to introduce simple

correlations that can help in calculating the mean heat

transfer coefficients in the cases of changing phase of

inside tubes flow. Eq. (12a) shows a non-dimensional

formula that can be used for any of these cases:

= C(Brm)m

(We m)n

It can be written as:

h

Um/m Cpm hfg= C[Um2

m/(kmTm)]m

[(l v)Um

2D / ]

n(14)

This correlation can be used to calculate the mean

values of heat transfer coefficients if the constants: C,

m, and n are known. The values of these constants

depend on the kind of gas underconsideration and can

be calculated using three experimental results. The

mean value of heat transfer coefficient thus calculated

will be called the correlation resulted value (h

corr).

The mean heat transfer coefficient (hcorr) calculated

using this analytical procedure will be:

hcorr

= C(m Cpm hfg/ Um) [Um

2m/(kmTm)]

m

[( l v)Um

2D/ ]

n(15)

Constants C, m, and n were evaluated.

Results were calculated for both cases: evaporation

and condensation. Values are shown in Table 3.

This gave the following two equations:

= 1.6 10

-6(Brm)

0.78(We m)

0.09for evaporation. (16)

And:


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Table 3 Constants of Eq. (14), for CO2.

Constant C m n

Evaporation 1.6 10-6 0.78 0.09

Condensation 1.02 10-2 1.55 -0.55

= 1.02 10

-2(Brm)

1.55(We m)

-0.55for condensation.

(17)

These equations may be used for any other gas, but

with less accuracy.

4. Analysis

Comparison between mean heat transfer coefficients,

experimental, hexp and analytical, h

corrwas conducted.

Using the correlations shown by Eqs. (15)-(17),

hcorr was calculated for the same experimental

conditions. They are presented in Figs. 4 and 5 for

comparison.

The range of the comparison covered the domain of

application which was mentioned before.

In Figs. 4 and 5, a good degree of conformity is

shown. They are conformed by the factors, 0.976 and

1.006 for both cases of evaporation and condensation,

respectively. The conformity factor used is hcorr/h

exp.

The uncertainty expected in the correlation values of

the coefficients of the heat transferh

corrwill be related

to those ofhexpby the relationsshown inFigs. 4 and 5

as:

For evaporation: hcorr= 0.976 h

exp and hcorr=

1.006 h

exp for condensation.

Fig. 4 hcorrcalculated versush

exp experimentally resulted for CO2 at evaporation process flow inside tubes.

Fig. 5 hcorr calculated versush

exp experimentally resulted for CO2 at condensation process flow inside tubes.

y = 0.976 x

R2 = 0.939

0

200

400

600

800

0 200 400 600 800 1000

h

corr

(W/m2K)

Evaporation Process

y =1.0061x

R2=0.9199

0

200

400

600

800

1000

0 200 400 600 800

hexp (W/m2K)

hcorr

(W/m2K)

Condensation Process

hexp (W/m2K)


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Fig. 6 h corr of this work versus h in other works of literature in evaporation case.

Fig. 7: h corrof this work versus h in other works of literature in condensation case.

Table 4 Correlation of this work.

CaseGeneralcorrelation

Specific for CO2correlation

Evaporation = C(Br)m (We)n = 1.6 10-6 (Brm)

0.78

(We m)0.09

Condensation = C(Br)m (We)n -= 1.02 10-2 (Brm)1.55

(We m)-0.55

For the typical condensation case mentioned before

in the uncertainty analysis, hcorr error will be 1.29

W/m2K. This is less than 1.5%.

This degree of conformity is considered as a close

agreement situation .

Comparing the results obtained in this work with

other works published in literature was conducted. The

other works used different form correlations as shown

in Table 1. The comparison is presented in Figs. 6 and 7.

A good degree of agreement is clear in the figures.

Limited differences are existing due to the fact that

literature correlations are common for use in different

gases applications.

5. Conclusions

The main goal of this work was achieved. The

authors ended up with a simple equation for mean heat

transfer coefficient for flow inside tubes during change

of phase. Eq. (11) elucidates the relation resulted from

this study. The resulted correlation was validated using

Evaporation Process

0

150

300

450

600

750

900

0 150 300 450 600 750 900

h corr(this work)

h

(Others)

[6]

[5]

[1]

0

50

100

150

200

250

300

350

0 50 100 150 200 250h corr (this work)

h(O

thers)

[6]

[5]

[4]

[12]

[1]


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Simplified Correlation Equations of Heat Transfer

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