Develop an unsteady-state model for a stirred batch reactor, using the nonlinear continuous reactor model presented in Example 4.8 as a starting point. For the parameters given below, compare the dynamics of the linearized models of the batch reactor and the continuous reactor, specifically the time constants of the open-loop transfer function between c_A and T_c, the concentration of A, and the jacket temperature, respectively. Assume constant physical properties and the following data:

Initial steady-state conditions for the continuous case are, the physical property data are, The reaction rate is first order with a rate constant (in min^?1). For the batch case, linearize the model around T = T.

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T= 150 °F, cAi = 0.8 mol/ft', g= 26 ft/min, V 1336 ft, T = 77 °F UA 142.03 kI min F k * 2.4 x 1015 e-20,000T (Tin °R).