Interfacial heat transfer

The terms dpG/dp, dpjdp, dXIdp, and dS/dp are determined by interfacial heat transfer, mass transfer, and momentum transfer, respectively. 

The final dimensionless group to be evaluated is the interfacial heat-transfer number, and therefore the interfacial heat-transfer coefficient and the interfacial area must be determined. The interface is easily described for this regime, and, with a knowledge of the holdup and the tube geometry, the interfacial area can be calculated. The interfacial heat trasfer coefficient is not readily evaluated, since experimental values for U are not available. A conservative estimate for U is found by treating the interface as a stationary wall and calculating U from the relationship... 

L Liquid mass flux (lbmass lit 2 U Interfacial heat transfer... 

The design of devices to promote cocurrent drop flows for heating or cooling a two-phase system, or for direct-contact heat transfer between two liquids, is difficult. The study by Wilke et al. (W7) is typical of the approach frequently used to analyze these processes. Wilke et al. described the direct-contact heat transfer between Aroclor (a heavy organic liquid) and seawater in a 3-in. pipe. No attempt was made to describe the flow pattern that existed in the system. The interfacial heat-transfer coefficient was defined by... 

The interfacial heat transfer coefficient can be evaluated by using the correlations described by Sideman (S5), and then the dimensionless parameter Ni can be calculated. If the Peclet numbers are assumed to be infinite, Eqs. (30) can be applied to the design of adiabatic cocurrent systems. For nonadiabatic systems, the wall heat flux must also be evaluated. The wall heat flux is described by Eqs. (32) and the wall heat-transfer coefficient can be estimated by Eq. (33) with... 

The fluidized bed characteristics of high solids heat capacity, large interfacial heat transfer area, and good solids mixing allow the assumptions of thermal equilibrium between the solids and the gas, uniform bed temperature and negligible heat capacitance of the gas. An additional assumption required to use equation (9) is that the reactions do not change the gas volume. 

Interfacial mass transfer is similar to interfacial heat transfer, but it is also different. 

To achieve the solvable temperature equation frequently used in reactor analysis it is necessary to assume that Cp k is constant within the averaging volume. In this particular case the continuity equation can be used manipulating the terms on the LHS of (3.176). In addition, in this approach the interfacial heat transfer term due to phase change has to be included by postulation. 

The interfacial heat transfer flux involves the interfacial heat which can be related to the heat of vaporization or condensation and the heat flux of the phase on the k side of the interface [112]. 

The time after volume averaged interfacial heat transfer is given by ... 

Similarly, the interfacial heat transfer (3.175), as related to (3.201), and the species transfer (3.179) due to phase change are normally expressed in terms of interfacial mass flux weighted quantities... 

Similar constitutive equations are used to approximate the integrals representing the interfacial heat transfer rates by convection and conduction through the stagnant films in the vicinity of a catalytic solid surface. Hence, the film model can be used to approximate the interfacial heat transport (3.167) by ... 

At the catalytic solid surfaces the interfacial heat transfer due to phase change vanishes, instead the convective contributions in (3.175) are incorporated into the convective heat transfer coefficients. 

To be consistent with the mass transfer model, the interfacial heat transfer flux through each of the films consists of both conductive (3.167) and convective (3.175) contributions. The heat transfer terms are approximated by... 

That is, the interfacial heat transfer per unit volume is modeled as the product of the interfacial area concentration and a mean normal interfacial flux which is proportional to the difference between the interfacial average and the intrinsic volume average of the temperature of phase k. 

The gas-particle interfacial heat transfer term in the temperature equations can be modeled by ... 

H(p, q, t) Hamiltonian function in Hamiltonian mechanics h Rep) dimensionless function in particle drag expression (—) interfacial heat transfer due to phase change (J/m s) h specific enthalpy of ideal gas mixture J/kg)... 

After the formal derivations, the energy equation for each phase ((T)f and (T) ) can be written in a more compact form by defining the following coefficients. Note that both the hydrodynamic dispersion, that is, the influence of the presence of the matrix on the flow (noslip condition on the solid surface), as well as the interfacial heat transfer need to be included. The total thermal diffusivity tensors Dff, D , D/s, and Dv/ and the interfacial convective heat transfer coefficient hsf are introduced. The total thermal diffusivity tensors include both the effective thermal diffusivity tensor (stagnant) as well as the hydrodynamic dispersion tensor. A total convective velocity v is defined such that the two-medium energy equations become... 

For ceramic foams, with air as the fluid, Yunis and Viskanta [86] have indirectly measured Nurf and obtained correlations with Re, as the variable. They obtain a lower value for the power Re,. The interfacial heat transfer is also discussed in detail by Kaviany [7],... 

In Eq. 9.152, (k) is the effective conductivity Dxx is the axial dispersion coefficient, and hs, is the interfacial heat transfer coefficient. 

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