Page 367 - Bird R.B. Transport phenomena

§11.4 Use of the Equations of Change to Solve Steady-State Problems 349

To determine that function, we have to make experimental measurements or solve Eqs.
3
11.4-44 to 4. In 1881, Lorenz obtained an approximate solution to these equations and found
С = 0.548. Later, more refined calculations gave the following dependence of С on Pr:
4
Pr 0.73 (air) 1 10 100 1000 00
С 0.518 0.535 0.620 0.653 0.665 0.670

These values of С are nearly in exact agreement with the best experimental measurements in
9
the laminar flow range (i.e., for GrPr < 10 ). 5
EXAMPLE 11.4-6 Develop equations for the relationship of local pressure to density or temperature in a stream
of ideal gas in which the momentum flux т and the heat flux q are negligible.
Adiabatic Frictionless
Processes in an SOLUTION
Ideal Gas
With т and q neglected, the equation of energy [Eq. (/) in Table 11.4-1] may be rewritten as
DT
-
nC — _{d In V\ D P (11.4-52)
d\x\ T) Dt
p
For an ideal gas, pV = RT/M, where M is the molecular weight of the gas, and Eq. 11.4-52
becomes
(11.4-53)
Dt Dt
Dividing this equation by p and assuming the molar heat capacity C p = MC p to be constant,
we can again use the ideal gas law to get

(11.4-54)

Hence the quantity in parentheses is a constant along the path of a fluid element, as is its an-
tilogarithm, so that we have
C,,/R -i instant (11.4-55)
T p =
This relation applies to all thermodynamic states p, T that a fluid element encounters as it
moves along with the fluid.
Introducing the definition у = C /C v and the ideal gas relations C p - C v = R and p =
p
pRT/M, one obtains the related expressions
(11.4-56)
and
pp y = constant (11.4-57)
These last three equations find frequent use in the study of frictionless adiabatic processes in
ideal gas dynamics. Equation 11.4-57 is a famous relation well worth remembering.

3
L. Lorenz, Wiedemann's Ann. der Physik u. Chemie, 13,422-^147, 582-606 (1881). See also U. Grigull,
Die Grundgesetze der Warmeiibertragung, Springer-Verlag, Berlin, 3rd edition (1955), pp. 263-269.
4 See S. Whitaker, Fundamental Principles of Heat Transfer, Krieger, Malabar Fla. (1977), §5.11. The
limiting case of Pr —» °° has been worked out numerically by E. J. LeFevre [Heat Div. Paper 113, Dept.
Sci. and Ind. Res., Mech. Engr. Lab. (Great Britain), Aug. 1956] and it was found that
0.5028 1.16
дУ) r,=0 /1/4
Equation 11.4-51 a corresponds to the value С = 0.670 above. This result has been verified experimentally
by C. R. Wilke, C. W. Tobias, and M. Eisenberg, /. Electrochem. Soc, 100, 513-523 (1953), for the analogous
mass transfer problem.
5
For an analysis of free convection in three-dimensional creeping flow, see W. E. Stewart, Int. ]. Heat
and Mass Transfer, 14,1013-1031 (1971).