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HOME EXAM 2022 GROUPS: ALL Subject: Numerical Analysis Unsteady heat conduction problem on a rectangular domain is to be analyzed by ADI method. Initially the temperature in the rectangular region is To= $$$ [Cº]. Then at the moment t = 0 temperature on the boundary sides S1, S2, and $3 is raised to *** Ər [Cº]. The side S4 the boundary condition is specified as -=0 and initially has temp. of ax To. Assume that Lx = Ly = 1 m, the thermal diffusivity a = 0.01 m²/h. The PDE which describes unsteady heat conduction process is given by the equation, where T is the temperature, x and y are the distances across the wall, and t is time. ƏT 8²T) + Ly Y ज at Solve the problem numerically for the temperature distribution as function of time using ADI method. *1 (1) ** MARKS Prof. Dr. Mostafa Essuri α 5₁ (8²T (357 ax² ay² A. Write down the finite difference equations for this problem in FTCS B. Explain ADI method and apply the boundary conditions.
C. Prepare your solution by writing the system of equations in matrix form for the first-time step using a 5x5 grid and time step of 0.01 s. (Write the system of equations only) D. Write a MATLAB code to solve the problem (you can modify the class notes example code) E. Use the code with Nx=11, Ny-11, At = 0.01 to find out the following a. The Temp. distribution after 1sec, 60 sec, 30 min ( plot 3 Figures) b. What is the temperature at the point (x, y) = (0.5m,0.5m) at the above three moments? (Table) c. Plot the temperature profile at the line defined by y=0.5 at the above three steps. (Figure) d. Calculate until the 12 norm of tempreture is less than 3%. G. Perform mesh sensitivity analysis for the steady state case (change number of grid points in x and y) and make sure that you are at/close to steady state. find out grid independent solution at the plate mid-point (a plot x-axis number of grids and y axis Temp. at mid point). Note: comment on every result Every figure should have title, x label and y label Tables should have clear title Your specific data will be sent to you via the google-class room