Transport Hirschfelder equation

Equation (56) states that the effect of a thermal gradient on the material transport bears a reciprocal relationship to the effect of a composition gradient upon the thermal transport. Examples of Land L are the coefficient of thermal diffusion (S19) and the coefficient of the Dufour effect (D6). The Onsager reciprocity relationships (Dl, 01, 02) are based upon certain linear approximations that have a firm physical foundation only when close to equilibrium. For this reason it is possible that under circumstances in which unusually high potential gradients are encountered the coupling between mutually related effects may be somewhat more complicated than that indicated by Eq. (56). Hirschfelder (BIO, HI) discussed many aspects of these cross linkings of transport phenomena. 

Chapman-Enskog theory provides the basis for the multicomponent transport properties laid out by Hirschfelder, Curtiss, and Bird [178] and by Dixon-Lewis [103]. The multi-component diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by the L matrix [103], seen below. It is convenient to refer to the L matrix in terms of its nine block submatrices, and in this form the system is given by... 

The equations presented thus far for diffusive transport are appropriate for low density, ideal gases. Additional terms are introduced for interdiffusion of dense gases or liquids. This topic is discussed in Hirschfelder, Curtiss, and Bird [178], and was more recently summarized by Curtiss and Bird [83]. 

Equations 11.2.5 and 11.2.6 are the complete forms of the constitutive relations for simultaneous mass and energy transport. The reader is referred to the treatise by Hirschfelder et al. (1964) and to the papers by Merk (1960) and Standart et al. (1979) for further background to these derivations. 

R. B. Bird, J. O. Hirschfelder, and C. F. Curtiss Theoretical Calculation of the Equation of State and Transport Properties of Gases and Liquids, Trans. ASME, 76 1011 (1954). 

Therefore, in this work a more physically consistent way is used by which a direct account of process kinetics is realised. This approach to the description of a column stage is known as the rate-based approach and implies that actual rates of multicomponent mass transport, heat transport and chemical reactions are considered immediately in the equations governing the stage phenomena. Mass transfer at the vapour-liquid interface is described via the well known two-film model. Multicomponent diffusion in the fdms is covered by the Maxwell-Stefan equations (Hirschfelder et al., 1964). In the rate-based approach, the influence of the process hydrodynamics is taken into account by applying correlations for mass transfer coefficients, specific contact area, liquid hold-up and pressure drop. Chemical reactions are accounted for in the bulk phases and, if relevant, in the film regions as well. 

The explicit evaluation of the transport coefficients of the gas is now performed in a series of steps detailed elsewhere (Hirschfelder etal. 1954 Chapman Cowling 1970 Ferziger Kaper 1972 Maitland 1987 McCourteta/. 1990,1991). First, using the local equilibrium solution, Euler s equations of hydrodynamics are constructed and... 

Turning now to transport properties, it is obvious that it is possible to extend the principle of corresponding states from the equation of state to the transport coefficients. This step was soon taken by engineers, using critical constants (Hirschfelder et al. 1964). For example, a dimensionless reduced viscosity could be defined as... 

The method recommended to calculate the viscosity and thermal conductivity of HeXe mixtures is the rigorous kinetic theory described by Hirschfelder (Hirschfelder, Curtiss, and Bird, 1954) with the addition of two higher order thermal conductivity correction factors provided by Kestin (Kestin et al., 1984) and Singh (Singh, Dham, and Gupta, 1992). This mixture method when coupled with the use of DIPPR pure component equations, is referred to as the Chapman (DIPPR) method. Of the mixture methods evaluated, the results of the Chapman (DIPPR) method compare best to the limited amount of experimental mixture data currently available. Different system calculations and component designs have different sensitivities to variations in transport properly values, but impacts are potentially significant. 

---