Dispersion coefficients axial-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... 

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. 

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. 

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... 

If the radial diffusion or radial eddy transport mechanisms considered above are insufficient to smear out any radial concentration differences, then the simple dispersed plug-flow model becomes inadequate to describe the system. It is then necessary to develop a mathematical model for simultaneous radial and axial dispersion incorporating both radial and axial dispersion coefficients. This is especially important for fixed bed catalytic reactors and packed beds generally (see Volume 2, Chapter 4). 

The determination of volumetric mass transfer coefficients, kLa, usually requires additional knowledge on the residence time distribution of the phases. Only in large diameter columns the assumption is justified that both phases are completely mixed. In tall and smaller diameter bubble columns the determination of kLa should be based on concentration profiles measured along the length of the column and evaluated with the axial dispersed plug flow model ( 5,. ... 

An alternative model for real flows is the dispersion model with the model parameters Bodenstein number (Bo) and mean residence time t, The Bodenstein number which is defined as Bo = uL/D characterises the degree of backmixing during flow. The parameter D is called the axial dispersion coefficient, u is a velocity and L a length. The RTD of the dispersed plug flow model ranges from PFR at one extreme (Bo = °) to PSR at the other (Bo = 0). The transfer function for the dispersion model with closed-closed boundaries is [10] ... 

Determinations of Peclet number were carried out by comparison between experimental residence time distribution curves and the plug flow model with axial dispersion. Hold-up and axial dispersion coefficient, for the gas and liquid phases are then obtained as a function of pressure. In the range from 0.1-1.3 MPa, the obtained results show that the hydrodynamic behaviour of the liquid phase is independant of pressure. The influence of pressure on the axial dispersion coefficient in the gas phase is demonstrated for a constant gas flow velocity maintained at 0.037 m s. 

Thermal axial dispersion must be treated with care. Even if axial dispersion of mass is negligible, the same may not be true for heat transport. The dispersion coefficient that appears in the thermal Peclet number is very different from the dispersion coefficient of the mass Peclet number. The combination of a plug-flow model for the mass balance and a dispersion... 

A further generalization of the Glueckauf approximation is suggested by comparison of the moments for the simple linear rate plug flow model (model la) and the general diffusion model with axial dispersion (model 46). One may define an overall effective rate coefficient (k ) which includes both the effects of axial dispersion and mass transfer resistance ... 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

Thus, we recover the Danckwerts model only if no distinction is made between the cup-mixing and spatial average concentrations (with this assumption, the effective axial dispersion coefficient is given by the Taylor-Aris theory). This derivation also shows that the concept of an effective axial dispersion coefficient and lumping the macro- and micromixing effects into one parameter is valid only at steady-state, constant inlet conditions and when the deviation from plug flow is small. [Remark Even with all these constraints, the error in the model because of the assumption (cj) — cym is of the same order of magnitude as the dispersion effect ]... 

When a fluid passes through a packed column, the flow is divided due to the packing. Modelling of these phenomena is carried out by superimposing a dispersion, characterized by a coefficient D on the convective plug flow of velocity U. This is the model for an axial dispersion reactor. This model allows characterisation of a flow with intermediate properties between those of the plug flow reactor and those of a continuous stirred reactor. 

Although the model equation included the axial dispersion coefficient (Dl), plug flow was approximated by assigning a very large value to the Peclet number (uL/Dl). This is because the effect of axial dispersion is quite negligible in a small column and the model with the second derivatives can give more stable numerical results. 

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. 

Experimental extraction curves can be represented by this type of model, by fitting the kinetic coefficients (mass transfer coefficient to the fluid, effective transport coefficient in the solid, effective axial dispersion coefficient representing deviations from plug flow) to the experimental curves obtained fi om laboratory experiments. With optimized parameters, it is possible to model the whole extraction curve with reasonable accuracy. These parameters can be used to model the extraction curve for extractions in larger vessels, such as from a pilot plant. Therefore, the model can be used to determine the kinetic parameters from a laboratory experiment and they can be used for scaling up the extraction. 

Here D, the axial dispersion (or diffusion) coefficient, is the parameter used to describe the deviations from ideal flow. If u is taken to be constant in the radial direction, the rightmost terms in equation (5-20) constitute the plug-flow mixing model and D (f Cld ) is a Fickian form of a diflusional correction term. 

The dispersion model assumes that the residence-time distribution of a real tubular reactor can be regarded as the superimposition of the plug flow that is characteristic of the ideal tubular reactor and diffusionlike axial mixing, characterized by an axial dispersion coefficient which has the same dimensions as, but can be much larger than, the molecular diffusion coefficient. The following effects can contribute to the axial mixing ... 

In Eq. 12.5a-l, u is taken to be the mean (plug flow) velocity through the vessel, and is a mixing-dispersion coefficient to be found from experiments with the system of interest. One important application is to fixed beds, as discussed in detail in Chapter 11, and then it is usually termed an effective transport model, with = Z> . However, the axial dispersion model can also be used to approximately describe a variety of other reactors. 

The following two models are frequently used to account for partial macromixing the dispersion model and the tanks-in-series model. In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. This model was anticipated in Chapter 12, and the governing equations for mass and heat are listed in Table 12.2 of that chapter. The derivation is similar to that for plug flow except that now a term is included for diffusive flow in addition to that for bulk flow. This term appears as -D ( d[A]/d ), where is the effective axial diffusion coefficient. When the equation is nondimensionalized, the diffusion coefficient appears as part of the Peclet number defined as = itd/D. A number of correlations for predicting the Peclet number for both liquids and gases in fixed and fluidized beds are available and have been reviewed by Wen and Fan (1975). 

In this model the effects of all mechanisms which contribute to axial mixing are lumped together into a single effective axial dispersion coefficient. More detailed models which include, for example, radial dispersion are generally not necessary and in many cases it is in fact possible to neglect axial dispersion altogether and assume ideal plug flow. 

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