Determine the steady state conditions at t lt 0 and

Question

1. The Power Method is a classic numerical algorithm for determining the largest eigenvalue and eigenvector associated with a matrix A.

Write a matlab function that returns the largest eigenvalue 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.

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(^°R))

V =2 ft³

Q = 1 ft³ /min

C_P) = 0.5 kcal / (lbmol ^°R)
CA,i_n = 10 lbmol / ft³
p = 10 lbmo//ft³

ΔHi_rxn = 200kcal /lbmol

Simulate this system for 20 minutes with an initial concentration of C_A(0) = 8 lbmol/°R and T(0) = 200°R or T(0) = 650°R. The feed temperature should be the same as the initial temperature, T_in = 200°R or Ti_n = 650°R. Compare and explain 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 1.2 kg/sec for t < 0 and 0.9 kg/sec for t > 0. The outflow rate depends upon the level, H, within the tank and is given by mout = k_v√R where k, = 0.8kg/(m^0.5s).

The tank is fitted with an electric heater that inputs Q = 90kW (constant), the water is being fed at 50°C, and the cross-sectional area of the tank is 1.1m². 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.