Constant-density system

Eor constant density system, the performance equation in terms of becomes... 

For a constant density system, the concentration of any species, K, Cp, during the course of reaction is given by... 

Closure normally begins by satisfying the overall mass balance i.e., by equating the input and outlet mass flow rates for a steady-state system. For the present case, the outlet flow was measured. The inlet flow was unmeasured so it must be assumed to be equal to the outlet flow. We suppose that A and B are the only reactive components. Then, for a constant-density system, it must be that... 

For the special case of a constant density system, Equation 5.41 can be substituted to give ... 

For the special case of constant density systems, substitution of Equation 5.50 gives ... 

Initial conditions are such that the reactants are equimolar with product concentrations of initially zero. From the experimental data, assuming a constant density system ... 

In this section we discuss the mathematical forms of the integrated rate expression for a few simple combinations of the component rate expressions. The discussion is limited to reactions that occur isothermally in constant density systems, because this simplifies the mathematics and permits one to focus on the basic principles involved. We will again place a V to the right of certain equation numbers to emphasize that such equations are not general but are restricted to constant volume batch reactors. The use of the extent per unit volume in a constant volume system ( ) will also serve to emphasize this restriction. For constant volume systems,... 

For variable density systems it is more convenient to work in terms of the fraction conversion (equation 8.2.9), but for constant density systems either equation 8.2.9 or equation 8.2.10 is appropriate. 

It is particularly convenient to choose the reference conditions at which the volumetric flow rate is measured as the temperature and pressure prevailing at the reactor inlet, because this choice leads to a convenient physical interpretation of the parameters and CA0 and, in many cases, one finds that the latter quantity cancels a similar term appearing in the reaction rate expression. Unless otherwise specified, this choice of reference conditions is used throughout the remainder of this text. For constant density systems and this choice of reference conditions, the space time t then becomes numerically equal to the average residence time of the fluid in the reactor. 

If the effluents from the two streams are to be identical and if equimolal feed rates and compositions are employed, the ratio of space times becomes equal to the ratio of total volume requirements. Thus, for constant density systems where CA out = CAN = CA0( 1 - fA),... 

For semibatch operation, the term fraction conversion is somewhat ambiguous for many of the cases of interest. If reactant is present initially in the reactor and is added or removed in feed and effluent streams, the question arises as to the proper basis for the definition of /. In such cases it is best to work either in terms of the weight fraction of a particular component present in the fluid of interest or in terms of concentrations when constant density systems are under consideration. In terms of the symbols shown in Figure 8.20 the fundamental material balance relation becomes ... 

For a constant density system concentrations are directly proportional to weight fractions. Thus equation 11.1.5 becomes... 

If we assume a constant density system and a constant mass flow rate and replace the integrals by finite sums, we have... 

VR/n is the volume of the CSTR used in the model Equations of this form were treated in Section 8.3.2.2. For a constant density system,... 

We derive the kinetics consequences for this scheme for reaction in a constant-volume batch reactor, the results also being applicable to a PFR for a constant-density system. The results for a CSTR differ from this, and are explored in Example 18-4. 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

Since this is a constant-density system, equation 12.3-33 applies. To use this, we require /A(f). From the rate law, and the material balance, equation 2.2-10,... 

For a constant-density system, several simplifications result. First, regardless of the type of reactor, the fractional conversion of limiting reactant, say fA, can be expressed in terms of molar concentration, cA ... 

The following example illustrates a simple case of optimal operation of a multistage CSTR to minimize the total volume. We continue to assume a constant-density system with isothermal operation. 

Isothermal Operation For a constant-density system, since... 

With these results, the general equations of Section 15.2.1 can be transformed into equations analogous to those for a constant-density BR. The analogy follows if we consider an element of fluid (of arbitrary size) flowing through a PFR as a closed system, that is, as a batch of fluid. Elapsed time (t) in a BR is equivalent to residence time (t) or space time (r) in a PFR for a constant-density system. For example, substituting into equation 15.2-1 [dWd/A - FAJ(-rA) = 0] for dV from equation 15.2-15 and for d/A from 15.2-13, we obtain, since FAo = cAoq0,... 

From the discussion in Section 15.2.2.1 comparing the performance of a BR and a PFR for a constant-density system, it follows that... 

The volume of a recycle PFR (V) for steady-state, isothermal operation involving a constant-density system is given by equation 15.3-4, and is a function of the recycle ratio R for given operating conditions (cAo, cAU q0). Show that, as R - V becomes equal to the volume... 

A performance comparison between a BR and a CSTR may be made in terms of the size of vessel required in each case to achieve the same rate of production for the same fractional conversion, with the BR operating isothermally at the same temperature as that in the CSTR. Since both batch reactors and CSTRs are most commonly used for constant-density systems, we restrict attention to this case, and to a reaction represented by... 

Table 18.1 Comparison of PFR and CSTR for series-reaction network A -4 B -+ C (isothermal, constant-density system K = kz/ki)...

For unsteady-state operation, equation 20.1-1 constitutes a set of N ordinary differential equations that must be solved simultaneously (usually numerically) to obtain the time-dependent concentration within each tank. For a constant-density system, dnAl/dr is replaced by Vt dcAi/dr. We focus on steady-state operation in this chapter. 

Equation 20.1-7 may also be used to calculate performance in terms of /A, since, for a constant-density system, fJAN = l-c /c. ... 

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