Page 353 - Bird R.B. Transport phenomena

§11.2 Special Forms of the Energy Equation 337

put in the substantial derivative form by using Eq. 3.5-4. This gives, with no further
assumptions

t^±= -(V-q)-p(V-v)-(T:Vv) (11.2-2)

Next it is convenient to switch from internal energy to enthalpy, as we did at the very
end of §9.8. That is, in Eq. 11.2-2 we set U = H - pV = H - (p/p), making the standard
assumption that thermodynamic formulas derived from equilibrium thermodynamics
may be applied locally for nonequilibrium systems. When we substitute this formula
into Eq. 11.2-2 and use the equation of continuity (Eq. A of Table 3.5-1), we get

(11.2-3)

Next we may use Eq. 9.8-7, which presumes that the enthalpy is a function of p and T
(this restricts the subsequent development to Newtonian fluids). Then we may get an ex-
pression for the change in the enthalpy in an element of fluid moving with the fluid ve-
locity, which is
Dp
DH ~ DT
Dt p ^ Dt Dt
~ DT
p L
? Dt Dt
' P .
Dp
^ Dt Dt (11.2-4)
Equating the right sides of Eqs. 11.2-3 and 11.2-4 gives

(11.2-5)

This is the equation of change for temperature, in terms of the heat flux vector q and the
viscous momentum flux tensor т. То use this equation we need expressions for these
fluxes:
(i) When Fourier's law of Eq. 9.1-4 is used, the term -(V • q) becomes +(V * kVT),
2
or, if the thermal conductivity is assumed constant, +kV T.
(ii) When Newton's law of Eq. 1.2-7 is used, the term — (T:VV) becomes дФ + к"Фу,
у
the quantity given explicitly in Eq. 3.3-3.
We do not perform the substitutions here, because the equation of change for tempera-
ture is almost never used in its complete generality.
We now discuss several special restricted versions of the equation of change for tem-
perature. In all of these we use Fourier's law with constant k, and we omit the viscous
dissipation term, since it is important only in flows with enormous velocity gradients:
-
(i) For an ideal gas, (d In p/д In T) = 1 , and
p
oC ^ - (11.2-6)
pC
?Dt ~ Dt
Or, if use is made of the relation C — C = R, the equation of state in the form
p v
pM = pRT, and the equation of continuity as written in Eq. A of Table 3.5-1, we get

2
= kV T - v) (11.2-7)