Effective radial thermal conductivities

Xer, er ers radial effective thermal conductivity lumped for fluid phase for solid phase (kcal/hm °K)... 

The choice of a model to describe heat transfer in packed beds is one which has often been dictated by the requirement that the resulting model equations should be relatively easy to solve for the bed temperature profile. This consideration has led to the widespread use of the pseudo-homogeneous two-dimensional model, in which the tubular bed is modelled as though it consisted of one phase only. This phase is assumed to move in plug-flow, with superimposed axial and radial effective thermal conductivities, which are usually taken to be independent of the axial and radial spatial coordinates. In non-adiabatic beds, heat transfer from the wall is governed by an apparent wall heat transfer coefficient. ... 

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 this heat transport model, the radial heat transfer is represented by a Fourier-like law, using an equivalent radial thermal conductivity of the medium with flowing fluids 4r = A dT/dr. The bed radial effective thermal conductivity was expressed as sum of two terms—conductive contribution and convective contribution A -Aso+ gf In Hashimoto et al. [94] and Mat-suura et al. s [95] approach, the theory of single-phase gas flow... 

Also, heat transfer in three-phase fixed-bed reactors has been investigated using a two-dimensional homogeneous model with two parameters [94, 96]—the bed radial effective thermal conductivity and the heat transfer coefficient at the wall ... 

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... 

Equation (6.11.43) considers the bed and the fluid as a pseudo-homogeneous medium, and the heat transfer within the bed up to the wall is represented by the radial effective thermal conductivity and the internal wall heat transfer coefficient aw.int- The model assumes a jump in temperature directly at the wall from Tw.int.i to The values of X ad and aw.im were calculated by... 

Figure 28.8. Effect of material properties and monolith void fraction on estimated radial effective thermal conductivity of honeycomb monohths with square channels. Adapted from Ref. 97.

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. 

The radial heat transport is complex, involving conduction, convection, and radiation between voids and solid and between solid particles. Possible modes of heat transfer in the radial direction are shown in Figure 14.3. Different physical models result depending on whether various resistances to the heat transport are in series, parallel, or a combination of both. Here an additive model is considered, which assumes that the radial effective thermal conductivity consists of static (conduction and radiation) and dynamic contributions, the latter caused by fluid motion. These two contributions are considered to be additive ... 

Figure 4.9 Overall heat transfer coefficient (a) and heat transfer parameters (b) of a highly conductive structured catalyst in methanol synthesis as a function of the syngas stoichiometric number in the fresh feed stream (Mp). (Squares) Commercial Lurgi multitubular packed-bed reactor (PB) (circles) copper honeycomb monoliths (HM) (triangles) open-cell foams (OF). In Figure 4.9b, the radial effective thermal conductivity is plotted with solid symbols and the wall heat transfer coefficient, h, with empty ones. Reprinted from Montebelli etal. [162], with permission from Elsevier.

Effective Thermal Conductivities of Porous Catalysts. The effective thermal conductivity of a porous catalyst plays a key role in determining whether or not appreciable temperature gradients will exist within a given catalyst pellet. By the term effective thermal conductivity , we imply that it is a parameter characteristic of the porous solid structure that is based on the gross geometric area of the pellet perpendicular to the direction of heat transfer. For example, if one considers the radial heat flux in a spherical pellet one can say that... 

This equation may be used as an appropriate form of the law of energy conservation in various pseudo homogeneous models of fixed bed reactors. Radial transport by effective thermal conduction is an essential element of two-dimensional reactor models but, for one-dimensional models, the last term must be replaced by one involving heat losses to the walls. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Heat transfer of packed bed has been the subject of numerous studies. For cylindrical packed columns, a solution for determining temperature distributions was given using Bessel functions. Here, it is important to find out exact effective thermal conductivity of bed because of flowing gas and relatively high temperatures. Radial temperature distributions are more important than that of axial direction because the latter can be measured and controlled during the operation. 

The temperature, T, is a function of radial position, r, /Cg is an effective thermal conductivity, and -AHrxn is the heat of reaction. Again, there is no flux at the center, and the temperature at the outer boundary is set at Tr. Now the reaction rate depends upon both concentration and temperature [and Eq. (9.3) has to be written with R(c, T)]. Such problems can be very difficult owing to the rapid change in the kinetic constants with temperature, and they provide a severe test of any numerical method. 

---