Rate Equations and Operating Modes

The equations of this section are summarized and extended in Table 17.2. The term rate of reaction used here is the rate of decomposition per unit volume, 

A rate of formation will have the opposite sign. When the volume is constant, the rate is the derivative of the concentration 

In homogeneous environments the rate is expressed by the law of mass action in terms of powers of the concentrations of the reacting substances 

When the reaction mechanism truly follows the stoichiometric equation 

The coefficient k is called the specific rate. It is taken to be independent of the concentrations of the reactants but does depend primarily on temperature and the nature and concentration of 

TABLE 17.1. Residence Times and/or Space Velocities in Industrial Chemical Reactors 

In practical cases reaction times vary from fractions of a second to many hours. The compilation of Table 17.1 of some commercial practices may be a basis for choosing by analogy an order of magnitude of reactor sizes for other processes. 

For ease of evaluation and comparison, an apparent residence time often is used instead of the true one it is defined as the ratio of the reactor volume to the inlet volumetric flow rate, 

On the other hand, the true residence time must be found by integration. 

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It is important to emphasize the general character of this definition for the volumetric productivity (Py), since it includes all phases of a production cycle in a bioreactor, allowing an evaluation of the impact of bioreactor preparation time and duration of growth and production phases on productivity. As can be observed from Equations 19 to 22, for an industrial bioreactor with a given volume and operation mode, the volumetric productivity depends basically on cell concentration in the production phase and on the specific product formation rate (qp). 

A kinetic distinction between the operation of the SN1 and SN2 modes can often be made by observing the effect on the overall reaction rate of adding a competing nucleophile, e.g. azide anion, N3e. The total nucleophile concentration is thus increased, and for the SN2 mode where [Nu ] appears in the rate equation, this will result in an increased reaction rate due to the increased [Nut]. By contrast, for the Stfl mode [Nut] does not appear in the rate equation, i.e. is not involved in the rate-limiting step, and addition of N3e will thus be without significant effect on the observed reaction rate, though it will naturally influence the composition of the product. 

For isothermal and adiabatic modes of operation the energy balance equations developed above will simplify so that the design calculations are not nearly as tedious as they are for the other modes of operation. In the case of adiabatic operation the heat transfer rate is zero, so equation 10.2.10 becomes... 

The overall approach relies on the determination of accurate rate constants for the reaction of the chiral C-0 carbon. The rate equation is dn/dl = -ufn, where the reaction cross section is ct (cm2). Straightforward integration yields n(Q/n(0) = e n - where c = Jf(t)dt is the photon exposure. Note that in general the photon flux fit) varies with time so the integral must be used. However, the APS operates in top-up mode where the beam current of the stored electrons is kept nearly constant by injection every few minutes. Thus/(f) = constant = / and the resulting equation is n(Qln( 0) = where/is the flux density and f is the time. The time... 

Enzymatic action can be defined on three levels operational kinetics, molecular architecture, and chemical mechanism. Operational kinetic data have given indirect information about cellulolytic enzyme mode of action along with important information useful for modeling cellulose hydrolysis by specific cellulolytic enzyme systems. These data are based on measurement of initial rates of enzyme hydrolysis with respect to purified celluloses and their water soluble derivatives over a range of concentrations of both substrate and products. The resulting kinetic patterns facilitate definition of the enzyme s mode of action, kinetic equations, and concentration based binding constants. Since these enable the enzymes action to be defined with little direct knowledge of its mechanistic basis, the rate equations obtained are referred to as operational kinetics. The rate patterns have enabled mechanisms to be inferred, and these have often coincided with more direct observations of the enzyme s action on a molecular level [2-4]. 

Thus, as in the isothermal mode of operation, appropriate quantities are monitored continuously or at short time intervals in the TS-BR, and concentrations and temperatures are recorded as raw data. Since there is no volume expansion in a constant volume BR, the relationship between conversion and concentration is simply X = 1 - CA / CA0. Then the following rate equation holds for a constant-volume BR ... 

Let us first consider the steady-state RD operation mode. The mass-balance equations of the traditional multicomponent rate-based model (see, e.g. [42, 43]) are written separately for each phase, and, as chemical reactions take place in the liquid phase, the steady-state liquid-phase balance equation should be extended to include the reaction source term ... 

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