1) The Power Method is a classic numerical algorithm for determining the largest eigen value and eigenvector associated with a matrix A. prepare a matlab function that returns the largest eigen value and eigenvector using the Power Method (see Wikipedia) for a given matrix A. Construct the algorithm so that it always runs 100 iterations { you should not implement a stopping criteria. Do not copy others codes. Test your algorithm on the matrix: A = [2.0, 0.2, 1.0, 0.2, 4.0, 1.3, 0.0, 1.3, 3.0] using an initial eigenvector guess of b = [1, 1, 1].
2) In class, we derived and solved a model of a non-isothermal, insulated, CSTR with an exothermic reaction (model6.m). We want to derive and solve a very similar model of a system with only a single feed stream and the following properties:
A→B; k(min^{-1}) = 100. exp(-2000/T (^{o}R))
V = 2 ft^{3}
Q = 1 ft^{3}=min
Cp = 0:5 kcal=(lbmol ^{o}R)
C_{A,in} = 10 lbmol=ft^{3}
ρ = 10 lbmol=ft^{3}
Hrxn = 200kcal=lbmol
Simulate this system for 20 minutes with an initial concentration of CA(0) = 10 lbmol=^{o}R and T(0) = 200oR or T(0) = 600^{o}R. Compare and describe the steady-state result for the two different initial conditions. What is the significance of this result for implementing a process control scheme?
3) Derive and solve a model of an insulated water tank with a changing level (i.e., changing tank volume). There is only one inflow stream with a flow rate of 1 kg/sec for t < 0 and 0.9 kg/sec for t 0. The out flow rate depends upon the level, H, within the tank and is given by m_{out} = kv√H where kv = 0.8kg=(m^{0.5}s). The tank is fitted with an electric heater that inputs Q = 100kW (constant), and the cross-sectional area of the tank is 1.1m^{2}. Derive both the mass and energy balances, determine the steady state conditions at t < 0, and determine via numerical simulation the impact of the changing inflow rate at t >=0.