Robust and General Model to Forecast the Heat Transfer Coefficient for Flow Condensation in Multi Port Mini/Micro-Channels

Abstract

A general correlation for predicting the two-phase heat transfer coefficient (HTC) during condensation inside multi-port mini/micro-channels was presented. The model was obtained by correlating the two-phase multiplier, ${\phi }_{tp}$ with affecting parameters using the genetic programming (GP) method. An extensive database containing 3503 experimental data samples was gathered from 21 different sources, including a broad range of operating parameters. The newly obtained correlation fits the broad range of measured data analyzed with an average absolute relative deviation (AARD) of 16.87% and estimates 84.73% of analyzed data points with a relative error of less than 30%. Evaluation of previous correlations was also conducted using the same database. They showed the AARD values ranging from 36.94% to 191.19%. However, the GP model provides more accurate results, AARD lower than 17%, by considering the surface tension effects. Finally, the effect of various operating parameters on the HTC was studied using the proposed correlation.

1. Introduction

Condensers play important roles in different systems, such as air conditioning [1,2], heat pumping [3], refrigeration [4,5], nuclear [6,7], chemical [8,9], etc. Nowadays, condensers with mini/micro-channels are used instead of conventional channels in many industries [10,11,12,13,14,15]. The main reasons are reducing the size of the condensers, refrigeration charge, required space, and overall cost of the process [16]. Therefore, a lower condenser’s weight and size are required for a given heat duty in this type of channel. The multi-port mini/micro-channels can raise the contact region between the condenser and the working fluids and leads to a higher evaporating rate as an internal fin [17]. Furthermore, the influence of surface tension becomes important in the smaller channels. That is, the surface tension can decrease the liquid film thickness during condensation in these channels. Furthermore, heat exchange between parallel channels can lead to a difference in HTC in multi-port channels from conventional ones.

1.1. Previous Works

1.1.1. Channels’ Classifications

Several classifications are suggested for channels based on their characterizations [18,19,20,21,22,23,24,25,26,27]. Some of these classifications are based on channel diameters, and the others are based on the Bond number (Bo). An extensive review of these classifications is presented in [28]. The most commonly used classification was developed by Kandlikar [19]. According to this classification, the channels by hydraulic diameters lower than 0.2 mm are classified as micro-channels. The mini-channels have a hydraulic diameter (HD) between 0.2 to 3 mm, and the conventional channels are ones with HD of more than 3 mm. Through the classification suggested by Mehendale et al. [26], channels with $D<6$ mm are mini-channels, and the micro-channels have an HD of less than 0.1 mm. Cheng and Wu [21] classified the channels based on the Bond number. They indicated that the buoyancy forces overcome the surface tension in the conventional channels, and these channels have a Bond number higher than 3. Furthermore, the surface tension effects could overcome the buoyancy forces in micro-channels. Therefore, the Bond number of less than 0.5 is defined for micro-channels. In addition, the mini-channels are those with a Bond number between 0.5 to 3. The classification suggested by Kandlikar [19] is used in the present study for defining the mini and micro-channels.

1.1.2. Experimental Research

Several experimental works have studied condensation heat-transfer in mini/micro-channels [16,29,30,31,32,33,34,35]. However, there are only limited works focusing on heat transfer in multi-port channels. Garimella et al. [36] investigated the condensation HTC of R134a in rectangular micro-channels with a HD between 0.1 to 0.16 mm and an aspect ratio between 1 to 4. They observed that similar to conventional channels, the HTC in multi-port channels increased with increasing vapor quality. The authors also compared measured data by the available correlations. All of the correlations showed large deviations and under-predicted the measured data. A similar study was conducted by Fronk and Garimella [37] for the HTC of carbon dioxide. Their result showed that the multi-port channel with a 0.1 mm HD had the highest HTC at a constant mass flux and saturation temperature. Jige et al. [38] studied the heat transfer of different refrigerants at different saturation temperatures in multi-port channels. They indicated that the HTC raises with raising the saturation temperature. In addition, their results showed that the HTC of R32 was more than R134a and R1234ze(E).

1.1.3. Previous Models for Condensation HTC

There are different models in the literature for approximating the HTC during condensation inside single and multi-channels. Some of these models have been improved by the experimental data for limited operating conditions [39,40,41,42,43]. The single-phase HTC equation developed by Dittus and Boelter [44] has been widely used as the basic equation for estimating the HTC in different studies. Based on this equation, the single-phase Nusselt number, $N{u}_{sp}$, is a function of single-phase Reynolds and Prandtl numbers:

$$N{u}_{sp}=\frac{h_{sp}D}{k_l}=0.023R{e}_l^{0.8}P{r}_l^{0.4}$$

(1)

In many previous studies, the two-phase Nusselt number, $N{u}_{tp}$, was obtained by multiplying a two-phase correction factor by the single-phase HTC,

$$N{u}_{tp}=N{u}_{sp}{\phi }_{tp}$$

(2)

Here, ${\phi }_{tp}$ is the two-phase correction factor. The values of the two-phase correction factor, ${\phi }_{tp}$, depend on several parameters, and different studies have been conducted to estimate this factor. Shah [45] presented the following simple expression for estimating the two-phase correction factor during film condensation in pipes,

$${\phi }_{tp}=1+3.8{\left[{\left(\frac{1}{x}-1\right)}^{0.8}{P}_{red}^{0.4}\right]}^{0.95}$$

(3)

Based on this correlation, only the vapor quality and the reduced pressure can affect the ${\phi }_{tp}$. Cavallini et al. [46] proposed a general correlation for various fluids. They indicated that the vapor quality, liquid $Pr$, viscosity, and density of the liquid/vapor are the important parameters for estimating ${\phi }_{tp}$. In addition, they reported that the HTC is related to the wall temperature difference, when $J_g>J_g^T$. The transition line for dimensionless vapor velocity, $J_g^T$ was defined as following equation,

$$J_g^T={\left[{\left(\frac{7.5}{4.3 X_{tt}^{1.111}+1}\right)}^{-3}+C_T^{-3}\right]}^{-\frac{1}{3}}$$

(4)

where $C_T$ value is 1.6 for hydrocarbons and 2.6 for other refrigerants.

Dobson and Chato [41] suggested the use of the Lockhart–Martinelli Parameter, $X_{tt}$ for predicting the two-phase correction factor in annular flow,

$${\phi }_{tp}=1+\frac{2.22}{X_{tt}^{0.89}}$$

(5)

There are several general correlations for estimating the two-phase HTC during condensation that was developed on the basis of large datasets, which are applicable for both single and multi-port channels [47,48,49,50,51]. Kim and Mudawar [52] proposed a correlation approach on the basis of 4045 data from 28 sources for single- multi port mini-channels with 0.424 to 6.22 mm hydraulic diameters. The analyzed database included 1964 data points for single-port channels and 2081 data points for multi-port channels. They suggested two separated correlations for slug/bubbly and annular flows. The transition line between these regimes was defined as follows,

$$W{e}^{*}>7X_{tt}{}^{0.2}$$

(6)

where modified Weber number, $W{e}^{*}$, can be calculated as Soliman [53] method,

$$W{e}^{*}=2.45\frac{R{e}_v^{0.64}}{S{u}_{vo}^{0.3}{\left(1+1.09X_{tt}^{0.039}\right)}^{0.4}} for R{e}_l\le 1250$$

(7)

$$W{e}^{*}=0.85\frac{R{e}_v^{0.79}{X}_{tt}^{0.157}}{S{u}_{vo}^{0.3}{\left(1+1.09X_{tt}^{0.039}\right)}^{0.4}}{\left[\left(\frac{{\rho }_l}{{\rho }_v}\right){\left(\frac{{\mu }_v}{{\mu }_l}\right)}^2 \right]}^{0.084} for R{e}_l>1250$$

(8)

In the general correlation developed by Dorao and Fernandino [49], only two parameters were used to estimate the $N{u}_{tp}$. The correlation was suggested for horizontal mini/macro-channels with 0.067 to 20 mm diameters. This model presented two different regimes, i.e., high and low mass fluxes (or vapor qualities) for HTC. They indicated that the pipe diameter effect becomes significant when the two-phase Reynolds number, $R{e}_{tp}$ is greater than transient value, $R{e}_{tp}^T$,

$$R{e}_{tp}^T={10}^8{D}^{1.5}$$

(9)

Among the data points analyzed by Dorao and Fernandino [49], only the data from Derby et al. [54] and Wang et al. [55] were for multi-port channels, and the model is more suitable for single-port channels. This point was confirmed by Hosseini et al. [56], which compared the model of Dorao and Fernandino [49] with a large data for single-port channels and observed a good agreement with the experimental values.

Shah [57] presented general correlations based on 5100 data points for different flow regimes and channel orientations. The analyzed data contained single and multi-port channels with HDs from 0.08 mm to 49 mm. This model provided an improvement for estimating the HTC of hydrocarbons. In addition, the author suggested predictive methods for low Froude and Reynolds numbers. It should be mentioned that the machine learning algorithms have been extensively used in various applications such as biomedical [58], clinical [59,60], solar power prediction [61], food production [62], etc., for sustainable solutions. Zhou et al. [63] used the machine learning algorithms for estimating the HTC in mini/micro-channels using 4882 data points. Among the different methods tried by them, the artificial neural network (ANN) with 15 hidden layers and 22 adjusted parameters showed the best agreement with measured data with a total AARD of 6.80%. Recently, Hosseini et al. [56] developed a general model using the genetic programming (GP) approach for predicting the condensation HTC in single channels, with 5728 data points for a broad range of operating conditions. The model predicted the analyzed data points with a good AARD value of 17.82%. In addition, they used a separate database including 712 data points from 6 independent sources for validating the developed model. The results showed that the proposed model provides the best accuracy for single-port channels.

1.1.4. Contributions of the Present Study

It is seen that the previous researchers used the same methods for estimating the HTC during condensation in single and multi-port channels. Therefore, these correlations estimate the same values for these cases. However, as discussed previously, the HTC in parallel channels differs from that for single-port ones because of the contact area effects, surface tension forces, and heat exchange between parallel channels. In the previous work [56], a general correlation was developed for the HTC inside single port mini/micro and macro-channels using an extensive database. The basic objective of this study is to extend the former attainments by developing a general model for the HTC in flow condensation. Therefore, an extensive database containing 3503 data points was gathered from 21 independent studies, covering a broad range of operating conditions and working fluids. In addition, the intelligent and powerful method of genetic programming (GP) was employed for improving a mathematical equation for HTC during the condensation. Furthermore, the performance of previous correlations in estimating the HTC in multi-port channels was also evaluated using the same database. Finally, the predictive trends of the new model were evaluated, and the impact of various parameters on the HTC was discussed.

2. Materials and Methods

2.1. Genetic Programming

Recently, using intelligent methods for modeling, the thermal and hydrodynamic characterizations of different systems have received much attention [63,64,65,66,67]. In the present paper, GP was used for obtaining a general nonlinear correlation for predicting the HTC during the condensation of fluids inside multi-port mini/micro-channels. The flowchart of the present GP model is shown in Figure 1.

Genetic Programming (GP) is based on genetic algorithms thus it comprises a population. An equation in the population, a member of the population, is presented as a tree structure, thus that its branches contain arithmetic functions (i.e., sin, cos, +, *, etc.), and also its leaves are terminal points (i.e., variables). The initial population has a set of equations that each represents an individual. Combining the individuals of the previous population generates the next population. As such, the new equations are produced by applying the crossover and mutation operations to the equations from the last generation in a way that combines the individuals of the previous generation and creates the next individuals of the new generation. A crossover process operates when two individuals are selected based on their evaluations determined by the objective function, and a mutation process operates on a small percentage of the individuals that result from a crossover. Once the individuals of the next generation meet the problem criteria, the GP process is halted.

This method presents an explicit mathematical model with high accuracy for predicting the experimental data. Our previous studies [68,69,70] used this method to develop general models with high accuracy for thermal and hydrodynamic characterizations of different systems.

2.2. Experimental Data Samples

An extensive database containing 3503 HTC data samples during condensation inside multi-port mini/micro-channels was collected from 21 independent studies, which covered a wide range of working fluids, HDs, mass fluxes, saturation temperatures, and thermophysical properties. The operating parameters for each analyzed source are summarized in

Table 1. The database sources for the HTC in multi-port mini/micro-channels.

References	Fluid	Channel Geometry	Hydraulic Diameter (mm)	Mass Flux $(\mathbf{k}\mathbf{g} {\mathbf{m}}^{-2}{\mathbf{s}}^{-1})$	Reduced Pressure (-)	Number of Points
Agarwal [72]	R134a	Rectangular, Square	0.1 to 0.16	300 to 800	0.19 to 0.42	291
Belchi et al. [73]	R290	Square	1.16	175 to 350	0.25 to 0.40	100
Belchi et al. [74]	R32, R410A	Square	1.16	470 to 710	0.33 to 0.63	163
Agarwal et al. [75]	R134a	Barrel, N-shape, Rectangular, Square, Triangular	0.424 to 0.839	150 to 750	0.37	152
Belchi et al. [76]	R32, R410A	Square	1.16	350 to 800	0.33 to 0.63	515
Fronk and Garimella [37]	R744	Rectangular	0.1 to 0.16	400 to 800	0.69 to 0.87	189
Bandhauer et al. [77]	R134a	Circular	0.506 to 1.524	150 to 750	0.41	128
Cavallini et al. [78]	R134a, R410A	Circular	1.4	200 to 1400	0.25 to 0.5	61
Belchi [79]	R1234yf, R134a	Square	1.16	470 to 710	0.25 to 0.49	81
Belchi et al. [80]	R32, R410A	Square	1.16	475	0.38 to 0.49	88
Derby et al. [54]	R134a	Square	1	75 to 450	0.22	80
Andresen [81]	R410A	Circular	0.76 to 1.52	200 to 800	0.80 to 0.90	198
Heo and Yun [82]	R744	Rectangular	0.68 to 1.5	400 to 800	0.41 to 0.54	203
Gomez et al. [83]	R1234yf	Square	1.16	350 to 945	0.23 to 0.43	162
Jige et al. [38]	R1234ze(E), R134a, R32	Rectangular	0.76 to 1.06	100 to 400	0.21 to 0.43	323
Park et al. [84]	R1234ze(E), R134a, R236fa	Rectangular	1.45	100 to 260	0.13 to 0.44	97
Pham et al. [85]	R22, R290, R32, R410A	Rectangular	0.83	50 to 500	0.37 to 0.60	79
Park and Hrnjak [86]	R744	Rectangular	0.89	200 to 800	0.23 to 0.40	112
Li et al. [87]	R1234ze(E), R134a, R32, R32/R134a (24.5/75/5%), R32/R134a (51/49%)	Circular	0.86	100 to 300	0.18 to 0.42	151
Wang et al. [55]	R134a	Circular	1.46	150 to 750	0.45	279
Rahman et al. [88]	R134a	Rectangular	0.81	50 to 200	0.19 to 0.22	51
Total						3503

. All thermophysical characteristics of the fluids are computed with the REFPROP 9.1 software from NIST [71] at the saturation conditions of the working fluids. The analyzed data were gathered from experimental studies of average Nu_tp numbers for refrigerants condensation in multi-port mini/micro-channels. Most of these data are available as figures from the database sources listed in Table 1, which have been extracted using the GetData Graph Digitizer.

In addition to the range of the non-dimensional parameters used,

Table 2. Operating conditions, geometric dimensions, and the range of dimensionless factors used in the present study.

Parameter	Type of Refrigerant/Operating Conditions/Channel Geometry/Dimensionless Factors
Fluids	R744, R1234yf, R1234ze(E), R134a, R22, R236fa, R290, R32, R32/R134a (24.5/75.5%), R32/R134a (51/49%) and R410A
$G, (\mathrm{k}\mathrm{g} {\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$)	50–1400
$D, \left(mm\right)$	0.1–1.524
Channel geometry	Rectangular, circular, square, barrel, N-shaped and Triangular mini/micro channels
Reduced pressure, (-)	0.13–0.90
Vapor quality, x, (-)	0.002–0.978
$R{e}_l$, (-)	11–16886
$P{r}_l$, (-)	1.75–4.69
$X_{tt}$, (-)	0.0088–105.26
$Bo$, (-)	0.015–31.36

sets out the refrigerant types, channel geometry and dimension, and the operating conditions for the analyzed data. The range of dimensional and dimensionless factors is presented in Table 2. The restrictions of applicability of the new correlation for approximating the HTC during flow condensation inside multi-port channels are shown in this table.

In this paper, the term ${\phi }_{tp}$ given by Equation (2) is obtained using the method of GP. The affecting parameters on ${\phi }_{tp}$ in multi-port channels, such as vapor quality, saturation pressure, mass flux, surface tension, channel diameter, and thermophysical properties of the working fluids are considered for developing the new model. These parameters are employed to improve a general correlation for approximating the HTC using the GP. The two-phase multiplier is proposed using the following dimensionless expression,

$${\phi }_{tp}=f\left(R{e}_l, P{r}_l, X_{tt}, Bo, x\right)$$

(10)

2.3. Error Analysis

AARD is used for evaluating the performance of different correlations in this study. Furthermore, the arithmetic average deviation (AAD) is used to assess the over-estimation or under-estimation of the models. These error estimates can be calculated as,

$$AAD=\frac{1}{n}{\sum }_{i=1}^n\left(\frac{N{u}_{tp, calc}-N{u}_{tp, exp}}{N{u}_{tp, exp}}\right)\times 100$$

(11)

$$AARD=\frac{1}{n}{\sum }_{i=1}^n\left|\frac{N{u}_{tp, calc}-N{u}_{tp, exp}}{N{u}_{tp, exp}}\right|\times 100$$

(12)

3. Results and Discussion

The former studies showed that the relationships between the two-phase multiplier and the affecting factors were nonlinear. In the present study, a powerful GP method was introduced for developing a nonlinear equation for estimating the two-phase multiplier. Among the many correlations generated by the GP, the best correlation is given by,

$$\begin{array}{ll}{\phi }_{tp}=0.87C_1& +0.63x\\ & +\tan \left(x\right)\sin \left(9783.5P_{red}\right)\\ & +0.25C_1\sin \left(9118P_{red}\right)x\\ & +\tan \left(\tan \left(x\right)\right)\sin \left(\sin (9640.4P_{red}\right))\end{array}$$

(13)

where

$$C_1=0.73\left(1+\frac{1}{X_{tt}}\right)+\frac{6.6}{X_{tt}R{e}_l}+\frac{2.93}{X_{tt}\exp \left(P{r}_l^{2}Bo\right)}+\frac{1380.9+1211.7\sin \left(0.78P{r}_l^2\right)}{R{e}_l}$$

(14)

We see that Equation (13) describes a nonlinear correlation, consisting of all affecting dimensionless factors.

To show the performance of the established model for estimating the HTC, the predicted Nusselt number values were compared with the measured values in Figure 2. This result verifies that the GP correlation predictions given by Equation (13) were in the desired range, and most of the data were bounded with an error of less than $\pm$30%. The calculations showed the suggested correlation estimates of the 3503 data points with an AARD value of 16.87%. In addition, predicted data with an error of less than 20% and 30% for the GP correlation were 70.37% and 84.73%, respectively. These results showed excellent accuracy of the correlation obtained by the GP for approximating the condensation HTC inside multi-port channels.

As noted before, there are various correlation approaches for predicting the condensation HTC. In most of the previous correlations, the authors did not consider the difference between single-port and multi-port channels in the prediction of the HTC. A list of several well-known correlations for estimating the HTC inside single and multi-port channels is presented in

Table 3. Correlations for predicting the HTC during condensation.

Reference(s)	Correlation	Remarks
Kim and Mudawar [52]	$for annular flow \left(W{e}^{*}<7X_{tt}\right):$ $N{u}_{tp}=0.048{\left(R{e}_l\right)}^{0.69}{\left(P{r}_l\right)}^{0.34}\frac{{\varphi }_v}{X_{tt}}$ $for slug and bubbly flow \left(W{e}^{*}>7X_{tt}\right):$ $N{u}_{tp}={\left[{\left(0.048{\left(R{e}_l\right)}^{0.69}{\left(P{r}_l\right)}^{0.34}\frac{{\varphi }_v}{X_{tt}}\right)}^2+{\left(3.2\times {10}^{-7}{\left(R{e}_l\right)}^{-0.38}{\left(S{u}_{vo}\right)}^{1.39}\right)}^2\right]}^{0.5}$ $W{e}^{*}=2.45\frac{R{e}_v^{0.64}}{S{u}_{vo}^{0.3}{\left(1+1.09X_{tt}^{0.039}\right)}^{0.4}} for R{e}_l\le 1250$ $W{e}^{*}=0.85\frac{R{e}_v^{0.79}{X}_{tt}^{0.157}}{S{u}_{vo}^{0.3}{\left(1+1.09X_{tt}^{0.039}\right)}^{0.4}}{\left[\left(\frac{{\rho }_l}{{\rho }_v}\right){\left(\frac{{\mu }_v}{{\mu }_l}\right)}^2 \right]}^{0.084} for R{e}_l>1250$ ${\varphi }_v=1+CX+X^2 S{u}_{vo}=\frac{{\rho }_v\sigma D}{{\mu }_v^2}$ C and X are calculated by Kim and Mudawar [89] method.	General correlations for condensation in mini/micro-channels.
Dorao and Fernandino [49]	$N{u}_{tp}={\left(N{u}_I^9+N{u}_{II}^9\right)}^{\frac{1}{9}}$ $N{u}_I=0.023 R{e}_{tp}^{0.8} P{r}_{tp}^{0.3}$ $N{u}_{II}=41.5 D^{0.6} R{e}_{tp}^{0.4} P{r}_{tp}^{0.3}$	General correlations for condensation in horizontal tubes.
Shah (2016) [90]	$h_{tp}=h_I For J_g\ge 0.98{\left({\left(\frac{1}{x}-1\right)}^{0.8}{P}_{red}^{0.4}+0.263\right)}^{-0.62}and W{e}_{vo}\ge 100$ $h_{tp}=h_{Nu} For J_g\le 0.95{\left(2.27{\left({\left(\frac{1}{x}-1\right)}^{0.8}{P}_{res}^{0.4}\right)}^{1.249}+1.254\right)}^{-1}$ $h_{tp}=h_{Nu}+h_I For other regimes$ $h_I=0.023R{e}_{lo}{}^{0.8}P{r}_l{}^{0.4}\left[1+1.128x^{0.8170}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.3685}{\left(\frac{{\mu }_l}{{\mu }_g}\right)}^{0.2363}\times {\left(1-\frac{{\mu }_g}{{\mu }_l}\right)}^{2.144}P{r}_l{}^{-0.100}\right]\left(\frac{k_l}{D}\right)$ $h_{Nu}=1.32R{e}_l{}^{\frac{-1}{3}} {\left[ \frac{{\rho }_l\left({\rho }_l-{\rho }_v\right)g{k}_l^3}{{\mu }_l^2} \right]}^{\frac{1}{3}}$ $J_g=\frac{xG}{{\left(gD{\rho }_v\left({\rho }_l-{\rho }_v\right)\right)}^{0.5}} W{e}_{vo}=\frac{G^{2}D}{{\rho }_v\sigma }$	General correlations for condensation in horizontal mini/micro-tubes.
Shah (2019) [57]	$For J_g\ge 0.98{\left({\left(\frac{1}{x}-1\right)}^{0.8}{P}_{red}^{0.4}+0.263\right)}^{-0.62},F{r}_l>0.012 and W{e}_{vo}\ge 100:$ $h_{tp}=h_I$ $For J_g\le 0.95{\left(2.27{\left({\left(\frac{1}{x}-1\right)}^{0.8}{P}_{res}^{0.4}\right)}^{1.249}+1.254\right)}^{-1}and F{r}_l>0.012$ $h_{tp}=h_{Nu}$ $For other regimes :$ $h_{tp}=h_{Nu}+h_I$ $F{r}_L=\frac{G^2}{{\rho }_l^{2}gD}$ $h_I$$, h_{Nu}$$, J_g$$\mathrm{and} W{e}_{vo}$ are calculated by Shah (2016) method.
Akers et al. [91]	$N{u}_{tp}={\mathit{CPr}}_{\mathrm{l}}^{\frac{1}{3}}{\left\{G\left[\left(1-x\right)+x{\left(\frac{{\rho }_{\mathrm{l}}}{{\rho }_V}\right)}^{0.5}\right]\frac{D}{{\mu }_{\mathrm{l}}}\right\}}^n$ $C=0.026, n=0.8{ \mathit{for} \mathit{Re}}_{\mathit{eq}}>50000$ $C=5.3, n=\frac{1}{3}{ \mathit{for} \mathit{Re}}_{\mathit{eq}}<50000$ $R{e}_{eq}=G\left[\left(1-x\right)+x{\left(\frac{{\rho }_l}{{\rho }_v}\right)}^{0.5}\right]\left(\frac{D}{{\mu }_l}\right)$	Condensations for horizontal plain tubes
Crosser [92]	$N{u}_{tp}=0.0265P{r}_l{}^{\frac{1}{3}} {\left(G\left(x{\left(\frac{{\rho }_l}{{\rho }_v}\right)}^{0.5}\right)\frac{D}{{\mu }_l}\right)}^n$ $n=0.8 for \frac{Gx}{{\mu }_l}>60000$ $n=0.2 for \frac{Gx}{{\mu }_l}<60000$
Bohdal et al. [93]	$N{u}_{tp}=25.084 R{e}_l{}^{0.258}P{r}_l{}^{-0.495} P_{res}{}^{-0.288}{\left(\frac{x}{1-x}\right)}^{0.266}$	R134a and R410A condensation in mini-channels
Hosseini et al. [56]	$\mathit{For} G\le 200 \mathrm{k}\mathrm{g} {\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$: $N{u}_{tp}=0.0022\times R{e}_{tp}\times \left(\frac{\rho l-\rho v}{\rho l}\right)+0.0342\times W{e}_{vo}\times {\left(\frac{\rho l-\rho v}{\rho l}\right)}^2+\frac{Sin\left(39.8963\times P_{red}\right)-Ln\left(W{e}_{vo}\right)}{-0.0298-0.2203\times F{r}_l}-P{r}_{tp}$  $\mathit{For} G>200 \mathrm{k}\mathrm{g} {\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$: $$\begin{gathered} N{u}_{tp}=ABS\left(0.0169\ast R{e}_{tp}^{0.8620}-0.00146\ast \frac{R{e}_{tp}}{P{r}_{tp}\ast \left(\frac{\rho l-\rho v}{\rho l}\right)}+ \\ \frac{0.0036+0.0171\ast W{e}_{vo}}{F{r}_l}+17.9480\ast \sin \left(\frac{0.0036+0.0171\ast W{e}_{vo}}{F{r}_l}\right)+ \\ \tan \left(27.6370106546243\ast \left(\frac{\rho l-\rho v}{\rho l}\right)\right)-\tan \left(369.8572+ \\ \sin \left(\frac{0.0036+0.01712\ast W{e}_{vo}}{F{r}_l}\right)\right)\right) \end{gathered}$$	General correlations for condensation in single-port horizontal plain tubes

.

For evaluating the capability and generality of the new correlation developed by the GP method, a comparative investigation is presented in this section.

Table 4. AARD and AAD analysis of different correlations for various data sources.

	Model $\mathit{A}\mathit{A}\mathit{R}\mathit{D} \mathit{\%}$ $\mathit{A}\mathit{A}\mathit{D} \mathit{\%}$
References	Akers et al. [91]	Crosser [92]	Bohdal et al. [93]	Shah (2016) [90]	Shah (2019) [57]	Dorao and Fernandino [49]	Hosseini et al. [56]	Kim and Mudawar [52]	GP correlation
Agarwal [72]	44.06 41.63	83.56 −83.56	40.09 36.40	68.35 −68.35	68.35 −68.35	70.93 −70.93	76.33 −76.33	72.94 −72.94	17.91 −4.55
Belchi et al. [73]	144.97 144.97	30.61 −30.48	128.72 128.72	21.23 18.65	21.23 18.65	15.00 10.95	24.72 1.19	16.38 11.37	17.54 −7.91
Belchi et al. [74]	184.29 184.29	20.74 −5.72	190.63 190.63	60.45 60.45	60.45 60.45	59.83 59.83	63.26 63.26	41.05 41.05	16.41 14.16
Agarwal et al. [75]	143.40 143.25	47.80 −47.80	100.93 100.27	22.77 −1.92	22.77 −1.92	23.80 −4.88	24.78 −5.76	23.85 −14.58	22.59 −4.98
Belchi et al. [76]	153.28 153.28	26.62 −20.03	170.31 170.31	35.13 34.54	35.13 34.54	34.78 34.16	39.60 39.50	23.50 21.82	9.96 −2.49
Fronk and Garimella [37]	273.45 273.45	57.49 −57.49	170.52 170.52	31.09 −30.41	31.09 −30.41	30.72 −30.44	34.61 −34.27	37.78 −37.78	14.59 2.43
Bandhauer et al. [77]	129.60 129.60	40.92 −40.92	87.58 86.65	12.08 5.42	12.08 5.42	11.94 3.60	15.61 4.01	12.66 −8.10	17.72 −4.01
Cavallini et al. [78]	63.59 57.97	43.90 −43.90	53.81 44.01	8.91 −4.03	8.91 −4.03	9.82 −5.27	12.25 1.68	19.04 −17.93	25.98 −25.98
Belchi [79]	315.23 315.23	34.81 21.10	224.80 224.80	117.50 117.50	117.50 117.50	105.43 105.43	122.73 122.73	92.85 92.85	52.30 51.82
Belchi et al. [80]	177.70 177.70	26.17 −25.28	200.52 200.52	32.07 32.07	32.07 32.07	31.03 31.03	38.50 38.50	22.41 21.95	8.00 −4.25
Derby et al. [54]	182.16 182.16	55.93 −55.93	149.56 149.56	11.79 9.57	11.79 9.57	14.55 −11.18	35.33 1.75	13.38 −11.13	17.07 16.75
Andresen [81]	201.77 201.77	17.66 −11.44	91.15 90.09	39.38 39.28	39.38 39.28	40.96 40.96	17.46 0.43	22.03 19.10	21.31 11.21
Heo and Yun [82]	384.25 384.25	55.87 42.53	382.29 382.29	132.05 132.05	132.05 132.05	134.14 134.13	123.45 123.38	116.22 116.22	21.74 10.60
Gomez et al. [83]	84.08 84.08	44.40 −44.33	43.28 43.28	7.65 −1.02	7.65 −1.02	10.42 −8.00	6.52 −0.57	15.81 13.42	13.46 −12.03
Jige et al. [38]	147.22 147.22	57.47 −57.47	123.73 123.73	17.26 −7.32	17.26 −7.32	30.92 −23.76	30.87 −18.22	26.94 −21.78	14.50 0.25
Park et al. [84]	244.99 244.99	42.44 −42.44	173.49 173.49	27.50 20.80	27.72 23.84	22.14 20.59	34.50 23.67	21.13 18.64	14.06 −0.97
Pham et al. [85]	652.27 652.27	46.23 1.14	680.51 680.51	179.40 179.40	179.40 179.40	103.09 102.63	116.14 102.92	111.40 111.22	42.09 28.05
Park and Hrnjak [86]	232.11 232.11	33.01 −33.01	251.06 251.06	38.77 38.77	38.77 38.77	27.22 23.67	34.06 33.46	24.47 24.04	13.07 −11.26
Li et al. [87]	338.36 338.36	31.81 −29.64	353..18 353.18	57.92 57.03	57.92 57.03	30.78 25.07	37.10 27.87	40.40 37.87	15.73 6.26
Wang et al. [55]	167.58 167.58	49.10 −49.10	89.72 89.72	17.11 11.22	17.67 12.87	15.43 12.93	28.34 19.57	9.90 −5.32	10.36 8.29
Rahman et al. [88]	108.70 107.50	72.09 −72.09	99.34 98.03	24.01 −21.90	24.01 −21.90	47.80 −47.80	49.91 −49.56	38.66 −38.42	20.98 −2.42
Total	191.19 190.87	44.07 −33.87	164.97 164.35	43.39 22.94	43.45 23.15	41.54 15.95	44.68 15.92	36.94 8.17	16.87 2.33

shows the AARD and AAD values of previous correlations (Table 3) for approximating the condensation HTC as well as the GP correlation for different analyzed data sources.

Based on the results shown in Table 4, the GP correlation (Equation (13)) shows the best results for predicting the condensation HTC in multi-port channels among the available correlation with AARD and AAD values of 16.87% and 2.33%, respectively, for all analyzed data. In addition, the AARD values of the new model for estimating the condensation HTC from most of the analyzed data sources were lower than 20%. These results for estimating the condensation HTC confirm the accuracy and generality of the correlation determined by the GP approach. Among the previous models, the best results were obtained by Kim and Mudawar [52] with a total AARD of 36.94%, which was two times greater than that obtained by the GP correlation. It should be noted that most of the data (2081 of 4045 data points) analyzed by Kim and Mudawar [52] were for multi-port channels. The Dorao and Fernandino [49], Shah (2016) [90], Shah (2019) [57] and Crosser [92] correlations showed almost the same results with total AARD values of 41.54%, 43.39%, 43.45%, and 44.07%, respectively. In addition, the Hosseini et al. [56] model, which was improved based on the experimental data for single port channels, also had fairly big deviations in estimating the HTC inside multi-port channels with an AARD of 44.68%. This observation verified that the HTC in multi-port mini/micro-channels differs from that in single port channels. The Bohdal et al. [93] and Akers et al. [91] had the highest deviations with the experimental data among the previous correlations, with the AARD values of, respectively, 164.97% and 191.19%. Therefore, these correlations did not provide reasonable predictions for multi-port mini/micro-channels. Although the previous models exhibit nice accordance with some data sources, none of the models can be used as a general correlation for predicting the condensation HTC in multi-port mini/micro-channels. The new correlation and all of the previous models result in large deviations from the experimental data reported by Belchi et al. [79] and Pham et al. [85] with AARD values of 52.30% and 42.09%, respectively.

To show the accuracy of the other models, the results estimated by Kim and Mudawar [52], Dorao and Fernandino [49], Shah (2019) [57], Shah (2016) [90], Crosser [92], and Hosseini et al. [56] correlations were examined in contrast to the measured values in Figure 3. This figure confirms the above assertion that the previous models have relatively large deviations from the measured values of the condensation HTC in multi-port mini/micro-channels.

The accuracy evaluation of previous models and our model for estimating the HTC inside multi-port micro and mini-channels based on Kandlikar [19] classification was given in

Table 5. Comparison of the accuracy of the available models for estimating the HTC in multi-port micro and mini-channels.

Channel Diameter	Micro-Channels $\left(\mathit{D}\le 0.2 \mathit{m}\mathit{m}\right),$ 480 Data Points			Mini-Channels $\left(0.2<\mathit{D}<3 \mathit{m}\mathit{m}\right),$ 3023 Data Points			All Data, 3503 Data Points
Models	Percentage of Data within the AARD of 20%	Percentage of Data within the AARD of 30%	AARD (%)	Percentage of Data within the AARD of 20%	Percentage of Data within the AARD of 30%	AARD (%)	Percentage of Data within the AARD of 20%	Percentage of Data within the AARD of 30%	AARD (%)
Akers et al. [91]	14.36	20.83	134.39	1.36	1.72	200.21	3.14	4.34	191.19
Crosser [92]	0.00	0.00	73.31	22.23	36.09	39.43	19.18	31.14	44.07
Bohdal et al. [93]	16.86	25.83	91.45	4.30	6.55	176.64	5.99	9.19	164.97
Shah (2016) [90]	10.21	16.67	53.68	40.13	55.94	41.76	36.03	50.56	43.39
Shah (2019) [57]	10.21	16.67	53.68	39.93	55.71	41.82	35.86	50.36	43.45
Dorao and Fernandino [49]	10.00	16.25	55.09	40.46	56.17	39.39	36.28	50.70	41.54
Hosseini et al. [56]	9.38	13.96	59.91	32.09	49.49	42.27	28.98	44.62	44.68
Kim and Mudawar [52]	4.17	7.29	59.10	49.19	63.44	33.42	43.05	55.75	36.94
New Correlation	68.75	84.58	16.60	70.66	84.75	16.91	70.37	84.73	16.87

. This table verifies that GP correlation (Equation (13)) can predict the HTC in both micro and mini-channels reasonably well and exhibits the best agreements with the experimental data among all available models with AARD of 16.60% and 16.91% for micro- mini channels, respectively. In addition, our approach predicts almost 85% of data for both micro- mini channels with an error of less than 30%. The Kim and Mudawar correlation [52] predicts 63.44% of the data for mini-channels with an error of lower than 30%. However, its overall accuracy for mini-channels is not satisfactory, with an AARD of 36.94%. In addition, its deviations for micro-channels are much more significant, and its AARD for micro-channels is 59.10%.

The models developed by Dorao and Fernandino [49], Shah (2016) [90], Shah (2019) [57], and Crosser [92] do not provide acceptable results for multi-port micro and mini-channels, and their AARD values for all cases are more than 35%. Furthermore, similar to Kim and Mudawar [52] model, their deviations for micro-channels are significantly higher than those for mini-channels. The models of Bohdal et al. [93] and Akers [91] show substantial deviations for both micro and mini-channels, and they cannot be used for multi-port channels.

In the condensation of the fluids in multi-port channels, the surface tension force plays a remarkable role in controlling the HTC, and its effect becomes more critical as channel diameter is reduced [94]. The effect of the surface tension force was explicitly considered in the new correlation given by Equation (12) using the Bond number. This dimensionless number (Bo) quantifies the relative importance of surface tension forces compared to the gravitational ones. Therefore, the Bo number is important for condensation HTC modeling of multi-port channels, especially for smaller diameters. However, in most of the previous correlations, there was no factor accounting for the effect of surface tension force.

It should be noted that some comparisons between the measured HTC reported by different sources were conducted at identical conditions in [94]. A high degree of consistency between analyzed experimental data was observed. It means that the accurate intelligent based models obtained for a broad range of operational and geometrical conditions capable of estimating HTC for new independent sources, properly. Moreover, the GP-based models outperform earlier models when examining data samples other than those used for developing GP models, as reported in the previous studies [56,94].

The physical trends of the established model and those of Kim and Mudawar [52], as the most accurate model among the previous correlations, are presented in Figure 4, Figure 5, Figure 6 and Figure 7. Here, the influence of the input parameters on the HTC in multi-port channels, are also discussed.

Figure 4 illustrates the impacts of the vapor quality and flow mass flux on the HTC during the condensation of the R744 in a multi-port channel with a HD of 0.13 mm at the saturation temperature of 15 °C. The experimental data shows that the condensation HTC increased with increasing the vapor quality and mass flux. This was in accordance with Sikora et al. [95] findings for the HFE 7000 refrigerant in microchannels. According to Coleman and Garimella [96] flow patterns, the flow regime changes from slug and bubbly with thick liquid film to annular with thin liquid film by raising the vapor quality. In addition, increasing the mass flux can increase the shear stress at the two phases interface, and the liquid film thickness reduces. Therefore, increasing the mass flux and vapor quality decreases the liquid film thickness and increases the HTC. Figure 4 shows that the GP correlation shows good accordance with the experimental data. However, the correlation of Kim and Mudawar [52] underestimates the measured values.

The predicted HTCs for R134a are examined in contrast to the real data for saturation temperatures of 50 and 60 °C in Figure 5. This result showed that the HTCs for the saturation temperature of 50 °C were higher than those for 60 °C.

Table 6. Thermophysical characteristics of R134a at 50 and 60 °C.

${\mathit{T}}_{\mathit{s}} (℃)$	${\mathit{\rho }}_{\mathit{l}} \left(\mathbf{k}\mathbf{g} {\mathbf{m}}^{-3}\right)$	${\mathit{\rho }}_{\mathit{v}} \left(\mathbf{k}\mathbf{g} {\mathbf{m}}^{-3}\right)$	${\mathit{k}}_{\mathit{l}} \left(\mathbf{w} {\mathbf{m}}^{-1}{\mathbf{K}}^{-1}\right)$
50	1102.3	66.27	0.070427
60	1052.9	87.38	0.066091

presents the thermophysical characteristics of the R134a at the saturation temperatures of 50 and 60 $℃$. Reducing the saturation temperature led to higher liquid density and lower vapor density. Therefore, the vapor velocity was increased, and the liquid velocity was reduced and led to a higher velocity difference at the two phases interface. Therefore, higher interfacial shear stress and thinner liquid film were obtained at the lower saturation temperature, and the HTC increased. In addition, R134a had a higher liquid thermal conductivity at the saturation temperature of 50 °C. Therefore, its thermal resistance in liquid film at this temperature was lower, which resulted in a higher HTC. Figure 5 shows that the GP and Kim and Mudawar [52] correlations showed the same trends for different saturation temperatures. However, the GP correlation led to a much better agreement with the experimental data.

Figure 6 compares the HTC of R744 at the saturation temperature of 0 $℃$ for two different channel diameters. The experimental data and empirical correlations show that the HTCs of R744 in the channels with 0.68 mm hydraulic diameter was slightly higher than those for the channel with 0.78 mm hydraulic diameters. As was noted previously, the surface tension force plays a more critical role in the smaller channels, and the interactions of the phases increase. Therefore, the liquid film becomes thinner, and the HTC increases. Figure 6 shows that the new GP correlation is consistent with the experimental data, while the correlation of Kim and Mudawar [52] overestimated the HTC.

Figure 7 shows the effect of the working fluids (R32 and R290) on the HTC. As shown in this figure, the condensation HTCs for R290 as the working fluid were higher than that for R32. The thermophysical properties of the R32 and R290 at the constant saturation temperature of 40 °C are compared in

Table 7. Thermophysical properties of R290 and R32.

Fluids	${\mathit{T}}_{\mathit{s}} (℃)$	${\mathit{\rho }}_{\mathit{l}} \left(\mathbf{k}\mathbf{g} {\mathbf{m}}^{-3}\right)$	${\mathit{\rho }}_{\mathit{v}} \left(\mathbf{k}\mathbf{g} {\mathbf{m}}^{-3}\right)$	${\mathit{H}}_{\mathit{l}\mathit{v}} \left(\mathbf{k}\mathbf{J} \mathbf{K}{\mathbf{g}}^{-1}\right)$
R290	40	467.46	30.165	307.06
R32	40	893.04	73.268	237.10

. Accordingly, R290 had lower liquid and vapor densities. Thus, the liquid and vapor velocities for R290 were higher than those for R32 at the same mass flux. Therefore, the shear stress at the interface increased, which resulted in a higher HTC. In addition, the latent heat of vaporization of R290 was higher than that of R32. This information also justifies the higher HTCs for R290. Figure 7 also shows that the GP and Kim and Mudawar correlations predict the same trends for different working fluids. However, the GP correlation results were in closer accordance with the real data.

4. Conclusions

A new general correlation for predicting the condensation HTC in multi-port mini and micro-channels was developed using the method of genetic programming. For validating the established model, 3503 experimental data points were collected from 21 published papers covering a broad range of working fluids and operating conditions. Key conclusions of this study are:

• The effects of all affecting parameters $\left(R{e}_l, P{r}_l, X_{tt}, Bo, x\right)$ on the HTC were considered in the GP correlation. Using these parameters led to a correlation that estimates the condensation HTC with reasonable accuracy. The new correlation estimated the HTC with a total AARD of 16.87% for a broad range of data samples. In addition, the percentages of all data with error lower than 20% and 30% for the new model were 70.04% and 84.43%, respectively.
• The previous models’ predictions of the HTC were also compared to the measured data. The previous correlations showed significantly higher deviations from the experimental data compared to the new correlation. The total AARD values for the correlations of Kim and Mudawar [52], Dorao and Fernandino [49], Shah (2016) [90], Shah (2019) [57], Crosser [92], Hosseini et al. [56], Bohdal et al. [93], and Akers et al. [91] were, respectively, 36.94%, 41.54%, 43.39%, 43.45%, 44.07%, 44.68%, 164.97%, 191.19%. It was found that the previous models can not be considered as general correlations for estimating the condensation HTC in multi-port channels. However, the new GP correlation provides much-improved estimates for the HTCs.
• The new model and previous correlations were used for estimating the HTC in channels with different sizes. It was shown that the new correlation estimated the data for micro and mini-channels with the AARD of 16.60% and 16.91%, respectively, and was the most accurate for all cases among all correlations. In addition, the new GP correlation estimated more than 80% of all data for both micro and mini-channels with an error of lower than 30%. This is because using the Bond number in the new model explicitly includes the effects of surface tension in different channel sizes. The previous correlations, however, showed relatively large deviations for all cases. Furthermore, their deviations become larger when the channel’s diameter decreases, and all correlations had AARD values of more than 50% for micro-channels.
• The new model developed by GP was shown to be suitable for estimating the condensation HTC in multi-port channels over a broad range of vapor qualities, mass velocities, saturation temperatures, channel diameters, and working fluids.

Knowledge of the parameters of greater effect on the target function is critical for developing further simple and precise correlations. Therefore, performing a Pearson’s correlation coefficient analysis to select the most influential parameters in controlling the condensation of two-phase HTC in multi-port channels and also to determine the weight of each factor in the model is suggested as future work.