# A Self-Consistent Physical Model of the Bubbles in a Gas Solid Two-Phase Flow
## 2 Theoretical Model
### PR Equation and the Heat Capacity Ratio of Bubbles  
None of the real thermodynamic systems conform completely with the law of ideal gas. The deviation depends on the pressure, the temperature, and the gas properties, especially with respect to the degree of difficulty of gas liquefactions. The gas-solid two-phase flow bubble systems contain not only gas interacting between air molecules, but also partial solid particles. With the change of ambient pressures and temperatures, the proportion of the particle phase and the gaseous phase varies. Therefore, the system is a non-ideal complex system, which needs the non-ideal state equation for study. The non-ideal gas state equation is widely used in engineering, and includes the  
Figure 1: Schematic for the gas-solid two-phase flows. Van der waals equation, the Redlich Kwong (RK) equation, the virial equation and so on. When the RK equation is used to calculate strong polar compounds, it causes large deviations and is thus rarely used to calculate liquid pressures, volumes, and temperatures. The virial equation is not a good match for the polar compounds, and gas-solid two-phase flow cannot be described by a set using the virial coefficient [20]. There are large deviations for the RK and Redlich Kwong Soave (RKS) equations in calculating the critical compressibility factor \(Z_{c}\) and liquid density. However, the Peng-Robonson (PR) equation makes up for the obvious deficiencies, with a better accuracy in the calculation of saturated vapor pressure, saturated liquid density, and other aspects, especially in calculations of the multiphase fluid systems, which are some of the most commonly used equations in engineering design calculations. In this paper, the PR equation is developed to investigate the gas-solid two-phase flow [21]. In this system, the PR equations, which are corrected by [PERSON] and [PERSON], are proposed to calculate the bubble properties of the gas-solid two-phase flow system [22]. The specific form of the PR equation is  
\[P=\frac{RT}{V-b}-\frac{a}{V^{2}+2 bV-b^{2}}, \tag{1}\]  
where \(a=a_{c}\alpha\), \(a_{c}=0.457R^{2}T_{c}^{2}/P_{c}\), \(\alpha^{1/2}=1+(0.375+1.542\omega-0.27\omega^{2})[1-(T/T_{c})^{1/2}]\), \(b=0.078 RT_{c}/P_{c}\), \(P\) is the system pressure \(P_{a}\), \(R\) is the gas constant (J\(\cdot\) mol\({}^{-1}\cdot\) K\({}^{-1}\)), \(V\) is the molar volume (m\({}^{3}\cdot\) mol\({}^{-1}\)), T is the absolute temperature (K), \(\omega\) is eccentric factor with the zero dimension, \(T_{c}\) is critical temperature, and \(P_{c}\) is the critical pressure \(P_{a}\)[22]. The gas-solid two-phase flow system is a thermodynamic system consisting of a large number of solid particles and bubble phases, and the internal energy of the thermodynamic system is a state function of the system. According to the internal energy differential formula, it is expressed as  
\[dU=C_{v}dT+[T(\frac{\partial P}{\partial T})_{v}+P]dV, \tag{2}\]  
where \(U\) is the internal energy of system, and \(C_{v}\) is the constant volume molar specific heat of the system. The differential form of the internal energy of the non-ideal thermodynamic system can be given by combining the thermodynamic state equations, which reads  
\[dU=C_{v}dT+\frac{a}{V^{2}+2 bV-b^{2}}dV. \tag{3}\]  
According to the first law of thermodynamics (\(dU=dW+Q\)), in the gas-solid two-phase flow system, the internal energy of gas changes through the quantitative non-ideal gas via the adiabatic free expansion process. This occurs outside for the gas in the adiabatic process with \(dW=-PdV\). Combined with the differential form of internal energy of non-ideal gas, we can obtain  
\[C_{v}dT+(\frac{a}{V^{2}+2 bV-b^{2}}+P)dV=0. \tag{4}\]  
In order to avoid tedious calculus, the PR equation can be simplified as \(PV=\zeta RT\), where the deviation from the non-ideal gas to the ideal gas is attributed to the deviation factor \(\zeta\). After solving the state equation for the differential coefficient, \(PdV+VdP=\zeta RadT\).