Dispersion model pipes

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

Simpler optimization problems exist in which the process models represent flow through a single pipe, flow in parallel pipes, compressors, heat exchangers, and so on. Other flow optimization problems occur in chemical reactors, for which various types of process models have been proposed for the flow behavior, including well-mixed tanks, tanks with dead space and bypassing, plug flow vessels, dispersion models, and so on. This subject is treated in Chapter 14. 

Fortunately, it is not always necessary to recover the system RTD curve from the impulse response, so the complications alluded to above are often of theoretical rather than practical concern. In addition, the dispersion model is most appropriately used to describe small extents of dispersion, i.e. minor deviations from plug flow. In this case, particularly if the inlet pipe is of small diameter compared with the reactor itself, the vessel can be satisfactorily assumed to possess closed boundaries [62]. An impulse of tracer will enter the system and broaden as it passes along the reactor so that the observed response at the outlet will be an RTD and will be a symmetrical pulse, the width of which is a function of DjuL alone. 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

Experiments show that the dispersion model well represents flow in packed beds and in pipes. Thus theory and experiment give lyiud for these vessels. We summarize them in the next three charts. 

General Dispersion Model for Symmetrical Pipe Flow... 

Fw. 9. Axial dispersion in pipes, dispersed plug flow model (L13). 

Fig. 13. Radial dispersion in pipes, general dispersion model (01).

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

The extension of the two-mode axial dispersion model to the case of fully developed turbulent flow in a pipe could be achieved by starting with the time-smoothed (Reynolds-averaged) CDR equation, given by Eq. (117), where the reaction rate term R(C) in Eq. (117) is replaced by the Reynolds-averaged reaction rate term Rav(C), and the molecular diffusivity Dm / is replaced by the effective diffusivity Dej- in turbulent flows given by... 

Water at room temperature is flowing through a 20-cm-lD pipe at Re = 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance ... 

The mechanism by which solids are distributed throughout a vessel once they are suspended is different from that leading to suspension. It might be expected that the solids distribution would again be affected by the bulk flow pattern, i.e. the mean velocities throughout the vessel as well as the turbulence structure. These flows oppose the gravitational downwards force. As will be shown later, the measured vertical concentration profiles are very complicated, much more so than in solids transport in pipe flow, for example, where a steady decrease in concentration occurs from top to bottom. This concentration decay in pipes can be modelled rather easily by a one dimensional sedimentation-dispersion model. A similar model has been proposed for stirred vessels for the region above the impeller. 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

The PFR model is based on turbulent pipe flow in the limit where axial dispersion can be assumed to be negligible (see Fig. 1.1). The mean residence time rpfr in a PFR depends only on the mean axial fluid velocity (U-) and the length of the reactor Lpfr ... 

Figures 13.15 and 13.16 show the findings for flow in pipes. This model represents turbulent flow, but only represents streamline flow in pipes when the pipe is long enough to achieve radial uniformity of a pulse of tracer. For liquids this may require a rather long pipe, and Fig. 13.16 shows these results. Note that molecular diffusion strongly affects the rate of dispersion in laminar flow. At low flow rate it promotes dispersion at higher flow rate it has the opposite effect.

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

The dispersed plug-flow model can be regarded as the first stage of development from the simple idea of plug flow along a pipe. The fluid velocity and the concentrations of any dissolved species are assumed to be uniform across any section of the pipe, but here mixing or dispersion in the direction of flow (i.e. in the axial z-direction) is taken into account (Fig. 2.10). The axial mixing is described by... 

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