Energy flux vector expression

We now turn to the energy flux vector expression for gases. The dimensionless, kinetic contribution to the energy flux vector follows from... 

Assuming spherical symmetry, the mass and energy flux vectors can be expressed in terms of scalar functions as... 

Species mass concentration Tensor in heat-flux vector expression Species contribution to extra stress tensor Potential energy for all molecules in liquid Tensor used in heat-flux expression Potential energy for single molecule Potential energy for single molecule in external field... 

In order to use the expressions for the mass flux vector, the stress tensor, and the energy flux vector in Table 1, it is necessary to know the singlet and doublet configuration-space distribution functions. For example, we need to solve Eq. (10.6) or Eq. (10.7) to get the distribution function for a single polymer molecule. This cannot, however, be done until something is inserted for the double-bracket... 

In order to solve the conservation or transport equations (mass, momentum, energy, and entropy) in terms of the dependent variables n, Vo,U, and , we must further resolve the expressions for the flux vectors— P, q, and s and entropy generation Sg. This resolution is the subject of closure, which will be treated in some detail in the next chapter. However, as a matter of illustration and for future reference, we can resolve the flux vector expression for what is called the local equilibrium approximation, i.e., we assume that the iV-molecule distribution function locally follows the equilibrium form developed in Chap. 4, i.e., we write [cf Eq. (4.34)]... 

Up to an error proportional to the square of quantities assumed to be small, these expressions are identical to those for perfectly elastic spheres as given, for example, in Section 16.34 of Chapman and Cowling (1970). Consequently, so also are the expressions for the pressure tensor and the energy flux vector calculated by employing the velocity distribution function (4.7) and Enskog s extension of the assumption of molecular chaos. 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

The external resistance to heat transfer is incorporated in reactor design simulations by expressing the normal component of interfacial flux in terms of a transfer coefficient and a driving force, the latter of which is sensitive to the direction of the unit normal vector n and the fact that Fourier s law and Pick s law require a negative sign to calculate the flux in a particular coordinate direction. These considerations produce the following expressions for the conductive energy flux ... 

In the specific conservation equations for mass, momentum, and energy given in (19.14)-(19.16), there are material-dependent expressions such as the Cauchy stress tensor t, the specific heat flux vector q, and the specific internal energy e. The expressions for these quantities are called the constitutive equations. 

Sect. 6 the hydrodynamic equation of continuity for each species and a formal expression for the mass flux vector of each species Sect. 7 the hydrodynamic equation of motion for the liquid mixture and a formal expression for the stress tensor Sect. 8 the energy equation for the liquid and a formal expression for the heat flux vector... 

The compressible Navier-Stokes equations are the governing conservation laws for mass, momentum, and energy. These laws are written assuming that the fluid is Newtonian and follows the Fourier law of diffusion, so that the stress tensor is a linear function of the velocity gradients and the heat flux vector is proportional to the temperature gradient. Adding Stokes hypothesis, which expresses that the changes of volume do not involve viscosity, the compressible Navier-Stokes equations may be written as... 

The well-known dual-phase-lag heat conduction model introduces time delays to account for the responses among the heat flux vector, the temperature gradient and the energy transport. The dual-phase-lag heat conduction model has been used to interpret the non-Fourier heat conduction phenomena. The onedimensional dual-phase-lag constitutive equation relating heat flux to temperature gradient is expressed as (Xu, 2011 Zhou et al., 2009)... 

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

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