Calculating and Applying K-Values

1 Calculating and Applying K-Values Page 1 of 15. Calculating and Applying K-Values By Abdulreda Al-Saygh and Mahmood Moshfeghian Department of Chemical Engineering University of Qatar, Doha, Qatar And Robert N. Maddox School of Chemical Engineering Oklahoma State University Stillwater, Oklahoma, ABSTRACT. Several methods for determination of K-Values were reviewed and sample results are presented. Several case studies were presented to demonstrate their application. Based on the VLE calculations results, for light hydrocarbon systems, an equation of state should be used and for polar systems, an activity coefficient model is recommended. Introduction Modeling and design of many types of equipment for separating gas and liquids such as flash separators at the well head, distillation columns and even a pipeline are based on the phases present being in vapor -liquid equilibrium. The criteria for thermodynamic equilibrium between vapor and liquid phases are equality of temperature in both phases, equality of pressure in both phases, and equality of fugacity of each component in both phases.

2 The mathematical expression for the last equality in terms of the fugacity of component i, is written as: (1). Equation (1) is the foundation of vapor -liquid equilibrium calculations, however, we rarely use it in this form for practical applications. For calculation purposes, Equation (1) is transformed to a more commonly used expression: 2/23/2004. Calculating and Applying K-Values Page 2 of 15. (2). In Equation (2), Ki is called the vapor liquid equilibrium constant. Equ. (2) is also called "Henry's law". and K is frequently referred to as Henry's constant. For the more volatile components in a mixture the K-Values are greater than , whereas for the less volatile components they are less than Depending on the system under study, any one of several approaches may be taken to determine K-Values . Obviously, experimental measurement is the most desirable; however, it is expensive and time consuming. Alternatively, there are several graphical or numerical tools that can be used for determination of K-Values .

3 This paper presents a history of the development of many of those graphical methods and numerical techniques. In general K-Values for all components in a mixture are function of the pressure , temperature, and composition of the vapor and liquid phases present. The components making up the system plus temperature, pressure , composition, and degree of polarity affect the accuracy and applicability, and hence the selection, of an approach to estimating the K values. The widely used approaches are K-value charts, Raoult's law, the - approach and the - approach [1-5]. The last two approaches involve using an equation of state. Methods for Determining K-Values K-Value Charts There are several forms of K-value charts. One of the earliest forms of the K-value charts for light hydrocarbons is presented in reference [1]. In these charts, K-Values for individual components are plotted on the ordinate as a function of temperature on the abscissa with pressure as a parameter. In each chart the pressure range is from 10 psia to 1000 psia and the temperature range is from 40 F to 500.

4 F. Early high pressure experimental work revealed that, if a hydrocarbon system of fixed overall composition was held at constant temperature and the pressure was increased, the K-Values of all components converged toward a common value of unity ( ) at some high pressure . This pressure was termed the Convergence pressure of the system and has been used to correlate the effect of composition on K-Values . Plotting this way permits generalized K-Values to be presented in a moderate number of charts. In more recent publications [2], the K-Values are plotted as a function of pressure on the abscissa with Convergence pressure and temperature as parameters. In order to use these charts, one should determine the Convergence pressure first. The determination of convergence pressure is a trial-and-error procedure. Illustrative example calculations can be found elsewhere [6]. In 1958, for computer use, these K-Value charts were curve fitted to the following equations by 2/23/2004. Calculating and Applying K-Values Page 3 of 15.

5 Academic and industrial experts collaborating through the aegis of the Natural Gas Association of America [7]. (3). Raoult's Law Raoult's Law is based on the assumptions that the vapor phase behaves as an ideal gas and the liquid phase is an ideal solution. Under these conditions the fugacities are expressed as (4). (5). Substituting from Equations (4) and (5) into Equation (1) gives (6). The vapor pressure may be read from a Cox chart or calculated from a suitable equation in terms of temperature. A typical Cox chart can be found in reference [8]. The antoine [5] equation is recommended for Calculating vapor pressure . Complex vapor pressure equations presented by Wagner [5], even though more accurate, should be avoided because they should not be used to extrapolate to temperatures beyond the critical temperature of each component. Also, Raoult's law is applicable to only to low- pressure systems (up to about 50. psia) and to systems in which the components are very similar such as benzene and toluene.

6 This method is simple but it suffers when the temperature of the system is above the critical temperature of one or more of the components in the mixture. At temperatures above the critical point of a component, one must extrapolate the vapor pressure , which frequently results in erroneous K-Values . In addition, this method ignores the fact that the K-Values are composition dependent. Correlation Method As mentioned earlier, determination of K-Values from charts is not suitable for computer calculations. Therefore, scientists and engineers have developed numerous curve fitted expressions for calculation of K-Values . However, these correlations have limited application because they are specific to a certain system or applicable over a limited range of conditions. Some of these are polynomial or exponential correlations in which K-Values are expressed in terms of pressure and temperature. One of these correlations presented by Wilson [9], is: 2/23/2004. Calculating and Applying K-Values Page 4 of 15.

7 (7). This correlation is applicable to low and moderate pressure , up to about MPa (500 psia), and the K- values are assumed to be independent of composition. - Approach The - approach is based on using an Equation Of State (EOS). The fugacity of each component is determined by using an EOS. The same EOS describes both phases. The - approach is a powerful tool and it is relatively accurate if used appropriately. This approach is widely used in industry for light hydrocarbon and non-polar systems. In this approach the fugacities are expressed by (8). (9). Substitution of fugacities from Equations (8) and (9) in Equation (1) gives (10). In equation (10), fugacity coefficients must be determined from a generalized chart or calculated using an EOS. In order to calculate the fugacity coefficients for a mixture by an EOS, the mole fractions in both phases are needed in addition to pressure and temperature. Normally not all of these variables are known. Therefore, calculation of K-Values using an EOS is a trial and error procedure.

8 The - . approach is applicable to non-polar systems and yields good results up to about 15,000 psia. - Approach The so-called - approach is also based on using an EOS, but requires that the vapor phase non- ideality be described through the fugacity coefficient, with an activity coefficient model being used to account for the non-ideality of the liquid phase. This approach is widely used in industry even for polar systems exhibiting highly non-ideal behavior. Using this model the fugacities are expressed by (11). (12). The saturation fugacity coefficient for a component in the system, , is calculated for pure component i at the temperature of the system but at the saturation pressure of that component. 2/23/2004. Calculating and Applying K-Values Page 5 of 15. Normally, an EOS is used to calculate both and . Substitution of fugacities from Equations (11). and (12) in Equation (1) gives (13). Activity coefficients are calculated by an activity coefficient model such as that of Wilson [11] or the NRTL (Non-Random Two Liquid) model [12].

9 The required parameters for a few binary systems are given in reference [13]. In order to calculate the K-Values by Equation (13), the mole fractions in both phases, in addition to pressure and temperature, are needed. Normally not all of these variables are known. As is the case for the - approach, calculations are trial and error. The - approach is applicable to polar systems such as water ethanol mixtures from low to high pressures. Approach Normally, for low pressures, we can assume that the vapor phase behaves like an ideal gas;. therefore both and are set equal to Under such circumstances, Equation (13) reduces to (14). Equation (14) is applicable for low- pressure non-ideal solutions and polar systems. Assuming the liquid phase is an ideal solution, i becomes unity and Equation (14) is reduced further to a simple Raoult's law. The Impact of Computers on Calculating K-Values The accuracy of any process simulation by computer software depends directly on the accuracy of the K-Values used.

10 The K-Values are the essential ingredient for design and simulation of a separation system involving distillation columns, flash separators, etc. As computers were developed for engineering calculations, scientists and engineers strived for the development of generalized and accurate EOSs. Since the K-value charts are limited and simplified and calculation of K-Values based an EOS is trial and error and, consequently tedious and time consuming for hand calculation, the computer is ideally suited for this task. The first computer generated K-Values were based on the Chao-Seader EOS [14]. An equation similar to Equation (13) was used for this purpose. Later Erbar and Maddox [15] developed the K&H. Mod II software marketed by the Gas Processors Association for generating K-Values and enthalpies of light hydrocarbons as well as some selected non-hydrocarbon compounds such as nitrogen, carbon dioxide and hydrogen sulfides. This program which was based on the SRK EOS was well received by industry due to its accuracy, reliability and flexibility.