Integration of Genetic Programing With Genetic Algorithm for Correlating Heat Transfer Problems

Yan LiuKey Laboratory of Thermal-Fluid Science

and Engineering,

Ministry of Education,

Xi’an Jiaotong University,

Xi’an 710049, Shaanxi, China

Jian YangKey Laboratory of Thermal-Fluid Science

and Engineering,




Jing XuSuzhou Nuclear Power Research Institute (SNPI),

Suzhou 215000, Jiangsu, China

Zhi-long ChengKey Laboratory of Thermal-Fluid Science

and Engineering,




Qiu-wang Wang1

Key Laboratory of Thermal-Fluid Science

and Engineering,




e-mail: [email protected]

Integration of GeneticPrograming With GeneticAlgorithm for Correlating HeatTransfer ProblemsIn the present paper, the genetic programing (GP) is integrated with the genetic algo-rithm (GA) for deriving heat transfer correlations. In the process of developing heattransfer correlations with the approach (GP with GA (GPA)), the GP is first employed toobtain some potential optimal forms. After that, the forms are further optimized with theglobal GA to reach minimum errors between the predicted values and experimental val-ues. With the proposed approach, three typical different heat transfer problems areapplied to the data reduction processes from published experimental data, which areheat transfer in a shell-and-tube heat exchanger (STHE) with continuous helical baffles,a single row heat exchanger with helically finned tubes and a finned oval-tube heatexchanger with double rows of tubes, respectively. The results indicate that the GPAapproach could improve the performance of heat transfer correlations obtained with theGP. Compared with the power-law-based correlations, the heat transfer correlationsobtained with the approach have higher predicted accuracies and more excellentrobustness. [DOI: 10.1115/1.4029871]

Keywords: genetic programing, genetic algorithm, heat transfer correlations, accuracy,robustness

1 Introduction

As a tool for searching and optimizing methodology, GA whichis based on principles of stochastic and Darwinian evolution, hasbeen widely used in optimization problems since proposed byHolland [1] in 1960 s. With the help of genetic operators, GA sol-ves complicated problems by self-evolution of computer pro-grams. In the process of evolution, GA adopts strings to expressproblems whose lengths are fixed. Therefore, it is difficult todescribe hierarchical problems due to lack of dynamic variability.Under these circumstances, GP was proposed by Koza [2] in1990 s, which employs a set of functions to obtain the optimumcorrelations for a given data set. GP adopts a dynamic tree struc-ture instead of fixed-length strings to describe complicated prob-lems. The dynamical changing of the tree structures makes itpossible to provide an answer that is better than others.

Different from traditional methods of parametric regression, GPcould search and determine the correlations’ forms and their con-stant terms simultaneously. All reports, in recent years, aboutapplications of GP in the field of heat transfer are listed as below.The first attempt was conducted by Lee et al. [3] to derive criticalheat flux correlations. Correlations obtained by Lee et al. [3] weremore accurate or comparable to other existing correlations. Caiet al. [4] employed two sets of published experimental data to testthe GP, which was modified by a penalty function to prevent largecorrelation functions. Zdaniuk et al. [5] correlated both Colburn j-factor and friction factor with linear combinations of five simplegroups of parameters and a constant. The correlations proposed by

Zdaniuk et al. [5] were proven to be more excellent. Thus, moreexcellent correlations could be obtained with GP, compared withtraditional regression methods.

GA was initially adopted in heat transfer research in the mid-1990 s, timidly at first, but more and more regularly nowadays [6].GA has been used to minimize the errors between a set of dataand a correlation that one wants to develop to obtain the appropri-ate fitting coefficients. Presently, application of GA on linear ornonlinear regression to a given-form fitting is already a well-developed approach. With a GA, Pacheco-Vega et al. [7] obtainedthe two heat transfer correlations in and over the tubes of a heatexchanger simultaneously. Based on the form of classical power-law correlation, Momayez et al. [8] derived and optimized corre-lations for the convective heat transfer coefficient in a boundarylayer over a concave wall by GA. Porto et al. [9] adopted GA tosearch the solution space, which consists of the input parametersplus the different functional forms, and obtained two-phase heattransfer correlations that cover a wide range of heat transferregimes. Lorenzini et al. [10] adopted GA in association withconstructal design for the geometric optimization of a convective,Y-shaped cavity, which were proved to be very effective. Zenget al. [11] and Wang et al. [12,13] correlated the Nusselt numberand the friction factor versus Reynolds number by GA of bothsides of heat transfer surface, for a high-efficiency plate-fin struc-ture used in an air preheater, three different types of STHEs and across-corrugated primary surface heat exchanger, respectively.

Considering that GA’s advantages in searching and optimizingfitting coefficients, applications of GA to correlations obtainedwith the GP could have the potential of improving performance.Therefore, in the present paper, an approach that integrates GPAis firstly proposed to derive heat transfer correlations in experi-mental data reduction processes. The optimal forms and fitting

1Corresponding author.Manuscript received July 6, 2014; final manuscript received December 22, 2014;

published online March 17, 2015. Assoc. Editor: Giulio Lorenzini.

Journal of Heat Transfer JUNE 2015, Vol. 137 / 061012-1Copyright VC 2015 by ASME

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coefficients are obtained simultaneously by the GPA approach.The GPA approach is carefully described and the assess criterionsfor correlation are also provided. With the proposed approach,three different heat transfer problems are put into applications inthe data reduction processes from published experimental data,which are heat transfer in a STHE with continuous helical baffles,a single row heat exchanger with helically finned tubes and afinned oval-tube heat exchanger with double rows of tubes,respectively. Based on both their accuracies and robustness, heattransfer correlations are carefully compared between differentapproaches, including the power-law-based regression, the GPand the GPA.

2 Description of the Approach and Assessments

2.1 Integration of GPA. This section describes the method-ology adopted to derive new correlations. Figure 1 illustrates flowchart of the approach integrating GPA. The first step in GPA is toprepare correlating data and generally, the greater the amount ofdata, the better. A set of major determinants could also be deter-mined before applying of the GP. As required, original parameterscould be operated and dimensionless parameters are alwaysadopted. Thus, a set of inputs that could be included in thetargeted correlations could be obtained. For example, U is the tar-geted variable, X (set of dimensionless variables) and Y (set ofvariables measured directly) are inputs for the application of theGP.

The second step in the process is to obtain correlations with oneor more potential optimal forms with the GP. The detailed imple-mentation steps of GP could be found in Refs. [3] and [4]. Mainparameters settings of the GP are based on Refs. [14] and [15].All of the forms are obtained without any assumptions in advancein this step. Potential optimal forms could be expressed asfollows:

Up ¼ f X;Y;Cð Þ (1)

where Up is the predicted targeted variable; f represents set ofpotential optimal forms; and C is the set of constants obtainedwith the GP.

The last step in GPA is to optimize fitting coefficients in corre-lations, which are obtained in the second step. In this step, theforms remain the same with Eq. (1). After performing the globalGA, several potential forms could be optimized to the finaloptimal correlation, which could be expressed as follows:

Up ¼ f X;Y;C0ð Þ (2)

where C0 is the final set of constants obtained with GPA.In the present paper, the modification and validation of the GP

are based on the GP toolbox for MATLAB (GPLAB, version 3.0). TheGA tools used in the present paper are the ones implemented inthe global optimization toolbox (version 6.3) in MATLAB R2013a.We spend about 150 min on deriving one correlation on averagebased on GPA approach. All computations are performed on apersonal computer with Intel

VR

CoreTM 2.40 GHz CPU and 8 GBRAM.

2.2 Procedures of GA. Figure 2 illustrates the flow chart ofGA [6,16]. The detailed procedures are as follows:

(1) Index of generation (G) is initialized to 0 and an initial pop-ulation of the problem is created randomly.

(2) Judge whether the termination criterion is satisfied. If thetermination criterion is reached, the optimal individuals andend the evolution are yielded. Otherwise, go to step 3.

(3) Evaluate values of fitness function for each individual. Foreach individual, applying of genetic operators is performedbased on their probabilities.

(4) Index of generation (G) counts when index of individuals(i) reaches total number of individuals (n) and new genera-tion could be created.

Two different fitness functions are adopted in the present paper[17], which could be expressed as follows:

F1 ¼XN

j¼1

Upði; jÞ � Ueði; jÞUeði; jÞ

�� (3)

F2 ¼XN

j¼1

Upði; jÞ � Ueði; jÞUeði; jÞ

��þ Upði; jÞ � Ueði; jÞ

Ueði; jÞ

� �2 !

(4)

where Upði; jÞ is predicted targeted variable of individual i for thejth input value; Ueði; jÞ is experimental targeted variable of indi-vidual i for the jth input value; i and j are index of individuals andinput values, respectively; and N is total number of inputs.

More detailed description about GA could be found in Refs.[18] and [19].

2.3 Assessments of Correlations. In the present paper, twobasic assess criterions, including average deviation rate (ADR)and maximum deviation rate (MDR), are employed to assess per-formance of different correlations. The two basic criterions couldbe expressed as follows:

ADR ¼ Upði; jÞ � Ueði; jÞUeði; jÞ

��

� �(5)

Fig. 1 Flow chart of approach integrating GPA Fig. 2 Flow chart of GA from Refs. [6] and [16]

061012-2 / Vol. 137, JUNE 2015 Transactions of the ASME


MDR ¼ maxUpði; jÞ � Ueði; jÞ

Ueði; jÞ

�� (6)

In order to obtain comprehensive assessment for correlations,both their accuracies and robustness could be taken into consider-ation. According to the description of uncertainties in experimen-tal results of Moffat [20] and Khartabil et al. [21], new assesscriterions are proposed. When input variables (X and Y) arechanged by a certain perturbation (dX and dY), the targeted vari-able U0p could be determined, which could be shown as follows:

U0p ¼ f X6dX; Y6dY;C0ð Þ (7)

Then two new assess criterions, including ADR0 and MDR0, couldbe obtained as follows:

ADR0 ¼ U0pði; jÞ � Ueði; jÞUeði; jÞ

��

� �(8)

MDR0 ¼ maxU0pði; jÞ � Ueði; jÞ

Ueði; jÞ

�� (9)

3 Application to a STHE With Continuous Helical

Baffles

The GPA approach now is applied to experimental data reduc-tion processes of a STHE with continuous helical baffles (referredas application 1). Experimental data of a STHE reported by Penget al. [22] is adopted in application 1. The detailed informationabout the experiments has been carefully described by Peng et al.[22]. Sketch of the tube bundle, side view of tube bundle, andshell configurations are illustrated in Fig. 3. For the test heatexchanger, 15 experimental points for dimensionless heat transferparameters of the STHE with continuous helical baffles werereported in Ref. [22]. Traditionally, the power-law forms areemployed for heat transfer correlations of the STHEs [23,24]. Apower-law correlation obtained by Peng et al. [22] could beexpressed as follows:

Nu ¼ 1:07 Re0:579Pr1=3 (10)

The following are two correlations obtained with the GP andthe GPA, respectively:

Nu ¼ 42:94þ 5:748 Reð Þ0:638þ Pr� 86:313 (11)

Nu ¼ 34:49þ 5:815 Reð Þ0:637þ Pr� 86:312 (12)

Table 1 shows a comparison of the deviation rates in Nuobtained from the GPA-based correlations with the publishedcounterparts. From Table 1, the correlations obtained with theGPA shows the smaller ADR and MDR than these obtained withthe GP and the power-law correlations. Though the correlationobtained with the GP has much more excellent performance thanthe power-law-based one, the predicted errors could still bereduced with the GPA approach.

A comparison between experimentally determined Nu and thatpredicted with GPA is illustrated in Fig. 4. The solid straight lineis also presented as the perfect prediction, which could be treatedas a reference. The predicted values of power-law-based correla-tion are also included. From Fig. 4, the values predicted with theGPA-based correlation are closer to the perfect predictions thanthat with a power-law correlation.

Figure 5 shows experimental and GPA-based correlation pre-dicted values of Nu plotted versus different experimental points.From Fig. 5, experimental values are in good agreement with theGPA predicted values. As a whole, the GPA-based correlationcould describe 14 experimental points within error of 62% andshows a 0.88% ADR and a 4.49% MDR, compared with power-law-based correlation describing nine experimental points withinthe error range, showing a 1.77% ADR and a 4.76% MDR.

Tables 2 and 3 show comparisons of the deviation rates of dif-ferent correlations for Nu of the STHE with continuous helicalbaffles when Re and Pr increase or decrease by 1%. From Tables2 and 3, the GPA-based correlation shows the smaller MDR0 thanthe GP-based correlation in all cases. However, for ADR0, thesetwo correlations perform equally. Overall, robustness of GPA-based correlation is more excellent than the GP-based correlation.

4 Application to a Single Row Heat Exchanger With

Helically Finned Tubes

The GPA approach now is applied to experimental data reduc-tion processes of a single row heat transfer exchanger with heli-cally finned tubes (referred as application 2). In application 2, theexperimental data reported by Huisseune et al. [25] is adopted totest the GPA approach. The physical map of the tested tube couldbe seen in Fig. 6. The experiments were carried on a test rig,which consisted of an open-air wind tunnel and a closed watercycle [25]. More detailed descriptions about the test rig have beencarefully conducted by T’Joen et al. [26,27]. The experimentaldata for friction factor (f) of Huisseune et al. [25] could be digi-tized with DIGXY software and 50 experimental points areobtained for f.

Figure 7 illustrates experimental data of friction factor (f) plot-ted versus Rec [25]. From Fig. 7, f decreases with increasing Rec

for a given Xt/da and f increases with decreasing Xt/da for a given

Fig. 3 A STHE with continuous helical baffles: (a) sketch ofthe tube bundle, (b) side view of tube bundle, and (c) shell con-figurations [22]

Table 1 Deviation rates of heat transfer correlations of theSTHE with continuous helical baffles

Deviation

Predicted equations ADR (%) MDR (%)

Eq. (10) 1.77 4.76Eq. (11) 0.96 4.68Eq. (12) 0.88 4.49

Journal of Heat Transfer JUNE 2015, Vol. 137 / 061012-3


Rec. According to Huisseune et al. [25], a power-law-based corre-lation was obtained via a least-mean-square fit as follows:

f ¼ 2:271 Re�0:325c

Xt

da

� ��1:849

(13)

The following are two correlations obtained with the GP andthe GPA, respectively,

ln f ¼ 1:2413:590� ln Recð Þ Xt=da þ 2:553ð Þ

ln Rec

(14)

ln f ¼ 1:2133:464� ln Recð Þ Xt=da þ 2:555ð Þ

ln Rec

(15)

Table 4 shows a comparison of ADR and MDR of correlations fof the single row heat exchanger with helically finned tubes. FromTable 4, the correlation obtained with GPA shows the smaller

Fig. 4 Experimental values from Ref. [22] versus GPA-basedcorrelation predicted Nu of the STHE with continuous helicalbaffles

Fig. 5 Experimental values from Ref. [22] and GPA-based cor-relation predicted values of Nu plotted versus different experi-mental points

Table 2 Deviation rates of heat transfer correlations of theSTHE with continuous helical baffles when Re increases ordecreases by 1%

Re increases by 1% Re decreases by 1%

Predicted equations ADR0 (%) MDR0 (%) ADR0 (%) MDR0 (%)

Eq. (10) 1.91 4.21 1.74 5.31Eq. (11) 1.29 5.22 0.96 4.15Eq. (12) 1.15 5.02 1.06 3.95

Table 3 Deviation rates of heat transfer correlations of theSTHE with continuous helical baffles when Pr increases ordecreases by 1%

Predicted equations Pr increases by 1% Pr decreases by 1%


Eq. (10) 1.84 4.44 1.73 5.08Eq. (11) 1.14 5.23 0.92 4.13Eq. (12) 0.99 5.04 0.95 3.94

Fig. 6 A single row heat exchanger with helically finned tubes:(a) side view and (b) cross-sectional view [25]

Fig. 7 Experimental values of f plotted versus Rec fromRef. [25]



ADR and MDR than any others, including the correlationobtained with the GP and the power-law-based correlation. It iscould also be observed that the correlation has been furtherimproved with GPA on the basis of the GP-based correlation.

A comparison between experimental and GPA-based correla-tion predicted f of the single row heat exchanger with helicallyfinned tubes is illustrated in Fig. 8. The predicted values of thepower-law-based correlation are also included. From Fig. 8, thevalues predicted with GPA are closer to the perfect predictionsthan that with a power-law-based correlation. Figure 9 showsexperimental and GPA-based predicted values of Nu plotted ver-sus Rec. It can be seen that the experimental values are in goodagreement with the GPA predicted values in most experimentpoints. As a whole, the GPA-based correlation could describe 49experimental points within error of 615% and shows a 4.52%ADR and a 16.25% MDR, compared with power-law-based corre-lation describing 45 experimental points within the same errorrange, showing a 6.87% ADR and a 22.16% MDR.

Comparisons of the deviation rates of correlations for f of thesingle row heat exchanger with helically finned tubes when Rec

increases or decreases by 1% could be seen in Table 5. FromTable 5, the correlation obtained with GPA shows the smallerADR0 and MDR0 than any others, including the GP-based correla-tion and the power-law-based correlation.

5 Application to a Finned Oval-Tube Heat Exchanger

With Double Rows of Tubes

The GPA approach now is applied to experimental data reduc-tion processes of a finned oval-tube heat exchanger with doublerows of tubes (referred as application 3). Experimental data of afinned oval-tube heat exchanger reported by Du et al. [28,29] isadopted in application 3. The experiments have been carefullydescribed by Du et al. [28,29] and Tang et al. [30]. Schematicview and a partial enlarged picture of test heat exchanger are illus-trated in Fig. 10. Figure 11 illustrates experimental data of k plot-ted versus va [28,29]. Traditionally, corresponding power-lawcorrelations of Nu versus Re are employed for description of heattransfer [28,29]. However, Pacheco-Vega et al. [31] reported thatit was more accurate to develop correlations to predict the heattransfer characteristic directly, rather than intermediate quantities.Additionally, the user of the information is usually interested inthe overall heat transfer characteristic instead of intermediate vari-ables, from which the heat transfer characteristic have to be fur-ther calculated. In the present paper, the overall heat transfercoefficient (k) is directly correlated based on fluid velocity onboth sides of heat transfer surface (va and vw). After performingthe GP, two potential optimal correlations could be obtained asfollows:

k

k0

� �¼ 2:453

va

v0

� �� vw

v0

� �þ vw

v0

� �þ va

v0

� �þ12:608

� �(16)

k

k0

� �¼ va

v0

� �þ 5:858

� �� vw

v0

� �þ 4:497

� �(17)

Based on Eqs. (16) and (17), a potential optimal form could beobtained:

k=k0ð Þ ¼ va=v0ð Þ � vw=v0ð Þ þ c1 va=v0ð Þ þ c2 vw=v0ð Þ þ c3. Basedon this potential form, the correlation could be obtained with GPAand the result is as follows:

k

k0

� �¼ va

v0

� �� vw

v0

� �þ 4:076

va

v0

� �þ 4:219

vw

v0

� �þ 28:864

(18)

where k0 and v0 are unit overall heat transfer coefficient andvelocity, respectively [3].

Table 6 shows a comparison of the deviation rates in k obtainedbetween the GPA-based correlation and the GP-based

Table 4 Deviation rates of correlations for f of the single rowheat exchanger with helically finned tubes

Deviation


Eq. (13) 6.87 22.16Eq. (14) 4.65 17.91Eq. (15) 4.52 16.25

Fig. 8 Experimental values from Ref. [25] versus GPA-basedcorrelation predicted f of the single row heat exchanger withhelically finned tubes

Fig. 9 Experimental values from Ref. [25] and GPA-based cor-relation predicted values of f plotted versus Rec

Table 5 Deviation rates of correlations for f of the single rowheat exchanger with helically finned tubes when Rec increasesor decreases by 1%

Rec increases by 1% Rec decreases by 1%


Eq. (13) 6.87 21.77 6.88 22.56Eq. (14) 4.63 17.48 4.68 18.36Eq. (15) 4.52 15.84 4.55 16.67



correlations. From Table 6, the correlation obtained with GPAshows the smaller ADR and MDR than these obtained with theGP. Though the GP-based correlations already have excellent per-formance, the predicted errors could still be reduced with GPA.

A comparison between experimentally determined Nu and thatpredicted from Eq. (18) is illustrated in Fig. 12. From Fig. 12, thecorrelation obtained with GPA could describe 100% of all datawithin 65%. Figure 13 shows experimental and GPA predictedvalues of k plotted versus va. From Fig. 13, experimental valuesare in good agreement with the GPA predicted values in mostexperiment points. As a whole, the GPA-based correlation coulddescribe 70 experimental points within error of 64% and shows a1.77% ADR and a 4.88% MDR, compared with the GP-based cor-relation describing 64 experimental points within the same errorrange, showing a 2.27% ADR and a 5.45% MDR.

Tables 7 and 8 show comparisons of the deviation rates of heattransfer correlations of the finned oval-tube heat exchanger with

Fig. 10 Schematic view of the finned oval-tube heat exchangerwith double rows of tubes [28,29]

Fig. 11 Experimental values of k plotted versus va from Refs.[28] and [29]

Table 6 Deviation rates of correlations for k of the finned oval-tube heat exchanger with double rows of tubes

Deviation


Eq. (16) 2.46 5.85Eq. (17) 2.27 5.45Eq. (18) 1.77 4.88

Fig. 12 Experimental values from Refs. [28] and [29] versusGPA-based correlation predicted k of the finned oval-tube heatexchanger with double rows of tubes

Fig. 13 Experimental values from Refs. [28] and [29] and GPA-based correlation predicted values of k plotted versus va

Table 7 Deviation rates of correlations for k of the finned oval-tube heat exchanger with double rows of tubes when va

increases or decreases by 1%

va increases by 1% va decreases by 1%


Eq. (16) 2.34 5.62 2.36 6.09Eq. (17) 2.30 5.23 2.27 5.67Eq. (18) 1.85 4.63 1.73 5.14

Table 8 Deviation rates of correlations for k of the finned oval-tube heat exchanger with double rows of tubes when vw

increases or decreases by 1%

vw increases by 1% vw decreases by 1%


Eq. (16) 2.36 5.71 2.33 6.00Eq. (17) 2.30 5.30 2.26 5.59Eq. (18) 1.80 4.77 1.75 5.00



double rows of tubes when va and vw increase or decrease by 1%.From Tables 7 and 8, the GPA-based correlation shows thesmaller ADR0 and MDR0 than the GP-based correlations in allcases. In other words, robustness of GPA-based correlation ismore excellent than the GP-based correlation.

6 Conclusions

In the present paper, an approach that integrates GPA for deriv-ing heat transfer correlation in the experimental data reductionprocesses is proposed. Aiming at developing more excellent corre-lations, the GP is first employed to obtain some potential optimalforms. Then, these potential forms are further optimized with theglobal GA to reach minimum errors between the predicted valuesand experimental values. Synthesizing preponderance of the GPand GA, the optimal forms and fitting coefficients are obtainedsimultaneously.

With the GPA approach, three different heat transfer problemsare put into applications in the data reduction processes from pub-lished experimental data, which are heat transfer in a STHE withcontinuous helical baffles, a single row heat exchanger withhelically finned tubes and a finned oval-tube heat exchangerwith double rows of tubes, respectively. Compared with thepower-law-based correlations, the heat transfer correlationsobtained with GPA have higher predicted accuracies and moreexcellent robustness (applications 1 and 2). In application 3, heattransfer correlations are correlated directly based on measuredparameters in experiments, rather than dimensionless parameters.The results of applications 1–3 also reveal that the GPA approachcould improve the performance of heat transfer correlationsobtained with the GP, although the GP-based correlations alreadyhave excellent performance.

Acknowledgment

We would like to acknowledge financial supports for this workprovided by China National Funds for Distinguished Young Sci-entists (No. 51025623), National Basic Research Program ofChina (973 Program No. 2012CB720402) and National NaturalScience Foundation of China (No. 51106126).

Nomenclature

ADR ¼ average deviation rateADR0 ¼ average deviation rate with a certain perturbation

c1, c2, c3 ¼ constantsC, C0 ¼ constant sets in Eqs. (1) and (2)

d ¼ tube diameter (m)f ¼ friction factorf ¼ set of potential optimal forms

F ¼ fitness functionG ¼ index of generationi ¼ index of individualsj ¼ index of input valuesk ¼ overall heat transfer coefficient (W �m�2 �K�1)

k0 ¼ unit overall heat transfer coefficient (W �m�2 �K�1)MDR ¼ maximum deviation rateMDR0 ¼ maximum deviation rate with a certain perturbation

n ¼ total number of individualsN ¼ total number of inputs

Nu ¼ Nusselt numberP ¼ probability of genetic operator

Pr ¼ Prandtl numberRe ¼ Reynolds number

Rec ¼ Reynolds number based on the velocity in the mini-mum cross section

v ¼ velocity (m � s�1)v0 ¼ unit velocity (m � s�1)

X, Y ¼ set of variablesXt ¼ transversal tube pitch (m)

Greek Symbols

U ¼ targeted variableU0 ¼ targeted variable with a certain perturbation

Subscripts

a ¼ air sidec ¼ crossovere ¼ experimental values

m ¼ mutationp ¼ predicted valuesr ¼ reproduction

w ¼ water side1, 2, 3 ¼ labels of variables

Abbreviations

GA ¼ genetic algorithmGP ¼ genetic programing

GPA ¼ integration of GP with GAGPLAB ¼ genetic programing toolbox for MATLAB

STHE ¼ shell-and-tube heat exchanger

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Integration of Genetic Programing With Genetic Algorithm for Correlating Heat Transfer Problems

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