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HEI 43 TC 2007-2008
Control of Temperature
.
The controlled temperature
θ
of indirect heating is represented by the following figure:
u(t) represents the voltage control of valve,
q(t )
the
inlet flow in exchanger and
θ
1
the output
temperature of exchanger.
Let the following relations:
(t)
k
=
dt
(t)
d
+
(t)
q(t)
k
=
dt
(t)
d
+
(t)
).d
u(
k
=
q(t)
1
2
2
1
1
1
1
t
0
o
θ
θ
τ
θ
θ
τ
θ
τ
τ
We assume that all initial conditions are null. Numerical applications
:
τ
1
=600s,
τ
2
=6000s,
k
2
=1, k
1
=20 S.I., k
o
=2.10
-4
S.I.
We take:
λ
= k
o.
k
1
.k
2
.
1
. Give the box diagram of the system and deduce the transfer function of the exchanger H(p)
(input u and
θ
).
2. Proportional Controller
u(t) =G
r
ε
(t)= G
r
.(
θ
ref
(t) -
θ
(t)) where
θ
ref
(t) is the setpoint or reference.
2.1
Give the box diagram of the closed loop system
2.2
To determine k, we neglect a pole of the system.
2.2.1
From the numerical values, which is the pole to be neglected?
2.2.2
Give the expression of the approximated open loop transfer noted W
Oa
.
2.2.3
Compute from W
Oa
the approximated closed loop transfer W
Fa
.
2.3.4
W
Fa
will be written as normalized second order:
1
2
1
1
0
2
2
0
+
+
p
p
ω
ξ
ω
. Give
ξ
and
ω
0
.
2.2.5
Compute the proportional gain G
r
in order to obtain a damping coefficient un coefficient
2
2
=
ξ
.Numerical Application..
2.2.6
Determine the settling time (5%) of the controlled system . Numerical Application.
θ
1
q
u
θ
Exchanger
pomp
Plate
valve