Vector, heat-flux

It is sometimes useful to write the rate of heat added to the control volume in terms of a heat flux vector, q", as... 

An instantaneous snapshot of the jet showing soot volume fraction contours and radiation heat flux vectors is shown in Fig. 10.3. The soot forms immediately downstream of the jet exit as a result of the mixing controlled soot formation model. The soot appears in thin streaks in physical space which is consistent with previous experimental observations [2]. The radiation heat flux vectors are seen... 

Figure 10.3 Instantaneous soot volume fraction contour plot with radiation heat flux vectors...

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,... 

L Geometric dimension 9k Kinetic heat flux vector... 

Tmf Characteristic length at minimum fluidization % Total heat flux vector of fluctuation energy... 

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

C Speed of light F Probability density function F Probability distribution function I Intensity ijk Cell indices q" Heat flux vector M Cell index... 

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... 

The turbulent viscosity i/j is determined using the WALE model [330], similar to the Smagorinski model, but with an improved behavior near solid boundaries. Similarly, a subgrid-scale diffusive flux vector Jfor species Jk = p (uYfc — uYfc) and a subgrid-scale heat flux vector if = p(uE — uE) appear and are modeled following the same expressions as in section 10.1, using filtered quantities and introducing a turbulent diffusivity = Pt/Sc], and a thermal diffusivity Aj = ptCp/Pr. The turbulent Schmidt and Prandtl numbers are fixed to 1 and 0.9 respectively. 

In Eqs. (6.2.8) and (6.2.9) JQ was identified as a heat flux vector, yet this quantity corresponds to e, the internal energy density. Consult Sections 1.8 and 1.16 and explain again why this particular designation is appropriate. 

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... 

The absolute specific enthalpy of the mixture is then= u- -p/p. The total heat-flux vector is defined as... 

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