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This causes the fluid phase temperature profile to experience a sharp gradient near the wall since the energy transport by diffusion substantially increases as compared to that by convection. The following characteristics can be depicted from the modeling of dispersion conductivity: (i) dispersion conductivity is modeled as a function of the pore velocity, which vanishes near the wall region leaving the effective conductivity equivalent to the stagnant component only, (ii) the magnitude of the dispersion conductivity reaches its maximum at the peak velocity location. Therefore, it is not surprising that discrepancies up to 1000% has been reported in the literature between different numerical studies and the experiments, see the review of literature by Chou et al. In addition, incorporating empirical parameters, such as η ∞ and γ, demands coupling the analytical models with experimental findings to account for such coefficients. Due to the wide pool of empirical correlations, caution must be exercised in choosing an appropriate one since some of these correlations are tailored to specific problems. have surveyed some of the most prominent methods in this regard.
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There are many different techniques associated with measuring the dispersion conductivity in both longitudinal and transverse directions. Their predictions were found to be in good agreement with experimental results. What is more, Koch and Brady presented a semi analytical solution for the longitudinal and transverse dispersion conductivity in a randomly arranged sphere-packed bed. (18) k d ) r k f = γ Da Pe p D h / ( ɛ d p ) suggested such measurements to occur after an effective length L e defined as Li and Finlayson criticized a significant number of the earlier conducted experimental work due to their failure to incorporate the entrance length effect on the measurements of dispersion conductivity. Moreover, such models were found to best fit a linear function of either the particle Reynolds number Re p or the particle Peclet number Pe p. The bulk of the existing experimental studies on determining dispersion conductivity were conducted using a cylindrical packed bed with air as the working fluid. The pioneering work of Yagi and his coworkers is widely considered in this regard in addition to other more recent experimental investigations reported by Gunn and Khalid and Vortmeyer. The solid phase consists merely of the stagnant component since it is stationary. Typically, the stagnant component is the product of the phase fractions and the individual thermal conductivities of the phases. Where k o and k d are the stagnant and the dispersion thermal conductivities, respectively. This ratio interacts in a quite complicated way with the other parameters. The main new feature is the way in which the Nusselt number varies with the velocity ratio.
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For the conjugate problem the Nusselt number also depends on a Biot number, while for the thermally developing convection problem it also depends on a suitably scaled longitudinal coordinate. For the case of uniform temperature boundaries, the analysis has been extended two directions, namely a conjugate problem and a problem involving thermally developing convection. For the case of fully developed convection, the analysis leads to expressions for the Nusselt number as a function of properties of the BDPM, namely a conductivity ratio, a permeability ratio (which for the Darcy case is equivalent to a velocity ratio), a volume fraction, and an internal heat exchange parameter. The hydrodynamic problem has been solved for the case of the Brinkman momentum equation, but work to date on the thermal problem has been confined to the case of the Darcy equation. This model has been applied to forced convection in a channel between two parallel walls, with either uniform temperature or uniform heat flux imposed at the walls. The present authors have proposed a new model involving two velocities as well as two temperatures. It has been noted that to date there has been little work done on heat transfer in BDPM, and efforts have been mainly confined to the measurement of permeability, thermal conductivity and dispersion. KUZNETSOV, in Transport Phenomena in Porous Media III, 2005 2.8 CONCLUSIONS