# HPB case study MALCOLM

Van Der Waals Equation of StateThe van der waals equation of state was first generated in 1873 by Johannes Diderik van der Waals. This equation of state was generated to change the Ideal Gas Law in order to adept to intermolecular interactions as gases have different behaviour when it is put under high pressure. This is because the gas law equation ideally describes gases at certain temperature and pressure according to de Visser 2011. The general equation of the Van Der Waals equation of state is listed below.

P = RTVm-b – aVm2 Figure 2.0: Correlation between the polarizability volume and van der waals value of a and b of 130 molecules (de Visser 2011).In 2011 Sam P. de Visser, proposed the parameters a and b based on fundamental physical constants and experimentally measurable coefficients. De Visser shows that the parameters a and b has a linear correlation to the polarizability volume, ± of liquid and gas of a substance through graphical means as shown in Figure 2.0 above. The figure shows that a= k1± and b=k2±.

Therefore, making the double parameter equation into a single parameter thermodynamic equation of state for diluted liquid and gas. Redlich Kwong Equation of State (RK)The equation was the first equation of state in the modern era. It is the modified version of the Van der Waal equation which takes temperature into consideration. Later on, in 1949 the equation is modified by Otto Redlich and Joseph Neng Shun Kwong. Temperature, pressure and volume of the gas are co-related in the equation. The Redlich Kwong Equation of state functions to improve the prediction of the liquid-phase molar volumes from the Van der Waal equation of state.

The application of RK is applied when modelling the solubility of methane with specific series of pressure and temperatures. The results are by estimating the binary interaction coefficient by data deterioration and then comparing those results on how accurately consistent the experimental results are (Markocic and Knez 2016).The general formula is shown below.P = RTV-b – aTV(V+b) , a = 0.42748 R2Tc2.5Pc and b = 0.08664 RTcPcThe following mixing rule equations are used for a mixture comprising of N – components.amix= i=1Nj=1Nzizjaij where aij = 0.42748 R2Tcij2.5PcijTcij= TciTcj 1-kij and Pcij= ZcijRTcijVcijZcij= Zci+ Zcj2 and Vcij=Vci1/3+ Vci1/3 23bmix= i=1Nzibi where bi= 0.0866 R Tci PciV is the molar volume, Tc is the temperature at critical point. Zc is the compressibility factor at critical point. The SRK equation of state will then be further modified in 1972 because of its limitation with non-polar hydrocarbons. This proposition is due its lacking accuracy with hydrogen bonding hydrocarbons. There was another improvement proposed by (Ratnawati 2010), is by modifying the co-volume parameter. Ratnawati quoted that the Redlich Kwong equation does not accurately predict density and phase-equilibria calculations including vapour pressures and solubility of solids in fluids with supercritical conditions. In 1993, Riazi proposed a modification which focused on the b parameter which is called the effective molecular volume or co-volume to have a more accurate prediction of density of fluids and supercritical fluids (Mohammad-Reza and Mansoori 1993). It is proven theoretically to be more effective in calculating for density as b represents the volume of molecules. The equation is referred to the RM equation. The RM equation has a modified parameter of b with inserting ћµ in the equation, in which ћµ is the function of Rm. Rmis the molar refraction which represents the molecules occupied by the volume. Figure 1.0: Relation between molecular weight and molar refraction (Ratnawati 2010)The RM equation predicted densities of 27 compounds with and average absolute deviation of 1.8% compared to the RK equation with an average absolute deviation of 11.6%. It shows improvement of predicted densities of dense gases and liquids using the RM equation. For heavy molecular weight compounds which does not have its parameter of molar refraction, ћµ is known as a function of molecular weight as molar refraction is correlated linearly to molecular weight nevertheless whichever compound according to Mohammad-Reza and Mansoori.Soave Redlich Kwong Equation of StateThe equation for Redlich Kwong is proposed and modified by Soave in 1972 and the equation is known to be the Soave Redlich Kwong equation of state. The modification recognizes the deformity in the molecules of the compound compared to the previous Redlich Kwong equation of state that does not recognize it. It gives a description of the vapour-liquid equilibrium behaviour of systems formed by light hydrocarbons. Part of the Soave Redlich Kwong equation’s frailty is that it has low accuracy when it comes to mixtures like CO2, H2S, CO, Nitrogen etc.The general formula is shown below.P = RTV-b – a(T)V(V+b)Where,aT=ac±T,±T=(1+m1-Tr)2 andm=0.480+1.574Ј-0.176Ј2 ac = 0.42747 R2Tc2Pc and b = 0.08664 RTcPcThe following mixing rule equations are used for a mixture comprising of N – components.amix= i=1Nj=1Nzizjaij Where, aij = a(T)ia(T)j 1-kijbmix= i=1NzibiFurthermore, another modification was done by Holderbaum and Gmehling based on the SRK equation in 1991. The equation is known as the PSRK, Predictive Soave Redlich Kwong equation of state. This equation uses UNIFAC (Universal Functional-group Activity Coefficients) system to calculate a. The PRSK can predict multi-component VLE data from binary information (Holderbaum and Gmehling 1991). The application of the Soave Redlich Kwong equation of state is to predict the binary interaction parameters of hydrocarbons such as methane, nitrogen, CO2, H2S and other light compounds according to Soave, Gamba and Pellegrini 2010. The equation’s mixing rule uses the Huron-Vidal mixing rules where the coefficients of limitless pressure are predicted by group contributions. The pressure coefficients are expressed by Scatchard-Hildebrand equations. According to the Soave, Gamba and Pellegrini’s method, it can obtain thermodynamic properties and phase behaviour of systems containing nitrogen, CO2, H2S. Using the proposed method, the results of the experimental data of VLE is more distinguishably accurate compared to the PRSK equation of state.ReferencesMarkocic, Elena, and Zeljko Knez. 2016. “Redlich”Kwong Equation of State for Modelling The Solubility of Methane in Water Over A Wide Range of Pressures And Temperatures”. Fluid Phase Equilibria 408: 108-114. doi:10.1016/j.fluid.2015.08.021.Soave, Giorgio, Simone Gamba, and Laura A. Pellegrini. 2010. “SRK Equation Of State: Predicting Binary Interaction Parameters Of Hydrocarbons And Related Compounds”. Fluid Phase Equilibria 299 (2): 285-293. doi:10.1016/j.fluid.2010.09.012.Ratnawati, Ratnawati. 2010. “IMPROVEMENT OF THE REDLICH-KWONG EQUATION OF STATE BY MODIFICATION OF CO-VOLUME PARAMETER” 13 (1): 58-65. doi:10.14710/reaktor.13.1.58-65.Mohammad-Reza, Riazi, and G.Ali Mansoori. 1993. “Simple Equation Of State Accurately Predicts Hydrocarbon Densities”. Oil And Gas Journal 91 (28): 4. doi:274429774.de Visser, Sam P. 2011. “Van Der Waals Equation Of State Revisited: Importance Of The Dispersion Correction”. The Journal Of Physical Chemistry B 115 (16): 4709-4717. doi:10.1021/jp200344e.Holderbaum, T., and J. Gmehling. 1991. “PSRK: A Group Contribution Equation Of State Based On UNIFAC”. Fluid Phase Equilibria 70 (2-3): 251-265. doi:10.1016/0378-3812(91)85038-v.