Effective thermal diffusivity

Solids-liquid-gas mixing 275 Solids-liquid mixing 275 Solids—solids mixing 275 Sonic velocity 150, 156,158, 189 Sorel effect, thermal diffusion 589 Spalding, D. B, 393,562 Sparrow, E. M. 465, 564 Specific energy, open channel flow 98... 

Regrettably, the particular effect of an additive is difficult to assess. A chosen diluent cannot lower the flame temperature without altering the effective thermal diffusivity of the fuel-diluent jet. Thus, in examining the possible effect of an additive it is imperative to consider how the measurements of the extent of sooting were made. With respect to physical parameters, the effect of temperature is clear. However, the data reported on pressure variation must be viewed with caution since, as stated, pressure variations cause variations in temperature, thermal diffusivity, flow velocity, and flame structure. 

Next one needs an expression for (xA — xA"). The difference in concentration between the two streams results from two effects thermal diffusion, which tends to increase the concentration difference, and convection, which tends to decrease it. Each of these effects is considered separately by obtaining an approximate integrated form of the steady state equation of continuity as applied to that particular process. If the only effect tending to produce a concentration difference were thermal diffusion, then according to Eq. (131) dxA/dx = — (kT/T)(d,T/dx) this expression may be written in difference form over the distance from x = — ( 4)a to x — + (M)° thus ... 

Russell, L.M., Donaldson, K.Y., Hasselman, D.P.H., Corbin, N.D. Petrovic, J.J. and Rhodes, J.F. Effect of vapor-liquid-solid and vapor-solid silicon carbide whiskers on the effective thermal diffusivity/conductivity of silicon nitride matrix composites , J. Am. Ceram. Soc., 74[4] (1991) 874-877. 

Here, ae is the effective thermal diffusivity of the bed and Th the bulk fluid temperature. We assume that the plug flow conditions (v = vav) and essentially radially flat superficial velocity profiles prevail through the cross-section of the packed flow passage, and the axial thermal conduction is negligible. The uniform heat fluxes at each of the two surfaces provide the necessary boundary conditions with positive heat fluxes when the heat flows into the fluid... 

It is thought, therefore, that the variables which characterize the reduced form of the F-K equation are not the heat generation data of a powdery chemical of the TD type but the heat transfer data of the chemical, i.e., the effective thermal diffusivity, the shape factor, and the radius, r. 

If, however, the powdery chemical of the TD type, placed in the atmosphere under isothermal conditions, has some one of several specific shapes including the so-called class A geometries as well as an arbitrary size, the calculation of the Tc for the chemical becomes very simple, because the values of for these specific shapes have already been calculated by T. Boddington et al, respectively [58], with the result that, as explained in Section 6.2, the heat transfer data required to calculate the for such a chemical as specified above is only the value of the effective thermal diffusivity, of the chemical. 

Equation to calculate the effective thermal diffusivity, of a powder... 

Effective thermal diffusivity of a powdeiy chemical of the TD t3 )e, including every gas-permeable oxidatively-heating substance [cm /min]. 

In the actual bed, solid exchange between the ascending and descending zone is significant, and the circulation is seen as a series combination of localized recirculation. If this scheme of solid mixing is more realistic, the temperature profile may be suitably expressed in terms of an effective thermal diffusivity, which will be used later for discussion of instability in the dilute phase. 

The heat-transfer rate is found to be substantially higher under conditions of agitation. The heat transfer is usually said to occur by combined conductive and convective modes. A discussion and explanation are given by Holt [Chem. Eng., 69(1), 110 (1962)]. Prediction of [/ by Eq. (11-48) can be accomplished by replacing a by a, the effective thermal diffusivity of the bed. To date so httle work has been performed in evaluating the effect of mixing parameters that few predictions can be made. However, for agitated liquid-phase devices Eq. (18-19) is applicable. Holt (loc. cit.) shows that this equation can be converted for solids heat transfer to yield... 

The first and the second terms on the right-hand side of Equation (2.8) represent the effect of electrotransfer of the diffusing ions and the Soret effect (thermal diffusion due to temperature gradients), respectively. The electrical force is given by... 

In the problem considered here, however, the velocity (3-220) is nonuniform, and hence the initial temperature pulse is convected downstream at different speeds, with a rate that is maximum at the centerline of the tube and zero at the tube wall. Thus the temperature pulse is rapidly distorted and spread out in the z direction, with the heated fluid initially carried downstream with a maximum velocity 2 U. Nevertheless, if we examine the system at large times, experimental evidence shows that the fluid far enough downstream (i.e., z 1) is heated almost uniformly across the complete cross section of the tube, and the temperature pulse has spread symmetrically with respect to a plane cross section of the tube that moves with the mean velocity of the fluid, U. This spreading process is qualitatively similar to axial conduction, but with an effective thermal diffusivity that is much larger than the molecular thermal diffusivity k. ... 

A remarkable property of the solution that we have derived, clearly evident in Eq. (3-244) for the cross-sectionally averaged temperature profile, is that the position in the tube where the cross-sectionally averaged temperature is maximum moves as if there were convection downstream at the mean velocity U. The temperature pulse also spreads about this plane as though there were axial conduction with an effective thermal diffusivity of... 

However, Eq. (3-248) is just Fourier s law, with an effective thermal conductivity k(azUz/48k2). Further, because there is no net convection contribution to the cross-sectionally averaged heat flux calculated with respect to the moving axis z, we see that the convective transport rate down the tube is just that due to the mean velocity U. Both of these results may at first seem surprising. For example, the maximum rate of transport that is due to convection acting alone is the centerline velocity 2 U, and it is not immediately obvious why the actual convective transport rate is slower. In addition, the effective thermal diffusivity is seen in (3-246) to be inversely proportional to the molecular thermal diffusivity k, and this may also seem to be counterintuitive. 

Fit . 6. Particle residence time requirements for isothermal oil shale reactor operating at 482°C. The residence time requirements for achieving the desired degree of conversion were obtained from the kerogen decomposition kinetics of Wallman et al. (1980) see Fig. 5. The heating-up requirements of the shale particles were calculated assuming an effective thermal diffusivity inside the particle of 2.7 X 10 7 [m2 s ] ... 

FIGURE 9.6 The prediction of Koch and Brady (for random arrangement of spheres) [28] for longitudinal and transverse total effective thermal diffusivity, compared with the experimental results of many investigators as presented in Fried and Combamous [29]. 

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... 

Ludwig-Soret effect Soret effect Thermal diffusion Thermodiffusion... 

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