Passive Mass Convection Network. We can generalize the above free convection case to obtain a convection network which is asymptotically stable, that is, passive in itself, as follows. Let us collect the mass in- and outflows of every balance volume into the vectors:
Vout = [v(1)out . . . v(C)out]T, Vin= [vin(1) . . . v(C)in]T and write Equation (4.17) as
Vin= (Cconv+I)Vout+VIN (4.23)
where I is the unit matrix. In order to make the overall mass subsystem stable, let us apply a full state feedback stabilizing controller (see Chapter 9) in the form of
Vout=KM (4.24)
with a positive definite square state feedback matrixK. By substituting Equa- tions (4.23) and (4.24) into the overall mass balance equations in Equations (4.21), a linear time-invariant state equation results, which has exactly the same form as Equation (4.22).
Definition 4.2.2 (Passive mass convection network)
An overall mass subsystem with the linear time-invariant state equation (4.22) and with a positive definite square state feedback matrix K is called a passive mass convection network.
4.3 Special Nonlinear Process Systems
The previous section shows that the state-space model of lumped process systems indeed exhibits a number of special and potentially useful properties.
This section is devoted to two interesting special cases within the class of lumped process systems, which are as follows:
• lumped process systems with no source terms, resulting in state equations in bilinear form,
• lumped process systems not obeying the “general modeling assumptions”, giving rise to non-standard DAE system models.
4.3.1 Bilinear Process Systems
We have seen in the previous section that lumped process models obeying the “general modeling assumptions” can be transformed into an input-affine nonlinear state-space model form. Moreover, a decomposition of the state equation of lumped process systems has been introduced in Section 4.2.3 in the form of Equation (4.15). There, it was assumed that the flow ratesv(j)
(and the conserved extensive quantities at the inlet) were the input variables (manipulable input variables or disturbances).
This decomposition shows that the input function g in the input-affine state equation ˙x = f(x) +g(x)u is always a linear function of the state vector x because of the properties of the convective terms from which it originates. The nonlinear state function f(x) is broken down into a linear term originating from transfer and a general nonlinear term caused by the sources (other generation and consumption terms, including chemical reac- tions, phase changes,etc.). These observations together result in the condi- tions for a lumped process system model to be in a bilinear form.
Lumped process models with no source term and obeying the “gen- eral modeling assumptions” can be transformed into a bilinear state-space model form, assuming that the flow rates (and pos- sibly the conserved extensive quantities at the inlet) are the input variables (manipulable input variables or disturbances).
4.3.2 Process Models in DAE Form
In this subsection we temporarily relax the constant pressure and constant physico-chemical properties assumptions in the set of “general modeling as- sumptions” in Section 4.1.1 in order to investigate their effect on the set of model equations. It is shown below in the example of a simple evaporator that the presence of the state equation (think of the ideal gas equation as an example), together with the dependence of specific heat capacity on the temperature and pressure, results in a model where the algebraic constitutive equations cannot be substituted into the differential ones. This means that one cannot transform the lumped process model into a canonical nonlinear state-space model form. It remains inherently in its original DAE form.
Example 4.3.1 (A simple evaporator model)
Consider a simple single component phase equilibrium system where vapor and liquid phases are present [31]. This is shown in Figure 4.1. Vapor (denoted by subscriptV) and liquid (L) are taken from the vessel, whilst energy is supplied via a heater. Inside the vessel we have two phases with respective hold-upsMV, ML
and temperaturesTV,TL. A feed (with mass flow rateF) enters the system. In this model representation, we consider two distinct balance volumes: one for vapor, the other for liquid. From this de- scription we can write the system equations in lumped parameter form, which describes the dynamic behavior:
4.3 Special Nonlinear Process Systems 53
1. Conservation balances Mass:
dMV
dt =E−V (4.25)
dML
dt =F−E−L (4.26)
Energy:
dUV
dt =EhLV −V hV +QE (4.27)
dUL
dt =F hF−EhLV −LhL+Q−QE (4.28) 2. Transfer rate equations
Mass:
E= (kLV +kV L)A(P∗−P) (4.29)
Energy:
QE= (uLV +uV L)A(TL−TV) (4.30) where the subscriptLV means liquid to vapor andV L stands for vapor to liquid in the mass and heat transfer coefficientski and ui. The coefficients for V LandLV are normally different.
3. Property relations
hV =hV(TV, P) (4.31)
hL=hL(TL, P) (4.32)
hLV =hLV(TL, P) (4.33)
hF =hF(TF, P) (4.34)
P∗=P∗(TL) (4.35)
P VV = MV
mw
RTV (4.36)
UV =MVhV(TV, P) (4.37)
U =M h (T , P) (4.38)
VL=ML
ρL
(4.39)
kLV =kLV(TL, TV, P) (4.40)
kV L=kVL(TL, TV, P) (4.41)
uLV =uLV(TL, TV, P) (4.42)
uV L=uVL(TL, TV, P) (4.43)
ρL=ρL(TL, P) (4.44)
4. Balance volume relations
VV =VT −VL (4.45)
where VT is the total volume occupied by the vapor and liquid balance volumes. (In this case, it is the vessel volume.)
5. Equipment and control relations
L=f1(ML, P) or L=f2(ML) (4.46) 6. Notation
Boldface variables denote thermodynamic property functions.
Other system variables are:
MV mass hold-up of vapor ML mass holdup of liquid UV vapor phase internal energy UL liquid phase internal energy
F feed flow rate V vapor flow rate
L liquid flow rate E inter-phase mass flow rate
TV vapor phase temperature TL liquid phase temperature Q energy input flow rate QE inter-phase energy flowrate
P system pressure P∗ vapor pressure
A interfacial area R gas constant
VV vapor phase volume VT vessel volume
hV vapor specific enthalpy hL liquid specific enthalpy hF feed specific enthalpy mw molecular weight
ρL liquid density hLV inter-phase vapor specific
VL liquid phase volume enthalpy