Isothermal First-order Reaction

In this section, reaction at isothermal conditions will be considered, i.e. the reaction proceeds at constant temperature. 

Process Dynamics and Control Modeling for Control and Prediction. Brian Roffel and Ben Betlem. 2006 John Wiley Sons Ltd. 

In most cases it will be desired to control the conversion of component A, which is directly related to the yield of component B, hence the response of c,4 to changes in CAm and F are important. The behavioral model of the reactor is shown in Fig. 12.2, 

Since there is no energy balance, nor an overall mass balance, the only dynamic balance that remains is a component balance for component A and/or B. To describe the outlet concentration of component A, the dynamic component balance of A is of interest It can be written as  

The derivative term on the left-hand side of the equal sign represents the accumulation of mass of component A in kg/s the first term on the right-hand side represents the difference between the inlet flow and outlet flow of component A and the second right-hand side term represents the disappearance of component A through reaction. 

---

Example 8.1 Find the mixing-cup average outlet concentration for an isothermal, first-order reaction with rate constant k that is occurring in a laminar flow reactor with a parabolic velocity profile as given by Equation (8.1). 

This technique should give reasonable results for isothermal, first-order reactions. It and other modeling approaches are largely untested for complex and nonisothermal reactions. 

The fed-batch scheme of Example 14.3 is one of many possible ways to start a CSTR. It is generally desired to begin continuous operation only when the vessel is full and when the concentration within the vessel has reached its steady-state value. This gives a bumpkss startup. The results of Example 14.3 show that a bumpless startup is possible for an isothermal, first-order reaction. Some reasoning will convince you that it is possible for any single, isothermal reaction. It is not generally possible for multiple reactions. 

For an isothermal, first-order reaction, the probability that a particular molecule reacts depends only on the time it has spent in the system ... 

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

For an isothermal first-order reaction taking place in a constant-volume BR but at varying density in a PFR, it can be shown that the times are also equal this is not the case for other orders of reaction (see problem 17-8). 

In the CRE literature, the residence time distribution (RTD) has been shown to be a powerful tool for handling isothermal first-order reactions in arbitrary reactor geometries. (See Nauman and Buffham (1983) for a detailed introduction to RTD theory.) The basic ideas behind RTD theory can be most easily understood in a Lagrangian framework. The residence time of a fluid element is defined to be its age a as it leaves the reactor. Thus, in a PFR, the RTD function E(a) has the simple form of a delta function ... 

Only for an isothermal, first-order reaction where Sa = will the chemical source... 

For an isothermal first-order reaction, the design equation may be integrated to give... 

The above ordinary differential equations (ODEs), Eqs. (19-11) and (19-12), can be solved with an initial condition. For an isothermal first-order reaction and an initial condition, C(0) = 0, the linear ODE may be solved analytically. At steady state, the accumulation term is zero, and the solution for the effluent concentration becomes... 

The effectiveness factor Tj is the ratio of the rate of reaction in a porous catalyst to the rate in the absence of diffusion (i.e., under bulk conditions). The theoretical basis for q in a porous catalyst has been discussed in Sec. 7. For example, for an isothermal first-order reaction... 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

Li, G., and Rabitz, H., Determination of constrained lumping schemes for non-isothermal first order reaction systems. Chem. Eng. Sci. 46,583 (1991a). 

For example, an isothermal, first-order reaction in a flat plate catalyst pellet has individual effectiveness factors that are ... 

A lack of significant intraphase diffusion effects (i.e., 17 > 0.95) on an irreversible, isothermal, first-order reaction in a spherical catalyst pellet can be assessed by the Weisz-Prater criterion [P. B. Weisz and C. D. Prater, Adv. Catal., 6 (1954) 143] ... 

The isothermal, first-order reaction of gaseous A occurs within the pores of a spherical catalyst pellet. The reactant concentration halfway between the external surface and the center of the pellet is equal to one-fourth the concentration at the external surface. 

Thus far, the overall effectiveness factor has been used in the mass and energy balances. Since 77 is a function of the local conditions, it must be computed along the length of the reactor. If there is an analytical expression for 17, for example for an isothermal, first-order reaction rate ... 

Consider first the isothermal, first order reaction for which the true reaction rate per unit volume of catalyst is pi,Sgicc, The observed rate of reaction, r (say), is a function of the observed concentration c, and is in fact... 

N. Wakao and J. M. Smith [Ind. Eng. Chem., Fund. Quart., 3, 123 (1964)] used the general equations (11-1) and (11-2) in developing expressions for intrapellet effects for isothermal first-order reactions. 

Only for an isothermal, first-order reaction where Sa = —k a will the chemical source term in (3.102) be closed, i.e., ++(<+ = h (u,(pa). Indeed, for more complex chemistry, closure of the chemical source term in the scalar-flux transport equation is a major challenge. However, note that, unlike the scalar-flux dissipation term, which involves the correlation between gradients (and hence two-point statistical information), the chemical source term is given in terms of u(x, t) and 0(x, t). Thus, given the one-point joint velocity, composition PDF /u,chemical source term is closed, and can be computed from... 

Figure 8.1 gives conversion curves for an isothermal, first-order reaction in various types of reactor. The curves for a PFR and CSTR are from Equations 1.38 and 1.49. The curve for laminar flow without diffusion is obtained from Equation 8.14 and the software of Example 8.2. Without diffusion, the laminar flow reactor performs better than a CSTR but worse that a PFR. Add radial diffusion and the performance improves. This is illustrated by the curve in Figure 8.1 that is between those for laminar flow without di ffusion and piston flow. The intermediate curve is one member of a family of such curves that depends on theparameter f// . IfL // is small, <... 

---