F, {z} Isothermal Flash Tank

Transcription

1 Exam III dministered: Thursday, November 7, points For each problem part: points i not attempted or no work shown, point or partial credit, i work is shown, 2 points or correct numerical value o solution Problem. ( points) Consider an isothermal lash tank: F, {z} Isothermal Flash Tank V, {y} {x} This unit takes a pressurized liquid, two-component eed stream and exposes it to a low pressure vessel maintained under isothermal conditions. The net result is that some o the luid is orized, while some luid remains liquid. The compositions o the liquid and or phase are determined by the combined analysis o mass balances and Raoult s Law or or-liquid equilibrium. The temperature in the lash tank is T 298K and the pressure in the tank is P kpa. F mol / hr z z.4.6 V 4mol / hr y y?? L F - V mol / hr x? x? You have our unknowns, the compositions o the liquid stream, stream, y and y. You also have our equations. x and x, and the composition o the or material balance on moles o : liquid mole raction constraint: or mole raction constraint: one equilibrium constraint: Fz Lx Vy x x y y xp yp (Raoults law or or-liquid equilibrium) where T 298K P (a) Is the system o equations linear or nonlinear in the unknowns? (b) Solve or the our unknown mole ractions. Show your work. Emphasize the steps in the procedure.

2 Solution (a) Equations are linear in the unknowns. Thereore, use linear algebra to solve. (b) Solve. Put equations in linear orm Lx Vy Fz x x y y x P y P Put equations in matrix orm matrix o coeicients,, (4x4) eqn/var x x y L V P P vector o right hand sides, b, (4x) eqn b Fz I then created a matlab script to solve this problem, xm3p_4.m. clear all; F = ; mol/hr V = 4; mol/hr L = F - V; mol/hr z =.4; z =.6; P = 6; kpa P = ; kpa = [L V P -P ] b =[F*z; ; ; ] det = det() x = \b; x = x() x = x(2) y 2

3 y = x(3) y = x(4) Upon execution, the output o this script was as ollows, >> xm3p_4 = b = 4 det = 846 x =.4775 x =.5225 y =.2837 y =.763 The mole ractions are provided above. 3

4 Problem 2. ( points) Consider the same system given in Problem. In a more realistic version o the problem, not only are the the compositions o the liquid stream, x and x, and the composition o the or stream, y and y, unknown, but the liquid and or stream lowrates, L and V respectively are also unknown. In this case, you have six unknowns and six equations. The equations are material balance on total moles: material balance on moles o : liquid mole raction constraint: or mole raction constraint: equilibrium constraint on : equilibrium constraint on : F L V Fz Lx Vy x x y y xp yp (Raoults law or ) xp yp (Raoults law or ) where P T 298K and P T 298K (a) Is this set o equations linear or nonlinear in the unknowns? (b) Come up with a good set o initial guesses or the solution. (c) Solve or the lowrates and compositions o the liquid and or streams. Show your work. Emphasize the steps in the procedure. Solution (a) Equations are nonlinear in the unknowns. Thereore, use a multivariate rootinding technique, such as the multivariate Newton-Raphson method with numerical derivatives to solve. (b) Come up with a good set o initial guesses or the solution. good set o initial conditions comes rom the solution to Problem. x =.4775 x =.5225 y =.2837 y =.763 L = 6 V = 4 (c) Solve or the lowrates and compositions o the liquid and or streams. Show your work. Emphasize the steps in the procedure. Put equations in appropriate orm x, x2, y, V F L V x, x2, y, V Fz Lx Vy x x, y, y, V x x, 2 2 x, x2, y, V y y x, x2, y, V x P yp x x, y, y, V x P y P , 2 2 4

5 I then created a matlab script to solve this problem, xm3p2_4.m. Note that this script irst solves problem again in order to obtain a consistent guess or the six variables. clear all; come up with good initial guess rom problem x =.4775; x =.5225; y =.2837; y =.763; L = 6; V = 4; create initial guess vector x=[x,x,y,y,v]; tol =.e-6; iprint = ; call multivariate Newton-Raphson with Numerical derivatives [x,err,] = nrndn(x,tol,iprint); identiy solutions x = x() x = x(2) y = x(3) y = x(4) L = x(5) V = x(6) The input ile or nrndn.m is given by unction = unkeval(x) n = max(size(x)); = zeros(n,); identiy variables x = x(); x = x(2); y = x(3); y = x(4); L = x(5); V = x(6); assign constants F = ; mol/hr z =.4; P = 6; kpa P = 4; kpa P = ; kpa write equations () = F - L - V; 5

6 (2) = F*z - L*x - V*y; (3) = x + x - ; (4) = y + y - ; (5) = x*p - y*p; (6) = x*p - y*p; Upon execution, the output o this script was as ollows, >> xm3p2_4 iter = iter = iter =, err = 2.48e+ = 3.25e- 2, err = 5.8e-2 = 7.e-3 3, err = 3.e-4 = 4.35e-5 x =.4875 x =.525 y =.2896 y =.74 L = V = The mole ractions are provided above. The lowrates, L and V, are also given with units o mol/hr. Problem 3. (2 points) n even more realistic version o this problem occurs when the lash tank is operated under adiabatic rather than isothermal conditions. In this case, the temperature o the system is unknown. Discuss briely how one might solve this problem. What kind o equation would be added to account or the new variable? What kind o additional parameters would be needed? What technique could you use to solve this? Solution The new equation would be an energy balance. We would need or pressures or pure components as a unction o temperature, not just at one point, rom something like the ntoine equation. The system o seven equations and seven unknowns (at least or a binary system) would be nonlinear and would require a technique like the multivariate Newton Raphson method with numerical derivatives as used in problem 2. 6