Modeling of heat transport

Overview of combined modeling of heat transport and air movement, AIVC Technical Note TN 40. Coventry Air Infiltration and Ventilation Centre, 1993. 

Slack [25] and Cahill et al. [26] explored the theoretical limits on k for solids within a phonon model of heat transport. Their work utilized the concept of the minimum thermal conductivity, Kj n- At this minimum value the mean free path for all heat carrying phonons in a material approaches the phonon wavelengths [25]. In this limit, the material behaves as an Einstein solid in which energy transport occurs via a random walk of energy transfer between localized vibrations in the solid. Experimentally, K an is often comparable to the value in the amorphous state of the same composition. In principle jc in can be achieved by the introduction of one or more phonon scattering mechanisms that reduce the phonon mean free path to its minimum value over a broad range of frequencies, and therefore reduces Kl over a broad range of temperatures. In practice, there are relatively few crystalline compounds for which this limit is approached. 

In this section, we construct a model of heat transport in the DMFC MEA, which takes into account the thermal effect of crossover. We derive the exact analytical solution to model equations. The solution is greatly simplified under open-circuit conditions. As in a PEFC, the respective relations suggest a method for in situ measurements of the thermal conductivities of the catalyst layers and membrane. 

We begin this chapter with the analjdical modelling of heat transport in a single element of an SOFC stack. This problem illustrates the advantages and limitations of ID models. 

Below we develop a simple model of heat transport in a stack element shown in Figure 5.2 (Kulikovsky, 2009d). An analysis of dimensionless equations reveals the dominant term the rate of heat exchange between the BP and air in the channel. This allows us to construct an asymptotic solution to a problem. 

The model of heat transport in the element (Figure 5.2) is based on the following assumptions. 

Daviskas, E., Gonda, I., and Anderson, S. D. (1990). Mathematical modeling of heat and water transport in human respiratory tract. /. Appl. Physiol. 69, 362-372. 

Hanna, L. M. (1983). Modeling of heat and water vapor transport in the human respiratory tract. Ph.D. Dissertation, University of Pennsylvania, Philadelphia. 

In technical furnaces the radiation from soot, coal and ash particles has to be considered as well as the gas radiation. Then the scattering of radiation by the suspended particles becomes important, alongside absorption and emission. P. Biermann and D. Vortmeyer [5.67], as well as H.-G. Brummel and E. Kakaras [5.68] have developed models for this. A summary can be found in [5.69] and in [5.37], p. 652-673. The calculations of heat transport in furnaces has been dealt with by W. Richter and K. Corner [5.70] as well as H.C. Hottel and A.F. Sarohm [5.48],... 

Gani, R., and Williams, A. (1992), Physical Modeling of Fires, Transport Phenomena in Heat and Mass Transfer, (J.A. Reizes, editor), Elsevier Publishers, 1215-1223. 

Zaichik, L. I. 1999 A statistical model of particle transport and heat transfer in turbulent shear flows. Physics of Fluids 11, 1521-1534. 

The steady state temperature of the catalyst surface under mass-transport-limited conditions can exceed the adiabatic flame temperature if the rate of mass transport of fuel to the surface is faster than the rate of heat transport from the surfaee. The ratio of mass diffusivity to heat dilTusivity in a gas is known as the Lewis number. Reactor models [9] show that for gases with a Lewis number close to unity, such as carbon monoxide and methane, the catalyst surface temperature jumps to the adiabatic flame temperature of the fuel/air mixture on ignition. However, for gases with a Lewis number significantly larger than unity the rate of mass transport to the surface is much faster than the rate of heat transport from the surface, and so the wall temperature can exceed the adiabatic gas temperature. The extreme case is... 

Traditional turbulence-diffusion models (based on the boundary layer adjoining a solid wall) imply that n = 2/3, but a value n = 1/2 is appropriate for a boundary layer adjoining a free surface (Jahne and Haussecker 1998). The appropriate value of n depends on the wind stress and the surfactant loading of the surface. Soloviev and Schluessel (1994) have described a procedure for estimating gas transfer velocities from measurements of heat transport, assuming that transport is adequately described by the classical (Danckwerts) surface renewal model. The key relationship can be written in the form ... 

V. R. Voller, A. D. Brent, and C. Prakash, The Modeling of Heat, Mass and Solute Transport in Solidification Systems, Int. J. Heat Mass Transfer, (32) 1719-1731,1989. 

H. Ramamurthy, S. Ramadhyani, and R. Viskanta, Modeling of Heat Transfer in Indirectly Fired Continuous Reheating Furnace, in P. J. Bishop et al. (eds.), Transport Phenomena in Materials Processing—1990, HTD-Vol. 146, pp. 37-46, ASME, New York, 1990. 

The good agreement between simulated and measured temperature indicates that thermal responses can be predicted with high confidence. This is a consequence of heat conduction being the dominant mode of heat transport. If the thermal conductivity and specific heat are accurately represented in the model, the temperature responses should be well predicted. 

In addition to this, the three-dimensional mathematical model of heat and mass transfer [3, 4] has been developed. Stephan-Maxwell equation was used for mass transport calculations in gas channels and gas diffusion layers. Proton transport in membrane and electrocatalytic layer was described by Nemst-Planck equation. The diffusion and electroosmosis of water were taken into account for membrane potential distribution. 

Spolek, G.A. and Plumb, O.A., 1980. A numerical model of heat and mass transport in wood during drying, in Second International Drying Symposium, Published in Drying 80, pp. 84-92. 

Winterberg, M., Tsotsas, E., Krischke, A. and Vortmeyer, D., 2000. A Simple and Coherent Set of Coefficients for Modeling of Heat and Mass Transport with and without Chemical Reaction in Tubes Filled with Spheres. Chemical Engineering Science, 55(5) 967-979. 

Modeling Heat Transfer Thermal particle dynamics (TPD) primarily introduced by Vargas and McCarthy [14] incorporated both the contact mechanics and contact conductance theories to model the flow dynamics and heat conduction through dry granular materials. The details of the model can be found in Sahni et al. [12]. Heat transport is simulated accounting for the initial material temperature, wall temperature, heat capacity, heat transfer coefficient, and flow properties using a linear model. The flux of heat transported across the mutual boundary between two particles i and j in contact is described as follows ... 

In the next section, we will see that the temperature variation across the CCL is small. For that reason, numerical models of cells and stacks usually ignore the details of heat transport in CLs and assume that these layers are infinitely thin interfaces generating heat (Berning et ah, 2002 Senn and Poulikakos, 2005 Liu et al., 2005 Freunberger et ah, 2006). This approach requires the expression for the heat flux from the CL. 

Bogdanis, E., 2001. Modelling of heat and mass transport during drying of an elastic-viscoelastic medium and resolution by the finite element methods. Diss. University Pau, France. 

For isothermal experiments, the reactor can now be modeled similarly to a ID fixed bed reactor with external mass transfer. The annular reactor allows the axial temperature profile to be measured easily as shown in Figure 36.5. As the thermocouple is in close contact with the catalyst layer, the actual catalyst temperature is measured. This temperature can be incorporated into the reactor model. This eliminates any need of heat transport terms in the model. 

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