Radial heat dispersion

From Equations 14 and 15 it follows that the error in the evaluation of the reaction rate for type II models is mainly due to the incorrect evaluation of the solid temperature, since the radial heat dispersion is not included in the solid phase heat balance. It is easy to deduce from Equation 15b the influence that the magnitude and sign of (Tg - Tw) will have on the solid phase temperature and hence on the error of type II models. 

Stanton Number = a /puCp) mass Peclet number = dpu/D) axial heat dispersion Peclet number = (dpu/Xa) radial heat dispersion Peclet number = dpu/A,.)... 

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

Equation 3.43g compares the timescale for radial heat dispersion in the solid phase with the one for internal heat conduction. For catalysts with good heat conduction properties and low particle-to-bed diameter ratios, A l. In this case, the surface boimdary condition is homogeneous and of Robin type, as given by the first terms on each side of (3.42b). A similar dimensionless number related with dispersion in the axial direction also appears, but its magnitude is considered much smaller than that of the other parameters in Equation 3.43, due to the geometrical reasons explained earlier. Note that Equations 3.32 and 3.34 are obtained by integrating Equation 3.41 with respect to over the pellet domain and using Equation 3.42 as boundary conditions. 

Assuming reasonably fast radial heat dispersion, the perturbation parameter can be written in this case as... 

Concerning the radial heat dispersion, the criterion for negligible interparticle thermal resistance (applicable at the hot spot) is given by [128] ... 

Dispersion of heat can be described in a similar manner as dispersion of mass if we use an effective thermal conductivity in the axial and radial direction (kax, 2-rad)- The corresponding dimensionless Pedet numbers are Pch ad (= MsCpp oidp/Xrad) and Pch,ax (= WsCpPmoi p/ ax)- Note that the superficial fluid velocity, Us, and not the interstitial velocity, Ws/e, is used in the definition of Pejj, as the effective heat conduction (reflecting both the effective heat conduction in the gas and solid phase) is not limited to the empty space of the packed bed as in the case of dispersion of mass (see also differential equations of a fixed bed reactor in Section 4.10.7). As a rule of thumb, we can approximately use the same values for the Pedet number for dispersion of heat for high Rep numbers (>100) as for the corresponding numbers for dispersion of mass, that is, Pe r d 10 and Pe 2. Details on the radial heat dispersion, which is important for wall-cooled reactors, are given in Section 4.10.7.3. 

This criterion for radial heat dispersion is independent of the length I (Da includes I) for example, for a first-order reaction, Eq. (4.10.163) reduces to ... 

Experimental fixed bed reactors are commonly heated electrically or cooled, and radial temperature gradients within the packed bed may occur. The criterion for exclusion of the influence of radial heat dispersion (negligible radial temperature profile) is ... 

Atwood et al, (1989) developed a reactor model that included axial and radial mass and heat dispersions to compare the performance of laboratory... 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

Radial dispersion of mass and heat Is Included. Axial dispersion of mass Is always negligible, but axial heat dispersion must be Included at low throughputs. The mass balance for each gaseous species Is of the form... 

Haidegger et al, 1989, have studied the total oxidation of ethane in a fixed-bed reactor and found a better agreement between experiment and simulation with the A (r)-model, compared to the model using a wall resistance (l/ w)-Simulation results for models with and without the radial porosity profile will be compared below. In the model neglecting the influence of the voidage profile, constant mass and heat dispersion and a wall resistance (I/Kw 0) will be applied. The effects of the radial voidage profile will be studied using the same correlation as used in Kiirten, 2003, for the mass and the A(r)-model for the heat dispersion. 

Fluid heat dispersion (axial) Fluid heat dispersion (radial) Solid heat dispersion (axial) Solid heat dispersion (radial) Fluid-solid mass transfer... 

The following current trends emanate from the analysis of the radial heat transfer two-phase downflow and upflow fixed-bed literature [98] (i) radial heat transfer is strongly influenced by the flow regime [96,99,100] (ii) the bed radial effective thermal conductivity always increases with liquid flow rate for both two-phase downflow and upflow [96, 100] (iii) Ar is very little dependent on gas flow rate in trickle flow, and it decreases with gas flow rate in pulsing flow regime and increases in dispersed bubble flow regime [99,100] (iv) Ar decreases with the increase of the liquid viscosity [101] (v) the inhibition of coalescence induces higher Ar values [101] (vi) Ar always increases with... 

As we will learn in Section 4.10.7.2, radial dispersion of heat is by far the most important dispersion effect in fixed bed reactors. Thus, Section 4.10.7.3 gives some more accurate equations to determine the effective radial heat conductivity XefF. 

These two equations may be rewritten in dimensionless form by introduction of the Pedet numbers for axial and radial dispersion of mass Pem.ax = Msdp/(fiDax), PAn.rad Msdp/( l7. ad)], and for heat dispersion [P. rad MgCpPmol /kradf P, ax M Cp/Omoi ip/kax)] (scc Section 4.10.6.4 for details). Together with the dimensionless axial and radial coordinates Z = z/L and P = r/rR, and the residence time with... 

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. 

The model equations of the fixed bed reactor given by Eqs. (4.10.125) and (4.10.126) are still rather complicated. Thus, criteria would be helpful to decide whether and which of the different dispersion effects can be neglected. In Section 4.10.7.2, these criteria are examined. In Section 4.10.7.3, we will give deeper insight into the modeling of wall-cooled fixed bed reactors and the problems related to the modeling of radial heat transport. 

Example 4.11.1 shows that the criterion to exdude radial dispersion of heat is the most severe problem of heat dispersion and dispersion in general. Thus, we have to limit ATad by dilution with inerts, if the criterion according to Eqs. (4.11.19) and (4.11.20) are not fulfilled. 

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. 

A detailed analysis of the radial heat transfer in the catalyst bed /4,5/ shows (see Fig. 3) that e.g. the dispersion parameters in the balance equations for the fluid phase in Fig. 1 are dependent on other parameters characteristic of more fundamental transport mechanisms and, moreover, the heat conduction through the porous particles itself consists again of four basic mechanisms (Fig. 4), each having its own characteristic parameter /6/. 

To sum up, it is sufficient to treat an adiabatic reactor as a plug-flow reactor. If the axial dispersion effect is to be included, only the heat dispersion term needs to be added. In the case of nonadiabatic, nonisothermal reactors, axial dispersion terms can be neglected in comparison to the radial dispersion terms. In addition, the radial dispersion terms can often be neglected if the radial aspect ratio is small. The conservation equations for various cases are summarized in Table 9.1. 

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