Modeling effective thermal conductivity

However, the models represent only crude approximate descriptions of the complex physical systems involved. Probably the most important phenomenon excluded is that of heat transfer. Suspended-bed operations are characterized by a high effective thermal conductivity, and thus represent a good approximation to isothermal behavior, and the above models should provide an adequate description of these systems. Fixed-bed operations will probably in many cases depart significantly from isothermal conditions, and in such cases models should be constructed that take heat transfer into... 

Aluminum foam can be used as a porous medium in the model of a heat sink with inner heat generation (Hetsroni et al. 2006a). Open-cell metal foam has a good effective thermal conductivity and a high specific solid-fluid interfacial surface area. 

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. 

Illustration 12.7 indicates how to estimate an effective thermal conductivity for use with two-dimensional, pseudo homogeneous packed bed models. 

This packet renewal model has been widely accepted and in the years since 1955 many researchers have proposed various modifications in attempts to improve the Mickley-Fairbanks representation. Several of these modifications dealt with the details of the thermal transport process between the heat transfer surface and the particle packet. The original Mickley-Fairbanks model treated the packet as a pseudo-homogeneous medium with a constant effective thermal conductivity, suggesting that... 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

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)]... 

Effective thermal conductivities and heat transfer coefficients are given by De Wasch and Froment (1971) for the solid and gas phases in a heterogeneous packed bed model. Representative values for Peclet numbers in a packed bed reactor are given by Carberry (1976) and Mears (1976). Values for Peclet numbers from 0.5 to 200 were used throughout the simulations. 

Having a validated model for the effective thermal conductivity of DPFs, it is possible to study segmented DPF designs. An example showing the influence of the segment s gluing material thermal conductivity on the inter-segment heat transfer is shown in Fig. 37. 

Thiele(I4>, who predicted how in-pore diffusion would influence chemical reaction rates, employed a geometric model with isotropic properties. Both the effective diffusivity and the effective thermal conductivity are independent of position for such a model. Although idealised geometric shapes are used to depict the situation within a particle such models, as we shall see later, are quite good approximations to practical catalyst pellets. 

The model shows that the non-isothermal uptake curve for an adsorbent mass which has low effective thermal conductivity (k ) is identical in form to that of the isothermal Fickian diffusion model for mass transport. 

Figure 12.11. Conceptual representation of models for evaluating the effective thermal conductivity of a fixed bed (from Kunii and Levenspiel, 1991) (a) Simple parallel path model (b) Modified model of Kunii and Smith (1960).

Values of effective thermal conductivity for a cell modeling tube sorber A,ef=5 0.5 W/(m-K) with a corrugated foil and 1.0-1.5 W/(m K) without it were found in experiments. Corrugated aluminium foil increases effective thermal conductivity of powder bed approximately 3-5 times. Value of effective thermal conductivity in mathematical model for calculations of tube sorbers was assumed to be f=5.8 W/(m-K). 

The structure of the disperse system (the gas phase content and its distribution) exerts considerable influence on thermal conductivity. A large number of models are proposed to clarify the effect of the structure of the disperse system on thermal conductivity. Among them one of Russell s models [87] corresponds most fully to the foam structure (the pores are isometric gas cubes, uniformly distributed in the disperse medium). According to this model the thermal conductivity is... 

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. ... 

Because it is difficult to account for changes in the properties of the reaction medium (e.g., permeability, thermal conductivity, specific heat) due to structural transformations in the combustion wave, the models typically assume that these parameters are constant (Aldushin etai, 1976b Aldushin, 1988). In addition, the gas flow is generally described by Darcy s law. Convective heat transfer due to gas flow is accounted for by an effective thermal conductivity coefficient for the medium, that is, quasihomogeneous approximation. Finally, the reaction conditions typically associated with the SHS process (7 2(XX) K and p<10 MPa) allow the use of ideal gas law as the equation of state. 

Kim et al. [22] modeled microchannel heat sinks as porous structures, while stud3ung the forced convective heat transfer through the microchannels. From the analytical solution, the Darcy number and the effective thermal conductivity ratio were identified as variables of engineering importance. 

Due to the Langrangian formulation applied to the solid phase, the use of an effective thermal conductivity as usually applied to porous media is not necessary. In a packed bed heat is transported between solid particles by radiation and conduction. For materials with low thermal conductivity, such as wood, conduction contributes only to a minor extent to the overall heat transport. Furthermore, heat transfer due to convection between the primary air flow through the porous bed and the solid has to be taken into account. Heal transfer due to radiation and conduction between the particles is modelled by the exchange of heat between a particle and its neighbours. The definition of the neighbours depends on the assembly of the particles on the flow field mesh. 

Total thermal conductivity (k In practice, the radiation and conduction contributions to the heat flux (Q) are interactive, and the interpretation of the combined conductive-radiative heat transfer is complex. Various models have therefore been proposed to simpify the theory of the heat transfer process. One widely-used model is the diffusion approximation which assumes that the heat flux (Q) is given by Equation 2, where k is the effective thermal conductivity and is defined by Equation 3 where x is the distance. Garden has pointed out that this model only applies strictly when (i) k is small and (ii) ad > 8. ... 

It is evident from the foregoing discussion that the effective diffusivity cannot be predicted accurately for use under reaction conditions unless surface diffusion is negligible and a valid model for the pore structure is available. The prediction of an effective thermal conductivity is even more difficult. Hence sizable errors are frequent in predicting the global rate from the rate equation for the chemical step on the interior catalyst surface. This is not to imply that for certain special cases accuracy is not possible (see Sec. 11-10). It does mean that heavy reliance must be placed on experimental measurements for effective diffusivities and thermal conductivities. Note also from some of the examples and data mentioned later that intrapellet resistances can greatly affect the rate. Hence the problem is significant. 

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