Rate equations, chemical plug flow reactor

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

Consider a plug flow reactor - a cylindrical tube with reactants flowing in one end and reacting as they flow to the outlet. A mole balance is made here for the case when the operation is steady. Let F be the molar flow rate of a chemical, which changes dF in a small part of the tube take the volume of that small part to be d V. The rate of reaction is expressed as the moles produced per unit of time per unit of volume. If a chemical specie reacts, then the r for that specie is negative. Thus, the overall equation for a smaU section of the tube is... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

The parameter p measures an activating influence of intermediate A, on the rate consttuit for the first step. This law has been followed by diverse chemical reactions [27]. Some biological reactions catalyzed by allosteric enzymes also give such reaction scheme [12]. The relevant differential equations are written for the isothermal plug flow reactor for carrying out the reaction given by Equation 1. 

Analytical solutions also are possible when T is constant and m = 0, V2, or 2. More complex chemical rate equations will require numerical solutions. Such rate equations are apphed to the sizing of plug flow, CSTR, and dispersion reactor models by Ramachandran and Chaud-hari (Three-Pha.se Chemical Reactors, Gordon and Breach, 1983). 

Solutions with other chemical rate equations are in P8.03.03, and some numerical cases in P8.03.04-P8.03.06. Such rate equations can be applied to the sizing of plug flow, CSTR and dispersion reactor models. 

For plug flow, only the flow and the processes other than mixing, diffusion, and conduction are considered. These have been studied in Chapter 4. In a plug flow tubular reactor model we consider only the convective one-dimensional flow and the chemical reaction as shown in Figure 5.1, where n is the convective molar flow rate for the constant volumetric flow rate g of component i. These two rates are connected by the equation rq = q Ci for the concentration Cj. 

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