Other Equilibrium Relationships Relative Volatility It will be convenient later on in separation problems to express the relative volatility in terms of the <a href="/keyword/mole-fraction/" >mole fraction</a> s and vice-versa For binary systems, the <a href="/keyword/mole-fraction/" >mole fraction</a> s are related by y B = 1 - y A and x B = 1 - x A and substituting these into Eq. (2-20) yields: Eq. (2-4) AB = y A (1 - x A ) (1 - y A )x A 1 - xA 1 - yA Eq. (2-22) or AB = K A Solving Eq. (2-22) for yA and xA yields: yA = AB x A 1 + ( AB - 1) x A yA AB - ( AB - 1) y A Eq. (2-23) and xA = Lecture 6 1 Calculation of BubblePoint and DewPoint Temperatures The bubblepoint temperature is the temperature at which a liquid mixture begins to boil. The dewpoint temperature is the temperature at which a vapor mixture first begins to condense. Lecture 6 2 Temperature-Composition Diagram for Ethanol-Water, P = 1 atm 100 95 Two Phase Superheated Vapor Phase 90 T( C) 85 80 Subcooled Liquid Phase o 75 0.0 0.2 0.4 zEtOH 0.6 0.8 1.0 3 xEtOH or yEtOH Lecture 6 Calculation of BubblePoint Temperatures If one is given a liquid mixture, one often needs to determine the bubblepoint temperature of the mixture. We have done this to date using equilibrium data for binary systems for example, from the saturated liquid line on a T vs. x,y plot for a given feed composition, zi. How do we handle multicomponent systems? Lecture 6 4 Calculation of BubblePoint Temperatures If the feed is in the liquid phase, the pressure p and the composition, xi's, of the liquid phase will be given. One then needs to determine the bubble point temperature of the mixture. Lecture 6 5 Calculation of BubblePoint Temperatures Where do we start? Well, we are given the liquidphase <a href="/keyword/mole-fraction/" >mole fraction</a> s, and we know that the <a href="/keyword/mole-fraction/" >mole fraction</a> relationships for both the liquid and vapor phase are given by Eq. (213): x i =1 N i = 1.0 y i =1 Lecture 6 N i = 1.0 6 Calculation of BubblePoint Temperatures Equilibrium Relationship We are given the liquidphase <a href="/keyword/mole-fraction/" >mole fraction</a> s, xi's, and we need to &quot;link&quot; the vaporphase <a href="/keyword/mole-fraction/" >mole fraction</a> s, yi's, to the liquidphase <a href="/keyword/mole-fraction/" >mole fraction</a> s to do this, one can use the definition for the equilibrium distribution coefficient K, Eq. (210) and solve for yi: y K= x i i or i yi = K x i i Lecture 6 7 Calculation of BubblePoint Temperatures Equilibrium Distribution Coefficient Relationship Substituting Eq. (210) into Eq. (213) yields: [ K x ] = 1.0 N i =1 i i K is a function of both temperature and pressure, Eq. (211): Ki = K(T, p) Lecture 6 8 Calculation of BubblePoint Temperatures Equilibrium Distribution Coefficient T,P Relationship One needs an expression for K as a function of T and P. One convenient expression for K as a function of T and P for light hydrocarbons is the DePriester equation, Eq. (212): a T1 a T2 a p2 a p3 K = exp 2 + + a T6 + a p1 ln p + 2 + T T p p Lecture 6 9 Calculation of BubblePoint Temperatures Analytical Expression Substituting the Depriester equation, Eq. (2 12), into [ K x ] = 1.0 N i =1 i i yields the analytical expression for bubble point calculations: a i,T1 a i,T2 a i,p2 a i,p3 + a i,T6 + a i,p1 ln p + 2 + xi = 1.0 exp 2 + T i =1 p p T N Lecture 6 10 Calculation of BubblePoint Temperatures One must solve the bubblepoint expression for T. There are several ways to solve for T: 1.) If the expression for K is simple enough, one may able to algebraically solve for T e.g., if some of constants in the DePriester equation are 0. 2.) One may use a trail and error method as outlined in 213, Wankat, p. 29. 3.) One may solve numerically. be the Fig. Lecture 6 11 Lecture 6 12 Calculation of DewPoint Temperatures If the feed is in the vapor phase, the pressure p and the composition, yi's, of the vapor phase will be given. One then needs to determine the dewpoint temperature of the mixture. Lecture 6 13 Calculation of DewPoint Temperatures Just as one may derive the bubblepoint temperature relationship, one can use a similar derivation using the equilibrium coefficient equation definition, Eq. (210), but this time solving for xi, since one would be given the vaporphase <a href="/keyword/mole-fraction/" >mole fraction</a> s, yi's: yi or Ki = x i Lecture 6 y x= K i i i 14 Calculation of DewPoint Temperatures Equilibrium Distribution Coefficient Relationship Substituting Eq. (210) into Eq. (213) yields: y = 1 .0 K N i =1 i i K is a function of both temperature and pressure, Eq. (211): Ki = K(T, p) Lecture 6 15 Calculation of DewPoint Temperatures Analytical Expression Substituting the Depriester equation, Eq. (212), into y = 1.0 K N i =1 i i yields the analytical expression for dewpoint calculations: yi i =1 N a i, T1 a i, T2 a i, p2 a i, p3 / exp 2 + + a i, T6 + a i, p1 ln p + 2 + = 1.0 T p p T Lecture 6 16 Calculation of BubblePoint and DewPoint Temperatures Numerical Solutions One can conveniently use numerical methods for these types of problems, using, for example, Mathcad. Mathcad uses nonlinear numerical methods such as the QuasiNewtonian or LevenbergMarquardt algorithms to solve equations. Lecture 6 17 A Final Note! While we will solve bubblepoint and dew point temperature problems for a given pressure, there is no reason why this same methodology cannot be applied to determining bubblepoint and dewpoint pressure problems for a given temperature! Lecture 6 18