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Problem 2. A chemical constituent flows between three well-mixed tanks, also called reactors, as depicted in Figure below. Steady-state mass balances can be written for a substance that reacts with first- order kinetics. For example, the mass balance in reactor 1 is where Qin volumetric inflow to reactor 1 (L/min), cin inflow mass concentration to reactor 1 (g/L), Qij volumetric flow from reactor i to reactor j (L/min), c = concentration in reactor i (g/L), k first-order decay rate (min), and V₁ = volume of reactor i (L). (a) Write the mass balances for reactors 2 and 3. (b) If k = 0.1 min, write the mass balances for all three reactors as a system of linear algebraic equations. (c) Compute the LU factorization for this system. (d) Use the LU factorization to compute the matrix inverse. (e) Use the matrix inverse to address the following questions: Q-10 -200 (i) What are the steady-state concentrations in the three reactors? 021-22 (ii) If the inflow in the second reactor is set to zero, what is the resulting reduction in the concentration in reactor 1? 12-5 V₁-100 V-150 Q13-117 10-110 Sia - 10 (iii) If the inflow concentration to reactor 1 is doubled, and the inflow concentration to reactor 2 is halved, what will be the concentration in reactor 3?

Problem 2. A chemical constituent flows between three well-mixed tanks, also called reactors, as depicted in Figure below. Steady-state mass balances can be written for a substance that reacts with first- order kinetics. For example, the mass balance in reactor 1 is where Qin volumetric inflow to reactor 1 (L/min), cin inflow mass concentration to reactor 1 (g/L), Qij volumetric flow from reactor i to reactor j (L/min), c = concentration in reactor i (g/L), k first-order decay rate (min), and V₁ = volume of reactor i (L). (a) Write the mass balances for reactors 2 and 3. (b) If k = 0.1 min, write the mass balances for all three reactors as a system of linear algebraic equations. (c) Compute the LU factorization for this system. (d) Use the LU factorization to compute the matrix inverse. (e) Use the matrix inverse to address the following questions: Q-10 -200 (i) What are the steady-state concentrations in the three reactors? 021-22 (ii) If the inflow in the second reactor is set to zero, what is the resulting reduction in the concentration in reactor 1? 12-5 V₁-100 V-150 Q13-117 10-110 Sia - 10 (iii) If the inflow concentration to reactor 1 is doubled, and the inflow concentration to reactor 2 is halved, what will be the concentration in reactor 3?

Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter11: Nonlinear Programming
Section11.10: Quadratic Programming
Problem 7P
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Problem 2. A chemical constituent flows between three well-mixed tanks, also called reactors, as depicted in Figure below. Steady-state mass balances can be written for a substance that reacts with first- order kinetics. For example, the mass balance in reactor 1 is where Qin volumetric inflow to reactor 1 (L/min), cin inflow mass concentration to reactor 1 (g/L), Qij volumetric flow from reactor i to reactor j (L/min), c = concentration in reactor i (g/L), k first-order decay rate (min), and V₁ = volume of reactor i (L). (a) Write the mass balances for reactors 2 and 3. (b) If k = 0.1 min, write the mass balances for all three reactors as a system of linear algebraic equations. (c) Compute the LU factorization for this system. (d) Use the LU factorization to compute the matrix inverse. (e) Use the matrix inverse to address the following questions: Q-10 -200 (i) What are the steady-state concentrations in the three reactors? 021-22 (ii) If the inflow in the second reactor is set to zero, what is the resulting reduction in the concentration in reactor 1? 12-5 V₁-100 V-150 Q13-117 10-110 Sia - 10 (iii) If the inflow concentration to reactor 1 is doubled, and the inflow concentration to reactor 2 is halved, what will be the concentration in reactor 3?
Transcribed Image Text:Problem 2. A chemical constituent flows between three well-mixed tanks, also called reactors, as depicted in Figure below. Steady-state mass balances can be written for a substance that reacts with first- order kinetics. For example, the mass balance in reactor 1 is where Qin volumetric inflow to reactor 1 (L/min), cin inflow mass concentration to reactor 1 (g/L), Qij volumetric flow from reactor i to reactor j (L/min), c = concentration in reactor i (g/L), k first-order decay rate (min), and V₁ = volume of reactor i (L). (a) Write the mass balances for reactors 2 and 3. (b) If k = 0.1 min, write the mass balances for all three reactors as a system of linear algebraic equations. (c) Compute the LU factorization for this system. (d) Use the LU factorization to compute the matrix inverse. (e) Use the matrix inverse to address the following questions: Q-10 -200 (i) What are the steady-state concentrations in the three reactors? 021-22 (ii) If the inflow in the second reactor is set to zero, what is the resulting reduction in the concentration in reactor 1? 12-5 V₁-100 V-150 Q13-117 10-110 Sia - 10 (iii) If the inflow concentration to reactor 1 is doubled, and the inflow concentration to reactor 2 is halved, what will be the concentration in reactor 3?
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