Write a MATLAB code to implement the solution. Plot the

ECH3119 Numerical Methods And Optimization Process

Question 1:
Write a computer program to compute the profile of the concentration in a given tank as shown in Figure 1.

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Figure 1: diagram of the tank

The system can be described based on the following partial differential equations:

D (∂2c/∂x + ∂2c/∂y2) - kc = 0

and the boundary conditions are as shown. Use suitable value for Δx and Δy to get a accurate results and use value of 0.63 for D and 0.09 for k. Plot the results using a three-dimensional plotting routine where the horizontal plan contains the x and y axes and the z axis is the dependent variable c.

Question 2:

A layer of oil-water that is initially not flowing is placed in between two plates 10 cm apart as shown in Figure 2. At t = 0, the top plate is moved at a constant velocity of 7.5 cm/s. The partial differential equations governing the motions of the fluids are

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Figure 2: Two plates with the oil-water layer

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Write a computer program to compute the velocity profile of the two fluid layers at t = 0.5, 1 and 1.5 s at distances x = 2, 4, 6 and 8 cm from the bottom plate. Note that μwater and μoil are 1.7 and 4.3 cp, respectively. Plot the velocity v for oil and water versus length x for various values of time t.

Question 3:

The Poisson equation as shown in the equation below governing the displacement of a uniform membrane subject to a tension and a uniform pressure:

2z/∂x + ∂2z/∂y2 = -P/T

Write a computer program to solve for the displacement of a 2-cm square membrane that has P/T = 0.6/cm and is fastened so that it has zero displacement along its four boundaries. Employ suitable value for Δx and Δy to obtain accurate results. Display your results as a contour plot. Simulate for P/T = 0.5/cm and P/T = 1.0/cm

Question 4:
Determine the temperature profile along an insulated composite rod that consist of two parts. The nondimensional transient heat conduction equations for the composite rod are given in the equations below:

2u/∂x = ∂u/∂t  0 ≤ x ≤ 1/3

r∂2u/∂x2 = ∂u/∂t  1/3 ≤ x ≤ 1

where u = temperature, x = axial coordinate, t = time, and r = ka/kb. ka and kb are the thermal conductivity for part a and b. The boundary and initial conditions are

 

Boundary Conditions

u (0, t ) = 1.5

u (1, t ) = 1.5

 

(∂u/∂x)a = (∂u/∂x)b   x = 1/3

Initial Conditions

u ( x, 0) = 0         0 ≤ x ≤ 1

Use second-order accurate finite-difference analogues for the derivatives with a Crank-Nicholson formulation to integrate in time. Write a MATLAB code for the solution, and select values of Δx and Δt for accurate results. Plot the temperature u versus length x for various values of time t. Generate a separate curve for the following values of parameter r = 1, 0.1 0.01, 0.001, and 0.0001.

Question 5:

Apply Alternate Direction Implicit (ADI) method to solve for the temperature profile in an insulated plate. The governing equation is

2u/∂x2 + ∂2u/∂y2 = ∂u/∂t

where u = temperature, x and y are spatial coordinates and t = time. The boundary and initial conditions are

 

Boundary Conditions

u ( x, 0, t ) = 0

u ( x,1, t ) = 1

 

u (0, y, t ) = 0

u (1, 0, t ) = 1

Initial Conditions

u ( x, y, 0) = 0    0 ≤ x < 1   0 ≤ y < 1

Write a MATLAB code to implement the solution. Plot the results using a three-dimensional plotting routine where the horizontal plan contains the x and y axes and the z axis is the dependent variable u. Construct several plots at various times, including the following: a) the initial conditions; b) one intermediate time, approximately halfway to steady state; and c) the steady-state condition.


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