## Basic Modeling Equations

The dynamic models used for process modeling and control can be mathematically represented by a set of balance equations (conservation equations). These may be supplemented by one or more constitutive equations that further define terms in the balance equations.

Every dynamic model will include at least one balance equation. The balance equations will have the same general form you've been using for several years now:

The accumulation term will supply the time derivative and produce a differential equation. If the model is distributed, the equation will also include position derivatives. Transport terms will typically be initially written in terms of the transport fluxes.

### Mass Balance

(a.k.a. total material balance, continuity equation)

A mass balance is needed whenever you are interested in the "holdup" of a system. Holdup is typically measured using level for liquid systems or pressure for gas/vapor systems. You should expect your mass balance equation to have either or both of these variables inside the accumulation term. Unless you are dealing with a nuclear reaction, the mass balance will not have a generation term.

A mass balance may be written over each system or subsystem that you can define within your process.

Constitutive equations may be needed to define system properties such as density in terms of composition, temperature, pressure, etc.

As an example, consider the mass balance on a mixing tank.

Common assumptions used in writing and simplifying mass balances include:

1. perfect mixing (eliminates position dependence)
2. constant densities (w.r.t. time)
3. equal densities (for similar components)

### Component Balance

(a.k.a. component continuity equation)

A component balance must be written whenever composition changes are to be examined. Almost all reactor or separator problems will involve a component balance. Compositions are usually expressed in terms of mole fractions.

You can write one balance for each component over each subsystem, but remember that the sum of all component balances is the total material balance, so normally will use one total mass balance and (NC-1) component balances.

Initially, transport terms in a component balance will be expressed in terms of the transport fluxes (both molecular and convective). Constitutive equations defining the fluxes will thus be needed. When reactors are modeled, generation terms will be required and will be written in terms of reaction rate expressions. These will also need to be defined by a constitutive equation.

A CSTR with an A-->B reaction provides an example:

Common assumptions include:

1. perfect mixing (CSTRs)
2. constant volume (gas systems, filled liquid reactors)

### Energy Balance

(a.k.a. equation of energy, enthalpy balance)

You will need to write an energy balance whenever the temperature within your system changes; temperature will almost always be inside the derivative. Reference temperatures for enthalpies can complicate things, so be careful.

One energy balance can be written for each separable system or subsystem.

Energy transport fluxes and thermodynamic property relations will require constitutive equations for fuller definition.

A heated, stirred tank serves as an example:

Modeling assumptions may include:

1. perfect mixing
2. constant densities, specific heats
3. equal densities, specific heats

### Constitutive Equations

All models will include one or more balance equations. Most will also use a set of constitutive equations to better define specific terms in the balance equations. Common constitutive relationships include:

• property relations and equations of state
• transport flux relations
• reaction rate expressions
• equilibrium expressions
• fluid flow relations

#### Property Relations / Equations of State

Physical and thermodynamic properties (density, heat capacity, enthalpy, etc.) vary with temperature, pressure, and composition. These relationships usually must be incorporated into dynamic models.

Enthalpy is typically expressed as a function of temperature

Equations of state are typically used to express vapor densities in terms of system temperature and pressure. Often, the ideal gas equation is adequate

#### Transport Flux Expressions

Transport flux expressions are usually used to quantify heat and mass transfer. When transport is purely molecular, these are nothing more than statements of Fick's Law, Fourier's Law, and Newton's Law of Viscosity. They then look like

When convective transport is significant, heat and mass transfer coefficients are typically used, leading to expressions like

#### Reaction Rate Expressions

The reaction rate expressions used in dynamic modeling are typically based on the principles of mass action. The Arrhenius expression must be incorporated directly when rate constants depend on temperature; otherwise, the energy balance won't adequately describe temperature changes.

#### Equilibrium Expressions

Phase equilibrium expressions are often needed when modeling separation systems. Raoult's law, equilibrium K-values, and relative volatility, all are used. The choice is the modelers.

Chemical equilibrium expressions are needed less often. If they are needed, they are usually incorporated as part of the reaction rate expression.

#### Fluid Flow Relations

Fluid flow relationships are typically used when it is necessary to relate pressure drop to flow rate. These usually take the form of a momentum balance (equation of motion) or mechanical energy balance. Momentum balances are typically required for gravity flow problems (where the balance may reduce to Torricelli's Law).

For systems involving flow through a weir or across a valve, a reduced form of the mechanical energy balance is often used. Rather than deriving these from the balance, it is usually reasonable to select an appropriate eqaution, for instance

References:

1. Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 17-38.
2. Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 57-61.
3. Riggs, J.B., Chemical Process Control (2nd Edition), Ferret Publishing, 2001, pp. 98-99.

R.M. Price
Original: 9/29/93
Modified: 1/6/98; 5/2/2003