10.34 Numerical Methods Applied to Chemical Engineering Fall Homework #3: Systems of Nonlinear Equations and Optimization

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1 10.34 Numercal Methods Appled to Chemcal Engneerng Fall 2015 Homework #3: Systems of Nonlnear Equatons and Optmzaton Problem 1 (30 ponts). A (homogeneous) azeotrope s a composton of a multcomponent mxture for whch the vapor phase n equlbrum wth a sngle lqud phase has the same composton as the lqud phase,.e., for a C-component mxture: y = x where y R C and x R C are vectors of vapor and lqud phase mole fractons, respectvely. Note that by ths defnton the pure component compostons are also azeotropes. Mxtures of collectons of components that can form azeotropes can be dffcult to separate usng dstllaton because the azeotropes can block separatons that would be possble n an deal mxture. The classcal example of ths s the water-ethanol azeotrope, whch prevents pure ethanol beng dstlled from the results of a fermentaton. When desgnng a separaton process for a mxture of a collecton of components that can form azeotropes, the frst step s to dentfy all the azeotropes that can be formed, from whch the blocked separatons can then be determned and possbly crcumvented by clever desgn. Ths week s homework s about computng all the azeotropes occurrng n mxtures of ethanol, methyl ethyl ketone and water at constant pressure P = 1 atm. Assumng a certan class of vaporlqud equlbrum model, ths amounts to fndng all physcally relevant solutons of the system of nonlnear equatons: y = x, = 1,..., C 1, γ (x, T )P SAT (T ) y = x, P = 1,..., C, (VE) C x = 1, =1 C y = 1. =1 Note that ths s a system of 2C + 1 equatons n the 2C + 1 varables (x, y, T ) (.e., at constant pressure there s a temperature assocated wth each azeotrope). For a soluton to be physcal t must also satsfy the nequaltes: x, y, T 0. Compute the actvty coeffcents γ (x, T ) usng the Wlson model: Λjx j ln γ = 1 ln Λ k Λ, jkx k Λ j = V j V j jx j j ( λ exp j λ RT ).

2 Component Molar Volume (m/mol) Interacton Energy Dfference (cal/mol) V λ E λ EE λ M λ MM λ W λ WW E M W Table 1: Parameter values needed to compute bnary parameters for the Wlson actvty coeffcent model. Takng E, M and W as ethanol, methyl ethyl ketone and water, respectvely, the necessary data are gven n Table 1. The pure component vapor pressures P SAT (T ) should be calculated usng the Antone equaton: ( ) P sat B log = A, (1) torr C + T ( o C) wth parameter values gven n Table 2. Component Antone Parameters A B C E M W Table 2: Antone parameters for Equaton (1). 1. Wrte functons antone_eq and wlson_actvty that calculate P sat (T ) and γ (x, T ), respectvely, for all three components (E, M and W). The functon antone_eq takes T as an nput and returns a vector of vapor pressure values that are calculated based on the Antone s equaton. Test that your antone_eq functon s workng correctly by checkng these values: at T = 100 o C, PE sat =1694 torr, PM sat =1400 torr, and PW sat =760 torr. Smlarly, wlson_actvty functon takes a vector of compostons (x) and a temperature value T as nputs and returns a vector of actvty coeffcents that are calculated based on the Wlson model. Test that your wlson_actvty functon are workng correctly by checkng these values: at T = 70 o C ( K), x E = 0.25, x M = 0.5 and x W = 0.25, γ E = , γ M = and γ W = Use the equatons y = x to elmnate y and derve a system of C + 1 equatons just n (x, T ) of the form: f(x, T ) = 0. Wrte a functon called functon_evaluator_wlson for your equatons f(x, T ) = 0. Ths s a functon that takes as nput a vector z = (x, T ) and returns a vector of the values of the functons f defnng your system of equatons at z. Test that your functon s functonng correctly by applyng t to the pure component compostons. 3. Wrte a MATAB scrpt that uses the bult-n routne fsolve to solve the equaton f(exp(xˆ), exp(t ˆ)) = 0 for xˆ and T ˆ, where the composton and temperature are defned n terms of the auxlary hat varables as x = exp(xˆ) and T = exp(t ˆ). Ths change of varables s recommended because t ensures that x and T are postve throughout the soluton process. However, wth

3 ths change, some approxmaton of the pure component solutons s requred. Why? Use your scrpt to solve for the azeotropes n ethanol, methyl ethyl ketone and water mxtures at 1 atm by provdng fsolve wth a selecton of dfferent ntal guesses. Report values for any numercal tolerances used. Prepare a table dsplayng nformaton on the ntal guesses used and the correspondng number of teratons and solutons found. HINT: There should be a total of 7 azeotropes: 3 pure compoments, 3 bnary mxtures, and 1 ternary mxture. 4. As an alternatve to randomly selectng ntal guesses, replace Equaton (VE) wth the followng physcally-nspred homotopy P SAT (T ) y = ((1 λ) + λγ (x, T )) x, = 1,..., C, P where the vapor-lqud equlbrum model s Raoult s law when λ = 0 and the Wlson model when λ = 1. Implement n a basc contnuaton method wth ths homotopy by copyng and modfyng the code you had prevously wrtten. Explan the steps you took to do ths. Plot T ( o C) versus 0 λ 1. for all three pure compoments to make sure that your code can track the soluton branches correspondng to the three pure component compostons from λ = 0 to λ = Along the pure component branches, your homotopy method wll encounter bfurcatons sgnaled by a change n the sgn of the determnant of the Jacoban of f(x, T ). In fact, t can be shown that any bnary azeotropes occur on soluton branches that bfurcate off the pure component branches at some 0 < λ < 1. Smlarly, any ternary azeotropes occur on soluton branches that bfurcate off the bnary azeotrope branches at some 0 < λ < 1. Copy and then modfy your homotopy method to detect these bfurcatons. At (or just before) a bfurcaton pont, your code should pck up a new branch of the soluton by perturbng your ntal guesses n a drecton ndcated by vectors n (or near) the null space of the Jacoban. Use your code to locate systematcally all azeotropes n ethanol, methyl ethyl ketone and water mxtures at 1 atm. Prepare a table wth the composton and temperatures of all the azeotropes. Prepare a bfurcaton dagram plottng the temperature of all soluton branches versus 0 λ 1. Problem 2 (20 ponts). In ths problem you wll optmze the desgn of a chemcal reactor usng MATAB s bult-n constraned optmzaton tools. We have a CSTR (contnuous strred tank reactor) whch produces speces C from reactants A and B usng the followng reacton network. Note that n ths problem the reacton rate coeffcents are gven as functons of temperature. A + B C + D C + B S 1 + D A + D S 2 A + B S 3 C + B S 4 r 1 = k 1 (T )C A C B r 2 = k 2 (T )C C C B r 3 = k 3 (T )C A C D r 4 = k 4 (T )C A C B r 5 = k 5 (T )C C C B We provde to the CSTR an nput stream wth a volumetrc flow rate of Q = 10 /s contanng speces A and B n a solvent. To provde better temperature control, the total concentraton of A and B s kept mnmal: mol C An + C Bn = 0.1.

4 Rate data has been collected and an Arrhenus dependence of the rate on temperature can be assumed for all k j (T ). k 1 (298K) = 0.01 k 1 (310K) = 0.02 k 2 (T ) = k 1 (T ) k 3 (298K) = k 3 (310K) = k 4 (298K) = k 4 (310K) = k 5 (T ) = k 4 (T ) We desgn the reactor to operate sothermally so as to maxmze the concentraton of our desred product C n the output stream. We have three desgn varables: the rato of reactant nlet concentratons α C An /C Bn, and the volume of the reactor whch s restrcted to the range 100 V 10, 000, and the operatng temperature 298 T 335 K. 1. Report the constants for all rate laws based upon the data provded. 2. Wrte a MATAB functon that computes the steady-state concentratons of A,B,C and D. Ensure your code returns stable steady state values. Descrbe how you calculate steady-state concentratons for ths system and how you confrm such results are stable steady states. 3. Usng your MATAB functon for the steady-state concentratons n the system utlze fmncon n MATAB to optmze the desgn of the reactor to maxmze the concentraton of C. Descrbe the optmal desgn pont, your objectve functon, and the mplementaton of the constrants n your report. Do you need to add addtonal constrants to keep the system physcally meanngful? Use a value of for the functon and x value tolerances (see optmoptons for syntax). 4. We may choose to desgn to maxmze the yeld of C nstead. In ths case we have a dfferent objectve functon. Report your objectve functon under ths scenaro and use fmncon n MATAB to optmze the desgn of the reactor. Descrbe the optmal desgn pont, and f t dffers from the prevous optmal desgn pont descrbe how and why. If you needed to add any constrants descrbe them and why they were added. 5. Produce a contour plot ndcatng the poston and objectve value of each desgn pont. Make your plots at constant T, wth the objectve on the z axs and α and V as the x and y axes Staff

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