Description

Homework Assignment #3

Use Matlab to solve these problems. Generate script which is organized and commented. Please make one m file for the entire assignment. Turn in the m files AND pdf files of the output (using the publish function).

Initial Value Problems

1. Michaelis-Menten kinetics is an established model for enzyme reactions. The reaction mechanism requires two steps, the formation of an enzyme-substrate structure, ES, that reacts to form a product, P. This two step reaction mechanism leads to nonlinear kinetics with a rate of reaction as given below. Assuming that there is initially no product, P, in the reactor, predict the concentration of product as a function of time for the situations below. Assume that the concentration of enzyme is constant and equal to 10% of the initial substrate concentration. For each set of conditions, generate a plot of the concentration of the four species as a function of time. Then plot results for P(t) to compare the impact of different parameters.

E  S ES  kcat E P d P[ ] Vmax[ ]S kcat[ ]E o [ ]S Reaction rate =  
dt Km [ ]S Km [ ]S
a) Pepsin catalyzed: kcat = 0.5 s-1; Km = 0.3 mM; [S]o = 1 mM
b) Fumarase catalyzed: kcat = 0.08 s-1; Km = 5 M; [S]o = 1 mM

2. The centerline temperature of a thin copper (=8933 kg/m3) plate of thickness d = 0.002 m placed in a 1200K furnace (assuming a uniform temperature in the plate and radiative heat transfer,  = 5.676 10-8 W/m2 K4) is given by:

dT  2d  c Tp ( )TF4 TT4dcp , dt 
dT

where TF is the furnace temperature and the specific heat of copper, cp(T), is given by:
c Tp( )  355.2 0.1004T ; cp [=] J/kg K and T [=] K.

The initial temperature of the plate is 300K and you have been asked to predict the temperature as a function of time. Assume that the plate dimensions and density are not functions of temperature. Scale the temperature as  = T/TF, develop a natural scaling for time using the variables given and define  = t/t* as a dimensionless time. Plot  for this problem. Be sure to provide the value of t* and a clear description of your method in the comments of your code.

3. The following system describes the trajectory (velocity and position) along a unit circle. This is the DAE system that I discussed in class. Solve this as a DAE using Matlab (see the documentation for ode15s) and plot x2 versus x1. Choose a value of o and solve the system first using an initial condition of x1(t=0) = 0 and a value of x2(0) that follows from the algebraic constraint. Then solve it again using x1(t=0) = 0 and x2(t=0) = 0.8; what does this tell you about how the solver computes an initial state that is consistent?

dx1 [x1 cos(o )] [ x2 sin(o )] dt
0  x12  x22 1

k2 C

A

k4 D

For the case that the rate constants are k = [ 1 1 0.5 1 ]T [=] 1/min and the reactor is initially charged with 10 M of reactant A, plot the concentration of each species as a function of time. Use a numerical time marching approach (i.e., do not solve this analytically). Plot each concentration on a separate subplot (four plots) and label the axes. Choose a time range so that the dynamics and the steady state can be seen.

5. Solve the reactor problem above but perform the calculations for a CSTR with a constant flow rate in and out, Q. Choose a flow rate to give a residence time in the reactor of 1 minute. Generate a similar plot of the concentration of each component in the reactor on four subplots. Also provide one plot comparing the batch and CSTR results for the concentration of A as a function of time.

6. A small container and its contents are at 150F. To cool both the container and the liquid to room temperature (70F, the container is immersed in a bath held at 32F. Balancing the rate of change of energy storage between the liquid and the container with the rate of convective heat transfer (liquid – container and container – bath) leads to a coupled system of equations, where L is the temperature of the liquid and C is the temperature of the container. Using the parameters given, plot the temperature of the liquid and the container as a function of time.
Be careful, the time constants for these two processes are very different!

Liquid Container
Mass density
[lbm/ft3] 62 139
Specific heat
[Btu/lbmF] 1.00 0.2
Volume
[ft3] 0.03 0.003
dL  Ahi (C  L) dt 1c Vp,1 1

dC  A ho (32 C) Ahi (L C) dt 2cp,2V2 2cp,2V2

Ai=0.4 ft2 and Ao=0.5 ft2
Assume h = 8.8 Btu/hr ft3 F

Boundary Value Problems

7. The following third order differential equation can be solved as a system of coupled first order differential equations. Write this equation as a system of three coupled first order equations u  f ( )u
y y 2y  2×2  2x

(a) Solve with the initial conditions y(0)  1; y(0)  0; y(0)  4
(b) Solve using the shooting method and y(1)  1; y(0)  0; y(0)  4

8. Find the solution to the boundary value problem below using the shooting method. Plot y(x) over the range 1 x 2 given the boundary conditions that y(1) = 6.308447 and y(2)=55.430436.

y”  y’ y 3e2x 2sin x

9. For linear boundary value problems of the form y ”  p x y q x y r x( ) ‘ ( )  ( ) there is a simple procedure that makes the shooting method very effective. The function
y( )x  y x1( )cy2( )x will be an exact solution to the problem where y1( )x is the solution to
to the IVP that corresponds to the nonhomogeneous BVP and y2( )x is the solution to the corresponding homogeneous IVP (i.e., with r x( )  0 ). For the following problem:

 u” 2u  2 2 sin( x)

u(0)  u(1)  0

(a) Convert this problem into two first order initial value problems to solve for u x1( ) and u2( )x . Show the systems that you plan to solve and the relevant initial value conditions.
(b) Use an RK4 method to estimateu x1( ) and u2( )x at xi  0,0.25,0.50,0.75,1.00 . Is the prediction for the Dirichlet condition at x1correct?
(c) Determine the value of c that will give the approximate solution for y x( ). Use this to point wise calculate w x( )  y x1( )cy2( )x given the values calculated in part (b).
(d) Since the analytical solution is u x( )  sin(x) calculate the exact error at each point.