The heat-flux vector

Figure 3.9 illustrates the spatial components of the heat-flux vector in spatial components that align with the cylindrical coordinates. Because the heat flux is a continuous, differentiable function, its variation throughout the control volume can be represented as a Taylor series expansion. In a procedure that is analogous to that in Section 3.6.2, the net heat conducted across the control surfaces is... 

The right-hand side of the expression above can be recognized as the divergence of the heat-flux vector,... 

For isotropic fluids the heat flux vector q takes the form... 

The transport relations, providing the closure of the system by expressing the viscous tensor and the heat flux vector in terms of the moments n, V and T and gradients thereof, are obtained as a result of the above mentioned Braginskii approximation for the distribution function as... 

In this form of equation (24), the effect of diffusion on the enthalpy flux is included implicitly in the first term, and the only term from the heat-flux vector that remains explicitly is that of heat conduction. The above equations serve to eliminate completely the diffusion velocities from the governing equations, replacing them by the flux fractions. The fact that equations (31) and (33) appear to be less complicated than equations (22) (with dYJdt = 0) and (24) often justifies this transformation. 

These results require further that w, v, R, 0, and Yi be continuous in the first approximation and rely on the assumptions that fi2, and 6 are continuous. The conditions obtained from equations (89), (90), and (93) in effect involve only the longitudinal components of the stress tensor and of the heat-flux vector. The first of the conditions quoted from equations (89) and (90) expresses a pressure discontinuity that balances the discontinuity in the viscous stress tensor, and the second states that the streamwise gradients of the components of velocity tangent to the sheet are continuous. 

Here the heat flux vector for species K, taken as positive for outward heat transport. Equation (37) can be transformed by the use of equation (20) with... 

In order to show that the model of independent, coexistent continua represents correctly a real mixture of gases composed of different chemical species, we must compare the results obtained from this model with those of the kinetic theory of nonuniform gas mixtures (see Appendix D). Quantities such as the density p, the mass-weighted average velocity v j, and the body force fj have obviously analogous meanings in both the kinetic theory and the coexistent-continua model. On the other hand, the precise kinetic-theory meaning of terms such as the stress tensor, the absolute internal energy per unit mass and the heat-flux vector qf is not immediately apparent. In view of the known success of continuum theory for one-com-... 

D-23). Then F becomes the energy per second transported across a surface of unit area that is, F is the heat-flux vector q [see equations (D-28) and (D-29) with neglected]. Hence equation (31) becomes... 

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 ... 

To obtain a general relation for Fourier s law of heat conduction, consider a medium in which the temperature distribution is three-dimensional. Fig. 2-8 shows an isothermal surface in that medium. The heat flux vector at a point P on this surface must be perpendicular to the surface, and it must point in the direction of decreasing temperature. If n is Ihe normal of the isothermal surface at point the rate of heat conduction at that point can be expressed by Fourier s law as... 

In terms of a continuum theory the heat flux vector represents the direction and magnitude of the energy flow at a position indicated by the vector x. ft can also be dependent on time t. The heat flux q is defined in such a way that the heat flow dQ through a surface element d.4 is... 

Fig. 1.2 Point P on the isotherm i = const with the temperature gradient gradi from (1.4) and the heat flux vector q from (1.5)...

When the second order approximations to the pressure tensor and the heat flux vector are inserted into the general conservation equation, one obtains the set of PDEs for the density, velocity and temperature which are called the Burnett equations. In principle, these equations are regarded as valid for non-equilibrium flows. However, the use of these equations never led to any noticeable success (e.g., [28], pp. 150-151) [39], p. 464), merely due to the severe problem of providing additional boundary conditions for the higher order derivatives of the gas properties. Thus the second order approximation will not be considered in further details in this book. 

However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.247) is used to determine expressions for the viscous-stress tensor o and the heat flux vector q. 

An analytical expression for the heat flux vector can be derived in a similar manner using the Enskog approach. That is, we introduce the first order approximation of the distribution function from (2.246) into the heat flux definition (2.72) and thereafter substitute the partial solution for flux vector integrand as follows [39] ... 

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