Axial conduction of heat

Taking into account the axial conduction of heat in the solid phase, the energy conservation equation for the gas is... 

When the axial conductivity of heat, i.e., the heat dispersion, is also considered, the situation becomes as depicted in Figure 5.4. 

Applying the same procedure for the heat balance with axial dispersion (axial conduction) of heat, we get... 

The model here assumes no radial variation of fluid velocity or temperature, physical properties independent of temperature, and no axial conduction of heat in the bed. This last assumption is not always justified, but we will worry about that later. 

Answer by author No attempt was made to determine the influence of the transient term in the differential equation. The basic differential equation governing the phenomenon studied by Schneider was of the form used in the present study, so the solutions are certainly analogous. Schneider was interested in a range of Peclet numbers somewhat higher than those which are shown in the present paper, but certainly the results demonstrate parallel trends with regard to the axial conduction of heat, or the back-diffusion of contaminants. 

The equation describing axial conduction of heat along the superheater wall is developed through a shell balance. Finally, a method of including the properties of the large endcaps of the superheater is developed. Including the endcaps in the model introduces discontinuous first spatial derivatives of the superheater wall temperature. 

The thermal mass of the endcaps is an important parameter in the thermal response of the superheater and will be preserved in the model. The wall thickness is directly related to the axial conduction of heat and will also be preserved. These two properties determine the final form of the model. The heat-transfer area could have been preserved at the expense of one of the other properties. This would not have been the best choice. The heat-transfer coefiicient in the neighborhood of the endcaps will vary with position in an unknown fashion. This coefficient will have to be estimated experimentally, so using an artificial thermal mass or wall thickness for a well-characterized property would not be appropriate. These calculations are presented in Table 7.3. 

A model developed earlier (4, ) used the collocation method to solve the equations for heat, mass and momentum transfer In a single, adiabatic channel of the monolith. The basic model Is the one described as Model II-A(5) a square duct with axial conduction of heat longitudinally In the solid walls, but with Infinitely fast conduction peripherally around the square, and Including the diffusion of heat and mass In the transfer direction In the fluid (See for a discussion of the Importance of Including this effec.) Nusselt and Sherwood numbers are not assigned priori, but are derived from the solution. The reaction rate expression P2 In (3) with a basic form... 

The importance of axial conduction of heat is emphasized by the other curves in Figure 3. Shown there are the ignited states for different solid thermal conductivities, and as the solid thermal conductivity decreases the ignition zone moves to the end of the reactor, and for a small enough value only the extinguished state is possible. The hysteresis is thus much enhanced by a large value of k, or by thicker walls. This is... 

Since the mechanism for hysteresis is due to the axial conduction of heat in the wall, the hysteresis should decrease as k or the wall thickness decrease, as illustrated by the parameter affecting the rate of axial conduction. 2... 

The first case corresponds to a case which has not "lit off." We note first that without axial conduction of heat in the wall the solution has to be unique ( ), and there is no hysteresis curve. For an inlet temperature of 600°F the wall temperature profile is shown in Figure 4 for different average velocities. 

He thus conclude that the effect of different velocities In adjacent channels is not great. This conclusion was reached, however, for a model which excludes axial conduction of heat. However, we have made calculations for three different types of transverse conduction different geometries of the duct (7 ), peripheral conduction around the duct (5), and now different velocities in adjacent ducts. In the first two cases the Inclusion or exclusion of axial conduction had little effect on the qualitative conclusion as to the importance of the effect, and there is no reason to assume that the multiple channel analysis will be any different. The inclusion of axial conduction will have a dramatic effect on the hysteresis but that hysteresis should not be greatly affected by any of the three transverse phenomena different geometries, peripherlcal conduction, or multiple channels with different velocities. We thus conclude that analyzing a single channel suffices. 

This model v/as used by Atwood et al (1989) to compare the performance of 12 m and 1.2 m long tubular reactors using the UCKRON test problem. Although it was obvious that axial conduction of matter and heat can be expected in the short tube and not in the long tube, the second derivative conduction terms were included in the model so that no difference can be blamed on differences in the models. The continuity equations for the compounds was presented as ... 

Now, the heat conducted from the cell will be considered to be controlled by the radial conductivity of the total cell contents and not by the cell walls alone. Furthermore, the axial conductivity of the cell will be ignored as its contribution to heat loss will be several orders of magnitude less than that lost by radial convection. 

The effect of axial conduction on heat transfer in the fluid in the micro-channel can be characterized by a dimensionless parameter... 

Looking at a little slice of the process fluid as our system, we can derive each of the terms of Eq. (2.18). Potential-energy and kinetic-energy terms are assumed negligible, and there is no work term. The simplified forms of the internal ener and enthalpy are assumed. Diffusive flow is assumed negligible compared to bulk flow. We will include the possibility for conduction of heat axially along the reactor due to molecular or turbulent conduction. 

The axial dispersion of heat (axial heat conduction) is described by Fourier s law (5.2)... 

The radial term in (3-211) adjusts itself to maintain the correct overall heat balance, plus the local balance between conduction of heat in the radial direction and convection in the axial direction. We shall see that the analysis in this section is very similar to that used for the solution of the Taylor dispersion problem, which is discussed in the next section. 

We have not specified l, except to say that l c a. Assuming this to be true, however, we see immediately that conduction of heat in the axial direction can be neglected compared with conduction in the radial direction because a2/l 2 < 0(1). 

Axial dispersion of heat In the case of strong exothermic or endothermic reactions, axial temperature profiles will occur even in intensively cooled reactors, and axial (longitudinal) dispersion smoothens these profiles. This dispersion can be described by an effective axial thermal conductivity Xax that combines heat conduction via the gas and solid phase. 

On this basis, the two-dimensional model (Froment et al., 1979) can be derived from the basic equations of continuity assuming that axial dispersion of heat can be neglected and that the heat transfer in the radial direction can be described by an effective thermal conductivity independent of radial distance. 

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