Tubular flow reactor describing equations

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

Example 9.11 Modeling of a nonisothermal plug flow reactor Tubular reactors are not homogeneous, and may involve multiphase flows. These systems are called diffusion convection reaction systems. Consider the chemical reaction A -> bB described by a first-order kinetics with respect to the reactant A. For a nonisothermal plug flow reactor, modeling equations are derived from mass and energy balances... 

Tubular flow reactors are usually operated under steady conditions so that, at any point, physical and chemical properties do not vary with time. Unlike the batch and tank flow reactors, there is no mechanical mixing. Thus, the state of the reacting fluid will vary from point to point in the system, and this variation may be in both the radial and axial direction. The describing equations are then differential, with position as the independent variable. 

Two key topics need to be addressed before developing the equations describing (individually) batch, CSTR, and tubular flow reactors. 

Solution. The describing equation for a tubular flow reactor is... 

The equation describing the concentration of a reactant in a tubular flow reactor is given by... 

The axial dispersion equation of the Fickian form is one of the most widely used models characterizing the residence time distribution and performance of a tubular flow reactor [27,28]. Deterministic models, based on population mass balance, have been used predominately in describing and modeling the dispersion of molecules or particles in the flow reactor. A probabilistic approach may be more appropriate than a mass balance approach especially when the rate of reactant flow through the reactor is high, when the flow involves more than one phase, or when the path of reactant flow is affected by internals or agitators [29]. 

So far in dealing with tubular reactors we have considered a spatial coordinate as the variable, i.e. an element of volume SV, situated at a distance z from the reactor inlet (Fig. 1.14), although z has not appeared explicitly in the equations. For a continuous flow reactor operating in a steady state, the spatial coordinate is indeed the most satisfactory variable to describe the situation, because the compositions do not vary with time, but only with position in the reactor. 

Tubular Reactor with Dispersion An alternative approach to describe deviation from ideal plug flow due to backmixing is to include a term that allows for axial dispersion De in the plug flow reactor equations. The reactor mass balance equation now becomes... 

The flow pattern in the tubular reactor (HOTCR) is described as ideal plug flow. The performance equation relates the conversion of the tar X igf and the gas residence time t ... 

Under steady-state conditions in a plug-flow tubular reactor, the onedimensional mass transfer equation for reactant A can be integrated rather easily to predict reactor performance. Equation (22-1) was derived for a control volume that is differentially thick in all coordinate directions. Consequently, mass transfer rate processes due to convection and diffusion occur, at most, in three coordinate directions and the mass balance is described by a partial differential equation. Current research in computational fluid dynamics applied to fixed-bed reactors seeks a better understanding of the flow phenomena by modeling the catalytic pellets where they are, instead of averaging or homogenizing... 

As noted in the Introduction, kinetic and design calculations for reactors require information on the rate of reaction and equations to describe the concentration and temperature within the reactor. Part I developed equations to describe the rate of a chemical reaction, and briefly discussed energy (temperature) effects and chemical equilibrium. The contents of Part I can be combined and the results applied to the three major classifications of reactors batch, tank flow, and tubular flow. These three classes of reactors are discussed in the subsections that follow. 

The PFR model assumes a flat velocity profile across the whole of the reactor cross-section in reality, this is impossible to achieve although in practice certain combinations of physical conditions are closely described by this assumption. If the Reynolds number, dupln, in a tubular reactor is less than about 2100, then the flow therein will be laminar and where the flow is fully developed, the velocity profile across the reactor will be parabolic in form. If one assumes that diffusion is negligible between adjacent radial layers of fluid, then it is relatively straightforward to derive the forms of E(t), E(0) and F(0) associated with this type of reactor [42]. These are given in the equations... 

Figure 2-9 Sketch showing correspondence between time in a batch reactor and position a in a plug-flow tubular reactor. The mass-balance equations describe both reactors for the constant-density situations.

We can therefore replace dt by dz/u in all of the preceding differential equations for the mass balance in the batch reactor and use these equations to describe reactions during flow through a pipe. This reactor is called the plug-flow tubular reactor, which is the most important continuous reactor encountered in the chemical industry. 

Then we can use the following system of equations (reduced to dimensionless variables) to describe the flow in a tubular reactor l l... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

A batch experimental reactor is used for slow reactions since species compositions can be readily measured with time. The determination of reaction rate expression is described in Chapter 6. A tubular (plug-flow) experimental reactor is suitable for fast reactions and high-temperature experiments. The species composition at the reactor outlet is measured for different feed rates. Short packed beds are used as differential reactors to obtain instantaneous reaction rates. The reaction rate is determined from the design equation, as described in Chapter 7. An experimental CSTR is a convenient tool in determining reaction rate since the reaction rate is directly obtained from the design equation, as discussed in Chapter 8. 

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