Steady-State Design Procedure

Using the assumption of equal eompositions of zc and zd in the reactor, the eoneentration Za can be calculated by rearranging Eq. (3.1). 

The following steps in the design procedure are employed, utilizing the material balances and specifying necessary variables. 

Fix the value of reactor holdup Vr at a small value (to be varied). 

Calculate the distillate product flowrate and compositions for the first column. 

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Traditional steady-state design procedures are used to specify the various pieces of equipment in the plant ... 

Given the fresh feed flow rate and composition, the steady-state design procedure is ... 

The steady-state design procedure is outlined for a given fresh feed flow rate F and fresh feed composition (zo = 1, pure reactant) and a specified conversion overall heat transfer coefficient U, and k 4Q. 

This process provides a dramatic example of the need for simultaneous design. Conventional steady-state design procedures would select temperature differentials for this process of 30 to 40 °F. Dynamic considerations show that much smaller AT s (an order of magnitude for low conversion reactors) must be used to provide good temperature control. 

Steady-state models are easily manipulated and are robust. This allows for the efficient generation of a large number of case studies necessary for steady-state design procedures. The obvious disadvantage of this procedure is that nothing is known about the dynamic response, and hence the dynamic disturbance rejection capability of alternative control schemes is also not known. These need to be evaluated using a dynamic simulator. 

The basic steady-state design procedure consists of the following five steps ... 

The design objective is to obtain 95% conversion for fixed fiesh feed flowrates (Fqa and Fob) of 12.6 mol/s and product purilies of both components C and D of 95 mol%. The assumptions, specifications, and steady-state design procedures used for both process flowsheets are the same as used earlier in this chapter. There arc three optimization variables for the conventional multiunit process molar holdup in the reactor Vr, composition of reactant B in the reactor zb. and reactor temperature Tr. 

With the exception of this method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column other conditions will exist at start-up, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behaviour of the column. 

In this chapter we have applied the plantwide control design procedure to the HDA process. The HDA process is typical of many chemical process with many chemical components, many unit operations, several recycle streams, and energy integration. The steady-state design of the HDA process has been extensively studied in the literature, but no quantitative study of its dynamics and control has been reported. 

To gain some understanding of the steady-state design aspects of the reactor by itself, the following procedure is used ... 

Temperature differentials of 20 to 30 °F have been chosen for the condenser designs shown in Table 4. These AT s are typical of what might be selected for steady-state design when no consideration of dynamics is incorporated in the design procedure. 

Several different specifications in items 2-5 will be used to investigate the effects of product quality, conversion, and recycle impurities on the economically optimal steady-state design. However, these do not affect the general stmcture of the design procedure. 

Patankar, S. V., and D. B. Spalding. 1974. A calculation procedure for the transient and steady-state behavior of shell-and-tube heat exchangers. In N. H. Afgan and E. V. Schliinder (eds.). Heat Exchangers Design and Theory Sourcebook. New York McGraw-Hill, pp. 155-176. 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

Some recent applications have benefited from advances in computing and computational techniques. Steady-state simulation is being used off-line for process analysis, design, and retrofit process simulators can model flow sheets with up to about a million equations by employing nested procedures. Other applications have resulted in great economic benefits these include on-line real-time optimization models for data reconciliation and parameter estimation followed by optimal adjustment of operating conditions. Models of up to 500,000 variables have been used on a refinery-wide basis. 

Even if the initial value of the level-set function (x,0) is set to be the distance function, the level set function r/j may not remain as a distance function at t >() when the advection equation, Eq. (3), is solved for . Thus, a redistance scheme is needed to enforce the condition of V0 = 1. An iterative procedure was designed (Sussman et al., 1998) to reinitialize the level-set function at each time step so that the level-set function remains as a distance function while maintaining the zero level set of the level-set function. This is achieved by solving for the steady-state solution of the equation (Sussman et al., 1994, 1998 Sussman and Fatemi, 1999) ... 

Obviously, the pairs (j4o,Bo) and (A, G) must be stabilizable and detectable, respectively. As we can see, controller (22) has the form of (5) and does not contain the mappings U (/x) and F (/x) thus, although the initial condition for 2 t) is not exactly known, the immersion observer (second expression in (22)) estimates the correct steady-state input and as a result, the controller is capable to drive the system towards the correct zero-error submanifold in spite of parametric variations. It can be seen from the first equation in (22) that as e t) approaches asymptotically zero, so does z. Notice also that the dynamics of Z2 is similar to immersion (21). It is important to point out that this design procedure does not require the exact calculation of mappings II (/x) and F (/x), but it suffices only to know the dimension of matrix S. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

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