Tubular reactor model assumptions

In addition, the PFR model assumes that mixing between fluid elements at the same axial location is infinitely fast. In CRE parlance, all fluid elements are said to be well micromixed. In a tubular reactor, this assumption implies that the inlet concentrations are uniform over the cross-section of the reactor. However, in real reactors, the inlet streams are often segregated (non-premixed) at the inlet, and a finite time is required as they move down the reactor before they become well micromixed. The PFR model can be easily... 

Amongst the assumptions we have made in developing the model are the following that Pick s law is applicable to the diffusion processes, the gel particles are isotropic and behave as hard spheres, the flow rate is uniform throughout the bed, the dispersion in the column Ccui be approximated by the use of an axial dispersion coefficient cuid that polymer molecules have an independent existence (i.e. very dilute solution conditions exist within the column). Our approach borrows extensively many of the concepts which have been developed to interpret the behaviour of packed bed tubular reactors (5). 

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... 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

Here R is expressed as kmoles converted per unit of time and per unit of mass of catalyst. The pseudo-homogeneous one-dimensional model of the cooled, tubular reactor used by us is based on the following assumptions ... 

The axial mixing in a tubular reactor can sometimes be described by a dispersion model. This model is based on the assumption that the RTD may be considered to result from piston flow on which is superimposed an axial dispersion. The latter is taken into account by means of a constant effective axial dispersion coefficient, Dax, which has the same dimensions as the molecular diffusion coefficient, Dm. Usually Dax is much larger than the molecular diffusion coefficient because it incorporates all effects that cause deviations from plug flow, such as variations in radial velocities, eddies, and vortices. 

The tubular reactor is so named because the physical configuration of the reactor is normally such that the reaction takes place within a tube or length of pipe. The idealized model of this type of reactor is based on the assumption that an entering fluid element moves through the reactor as a differentially thin plug of material that fills the reactor cross section completely. Thus, the terms piston flow or plug flow reactor (PFR) are often employed to describe the idealized model. The contents of a specific differential plug are presumed to be uniform in temperature and composition. This model may be used to treat both the case where the tube is packed with a solid catalyst (see Section 12.1) and the case where the fluid phase alone is present. 

The plug flow reactor is an idealised model to which certain types of actual reactors approximate. In many cases a tubular reactor of some sort is visualised but the plug flow assumption is couched in general terms. It is assumed that (a) at any cross-section normal to the fluid flow the velocity is constant, also pressure, temperature and composi-... 

The assumption of plug flow is not always correct. The plug flow assumes that the convective flow (flow by velocity q/A, = v, caused by a compressor or pump) is dominating over any other transport mode. In fact, this is not always correct, and it is sometimes important to include the dispersion of mass and heat driven by concentration and temperature gradients. However, the plug flow assumption is valid for most industrial units because of the high Peclet number. We will discuss this model in some detail, not only because of its importance but also because the techniques used to handle these two-point boundary-value differential equations are similar to that used for other diffusion-reaction problems (e.g., catalyst pellets) as well as countercurrent processes and processes with recycle. The analytical analysis as well as the numerical techniques for these systems are very similar to this axial dispersion model for tubular reactors. 

The methods listed in Sections 5.1.1-5.1.7 are illustrated by a tubular reactor with a first-order reaction and laminar flow. Models for species, heat, and momentum have been formulated and simplified. In addition to showing the methods, we discuss the assumptions in a traditional 1D lumped-parameter model for a tubular reactor with axial dispersion,... 

The assumptions in a lumped-parameter model are not always transparent. For example, in the ID model for a tubular reactor with axial dispersion (Equations (5.36) and (5.37), repeated here for convenience)... 

This chapter has discussed the analysis of reactors for step-growth polymerization assuming the equal reactivity hypothesis to be valid. Polymerization involves an infinite set of elementary reactions under the assumption of this hypothesis, the polymerization can be equivalently represented by the reaction of functional groups. The analysis of a batch (or tubular) reactor shows that the polymer formed in the reactor cannot have a polydispersity index (PDI) greater than 2. However, the PDI can be increased beyond this value if the polymer is recycled or if an HCSTR is used for polymerization. A comparison of the kinetic model with experimental data shows that the deviation between the two exists because of (1) several side reactions that must be accounted for, (2) chain-length-dependent reactivity, (3) unequal reactivity of various functional groups, or (4) comphca-tions caused by mass transfer effects. 

A reactor for thin-film deposition can have a tubular structure with a flowing chemically active gas. In this case, the PFR model can be used. This one-dimensional model uses the assumption of a uniform component distribution in a reactor section. This means that the boundary layer formation processes are not taken into account. 

In Chap. 4 the plug-flow model was used as a basis for designing homogeneous tubular ow reactors. The equation employed to calculate the conversion in the effluent stream was either Eq. (3-13) or Eq. (4-5). The same equations and the same calculational procedure may be used for fixed-bed catalytic reactors, provided that plug-flow behavior is a vahd assumption. AH that is necessary is to replace the homogeneous rate of reaction in those equations with the global rate for the catalytic reaction. 

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