Estimate Thermal Conductivity of a Mixture

Example 30 Estimate thermal conductivity of a mixture of 0.23 mole fraction dimethylether (1) and 0.77 mole fraction methyl chloride (2) at... 

In general, the thermal conductivities of liquid mixtures, and gas mixtures, are not simple functions of composition and the thermal conductivity of the components. Bretsznajder (1971) discusses the methods that are available for estimating the thermal conductivities of mixtures from a knowledge of the thermal conductivity of the components. 

The thermal conductivity of a multicomponent mixture of monatomic species therefore requires a knowledge of the diermal conductivity of the pure components and of three quantities characteristic of the unlike interaction. The final three quantities may be obtained by direct calculation from intermolecular potentials, whereas the interaction thermal conductivity, Xgg, can also be obtained by means of an analysis of viscosity and/or diffusion measurements through equations (4.112) and (4.125) or by the application of equation (4.122) to an analysis of the thermal conductivity data for all possible binary mixtures, or by a combination of both. If experimental data are used in the prediction it may be necessary to estimate both and This is readily done using a realistic model potential or the correlations of the extended law of corresponding states (Maitland et al. 1987). Generally, either of these procedures can be expected to yield thermal conductivity predictions with an accuracy of a few percent for monatomic systems. Naturally, all of the methods of evaluating the properties of the pure components and the quantities characteristic of binary interactions that were discussed in the case of viscosity are available for use here too. 

The first term of equation (4.127) is an approximation to the translational contribution to the thermal conductivity of the mixture. It is obtained by making use of equations (4.122)-(4.125) for the thermal conductivity of a monatomic gas mixture. For this purpose approximate translational contributions to the thermal conductivity of each pure component X, tr and an interaction thermal conductivity for each unlike interaction Xqq are evaluated by the heuristic application of equation (4.125) for monatomic species to polyatomic gases. Thus, the technique requires the availability of experimental viscosity data for pure gases and the interaction viscosity for each binary system or estimates of them. As the discussion of Section 4.2 makes clear, the use of... 

The remaining quantities required to evaluate the approximate translational contribution to the thermal conductivity of the mixture are /l, and for each unlike interaction. Almost invariably these quantities must be estimated by means of spherically symmetric potential models or correlations based upon corresponding states (Maitland et al. 1987). Recent evidence (Heck Dickinson 1994a,b) suggests that while such means of evaluating these cross section ratios are not exact, the errors incurred amount to a few percent at most. 

The second term of equation (4.127) represents an approximation to the contribution of internal energy transport to the thermal conductivity of the mixture. The numerator of each term in the summation contains the difference between the total thermal conductivity of a pure gas and its translational part, estimated as discussed above. Thus, inasmuch as the latter quantity is approximate, the internal contribution to the thermal conductivity is merely an estimate. However, the combination of the first and second terms of equation (4.127), in the limit of any one mole fraction approaching unity, ensures that the total thermal conductivity of each pure component is reproduced. 

In order to calculate the thermal conductivity of a dense multicomponent gas mixture using the procedure outlined above, no information on the behavior of the gas mixture at elevated densities is required. However, the pure component thermal conductivity as a function of density must be available, together with three quantities characteristic of the zero-density state namely A, B and They can be easily obtained, for a large number of binary interactions, by the methods described in Chapter 4. Again, the procedure automatically reproduces the behavior of all of the pure components in the mixture and acts as an interpolatory formula between them. If the thermal conductivity of one of the pure components is not available as a function of density at the temperature of interest, it can be estimated by one of the methods described in Section 5.3, preferably that which makes use of the concept of a temperature-independent excess property. 

Methods to estimate the thermal conductivity of liquid mixtures have been reviewed by Reid et al. (1977, 1987) and Rowley et al. (1988). Five methods are summarized by Reid et al. (1987), but three of these can be used only for binary mixtures. The two that can be extended to multicomponent mixtures are the Li method (Li 1976), and Rowley s method (Rowley et al. 1988). According to the latter the Li method does not accurately describe ternary behavior. Furthermore, it was indicated that the power law method (Reid et al. 1977 Rowley et al. 1988) successfully characterizes ternary mixture behavior when none of the pure component thermal conductivities differ by more than a factor of 2. But, the power law method should not be used when water is present in the mixture. Rowley s method is based on a local composition concept, and it uses NRTL parameters from vapor-liquid equilibrium data as part of the model. These parameters are available for a number of binary mixtures (Gmehling Onken 1977). When tested for 18 ternary systems, Rowley s method gave an average absolute deviation of 1.86%. 

Transport properties such as viscosity and thermal conductivity of a gas mixture are estimated based on the mixture rules. A simplified mixfure model developed based on fhe kinefic theory model is widely used (Bird et. al., 1960 Mills, 2001 Wilke, 1950). These formulae are given as follows ... 

Although direct measurement of reactant temperatures have enabled more quantitative assessment of such reactions, precise tests of thermal explosion theory require a reaction for which the mechanism and Arrhenius parameters are sufficiently well established to give accurate estimates of rates under explosive conditions. Typically the reaction rates involved will be around ten times those determined by static kinetic methods. In addition the thermal conductivity of each gas mixture used and the stoicheiometry and heat of reaction must be known. Pritchard and Tyler suggest the thermal isomerization of methyl isocyanide as a suitable candidate. They report temperature-time records for diluted mixtures in which temperature excesses of 70—80 K occur without explosion. However, the roll-call of missing data—improved heats of formation, isothermal kinetic data at higher temperatures, thermal conductivity measurements up to 670 K, and the recognition and elucidation of side reactions (if any) indicate the extent of further investigations necessary if their proposal is to be fully realized. 

For heterogeneous materials, the effect of geometry must be considered using structural models. Utilizing Maxwell s and Eucken s work in the field of electricity, Luikov et al. [105] initially used the idea of an elementary cell, as representative of the model structure of materials, to calculate the effective thermal conductivity of powdered systems and solid porous materials. In the same paper, a method is proposed for the estimation of the effective thermal conductivity of mixtures of powdered and solid porous materials. 

Thermophysical Property Estimation. For pure components there is not a large difference in thermophysical properties estimated by different methods except for thermal conductivity of water. The main problem is the thermal conductivity estimation of mixtures.The method of Friend and Adler (15) is suitable due to its simplicity. However, the method of Mason and Saxena as given by Touloukiau (26) estimates higher thermal conductivities and requires larger computation time than the previous method but seems to be more accurate. 

This identification means that it is possible to use experimental values of diffusion coefficients or the viscosities of binary mixtures and pure components to estimate the internal energy diffusion coefficients through equation (4.125). What evidence there is for both pure gases (Section 4.2) and gas mixtures (Vesovic etal. 1995) suggests that the mass and internal energy diffusion coefficients seldom differ substantially, so that this is a reasonable approximation. In any event, owing to the fact that the approximate theory is used in an interpolatory manner in this formulation, it has usually been possible to predict the thermal conductivity of binary and multicomponent gas mixtures with errors of a few percent. 

Fig. 14.13 (a) Bubble temperatures estimated using the MRR method as a function of thermal conductivity for the rare gases, (b) Hydrogen peroxide concentration following sonication of pure water as a function of gas solubility in different rare gases ( ) He ( ) Ne (a) Ar ( ) Kr ( ) Xe ( ) He/Xe mixture [42] (reprinted with permission from the American Chemical Society)... 

Key material properties for SOFC, such as the ionic conductivity as a function of temperature, are available in refs 36—39. In addition, Todd and Young ° compiled extensive data and presented estimation methods for the calculation of diffusion coefficients, thermal conductivities, and viscosities for both pure components and mixtures of a wide variety of gases commonly encountered in SOFCs. Another excellent source of transport properties for gases and mixtures involved in a SOFC is the CHEMKIN thermodynamic database. ... 

How efficient is the described representation of the ArCC>2 potential To answer this question the above PES along with a few empirical potentials have been used to derive a number of properties, such as the ground vibrational state and dissociation energy of the complex, ground state rotational constants, the mean square torque, the interaction second virial coefficients, diffusion coefficients, mixture viscosities, thermal conductivities, the NMR relaxation cross sections, and many others [47]. Overall, the ab initio surface provided very good simulations of the empirical estimates of all studied properties. The only parameters that were not accurately reproduced were the interaction second virial coefficients. It is important that its performance proved comparable to the best empirical surface 3A of Bohac, Marshall and Miller [48], This fact must be greeted with satisfaction since no empirical adjustments were performed for the ab initio surface. 

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