Tubular reactor, simplest model

One of the simplest models used to describe the performance of tubular reactors is the well-known isothermal one-dimensional plug flow tubular reactor (PFTR) model. The mass balance of this model for steady-state conditions, the simultaneous occurrence of M reactions and a constant volumetric flow rate V is ... 

THE SIMPLEST DISTRIBUTED MODEL Example 1 The Tubular Reactor... 

To illustrate we consider a homogeneous tubular reactor. The simplest model is given by plug flow and the design equations are obtained from the mass-balance equations by taking the mass balance over an element of length Al. This is expressed in the formula... 

Tubular Reactors. The simplest model of a tubular reactor, the plug-flow reactor at steady state is kinetically identical to a batch reactor. The time variable in the batch reactor is transformed into the distance variable by the velocity. An axial temperature gradient can be imposed on the tubular reactor as indicated by Gilles and Schuchmann (22) to obtain the same effects as a temperature program with time in a batch reactor. Even recycle with a plug flow reactor, treated by Kilkson (35) for stepwise addition without termination and condensation, could be duplicated in a batch reactor with holdback between batches. 

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... 

As we observed earlier, the adiabatic reactor is not so much a type, more a way of operation. We shall therefore refer to both stirred tank reactors and to tubular ones, and this chapter forms a suitable bridge between the two. We shall introduce the simplest model of the tubular reactor, but this is so elementary that the anticipation of the following chapter will cause no difficulty. 

The simplest model of the tubular reactor is that the stream flows uniformly through the reactor with... 

We may first divide tubular reactors into those designed for homogeneous reactions, and therefore basically just an empty tube, and those designed for a heterogeneously catalyzed reaction, and hence to be packed with a catalyst. Both types can of course be operated adiabatically, and it was the simplest model of these that we discussed in the last chapter. If the temperature of the reactor is to be controlled this is through the wall, and the associated problems of heat transfer now arise. These include transfer at the wall and subsequent radial diffusion across the flowing reactants. In the empty tubular reactor there may be considerable variations in flow rate across the tube. For example, in the slow laminar flow the fluid... 

In this section we shall be concerned with more realistic models of tubular reactors. The isothermal reactor is obviously the simplest type, but it implies that either there are no large heat effects or that they can be completely dominated by temperature control. The reactor with an optimal temperature profile is clearly the most desirable, but this means that the rate of heat exchange can be regulated precisely at each point. Between these two extremes there is a range of designs about which something should be said. We shall not always solve the equations in detail but we shall try to show the important features of the behavior of the reactor by means of examples. 

Chemical engineers have used various models to represent reactions in open systems. The simplest two, representing extreme viewpoints, are the continuous-flow, stirred-tank reactor (CSTR) and the plug-flow, tubular reactor (PFTR). The... 

The simplest realistic model is the TRAM (tubular reactor with axial mixing), in which average temperatures and concentrations are used across the reactor but the finite rates of heat conduction and mass diffusion along the reactor are admitted. 

For noncatalytic homogeneous reactions, a tubular reactor is widely used because it cai handle liquid or vapor feeds, with or without phase change in the reactor. The PFR model i usually adequate for the tubular reactor if the flow is turbulent and if it can be assumed tha when a phase change occurs in the reactor, the reaction takes place predominantly in one o the two phases. The simplest thermal modes are isothermal and adiabatic. The nonadiabatic nonisothermal mode is generally handled by a specified temperature profile or by heat transfer to or from some specified heat source or sink and a corresponding heat-transfer area and overall heat transfer coefficient. Either a fractional conversion of a limiting reactant or a reactoi volume is specified. The calculations require the solution of ordinary differential equations. 

For the homogeneous tubular reactor (the simplest model is the plug flow), the design equations are again obtained from the mass balance equations... 

For the modeling of a reactor we need solutions of the equations of the balances of mass, energy, and impulse. For isothermal operation the energy balance is not needed. The impulse balance mostly only serves to calculate the pressure drop of a reactor. The definition of a suitable control space for balancing is important. In the simplest case, the variables - such as temperature and concentrations - are constant within the control space (stirred tank reactor). However, in many cases the system variables depend on the location, for example, in the axial direction in a tubular reactor. Then infinitesimal balances (differential equations) have to be solved to obtain integral data. 

Equations 12.7.48 and 12.7.39 provide the simplest one-dimensional mathematical model of tubular fixed bed reactor behavior. They neglect longitudinal dispersion of both matter and energy and, in essence, are completely equivalent to the plug flow model for homogeneous reactors that was examined in some detail in Chapters 8 to 10. Various simplifications in these equations will occur for different constraints on the energy transfer to or from the reactor. Normally, equations 12.7.48 and 12.7.39... 

Membrane bioreactors have been modelled using approaches that have proven successful in the more conventional catalytic membrane reactor applications. The simplest membrane bioreactor system, as noted in Chapter 4, consists of two separate units, a bioreactor (typically a well-stirred batch reactor) coupled with an external hollow fiber or tubular or flat membrane module. These reactors have been modelled by coupling the classical equations of stirred tank reactors with the mathematical expressions describing membrane permeation. What makes this type of modelling unique is the complexity of the mecha-... 

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