Page 309 - Intro to Tensor Calculus

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Fourier law of heat conduction

The Fourier law of heat conduction can be written q i = −κT ,i for isotropic material and q i = −κ ij T ,j
c pµ ∗
for anisotropic material. The Prandtl number is a nondimensional constant defined as Pr = so that
κ
the heat flow terms can be represented in Cartesian coordinates as
∗
∗
∗
c p µ ∂T c p µ ∂T c p µ ∂T
q x = − q y = − q z = −
Pr ∂x Pr ∂y Pr ∂z
γR γRT
Now one can employ the equation of state relations P = %e(γ − 1), c p = , c p T = and write the
γ−1 γ−1
above equations in the alternate forms

µ ∗ ∂ γP µ ∗ ∂ γP µ ∗ ∂ γP
q x = − q y = − q z = −
Pr(γ − 1) ∂x % Pr(γ − 1) ∂y % Pr(γ − 1) ∂z %
s
γP p γP
2
The speed of sound is given by a = = γRT and so one can substitute a in place of the ratio
% %
in the above equations.
Equilibrium and Nonequilibrium Thermodynamics
High temperature gas flows require special considerations. In particular, the specific heat for monotonic
and diatomic gases are different and are in general a function of temperature. The energy of a gas can be
written as e = e t + e r + e v + e e + e n where e t represents translational energy, e r is rotational energy, e v is
vibrational energy, e e is electronic energy, and e n is nuclear energy. The gases follow a Boltzmann distribution
for each degree of freedom and consequently at very high temperatures the rotational, translational and
vibrational degrees of freedom can each have their own temperature. Under these conditions the gas is said
to be in a state of nonequilibrium. In such a situation one needs additional energy equations. The energy
equation developed in these notes is for equilibrium thermodynamics where the rotational, translational and
vibrational temperatures are the same.
Equation of state
It is assumed that an equation of state such as the universal gas law or perfect gas law pV = nRT
3
2
holds which relates pressure p [N/m ], volume V [m ], amount of gas n [mol],and temperature T [K]where
R [J/mol − K] is the universal molar gas constant. If the ideal gas law is represented in the form p = %RT
3
where % [Kg/m ] is the gas density, then the universal gas constant must be expressed in units of [J/Kg−K]
(See Appendix A). Many gases deviate from this ideal behavior. In order to account for the intermolecular
forces associated with high density gases, an empirical equation of state of the form

M 1 M 2
X −γ 1 ρ−γ 2 ρ 2 X
p = ρRT + β n ρ n+r 1 + e c n ρ n+r 2
n=1 n=1
involving constants M 1 ,M 2,β n ,c n ,r 1 ,r 2 ,γ 1 ,γ 2 is often used. For a perfect gas the relations
c p R γR
e = c v T γ = c v = c p = h = c p T
c v γ − 1 γ − 1
hold, where R is the universal gas constant, c v is the specific heat at constant volume, c p is the specific
heat at constant pressure, γ is the ratio of specific heats and h is the enthalpy. For c v and c p constants the
relations p =(γ − 1)%e and RT =(γ − 1)e can be verified.