Dispersion plug flow model

This model is referred to as the axial dispersed plug flow model or the longitudinal dispersed plug flow model. (Dg)j. ean be negleeted relative to (Dg)[ when the ratio of eolumn diameter to length is very small and the flow is in the turbulent regime. This model is widely used for ehemieal reaetors and other eontaeting deviees. 

Comparison of solutions of the axially dispersed plug flow model for different boundary conditions... 

The axial dispersion plug flow model is used to determine the performanee of a reaetor with non-ideal flow. Consider a steady state reaeting speeies A, under isothermal operation for a system at eonstant density Equation 8-121 reduees to a seeond order differential equation ... 

The dispersed plug flow model has been successfully applied to describe the flow characteristics in the Kenics mixer. The complex flow behavior in the mixer is characterized by the one-parameter. The Peclet number, Np, is defined by ... 

The UASB tractor was modeled by the dispensed plug flow model, considering decomposition reactions for VFA componaits, axial dispersion of liquid and hydrodynamics. The difierential mass balance equations based on the dispersed plug flow model are described for multiple VFA substrate components considaed... 

A pilot scale UASB reactor was simulated by the dispersed plug flow model with Monod kinetic parameters for the hypothetical influent composition for the three VPA ccmiponents. As a result, the COD removal efflciency for the propionic acid is smallest because its decomposition rate is cptite slow compared with other substrate components their COD removal eflSciencies are in order as, acetic acid 0.765 > butyric acid 0.705 > propionic acid 0.138. And the estimated value of the total COD removal efficiency is 0.561. This means that flie inclusion of large amount of propionic acid will lead to a significant reduction in the total VFA removal efficiency. 

Explain carefully the dispersed plug-flow model for representing departure from ideal plug flow. What are the requirements and limitations of the tracer response technique for determining Dispersion Number from measurements of tracer concentration at only one location in the system Discuss the advantages of using two locations for tracer concentration measurements. 

Figure 13.3 Representation of the dispersion (dispersed plug flow) model.

Even with constant dispersion coefficients, accounting for the velocity profile still creates difficulties in the solution of the partial differential equation. Therefore it is common to take the velocity to be constant at its mean value u. With all the coefficients constant, analytical solution of the partial differential equation is readily obtainable for various situations. This model with flat velocity profile and constant values for the dispersion coefficients is called the dispersed plug-flow model, and is characterized mathematically by Eq. (1-4). The parameters of this model are Dr, Dl and u. 

When there is no radial variation in composition in the fluid flowing in the cylindrical vessel, the only observable dispersion takes place in the direction of fluid flow. In this situation Eq. (1-4) reduces to Eq. (1-5), and we get the axial-dispersed plug-flow model with parameters D r and u. 

Experimental Schemes Used in Relation to the Axial-Dispersed Plug-Flow Model... 

Fig. 8. Axial dispersion in packed beds, dispersed plug flow model (L13).

Laminar Flow in Empty Tubes. As will be discussed in Section II, Dij, the radial coefiicient for the dispersed-plug flow model for laminar flow is merely the molecular diffusivity. 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

Thus we may drop the primed notation on the coefiicient for the axial-dispersed plug-flow model and identify this coeflScient with the one for the dispersed plug-flow model. 

Next we compare the dispersed plug-flow model with the general model by equating Eqs. (67d) and (69d). Thus, as found by Aris (A6),... 

This equation enables us to calculate the value of Pl from the velocity profile using mean values of the coefficients of the general dispersion model. The constant radial coefficient used in the dispersed plug-flow model is the same as the mean value of the varying radial coefficient in the general dispersion model. 

Fig. 17. Restrictions on length to diameter ratio for dispersed plug flow models to be valid (B14).

Fix the dispersion coefficients of the dispersed plug flow model, Di = Di or D2, at inflnity or zero to obtain backmix or plug flow in the individual regions. 

Fig. 28. Comparison of performance of reactors for the plug flow and dispersed plug flow models. Reaction is of first order, aA- products, and constant density, occurring in a closed vessel (L14, L15).

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

Kjaer (K9) gives a very comprehensive study of concentration and temperature profiles in fixed-bed catalytic reactors. Both theoretical and experimental work is reported for a phthallic anhydride reactor and various types of ammonia converters. Fair agreement was obtained, but due to the lack of sufficiently accurate thermodynamic and kinetic data, definite conclusions as to the suitability of the dispersed plug flow model could not be reached. However, the results seemed to indicate that the... 

Mean concentration of pulse of tracer if uniformly distributed in experimental section of vessel of length L = C/C°. Dimensionless concentration = C/Cava. Dimensionless concentration = C/C Eve Dimensionless concentration Effective diameter, defined by Eq. (50) Particle diameter Tube diameter Dispersion coefficient Axial dispersion coefficient, dispersed plug flow model... 

Axial dispersion coefficient, axial-dispersed plug, flow model shown equal to Dl in Eq. (72) Mean value of Dl(R) Axial dispersion coefficient, uniform dispersion model... 

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