Continuous casting in steel production is essentially a solidification process by which molten steel is solidified into a steel slab after passing through a mold, as shown in Figure P5.41(a). Final product dimensions depend mainly on the casting speed V_p (in m/min), and on the stopper position X(in %) that controls the flow of molten material into the mold (Kong, 1993). A simplified model of a casting system is shown in Figure P5.41 (b) (Kong, 1993) and (Graebe, 1995). In the model, H_m =mold level (in mm); H_t =assumed constant height of molten steel in the tundish; D_z= mold thickness = depth of nozzle immerged into molten steel; and W_t= weight of molten steel in the tundish.

For a specific setting let A_m = 0.5 and

Also assume that the valve positioning loop may be modeled by the following second-order transfer function:

and the controller is modeled by the following transfer function:

The sensitivity of the mold level sensor is β = 0.5 and the initial values of the system variables at t= 0- are: R(0-) = 0; Y_C (0-) = X(0-) = 41.2; ΔH_m = (0-) = 0; H_m(0-) = -75; ΔV_p (0-) = 0; and V_p (0-) = 0. Do the following:

a. Assuming v_p(t) is constant [v_p = 0] , find the closed-loop transfer function T(s) = ΔH_m(s)/R(s).

b. For r(t) = 5 u(t); v_p(t) = 0.97 u(t); and H_m(0-) = 75 mm, use Simulink to simulate the system. Record the time and mold level (in array format) by connecting them to Workspace sinks, each of which should carry the respective variable name. After the simulation ends, utilize MATLAB plot commands to obtain and edit the graph of h_m(t) from t = 0 to 80 seconds.

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