A modeling task is specified by giving the description of the process system to be modeled together with the modeling goal,i.e.the intended use of the model. The modeling goal largely determines the model, its variables, spa- tial and time characteristics, as well as its resolution or level of detail and precision. Process control as a modeling goal does not require very accurate models, we only aim at about 5 percent precision in values, but process con- trol requires dynamic models that capture the time characteristics (dead time and time constants) of the model well. Moreover, we usually use lumped pa- rameter dynamic process models for control purposes because the resulting finite dimensional system models are much easier to handle. Therefore we usually use some kind of lumping, most often the so-called “method of lines”
procedure (see,e.g.[32]) to obtain a lumped parameter approximation from a distributed parameter system model.
Definition 4.1.1 (Balance volume) Parts of a lumped process system which
• contain only one phase or pseudo-phase,
• can be assumed to be perfectly mixed, will be termed balance volumes or lumps.
Balance volumes are the elementary dynamic units of a lumped process sys- tem for which dynamic balances can be constructed and assumptions can be made.
4.1.1 General Modeling Assumptions
We restrict ourselves to the following class of systems and system models throughout the book:
1. Only lumped process models that result in a model in ordinary differential-algebraic equation (DAE) form are considered.
2. We only treat initial value problems.
3. All physical properties in each phase are assumed to be func- tions of the thermodynamic state variables (temperature, pressure and compositions) of that phase only.
These general assumptions ensure that we always have an index 1 model with the possibility of substituting the algebraic constitutive equations into the differential ones. This is always possible if we choose the free variables and parameters of the model, the so-calledspecification, in a proper way.
In order to get relatively simple models where the algebraic equations can all be substituted into the differential ones, we use the following two additional general assumptions in the majority of our examples:
4.1 Process Modeling for Control Purposes 41
4. Constant pressure is assumed in the whole process system.
5. All physical properties in each phase are assumed to be con- stant.
Note that the last general assumption overrides assumption 3 by stating that all physical properties do not depend on any other variable, but that they are constant.
4.1.2 The Principal Mechanisms in Process Systems
Mechanisms in a process system describe different means of material or en- ergy transport or transformation. Therefore we encounter a great variety of possible mechanisms in a process system. In order to complete the steps of constructing a mathematical model of a process system (see Section 4.1.4 for more details) we need to analyze the modeling problem statement and decide which mechanisms should be included in the model. For lumped pa- rameter process systems these mechanisms include, but are not limited to, the following:
1. Convection:
A material and energy transport mechanism in which the conserved extensive quantities (overall mass, component masses and energy) are carried by the transport of the overall mass, i.e. by flows. The inflows and outflows of the balance volumes can be regarded as convection in lumped process systems.
2. Transfer:
A component mass or energy transport mechanism between two phases in contact when there are no convective flows involved. The driving force for transfer is the difference between the thermodynamical state variables (temperature, pressure and compositions) in the two phases.
3. Chemical reaction:
A component mass transformation mechanism, which generates the products of a chemical reaction from the reactants. It usually also in- volves enthalpy (energy) transformation: generation or consumption.
4. Phase changes:
A phase transformation mechanism, such as evaporation, condensation, melting, boiling, crystallization,etc.where the chemical composition re- mains unchanged. It also involves enthalpy (energy) transformation.
A basic property of the mechanisms of a process system is that they are assumed to bestrictly additive, that is, they give rise to additive terms in the conservation balance equations of a process system.
4.1.3 The Basic Ingredients of Lumped Process Models
The equations of a particular model satisfying the general modeling assump- tions” in Section 4.1.1 are of two types:
• differential equations (explicit first-order nonlinear ODEs with initial con- ditions),
• algebraic equations.
The differential equations originate from conservation balances, therefore they can be termedconservation balance equations. The algebraic equations are usually of mixed origin: they will be calledconstitutive equations. Along with the above equations we have other model elements associated with them such as:
• modeling assumptions,
• variables and parameters,
• initial conditions,
• data (specification) of model parameters and constants.
Variables are time-varying or time-dependent quantities in process model equations. They are also called signals in system theoretical terminology. A variablexis calleddifferential if its time derivative (dxdt) is explicitly present in the DAE model. A variable is termedalgebraic if it is not differential.
Parameters, on the other hand, are quantities which are either constant or are regarded to be constant in a particular process model.
4.1.4 The Model Construction Procedure
In order to construct a process model satisfying a given modeling goal, the problem statement of the modeling, that is, the process system description together with the goal, should be carefully analyzed first to find the relevant mechanisms together with a suitable level of detail. These drive the construc- tion of the process model equations, which is carried out in steps forming the model construction procedure. Good modeling practice requires asystematic way of developing the model equations of a process system for a given purpose.
Although this procedure is usually cyclic, in which one often returns back to a previous step, the systematic procedure can be regarded as a sequence of modeling steps as follows:
Step 0. System and subsystem boundary and balance volume definitions The outcome of this step is the set of balance volumes for mass, energy and momentum. These are the conserved extensive quantities normally considered in process systems. Moreover, the number of components is also fixed for each mass balance volume.
4.1 Process Modeling for Control Purposes 43
Step 1. Establish the balance equations
Here we set up conservation balances for mass, energy and momentum for each balance volume.
Step 2. Transfer and reaction rate specifications
The transfer rate expressions between different balance volumes in the conser- vation balances are specified here usually as functions of intensive quantities.
The reaction rates within balance volumes are also specified.
Step 3. Property relation specifications
Mostly algebraic relationships expressing thermodynamic knowledge, such as equations of state and the dependence of physico-chemical properties on thermodynamic state variables, are considered here.
Step 4. Balance volume relation specifications
Equipment with a fixed physical volume is often divided into several balance volumes if multiple phases are present. A balance volume relation describes a relation between balance volumes and physical volumes.
Step 5. Equipment and control constraint specifications
There is inevitably the need to define constraints on process systems. These are typically in the form of equipment-operating constraints (in terms of tem- peratures, pressures, etc.) and in terms of control constraints, which define relations between manipulated and controlled variables in the system.
Step 6. Selection of design variables
The selection of design variables is highly dependent on theapplication area or problem and is not necessarily process- specific. The process itself only provides constraints on which variables are potentially relevant. The selection of design variables may greatly influence the mathematical properties of the model equations, such as the differential index.
4.1.5 Conserved Extensive and Intensive Potential Variables Any variable characterizing a process system can be classified as eitherexten- sive or intensive depending on how this variable behaves when joining two process systems together.
Definition 4.1.2 (Extensive variable)
A variable which is proportional to the overall mass of the system, that is, which is strictly additive when joining two process systems, is termed an extensive variable.
Dictated by the basic principles of thermodynamics, there is acanonical set of extensive variableswhich is necessary and sufficient to describe uniquely a single phase process system. This set includesoverall mass, component masses and energy for a perfectly stirred (lumped) balance volume. It is important to note that these extensive variables are conserved, therefore conservation balances can be constructed for each of them (see later in Section 4.1.6).
Potentials or intensive quantities are related to any of the above extensive conserved quantities.
Definition 4.1.3 (Intensive variable (potential))
Intensive variable (or potential) difference (both in space and between phases in mutual contact) causes transport (transfer or diffusion depending on the circumstances) of the related extensive quantity.
The following intensive variable (potential) – extensive variable pairs are normally considered in process systems:
• temperature to internal energy or enthalpy,
• chemical potential, or simply concentration, to mass of a component,
• pressure to overall mass (not relevant for our case due to the constant pressure general assumption in Section 4.1.1).
4.1.6 Conservation Balances
Conservation balances can be set up for any conserved extensive variable in any balance volume of a process system. Recall that overall mass, compo- nent masses and internal energy form the canonical set of conserved extensive quantities. If we consider an open balance volume with in- and outflows (con- vection), transport (an inter-phase mechanism) and other intra-phase trans- formation mechanisms, the verbal form of a conservation balance equation is as follows:
net change in ext. quantity
= in-
f lows
− out-
f lows
+
generation consumption
(4.1) Note that there is no source term for the overall mass balance because of the mass conservation principle. Chemical reaction appears in the generation- consumption term of the component mass and the energy balances, while phase transition gives rise to a generation-consumption term in the energy balance. Inter-phase transfer also appears in the generation-consumption term of both the component mass and energy balance equations. There is no overall mass transfer between the phases because of the constant pressure assumption.
The basic equation which drives all the other conservation balances is the overall mass balanceof the perfectly stirred balance volumej:
dm(j)
dt =vin(j)−vout(j) (4.2)
wherevin(j)andvout(j) are the mass in- and outflow rates respectively.
Under the above conditions the general form of a differential balance equation of a conserved extensive quantityφ for a perfectly stirred balance volumej takes the form:
4.1 Process Modeling for Control Purposes 45 dφ(j)
dt =vin(j)φ(j)in −vout(j)φ(j)out+q(j)φ,transf er+q(j)φ,source (4.3) Observe that the overall mass balance (4.2) is a special case of the general balance equation (4.3) with
q(j)m,transf er= 0, q(j)m,source= 0 (4.4)
Note that the conserved extensive quantityφ(j) of balance volumej can be any variable from the following set:
φ∈ {E,(mk, k= 1, . . . , K)} (4.5)
where E is the energy, mk is the component mass of the k-th component, withK being the number of components in the balance volume. The related intensive variable (potential) is taken from the set:
Φ∈ {T,(ck, k= 1, . . . , K)} (4.6)
whereTis the temperature andckis the concentration of thek-th component.
4.1.7 Constitutive Equations
Some of the terms in the general conservation balance equations (4.3) above call for additional algebraic equations to complete the model in order to have it in a closed solvable form. These complementary algebraic equations are calledconstitutive equations. Constitutive equations describe
• extensive–intensive relationships,
• transfer rate equations for mass transfer and heat/energy transfer,
• reaction rates,
• property relations: thermodynamical constraints and relations, such as the dependence of thermodynamical properties on the thermodynamical state variables (temperature, pressure and compositions), equilibrium relations and state equations,
• balance volume relations: relationships between the defined mass and en- ergy balance volumes,
• equipment and control constraints.
Extensive–intensive Relationships. Recall that potentials, being inten- sive variables, are related to their extensive variable counterparts through algebraic equations. These involve physico-chemical properties, which may depend on other potentials or on the differential variables, because the ther- modynamical state variables, temperature, pressure and concentrations are intensive quantities. An example is the well-known intensive–extensive rela- tionship of theU–T pair
U =mcVT
where U is the internal energy, m is the total mass, T is the temperature and cV is the specific heat capacity (a physico-chemical property) of the material in a balance volume. Note that because of the above properties of the intensive quantities and their relation to the extensive conserved quantities, the potential form of the model equations is usually derived by additional assumptions on the physical properties. In the above example
cV =cV(T, P,(ci, i= 1, ..., K))
but because of the “general modeling assumptions” in Section 4.1.1 cV =constis assumed.
Transfer Rate Equations. These are used to describe the algebraic form of the transfer termqφ,transf er(j) in the general balance equation (4.3). Dictated by the Onsager relationship from non-equilibrium thermodynamics, this term has the following general linear form:
q(j)φ,transf er=Kφ,transf er(j,k)
Φ(j)−Φ(k)
(4.7) Here the transfer coefficients Kφ,transf er(j,k) are generally assumed to be con- stants and the driving force for the transfer between balance volumesj and kis the difference of the potential variablesΦ(j) andΦ(k).