Fluid heat dispersion coefficients

Fluid and solid heat dispersion coefficients. The description of heat transfer follows the same lines, with two main distinctions concerning mostly the processes occurring in... 

Wakao N, Kaguei S, Funazkri T. Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed beds. Correlation of nusselt numbers. Chemical Engineering Science 1979 34(3) 325-336. 

The pore diffusivity used in this analysis was determined by the Renkin equation4, the axial dispersion coefficient calculated by assuming a constant Peclet number of 0.2, and the mass transfer coefficient from the bulk to the particle surface calculated by the correlation of Wakao and Kaguei. The product of the heat capacity and density of the solid phase was taken to be the same as that used by Raghavan and Ruthven17. The density of the fluid phase was assumed to be that of pure C02 and was calculated from data provided by the Dionix Corporation in their AI-450 SFC software. Constant pressure heat capacities for the mobile phase were also assumed to be that of pure C02 and were taken from Brunner3. 

Liquid holdup is defined as the volume of liquid contained in the bed per unit bed volume. It is a function of the physical properties of the fluid phases and the bed characteristics. It is a basic parameter for reactor design, because it is related to other important parameters, namely, pressure gradient, gas-liquid interfacial area, the mean residence time of the liquid phase, catalyst loading per unit volume, axial dispersion coefficient, mass transfer characteristics, and heat transfer coefficient at the wall, etc. The optimal value of liquid holdup is desirable for better performance of TBR as a high value of liquid holdup will increase mass transfer resistance while too low a value of liquid holdup will decrease the proper utilization of the catalyst bed. Sometimes, the term total liquid saturation (j t) is used to describe the amount of liquid in the bed. It is defined as the volume of liquid present in a unit void volume of the reactor. Thus, the liquid holdup and total liquid saturation are related as ... 

The concept of nanofluids to further intensify microreactors has been discussed by Fan et al. [17]. The nano-fluids are suspensions of solid nano-partides with sizes typically of 1-100 nm in traditional liquids such as water, glycol and oils. These solid-liquid composites are very stable and show higher thermal conductivity and higher convective heat transfer performance than traditional liquids. They can thus be used to enhance the heat transfer in nanofluids in compact multifunctional reactors. A nanofluid based on Ti02 material dispersed in ethylene glycol showed an up to 35% increase in the overall heat transfer coefficient and a... 

For sound process design, we need values of numerous design parameters such as fractional phase holdups, pressure drop, dispersion coefficients (the extent of axial mixing) of all the compounds, heat and mass transfer coefficients across a variety of fluid-fluid and fluid-solid interfaces depending on the type of multiphase system, type of reactor, and the rate-controlling steps. To clarify the scope of the case studies selected, their salient features are next listed. 

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 ... 

The nonuniformities in the phase distributions at and near the bounding surface and their effects on the fluid flow and heat transfer are most significant if the primary heat transfer is through these surfaces. In analyzing the variation of the dispersion coefficient at and near these surfaces, the following should be considered. 

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

Woodward (W12, T2) studied the operation of single, sectionalized, and double-column exchangers. Shell Fluid A and Shell Deodorized Spray Base were dispersed in water and in sea water. Data reported for the Fluid A-water system in a single column are summarized in Table V. From these, the dependence of the overall heat-transfer coefficient on dispersed-phase holdup... 

In Eq. 12.S.b-l, 2, the concentrations, C,-, could be mass concoitrations, or for heat transfer, Cj — p.-c, 7. Also, the axial dispersion coefficients for heat transfer would be D - = KJPiCpi, and the transfer coefficient would be k = h/PiCp, (for region 1 normally being the fluid, and then K = P2Cp2/PiCpt)-... 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

Throughout this book various transport properties and transfer coefficients have been used. These include effective diffusivity and thermial conductivity for mass and heat transport in catalyst pellets, film transfer coefficients for mass and heat transfer across the pellet-bulk fluid interface, transport properties for the degree of dispersion of mass and heat in the reactor, and heat transfer coefficients for heat exchange between the cooling medium and the reactor. In this chapter these transport properties and transfer coefficients are treated in detail, including experimental methods for obtaining these properties. 

Most studies on heat- and mass-transfer to or from bubbles in continuous media have primarily been limited to the transfer mechanism for a single moving bubble. Transfer to or from swarms of bubbles moving in an arbitrary fluid field is complex and has only been analyzed theoretically for certain simple cases. To achieve a useful analysis, the assumption is commonly made that the bubbles are of uniform size. This permits calculation of the total interfacial area of the dispersion, the contact time of the bubble, and the transfer coefficient based on the average size. However, it is well known that the bubble-size distribution is not uniform, and the assumption of uniformity may lead to error. Of particular importance is the effect of the coalescence and breakup of bubbles and the effect of these phenomena on the bubble-size distribution. In addition, the interaction between adjacent bubbles in the dispersion should be taken into account in the estimation of the transfer rates... 

It should be emphasized that Oh and Cavendish assumed that the reactions only occur on the surface of the channel wall. This assumption is less realistic for a layer of washcoat (typically y-alumina) dispersed with catalyst applied on to the wall surface. Ramanathan, Balakotaiah, and West showed that the diffusion in the washcoat has a profound influence on the light-off behavior of a monolith converter. They derived an analytical light-off criterion based on a onedimensional two-phase model with position-dependent heat and mass transfer coefficients. The derivation of this criterion is based on the two key assumptions a positive exponential approximation (i.e., the Frank-Kameneskii approximation) and negligible reactant consumption in the fluid phase. The light-off is defined as the occurrence of multiple steady states with the attainment of the ignited steady state. Here, we discuss only the results of their analysis, without going into the details of their derivation. 

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