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Initial value problems reactor

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. [Pg.472]

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. [Pg.82]

To find u, it is necessary to use some ancillary equations. As usual in solving initial value problems, we assume that all variables are known at the reactor inlet so that (Ac)i UinPin will be known. Equation (3.2) can be used to calculate m at a downstream location if p is known. An equation of state will give p but requires knowledge of state variables such as composition, pressure, and temperature. To find these, we will need still more equations, but a closed set can eventually be achieved, and the calculations can proceed in a stepwise fashion down the tube. [Pg.86]

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... [Pg.327]

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)-... [Pg.774]

Initial Value Problem. If the conditions at the inlet of the reactor are known, the problem is classified as an initial value problem. One example is the simulation of an operating reactor (11). In this case, the temperature, the pressure, the feed composition at the reactor inlet, and the heat flux to each section of the reactor are known. The yield structure and fluid temperature at the end of the... [Pg.380]

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. [Pg.29]

In section 3.2.7, boundary value problems were solved as initial value problems. This methodology is especially useful for predicting the performances in chemical reactors. Maple s stop condition was used in this section to obtain t] vs. O curves. This is very useful because, it is generally easier to solve an initial value problem than a boundary value problem. This technique was then used in section 3.2.8 to predict multiple steady states in a catalyst pellet in section 3.2.8. This methodology is extremely useful for predicting the hysteresis curves in multiple steady state problems. [Pg.287]

Chapters 8-11 treat problems that are governed by differential equations. Chapter 8 provides methods to model chemical reactors. These are usually initial value problems, which are illustrated in Eq. (1.1). [Pg.3]

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. [Pg.111]

Note to experienced users Sometimes FEMLAB has difficulty with plug flow reactor problems, when MATLAB did not, because FEMLAB is solving them as boundary value problems, whereas MATLAB solves them as initial value problems. They are actually initial value problems you fool FEMLAB by setting the diffusion coefficient to zero, and the thermal conductivity to zero. The other difficulty is that the kinetic expression makes huge changes when the temperature changes. Anytime you have to iterate to find... [Pg.129]

Note that all the conditions are known at one time, t = 0. Thus it is possible to calculate the function on the right-hand side at f = 0 to obtain the derivative there. This makes the set of equations initial value problems. The equations are ordinary differential equations because there is only one independent variable. Any higher-order ordinary differential equation can be turned into a set of first-order ordinary differential equations they are initial value problems if all the conditions are known at the same value of the independent variable [Finlayson, 1980, 1997 (p. 3-54), 1990 (Vol. BI, p. 1-55)]. The methods for initial value problems are explained here for a single equation extension to multiple equations is straightforward. These methods are used when solving plug-flow reactors (Chapter 8) as well as time-dependent transport problems (Chapters 9-11). [Pg.310]

As shown by Equations 2.9 and 2.10, the design equations for batch reactors are sets of first-order ODEs of the type known as initial-value problems. Time t is the independent variable. The dependent variables are the component concentrations a, b,c,... and, in subsequent sections, the state variables of temperature and pressure. These ODEs all have the form... [Pg.46]

The microreactor was described by the most simple mathematical model the onedimensional pseudohomogeneous reactor model. The different rate expressions were tested for use by calculation. The initial value problem... [Pg.432]

Figure 6.38 shows the results for the parameter values listed in Table 6.6, which are based on those used by van Heerden [25J. For given values of Ta 0), we solve the initial-value problem, Equation 6.60, and plot the resulting TaiVn) as the solid line in Figure 6.38. The intersection of that line with the feed temperature Taf = 323 K indicates a steady-state solution. Notice three steady-state solutions are indicated in Figure 6.38 for these values of parameters. The profiles in the reactor for these three steady states are shown in Figures 6.39 and 6.40. It is important to operate at the upper steady state so that a reasonably large production of ammonia is achieved. o... Figure 6.38 shows the results for the parameter values listed in Table 6.6, which are based on those used by van Heerden [25J. For given values of Ta 0), we solve the initial-value problem, Equation 6.60, and plot the resulting TaiVn) as the solid line in Figure 6.38. The intersection of that line with the feed temperature Taf = 323 K indicates a steady-state solution. Notice three steady-state solutions are indicated in Figure 6.38 for these values of parameters. The profiles in the reactor for these three steady states are shown in Figures 6.39 and 6.40. It is important to operate at the upper steady state so that a reasonably large production of ammonia is achieved. o...
Notice that this is an initial-value problem, but, in general, we require the solution at d = oo to determine the effluent concentration of the reactor. Differential equations on semi-infinite domains are termed singular, and require some care in their numerical treatment as we discuss next. On the other hand, if the residence-time distribution is zero beyond some maximum residence time, dmax then h ts straightforward to integrate the. initial-value problem on 0 < d < dmax<... [Pg.561]

Eoint, then the ordinary differential equations become two-point oundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. [Pg.476]

Now let us see what this model gives us in terms of F t) or (/) responses. T o solve this equation (which, incidentally, is no longer of the form of an initial-value problem but is a boundary-value problem), it is convenient to make a change of variables. We let V represent the position of the moving interface represented by all elements of fluid introduced into the reactor at some given time, analogous to the downstream volume y used in Chapter 4. In terms of the length variable, this transformation is... [Pg.342]

The calculations do not necessarily proceed according to the direction of the process flow. This is only so for a final-value problem (i.e., when the conditions at the exit of the reactor are fixed). For an initial-value problem, whereby the inlet conditions are fixed, the direction of computation for the optimization is opposite to that of the process stream. In what follows an initial-value problem is treated. First consider the last bed. No matter what the policy is before this bed the complete policy cannot be optimal when the last bed is not operating optimally for its feed. The specifications of the feed of the last bed are not known yet. Therefore, the optimal policy of the last bed has to be calculated for a whole set of possible inlet conditions of that bed. [Pg.495]

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). [Pg.238]

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. [Pg.282]

If one is ill at ease with the N-CSTRs in series or the gamma density model as a physical picture of the packed bed (although both have a physical basis) there are other good models to choose from. The DeanS cell model (103) is a proven representation of packed beds which can also be arrived at by probability arguments (47). It leads to an initial value problem of the tanks in series type for reactor performance calculations. From the physical point of view Levich et. al. [Pg.143]

This allows "a priori" use of the model and results in an initial value problem for reactor performance calculations. This model is recommended when deviations from plug flow are not overly large. [Pg.145]


See other pages where Initial value problems reactor is mentioned: [Pg.89]    [Pg.139]    [Pg.173]    [Pg.639]    [Pg.42]    [Pg.651]    [Pg.391]    [Pg.498]    [Pg.472]    [Pg.253]    [Pg.625]    [Pg.144]    [Pg.517]   
See also in sourсe #XX -- [ Pg.199 , Pg.200 , Pg.201 ]




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