Consider the design of a continuous-stirred tank reactor (CSTR) for the chlorination of benzene (Fig. 1) with the following reactions taking place: k1 C6H6 + Cl2 C6H5Cl + HCl k2 C6H5Cl + Cl2 C6H4Cl2 + HCl where the rate constants k1 and k2 [h1] are known and the feed flow rate F1 [m3/h] and reactor volume V [m3] will be design variables. Stream 1 Stream 2 Figure 1: The continuous-stirred tank reactor for the chlorination of benzene. For simplicity, the reactions will be first-order (no dependence on Cl2) with respect to benzene (A) and chlorobenzene (B). The molar volumes of each specie are: VA = 8.937 102m3/kmol, VB = 1.018 101m3/kmol, and VC = 1.13 101m3/kmol with dichlorobenzene as C. The feed is considered to be pure A (ignoring Cl2) with a concentration of 1 kmol/m3. The rate constants are k1 = 0.40h1 and k2 = 0.055h1. The steady-state model equations are: yA,0x4 = x1x5 + x6r1 (1) yB,0x4 = x2x5 + x6(r2 r1) (2) yC,0x4 = x3x5 x6r2 (3) x1 +x2 +x3 =yA,0 +yB,0 +yC,0 (4) where yi,0 is the concentration of i {A, B, C} [kmol/m3] in the feed and r1 and r2 are the reaction rates [kmol/(m3h)] defined as: r1 =k1x1/(x1VA +x2VB +x3VC) r2 =k2x2/(x1VA +x2VB +x3VC). and we are using the definition of the variable vector x R6: x = (yA , yB , yC , F1 , F2 , V ) with yi as the concentration of i {A, B, C} in the tank. 1 a) Lets suppose we want to minimize the reactor volume and ensure that we produce at least 22kmol/h of B. Further, from another study, we found that the residence time for the reactor must be at least 45 seconds. Formulate the constrained optimization problem.

Please solve all parts. Thank you in advance!

Problem 1 Consider the design of a continuous-stirred tank reactor (CSTR) for the chlorination of benzene (Fig. 1) with the following reactions taking place: C6H6+Cl2k1C6H5Cl+HClC6H5Cl+Cl2k2C6H4Cl2+HCl where the rate constants k1 and k2[h1] are known and the feed flow rate F1[m3/h] and reactor volume V[m3] will be design variables. Figure 1: The continuous-stirred tank reactor for the chlorination of benzene. For simplicity, the reactions will be first-order (no dependence on Cl2 ) with respect to benzene (A) and chlorobenzene (B). The molar volumes of each specie are: VA=8.937102m3/kmol, VB=1.018101m3/kmol, and VC=1.13101m3/kmol with dichlorobenzene as C. The feed is considered to be pure A (ignoring Cl2 ) with a concentration of 1kmol/m3. The rate constants are k1=0.40h1 and k2=0.055h1. The steady-state model equations are: yA,0x4yB,0x4yC,0x4x1+x2+x3=x1x5+x6r1=x2x5+x6(r2r1)=x3x5x6r2=yA,0+yB,0+yC,0 where yi,0 is the concentration of i{A,B,C}[kmol/m3] in the feed and r1 and r2 are the reaction rates [kmol/(m3h)] defined as: r1=k1x1/(x1VA+x2VB+x3VC)r2=k2x2/(x1VA+x2VB+x3VC). and we are using the definition of the variable vector xR6 : x=(yA,yB,yC,F1,F2,V) with yi as the concentration of i{A,B,C} in the tank. a) Let's suppose we want to minimize the reactor volume and ensure that we produce at least 22kmol/h of B. Further, from another study, we found that the residence time for the reactor must be at least 45 seconds. Formulate the constrained optimization problem. b) Write a MATLAB program to solve the optimization problem from Part a using fmincon with the SQP solver. Solve the optimization problem. What are the active inequality constraints, if any, and their respective Lagrange multipliers? c) Now, let's assume we're given an existing reactor with volume 10m3 and we want to maximize the concentration of B in the outlet stream. Formulate the new optimization problem using the original productivity and residence time constraints from Part a. d) Solve the optimization problem from Part c. What are the active inequality constraints, if any. and their respective Lagrange multinliers