A New Cubic Equation of State for Improved Liquid Density Prediction

Introduction

By way of definition, an equation of state (EOS) is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Cubic equations of state are a class of equations of state that may be represented by a polynomial when referencing the volume or compressibility factor, in such a way that the highest power in the polynomial is to the third degree. The states of matter of interest for which natural gas and gas condensates are handled in the industry involve only two (vapour and liquid) phases for which a cubic equation is suitable. Cubic equations of state (EOS) when solved for molar volumes (or compressibility factor) give three values; the highest value corresponds to the vapour phase property, the lowest value corresponds to the liquid phase property while the intermediate value has no known significance. The roots of cubic equations of state can be obtained analytically without the need for an iterative solution procedure, which simplifies the solution method. The popularity of cubic EOSs is further enhanced by their structural simplicity, requirement of only a few parameters for implementation and little computer resources, thus assuring low computational overhead, while providing good phase equilibrium correlations and saturated phase volumes and densities of acceptable accuracy.

Literature Review

The simplest known equation of state is the perfect gas law, which is used for thermodynamic calculations for ideal gases, also called perfect gases. The perfect or also called, ideal gas is the simplest kind of gas and is in fact, an idealization of real gases as no gas is truly ideal. Most of the early works with gases were carried out at conditions near standard conditions of temperature and pressure; that is 𝑇= 60𝑜𝐹= 520𝑜𝑅= 288.72𝐾 and 𝑃= 14.7 𝑝𝑠𝑖𝑎= 101.325 𝐾𝑃𝑎. At these conditions, gas behavior approaches the behavior of the hypothetical ideal gas. The analytical expression of the PVT behavior of the hypothetical perfect gas behavior is written as:

PV=RT (1.0) Where, V is the molar volume of the container containing the fluid, P is the pressure of the fluid, T is the absolute temperature and R is the universal gas constant evaluated from R= PV/nT. The value of R depends on the units in which the parameters P, V, and T are evaluated. For example, at standard conditions of 14.7psia and 60oF (i.e.

𝑓𝑡3 𝑝𝑠𝑖

520 oR), 𝑅= 10.73159

𝑚𝑜𝑙 𝑅 𝑜..

At low pressures (≤400 𝑝𝑠𝑖𝑎) and moderately high temperatures (low densities), most real gases exhibit an almost ideal behavior, such that the volume varies directly with the absolute temperature and inversely with the pressure. Therefore, the ideal gas law would be able to approximately predict its PVT behavior. The closer, however, a gas is to a phase change, or when at high pressures (above about 400 psia) and at moderate temperatures; the more significant the deviation from ideal gas behavior. Also, the ideal gas law fails to predict condensation from a gas to a liquid. The deviation of real gases from ideal behaviour is captured by introducing a factor called the gas deviation factor, or gas compressibility factor or z factor since it is denoted by the letter, z. The resulting real gas equation is:

PV=zRT (1.1) By definition, z-factor is the ratio of the actual volume occupied by a mass of gas at some pressure and temperature to the volume the gas would occupy if it behaved ideally. The compressibility factor of an ideal gas is 1.0, i. e. for an ideal gas, z=1.0. In real applications, deviations from ideal behavior can be as large as 30% and such deviations must be corrected to improve accuracy. The value of the correction factor Z generally increases with pressure and decreases with temperature. At high pressures molecules are colliding more often. This allows repulsive forces between molecules to have a noticeable effect, making the molar volume of the real gas (𝑉𝑚)𝑟𝑒𝑎𝑙 𝑔𝑎𝑠 greater than the molar volume of the corresponding ideal gas(𝑉𝑚)𝑖𝑑𝑒𝑎𝑙 𝑔𝑎𝑠,, which causes Z to exceed one.[56] When pressures are lower, the molecules are free to move. In this case attractive forces dominate, making Z<1.0. The closer the gas is to its critical or boiling points, the more Zdeviates from the ideal case. If the gas deviation factor is accurately determined, the actual gas law can give tolerable estimates of gas thermodynamic behavior, but like the perfect gas law, it too, fails to predict the condensation of liquid from gas. This was one of the motivating factors in early equations of state research. The earliest attempt to correct for the departure of real gases from ideal gas behavior and extend the use of the ideal gas equation of state (EOS) to account for vapor- liquid co-existence was made by Johannes Diderick van der Waals in 1873, with his work on "the equation of state for gases and liquids" which won a Nobel prize in 1910. Van der Waal's (vdW) EOS is written as 𝑅𝑇 𝑎 𝑉2 (1.2) 𝑃= (𝑉−𝑏) − Where, the parameters, a is the "specific attraction" and b is the "hard sphere volume" or co-volume, so that molar volume cannot be smaller than b.Equation (1.2) is based on the fact that the pressure on the walls of a container on which gas molecules collide is decreased because of attraction by the molecules in the bulk gas and the volume in which the molecules move is less than the total volume by the excluded volume, b, due to the size of the molecules themselves. Therefore, the constants a and b have positive values and are characteristic of the individual gas. The vdW's EOS expressed in pressure-explicit form, as in Eq. (1.2), has the repulsive term as 𝑅𝑇 (𝑉−𝑏)and the attractive 𝑎 term as 𝑉2thus indicating that pressure can be regarded as the sum of two terms: a hard sphere term and an attraction term that is [1]

𝑃= 𝑃𝑟𝑒𝑝𝑢𝑙𝑠𝑖𝑜𝑛+ 𝑃𝑎𝑡𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 (1.3) Which by implication, implies that compressibility factor can be expressed as?

𝑍= 𝑍𝑟𝑒𝑝𝑢𝑙𝑠𝑖𝑜𝑛+ 𝑍𝑎𝑡𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 (1.4) The phase diagrams using the vdW EOS to model hydrocarbon fluid behavior reveals that, at temperatures equal to or greater than the critical temperature (𝑇𝑐), only single roots exist. However, at temperatures below 𝑇𝑐, three roots exist. The smallest root represents liquid-like volume, the intermediate root has no known significance and the largest root represents the vapour-like volume. The vdW EOS has limitations which include the facts that it provides inaccurate vapour pressure predictions because the attraction term parameter, a, is not optimized to fit vapour pressure. Also, it provides inaccurate critical point predictions. The critical compressibility factor, 𝑧𝑐 from the vdW EOS has a fixed value of 𝑧𝑐= 𝑃𝑐𝑉𝑐 𝑅𝑇𝑐= 0.375.

This value is quite greater than that obtained for real hydrocarbon fluids (0.24 to 0.29). The vdW's EOS however, has significant historical relevance since it provides the basic foundation over which most other researchers have built their framework by either modifying the attractive term or repulsive term of the original van der Waals Equation or both.

Redlich and Kwong's (RK) EOS made the first significant improvement on the vdW's EOS [2]. By modifying the volume dependence on the attraction parameter, a slightly improved critical compressibility factor value of 𝑧𝑐=0.3333 was obtained. However, the predictions on vapour pressure and liquid density values were still inaccurate. The RK EOS has the form $$ P = \frac {R T}{(V - b)} - \frac {a}{T ^ {1 / 2} V (V)} $$

⁄ 𝑉(𝑉+ 𝑏) To further improve prediction performance of EOSs, Wilson in 1964 introduced temperature dependency of the a parameter. Wilson's EOS has the form: 𝑃= 𝑅𝑇 (𝑉−𝑏) − 𝑎(𝑇) 𝑉(𝑉+𝑏) Where, 𝑎(𝑇) = 𝑎𝑐𝛼(𝑇) , 𝛼= 𝑇𝑟[1 + (1.57 + 1.62𝜔) (

1 𝑇𝑟−

1)] and 𝜔 is the acentric factor, introduced by Pfizer and defined as 𝜔= −1.0 −𝑙𝑜𝑔(𝑃𝑟 𝑠𝑎𝑡)(𝑎𝑡 𝑇𝑟= 0.7). Soave provided a more refined temperature dependency of the parameter of the original RK EOS [3]. The resulting effort, called, Soave-Redlich-Kwong (SRK) EOS improved results of vapour-pressure calculations. The SRK EOS has the same structural form as the RK EOS, however, the a parameter is defined as;

2 𝑎(𝑇) = 𝑎𝑐𝛼(𝑇) , 𝛼(𝑇) = [1 + 𝑚(1 −√𝑇𝑇𝑐 ⁄ )]

and 𝑚= 0.48509 + 1.55171𝜔−0.15613𝜔2. Peng and Robinson proposed additional modifications to more accurately predict the vapor pressure, liquid density, and equilibria ratios [4]. Peng and Robinson's (PR) EOS has the form: 𝑃= 𝑅𝑇 𝑎(𝑇) 𝑉(𝑉+𝑏)+𝑏(𝑉−𝑏).

(𝑉−𝑏) − Patel and Teja presented a three parameter EOS called Patel-Teja (PT) EOS in 1982, which further improved prediction of phase equilibria and volumetric properties [5]. The EOS has the form:

$$ P = \frac {R T}{(V - b)} - \frac {a (T)}{V (V + b) + c (0)} $$

𝑉(𝑉+ 𝑏) + 𝑐(𝑉−𝑏) Equations of state based on modifications of the attraction term of the original vdW's EOS, such as those described above and several others in literature are called van der Waal's family of EOSs and usually can be shown to be cubic polynomials when expressed in terms of molar volume or compressibility factors. They are therefore categorized as cubic EOSs. Cubic EOSs have been known to predict fluid phase equilibria data with appreciable accuracy though their ability to predict volumetric data, especially liquid densities has remained poor. To assure high accuracy, researchers sometimes have to use two or more equations to calculate densities and phase equillibria separately. Thieryet used the Soave-Redlich-kwong cubic equation (SRK) (1972) to model phase equilibria and a virial type equation to calculate equilibrium volumes [6, 7, 8]. This approach has the obvious disadvantage of inconsistency of various fluid parameters as first pointed out by Bakker and Diamond [9]. Some authors (Carnahan and Starling, Guggenheim, and Boublik) have also modified the repulsion term of vdW's EOS to obtain more accurate expressions for hard body repulsion [10, 11, 12]. Other researchers like Chen and Kreglewski, Christoforakos and Franck, Heilig and Franck modified both the attractive and repulsive terms of the van der Waals Equation of state to obtain improved thermodynamic property estimations for hard convex geometries [13, 14]. Summaries of some popular equations of state based on repulsion term modifications are shown in Tables 1 below:

4𝑉= 𝑝𝑎𝑐𝑘𝑖𝑛𝑔 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛; 𝑏= 𝑐𝑜𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑣𝑑𝑊. Other forms of equations of state exist besides those based on modifications of vdW's EOS and based on formulation and functionality can be categorized as virial or complex. A tree of EOS relationships based on intermolecular interactions is shown in Figure 1 below:

Click to enlarge