Design equation solving

Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

Example 2.12 illustrates a general result. If local stoichiometry is preserved, no more than M reactor design equations need to be solved to determine all... 

Computational Scheme for Gas-Phase PFRs. A general procedure for solving the reactor design equations for a piston flow reactor using the marching-ahead technique (Euler s method) has seven steps ... 

The design equations for reactor 1 are solved and used as the input to reactor 2. 

Pick a rate expression and assume values for its parameters. Solve the reactor design equations to predict the response. Call this prediction. ... 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

It is readily apparent that equation 8.3.21 reduces to the basic design equation (equation 8.3.7) when steady-state conditions prevail. Under the presumptions that CA in undergoes a step change at time zero and that the system is isothermal, equation 8.3.21 has been solved for various reaction rate expressions. In the case of first-order reactions, solutions are available for both multiple identical CSTR s in series and individual CSTR s (12). In the case of a first-order irreversible reaction in a single CSTR, equation 8.3.21 becomes... 

For constant fluid density the design equations for plug flow and batch reactors are mathematically identical in form with the space time and the holding time playing comparable roles (see Chapter 8). Consequently it is necessary to consider only the batch reactor case. The pertinent rate equations were solved previously in Section 5.3.1.1 to give the following results. 

In the case of isothermal operation the material and energy balance equations are not coupled, and design equations like 10.2.1 can be solved readily, since the reaction rate can be expressed directly as a function of the fraction conversion. For operation in this mode, an energy balance can be used to determine how the heat transfer rate should be programmed to keep the system isothermal. For this case equation 10.2.12 simplifies to the following expression for the heat transfer rate... 

Equations B and C may now be combined with design equation A to solve for the required holding time. 

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. 

Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http //www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http //www.comsol.com/), and ANSYS (http //www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing... 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Example 18-1 can also be solved by means of the E-Z Solve software (file exl8-l.msp). In this case, the design equations for each stage are written for species A and B in terms of cA, CB> Cc> together with cB = cc. The resulting nonlinear equation set is solved by the software. In this approach, there is no need to introduce /A, nor to relate the concentrations to /A, although /A can be calculated at the end, if desired. 

Similarly, the design equation may be used to solve for c,(t). In this case, it is simpler to use the implicit relationship between cA(f) and cD(f), as follows ... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

The solution is complicated by the fact that many of the parameters in the design equations depend on the vessel dimensions (h or D or V). In addition to the design equations, we give expressions for these parameters in terms of D. The result is a set of nonlinear algebraic equations to be solved for the unknown quantities, including D. We solve these by means of the E-Z Solve software (file ex24-3.msp). 

Equation (35) must now be solved simultaneously with the design equation (5). 

This expression links Xa, T and V and must be solved simultaneously with the design equation (63) in which r = f(T, jCa) to give the reactor volume needed to achieve a given conversion. 

In this section we have presented modeling results for industrial type IV FCC units that produce high octane number gasoline from gas oil. Such units consist of two connected bubbling fluidized beds with continuous circulation of the catalyst between the two vessels, the reactor and the regenerator. The steady-state design equations are nonlinear transcendental equations which can be solved using the techniques described in the earlier chapters of the book. 

To gain the maximum benefit from the use of a flowsheet program, the operator/designer must be adequately trained. A suitable program will have 20-30 standard units available, numerous equation-solving procedures, control facilities and probably optimization facilities. The unit-equipment subroutine must adequately represent the process equipment, recycle streams need to be specified, and suitable solution convergence is required. For the effective use of CAD packages, it... 

Formulate the die design equation for an end-fed sheeting die such as the one solved for in this chapter. Unlike that die, the new die should have a variable radius manifold as presented in Fig. 6.73. In this example, the axial distance from the manifold center to the die lips must be maintained constant. Assume a Newtonian viscosity, //,. 

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