# Thermal Conductivity of Low-GWP Refrigerants Modeling with Multi-Object Optimization
## 1 Introduction  
The thermal conductivity (\(\lambda\)) of refrigerants is an essential property to describe the heat transfer in HVAC&R applications. For example, it directly affects the convection heat transfer coefficient in the heat exchangers. Therefore, this transport property is crucial for the optimization and design of these components. However, the experimentally determined \(\lambda\) data of refrigerants available in the literature do not cover all the temperature and pressure ranges needed for industrial applications. This is especially true for the fluids of low global warming potential (GWP) [1], also known as \"fourth-generation\" refrigerants [2, 3, 4], which have been identified as potential alternatives to high-GWP working fluids, such as several hydrofluorocarbons (HFCs). In fact, a limited number of experimental data for \(\lambda\) of these alternative refrigerants, mainly hydrofluoroolefins (HFOs) and hydrochlorofluoroolefins (HCFOs), were presented in the literature [5]. For this reason, several models have been developed to describe the temperature and pressure dependence of the thermal conductivity of refrigerants both in the vapor and liquid phases. Considering that no rigorous theory can accurately describe the liquid thermal conductivity [6], different semi-empirical and empirical models were proposed to calculate the \(\lambda\) of liquid refrigerants. Some of the main ones are extended corresponding states models, such as the ones used in REFPROP 10.0 [7]; models based on the entropy-scaling concept [8, 9, 10, 11]; models based on equations of state [12, 13]; semi-empirical correlations that describe the \(\lambda\) dependence on temperature [14, 15, 16]; and empirical equations that use specific fixed parameters [17, 18, 19, 20, 21]. Among them, some models have been specifically developed to describe the liquid thermal conductivity of low-GWP refrigerants, such asHFOs and HCFOs [11; 14; 20; 21]. In particular, our research group recently proposed a modified version of the [PERSON] et al. [19] correlation to calculate the liquid thermal conductivity dependence of low-GWP refrigerants on temperature and pressure [20]. It has the following expression:  
\[\frac{\lambda_{L}}{\lambda_{0}}=\left\lfloor aT_{r}+bp_{c}+c\omega+\left(\frac{ 1}{M}\right)^{d}\right\rfloor\left[1+\left(f_{0}+fT_{r}{}^{2}\right)p_{c} \right] \tag{1}\]  
where \(\lambda_{L}\) is the liquid thermal conductivity in W m\({}^{-1}\) K\({}^{-1}\), \(T_{r}=T\cdot T_{c}{}^{-1}\) is the reduced temperature, \(T_{c}\) is the critical temperature in K, \(p_{r}=p\cdot p_{c}{}^{-1}\) is the reduced pressure, \(p_{c}\) is the critical pressure in bar, \(\omega\) is the acentric factor, \(M\) is the molecular mass in kg kmol\({}^{-1}\), and \(\lambda_{0}\), \(a\), \(b\), \(c\), \(d\), \(f_{0}\), \(f\) and \(g\) are coefficients regressed on experimental data. In the original work, the following values for the studied alternative refrigerants were regressed by using a Random Search method [22] by minimizing the average absolute relative deviation of the liquid thermal conductivity: \(\lambda_{0}=0.43693\) W m\({}^{-1}\) K\({}^{-1}\), \(a=-0.28725\), \(b=0.00372\) bar\({}^{-1}\), \(c=0.26967\), \(d=0.36436\), \(f_{0}=-0.00135\), \(f=0.05484\) and \(g=0.88049\). In particular, the same dataset containing a total of 2073 experimental data for 6 low GWP refrigerants, namely R1233 zd(E), R1234 yf, R1234 ze(E), R1234 ze(Z), R1224 yd(Z), and R1336 mzz(Z), defined in the study [20] have been considered in the calculations reported in this work. In this paper, a new methodology was employed to find Equation (1) coefficients using the Multi-Objective Optimization (MOO) approach. The MOO approach allows to solve complex problems [23; 24] in which multiple and even conflicting objective functions exist. The classical approach of an optimization problem involves a single objective function and a single best solution.