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CHAPTER - VII

NUMERICAL MODEL FOR HEAT TRANSFER

Background

The objective of the chapter is to develop a predictive procedure

with a combination of CFD analysis and numerical model and to

compare the so obtained numerical heat transfer coefficients with that

of experiment and correlations.

An integral equation for heat transfer coefficient is established

using analytical model based on separated flow approach with

annular regime as the physical model. Using turbulent Prandtl

number, momentum and heat transfer equations are coupled. The

pressure gradient and wall shear stress needed in the establishment

of numerical model is taken from CFD simulations. A code in ‘C’

language is developed to evaluate the two phase or local heat transfer

coefficient. The so obtained heat transfer coefficients are compared

with the experimental data and correlations.

7.1 Analytical and Numerical Modeling

It is difficult to model condensing flows using CFD analysis.

Hence the CFD simulations are performed under adiabatic conditions

to predict the flow regimes and pressure drop. The contours obtained

in the CFD simulations show an excellent agreement with the

predictions of Thome et al. flow regime maps for all refrigerants

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considered in the present study. Also, the pressure gradient obtained

from CFD simulations predicted the experimental data better than the

correlations available in the literature. Hence, the pressure gradient

obtained from CFD analysis is used in modeling two phase heat

transfer.

As mentioned in Chapter – III, most of the analytical

correlations for heat transfer coefficient are developed for annular flow

regime as it occupies 60 – 70% of the heat exchanger length

depending on the mass flux. Fig 7.1 represents the physical model

based on which numerical model is developed.

From the extensive literature survey as mentioned in Chapter –

III, it is observed that the correlations for heat transfer coefficient are

developed by modifying the existing correlations using the

experimental data or using homogeneous model or using two phase

multiplier approach. In the later method, single phase Dittus – Boelter

equation or other similar equations are used and the two phase

Fig 7.1 Physical Model Representing Annular Flow Regime

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multiplier is determined using experimental data. The heat transfer

coefficient correlations obtained using these methods have limited

applications to the extent that they can predict the experimental data

only for the operating conditions based on which they are developed.

In addition, Park et al. [2008] had reported that the widely used and

recently developed correlations viz., Cavallini et al. [2002], Thome et

al. [2003a, 2003b], Shah [1979] and Dobson et al. [1998] etc.,

exhibited a deviation up to 300% with the experimental data of

ammonia. Jiang et al. [2006] had reported that most of the

correlations exhibited higher deviations from their experimental data

obtained at high reduced pressures. This represents the semi

empirical nature of the correlations and their limited applications.

The chapter focusses on the development of numerical

procedure based on CFD simulations. Since the CFD simulations will

take into account different operating conditions and properties of

different fluids, the resulting numerical model will also predict the

heat transfer coefficient for any fluid and at any operating pressure.

In general, two types of analytical models are used to predict the

heat transfer coefficient for condensing flows using seperated flow

model. The first one is evaluation of eddy viscosity using Prandtl

mixing length theory in combination with Van Driest’s hypothesis and

using analogy, heat transfer coefficient is evaluated. Chitti and Anand

[1995] used this method, together with an iterative procedure to

evaluate liquid film thickness, eddy viscosity and hence local heat

transfer coefficient. Sarma et al. [2005], used their own eddy viscosity

170

expression they developed in the lines of Van Driest model while

investigating the phenomenon of transition from laminar to turbulent

boundary layers for external flows and reported analytical model for

condensing flows.

The second method is a laborious process involving the

evaluation of liquid film thickness, shear stress and eddy viscosity

distribution using Von Karman universal velocity profiles in the liquid

film. Based on this data, using turbulent Prandtl number, heat

transfer coefficient is evaluated. Li et al. [2000] used this method, to

evaluate condensation heat transfer coefficient. They used Lockhart

and Martinelli correlation for the prediction of pressure drop which

exhibited a deviation of more than 100% with the experimental

pressure drop data as reported in the previous chapters. In the

present study, the second method is used to evaluate the heat

transfer coefficient using the pressure drop and wall shear stress

obtained from CFD simulations. The analytical model is presented as

follows.

7.1.1 Analytical Model

The physical model shown in Fig 7.1 represents the annular

flow regime. Uniform liquid film along the circumference of the tube is

assumed. The entrainment of liquid droplets in the vapor core is

neglected, thus assuming a smooth vapor – liquid interface. The sub

cooling of the liquid film is neglected and liquid and vapor properties

are considered to be constant corresponding to the condensing

temperature. Axial heat conduction is neglected. The flow of liquid

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film is considered to be steady and turbulent. The momentum and

energy equations used are same as that of single phase internal

turbulent boundary layer flows [1999].

The momentum and energy equations are written for the

differential element, of the liquid film as shown in Fig 7.1. For the

two dimensional turbulent flow as shown in Fig 7.1, the total shear

stress and heat transfer rate are made up of sum of molecular and

turbulent contribution [1999], given by Eqs. (7.1) and (7.2).

(7.1)

(7.2)

where, is the heat transfer rate per unit area perpendicular to heat

transfer in direction. Introducing turbulent Prandtl number,

given by Eq. (7.3), the heat transfer rate is written as Eq. (7.4).

(7.3)

(7.4)

where is the circumferential area of the differential element at a

radius, . Seperating the variables in Eq. (7.4) and integrating for the

thickness of liquid film, Eq. (7.5) is obtained.

(7.5)

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The condensation heat transfer coefficient is defined by Eq. (7.6).

(7.6)

Substituting Eq. (7.6) in Eq. (7.5)

(7.7)

In Eq. (7.7), eddy viscosity is needed to evaluate heat transfer

coefficient. From Eq. (7.1), expression for eddy viscosity can be written

as,

(7.8)

Introducing shear velocity, , the dimensionless flow

velocity and distance, are written as,

and (7.9)

Substituting Eq. (7.9) in Eq. (7.8) and in Eq. (7.7)

(7.10)

(7.11)

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Using Von Karman universal velocity distribution for liquid layer,

given by Eq. (7.12), expression for the eddy viscosity is obtained as

Eq. (7.13).

for (7.12a)

for (7.12b)

for (7.12c)

and for (7.13a)

for (7.13b)

for (7.13c)

In Eq. (7.13), the wall shear stress, is taken from the CFD

simulations. The local shear stress, is obtained by doing force

balance to the shaded portion shown in Fig. 7.1.

(7.14)

where and are the cross sectional area of the liquid film at a

given radius, and cross sectional area of the liquid film respectively.

and are perimeter at the interface and perimeter at the given

radius, respectively. From Eqs. (7.14) and (7.9), expression for local

shear stress, is written as Eq. (7.15).

174

(7.15)

The interfacial velocity, is given by Soliman et al. [1968] in their

analytical model as represented by Eq. (7.16).

(7.16)

The expression for wall shear stress is obtained from Eq. (7.15) by

substituting, .

(7.17)

Rearranging Eq. (7.17), expression for interfacial shear stress, is

obtained.

(7.18)

The phase velocity of liquid and mass flow rate are obtained using Eq.

(7.19a), substituting , from Zivi void fraction formula [1964] given by

Eq. (7.19b).

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; (7.19a)

(7.19b)

Substituting Eq. (7.19) in Eq. (7.15), the expression for local shear

stress, is obtained.

(7.20)

Similarly, substituting Eq. (7.19) in Eq. (7.18), expression for

interfacial shear stress is obtained.

(7.21)

With known wall shear stress and pressure gradient from CFD

simulations, the only unknown quantity in Eqs. (7.20) and (7.21) is

dimensionless liquid film thickness, . This is obtained from the

known mass flow rate as represented by Eq. (7.22) using Eq. (7.9).

(7.22a)

(7.22b)

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And from the definition of liquid Reynolds number, , Eq.

(7.22) is written as Eq. (7.23).

(7.23)

Substituting from Eq. (7.12), the relation between liquid Reynolds

number and dimensionless liquid film thickness is obtained.

for (7.24a)

for (7.24b)

for (7.24c)

From Eq. (7.24), with the known liquid Reynolds number, is

evaluated. Hence, numerical procedure is established as follows.

7.1.2 Numerical Procedure

The local heat transfer coefficient is obtained by performing

numerical integration across the thickness of liquid film. The

sequence of steps involved in the numerical procedure is as follows.

1. From the CFD simulations, the wall shear stress data

is obtained and hence shear velocity, is calculated.

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2. Using , the ranges of liquid Reynolds number from

Eq. (7.24) are obtained.

3. With the known , appropriate expression for is

selected among Eqs. (7.24) and hence dimensionless liquid

film thickness, is calculated.

4. From the CFD simulations, pressure gradient,

and wall shear stress, is obtained.

5. With known and , local and interface shear

stress is calculated using Eqs. (7.20) and (7.21).

6. With known and , eddy viscosity is obtained at

different locations across liquid film using Eq. (7.13).

7. Eq. (7.11) is numerically integrated and hence local

heat transfer coefficient is evaluated. The turbulent Prandtl

number is assumed unity.

A code is developed in the programming language, ‘C’ and is

enclosed in Appendix VI. The resulting local heat transfer coefficient

data is compared with the experimental data and correlations.

7.2 Comparison with Experimental Data

The heat transfer coefficient obtained from the numerical model

for R22, R134a and R407C is compared with the experimental data

and presented in Figs 7.2 and 7.3.

The comparison in general shows that the numerical model

predicts the experimental data well for high pressure refrigerants, R22

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and R407C as shown in Figs 7.2 a) and 7.2 c) compared to a low

pressure refrigerant, R134a as shown in Fig 7.2 b). This is primarily

due to R134a being a low pressure refrigerant, its reduced pressure

values are lower than R22 and R407C as given in Appendix III. The

separated flow approach considered in the present numerical model

as mentioned in Chapter – II, gives better results for fluids with high

reduced pressure. Similar results are observed with annular flow

based Shah correlation that takes the reduced pressure of fluids into

account while comparing the experimental data with correlations in

Chapter V.

a)

c)

b)

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Fig 7.2 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment for a) R22 b) R134a and c) R407C

a)

b)

180

Fig 7.3 b) shows that most of the experimental data falls within

a deviation band of ±20% including that of R407C. Figs 7.2 a) and 7.3

a) shows that for R22, the numerical model closely follows the

experimental heat transfer coefficient at the mass fluxes considered,

exhibiting a deviation of 8% at low mass flux, 4 - 9% at medium and

high mass flux.

For R134a, the numerical model over predicts the experimental

data for low and medium mass fluxes with a deviation of 11% and

22% respectively as shown in Fig 7.2 b) and 7.3 a). The numerical

model under predicts upto medium qualities for high mass flux with a

deviation of 15%.

As shown in Fig 7.2 c), for R407C the numerical model over

predicts at medium and high mass fluxes with a deviation of 14% at

medium mass flux and 3% at high mass flux as shown in Fig 7.3 a). It

is observed that even for a mixture refrigerant, the numerical model

predicted the heat transfer coefficients well.

7.3 Comparison with Correlations

Fig 7.3 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment for a) Deviation Graph b) Parity Graph

181

The numerical heat transfer coefficients are also compared with

the correlations to evaluate their relative performance in predicting

the experimental data from present study.

7.3.1 Comparisons for R22

As shown in Fig 7.4 a), the heat transfer coefficients predicted

using numerical model are as good as that of Cavallini et al.

correlation at a low mass flux of 200 kg/m2s, both exhibiting a

deviation of 8% from the experimental heat transfer coefficients, while

Dobson et al correlation exhibits lowest deviation of 3% among the

other correlations as shown in Fig 7.4 d).

a)

f)

e)

c)

b)

d)

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At a medium and high mass fluxes, the numerical model

predicts the experimental data better than correlations with a

deviation in the range of 4 – 9% as shown in Figs 7.4 e) and 7.4 f).

Cavallini et al. correlation under predicts with a deviation of 9%, while

the numerical model over predicts the experimental data as shown in

Figs 7.4 b) and 7.4 c). Other correlations exhibited further larger

deviations.

In general for R22, the predictions of numerical model are better

than the correlations used for comparison. The numerical model over

predicts the experimental data with an average deviation of 5 - 9%,

where as Cavallini et al. correlation under predicts the experimental

data with an average deviation of 9%.

7.3.2 Comparisons for R134a

As shown in Fig 7.5 a), all the correlations under predict the

experimental data at low mass flux while the numerical model over

predicts at medium and high qualities. Cavallini et al. correlation

Fig 7.4 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment and Correlations for R22 at a) G= 200 b) 400 andc) 600 kg/m2s and Corresponding Deviation Graphs

183

exhibits a minimum deviation of 5%, followed by the numerical model

and other correlations with a deviation of 11%.

A minimum deviation of 4% is exhibited by Cavallini et al.

correlation at medium mass flux and by Shah correlation at high

mass flux as shown in Figs 7.5 e) and 7.5 f). The deviation of

numerical model is higher than that of correlations at medium mass

flux with 22%, while Shah and Dobson et al. correlations deviate by

15 – 16%.

a)

f)c)

e)

d)

b)

184

At high mass flux, numerical model is as good as Cavallini et al.

correlation showing a deviation of 15%, while Dobson et al. correlation

exhibits better predictions with a deviation of 10% from the

experimental data as shown in Figs 7.5 b) and 7.5 c).

For R134a, on an average Cavallini et al. correlation exhibits a

deviation of 9% followed by Shah correlation with a deviation of 10%

and numerical model with a deviation of 15% from the experimental

data.

7.3.3 Comparisons for R407C

Fig 7.5 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment and Correlations for R134a at a) G= 200 b) 400 andc) 600 kg/m2s and Corresponding Deviation Graphs

a)

d)

c)

b)

185

All the correlations and numerical model over predict the

experimental data at a medium mass flux of 400 kg/m2s as shown in

Figs 7.6 a) and 7.6 c) with Cavallini et al. correlation exhibiting a

minimum deviation of 6 - 7% and the numerical model exhibiting a

deviation of 14%. The predictions of the numerical model are in

excellent agreement with the experimental data at high mass flux with

a deviation of 3% exhibiting an over prediction. Cavallini et al.

correlation is equally good, but exhibits under prediction. Thus, the

predictions of the numerical model exhibit excellent agreement even

with mixture refrigerant, compared to the correlations.

The maximum deviation exhibited by the numerical model is

22% for low pressure refrigerant, R134a and the minimum deviation

is 3% for mixture refrigerant, R407C. On an average, the predictions

of the numerical model are better than that of correlations for the

refrigerants considered in the present study, as the numerical model

exhibits over prediction of experimental data in general.

Fig 7.6 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment and Correlations for R407C at a) G= 400 b) 600 andCorresponding Deviation Graphs

186

7.4 Conclusions

The objective of the chapter is to develop a predictive procedure

using a combination of CFD analysis and numerical model to predict

the heat transfer coefficient for two phase flow of any fluid and at any

operating pressure.

The chapter describes the development of integral heat transfer

coefficient equation using separated flow approach by considering the

annular flow regime with uniform liquid film as physical model. A code

in ‘C’ language is written for the evaluation of dimensionless liquid

film thickness, interface shear stress and for the evaluation of local

shear stress at different locations in the liquid film using Von Karman

universal velocity profiles. The wall shear stress and pressure gradient

needed in the evaluation of local and interface shear stress is taken

from CFD simulations.

The heat transfer coefficients obtained from the numerical

model predicts the experimental data of all the refrigerants considered

in the present study within a deviation of ±20%. The model predicts

the heat transfer coefficients of high pressure refrigerants, R22 and

R407C with less deviation compared to R134a. This is due to the

separated flow approach used in the numerical model which gives

better results for fluids with high reduced pressure.

The numerical heat transfer coefficients predicted the

experimental data of R407C with a minimum deviation of 3% at a

mass flux of 600 kg/m2s and a maximum deviation of 14% at medium

187

mass flux. The predictions of numerical model are better than

correlations that exhibited a minimum deviation of 6-7% by Cavallini

et al. correlation and a maximum deviation of 30% by Shah

correlation. These results meet the objective of the present chapter.

Cavallini et al. correlation generally under predicts the

experimental data, while numerical model over predicts with an

average deviation of 11%. Therefore numerical model is suggested as a

better option compared to Cavallini et al. correlation for conservative

thermal design of condensers.

CHAPTER - VIII

CONCLUSIONS

The objectives of the present study are achieved primarily by

developing a test facility to evaluate the two phase heat transfer and

pressure drop at high pressures using refrigerants, R22, R134a and

R407C. Secondly, by developing a predictive procedure for heat

transfer coefficient and pressure drop using a combination of CFD

analysis and numerical model.

8.1 Experimental Study

The experimental analysis is performed for mass flux of 200,

400 and 600 kg/m2s for a pressure range of 10 – 16 bar at a

condensing temperature of 400C. The experiments are conducted with

average vapor quality of test section in the range of 0.28-0.71.

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The experimental set up has shown good repeatability by

exhibiting a heat balance error of less than 1% for 98% of the

runs, in spite of higher heat duties involved for the refrigerants,

R22, R134a and R407C considered in the present study.

The uncertainty analysis performed on experimental heat

transfer coefficient results in an average uncertainty of

±10.85%. The uncertainty involved with pressure drop is

±0.065% or ±0.081 kPa.

The validation of test facility with R22 shows that the

performance parameters are following the physics involved and

shows a definite behavior for runs at low, medium and high

mass fluxes.

The experimental data points plotted on Thome et al. [2003a]

flow regime map shows that it spreads in stratified wavy, slug

(Intermittent) and annular flow regimes.

8.2 Experimental Heat Transfer Coefficient

At low mass flux of 200 kg/m2s, the variation of two phase or

local heat transfer coefficient is not significant with vapor

quality. At high mass fluxes of greater than 400 kg/m2s, a

linear variation of heat transfer coefficient with quality is

observed. These trends corroborate with the heat transfer

characteristics of the flow regimes predicted by Thome et al.

[2003a] flow regime map, viz., stratified wavy and annular.

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The experimental heat transfer coefficients of R134a are almost

same of that of R22 as their liquid property combination

parameter, which is the non dimensional combination of

thermal and transport properties, is approximately same for

both the fluids.

The experimental heat transfer coefficients of R407C are

approximately 13% lower compared to that of R22 due to

dependency of vapor heat transfer coefficient on its two phase

heat transfer coefficient though its liquid property combination

parameter is 16% higher than that of R22.

8.2.1 Comparison with Correlations

The comparison of experimental data obtained in the present

study with Shah [1979], Dobson et al. [1998] and Cavallini et al.

[2002] correlations resulted into following conclusions.

Cavallini et al. [2002] correlation predicted the experimental

data falling in stratified, slug and annular flow regimes with a

minimum deviation of 5% and a maximum deviation of 18%.

Other correlations predicted the experimental data with good

agreement for only a particular flow regime. Shah correlation

exhibited better predictions for annular regime and for fluids

with high reduced pressure. Similarly Dobson et al. correlation

exhibited better predictions for stratified wavy regime.

8.3 Experimental Pressure Drop

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Low pressure refrigerant, R134a exhibits high pressure drop

penalty as its liquid to vapor density ratio and liquid viscosity

being higher compared to the other two refrigerants.

On an average, the pressure drop of R407C is almost same as

that of R22, as both have same value of liquid to vapor density

ratio.

8.3.1 Comparison with Correlations

The comparison of experimental pressure drop data with

Lockhart-Martinelli [1947], Grönnerud [1979], Chisholm [1973],

Friedel [1979] and Müller-Steinhagen and Heck [1986] correlations

resulted into following conclusions.

Lockhart and Martinelli correlation which is widely used in the

modeling of condensing flows exhibits a deviation of more than

100% from the experimental data of all the three refrigerants

considered in the present study.

Among the pressure drop correlations considered for

comparison, only Friedel correlation predicts the experimental

pressure drop with good agreement with a minimum deviation

of 5% and maximum deviation of 22%.

All most all the pressure drop correlations exhibited higher

deviations for low pressure refrigerant, R134a, thus leaving

scope for the development of better predictive procedure for

pressure drop at high pressures.

191

8.4 CFD Analysis

The fulfillment of the objectives of CFD analysis lies in proper

selection of multi phase model, appropriate mesh model of the test

section as the conditions near wall are to be captured and solver

controls according to the type of simulations.

The VOF model in the commercial CFD software, FLUENT

perfectly tracked the geometry of vapor – liquid interface of

refrigerants as the resulting mixture density contours are in

excellent agreement with the flow regimes predicted using

Thome et al. [2003a] flow regime map.

The work of Schepper et al. [2008] for air-water and gas-oil

mixtures at atmospheric pressures is successfully extended to

vapor-liquid flow of refrigerants at high pressures as all the flow

regimes are simulated in the CFD analysis.

Pressure drop data estimated using VOF model is in good

agreement with the experimental data compared to the pressure

drop correlations with a minimum deviation of 3% and a

maximum deviation of 16%.

8.5 Numerical Heat Transfer Model

The objective of developing the numerical model in combination

with the CFD analysis is to predict the heat transfer coefficient and

pressure drop for any fluid and at any pressure.

192

The numerical model over predicts the experimental data with

an average deviation of 11%.

The comparison of the heat transfer coefficient predictions by

the numerical model and by the correlations establishes that

numerical model is a better tool for conservative thermal design

of condensers.

8.6 Scope for Future Work

The test facility used in the present study is designed with

flexibility to use different tube geometries for the inner tube of the test

section. In addition, the scroll compressors used with different

displacement volumes and pressure ratios will enable the testing of

performance parameters of different future refrigerants. The scope for

the future work using the test facility and using CFD analysis is

presented as follows.

To test the performance of non azeotropic blends of HFCs and

HCs by modifying the test facility to visually observe the flow

regimes

To use different enhancement methods on refrigerant side

and to experimentally evaluate enhancement factors. Also to

develop a CFD model for enhanced surfaces and observe the

corresponding flow regimes

To experimentally evaluate the volumetric void fractions as

provision is made to isolate the test section from the rest of

experimental setup and tap the refrigerant

193

Prediction of flow regimes under diabatic conditions using

CFD analysis.

CFD analysis of double pipe heat exchanger with

condensation inside the inner tube.

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Horizontal Tube, Chemical Engineering Progress Symposium

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2. Aprea, C., Greco, A., Vanoli, G.P., Condensation Heat Transfer

Coefficients for R22 and R407C in Gravity Driven Flow Regime

within a Smooth Horizontal Tube, Int. J. of Refrigeration,

Vol.26, pp.393-401, 2003.

3. Baker, O., Simultaneous Flow of Oil and Gas, The Oil and Gas

Journal, Vol.53, pp.185-195, 1954.

4. Baroczy, C.J., Correlation of Liquid Fraction in Two Phase

Flow with Application to Liquid Metals, Chemical Engineering

Progress Symposium Series, Vol.61, No.57, pp.179-191, 1965.

5. Bell, K.J. and Ghaly, M.A., An Approximate Generalized

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American Institute of Chemical Engineers Symp. Ser., Vol.69,

pp.72-29, 1973.