Analytically Approximate Solution to the VLE Problem with the SRK Equation of State

Analytically approximate solution to the VLE problem with the SRK equation of state Hongqin Liu* Integrated High Performance Computing Branch, Shared Services Canada, Montreal, QC, Canada Abstract Since a transcendental equation is involved in vapor liquid equilibrium (VLE) calculations with a cubic equation of state (EoS), any exact solution has to be carried out numerically with an iterative approach [1,2]. This causes significant wastes of repetitive human efforts and computing resources. Based on a recent study [3] on the Maxwell construction [4] and the van der Waals EoS [5], here we propose a procedure for developing analytically approximate solutions to the VLE calculation with the Soave-Redlich-Kwong (SRK) EoS [6] for the entire coexistence curve. This procedure can be applied to any cubic EoS and thus opens a new area for the EoS study. For industrial applications, a simple databank can be built containing only the coefficients of a newly defined function and other thermodynamic properties will be obtained with analytical forms. For each system there is only a one-time effort, and therefore, the wastes caused by the repetitive efforts can be avoided. By the way, we also show that for exact solutions, the VLE problem with any cubic EoS can be reduced to solving a transcendental equation with one unknown, which can significantly simplify the methods currently employed [2,7]. *Emails: [email protected]; [email protected]. 1 Introduction contrast to the volumes of saturated liquid and vapor phases. By taking advantages of special features of the Vapor-liquid equilibrium (VLE) calculation plays a M-line, we will propose a procedure for developing central role in thermodynamics and process simulations analytically approximate solutions to the VLE problem [1,7]. With an equation of state (EoS), the chemical for a given cubic EoS. We demonstrate the procedure potential equilibrium condition, or the Maxwell with the Soave-Redlich-Kwong (SRK) EoS [6] by using the constructions [4], generates a transcendental function information at the critical point from Singley, Burns and with logarithm terms. As a result, exact solutions have Misovich (1997) [9]. to be carried out numerically by some iterative methods. The pressure equilibrium condition and the Exact solution to the VLE problem by a cubic EoS transcendental equation are usually solved together, Before targeting an analytical solution, here we first which is not a simple task and becomes a research topic show that the VLE problem with a cubic EoS can always by itself [2]. Worst of all, the efforts made by one be reduced to solving a transcendental equation with individual will have to be repeated by another even for one unknown. We use the SRK EoS as the example to the same system at the same conditions, unless prove the statement and it is applicable to any cubic numerical data are published, which is not practical. The EoS. The SRK EoS reads [6]: costs of computing resources are significant as well. As 푅푇 푎(푇 ) reported [7], the phase equilibrium computation is the 푃 = − (1) most expensive step in chemical process simulations. 푣 − 푏 푣(푣 + 푏) Some efforts have been made for developing analytical where, 푃 is pressure, 푇, temperature, 푣, molar volume, solutions, but mostly in the neighbourhood of the 푅, the gas constant, 푎(푇) is a function of temperature critical point [8,9]. The method aimed at the diameter (see the Appendix) and 푏 is a constant. For the SRK EoS, [10,11] of the coexistence curve seems pointing to a due to the adoption of a third parameter (species- right direction, but it is limited by the fact that both dependent), the acentric factor,휔, all the calculations saturated liquid and vapor volumes (densities) are have to be based on individual substance cases. From complex functions of temperature, which makes even the chemical potential equilibrium condition, we have: empirical correlations difficult. In practical applications, 푣 푣 휃푏 휃푏 − − + = engineers have to use empirical correlations for those 푣 − 푏 푣 − 푏 (푣 + 푏) (푣 + 푏) most useful properties. A well-known example is the 푣 − 푏 푣 + 푏 푣 + 푏 Antoine equation [13] for vapor pressure (the 푙푛 + 휃 푙푛 − 푙푛 (2) 푣 − 푏 푣 푣 equilibrium pressure of a pure system) correlations. where 휃 = 푎(푇 )⁄ (푅푇푏), 푣 and 푣 are molar volumes Others properties, including heat capacity, saturated for liquid and vapor phases, respectively. For the exact densities etc., have to be correlated separately [13]. solutions, we will generate a quadratic equation first. Theoretically, if an EoS works good enough for a given Throughout of this work, we only consider the VLE case system, the saturated volumes of the liquid and vapor where temperature is given. Applying the pressure phases will provide all the information needed for any equilibrium condition to liquid and vapor phases (see thermodynamic properties along entire the the Appendix, Eq.(S1)), we get: coexistence. 퐷 푣 − 푣 − (퐷 푏 +1− 휃)푏푣 − 휃푏 =0 (3) Moreover, for some applications where derivative properties, such as the vapor pressure vs temperature, where the liquid volume, 푣, has been replaced by a numerical solutions face difficulties. But with analytical dummy variable, 푣, and solutions, this task usually becomes minor, if not trivial. 1 휃푏 Consequently, analytical solution to the VLE problem is 퐷 = − (4) 푣 − 푏 푣(푣 + 푏) the ultimate goal for engineers and researchers. Eq.(3) is a cubic function. It is easy to prove that at When a cubic EoS is applied to the VLE calculation, temperature, 푇 ≥ 푇, there is only one root and as 푇 < subjected to the Maxwell construction, which is 푇 there are always three real (> 0) roots, and we denote equivalent to the combination of the pressure and them as 훼, 훽, 훾. According to the root-coefficient chemical potential equilibrium conditions, two real relations for a generic function, 푝푥 + 푞푥 + 푟푥 + 푠 = roots are for saturated volumes of liquid and vapor 0, we know that: phases, respectively, and the third “unphysical” root is 푞 1 the Maxwell crossover or the M-line [3]. It is found that 훼 + 훽 + 훾 =− = (5) 푝 퐷 the M-line is a smooth function of temperature, in 2 푠 휃푏 1 훼훽훾 =− = (6) Cubic EoS 푝 퐷 푮 M-line 푟 퐷푏 +1− 휃 0.8 훼훽 + 훽훾 + 훼훾 = =− (7) 푪 푝 퐷 풆 푷 푻 푫 푩 Now we assume that 훼, 훽 represent two volumes: 푣, 푭 푷 0.6 푣, respectively, where 푣 is the liquid molar volume 푨 and 푣, the unphysical intermediate volume named as Liquid the volume of M phase. From Eq.(5)-(7), we can easily 0.4 푬 Vapor prove that 훾 = 푣, namely the vapor volume. Now we can construct a quadratic equation that has the same 풗푳 풗푴 풗푮 0.2 two roots (푣, 푣) as those from Eq.(3): 0 0.5 1 1.5 2 2.5 3 푣 + 푢푣 + 푤 =0 (8) 풗 where 1 Figure 1. The Maxwell construction and the M-line. At ( ) 푢 =− + 푣 9 equilibrium condition: area FEDF = area DBCD. 퐷 휃푏 The Maxwell can be written analytically as: 푤 = (10) 퐷 푣 1 푃 = 푃푑푣 (12) 푣 − 푣 Then we have the solution for liquid volume: −푢 − √푢 −4푤 where 푃 is the equilibrium pressure at a given 푣 = = 푓(푣 ) (11) 2 temperature. By re-arrangement, we have: where 푓(푣) is a given function of 푣. In Eq.(2), (푃 − 푃)푑푣 = (푃 − 푝 )푑푣 (13) replacing 푣 with Eq.(11), we have a transcendental equation with only one unknown, 푣, and this ends our which is the EQR. Eq.(13) means that area FEDF = area proof for previous statement. ∎ DBCD. Therefore, the EQR generates an intermediate With traditional method [3], the pressure and chemical volume, 푣. It should be mentioned that the Maxwell equilibrium conditions are solved at the time and high construction applies to any EoS, not necessarily a cubic quality initial guesses are demanded. The new method one. As a cubic EoS is adopted, the value of 푣 is the significantly simplifies the solution process. It is found “unphysical” third root since there is only one that the transcendental equation, Eq.(2) and (11), is intersection between the pressure curve 푃(푣) and the stable given a reasonable initial volume. It can be easily equilibrium pressure, 푃 . From Figure 1, we also see solved with the Excel Solver. All the exact solutions used that: in this work are obtained this way. 푃(푣) = 푃(푣) = 푃(푣) = 푝 (14) The Maxwell construction and the M-line The state at 푣 represents an unstable state, since 휇(푣 ) ≠ 휇(푣 ) or 휇(푣 ), where 휇 is the chemical A full discussion on the Maxwell construction (known as potential. This means that a thermodynamic property at the equal-area rule, EAR) and the crossover is presented in [3] with the van der Waals EoS as an example. Figure this state is not a equilibrium property. Since we are only interested in the volume itself, which turns out to 1 illustrates the Maxwell construction and related be physically meaningful, the treatments below are quantities to help our presentations. physically sound. The trajectory of the intermediate volume, 푣, is called the Maxwell crossover or the M- line (Figure 1). The M-line can be easily calculated by the SRK EoS. In fact, the “unphysical” root of Eq.(8) is the intermediate volume as function of 푣: −푢 + √푢 −4푤 푣 = (15) 2 Therefore, as discussed above, the solution to the transcendental equation, Eq.(2), will give both 푣 and 푣 at a given temperature by Eq.(11) and (15), 3 respectively, as 푣 is known.

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