Initial value problems reactions

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

The outlet concentration from a maximum mixedness reactor is found by evaluating the solution to Equation 9-34 at X = 0 CAout = CA(0). The analytical solution to Equation 9-34 is rather complex for reaction order n > 1, the (-rA) term is usually non-linear. Using numerical methods, Equation 9-34 can be treated as an initial value problem. Choose a value for CAout = CA(0) and integrate Equation 9-34. If CA(X) achieves a steady state value, the correct value for CA(0) was guessed. Once Equation 9-34 has been solved subject to the appropriate boundary conditions, the conversion may be calculated from CAout = Ca(0)-... 

The formulation of the model as above has the advantage that mathematically it picturizes the bed as an initial value problem in contrast to the more complicated boundary value representation of the Fryer-Potter model. The implications of this reduced complexity become more evident (and considerably more important) when the reactions involved are nonlinear. 

Simultaneous to the graph creation, kinetic properties in each vRxn are used to create the appropriate reaction rate equations (ordinary differential equations, ODE). These properties include rate constants (e.g., Michaelis constant, Km, and maximum velocity, Vmax, for enzyme-catalyzed reactions, and k for nonenzymatic reactions), inhibitor constants, A) and modes of inhibition or allosterism. The total set of rate equations and specified initial conditions forms an initial value problem that is solved by a stiff ODE equation solver for the concentrations of all species as a function of time. The constituent transforms for the each virtual enzyme are compiled by carefully culling the literature for data on enzymes known to act on the chemicals and chemical metabolites of interest. 

When it is necessary to include these effects - slow reaction rates, catalysts, heat transfer, and mass transfer - it can make an engineering problem extremely difficult to solve. Numerical methods are a must, but even numerical methods may stumble at times. This chapter considers only relatively simple chemical reactors, but to work with these you must leam to solve ordinary differential equations as initial value problems. 

We can solve Eq. 6.2.4 when the reaction rate r is expressed in terms of t and Z. This is an initial value problem to be solved for the initial value, Z(0) = 0, and Z(t) indicates the reaction extent during the operation. When the rate expression depends only on Z, the design equation can be solved by separating the variables and integrating... 

The set of ODEs represented by Equations 8.30 can be solved by various means. They are first-order, initial-value problems of the type introduced in Chapter 2 for multiple reactions. We use Euler s method. Appling it to Equations 8.31 and 8.33 gives... 

An analytical solution to Equation 15.48 can also be obtained for a first-order reaction. The solution is Equation 15.43. Beyond these cases, analytical solutions are difficult since the 5 is usually nonlinear. For numerical solutions. Equation 15.48 can be treated as though it were an initial-value problem. Guess a value for dout = (0). Integrate Equation 15.48. If a k) remains finite at large X, the correct d(0) has been... 

As the next step up in complexity, we consider the case of multiple reactions. Some analytical solutions are available for simple cases with multiple reactions, and Axis provides a comprehensive list [2], but the scope of these is limited. We focus on numerical computation as a general method for these problems. Indeed, we find that even numerical solution of some of these problems is challenging for two reasons. First, steep concentration profiles often occur for realistic parameter values, and we wish to compute these profiles accurately. It is not unusual for species concentrations to change by 10 orders of magnitude within the pellet for realistic reaction and diffusion rates. Second, we are solving boundary-value problems because the boundary conditions are provided at the center and exterior surface of the pellet. Boundary-value problems (BVPs) are generally much more difficult to solve than initial-value problems (IVPs). 

This conforms to the initial-value problem posed by the nonisothermal PFR, and the solution is always unique if /i and fj have continuous first partial derivatives. Functions such as the reaction rate and heat-transfer terms appearing in equations (6-109) and (6-110) normally satisfy this requirement. 

The optimization procedure is illustrated for a particular case. The case considered is that of an exothermic, reversible reaction. The cooling between the beds is realized by means of heat exchangers. With N stages 2N decisions have to be taken N inlet temperatures to the beds and N conversions at the exit of the beds. The beds are numbered in the opposite direction of the process flow and the computation proceeds backward since the case considered is an initial value problem. The symbols are shown in Fig. 11.5.d-2. Xf, i and 7] +1 are given. The conversion is not affected by the heat exchanger so that Xj = Xj+The choice of the inlet temperature to bed j together with the exit temperature of bed j -I- 1 determines the heat exchanger between j -t- 1 and j the choice of Xy the amount of catalyst in j. 

With the exception of a few cases (simple reaction orders, isothermal condition), the solution of the above system of ordinary differential equations can be obtained only numerically. For this, there are several numerical software tools available, (e.g., Matlab, Mathematica, GNU Octave, etc.) that provide advanced solvers for such initial value problems. A good overview can be found, e.g., in Press et al., 2002. 

The last equation is the condition that we impose on the differential condition at t = 0, that is, before the reactions are allowed to start. These are called the initial conditions, and the mathematical problem attendant to such conditions is called the initial value problem (IVP). 

Alper.E. "Comments on "Gas-liquid reactions Formulation as initial value problems." Chem.Engng.Sci. 34 (1979) 1076-1078. 

The rate of an overall reaction is a composite of the rates of the elementary reactions in the mechanism, which form a set of ordinary differential equations coupled through the concentrations of chemical species, and can be expressed as the following initial value problem ... 

The above analyses of species concentrations and net reaction rates clearly indicate which reactions and which chemical species are most important in this reaction mechanism, under the particular conditions considered. However, for purposes of refining a reaction mechanism by eliminating unimportant reactions and species and by improving rate parameter estimates and thermochemical property estimates for the most important reactions and species, it would be helpful to have a quantitative measure of how important each reaction is in determining the concentration of each species. This measure is obtained by sensitivity analysis. In this approach, we define sensitivity coefficients as the partial derivative of each of the concentrations with respect to each of the rate parameters. We can write an initial value problem like that given by equation (35) in the general form... 

If the parameters Aij, aij and Eij are all known, the initial concentrations and a temperature profile are given, the rate equations would predict the behaviour of the reaction. For very large systems a program LARKIN that integrates the, in general stiff, system of equations [27]. The initial value problems may be solved by routines like METANl [29] or SODEX [30, 31]. Both methods are based on a semi-implicit midpoint rule. 

Considering chemical application problems, a large number of them yields mathematical models that consist of initial-value problems (IVPs) for ordinary differential equations (ODEs) or of initial-boundary-value problems (IBVPs) for partial differential equations (PDEs). Special problems of this kind, which we have treated, are diffusion-reaction processes in chemical kinetics (various polymerizations), polyreactions in microgravity environment (photoinitiated polymerization with laser beams) and drying procedures of hygroscopic porous media. 

In its most general form, the problem can be solved with N + 2 N = number of reactions) differential equations (column reactor and BRs) or algebraic equations (CSTR). If the column reactor operates in a countercurrent mode, the mass balances pose a boundary value problem. For concurrent column reactors and BRs, the mass balances are solved as initial value problems. 

For an arbitrary kinetic model, the surface concentrations c are solved by Equation 8.130 and the flux thus obtained, N, is inserted into the differential Equation 8.129. This equation is then solved numerically as an initial value problem (Appendix 2). A simplified solution procedure is possible for a first-order reaction. 

The solution of a set of coupled differential equations (DEs) describing a reaction is an initial value problem, which means that all c,s are known at some starting value Cj t=0). The basic principle of numerical solutions of DEs can be illustrated using Euler s stepping method on the three state... 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

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