Fall 2002 10 34 Numerical Methods Applied to Chemical Engineering Exam I 10 4 2002 There are two questions to this exam You have the full hour to work on the exam and may use all class material notes books etc Submit your results for each question 1 and 2 on a separate blue book You may keep your copy of the exam questions Make sure that your name is on each blue book Question 1 60 points total Consider the following set of three nonlinear algebraic equations 4 2 f 1 x 1 x 2 x 3 4x 1 3x 1 x 2 x 2 x 3 57 0 4 f 2 x 1 x 2 x 3 x 2 5x 1 x 3 9 0 EQ 1 4 f 3 x 1 x 2 x 3 3x 3 6x 1 x 2 x 3 15 0 1 A 15 pts Calculate the analytical form of the Jacobian matrix of this system expressed in terms of the unknowns x 1 x2 x3 1 B 10 pts Starting from an initial guess of x1 1 x2 1 x3 1 derive the set of linear algebraic equations that must be solved for the first iteration of Newton s method 1 C 20 pts Solve this set of equations for the full Newton update x 0 using Gaussian elimination with partial pivoting Show all of your calculations by hand and use the exact solution process followed by a computer no human intuition please You will not receive full credit unless you show all of your calculations as performed by hand This is to avoid giving an unfair advantage to those with fancy calculators You may use a calculator however to add or multiply simple numbers e g 5 4 2 1 D 15 pts We have seen that Newton s method can be erratic when the initial guess is far away from the solution If you were using a modification of Newton s method that was more robust to the choice of initial guess would you accept the Newton update x 0 that you have calculated above If not explain briefly how you would go about selecting the new estimate of the solution 1 x QUESTION 2 ON NEXT PAGE October 2 2002 1 Question 2 40 points total As we discussed in class eigenvalue analysis can be used to study the stability of a dynamic system If we have a system governed by a system of differential equations dx 1 x 1 f 1 x 1 x 2 x N dt dx 2 x 2 f 2 x 1 x 2 x N dt EQ 2 dx N x N f N x 1 x 2 x N dt with a steady state at x x that is f j x 1 x 2 x N 0 j 1 2 N EQ 3 then the condition that the steady state be stable is that ALL eigenvalues of the Jacobian J jk x f j xk EQ 4 x x must have real parts less than zero Jw j j w j Re j 0 j 1 2 N EQ 5 The full details of the derivation are repeated for your review should not be needed to solve the problem in the optional background section following the problem statement Consider the case of a CSTR with the single chemical reaction A B 2A r kc A c B EQ 6 For example this model may describe the increase in cell concentration species A in a bioreactor containing a growth factor species B Since we see that the reaction is selfaccelerating we might be concerned about the stability of this reactor system The mass balances for the reactor are d Vc Qc in Qc Vkc c A A A A B dt d Vc Qc in Qc Vkc c B B B A B dt October 2 2002 EQ 7 2 Q is the volumetric flow rate through the reactor V in is the total reactor volume and cA and c Bin are the inlet concentrations of A and B respectively Defining the reactor mean residence time as V Q the two nonlinear equations for the steady state are in 0 cA 0 in cB c A kc A c B EQ 8 c B kc A c B We can easily solve these equations by adding them to obtain the relation in 0 cA in cB in cB cA cA cB in cB EQ 9 cA Substitution into the balance for A yields in 0 cA in 0 cA in c A kcA c A in c A kc A c A in 2 kc A 1 k c A in cB cA in 2 c B kc A in in c B c A c A EQ 10 0 This quadratic equation is solved easily for the concentration of A For the process conditions 10 k 1 in cA 1 in cB 10 EQ 11 the concentrations of A and B in the reactor at steady state are c A 10 9092 c B 0 0908 EQ 12 2 A 20 pts Is the reactor stable under these process conditions 2 B 15 pts This application of eigenvalue analysis assumes that the eigenvectors of the Jacobian are linearly independent so that any vector may be written as a linear combination of the eigenvectors While this assumption is usually OK we can only prove that it is valid for special cases of the Jacobian matrix J and of its eigenvalues 1 2 N What are the most general conditions on J and eigenvectors of J to be linearly independent 1 2 N that you can impose for the 2 C 5 pts For the Jacobian matrix that you calculate in 2 A are the eigenvectors expected to be orthogonal to each other October 2 2002 3 Background should not need to solve problem A common use of eigenvalue analysis is to study the stability of a dynamic system Consider a system governed by the following set of N linear first order differential equations dx 1 x 1 a 11 x 1 a 12 x 2 a 1N x N dt dx 2 x 2 a 21 x 1 a 22 x 2 a 2N x N dt EQ 13 dx N x N a N1 x 1 a N2 x 2 a NN x N dt Defining the vectors and matrix x1 x x2 x xN a 11 a 12 a 1N x 1 x 2 A a 21 a 22 a 2N x N EQ 14 a N1 a N2 a NN this system of differential equations takes the form x Ax EQ 15 We see that x 0 is a steady state since the time derivatives are zero x 0 But will it be a stable steady state That is if we change x by a small value away from x 0 will it return to the steady state x 0 is stable or will it move away x 0 is unstable To answer that question we define the eigenvalues and eigenvectors of A as Aw j j w j …
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