UNIVERSIDADE DE AVEIRO
DEPARTAMENTO DE ELECTRÓNICA TELECOMUNICAÇÕES E INFORMÀTICA
Sistemas e Controlo I – Aulas Práticas (2015/16)
Trabalho prático nº 7
Objectives: Define a simplified mathematical model of a linear thermodynamic system. System analysis in
time domain. 1st order system response. Transient and stationary components of the system response.
Ex. 1 Consider the liquid heating tank (see Fig.1). The tank is well isolated, the heat loses outside the vessel
and the heat accumulation on the vessel walls are insignificant, the liquid is perfectly mixed and the
temperature is uniform.
The input liquid has a constant temperature Ti . At the moment t=0, the heating is on and the heating rate
is q. The temperature of the output liquid increases to Tout > Ti . Thus, the temperature variation in the
vessel is ´Tq (t ) = Tout (t ) − Ti .
a) Determine the transfer function Tq (s) / Q(s) .
b) Determine analytically the system response ( Tq ) when the heating rate q is a unit step. Determine
d)
e)
f)
Temp of heating tanque
60
50
40
Temp [ºC]
c)
the transient and the stationary components of the system response.
Determine how much constant heating is required to keep the output temperature Tout = 80 [ºC], if
Ti =20 [ºC]. Let Rt =2 [ºC/W], C t =10 [J/ºC].
Implement the system transfer function in a matlab script and confirm the previous result.
At the moment t=0, a constant heating q was applied, and the variation of the output temperature
was registered on Fig. 2. Let C t =10 [J/ºC], determine the heat resistance Rt of the liquid .
Obtain the equivalent electrical circuit of the thermal system.
30
20
10
0
Fig. 1 Liquid heating tank
0
20
40
60
80
100
Time (seconds)
120
140
Fig. 2 Registered temperature
160
180
Appendix : Thermodynamic systems and lows
Normally, the thermodynamic systems have distributed parameters; therefore they are mathematically
described by nonlinear differential equations with partial derivatives. Only very limited number of
thermodynamic systems can be described by linear differential equations. The principal condition for the
system linearity is that the temperature must be uniform, and particularly for liquids, they must be
homogeneous and well mixed, which is not the case in most of the real world thermodynamic systems.
These constraints are approximately valid only for small bodies, small liquid vessels.
Examples for thermodynamic systems: oven, fridge, car refrigeration, water heating, thermometer of
mercúrio, etc.
The heat qi is a thermal energy that flows over time as a result of temperature difference. It is measured
by calorie for second 1 cal/seg=4.1868 W.
1. 1st thermo-dynamical low (the energy conservation low):
The applied heat to a system = conserved heat + dissipated heat
2. 2nd thermo-dynamical low – the heat transfer goes only in direction from more to less hot body.
3. The conserved heat (the conserved thermal energy) of the system can be obtained from:
t
T (t ) =
1
q armazenado (τ )dτ + T (0)
C t ∫0
q armazenado (t ) = C t
dT
,
dt
C t is the heat capacity, T(t) –body temperature
4. The dissipated heat (the dissipated thermal energy) of the system is:
q dissipado (t ) =
Tout (t ) − Ti (t )
Rt
, Tout (t ) > Ti (t )
Rt is the heat resistance [ºK (Kelvin) /W], K = ºC+273.15.