Steady-State Solution Methods

The equations that make up the generalized column model. Equations 13.2 through 

The differences among the solution methods result from several considerations, including the choice of independent variables, the grouping and arrangement of the equations, and the convergence path. Some methods are based on equation decoupling, while others solve the entire system of equations simultaneously. 

In the equation decoupling methods, the model equations 13.2 through 13.9 are 

Each algorithm has its unique way of sequencing the calculations. Within this group 

Following is an overview of some of the methods, considered representative of the 

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This is a steady-state competitive method, applicable when a solute is capable of fluorescing. We consider the simplest case. The solute A undergoes excitation to the excited singlet state A upon absorption of radiation of frequency... 

We have given an analytical method of deriving a time-dependent solution to our problem that is complicated but illustrates an important method. Frequently, steady state solutions are all that is needed. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

Solution The reactions are the same as in Example 12.5. The steady-state performance of a CSTR is governed by algebraic equations, but time derivatives can be useful for finding the steady-state solution by the method of false transients. The governing equations are... 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

Enzymes that catalyze redox reactions are usually large molecules (molecular mass typically in the range 30-300 kDa), and the effects of the protein environment distant from the active site are not always well understood. However, the structures and reactions occurring at their active sites can be characterized by a combination of spectroscopic methods. X-ray crystallography, transient and steady-state solution kinetics, and electrochemistry. Catalytic states of enzyme active sites are usually better defined than active sites on metal surfaces. 

The comparison is made by calculating the residence times and volumes required for a particular value of fraction conversion. This allows a comparison of the reactor volumes for a given flow rate. Levenspiel (1999) presents design graphs for the steady-state solution, similar to those generated by this program. One advantage of this method is an easy use of arbitrary kinetic functions. 

When <0, the bifurcation diagram is as in Fig. 13. There exists a subcritical region in which three stable steady-state solutions may coexist simultaneously the thermodynamic branch and two inhomogeneous solutions. It must be pointed out that the latter are necessarily located at a finite distance from the thermodynamic branch. As a result, their evaluation cannot be performed by the methods described here. The existence of these solutions is, however, ensured by the fact that in the limit B->0, only the thermodynamic solution exists whereas for B Bc it can be shown that the amplitude of all steady-state solutions remains bounded. 

In the preceding sections we have analyzed the new solutions that appear at a point of instability and have shown that they can be calculated by the methods of bifurcation theory as long as their amplitude is small. In this section we consider a system of the form (2) whose steady-state solutions can be evaluated straightforwardly without implying any restriction on the parameters value. This allows a complete analysis of these branches of solutions.32... 

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

In Fig. 15.8 notice that during the time integration, the steady-state residuals increased for a period as the transient solution trajectory climbed over a hill and into the valley where the solution lies. This behavior is quite common in chemically reacting flow problems, especially when the initial starting estimates are poor. In fact it is not uncommon to see the transient solution path climb over many hills and valleys before coming to a point where the Newton method will begin to converge to the desired steady-state solution. 

It is often important to predict and understand the flame extinction phenomenon in stagnation or opposed flows. As discussed briefly in Sect. 17.5 and illustrated in Fig. 17.11, the extinction point represents a bifurcation where the steady-state solutions are singular. Thus direct solution of the discrete steady problem by Newton s method necessarily cannot work because the Jacobian is singular and cannot be inverted or factored into its LU products. Moreover, in some neighborhood around the singular point, the numerical problem becomes sufficiently ill-conditioned as to make it singular for practical purposes. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

Our final code generates the (a, y) 2D data of the zero contour for F in (3.6). This data is then interpolated for a user-specified input at ao, and the program decides whether the given adiabatic CSTR has one or three steady state solutions for the given parameters / and 7 at ao- Moreover, it computes the y values for all steady states and the associated function deviations of / in (3.4) from zero, called Fy. By all appearances this method is far more reliable near the bifurcation points and surpasses and supersedes our initial more generic root-finding code solveadiabxy.m from page 72. 

Figure 3.11 makes it obvious that the level-set method for equation (3.6) gives much more meaningful numerical results and clearer graphical representations of the multiple steady state solutions of the CSTR problem (3.3). 

For steady-state solutions, the set of the four differential equations (7.189) to (7.192) (or equivalently the DEs (7.190) to (7.193)) reduces to a set of four coupled rational equations in the unknown variables E (or e), Cx, Cs, and Cp. To solve the corresponding steady-state equations, we interpret the equations (7.189) to (7.192) as a system of four coupled scalar homogeneous equations for the right-hand sides of the DE system in the form F(E, Cx, Cs, Cp) =0. The resulting coupled system of four scalar equations is best solved via Newton s22 method after finding the Jacobian23 DF by partial differentiation of the right-hand-side functions / of the equations (7.189) to (7.192). I.e.,... 

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

Exchange of Stability. The TDGL method provides a procedure for determining the stability of the patterned states near the bifurcation point. Let us sketch the main ideas for the = 1 steady state patterns under the simplification that the developmental time scale is very long compared to the time to generate patterns, i.e. we neglect the time dependence of a and b in (36). With this (36) yields steady state solutions Vi obeying... 

In summary, the mathematical similarity of different steady-state solute zones, critical to the success of seemingly unrelated separation methods, demonstrates the impressive unifying power of the basic transport approach to chemical separations. This unity is emphasized again in the next chapter, where we delve into the classification and comparison of separation methods. 

Any of the global Newton methods can be converted to a relaxation form in Ketchum s method by making both the temperatures and the liquid compositions time dependent and by having the time step increase as the solution is approached. The relaxation technique should be applied to difflcult-to-solve systems and the method of Naphtali and Sandholm (42) is best-suited for nonideal mixtures since both the liquid and vapor compositions are included in the independent variables. Drew and Franks (65) presented a Naphtali-Sandholm method for the dynamic simulation of a reactive distillation column but also stated that this method could be used for finding a steady-state solution. 

Structure and mechanism in photochemical reactions. The reactions of geminal radical pairs created in bulk polymers are presented by Chesta and Weiss in Chapter 13. Of the many possible chemical reactions for such pairs, they are organized here by polymer and reaction type, and the authors provide solid rationalizations for the observed product yields in terms of cage versus escape processes. Chapter 14 contains a summary of the editor s own work on acrylic polymer degradation in solution. Forbes and Lebedeva show TREPR spectra and simulations for many main-chain acrylic polymer radicals that cannot be observed by steady-state EPR methods. A discussion of conformational dynamics and solvent effects is also included. 

Relaxation methods in which the MESH equations are cast in unsteady-state form and integrated numerically until the steady-state solution has been found... 

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