Design and Implementation of Object-Oriented ... - Automatic Control

4.11 Moving Boundary Models
Integrating the second term and differentiating the resulting equation
gives for a constant area pipe:
A d
L1
ρdz − Aρ(L1)
dt 0
dL1
dt + ˙m12 − ˙min = 0.
The density at the interface ρ(L1) is equal to the saturated liquid density
ρl. Pressure and mean enthalpy are chosen as the states in the subcooled
region. The mean enthalpy is defined as
¯h1 = 1
2 (hin + hl)
where hin is known from the boundary conditions and hl is a function of
the pressure. The mean density and temperature in the subcooled region
is approximated by
¯ρ1 = 1
L1
L1
0
ρdz ρ(p, ¯ h1)
¯ T1 T(p, ¯ h1).
Using the above expressions, the mass balance for the subcooled region
can be written as
dL1
A ¯ρ1 − ρl + L1
dt
The term d ¯
ρ1/dt is calculated using the chain rule:
d
¯ρ1 ¯ρ1 dp ¯ρ1
= +
dt p h dt ¯
d
h1 p
¯ h1
dt
¯ρ1
= +
p h
1 ¯ρ1
2 ¯
dhl dp
h1 p dp dt
d
¯ρ1
= ˙min − ˙m12. (4.53)
dt
+ 1
2
¯ h1
¯ρ1 dhin
p dt
The term dhin/dt is determined from the boundary conditions to the
evaporator model. Inserting this expression into the mass balance (4.53),
gives the final version of the mass balance for the subcooled region
A ( ¯ρ1 − ρl) dL1
¯ρ1
+ L1
dt p
+ 1
2 L1
¯ρ1
¯
h1
+
h 1 ¯ρ1
2 ¯
dhl dp
h1 p dp dt
= ˙min − ˙m12.
p
dhin
dt
(4.54)
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