Solving the Design Equation

Solving the Design Equation Equations (4-7)-(4-9) are equivalent forms of the design equation for an ideal, constant-volume, batch reactor. The only difference between them is that three different variables, Ca, xa, and have been used to describe the composition of the system at any time. 

The time and composition variables are separable in Eqns. (4-7)-(4-9). If the rate constant k is constant, or if it can be written as a function of either time or composition, the equations can be integrated directly. 

Integration of the design equation will be illustrated by working with Eqn. (4-7). However, you should convince yourself that the same operations can be earned out beginning with Eqns. (4-8) and (4-9), and that the same results are obtained. 

The left-hand side of Eqn. (4-10) can be evaluated by using a standard table of integrals. 

If the reactor is isothermal, k does not vary with time and Eqn. (4-11) becomes 

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Equation 10.3.6, the reaction rate expression, and the design equation are sufficient to determine the temperature and composition of the fluid leaving the reactor if the heat transfer characteristics of the system are known. If it is necessary to know the reactor volume needed to obtain a specified conversion at a fixed input flow rate and specified heat transfer conditions, the energy balance equation can be solved to determine the temperature of the reactor contents. When this temperature is substituted into the rate expression, one can readily solve the design equation for the reactor volume. On the other hand, if a reactor of known volume is to be used, a determination of the exit conversion and temperature will require a simultaneous trial and error solution of the energy balance, the rate expression, and the design equation. 

From the above mole balance equation we can develop the design equation for various reactor types. By solving the design equation we can then determine the time required for a batch reactor system or a reactor volume for a continuous flow system to reach a specific conversion of the reactant to products. 

When the rate of the reverse reaction is significant (i.e., when equilibrium is approached in the reactor) or when more than one reaction is involved, the mechanics of solving the design equation may become more complex, but the principles are the same. Equation (4-2) is applicable, but the more complicated nature of the rate function may make the mathematical integration difficult. 

To solve the design equations, we have to express the rates of the individual reactions in terms of the dimensionless extents of the independent reactions, Zi, Z2, and Zj. 

Note that, in this case, the design equations have seven terms (seven rate expressions), whereas in the formulation in part (a), design equations (d), (e), and (f), have only five terms. This illustrates that, by adopting the heuristic rule, we minimize the number of terms in the design equations. To solve the design equations, we have to express the rates of the individual reactions in terms of the dimensionless extents of the independent reactions, Z, Z5, and Ze. 

In this chapter, we anidyze the operation of ideal batch reactors. In Section 6.1, we review how the design equations are utilized and discuss the auxiliary relations that should be incorporated in order to solve the design equations. In the rest of the... 

As discussed in Chapter 4, in order to describe the operation of a reactor with multiple chemical reactions, we have to write the design equation (Eq. 6.1.1) for each independent chemical reaction. Also, to solve the design equations (to obtain relationships between Z s and t), we have to express V/ (t) and the rates of the chemical reactions, r s and r s, in terms of Z s, and t. The auxiliary relations needed to express the design equations explicitly in terms of Z s, and T, are derived next. 

To solve the design equation, we have to express the reaction rate r in terms of Z and, to do so we relate the species concentrations to the dimensionless extent From Eq. 6.1.11, for isothermal operations with single reactions, and when the reference state is the initial state ... 

Solve the design equations (Z s as functions of t) and obtain the reaction operating curves. 

Once we solve the design equation, we use Eq. 2.7.4 to obtain the species curves ... 

Solve the design equations simultaneously with the energy balance equation, and, if necessary, the energy balance equation of the heating/cooling fluid to obtain Z s, 9, and 9 as functions of the dimensionless space time, t. 

When more than one chemical reaction takes place in the reactor, we have to determine how many independent reactions there are (and how many design equations are needed) and select a set of independent reactions. Next, we have to identify all the reactions that actually take place (including dependent reactions) and express their rates. We write Eq. 8.1.1 for each independent chemical reaction. To solve the design equations (obtain relationships between Z s and t), we express the rates of the individual chemical reactions in terms of the Zm Js and t. Since the temperature is constant, the energy balance equation is used to determine the heating load. The procedure for designing isothermal CSTRs with multiple reactions goes as follows ... 

The design formulation of nonisothermal CSTRs with multiple reactions follows the same procedure outlined in the previous section—we write the design equation, Eq. 8.1.1, for each independent reaction. However, since the reactor temperature, out> is not known, we should solve the design equations simultaneously with the energy balance equation (Eq. 8.1.14). 

We solve the design equations simultaneously with the energy balance equation, subject to the initial condition that at t = 0, the extents of aU the independent reactions and the dimensionless temperature are specified. Note that we solve fiiese equations for a specified value of T t (or reactor volume). The reaction operating curves of plug-flow reactors with side injection are the final value of Z s and 9 for different values of Ttot-... 

Solve the design equations and calculate the molar flowrates of ethane, ethylene and NO as a function of reactor volume for a reactor temperature of 1050 K. 

If the rate equation for the reaction is known, and if the reactor behaves as an ideal batch reactor, these quantities can be calculated by solving the design equation, as illustrated below. 

We know that the reaction rate depends on temperature and concentration. If the temperature and concentration differences between the interior of the catalyst particles and the bulk fluid are significant, then these differences must be taken into account in solving the design equation. In essence, this would require simultaneously solving the design equation and equations that describe heat transport, mass transport, and reaction kinetics in the interior of the catalyst particle, using the equations for transport through the boundary layer as boundary conditions. 

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