Effective-thermal-conductivity Data

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

Figures 7 and 8 show thermal conductivity data for CBCF after exposure to temperatures of 2673, 2873, 3073, and 3273 K, for 5.7 and 15 7 seconds, respectively. The symbols in the Figs. 7 and 8 represent measured thermal conductivity values, and the solid lines are the predicted behavior from Eqs. (5) through (8) The model clearly accounts for the effects of measurement temperature, exposure tune, and exposure temperature The fit to the data is good (typically within 10%). However, the fit to the as fabricated CBCF data (Fig 6) was less good (-20%), although the scatter in the data was larger because of the much lower heat treatment temperature (1873 K) in that case.

Some workers have correlated experimental data in terms of k at the arithmetic mean temperature, and some at the temperature of the bulk plasma. Experimental validation of the true effective thermal conductivity is difficult because of the high temperatures, small particle sizes and variations in velocity and temperature in plasma jets. 

The last equation has been used by numerous investigators to evaluate effective thermal conductivities from experimental data. Figure 12.18 reproduced from Froment (94) indicates the... 

In Illustrations 12.5 and 12.6, some data on the catalytic oxidation of S02 were used to determine composition and temperature differences between the bulk fluid and the fluid at the pellet-gas interface. Use the data and results of these illustrations and the new data given below to predict the effective thermal conductivity of the bed. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Salamone and Newman (SI) recently studied heat transfer to suspensions of copper, carbon, silica, and chalk in water over the concentration range of 2.75 to 11.0% solids by weight. These authors calculated effective thermal conductivities from the heat transfer data and reached conclusions which not only contradicted Eqs. (35) and (36), but also indicated a large effect of particle size. However, if one compares the conductivities of their suspensions at a constant volume fraction of solids, the assumed importance of particle size is no longer present. It should also be noted that their calculational procedure was a difficult one in that it placed all undefined errors present in the heat transfer data into the thermal conductivity term. For example, six of the seven-... 

TABLE 17.15. Data for the Effective Thermal Conductivity, Kr (kcal/mh°C), and the Tube Wall Rim Coefficient, (kcal/m2h°C), in Packed Beds3... 

Liquid thermal conductivity data for cyclopcntane at 37.JTC and cyclohexane at 20 C and 37.8 C are available.13 Tabulated data are available.1 7 The effect of pressure on the thermal conductivity of cyclohexane is shown in Figure 40-11 API11 provides data on cyclohexane from the freezing point to the boiling point. The data lor cyclopropane and cy-rlobuianc were calculated by the method of Robbins and Kin grea." The data for cyclopcntane were extended by the method of Riedel ... 

Further advancements in the theory of fixed bed reactor design have been made(56,57) but it is unusual for experimental data to be of sufficient precision and extent to justify the application of sophisticated methods of calculation. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on the reaction rate. 

No experimental data on the effective thermal conductivity of an assemblage of micron size zeolite crystals under the conditions of sorption tests used in the examples above could be found in the literature. However, several methods are available for the calculation of k for an assemblage of particles with void fraction f. The thermal conductivities of the solid phase (k ) and the gas phase (k ) in the voids are needed [29]. We Bsed the method develop d by Maxwell. 

To use the various criteria given in the previous section, some experimental data on the reacting system are necessary. These are the effective diffusivity of the key species in the pores of the catalyst, the heat and mass transfer coefficients at the fluid-solid interface, and the effective thermal conductivity of the catalyst. The accuracy of some of these parameters, which are usually obtained from known correlations, may sometimes be subject to question. For example, under labo-... 

Measurements of A, are scarce. The available data are reviewed by Satterfield [2], The effective thermal conductivity of a porous catalyst can be estimated from the correlation of Russell [30] ... 

It is evident from the foregoing discussion that the effective diffusivity cannot be predicted accurately for use under reaction conditions unless surface diffusion is negligible and a valid model for the pore structure is available. The prediction of an effective thermal conductivity is even more difficult. Hence sizable errors are frequent in predicting the global rate from the rate equation for the chemical step on the interior catalyst surface. This is not to imply that for certain special cases accuracy is not possible (see Sec. 11-10). It does mean that heavy reliance must be placed on experimental measurements for effective diffusivities and thermal conductivities. Note also from some of the examples and data mentioned later that intrapellet resistances can greatly affect the rate. Hence the problem is significant. 

The two-dimensional method has resulted in better agreement than the simplified approach, but the computed conversions are still less than the experimental results. In view of the problems in estimating the radial heat- and mass-transfer rates, and possible uncertainties in kinetic rate data, the comparison is reasonably good. The net effect of allowing for radial heat and mass transfer is to increase the conversion. The computed results are sensitive to rather small variations in the effective thermal conductivities and diffusivities, which emphasizes the need for the best possible information concerning these quantities. 

There are many publications on predictive models for effective thermal conductivity of nanofluids [5, 11, 23, 25-27], some of these publications make an overview of the existing models, and some drives their own model and compares with experimental data. None of the models is able to explain and predict an effective thermal conductivity value for the nanofluids. 

Figure 9.2 shows existing data for the effective thermal conductivity of packed beds. These data include both ceramic and metallic packings. More accurate results can be obtained from the semitheoretical predictions of Dixon and Cresswell (1979). Once Kr is known, the wall heat transfer coefficient can be calculated from... 

Data for porosity and tortuosity of a Pd/Al203 catalyst are presented in Table 7.4. Using these data and the value of effective thermal conductivity of boehmite given in Table 7.5, estimate a value of the parameter (3 for the hydrogenation of ethylene at 50 °C, 1 atm total pressure, and an ethylene/ hydrogen (molar) ratio of 1 2. How does this compare with the values given in Table 7.1 for this reaction ... 

Predict the effective thermal conductivity of a fixed-bed reactor to be used for SO2 oxidation with the following conditions reactor is a 2.06-in i.d. tube through which a mixture of air (93.5 nol%) and SO2 (6.5 mol%) flow at a superficial velocity of 3501b/ft -h. The catalyst pellets, egg-shell Pt/Al203, (0.2 wt%), are in the form of 1/8-in spheres. Additional data ... 

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