Dispersion coefficients dispersed plug-flow model

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... 

Radial dispersion coefficient, 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. 

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. 

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. 

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. 

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. 

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... 

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). 

Trickle-bed reactors are widely used in the oil industry because of reliability of their operation and for the predictability of their large-scale performance from tests on a pilot-plant scale. Further advantages of trickle-bed reactors are as follows The flow pattern is close to plug flow and relatively high reaction conversions may be achieved in a single reactor. If warranted, departures from ideal plug flow can be treated by a dispersed plug-flow model with a dispersion coefficient for each of the liquid and gas phases. 

In deriving the material balance equations, the dispersed plug flow model will first be used to obtain the general form but, in the numerical calculations, the dispersion term will be omitted for simplicity. As used previously throughout, the basis for the material balances will be unit volume of the whole reactor space, i.e. gas plus liquid plus solids. Thus in the equations below, for the transfer of reactant A kLa is the volumetric mass transfer coefficient for gas-liquid transfer, and k,as is the volumetric mass transfer coefficient for liquid-solid transfer. 

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] ... 

Empty tubular reactors often are simulated by the simple plug flow model or by a dispersion model with a small value of the dispersion coefficient. 

We have discussed methods for experimentally finding dispersion coefficients for the various classes of dispersion models. Although the models were treated completely separately, there are interrelations between them such that the simpler plug-fiow models may be derived from the more complicated general models. Naturally, we would like to use the simplest possible model whenever possible. Conditions will be developed here for determining when it is justifiable to use a simpler plug-flow model rather than the more cumbersome general model. 

The shrinking-core model (SCM) is used in some cases to describe the kinetics of solid and semi-solids-extraction with a supercritical fluid [22,49,53] despite the facts that the seed geometry may be quite irregular, and that internal walls may strongly affect the diffusion. As will be seen with the SCM, the extraction depends on a few parameters. For plug-flow, the transport parameters are the solid-to-fluid mass-transfer coefficient and the intra-particle diffusivity. A third parameter appears when disperse-plug-flow is considered [39,53],... 

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. 

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 principal aspect that reflects flow distribution in fixed beds is dispersion. Flow through a packed bed is commonly represented by dispersed plug flow in which all mechanisms contributing to mixing are lumped together in effective dispersion coefficients, and A model analogous to Pick s law, as given by Equation (14.15), is applicable ... 

The way we have presented the one-dimensional dispersion model so far has been as a modification of the plug-flow model. Hence, u is treated as uniform across the tubular cross section. In fact, the general form of the model can be applied in numerous instances where this is not so. In such situations the dispersion coefficient D becomes a more complicated parameter describing the net effect of a number of different phenomena. This is nicely illustrated by the early work of Taylor [G.I. Taylor, Proc. Roy. Soc. (London), A219, 186 (1953) A223, 446 (1954) A224, 473 (1954)], a classical essay in fluid mechanics, on the combined contributions of the velocity profile and molecular diffusion to the residence-time distribution for laminar flow in a tube. 

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 ... 

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

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