Constitutive equations heat flux vector

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 ... 

G. THE CONSTITUTIVE EQUATION FOR THE HEAT FLUX VECTOR - FOURIER S LAW... 

G. The Constitutive Equation for the Heat Flux Vector - Fourier s Law... 

Here, K is a second-order tensor that is known as the thermal conductivity tensor, and the constitutive equation is known as the generalized Fourier heat conduction model for the surface heat flux vector q. The minus sign in (2-65) is a matter of convention the components of K are assumed to be positive whereas a positive heat flux is defined as going from regions of high temperature toward regions of low temperature (that is, in the direction of —V0). 

It has been emphasized repeatedly that continuum mechanics provides no guidance in the choice of a general constitutive hypothesis for either the heat flux vector q or the stress tensor T. On the other hand, it was noted earlier that (2 41) and (2-56), derived respectively from the law of conservation of angular momentum and the second law of thermodynamics, must be satisfied by the resulting constitutive equations. It thus behooves us to see whether... 

Finally, it was stated previously that fluids that satisfy the Newtonian constitutive equation for the stress are often also well approximated by the Fourier constitutive equation, (2-67), for the heat flux vector. Combining (2-67) with the thermal energy, (2-52), we obtain. 

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. 

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)... 

Using the subsequent first-order perturbation solution for ft, the stress tensor and heat-flux vector in the gas can be evaluated explicitly in terms of F, A and B, and thus the transport coefficients, which are the proportionality factors in the phenomenological constitutive equations for the gas, may be determined. It turns out that F is related only to the bulk viscosity of the gas, A to the thermal conductivity and B to the viscosity. Specifically the bulk viscosity is given by... 

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