Developing a model for calculating pure gas thermal conductivity at P=1bar using particle swarm optimization algorithm

1. Introduction
Conductive form of heat transfer is the transmission of heat using molecular motion in a stationary substance and occurs mainly in solids or stationary environments such as stationary liquids. Conductive heat transfer of gases and liquids is because of the collision and diffusion during the random mobility of molecules. On the other hand, energy transfer by free electrons and molecular network vibrations combine to create heat transfer in solids [1]. Thermal conductivity is a property of a material that proclaims its ability to conduct heat [2]. 
Even at high pressures, accurate measurement equipment can determine the thermal conductivity of gases with an accuracy of less than 1%. The measurement of thermal conductivity for the noble gases He and Ar at low densities is consistent with the kinetic theory's predictions, and similar outcomes are anticipated for Ne, Kr, and Xe [3]. Numerous gases have had their thermal conductivity measured over history and compiled. Table 1 shows an overview of the background of the studies conducted on the measurement and prediction of the conductivity coefficient of pure fluids and mixture fluids.

 
Table 1. A list of techniques and correlations used to predict thermal conductivity
Source    Comment    Year    Reference
Ubbelohde    He believed that chemical species might be thought of as dilute gas molecules in various energy states and that the diffusion of these species is what causes an energy flux.    1935    [4]
Pidduck    Pidduck was the first to use the Chapman-Enskog method for an infinitely diluted gas of spherical molecules in the case of ideally rough stiff elastic spherical molecules having rotational energy that could be converted into translational energy. This was the first instance when a polyatomic molecule scenario had been fully investigated.    1953    [5]
The Eucken approximation    He understood how the internal degree of freedom affected the conductivity of a diluted gas containing polyatomic molecules and how the specific heat ratio relates to this.    1954    [6]
Theory of Longuet-Higgins and Pople    Using the single collisional mechanism assumption, Longuet-Higgins and Pople estimated the conductivity in a dense gas of hard spheres.    1956    [7]
Theory of Longuet-Higgins and Valleau    They thought about how a square-well potential affects the interactions between molecules.    1958    [8]
Choh and Uhlenbeck    These theories produce formulas for conductivity in terms of a series expansion in the density and are frequently based on density expansions of a generalized Boltzmann equation.    1958    [9]
Srivastava and Barua    The thermal conductivity of binary mixtures including O2-He, O2-Ne, O2-Kr, and O2-Xe at 30 and 45 °C for different compounds was measured through a thick-set hot cord. For pure O2, the experimental amount of thermal conductivity is less than that proposed by Hirschfelder's recent theory based on the supposition of local chemical equilibrium.    1960    [10]
Theory of Davis et al.    In a more thorough analysis of the transport characteristics of a square-well fluid, Davis et al. computed the convective and collisional contributions using a modified Boltzmann equation.    1961    [11]
Bogolubov    These theories produce formulas for conductivity in terms of a series expansion in the density and are based on density expansions of an extended Boltzmann equation.    1962    [12]
Theory of Horrocks and McLaughlin    Using frequency as a measure of conductivity, they proposed the face-centered cubic lattice shape.    1963    [13]
Cohen    These theories generate formulas for conductivity that are based on density expansions of a generalized Boltzmann equation in terms of a series.    1965    [14]
Theory of Sengers    His analysis takes into account Longuet-Higgins and Valleau's omitted effect of the distribution function disturbance from the local equilibrium value.    1966    [15]
Mathur et al.    Thermal conductivity data for  - ,  - ,  - , and  -  systems have been recited as a dependent of percentage composition at temperatures of 40, 65, and 90 °C. Similar results have been obtained for the   -  - ,  -  -  triple systems, and an   -   -  -  quaternary system. In this method, a conductive cell of a kind of thick hot wire was used. A similar study has been done for triple and quadruple mixtures and good results have been obtained about the partial suitability of various methods for calculating the thermal conductivity of multicomponent mixtures of gases    1967    [16]
Theory of Enskog    Enskog considered the example of hard spheres to make the first significant contribution to the calculation of the conductivity of a dense gas.    1970    [17]
De Groot et al    They described new precision and estimation of the thermal conductivity of gases in the pressure and temperature ranges of  1 to 400 atm and 25 to 800 °C, respectively    1974    [18]
Healy et al.    They investigated the thermal conductivity of four different pure monoatomic gases including He, Ne, Ar, and Kr. The evaluation was made at room temperature on a high-accuracy instrument with a hot wire. The thermal conductivity of Ne, Ar, and Kr has been analyzed as a function of density    1976    [19]
Plinski    A correlation was used to calculate the thermal conductivity of CO,  ,  , He, Xe, CO,  , and Ar as a function of temperature.    2001    [20]
Mao-Gang He et al.    For halogenated hydrocarbon refrigerants, they proposed a new correlation to measure the thermal conductivity of the dense fluid.    2002    [21]
Eslamloueyan and Khademi.    The variables of the proposed  model were molecular weight, critical temperature, and critical pressure. The use of this model for the tested data suggests that the thermal conductivity of pure gases at atmospheric pressure can be predicted by a significant relative error and less than other alternative correlations.    2009    [22]
Gauthier et al    They developed a novel experimental method using a three-setup to measure the thermal conductivity of a gas. The purpose of this study was to confirm the 3 method's simplicity and convenience of use. It was possible to measure the thermal conductivities of both pure gases and mixtures of gases with high reproducibility and a variability of less than 5%. When evaluated at 298 K and atmospheric pressure, the thermal conductivity of various pure gases, including nitrogen, helium, and carbon dioxide, was compared to reference levels with an error of less than 5%.    2013    [23]
Ribeiro et al.    A novel method of estimating heat transfer coefficients for a gas-solid interface was proposed, in which the photon, even when radiation with a low frequency is transmitted, consists of electronic transitions rather than passing instantly through the wall of gases and solids. The Landau-Teller model determines the delay necessary for electron transmission, which is crucial to the process of heat transfer.    2017    [24]
Lang et al.    They created a technique for determining how thermally conductive combinations of pressurized gases are. They employ a mixture of carbon dioxide and PDMS, a linear polymer with a molecular weight of 11000 g/mol. Experiments were conducted at 25, 40, and 60  and up to 16 MPa.    2019    [25]
Cardona and Valderrama    They recently suggested a generalized equation of state that correlates and predicts various physical and transport properties of ionic liquids, such as thermal conductivity and speed of sound, and only has one changeable parameter.    2020    [26]
Li et al.    The thermal conductivity of  binary mixtures were measured using a system based on the short-hot-wire method at 323.15-620.05 K and 2.14-9.37 MPa.    2020    [27]
Kim et al.    Using a transient hot-wire technique, thermal conductivity measurements of pure R1243zf and binary mixes of R32 + R1243zf and R32 + R1234yf were made in the homogenous liquid and vapour phases. The relative differences between the calculated thermal conductivities using the extended corresponding states (ECS) model and the measured thermal conductivities.    2021    [28]
Liu et al.    They simulate the thermal conductivity of HFCs, HFOs, and their binary mixtures using the RES method and the CPA equation of state. Except the near-critical zone, the dependence of the thermal conductivity on the thermodynamic state is reduced to a univariate function of residual entropy throughout a wide range of temperatures and pressure.    2021    [29]
Dehlouz et al.    Rosenfeld's original entropy scaling method was modified to compute the thermal conductivities of pure fluids. A function of the Tv-residual was connected to a specially developed expression for decreased thermal conductivity.    2022    [30]
Yu Liu et.al    To represent the thermal conductivity of HFC, HFO, and HCFO refrigerants in a liquid phase, a semi-empirical model was developed. The Peng-Robinson Equation of State (PR equation) and the Modified Enskog Theory (MET) model served as the foundation for the suggested thermal conductivity model, which also included an empirical adjustment to the MET model's cross-contribution factor to increase precision.    2022    [31]
Niksirat et al.    Their model was used for binary and multicomponent liquid phase combinations and was tested on pure HCFCs, HFCs, and HFOs.    2022    [32]
Rottmann and Beikircher    The effective thermal conductivity of pure and opaque expanded perlite was investigated at gas pressures between 104 hPa and 103 hPa and temperatures between 293 K and 1073 K using guarded hot plates and transient hot wire measurements. The description of the coupling effect between solid-body and gas heat conduction was improved by the addition of a single novel coupling parameter to the model, which expresses the exceedance of the gas pressure-dependent contribution of the effective thermal conductivity relative to the thermal conductivity of the gas inside the pores.    2022    [33]

 
When the complications of issues arise, the experimental data must be estimated. Traditional optimization algorithms may find it hard to meet the supplies of the problems which leads to new influential algorithms [34]. Over the past few years, various traditional novelty algorithms have been developed as impressive and practical approaches to problem optimization [35, 36]. due to the continuous improvement of computing power. By using numerical integration, they aimed to quickly optimize the reaction rate, and conversion range, and accommodate any collection of differential rate equations [37]. Particle swarm optimization (PSO) and genetic algorithm (GA) are the most hopeful algorithms for network optimization [38]. PSO as an evolutionary random algorithm, that is nature-inspired, is evoked by the public behavior of organisms, which warrants a coordinated swarm to achieve the ideal result extended by Kennedy and Eberhart [39]. It is randomly placed in the workspace, and each particle's objective function quantity is assessed [40]. Like GA, PSO is an optimization tool which is based on population. But, GA and PSO are different in some ways: (1) PSO has various evolutionary mechanisms, with the exception of genetic agents such as crosses and mutations, which update their PSO particles at internal speeds. (2) At the same time, particles of PSO have a memory that is vital for the algorithm [41].
 In recent years, the PSO optimization was used in the nonlinear Regression in problems such as model design and parameter estimation for the thermodynamics, kinetics, and hydrodynamics of mixed salt precipitation in porous media [42], interacting parameter correlation in the Wilson, NRTL, and UNIQUAC models [43], toxic vapours' kinetic adsorption on activated carbon in the batch reactor [44], predict crude oil properties [45], for the gas cross flow in packed bed reactors, a novel Sauter mean diameter correlation has been developed [46], and an innovative theoretical and practical approach based on friction volume theory and friction theory parameter tweaking for viscosity-sensitive Iranian heavy crude oils [47].
In previous studies, relationships have been proposed to obtain thermal conductivity, which are mostly complex and detailed, and there are some difficulties in using them. In this study, an optimum and simple model with a low error rate of specific gases is proposed by PSO using data related to the physical properties of gas in 1 bar.

2. Methodology
2.1. PSO algorithm
James Kennedy and Russell Eberhart [48] presented the PSO technique in 1995. This algorithm is adapted from the collective performance of a collection of animals such as fliers and fish. PSO is an optimization approach based on population inspired by the public treatment of birds or fish training. It sometimes bears many similarities to Evolutionary Calculation techniques (EC), Genetic Algorithms (GA), and Evolutionary Strategies (ES). But there are also many contrasts between these methods [49]. The PSO starts with a collection of chance iotas (solutions) and therefore takes optimal search by keeping generations up to date by coursing the best valuations in every iterance, in which every particle is kept up to date. The first of these values is the foremost fit ( ). This foremost value is the best in the world and is entitled (  ). After gaining the two foremost values, the particle keeps its velocity and position up to date  using the following equations:
     (1)
     (2)
The   and   are the position and velocity of the particle “i” in the new iteration. The   shows the foremost position of particle” i”, and   represents the foremost position among the whole available particles.   and   are the random numbers between zero and one.   and   are positive constant parameters entitled acceleration coefficients. w is the inertia weight that is used to ensure convergence.
Figure 1 shows the flowchart algorithm for PSO [50]. 

 
 
Figure1. Flowchart algorithm for PSO

 
The optimal amounts of the parameters w,   and   are obtained from the following relations. Table 2 specifies the implementation parameters of the PSO algorithm. 
(3)     
(4)     
(5)     
(6)     = 
(7)     = 
In the above equations,   and   are constant values greater than zero, respectively, which must be adjusted to achieve the optimal value of the parameters of the PSO algorithm, including inertia weight (w), and learning coefficients (  and  ). Clerk suggested that if the values   and   were equal to 2.05 and the value of   was considered equal to one, the optimal value of 0.73 for the inertia weight would be obtained [51]. The PSO is employed for data fitting issues, at which the variables are the required coefficients of the regression model for data fitting. In the case of a randomly initialized solution at the first iteration, the minimized error between the actual output value and predicted value can be calculated from the initialized solution to compute the fitting function.  ,  , ….,  can be considered as the results of  the PSO for i iteration and k population. The predicted output value related to  ,  , ….,   for both the linear and non-linear regression models can be obtained using the following equations [52]:
      (8)
     (9)

To generate the fitting function, one can use the minimizing of the equation 10 (cost function) by the PSO algorithm [53]:
     (10)

Table 2. Parameters of the PSO algorithm
Parameter    Value
Number  of  Iterations    6000
Population size    300
Mutation    0.04
Inertia weight (w)    0.7298
Personal learning coefficient( )
1.4962
General learning coefficient( )
1.4962

2.2. Data acquisition and analysis
61 experimental data for gases at a pressure of 1 bar were gathered for this investigation [54-61]. The physical properties of pure gases including temperature, critical temperature, critical pressure, molecular weight, viscosity, and heat capacity at constant volume were obtained for pure components and used for prediction of the conductivity of these gases. The Aspen Hysys V.11 software database was used to calculate viscosity and heat capacity at constant volume. The data collection used to create the model includes a variety of hydrocarbon and non-hydrocarbon compounds, some of which are listed in Table 3. 
 

Table 3. A list of the compounds utilized in the model's development
No.    Component     (K)
 (K)
 (bar)          (N.S. 
 
1    Acetone    400    508.2    47.01    58.079    88.13    83.63
2    Acetylene    293    308.3    61.38    26.037    97.07    36.8
3    Acetylene    400    308.3    61.38    26.037    135.7    41.73
4    Ammonia    353    405.65    112.8    17.031    100.4    28.78
5    Ammonia    400    405.65    112.8    17.031    115.4    30.09
6    Argon    273.2    150.86    48.98    39.948    215.6    12.47
7    Argon    491.2    150.86    48.98    39.948    334.8    12.47
8    Benzene    400    562.05    48.95    78.112    95.9    103.8
9    Carbon dioxide    400    304.21    73.83    44.01    203.1    33.06
10    Carbon dioxide    473    304.21    73.83    44.01    244.2    35.08
11    Carbon dioxide    600    304.21    73.83    44.01    313    38.22
12    Carbon dioxide    1100    304.21    73.83    44.01    487.2    46.87
13    Carbon dioxide    1500    304.21    73.83    44.01    619.8    50.57
14    Carbon monoxide    91.88    132.92    34.99    28.01    59.52    21.16
15    Carbon monoxide    273    132.92    34.99    28.01    170    20.74
16    Carbon monoxide    291    132.92    34.99    28.01    178.6    20.76
17    Carbon tetrachloride    456.88    556.35    45.6    153.823    150.2    87.2
18    Chlorine    300    417.15    77.1    70.906    137.8    25.76
19    Ethane    239.11    305.32    48.72    30.069    75.54    38.18
20    Ethane    300    305.32    48.72    30.069    94.65    44.75
21    Ethane    500    305.32    48.72    30.069    156.8    69.3
22    Ethyl alcohol    373    514    61.37    46.068    76.78    69.15
23    Ethylene    250    282.34    50.41    28.053    85.51    32.12
24    Ethylene    273    282.34    50.41    28.053    93.27    33.75
25    Ethylene    300    282.34    50.41    28.053    100.24    35.78
26    Ethylene    400    282.34    50.41    28.053    136    44.05
27    Ethylene    500    282.34    50.41    28.053    167.6    52.88
28    Ethylene    600    282.34    50.41    28.053    195.7    61.59
29    Helium    144    5.2    2.275    4.003    109.9    12.48
30    Helium    273.2    5.2    2.275    4.003    598    12.48
31    Heptane    500    540.2    27.4    100.202    95.25    242.5
32    Hydrogen    250    33.19    13.13    2.016    74.66    20.01
33    Hydrogen    373    33.19    13.13    2.016    105.8    20.29
34    Hydrogen    450    33.19    13.13    2.016    122    20.48
35    Hydrogen    600    33.19    13.13    2.016    146.2    20.9
36    Hydrogen    800    33.19    13.13    2.016    178.2    21.52
37    iso-Butane    273    407.8    36.4    58.122    67.92    82.2
38    iso-Butane    373    407.8    36.4    58.122    94.13    109.2
39    iso-Pentane    373    460.4    33.8    72.149    84.32    136.4
40    Methane    200    190.564    45.99    16.042    78.24    25.06
41    Methane    300    190.564    45.99    16.042    113.4    27.74
42    Methane    400    190.564    45.99    16.042    144    32.33
43    n-Butane    273    425.12    37.96    58.122    65.85    82.7
44    n-Butane    400    425.12    37.96    58.122    98.6    116.2
45    Neon    373.2    44.4    26.53    20.18    425.7    12.49
46    Nitric oxide    200    180.15    64.8    30.006    134.7    21.28
47    Nitrogen    273    126.2    34    28.013    171.4    20.7
48    Nitrogen    400    126.2    34    28.013    226.4    21.37
49    Nitrogen    900    126.2    34    28.013    430.3    23.83
50    Nitrous oxide    273    309.57    72.45    44.013    130.7    29.31
51    Oxygen    173    154.58    50.43    31.999    128.8    19.56
52    Oxygen    200    154.58    50.43    31.999    147.5    19.84
53    Oxygen    350    154.58    50.43    31.999    237.7    21.38
54    Oxygen    500    154.58    50.43    31.999    307.7    22.84
55    Propane    250    369.83    42.48    44.096    68.22    57.38
56    Propane    273    369.83    42.48    44.096    74.7    61.79
57    Propane    300    369.83    42.48    44.096    82.27    66.94
58    Propane    373    369.83    42.48    44.096    103    80.6
59    Propane    400    369.83    42.48    44.096    110.9    85.53
60    Propane    500    369.83    42.48    44.096    140    103.1
61    Sulfur dioxide    273    430.75    78.84    64.064    107.4    30.46

 
2.3. Selection of optimal configuration
Three crucial characteristics that affect how well the constructed model performs are defined in this section. Each input data's correlation coefficient serves as the parameter [62]:
     (11)
The second is the mean square error (MSE) [63]:
     (12)
The third parameter is the mean relative error (MRE) [64]:
     (13)
n is the number of data points,   is the experimental conductive heat transfer coefficient,   is the average value of the experimental values and   is the conductive heat transfer coefficient obtained from the modeling.

3. Results
Equation 14 was proposed to calculate the thermal conductivity coefficient of pure gases by the PSO algorithm:
     (14)
     (15)
     (16)
     (17)
     (18)
Table 4. Values for Calculation of  
                          
1    -0.4470    8    0.3294    15    -0.4465
2    -0.1780    9    0.3504    16    -0.8012
3    0.0559    10    0.7352    17    0.2474
4    -0.1045    11    0.1625    18    0.4298
5    -0.0451    12    0.0490    19    0.6130
6    -0.1607    13    0.0675    20    -0.6132
7    0.5836    14    0.2522    21    -0.1012

The cost function (equation 10) can be changed with iteration for the best model, as shown in Figure 2.  It is clear that the cost function is almost equal to zero, which should be minimized.

 
Fig 2. A schematic of error variation during optimization

The correlation between the simulation results and experimental data is illustrated in Figure 3.
 
Fig 3. Correlation of experimental data versus PSO predictions

Table 5 shows the MRE, MSE and   values calculated.

Table 5. The MRE, MSE and   values for the PSO configuration
MRE 100
MSE 100
 
4.67         0.9995

Table 6 shows the comparison of the proposed model with other correlations. The advantage of the proposed equation compared to other models is its simplicity.

Table 6. Comparison of the proposed model with other correlations
No.    Component         Proposed Equation(RE )
Other correlations(RE )
Reference
1    Acetone    400    3.23    6.96    [65,66]
2    Acetylene    293    6.44    2.29    [65,66]
3    Acetylene    400    4.28    0.00    [65,66]
4    Ammonia    353    2.33    17.94    [68]
5    Ammonia    400    5.08    16.48    [68]
6    Argon    273.2    10.50    1.84    [67]
7    Argon    491.2    16.44    0.74    [67]
8    Benzene    400    5.14    3.07    [65,66]
9    Carbon dioxide    400    4.60    0.81    [67]
10    Carbon dioxide    473    0.92    2.23    [67]
11    Carbon dioxide    600    2.93    6.77    [67]
12    Carbon dioxide    1100    1.90    12.23    [67]
13    Carbon dioxide    1500    0.49    0.61    [22]
14    Carbon monoxide    91.88    8.21    1.25    [22]
15    Carbon monoxide    273    8.04    7.23    [67]
16    Carbon monoxide    291    6.10    5.06    [67]
17    Carbon tetrachloride    456.88    14.62    1.40    [68]
18    Chlorine    300    7.56    5.61    [67]
19    Ethane    239.11    10.02    3.35    [65,66]
20    Ethane    300    5.06    1.83    [65,66]
21    Ethane    500    1.02    1.35    [65,66]
22    Ethyl alcohol    373    9.89    10.23    [65,66]
23    Ethylene    250    13.50    1.31    [65,66]
24    Ethylene    273    6.33    4.91    [65,66]
25    Ethylene    300    2.91    5.14    [65,66]
26    Ethylene    400    0.83    3.80    [65,66]
27    Ethylene    500    3.30    4.68    [65,66]
28    Ethylene    600    4.87    5.81    [65,66]
29    Helium    144    0.07    14.11    [67]
30    Helium    273.2    0.00    2.67    [67]
31    Heptane    500    14.13    0.61    [65,66]
32    Hydrogen    250    0.64    2.75    [67]
33    Hydrogen    373    1.38    3.44    [67]
34    Hydrogen    450    1.88    3.66    [67]
35    Hydrogen    600    0.22    5.77    [67]
36    Hydrogen    800    0.16    2.50    [67]
37    iso-Butane    273    7.06    5.79    [65,66]
38    iso-Butane    373    2.20    2.90    [65,66]
39    iso-Pentane    373    0.66    0.45    [65,66]
40    Methane    200    0.23    5.04    [65,66]
41    Methane    300    0.42    6.12    [65,66]
42    Methane    400    1.47    4.13    [65,66]
43    n-Butane    273    7.29    5.18    [65,66]
44    n-Butane    400    2.19    6.06    [65,66]
45    Neon    373.2    0.35    18.96    [67]
46    Nitric oxide    200    2.44    8.60    [67]
47    Nitrogen    273    6.18    5.21    [67]
48    Nitrogen    400    0.33    2.10    [67]
49    Nitrogen    900    0.11    7.24    [67]
50    Nitrous oxide    273    0.95    4.40    [68]
51    Oxygen    173    8.21    15.24    [67]
52    Oxygen    200    4.22    7.14    [67]
53    Oxygen    350    2.26    2.93    [67]
54    Oxygen    500    0.71    0.71    [67]
55    Propane    250    11.97    6.20    [65,66]
56    Propane    273    10.05    3.31    [65,66]
57    Propane    300    5.52    3.27    [65,66]
58    Propane    373    4.62    3.83    [65,66]
59    Propane    400    3.45    4.74    [65,66]
60    Propane    500    3.87    9.11    [65,66]
61    Sulfur dioxide    273    13.01    10.34    [68]
62    Grand average    91.88-1500    4.67    5.24    -

 
Table 7 shows the results of the equation test for several materials other than those listed in Table 3. The results of comparing the proposed equation with other equations show the high accuracy of the proposed equation for predicting the thermal conductivity coefficient for a wide range of materials.
 

Table 7. Test of the proposed model
No.    Component         Proposed Equation(RE )
Other correlations(RE )
Reference
1    Acetonitrile    353    18.62    22.00    [55]
2    Acetonitrile    393    14.29    19.00    [55]
3    Cyclohexane    353    2.75    4.20    [55]
4    Cyclohexane    433    6.40    9.00    [55]
5    Ethyl ether    273    6.32    3.10    [55]
6    Ethyl ether    373    6.43    11.00    [55]
7    Ethyl ether    486    12.62    13.00    [55]
8    n-Hexane    373    2.22    6.80    [55]
9    n-Hexane    433    8.38    9.40    [55]
10    Grand average    273-486    8.67    10.83    -

 
4. Conclusion
The thermal conductivity coefficient as one of the important parameters in the field of heat transfer from about 50 years ago up to now has been considered by scientists and researchers and has always tried to determine methods for measuring this parameter. In this work, 61 experimental data for pure gases at P=1 bar and variable temperature (91.88-1500 K) are collected. Then, using the PSO algorithm and MATLAB V2015 software, a simple equation for approximating the thermal conductivity is processed. During the validation phase, the suggested model attained the most accurate prediction with  , MRE=4.67% and MSE=2.4210 10-4%. The result of our study is advantageous for the simulation of different chemical processes based on gases through appropriate prediction of the thermal conductivity coefficient. It is important to mention that most of the previous models that were presented to determine the double thermal conductivity coefficient of gases have variables that require a lot of background knowledge and the number of their input variables is large, and they also have a lot of complexity to calculate. The advantage of the equation proposed in this research compared to other models is its simplicity.

Notation
     Heat capacity at constant volume  
k    Conductive heat transfer coefficient ( )
     Molecular Weight
     Pressure (bar)
     Critical pressure (bar)
T    Temperature (K)
     Critical temperature (K)
     Viscosity (N.S.