Multicomponent mixtures, thermal conductivity

The task of the problem-independent chemistry software is to make evaluating the terms in Equations (6-10) as straightforward as possible. In this case subroutine calls to the Chemkin software are made to return values of p, Cp, and the and hk vectors. Also, subroutine calls are made to a Transport package to return the ordinary multicomponent diffusion matrices Dkj, the mixture viscosities p, the thermal conductivities A, and the thermal diffusion coefficients D. Once this is done, finite difference representations of the equations are evaluated, and the residuals returned to the boundary value solver. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

Pure species thermal conductivities are usually only needed for the purpose of later evaluating mixture-averaged thermal conductivities the conductivity in the multicomponent case presented in Section 12.5.6 does not depend on the pure species formulas stated in this section. 

The physical properties of the multicomponent mixture, such as viscosity, specific heats at constant volume and at constant pressure, and laminar thermal conductivity, are usually calculated under the assumption of an ideal mixture. Data and... 

For binary and multicomponent mixtures, the thermal conductivity depends on the concentrations as well as on temperature, and the formulas of the accurate kinetic theory are quite complicated [5]. Empirical expressions for X are therefore more useful for both binary [9] and ternary [6], [26] mixtures, although few data exist for ternary mixtures. Tabulations of available experimental and theoretical results for thermal conductivities may be found in [5], [6], [13], and [18]-[21], for example. The thermal diffusivity, defined as 2p/Cp, often arises in combustion problems its pressure and temperature dependences in gases are XjpCp T7p ( < a < 2), and its typical values in combustion lie between 10 cm /s and 1 cm s at atmospheric pressure. 

While equation (42) is valid for one-component systems without radiant transport, for binary and multicomponent mixtures there are other effects besides thermal conduction that contribute to the heat flux q. 

The identification of the chemical forms of an element has become an important and challenging research area in environmental and biomedical studies. Two complementary techniques are necessary for trace element speciation. One provides an efficient and reliable separation procedure, and the other provides adequate detection and quantitation [4]. In its various analytical manifestations, chromatography is a powerful tool for the separation of a vast variety of chemical species. Some popular chromatographic detectors, such flame ionization (FID) and thermal conductivity (TCD) detectors are bulk-property detectors, responding to changes produced by eluates in a characteristic mobile-phase physical property [5]. These detectors are effectively universal, but they provide little specific information about the nature of the separated chemical species. Atomic spectroscopy offers the possibility of selectively detecting a wide rang of metals and nonmetals. The use of detectors responsive only to selected elements in a multicomponent mixture drastically reduces the constraints placed on the separation step, as only those components in the mixture which contain the element of interest will be detected... 

Corresponding considerations are also valid for the thermal boundary layer in multicomponent mixtures. The energy transport through conduction and diffusion in the direction of the transverse coordinate x is negligible in comparison to that through the boundary layer. The energy equation for the boundary layer follows from (3.97), in which we will presuppose vanishing mass forces k Ki-... 

The differential form of the energy balance for a multicomponent mixture can be written In a variety of forms.1 6 It would contain terms reprenenting heat conduction and radiation, body forces, viscous dissipation, reversible work, kinetic energy, and the substantial derivative of the enthalpy of die mixture. Its formulation is beyond the scope of this chapter. Certain simplifled forms will be used in later chapters in problems such as simultaneous heal and mass transfer in air-water operations or thermal effects in gas absorbent. 

The thermal conductivity of a gas mixture which is measured directly is not the quantity X introduced in equation (4.79), because measurements are always performed in the absence of a net diffusive flux. In order to evaluate the measured thermal conductivity in the zero-density limit X, the multicomponent diffusion coefficients are employed (Ross et al. 1992) and then one obtains, in a consistent first-order approximation. 

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. 

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. 

In order to evaluate the thermal conductivity of a multicomponent mixture the functions gij are generated from those for the pure gases by means of the combination rule of equation (5.57) while the mean free-path parameters are generated by a combination... 

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. 

The thermal conductivity of a multicomponent mixture of noble gases also involves only binary-interaction quantities, A j, Bfj, Xfj and fij, where Bfj is a ratio similar to... 

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%. 

The GAMMA-F code (Lim, 2014) has been developed by KAERI for system and safety analysis of VHTR. The code has the capabilities for multidimensional analyses of the fluid flow and heat conduction as well as the chemical reactions related to the air or steam ingress event in a multicomponent mixture system. As a system thermo-fluid and network simulation code, GAMMA-F includes a nonequilibrium porous media model for pebble-bed and prismatic reactor core, thermal radiation model, point reactor kinetics, and special component models such as pump, circulator, gas turbine, valves, and more. 

Saxena, S. C., and Gandhi, J. M., Thermal Conductivity of Multicomponent Mixtures of Inert Gases, Reviews of Modem Physics, 35 (4), 1022-1023,1963. 

Equations (3.38) for the heat flux vector remain unchanged. However, in the approximate formulation it becomes necessary to calculate the thermal conductivities of the separate components of a multicomponent mixture, and then to combine these in an appropriate manner. The equations are ... 

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