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Continuous stirred tank reactor models
Dr. M.J. Willis Dept. of Chemical and Process Engineering, University of Newcastle. e-mail: Tel. Written: Updated: mark.willis@ncl.ac.uk 0191 222 7242 November, 1998 April, 1999; March, 2000

Aims and objectives
Chemical reactors are the most influential and therefore important units that a chemical engineer will encounter. To ensure the successful operation of a continuous stirred tank reactor (CSTR) it is necessary to understand their dynamic characteristics. A good understanding will ultimately enable effective control systems design. The aim of these notes is to introduce some basic concepts of chemical reaction systems modelling and develop simulation models for CSTR's . Non-linear and linear systems descriptions are derived.

Introduction
To describe the dynamic behaviour of a CSTR mass, component and energy balance equations must be developed. This requires an understanding of the functional expressions that describe chemical reaction. A reaction will create new components while simultaneously reducing reactant concentrations. The reaction may give off heat or my require energy to proceed. The mass balance (typical units, kg/s) Without reaction, the basic mass balance expression for a system (e.g tank) is written: Rate of mass flow in – Rate of mass flow out = Rate of change of mass within system Writing the mass balance expression for a stirred tank Consider a well-mixed tank of liquid (figure 1). The inlet stream flow is Fin (m3/s) with density ρin (kg/m3). The volume of the liquid in the tank is V (m3) with constant density ρ (kg/m3). The flow leaving the tank is F (m3/s) with liquid density ρ (kg/m3).

Table 1 summarises each term that appears in the mass balance.

Figure 1. Mixed Tank of Liquid

Rate of mass flow in inlet flowrate × density

Rate of mass flow out exit flowrate × density

Rate of change of mass within system

d(volume × density) dt d(Vρ ) Finρin Fρ dt Table 1. The terms in the mass balance for the stirred tank system.

Referring to table 1 the mass balance is, Fin ρ − Fρ = d(Vρ ) dt (1)

For liquid systems equation (1) normally can be simplified by making the assumption that liquid density is constant. Additionally as V = Ah then, Fin − F = Ad(h) dt (2)

The component balance (typical units, kg/s) To develop a realistic CSTR model the change of individual species (or components) with respect to time must be considered. This is because individual components can appear / disappear because of reaction (remember that the overall mass of reactants and products will always stay the same). If there are N components N – 1 component balances and an overall mass balance expression are required. Alternatively a component balance may be written for each species. A component balance for the jth chemical species is,

Rate of flow of jth component in – rate of flow of jth component out + rate of formation of jth component from chemical reactions = rate of change of jth component Adding a chemical reaction to the stirred tank model Assume that the reaction may be described as, A → B, i.e. component A reacts irreversibly to form component B. Further, assume that the reaction rate is 1st order. Therefore the rate of reaction with respect to CA is modelled as, d(C A ) (3) - kC A = dt The negative sign implies that CA disappears because of reaction (the specification of kinetic expressions describing chemical reactions is explained in more detail in Appendix 1). Writing the component balance for the stirred tank model If the concentration of A in the inlet stream is CAin (moles/m3) and in the reactor CA (moles/m3). Table 2 summarises the terms that appear in the component balance for reactant A. Rate of flow of ‘A’ in Rate of flow of ‘A’ out Rate of change of ‘A’ caused by chemical reaction ∝ {Conc. of A} × Molecular weight Rate of change of ‘A’ inside the tank Molecular weight × d(volume× Conc. of A) dt

Molecular weight × inlet flowrate × conc....
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