Vapor-liquid Equilibrium of the binary mixture ethanol-water at moderate pressures from the concept of Excess Properties What is Expected Submit your results in form of a typed report. Font size is single spaced 12 pt Calibri (body). Margins all around must not be less than 1 inch. An introduction is not necessary. The repox must be as brief and precise as possible. Use shox sentences. Identify all quantities used in the formulas. You can talk to other students and exchange ideas. But emphasis will be placed on individuality of the repoits. Signs of plagiarism will be severely penalized. Problem Statement The process design division of your company has asked you to mathematically model to describe vapor-liquid equilibria to enable them to design large scale separation columns for binary systems. In paiticular, your company wants to separate ethanol from water. They need a guideline for relating xi, yi, T and p necessary for process simulation. Alternatively, they wish to be able to set up T-x-y, p-x-y and x-y diagrams. Procedure The procedure we will follow is the one we have used in the class. 1. We will first find experimental data for vapor liquid equilibria for ethanol-water system. Do the internet search and locate some experimental data T-x-y at a pressure of 1 atm. Since most distillation processes are performed at atmospheric pressure, experimental T-x-y data at 1 atm pressure will be ideal for our work. 2. We will use these data to calculate experimental activity coePicients Y1 and Y2 at different xs and ys. Remember y$ p(T,p,x x y) $1 (T ,P,X1,X2) = /2 (T,P ,+1 X2) _y 2 p(T,p, , 2) p2 sT) In the above equations, we assume that the vapor phase behaves as an ideal gas. 3. Use just determined ys to calculate experimental molal excess Gibbs Free Energy GE (T,P,X1, 2) = RT [xi In y1 (T,p,xi, x2)] + x2 In y2 (T,P,X1,X2)] 4. In addition to tabulating the values of ys, x, y and T, you should also tabulate the values of GE/RTx1x2 and RTx1X2/GE. 5. Now you have experimental values of GE. You wish to fit GE to three mathematical models. These are 1) one parameter Margules model ( GE/RTx1X2 = A), 2) two-parameter Margules model [GE/RTx1x2 = A + B(x1 -x2)] and 3) two-parameter van Laar model [RTx1X2/GE = A + B(x1 X2) 6. Now perform the least square fits and determine the best values of A for one parameter Margules model, A and B for two parameter Margules model and A and B for van Laar model. 7. The above models for GE do not contain pressure and temperature explicitly. In general, the effect of pressure is negligible. However, temperature does have a significant impact. The parameters in the above models are indeed temperature dependent. Foxunately, the temperature eflect on vapor liquid equilibria primarily comes through the temperature dependence of the pure component vapor pressures in the above equations. Typically, a temperature increase of 1OC raises the vapor pressure by a factor of 2. However, the activity coefiicients over the same temperature change by only a few percentage points which is within the experimental uncertainty. As a result, the temperature dependence of GE is neglected too. Please keep in mind that this will not apply to liquid-liquid equilibria as no vapor pressures would be involved there. 8. Now you are ready to predict vapor liquid equilibria for a wide range of conditions. Please remember, you cannot expect very good results for conditions that are far removed from the conditions of underlying experimental data. Also please remember, these models do not explicitly contain T and P. 9. The procedure is outline below. a. Choose a mathematical model for GE (T,p,X1,X2). b. Determine the parameters of that model from the limited experimental data as follows. C. Use modified Raoults law to determine the numerical values of yi and y2. d. Use Equation (1) to calculate numerical values of GE at different experimental values of liquid compositions. e. Try and fit these values to three models: 1) Margules one constant model, 2) Margules two-constant model and 3) two-constant van Laar model. f. Determine the associated parameters from these linear fits or least squares analysis. g. Construct p-x-y and T-x-y diagrams. 10. Mathematical models for GE a. One parameter Margules Model The model contains only one parameter, A. c E b. Two parameter MargulesModel This model has two parameters A and B. G A -I- B( 1 2) (1) Jt7x i 2 = A( 1+ 2)- I- B ( 1 " 2) (2) C RT 2 {A -F B) * i -*- (fi B)x 2 (3) C ' A2 1 1+ ' A12 ) 2 (4) z x,2 (L iz- I 2( fi zi L iz) i) (5) Iny = x (A -*- 2( A12 A21) 2) (6) c. Two parameter van Laar Model The model has two parameters A and B. ATii*2 c A -1- B'( 1 2 ) (7) A'(= + xz) -F B(xc =2 ) {8) (A -F B)x + (A B)xz) (9) 1 A 12 2 An A1 2 $1 -J- A 2 x 2 fny2 l ii $1 -F (10) 2 (11) (12) 11. Fit the GE/RTx1X2 data to all three models and determine the best values of the parameters for each case. Then you can use the corresponding equations to evaluate ys over the entire composition range. This will enable you to calculate T-x-y and P-x- y diagrams over the full composition range. 12. Adopt the usual convention that component 1 is the more volatile component than 2. It implies that it has a higher vapor pressure at a given temperature than the component 2. You can calculate the vapor pressure of any chemical specie from Antoine equation B; Here the values of A, B, C can be looked up from any reference book. Just be certain to use appropriate units as specified by the source. 13. In cases where the binary mixture forms an azeotrope, one may be able to use the azeotrope information (T, P, azeotropic composition) to ones advantage to calculate the model parameters. At azeotropic composition, xi = y1 and x2 = y2. Hence yz*{T, P, *i. x2 ) p@{T, P,*i. xz) = Sat 1 Hence if you know the azeotropic composition and T and P of the azeotrope, one can easily calculate the activity coefficients of both components at the azeotropic conditions. One can then use these ys to calculate the parameters. One parameter Margules Model A 1 or 2 A y2 If the As turn out to be difierent, which one would you choose? Since the formation of an azeotrope is a highly non-ideal efiect, and the activity coefiicient is a measure of non-ideality, one may assume that the larger value of I!rtr: Icarries more significant information. Hence choose A coming from that component. Two parameter Margules Model Here the model parameters can be easily computed from the azeotropic point information. As' 2 xz) A21 j ) 1 Once A12 and A21 are known, you can use Equations 5 and 6 to calculate ys at any composition. Two parameter van Laar Model Here the model parameters can be easily computed from the azeotropic point information. A1 2 ln y 1 -1- A2 = In yg* 1 -I =2 ln yg Once these parameters are known, you can use Equations (11) and (12) to compute ys at any value of x. 14. Construction of T-x-y diagram at a fixed P and P-x-y diagram at a fixed T These will be discussed in class. 15. Check for thermodynamic consistency of the experimental The measured experimental data must conform to data. The Gibbs-Duhem Equation has to be satisfied. x[d in/ (T, P, . 2)] at constnnt T rind P 0 Of course, for the binary system, the above equation reduces to x1 [d In y1 (T,P,x1,x2)] + x2 [d In v2 (T,P,x1, x2)] =O The activity coefiicients yi and Y2 in their common mixture are not completely independent of one another. You can integrate the above equation numerically to check if the sum of the two terms is equal to zero or close to zero.