a. Develop a Simulink model of a thermostatic control system in which the temperature model is

 

where T is the room air temperature in °F, T_α is the ambient (outside) air temperature in )F, time t is measured in hours, q is the input from the heating system in lb ft/hr, R is the thermal resistance, and C is the thermal capacitance. The thermostat switches q on at the value q_max whenever the temperature drops below 69°F and switches q to q = 0 whenever the temperature is above 71°F. The value of q_max indicates the heat output of the heating system.

Run the simulation for the case where T(0) = 70°F, and T_α(t) = 50 + 10 sin (πt/12). Use the values R = 5 × 10^-5 °F ∙ hr/lb ∙ ft and C = 4 × 10⁴ lb ∙ ft/°F. Plot the temperatures T and T_a versus t on the same graph, for 0 ≤ t ≤ 24 hr. Do this for two cases: q_max = 4 × 10⁵ and q_max = 8 × 10⁵ lb ∙ ft/hr. Investigate the effectiveness of each case.

b. The integral of q over time is the energy used. Plot ∫q dt versus t and determine how much energy is used in 24 hr for the case where q_max = 8 × 10⁵.

dT RC + T = Rq + T.(t) dt