Isothermal reactors analytical solution

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

The next example treats isothermal and adiabatic PFRs. Newton s method is used to determine the throughput, and Runge-Kutta integration is used in the Reactor subroutine. (The analytical solution could have been used for the isothermal case as it was for the CSTR.) The optimization technique remains the random one. 

Equation (9.14) is a linear ODE with constant coefficients. An analytical solution is possible when the reactor is isothermal and the reaction is first order. The general solution to Equation (9.14) with = —ka is... 

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). 

Although semi-analytical solutions are available in some cases [5], these are cumbersome and it is more usual to employ a numerical method. A simple example is presented below which illustrates the solution of the design equation for a batch reactor operated isothermally the adiabatic operation of the same system is then examined. 

In this chapter we consider the performance of isothermal batch and continuous reactors with multiple reactions. Recall that for a single reaction the single differential equation describing the mass balance for batch or PETR was always separable and the algebraic equation for the CSTR was a simple polynomial. In contrast to single-reaction systems, the mathematics of solving for performance rapidly becomes so complex that analytical solutions are not possible. We will first consider simple multiple-reaction systems where analytical solutions are possible. Then we will discuss more complex systems where we can only obtain numerical solutions. 

First we introduce the reader to the principles of such problems and their solution in Sections 5.1.2 and 5.1.2. As an educational tool we use the classical axial dispersion model for finding the steady state of one-dimensional tubular reactors. The model is formulated for the isothermal case with linear kinetics. This case lends itself to an otherwise rare analytical solution that is given in the book. From this example our students can understand many characteristics of such systems. 

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

Consider (so that an analytical solution can be obtained) an isothermal, axially-dispersed PFR accomplishing a first-order reaction. The material balance for this reactor can be written as ... 

For the case of isothermal operafion with no pressure drop, we were able to obtain an analytical solution, given by equation B, which gives the reactor volume necessary to achieve a conversion X for a gas-phase reaction carried out isothermaliy in a PFR, However, in the majority of situations, analytical solutions to the ordinary differential equations appearing in (he combine step are not possible. Consequently, we include POLYMATH, or some other ODE solver such as MATLAB, in our menu in that it makes obtaining solutions to the differential equations much more palatable, ... 

The concentrations of bromine cyanide (A), and methyl amine are shown as a function of time in Figure EX-11.1, and the rate is shown in Figure E4-11.2. For first- and zero-order reactions we can obtain analytical solutions for semibatch reactors operated isothermally. 

Table 7-2 and Figs. 7-3 and 7-4 show the analytical solution of the integrals for two simple first-order reaction systems in an isothermal constant-volume batch reactor or plug flow reactor. Table 7-3 shows the analytical solution for the same reaction systems in an isothermal constant-density CSTR. 

Below, we describe tbe design formulation of isothermal batch reactors with multiple reactions for various types of chemical reactions (reversible, series, parallel, etc.). In most cases, we solve the equations numerically by applying a numerical technique such as the Runge-Kutta method, but, in some simple cases, analytical solutions are obtained. Note that, for isothermal operations, we do not have to consider the effect of temperature variation, and we use the energy balance equation to determine tbe dimensionless heat-transfer number, HTN, required to maintain the reactor isothermal. 

In complex and realistic situations, the material balance for the batch reactor must be solved numerically. However, if the reactor is isothermal, and the rate laws are assumed to be quite simple, then analytical solutions of the material balance are possible. Analytical solutions are valuable for at least two reasons. First, due to the closed form of the solution, analytical solutions provide insight that is difficult to achieve with numerical solutions. The effect of parameter values on the solution is usually more transparent, and the careful study of analytical solutions can often provide insight that is hard to extract from numerical computations. Secondly, even if one must compute a numerical solution for a problem of interest, the solution procedure should be checked for errors by comparison to known solutions. Comparing a numerical solution procedure to an analytical solution for a simplified problem provides some assurance that the numerical procedure has becm constructed correctly. Then the yerified numerical procedure can... 

When an analytical solution is not available for r, there is no gain in the use of t from the reactor design and computational viewpoint. The only advantage of r then is its possibility of characterizing the situation inside the particle by means of a single number. For reactions with an order different from 1 but isothermal... 

If the reactor is assumed to be isothermal, Equation 12.2 does not exist, and only Equation 12.1 needs to be solved. An analytical solution can be found, which for a first-order reaction (-fyA = ky[A ) is given by... 

Analytical solution of the mole balance equations is only likely to be possible when a number of simplifying assumptions can be made such as those adopted previously where we assumed a single irreversible first-order reaction, no change in molar flow due to reaction, isothermal reactor, negligible variation in pressure, plug flow of gas in the bubble phase, and either perfect mixing or plug flow in the dense phase (see Ref. [46]). Assumptions must also be made with respect to the respective... 

An analytical solution of the mass balance equations for three-phase reactors is possible in the case of isothermal reactors and reactions of first-order only. Analytical solutions [1,8] are rather cumbersome even in these cases. Some analytical solutions for effectiveness factors are listed in Table 6.4. This is why a numerical solution is preferred. Case studies will briefly... 

When using an ordinary differential equation (ODE) solver such as Polymath or MATLAB, it is usually easier to leave the mole balances, rate laws, and concentrations as separate equations, rather than combining them into a single equation as we did to obtain an analytical solution. Writing the equations separately leaves it to the computer to combine them and produce a solution. The fonnulations for a packed-bed reactor with pressure drop and a semibatch teactor are given below for two elementary reactions carried out isothermally. 

Equation 3.23 together with the kinetic model expressions gives a system of first-order ordinary differential equations (ODEs), which can be solved numerically with respect to reactor length using a fourth-order Runge-Kutta method. For the case of isothermal operation, this solution can also be done in analytic manner. For instance, for the 5-lump kinetic model reported by Singh et al. (2005) at the following initial conditions ... 

The various differential equations of Table 6.1 are nonlinear and eoupled, and, in principle, they must be solved numerically, which takes exeessive computational time. For isothermal reactors for time-invariant rate constants, it is possible to derive a complete analytical solution, which is given in Appendix 6.1. However, actual reactor performance is always nonisothermal in addition, rate constants (particularly kp and k ) are dependent on reaction parameters in a very complex way. Tables 6.2 and 6.3 show the physical properties and rate constants for polystyrene and polymethyl methacrylate systems. Several researchers have attempted to solve for the reactor performance for these systems, and all of them have reported that the differential equations of Table 6.1 (along with the energy balance relation) take excessive computational time. The following discussion minimizes this problem by using the isothermal solution presented in Appendix 6.1. 

The rational design of a reaction system to produce a desired polymer is more feasible today by virtue of mathematical tools which permit one to predict product distribution as affected by reactor type and conditions. New analytical tools such as gel permeation chromatography are beginning to be used to check technical predictions and to aid in defining molecular parameters as they affect product properties. The vast majority of work concerns bulk or solution polymerization in isothermal batch or continuous stirred tank reactors. There is a clear need to develop techniques to permit fuller application of reaction engineering to realistic nonisothermal systems, emulsion systems, and systems at high conversion found industrially. A mathematical framework is also needed which will start with carefully planned experimental data and efficiently indicate a polymerization mechanism and statistical estimates of kinetic constants rather than vice-versa. 

Consider the reaction 2A — B. Derive an analytical expression for the fraction unreacted in a gas phase, isothermal PFR of length L. The pressure drop in the reactor is negligible. The reactor cross section is constant. There are no inerts. The feed is pure A and the gases are ideal. Test your mathematics with a numerical solution. 

Nonanalytical Methods and Approximations for Adiabatic Reactions. A wide variety of techniques and associated numerical methods exist for the solution of non-isothermal reactions and reactor problems which are not tractable to analytical... 

---