Rate equations, chemical stirred tanks

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system (Dunn et ah, 2003). 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

When a series of stirred-tanks is used as a chemical reactor, and the reactants are fed at a constant rate, eventually the system reaches a steady state such that the concentrations in the individual tanks, although different, do not vary with time. When the general material balance of equation 1.19 is applied, the accumulation term is therefore zero. Considering first of all the most general case in which the mass density of the mixture is not necessarily constant, the material balance on the reactant A is made on the basis of FA moles of A per unit time fed to the first tank. Then a material balance for the rth tank of volume V (Fig. 1.17) is, in the steady state ... 

The general equations for chemical reaction in a turbulent medium are easy to write if not to solve (2). In addition to those for velocities (U = U + uJ and concentrations (Cj = Cj + Cj), balance equations for q = A u, the segregation ( , and the dissipations e and eg can be written (3). Whatever the shape of the reactor under consideration (usually a tube or a stirred tank), the solution of these equations poses difficult problems of closure, as u S, 5 cj, cj, and also c cj, c Cj in the reaction terms have to be evaluated. The situation is even more complicated when the temperature and the density of the reacting mixture are also fluctuating. Partial solutions to this problem have been proposed. In the case of instantaneous reactions (t << Tg) the "e-quilibrium assumption" applies the mixed reactants are immediately converted and the apparent rate of reaction is simply that of the decrease of segregation, with Corrsin s time constant xs. For instance, with a stoichiometric proportion of reactants, the extent of reaction X is given by 1 - /T ( 2), a simple result which can also be found by application of the IEM model (see (33)). 

We turn now to consider the principal types of reactors and derive a set of equations for each that will describe the transformation 5 of the state of the feed into the state of the product. The continuous flow stirred tank reactor is one of the simplest in basic design and is widely used in chemical industry. Basically it consists in a vessel of volume V furnished with one or more inlets, an outlet, a means of cooling and a stirrer which keeps its composition and temperature essentially uniform. We shall assume that there is complete mixing on the molecular scale. It would be possible to treat of other cases following the work of Danckwerts (1958) and Zweitering (1959), but the corresponding transformation is much less wieldy. If the reactants flow in and out at a constant rate q, the mean residence time T/g is known as the holding time of the reactor. 

It is possible to express Eq. (3-10) for isothermal operation in simpler forms when assumptions such as constant density are permissible. These will be considered in Chap. 4. The constant-density form of Eq. (3-10) was used in Chap. 2 to calculate rate constants from measured conversions or concentrations as a function of time (see, for example. Sec. 2-7). It is important to recall that we could determine the rate equation for the chemical step from a form of Eq. (3-10) because the reactor is assumed to be an ideal stirred-tank type, with no physical resistances involved. 

For a closed chemical system with a mass action rate law satisfying detailed balance these kinetic equations have a unique stable (thermodynamic) equilibrium, lim c( )=Cgq. In general, however, we shall be concerned with chemical reactions that are maintained far from chemical equilibrium by flows of reagents intoand out of a continuously stirred tank reactor (CSTR). In this case the chemical kinetic equation (C3.6.1) must be supplemented with flow terms... 

Another class of equations subsumes kinetic and phase transformations of all involved reactants. Such equations describe how the reactants molecules are transformed and distributed in the reactor depending on time and other environmental parameters. Depending on the kind of chemical processes under consideration, both classes of equations are of varying importance for modelling. E.g. for catalytic packed-bed reactors the chemicals reaction rates heavily depend on local physical conditions at the (solid) catalyst material. The precise modelling of the local physical conditions and the mixture of chemicals flowing is important and complex in this case. In contrast, for classic stirred-tank reactors kinetic and phase transformations are comparatively easy to model. 

Continuous flow stirred-tank reactors are normally just what the name implies tanks into which reactants flow and from which a product stream is removed on a continuous basis. CFSTRs, CSTRs, C-star reactors, and backmix reactors are only a few of the names applied to the idealized stirred-tank flow reactor model. We will use the letters CSTR in this book. The virtues of a stirred-tank reactor lie in its simplicity of construction and the relative ease with which it may be controlled. These reactors are used primarily for carrying out liquid phase reactions in the organic chemicals industry, particularly for systems that are characterized by relatively slow reaction rates. If it is imperative that a gas phase reaction be carried out under efficient mixing conditions similar to those found in a stirred-tank reactor, one may employ a tubular reactor containing a recycle loop. At sufficiently high recycle rates, such systems approximate the behavior of stirred tanks. In this section we are concerned with the development of design equations that are appropriate for use with the idealized stirred-tank reactor model. 

This section contains several models whose spatiotemporal behavior we analyze later. Nontrivial dynamical behavior requires nonequilibrium conditions. Such conditions can only be sustained in open systems. Experimental studies of nonequilibrium chemical reactions typically use so-called continuous-flow stirred tank reactors (CSTRs). As the name implies, a CSTR consists of a vessel into which fresh reactants are pumped at a constant rate and material is removed at the same rate to maintain a constant volume. The reactor is stirred to achieve a spatially homogeneous system. Most chemical models account for the flow in a simplified way, using the so-called pool chemical assumption. This idealization assumes that the concentrations of the reactants do not change. Strict time independence of the reactant concentrations cannot be achieved in practice, but the pool chemical assumption is a convenient modeling tool. It captures the essential fact that the system is open and maintained at a fixed distance from equilibrium. We will discuss one model that uses CSTR equations. All other models rely on the pool chemical assumption. We will denote pool chemicals using capital letters from the start of the alphabet. A, B, etc. Species whose concentration is allowed to vary are denoted by capital letters... 

Consider an irreversible, exothermic chemical reaction, A B, taking place in a continuously stirred tank reactor (CSTR) thermostated by a heat bath. Assuming that the rate coefficient of the reaction has an Arrhenius form, the macroscopic kinetic equations for the (dimensionless) composition, c, and temperature, T, can be written as. 

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