Assignment Tasks Consider a system with two thermal capacitances ( 1C and 2C ). Heat is supplied to the first capacitance at the rate 1( )q t by a heater, and heat is lost at the left end to the environment. The first capacitance is connected to the second one through the thermal resistance 2R . The second capacitance is connected on the right side to the environment that has the temperature a . Except for the thermal resistances 1R , 2R and 3R , the enclosure is assumed to be perfectly insulated. The system model is given by 1 1 1 2 1 12 1 1 1 1 2 2 2 1 2 23 2 3 2 2 1 1 1 1 1 1 1 , a a q C R C C R C R C R C R C R where 1 2 12 1 2 , R R R R R and 2 3 23 2 3 . R R R R R Hence, using the substitutions 1 1 ˆ a and 2 2 ˆ a , the following incremental model is obtained 1 1 1 2 1 12 1 1 2 2 2 1 2 23 2 2 1 1 1ˆ ˆ ˆ 1 1ˆ ˆ ˆ , q C R C C R C R C R which corresponds to the following equations in the Laplace domain 1 1 2 1 12 1 1 2 2 1 2 23 2 2 1 1 1ˆ ˆ( ) ( ) ( ) 1 1ˆ ˆ( ) ( ). ( ) ( ) s s Q s s C R C C R s s s C R C R Using these equations, the following transfer functions are derived 2 1 2 2 2 2 2 12 231 2 1 12 2 23 1 2 12 23 2 1 ˆ ( ) 1 1( ) ( ) s C C R R R RQ s s s C R C R C C R R R and 2 2 2 1 1 12 1 2 23 1 2 12 231 2 2 3 2 2 12 23 2 12 231 2 2 2 2 2 2 1 12 2 23 1 12 1 2 12 23 1 2 12 23 2 1 2 12 23 2 1 1 1 1 ˆ ( ) 2 1 1 1( ) ( ) ( ) ( ) s s C C R C C R C C R Rs R R R R R RQ s s s s C R C R C R C C R R C C R R R C C R R R Task 1 Consider the following numerical values for the model described above with the following default values. 1 2 1 2 3 50 / 60 / 10 / 10 / 10 / 293.15 kelvin (20 Celsius)oa C J K C J K R Ks J R Ks J R Ks J Implement the model in Matlab and Simulink, using Simulink for the model and Matlab to set parameters, call the model and plot the simulation results. Simulate the system with 1 0q for ten minutes then apply a step input of amplitude 3 . Allow the system to reach steady state and plot in a single figure: the temperatures 1( )t , 2 ( )t and the input heat flow 1( )q t . Plot the time in minutes on the X-axis. For each temperature ( 1 and 2 ), report the steady state value and the time required to reach 2% of the steady state value? Explain your reasoning. Task 2 Create a Matlab GUI that allows a user to explore different values for the model parameters and the corresponding response from the system. It is also your task to explore the system enough to be able to understand the impact of each parameter in the response of the system and to be able to describe the significance of that parameter in the actual thermal system modelled. For example: to begin exploring the system, increase the value of the second thermal resistance to 2 50 /R Ks J and plot the system’s response. What are the main differences compared to the response in Task 1? Relate the differences in the system’s response with the actual thermal system described in the assignment. Reset the parameters to the default case, then repeat this process for three more cases with different values of 1) 1R or 3R , 2) 1C or 2C , and 3) a . .
