Radial heat dispersion coefficient

The first approach presumes heat dispersion with different but radially constant coefficients in both directions. For the radial heat dispersion one can write ... 

Radial dispersion coefficient for heat in a packed-bed 9.3 Axial dispersion coefficient for temperature in PDE Sec. 9.1 model... 

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

For non-adiabatic reactors, along with radial dispersion, heat transfer coefficient at the wall between the reaction mixture and the cooling medium needs to be specified. Correlations for these are available (cf. % 10) however, it is possible to modify the effective radial thermal conductivity (k ), by making it a function of radial position, so that heat transfer at the wall is accounted for by a smaller k value near the tube-wall than at the tube center (11). 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

Flow through the porous bed enhances the radial effective or apparent thermal conductivity of packed beds [10, 26]. Winterberg andTsotsas [26] developed models and heat transfer coefficients for packed spherical particle reactors that are invariant with the bed-to-particle diameter ratio. The radial effective thermal conductivity is defined as the summation of the thermal transport of the packed bed and the thermal dispersion caused by fluid flow, or ... 

In particular, equation (7-146) expresses that there is no mass transfer at the wall, since the concentration derivative is zero, and that heat transfer occurs with a constant wall temperature, Tw, and a local heat-transfer coefficient, This heat-transfer coefficient is now appearing in a boundary condition and is not equivalent to the overall heat-transfer coefficient used in nonisothermal axial dispersion models. The radial dispersion coefficient, Z) is, as the name implies, the radial counterpart to the axial dispersion coefficient, and while we expect a different correlation for it there are no new conceptual boundaries set here. The effective bed thermal conductivity, A however, is another matter altogether and we will worry about it more later. 

If one were to attempt to determine any communality in the discussion of models given in this chapter, about the best would be to say that the parameters invoked are derivatives of the model, as would be inferred from the titles of the previous sections. For example, there is the overall heat-transfer coefficient, h, that appears in the nonisothermal, one-dimensional axial dispersion model, which is not to be confused with the wall heat transfer coefficient, a y, that belongs to the radial dispersion model. Similarly, would the bed thermal conductivity be the same in an axial dispersion model as in a radial dispersion model What is the difference between a mass Peclet number and a thermal Peclet number and so on. In fact, let us take a moment... 

Typically, there are two ways to inject tracers, steady tracer injection and unsteady tracer injection. It has been verified that both methods lead to the same results (Deckwer et al., 1974). For the steady injection method, a tracer is injected at the exit or some other convenient point, and the axial concentration profile is measured upward of the liquid bulk flow. The dispersion coefficients are then evaluated from this profile. With the unsteady injection method, a variable flow of tracer is injected, usually at the contactor inlet, and samples are normally taken at the exit. Electrolyte, dye, and heat are normally applied as the tracer for both methods, and each of them yields identical dispersion coefficients. Based on the assumptions that the velocities and holdups of individual phases are uniform in the radial and axial directions, and the axial and radial dispersion coefficients, E and E, are constant throughout the fluidized bed, the two-dimensional unsteady-state dispersion model is expressed by... 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

As a rule of thumb, axial dispersion of heat and mass (factors 2 and 3) only influence the reactor behavior for strong variations in temperature and concentration over a length of a few particles. Thus, axial dispersion is negligible if the bed depth exceeds about ten particle diameters. Such a situation is unlikely to be encountered in industrial fixed bed reactors and mostly also in laboratory-scale systems. Radial mass transport effects (factor 1) are also usually negligible as the reactor behavior is rather insensitive to the value of the radial dispersion coefficient. Conversely, radial heat transport (factor 4) is really important for wall-cooled or heated reactors, as such reactors are sensitive to the radial heat transfer parameters. 

Especially in the case of strongly exothermic reactions, radial temperature gradients appear in the reactor tube. The existence of these gradients implies that the chemical reaction proceeds at different velocities in various radial positions and, consequently, radial concentration gradients emerge. Because of these concentration gradients, dispersion of the material is initiated in the direction of the radial coordinate. Dispersion of heat and material can be described with radial dispersion coefficients, and the mathematical formulation of dispersion effects resembles that of Pick s law (Chapter 4) for molecular diffusion. 

If the heat effect that is caused by the chemical reactions is considerable and if the heat conductivity of the catalyst material is low, radial temperature gradients emerge in a reactor tube. This implies, accordingly, that the rate of the chemical reaction varies in the radial direction, and, as a result, concentration gradients emerge in a reactor tube. This phenomenon is illustrated in Figure 5.28. Radial heat conduction can be described with the radial dispersion coefficient as will be shown below. 

The process result of heat transfer is a heat transfer coefficient. For dispersion it is a drop or particle size and size distribution. For blending in tanks it is blend time to achieve a certain degree of mixing. The equivalent for mixing in pipeline flow is not as clear. Alloca and Streiff (1980) proposed using a radial coefficient of variation, and this concept is now widely accepted. Since it is unique in the process industries to pipeline flow, it merits some extended discussion. 

Dr radial dispersion coefficient for mass Kr — radial dispersion coefficient for heat r — normalized radial distance, r = 0 at the center Rt = tube radius... 

In this section, consideration is given to effective thermal conductivities and diffusiv-ities, in both the radial and axial directions, for the dispersion of mass and heat in a fixed-bed. The flux relationships for the mass and heat transport in the bed may be considered to define these transport properties. Also considered are the heat transfer coefficients at the tube wall. 

Many models and correlations have been developed for the radial effective thermal conductivity (see the review by Kulkarani and Doraiswamy 1980). The thermal dispersion coefficient K used in Chapter 9 is related to the effective conductivity by X = pCpK. For nonadiabatic fixed-beds, the main heat conduction is in the radial direction, and thus, the radial conductivity is much more important than the axial conductivity. The axial conductivity represents the conduction superimposed on the bulk flow, which is quite small relative to the heat transport by the bulk flow. Therefore, most work has been directed to the radial conductivity. 

Heat transfer coefficient for a one-dimensional model (ht) Wall coefficient for a two-dimensional model (/ ,) Radial Peclet number for mass dispersion ((Pe)r)... 

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective mixing have been lumped into the dispersion coefficient. With these model simplifications the semi-empirical correlations for the dispersion coefficients will be system- and scale specific and far from general. 

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