Optimized heat transfer correlations for pure and blended refrigerants

Matheus P. Porto a, Hugo T.C. Pedro b, Luiz Machado a, Ricardo N.N. Koury a, Enio P. Bandarra Filho c,Carlos F.M. Coimbra b,⇑a Programa de Pós Graduação em Engenharia Mecânica, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazilb Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, and Center for Energy Research, University of California San Diego, La Jolla, CA, USAc Faculdade de Engenharia Mecânica, Federal University of Uberlândia, Brazil

a r t i c l e i n f o

Article history:Received 9 September 2014Received in revised form 20 January 2015Accepted 20 January 2015Available online 20 February 2015

Keywords:Refrigerant blendsGenetic algorithm (GA) optimizationInternal flowsTwo-phase flowHeat transfer coefficient

a b s t r a c t

Refrigerant blends and pure refrigerants have wide applicability in thermal engineering. One of the cri-tical parameters in the design and evaluation of thermal equipment is the heat transfer coefficient, whichcan be difficult to determine for refrigerants that undergo phase change within the equipment. For purerefrigerants, classical experimental relations developed by Gungor and Winterton (GW87) are known toexhibit errors around 15% on average, and reaching more than 40% in some cases. For refrigerant blendslarger uncertainties are expected due to a complex number of factors such as nucleate boiling degrada-tion, particularly when using functional forms previously developed for pure refrigerants. This workprovides a comprehensive experimental study on the determination of heat transfer coefficients forR-22, R-134a, and the predefined refrigerant blends R-404A and R-407C. Genetic optimization is usedto obtain more accurate semi empirical relations based on the classical GW87 correlation, and resultsof the optimization analysis show large improvement for pure refrigerants. The use of a degradation fac-tor in the optimized correlation for R-407C allows for substantial error reduction for refrigerant blends.

! 2015 Elsevier Ltd. All rights reserved.

1. Introduction

General two-phase flow correlations for heat transfer coeffi-cient (HTC) in smooth tubes have been studied for several decades[1–9]. In these works, researchers have determined the HTC usingapproaches such as: (1) strictly convective, (2) strictly empirical,(3) superposition effects, (4) flow pattern map based types, andothers (see [10]). The explanation for the variety of approachesand the abundance of different correlations lies on the fact thatnone of the correlations are uniformly valid for a wide range ofoperating conditions.

As demonstrated by Porto et al. [10], comprehensive numericaloptimization of empirical constants and parameters in these corre-lations allows for reduction of uncertainty and add substantialrobustness to the generic correlations. Correlations from the firstgroup, which assume convective boiling effects alone, are basedon Martinelli’s parameter. Optimization techniques have beenused to adjust the relevant coefficients in the proposed expressionand improve their accuracy (see [11] for example). Improvementshave also been obtained for the third group of correlations. In thiscase both suppression and enhancement factors can be optimized

simultaneously [5,12,13]. The fourth group uses correlationsderived from the HTC asymptotic behavior [7], which is based onfunctional form derived from superposition effects. The strictlyempirical relations, or semi empirical relations, generally use morerobust optimization techniques so to include dimensionless factorsfor the convective and nucleate HTC terms, while at the same timeproviding values for multiplicative and power coefficients[4,14,15]. The best known family of correlations for this lattergroup were developed by Kandlikar [4], who presented a methodto optimize constants (C1 to C4) for the following correlation:

hTP ¼ C1CoC2 hl þ C3BoC4 hl; ð1Þ

where Co is the convective number and Bo is the boiling number. Inthis optimization C2 and C4 were estimated and C1 and C3 wereevaluated using least-square routines that minimize the error ofthe correlation against experimental data. Additionally, Kandlikarstudied the benefits of using product factors comprised of dimen-sionless numbers simulating enhancement and suppression effectson convective and nucleate boiling, respectively. For the convectiveterm Kandlikar used Bom, x=ð1% xÞ)m, Rem

l ; and for the nucleate termhe used: Com;ComBon;vm

tt ; ðx=ð1% xÞÞm; xm;Reml and (convective

term)m. However, Kandlikar did not observe improvements by thisexpedient, thus maintaining the base relation unchanged (Eq. (1)).

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.01.1020017-9310/! 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: ccoimbra@ucsd.edu (C.F.M. Coimbra).

International Journal of Heat and Mass Transfer 85 (2015) 577–584

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In 1990, due to computational limitations, Kandlikar tested only10 different product factors, with current computational tollsmuch larger number of combinations can be tested. Porto et al.[10], for example, presented a methodology based on GeneticOptimization to find coefficients based on 28 dimensionlessnumbers. In this work, more than 109 different combinations ofmultiplication factors for both nucleate and convective terms weretested. The results in Porto et al. [10] have shown agreement withexperiments with an error of 6%, on average. Following the effortsby Porto et al. [10], in this work, we employ a similar methodologyto optimize Gungor and Winterton (GW87) HTC correlation, usingthe large experimental data bank that is described in the nextsection.

2. Raw data

The data set used in this work is comprised of 1888 experimen-tal points for R-22, R-134a, R-404A and R-407C. The data wereobtained from three different experiments, all of them using asimilar test set up, as can be seen by comparing Bandarra Filho[11], Porto et al. [10] and Wattelet [5,16]. The data bank character-istics are summarized in Table 1, which lists inner diameters, massflow velocity, heat flux, quality, boiling temperature and a list offluids for each experiment.

Fig. 1 shows the experimental data for R-22, R-134a, R-404Aand R-407C, overlaid on top of the different pattern regions. Thefigure shows that almost no data points were collected for the mistand dry-out flow regions. This is recommended because theexperimental HTC values were evaluated using a Joule Effectheating source (see Ref. [9]). Another relevant observation is thatall fluids have experimental data points for 100, 200 and300 kg m%2 s%1 mass flow rates. For more information about theflow pattern maps presented in Fig. 1 see [6–10].

3. Experimental HTC for pure and blended refrigerants

The HTC in a horizontal tube can be determined experimentallyfor a pure or azeotrope fluid through the convective heat transferdefinition:

h ¼ Q 00=ðTw % TsÞ; ð2Þ

and,

Q 00 ¼ Q=S; ð3Þ

where h is the HTC ðW m%2 K%1Þ;Q 00 is the heat flux ðW m%2Þ; Tw isthe tube internal surface temperature ("C), Ts is the bulktemperature of the fluid ("C), and S is the tube surface area ðm2Þ.

For a zeotrope blend, as R-407C, Thome [17] and Cavallini et al.[18] highlight that heat flux provided from counter-current orJoule effect to the test section causes phase changing and sensibleheating at the same time, due to the vapor and liquid temperatureincreasing. Also, Thome [17] recommends not to use local mea-surements of temperature in an evaporating blend to determineenthalpy difference, because flow temperature often is higher thancalculated bubble-point. This difference of local measurements andactual bubble-point occurs as a consequence of nonequilibriumeffects caused by the temperature glide and velocity differencesin vapor and liquid phases. Boiling temperature (dTb) and quality(x) can be calculated using the equations below:

di ¼ ilvdxþ ð1% xÞcpldTb þ xcpvdTb; ð4Þ

and

dTb ¼ dTglide f ðxÞ % f ðx%dxÞ

h i; ð5Þ

where f ðxÞ is an empirical function of quality which depends onenthalpy difference. The two unknowns (dTb and x) can be deter-mined using Eqs. (4) and (5). Considering a pre-heater in whichfluid enters at saturated state (x ¼ 0) and exits as a two-phase flow,also approximating f ðxÞ by x [18], quality and boiling temperaturecan be obtained by the following relations:

x ¼%cpldTglide % ilv þ cp2

l dT2glide þ i2

lv þ 2dTglideilvð2cpv % cplÞh i0:5

2dTglideðcpv % cplÞ;

ð6Þ

and

dTb ¼ dTglidex; ð7Þ

Nomenclature

cp heat capacity at constant pressure, kJ kg%1 K%1

d diameter, mg gravitational acceleration, m s%2

G mass flux, kg m%2 s%1

h heat transfer coefficient (HTC), W m%2 K%1

i enthalpy, kJ kg%1

k thermal conductivity, W m%1K%1

Q heat flow, WQ 00 heat flux, W m%2

U mass flow or bulk velocity, m s%1

x Quality

Greek symbolsv Martinelli parameterl viscosity, Pa sq density, kg m%3

r surface tension, N m%1

Dimensionless numbersFr Froude number, G2 q%2 g%1 d%1

Pr Prandtl number, l cp K%1

Re Reynolds number, q Ud=lWe Weber number, G2 dq%1 r%1

Subscriptsb boilingcb convective boilingcrit critical propertieseq equivalent property – linear proportion between satura-

tion properties and qualityl saturated liquid, x = 0lv difference between vapor from liquid saturated proper-

tiesnb nucleate boilingtp two-phase flowtt liquid/vapor interfaces saturatedv saturated vapor, x = 1w tube wall

578 M.P. Porto et al. / International Journal of Heat and Mass Transfer 85 (2015) 577–584

and

h ¼Q 00

Tw % Ts þ dTbÞ½ ' ; ð8Þ

where Ts can be determined by measuring pressure at the testsection.

Another advantage of using heat balance to determine quality isrelated to the results accuracy. Uncertainty is very low when qual-ity is determined using heat balance instead of using pressure andtemperature measurements at the test section. This is because alittle difference in temperature causes a great error in quality,due to the temperature glide.

4. Classical correlations

The relations of Wojtan et al. [8,9], Gungor and Winterton 1987[2], Kandlikar [4], and Wattelet [5] were used to test the correla-tion with the experimental data, and these relations can be foundin Table 2, with the exception of Wojtan’s methodology, which istoo long to be presented here (see Refs. [8,9]).

Wojtan’s correlation is based on flow pattern map regions, andSteiner and Taborek [19] is used as the basic form for the HTC cor-relations. Wojtan et al. [8,9] presented results for R-22 and R-410A,and they predicted 93% of experimental results at five different

mass velocities and two different initial heat fluxes within (15%of error. Given that Wojtan et al. [9] use a different method forthe test section heating (counter-current flow instead of JouleEffect), it is not expected a good correlation with the experimentaldata provided here, specially for high quality values. On the otherhand, for low and intermediate quality values, Wojtan hasobtained satisfactory results, as shown in Ref. [9] (when he com-pares his data with Lallemand et al. [20] – which uses electricalheating).

Gungor and Winterton in 1986 [1] used 3693 experimentalpoints for Water, R-11, R-12, R-114 and Ethylene Glycol to elabo-rate their first correlation. Later Gungor and Winterton provideda simplified correlation, based just on the convective term of boil-ing (GW87) [2], achieving a large improvement with respect to thefirst one.

Kandlikar used 5246 data points considering Water, R-11, R-12,R-1281, R-22, R-113, R-114, R-152A, Nitrogen and Neon, and hisrelation was based on Convective (Co) and Boiling (Bo) number, alsothe influence of flow stratification was considered by using theFroude number. Before proposing a new correlation, Kandlikarstudied the data bank to investigate the accuracy of the experimen-tal results, and he listed some common errors, such as: incorrectcorrelation between quality and heat balance at the pre-heaterdue to heat losses; absence of wall temperature and pressure simul-taneously measured along the test section; very low temperature

Table 1Experimental data set presented by Bandarra [11], Porto et al. [10] and Wattelet [16,5].

Data[reference]

Tube diameter(mm)

Mass flow(kg m%2 s%1)

Heat flux(kW m%2)

Quality Boiling temperature()C)

Fluids Number of exp.points

From thispaper

7.17 and 12.7 46–510 4.67–20.7 0.05–0.99

3, 5, 8 and 15 R-22, R-134a, R-404A, R-407C

666

Porto et al.[10]

12.7 46–516 4.69–20.46 0.05–0.99

8 and 15 R-22, R-134a, R-404A 682

Wattelet[16,5]

7, 7.7, 10.2 and10.9

254–544 1.9–40.58 0.07–0.97

%15, %5, 5, 8, 15 and20

R-22, R-134a 540

Fig. 1. Experimental data set plotted on a flow pattern map for R-22, R-134a, R-404A and R-407C. Upper left: R-22 experimental points, dashed line (flow pattern map): Sat.Press. = 600 kPa, Mass flow vel. = 300 kg m%2 s%1, tube diam. = 10 mm, heat flux = 21 kW m%2, upper right: R-134a experimental points, dashed line (flow pattern map):Sat.Press. = 370 kPa, Mass flow vel. = 300 kg m%2 s%1, tube diam. = 10 mm, heat flux = 21 kW m%2, lower left: R-404A experimental points, dashed line (flow pattern map): Sat.Press. = 780 kPa, Mass flow vel. = 300 kg m%2 s%1, tube diam. = 13 mm, heat flux = 13 kW m%2, lower right: R-407C experimental points, dashed line (flow pattern map): Sat.Press. = 730 kPa, Mass flow vel. = 300 kg m%2 s%1, tube diam. = 13 mm, heat flux = 13 kW m%2.

M.P. Porto et al. / International Journal of Heat and Mass Transfer 85 (2015) 577–584 579

difference for fluid and wall when compared to the instrument’saccuracy; errors in the properties equations of state; presence ofoil in the refrigerant; and peculiar conditions in the entry regionof the test section (sharp bends and flow disturbance due toprobes).

Looking for better experimental conditions, Wattelet [5]designed a test bench in which some of the problems cited by Kan-dlikar [4] were avoided. A micropump with non metallic gear, forexample, was used to avoid contamination from oil. To increasethe range of evaporation temperature, Wattelet used a chiller sys-tem comprised of an ethylene glycol loop and a R-502 refrigerationsystem directly coupled with water supply [3]. Experimentalvalidations in the test section and pre-heater were also performed[5]. Using the improved experimental set up Wattelet collected520 experimental points for R-22 and R-134a [5,16]. In order toprovide a straightforward correlation with Wattelet’s data bank,

this work uses data from two similar apparatuses, which also usea chiller system and a micropump.

Fig. 2 shows the absolute percentage errors for all quality range,also presenting the CDF (Cumulative Distribution Function) forGW87, Wojtan, Wattelet and Kandlikar. It is possible to see thatGW87 has the best accuracy for both provided and independentdata, showing that 0.73 (73%) of the data have less than 20% ofabsolute percentage error; GW87 also provides the most homoge-neous distribution among all data. Wojtan does not predict datavery well, even for low quality. Similarly, Wattelet does not predictexperimental values from this work correctly, probably becausethey are out of the range he used in his work. However, consideringjust the data provided by Wattelet, good results are obtained forhis correlation, as expected. In terms of CDF results, Wattelet’s cor-relation for all data remains worse than Kandlikar, which is thesecond best.

These figures show that Gungor and Winterton (GW87) is thebest correlation for the data bank studied here, thus we use thiscorrelation as the starting point for the pure fluid HTC optimizationand HTC correction for blended refrigerants. Returning to the intro-duction section, Kandlikar also tried to optimize GW87 correlationwith no success [4]; in that opportunity, Kandlikar had little com-putational resources, a problem that does not limit the currentresearch.

5. Proposed correlations

This work proposes two new correlations that can be applied topure fluids and blended refrigerants. Given that GW87 presents thebest fitting results, as shown above, the new correlations follow asimilar functional form to GW87 modified by a factor A

htp ¼ hl Aþ 3000 Bo0:86 þ 1:12x

1% x

! "0:75 ql

qv

# $0:41" #

; ð9Þ

where the factor A is explained below in Section 6.In order to develop a HTC correlation for blended refrigerants

one must consider the HTC degradation. HTC degradation resultsfrom mass transfer resistance (a phenomenon discussed in greaterdetails by [21–36]. During phase change, the more volatile fluid

Table 2Heat transfer coefficient correlations used in this work.

Author Correlation

Gungor andWinterton [2]

htp ¼ hl 1þ 3000 Bo0:86 þ 1:12 x1%x

% &0:75 qlqv

! "0:41' (

Kandlikar [4] htp ¼ hl C1CoC2 ð25FrlÞC5 þ C3BoC4 Ffl

h i

Convective region:C ¼ 1:136;%0:9;667:2;0:7;0:3½ 'Nucleate boiling region:C ¼ 0:6683;%0:2;1058:0;0:7;0:3½ 'Ffl = 2.2 (R-22), 1.63 (R-134a), 1.55 (R-404A), 1.50(R-407C)

Wattelet [5]htp ¼ h2:5

ec þ h2:5nb

! "%2:5, where,

Fw ¼ 1þ 1:925v%0:83tt

if Frl 60.25, Rw ¼ 1:32Fr0:2l

if Frl > 0.25, Rw ¼ 1hec ¼ Fw Rw hl

Additionaldefinitions:

hl ¼ 0:023Re0:8l Pr0:4

lkld

Rel ¼ G dllð1% xÞ

hnb ¼ 55ðPrÞ0:12ð%log10ðPrÞÞ%0:55M%0:5ð/Þ0:67

Fig. 2. Quality and CDF (Cumulative Distribution Function) versus Absolute Percentage Error for the three different data set, from this work, from Wattelet [5] (Independentdata set (1) and from Wattelet [16] (Independent data set (2). Correlations used are: Gungor and Winterton [2] (upper left), Wojtan [8,9] (upper right), Wattelet [5] (lower left)and Kandlikar [4] (lower right).

580 M.P. Porto et al. / International Journal of Heat and Mass Transfer 85 (2015) 577–584

evaporates first and at a higher velocity rate than the less volatile.As consequence of this, the concentration of the higher boilingpoint fluid in the saturate blend increases. This change in compo-sition of the blended refrigerant causes an increase on the wallsuperheating for the same heat flux, resulting in a lower HTC.According to Venter [21] the degradation becomes more pro-nounced for higher pressure and heat flux.

In order to obtain the HTC correlations for blends that take intoaccount the HTC degradation authors [17,22,35,36] have used amethodology in which a reduction factor is applied to the correla-tions for pure fluids. The reduction factor applied on the nucleateboiling has the following functional form:

hnb;m

hnb;p¼

11þ K

; ð10Þ

where, hnb;m is the blend nucleate boiling HTC, hnb;p is the pure fluidnucleate boiling HTC, and K is the reduction or degradation factor.

Gorenflo [22] presented a review on pool boiling, showingaspects related to the degradation caused by blends on the nucle-ate boiling HTC, also recommending the following equations todetermine the reduction factor [37]:

K ¼ Tint % Tsð Þ=DTid; ð11Þ

and,

DTid ¼ _Q 00=hnb;p ¼X

xjDTj; ð12Þ

and,

Tint % Ts *X

Tsjðxj % yjÞð1% expð%B _Q 00=qbDhlvÞÞ; ð13Þ

where x and y are the molar fractions of liquid or vapor, b is themass transfer coefficient, q and Dhlv are density and heat of vapor-ization, B is a fitting parameter, and b is the mass transfer coeffi-cient. Approximating the term

PTsjðxj % yjÞ in Eq. (13) by the

temperature glide (dTglide) and evaluating B equals to the unity,we obtain the following relation [17]:

K ¼hnb;p

_Q 00dTglide 1% exp

% _Q 00

ql hlvb

!; ð14Þ

where b is fixed at 0.0003 m/s and DTbp is the total glide. The relationabove is presented by Thome in Ref. [17], and will be used here toevaluate the reduction on the nucleate boiling HTC for the R-407C.

Following this reasoning, the new correlation for blended refrig-erants proposed here, will follow almost the same form as the onein Eq. (9) modified to account for the HTC degradation

htp ¼ hl Aþ 3000Bo

1þ K

# $0:86

þ 1:12x

1% x

! "0:75 ql

qv

# $0:41" #

: ð15Þ

6. Model optimization

Following the discussion presented above and the availabledata, two HTC correlations for pure and blended refrigerants areproposed:

htp;1 ¼hl A11þ3000Bo0:86þ1:12 x

1%x

% &0:75 qlqv

! "0:41' (

; if pure refrigerant

hl A12þ3000 Bo1þK

! "0:86þ1:12 x

1%x

% &0:75 qlqv

! "0:41' (

; if blended

8>>><

>>>:

ð16Þ

htp;2¼hl A2þ3000Bo

1þK

# $0:86

þ1:12x

1%x

! "0:75 ql

qv

# $0:41" #

; ð17Þ

with K ¼ 0 for pure fluids.

The factors A11;A12 and A2 are optimized using a genetic algo-rithm as detailed in [10]. Following the methodology presentedin that work these factors can involve several dimensionless para-meters. The first group of dimensionless numbers are the para-meters used in Lima [13], as the enhancing factor (see correlationnumber 4, Table A.1). The second group of parameters takes theform of dimensionless numbers based on several propertiesd;U; k;l and ilv , and are determined through the Buckinham’s Ptheorem [12]. Other parameters consist of the dimensionless num-bers given by Wojtan et al. [8,9]. The parameters C1;C2 to C5 arealso used in order to consider the influence of the fluids’ criticalthermophysical properties in the HTC correlation. Table A.1 listsall the dimensionless parameters considered in this optimization.

Using these parameters and following [10] the factors A11 , A12

and A2 have the general functional form

a0

Xn1

i¼1

aiXbii

Yn2

j¼1

Xcjj ; ð18Þ

where the X’s are a subset of the dimensionless parameters inTable A.1, and the coefficients ai; bi and cj are real numbers deter-mined using an unconstrained line-search method [38] implement-ed in Matlab R2013b and accessible through the Optimizationtoolbox via the function fminunc. The optimization method searchesfor the combination of dimensionless parameters and the corre-sponding coefficients that minimize the root mean squared error(RMSE) between the experimental data, Eqs. (16) and (17):

argmin~x;~y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i¼1

htp;exp;i % htp;ið~x;~yÞ% &2

vuut ; ð19Þ

where htp;exp is the experimental HTC, and htp is the HTC computedby the new correlations defined by the vectors~x and~y. The vector~ycontains the coefficients ai; bi and ci, and the vector~x is a binary vec-tor whose elements determine the inclusion (xi ¼ 1) or exclusion(xi ¼ 0) of the dimensionless parameters Xi. Unlike for ~y, gradient-based optimization methods are not suited for determining theoptimal ~x because it is not possible to compute the gradient of Eq.(19) with respect to ~x. For this reason, we used GA to determinethe best parameter selection, i.e, the best ~x.

The GA is a space search technique based on the mechanism ofnatural selection and survival of the fittest. In this algorithm theevolution starts with a random population of individuals (deter-mined by vectors ~x). For each GA individual given by ~x we deter-mine the coefficients in ~y using the unconstrained line-searchmethod and compute the individual’s fitness as the RMSE betweenthe new correlation and the experimental data. The initial popula-tion is evolved based on the selection, crossover and mutationoperators with the objective to minimize RMSE. Crossover operateson individuals (parents) determined by the selection operator thatselects individuals for crossover based on fitness.

Table 3 lists the GA settings used in this work and describes theGA search space, i.e., the components of vector~x. The GA tools usedin this work are the ones implemented in the Global OptimizationToolbox (Version 3.2.2) in Matlab R2013b. Further details aboutthe GA optimization can be found in [10].

7. Results

The procedure presented above yields the following expressionsfor the factors A11;A12 and A2:

A11 ¼ a0 a1vb1tt þ a2Frb2

l þ a3Pb35 þ a4Cb4

2 þ a5Cb54 þ a6Bob6

! "

+ vc1tt Frc2

l Rec3l;2Prc4

eqPc59 Cc6

1 ; ð20Þ

M.P. Porto et al. / International Journal of Heat and Mass Transfer 85 (2015) 577–584 581

A12 ¼ a0 a1Pb13 þ a2Pb2

6 þ a3Cb32 þ a4Bob4

! "

+ vc1tt UCc2

1 Pc32 Pc4

3 Pc57 Pc6

9 Boc7 ; ð21Þ

A2 ¼ a0 a1vb1tt þ a2Pb2

5 þ a3Pb310 þ a4Cb4

2 þ a5Cb53 þ a6Cb6

5

! "

+ vc1tt Pc2

2 Pc39 Pc4

10Cc53 ec6 Boc7 : ð22Þ

With the corresponding coefficients listed in Table 4.It is an interesting fact that Martinelli’s parameter (vtt) appears

in all dimensionless terms proposed (Eqs. (20)–(22)), showing that

even considering this optimization process as a black box method,the theoretical relevance of Martinelli’s parameter is clearly evi-dent and demonstrated. Other parameters that are constantly pre-sent are the boiling number (Bo), critical property C2 (which isqv=qcrit) and the dimensionless term P9 (which is dg=ilv).

Fig. 3 presents results for the two proposed correlations (Eq.(16) in the left side and Eq. (17) in the right side) againstexperimental data (from this work and independent), in terms ofabsolute percentage error and cumulative distribution function.Comparing Fig. 3 with results presented in the Fig. 2 it is possibleto see that much better results are reached with the optimized

Table 3Description of GA settings and search space ~x.

GA settings

Selection Stochastic uniform

Cross-over Scatter Randomly selects entries from the parentsMutation Uniform 5% mutation probabilityElitism Yes 2 individualsPopulation size 200Max. generations 100Initial population Random

GA search space (description of ~x)Coef. No. elements Description Data used

A11 52 Binary values for input selection R-22, R-134a, R-404AA12 52 Binary values for input selection R-407CA2 52 Binary values for input selection R-22, R-134a, R-404A, R-407C

Table 4Optimized coefficients for Eqs. (16) and (17).

Eq. Coef. 0 1 2 3 4 5 6 7

ai 3:044+ 10%1 1:828+ 10%3 3:262+ 10%1 %7:465+ 10%1 %3:563+ 10%1 2:728+ 10%1 4:002 –Eq. (20) bi – 1:137 5:467+ 10%2 1:730+ 10%1 %1:145+ 10%2 %5:891+ 10%2 1:035 –

ci – %1:021 %9:987+ 10%1 1:653 3:981+ 10%1 9:560+ 10%1 %8:029+ 10%1 –ai 1:318 %8:965 8:234 1:264+ 10%1 1:263+ 10%6 – – –

Eq. (21) bi – %1:181+ 10%1 5:200+ 10%2 1:750 %1:651 – – –ci – 2:605 5:669 8:347+ 10%1 %4:439 %6:350 1:369 1:667ai 6:425+ 10%1 3:848+ 10%1 %1:730 1:559 1:782 2:327+ 10%1 %2:713 –

Eq. (22) bi – %5:732+ 10%2 4:217+ 10%1 %6:662+ 10%1 %1:199+ 10%2 %2:128+ 10%1 2:737+ 10%1 –ci – %6:839+ 10%1 %4:231+ 10%1 %9:379+ 10%1 %8:425+ 10%1 %4:572+ 10%1 %1:722 5:173+ 10%1

Fig. 3. Quality and CDF (Cumulative Distribution Function) versus Absolute Percentage Error for the three different data set, from this work, from Wattelet [5] (Independentdata set (1) and from Wattelet [16] (Independent data set (2). Correlations used are: (left) Eq. (16), and (right) Eq. (17).

582 M.P. Porto et al. / International Journal of Heat and Mass Transfer 85 (2015) 577–584

correlations: 73% of experimental points have less than 20% ofabsolute deviation for GW87, whereas 94% of experimental pointshave less than 20% of absolute error for the Eq. (16) and 87% in 20%for the Eq. (17). It was expected a better accuracy for Eq. (16) com-pared to Eq. (17), because the first uses different correlations forpure and blended refrigerants, and the latter uses the same forboth.

Fig. 4 indicates the same results presented in Fig. 3, but in thisfigure the results are presented separately for the different fluids.It is not difficult to see in Fig. 4 that Eq. (16) works better forR-407C than Eq. (17). Mainly for low quality, Eq. (16) is much moreaccurate than Eq. (17) for all fluids. Fluids R-22 and R-134a showsimilar results for Eqs. (16) and (17). This fact is explained by thegreater proportion in experimental data for these two fluids whencompared to R-404A and R-407C.

Fig. 5 shows the scatter plots for all experimental HTC (leftside), and for R-407C, only (right side). In either graph, Eq. (16)and Eq. (17) provide a deviation within 20% for great part of thedata set.

8. Conclusions

This work introduces optimization techniques for pure andmixed refrigerants to improve the absolute error of classical HTCcorrelations. A total of 1888 experimental data points from this

work and from other independent sources were used to developedthe optimized correlations presented here.

The classical correlations of Gungor and Winterton (GW87),Wattelet, Kandlikar and Wojtan were tested against theexperimental data set. Our analysis showed that GW87 is the mostaccurate correlation overall, and therefore the GW87 functionalform was selected as the basis for the optimized correlations pre-sented here. In the case of pure fluids, the optimization used theclassical form of GW87, but for the blended refrigerant (R-407C)the GW87 correlation was modified to include a reduction factorlinked to the boiling number. We also added a dimensionless groupnumber to complement the modified version of GW87. This groupwas optimized for two distinct approaches. The first approach usesdifferent correlations for pure and blended refrigerants, and thesecond approach uses the same correlation for both.

Comparing with the GW87 classical relation, both approachesdescribed above yield substantial reductions in error. The firstmodified relation yields 94% of experimental data points with lessthan 20% of absolute error, whereas the classical GW87 correlationonly captures 73% of the data points for the same deviation. Thesecond approach (common correlation for pure and mixed refriger-ants) captures 87% of experimental data points with within the 20%of absolute error deviation threshold.

The first approach yields much less dispersion in the results forall fluids, whereas the second is more dispersive mainly for theregimes characterized by low vapor quality, where nucleate boilingis more relevant, and the factor K (heat transfer degradation) ismore relevant. The nucleate boiling degradation factor (K) alsoinfluences the selection of dimensionless terms in Table A.1, whichin turn increase the error for pure refrigerants.

Conflict of interest

None declared.

Acknowledgements

This project has been supported financially by Fapemig – Fun-dação de Amparo à Pesquisa do Estado de Minas Gerais.

Appendix A. Input parameters

The dimensionless parameters considered in the optimizationof Eqs. (16) and (17) are listed in Table A.1.

Fig. 4. Quality and CDF (Cumulative Distribution Function) versus Absolute Percentage Error for the four fluids used in this work: R-22, R-134a, R-404A and R-407C.Correlation used are: (left) Eq. (16), and (right) Eq. (17).

Fig. 5. Scatter plots of experimental HTC versus calculated HTC. Left, all data set forR-22, R-134a, R-404A and R-407C. Right, data set for R-407C, only.

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Table A.1Input parameters considered in the search for the best HTC correlation.

Xi Relation Range Xi Relation Range

1 Rel;1 ¼ Gð1% xÞ dll

½83:0134;3:7733 104' 15 P8 ¼ di0:5lv

qvleq

½1:0818 105;9:9478 106'

2 vtt ¼ 1%xx

% &0:875 qvql

! "0:5 lllv

! "0:125 ½0:0031;3:0428' 16 P9 ¼ dgilv

½3:2746 10%7;8:0545 10%7'

3 Frl ¼ G2

q2l g d

½0:0051;8:7261' 17 P10 ¼ UvUl

½27:8290;163:9678'

4 UC1 ¼ 1þ 1:893v%0:77tt ½1:8036;163:6965' 18 C1 ¼ P

Pcrit½0:0399;0:2106'

5 Rel;2 ¼ qlUld

leq½1:1147 103;2:7533 105' 19 C2 ¼

qvqcrit

½0:0140;0:0705'

6 Preq ¼ leqcpeqkeq

½0:2087;24:8271' 20 C3 ¼qeqqcrit

½0:1364;2:6029'

7 P1 ¼ ilvU2

l½2:8933 105;6:0352 108' 21 C4 ¼ ql

qcrit½1:9623;2:7090'

8 P2 ¼ rUlleq ½118:8090;7:2535 103' 22 C5 ¼ Tl

Tcrit½0:6890;0:8151'

9 Rel;3 ¼ qlUvd

leq½1:0051 105;1:2914 107' 23

! ¼ xqv½1þ 0:12ð1% xÞ' x

qvþ 1%x

ql

! "+ 1:18ð1%xÞ½grðql%qv Þ'

0:25

Gq0:5l

* +%1 ½0:4588;0:9986'

10 P3 ¼ dgU2

l

½0:1146;197:9605' 24xIA ¼ f½0:291ðqv

qlÞ%0:571ðlv

llÞ%0:143' þ 1g

%1 ½0:2336;0:4259'

11 P4 ¼di0:5lv qlleq

½1:6403 107;3:2880 108' 25 Qcrit ¼0:131q0:5

v ilv ½grðql%qv Þ'0:25

Wm%2½2:8843 105;4:8149 105'

12 P5 ¼ Ul

i0:5lv½4:0706 10%5;0:0019' 26 Bo ¼ /

Gilv½4:7755 10%5;0:0013'

13 P6 ¼ ri0:5lv leq

½0:0944;1:0697'

14 P7 ¼ Uvi0:5lv

½0:0030;0:1159'

1–6 dimensionless parameters from [14].7–17 see Ref.[12].18–22 critical properties from [10].23–25 see Ref.[9].26 boiling number.

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