4. Physical Models for Thermo-Hydraulics
4.11 Moving Boundary Models
The models described in the previous sections are general purpose models for fluid flow using lumped or distributed parameters. The generality of the models makes them well suited for model libraries, but in many situa-tions these models are far from optimal for dynamic purposes. Distributed parameter models are in general not well suited for control design, where low order models are preferable. Steady state oriented models of thermo-fluid systems put large emphasis into getting pressure drop and heat transfer equations correct: the accuracy of those often does not matter very much for control purposes, except when feed forward control is used.
A controller with integral action will bring the system to the desired oper-ating point even if the model on which the controller is based is not very
4.11 Moving Boundary Models accurate at steady state. Physical phenomena at much higher frequencies than the bandwidth should be neglected altogether for the sake of simplic-ity. Model accuracy is needed most for the eigenvalues of the linearized system in the vicinity of the desired closed loop bandwidth. The important lesson from modeling of fluid systems for control is that this feature can often be achieved with models which are lower order and simpler than distributed parameter models. Even worse, the standard distributed pa-rameter models are in some cases particularly unsuited for describing the model in the frequency range of the control. Plant design engineers know that distributed parameter models are good at predicting steady state performance but fail to recognize their shortcomings for control design.
A particular example of low order models which are superior to dis-tributed models for control design are a class of models which are usually classified as moving boundary models of two phase flows. The main idea of these models is that they make use of the observation that in two phase flows, the physical behavior differs a lot between the liquid single phase, two phase and gas single phase regions. A modeling idea using this ob-servation uses control volumes with variable sizes. These models capture the behavior in the volume and the boundaries of the regions.
Moving boundary models as a low order alternative to distributed pa-rameter models have been used by other authors before, e. g., a model for incomplete vaporization [Beck and Wedekind, 1981], several variants of two region dry expansion evaporator models in [He and Liu, 1998], [He et al., 1997], [He et al., 1994] and [He et al., 1995] and a three region model by Bittanti et al.[Bittantiet al., 2001].
Moving Boundary Equations
The distribution of liquid and gas in a typical evaporator looks approxi-mately like in Figure 4.7. Some technically relevant variants of this gen-eral form are:
Once through boilers. The flow patterns are illustrated in Figure 4.7 with subcooled inflow and superheated outflow, but the flow is usu-ally vertical and upward.
Risers in drum boilers. They always have vertical up-flow in the risers and incomplete evaporation with subcooled inflow and two phase flow at the outlet.
Dry expansion evaporators are typical for household and commercial refrigeration systems. The inflow is usually in the two-phase region and the control variable of interest is the superheat, the tempera-ture difference between the outflow temperatempera-ture and the saturation temperature.
Chapter 4. Physical Models for Thermo-Hydraulics
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.
˙ .
Hin
˙ min
H˙out
˙ mout Q˙
Figure 4.7 Horizontal flow of gas and liquid in an evaporator.
The physical phenomena found in two phase flows are complex and the special literature about it provides many very detailed models with fine-grained classification of flow patterns. For low order models it is natural to choose three regions: subcooled, two phase and superheated. The three model types that correspond to the three applications listed above are:
Three-zone models consist of a subcooled, a two phase and a super-heated region.
Two-zone flooded evaporator models have a subcooled and a boiling two phase region.
Two-zone dry expansion evaporators start with two phase flow at the inlet, followed by a superheated region.
The detailed derivation of these models follow the same pattern: the mass- and energy balance equations are integrated over a control volume of variable size.
For the fluid mass balance we start from(4.6) and for the fluid energy balance with (4.15). The energy balance for the metal is also based on (4.15) with the additional assumption of a constant heat capacity. The momentum balances are replaced by a static relation of pressure drop at the outflow because their time constants are outside the bandwidth of interest for control. The result is a set of differential algebraic equations for the boundaries of the regions and for the averages of the variables that characterize the storage of mass and energy in the central volume.
For the convenience of reading the next section, notation for the moving boundary model equations is provided in Table 4.2.
To integrate the equations over the central volume it is necessary to make assumptions about the distributions of mass and energy inside that volume. Reasonable distributions are obtained by using essential param-eters from a detailed investigation of inhomogeneous, distributed models.
The distribution of mass depends on the void fraction, presented in Sec-tion 4.12.
4.11 Moving Boundary Models
Roman and Greek Letters
A area h enthalpy
Cw heat cap. of wall m˙ mass flow
t time q heat flux
D diameter L length
v velocity z length coordinate
S slip ratio x mass fraction
α heat transfer coeff. ρ density
γ void fraction ω pump speed
µ density ratio Φ dissipation function η liquid fraction Ψ vapor generation rate
Subscripts
1 subcooled i inner
2 two-phase in inlet
3 superheated l saturated liquid
12 interface 1-2 o outer
23 interface 2-3 out outlet
amb ambient r refrigerant
g saturated gas w wall
w1 subcooled wall w2 two phase wall w3 superheated wall f fluid
Double subscripts are used for some flows.
qPw11means: heat flux from subcooled wall to region 1 fluid Superscripts
P flux per length ∗ normalized variable
Table 4.2 Notation for moving boundary models.
By neglecting work terms like viscous stresses, axial conductance and assuming a single heat transfer interaction the energy balance(4.15) can
Chapter 4. Physical Models for Thermo-Hydraulics be simplified to
V(Aρh− Ap)
Vt +V ˙mh
V z = qPw f (4.48)
A simplified energy balance for the wall is obtained by setting all convec-tion terms in(4.48) equal to zero, assuming two heat transfer terms and neglecting the axial conductance, hence
CwρwAwVTw
Vt = −qPw f + qPambw (4.49) The heat flows per pipe length, qP, can usually be calculated using a con-stant heat transfer coefficient. For the subcooled wall section this gets:
qPw11=αiπDi(Tw− Tf) (4.50) qPambw1=αoπDo(Tamb− Tw) (4.51) Equations(4.6), (4.48) and (4.49) are the balance equations, which will be integrated over the three regions to give the general three region lumped model for a two-phase heat exchanger, see Figure 4.8. The detailed deriva-tion for the three-zone model is presented below. The derivaderiva-tion of the other models is similar.
Tw1 Tw2 Tw3
L1 L2 L3
m˙out
m˙in
hout
hin
m˙12, hl ρl m˙23, hgρg
T1, h1ρ1 T2= Tg= Tl T3, h3ρ3
Tamb
Figure 4.8 Schematic of the three region moving boundary model.
Mass Balance for the Subcooled Region
Consider the three region model outlined in Figure 4.8. This model is representative for a once-through boiler. Integration of the mass balance (4.6) over the subcooled region gives
Z L1
0
V(Aρ) Vt dz+
Z L1
0
V ˙m
V zdz= 0 (4.52)
4.11 Moving Boundary Models Integrating the second term and differentiating the resulting equation gives for a constant area pipe:
Ad dt
Z L1
0
ρdz− Aρ(L1)dL1
dt + ˙m12− ˙min= 0.
The density at the interfaceρ(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
ρ¯1= 1 L1
Z L1
0
ρdzρ(p,¯h1) T¯1 T(p,¯h1).
Using the above expressions, the mass balance for the subcooled region can be written as
A h
ρ¯1−ρl
dL1 dt + L1
d ¯ρ1
dt
i= ˙min− ˙m12. (4.53)
The term d ¯ρ1/dt is calculated using the chain rule:
d ¯ρ1
dt =V ¯ρ1
V p
h
dp dt +V ¯ρ1
V ¯h1
p
d ¯h1
dt
=V ¯ρ1
V p
h+1 2
V ¯ρ1
V ¯h1
p
dhl
dp
dp dt +1
2 V ¯ρ1
V ¯h1
p
dhin
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
h( ¯ρ1−ρl)dL1
dt + L1V ¯ρ1
V p
h+1 2
V ¯ρ1
V ¯h1
p
dhl
dp
dp dt +1
2L1V ¯ρ1
V ¯h1
p
dhin dt
i= ˙min− ˙m12. (4.54)
Chapter 4. Physical Models for Thermo-Hydraulics Energy Balance for the Subcooled Region
Integration of the energy balance(4.48) over the subcooled region gives Z L1
0
V(Aρh− Ap)
Vt dz+
Z L1
0
V ˙mh V z dz=
Z L1
0
qPw1ldz. (4.55)
For a constant area pipe and heat flow per length qP, integration over the length and subsequent differentiation result in:
Ad dt
Z L1
0
ρhdz− Aρ(L1)h(L1)dL1
dt − AL1
dp dt
= ˙minhin− ˙m12hl+ L1qPw11.
(4.56)
Using
ρ¯1¯h1ρ1h1= Z L1
0
ρhdz
and expanding equation(4.56) gives the expression for the energy balance of the subcooled region:
1 2A
"
ρ¯1(hin+ hl) − 2ρlhl
dL1
dt +
ρ¯1L1+V ¯ρ1
Vh
p
dhin
dt + L1
nρ¯1
dhl
dp + (hin+ hl)⋅V ¯ρ1
V p
h+1 2
V ¯ρ1
Vh
p
dhl
dp − 2odp dt
#
= ˙minhin− ˙m12hl+ L1qPw11
(4.57)
Superheated and Two Phase Zones
The derivation of the mass- and energy balances for the superheated and two phase regions follow the same pattern. The integrals are expanded, mean values are introduced which are defined by the integrals and deriva-tives of the density are expanded into derivaderiva-tives of pressure and specific enthalpy. The details of the derivation are presented in Appendix C. The final results for the superheated region becomes:
A h
L3
1 2
V ¯ρ3
V ¯h3
p
dhg dp +V ¯ρ3
V p
h
dp
dt + (ρg− ¯ρ3)dL1
dt + (ρg− ¯ρ3)dL2
dt +1
2L3V ¯ρ3
V ¯h3
p
dhout
dt
i= ˙m23− ˙mout.
(4.58)
4.11 Moving Boundary Models The energy balance for the superheated region reads
A
hρghg−1
2ρ¯3(hg+ hout)dL1 dt +dL2
dt
+ L3
h1
2(hg+ hout)1 2
V ¯ρ3
V ¯h3
p
dhg dp +V ¯ρ3
V p
h
+1 2ρ¯3
dhg
dp − 1idp dt +1
2ρ¯3L3+1 4
V ¯ρ3
V ¯h3
p(hg+ hout)L3
dhout dt
i
= ˙m23hg− ˙mouthout+ L3qPw33
(4.59)
The mean properties of the superheated region are calculated in the same way as in the subcooled region. Thus
¯h3= 0.5(hg+ hout),ρ¯3ρ(p,¯h3) and ¯Tr3 T(p,¯h3).
The heat fluxes between the pipe wall and the two phase zone respec-tively superheated zone are calculated as in(4.50) with the appropriate substitutions in the variables:
qPw33=πDiαi3(Tw3− ¯Tr3) (4.60) qPw22=πDiαi2(Tw2− Tr2) (4.61) The flow in the two-phase region is assumed to be at equilibrium con-ditions with a mean density of ¯ρ= ¯γ ρg+(1− ¯γ)ρl, where the void fraction is defined asγ = Avap/A. The average void fraction is defined as
γ¯= 1 L2
Z L1+L2
L1
γdz.
The central assumption for the following derivation is that ¯γ changes much slower in time than the other variables such that it can be treated as a constant when differentiated with respect to time. This precludes using the model for fast pressure transients. A detailed model of the calculation of the void fraction is derived in section 4.12. The mass balance for the two-phase region becomes
A
n(ρl−ρg)L1
dt + (1 − ¯γ)(ρl−ρg)dL2
dt + L2
γ¯dρg
dp + (1 − ¯γ)dρl
dp
dp dt
o= ˙m12− ˙m23. (4.62)
Chapter 4. Physical Models for Thermo-Hydraulics
and the energy balance for the two-phase region is given by A
( L2
h
γ¯d(ρghg)
dp + (1 − ¯γ)d(ρlhl)
dp − 1idp dt +h
γ ρ¯ ghg+ (1 − ¯γ)ρlhl idL1
dt +h
(1 − ¯γ) ρlhl−ρghgi dL2 dt
)
= ˙m12hl− ˙m23hg+ L2qPw2p
(4.63) The derivative of the properties at the phase boundaries are written in a short notation and can be rewritten as e. g., d(ρghg)/dp = hg(dρg/dp) + ρg(dhg/dp). Both d(ρghg)/dp and d(ρlhl)/dp are functions of only pres-sure because they are on the phase boundary.
Energy Balance for the Wall Regions
Integration of the wall energy equation(4.49) fromα toβ gives Z β
α CwρwAwVTw
Vt dz= Z β
α αiπDi(Tr− Tw)dz +
Z β
α αoπDi(Tamb− Tw)dz
(4.64)
Integrating, assuming constant wall properties and rearranging gives the general energy balance for a wall region:
CwρwAw
(β −α)dTw
dt + Tw(α) − Tw
dα
dt + Tw− Tw(β) dβ dt
= (β−α)qPw f − (β−α)qPambw.
(4.65)
For the wall region adjacent to the subcooled regionα = 0 andβ = L1, which gives
CwρwAw
L1
dTw1
dt + Tw1− Tw(L1) dL1 dt
= L1qPw1l− L1qPambw1.
(4.66)
The wall temperature in the model is discontinuous at L1giving Tw(L1) = Tw2 for dL1
dt > 0 Tw(L1) = Tw1 for dL1
dt ≤ 0
(4.67)
Similar expressions are derived for the walls adjacent to the two-phase and the superheated regions.
4.12 Void Distribution