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 .


.