Page 302 - Bird R.B. Transport phenomena

286 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

Table 9.8-1 Summary of Notation for Energy Fluxes
Symbol Meaning Reference
(W + pU)v convective energy flux vector Eq. 9.7-2

q molecular heat flux vector Eq. 9.1-6
[TT-V] molecular work flux vector Eq. 9.8-4
e = q + [тг • v] + (\pv + pU)v combined energy flux vector Eq. 9.8-5, 6
2
= q + [т • v] + (Ipv + pH)v
2

In Table 9.8-1 we summarize the notation for the various energy flux vectors intro-
duced in this section. All of them have the same sign convention.
To evaluate the enthalpy in Eq. 9.8-6, we make use of the standard equilibrium ther-
modynamics formula
dh r + = dT+ т р m 7)
- (#)/ (f h ^ [* - Ш} -

When this is integrated from some reference state p°, T° to the state p, T, we then get 1

(9.8-8)

in which H° is the enthalpy per unit mass at the reference state. The integral over p is
zero for an ideal gas and (l/p)(p — p°) for fluids of constant density. The integral over T
becomes C (T — T°) if the heat capacity can be regarded as constant over the relevant
p
temperature range. It is assumed that Eq. 9.8-7 is valid in nonequilibrium systems,
where p and T are the local values of the pressure and temperature.

QUESTIONS FOR DISCUSSION
1. Define and give the dimensions of thermal conductivity k, thermal diffusivity a, heat capacity
C , heat flux q, and combined energy flux e. For the dimensions use m = mass, / = length, T =
p
temperature, and t = time.
2. Compare the orders of magnitude of the thermal conductivities of gases, liquids, and solids.
3. In what way are Newton's law of viscosity and Fourier's law of heat conduction similar? Dis-
similar?
4. Are gas viscosities and thermal conductivities related? If so, how?
5. Compare the temperature dependence of the thermal conductivities of gases, liquids, and
solids.
6. Compare the orders of magnitudes of Prandtl numbers for gases and liquids.
20
7. Are the thermal conductivities of gaseous Ne and Ne the same?
22
8. Is the relation C p - C v = R true only for ideal gases, or is it also true for liquids? If it is not
true for liquids, what formula should be used?
9. What is the kinetic energy flux in the axial direction for the laminar Poiseuille flow of a New-
tonian liquid in a circular tube?
10. What is [тг • v] = pv + [т • v] for Poiseuille flow?

1 See, for example, R. J. Silbey and R. A. Alberty, Physical Chemistry, Wiley, 3rd edition (2001), §2.11.