Page 439 - Bird R.B. Transport phenomena

Problems 421

Sh = 0.0575 + 0.1184Sc -1/3 (13.6-26)
1/3
ReSc Vf/2
in which/(Re) is the friction factor defined in Chapter 6. Equation 13.6-26 combines the
observed Re number dependence of the Sherwood number with the two leading terms
of Eq. 13.6-25 (that is, the coefficients a b a ,.. . are proportional to ReVf72). Equation
2
13.6-26 lends itself to clear physical interpretation: The leading term corresponds to a
diffusional boundary layer so thin that the tangential velocity is linear in у and the wall
curvature can be neglected, whereas the second term accounts for wall curvature and the
y 1 terms in the tangential velocity expansions of Eqs. 13.6-1 and 2). In higher approxima-
7
tions, special terms can be expected to arise from edge effects as noted by Newman and
Stewart. 3

QUESTIONS FOR DISCUSSION

1. Compare turbulent thermal conductivity and turbulent viscosity as to definition, order of
magnitude, and dependence on physical properties and the nature of the flow.
2. What is the "Reynolds analogy/' and what is its significance?
3. Is there any connection between Eq. 13.2-3 and Eq. 13.4-12, after the integration constants in
the latter have been evaluated?
4. Is the analogy between Fourier's law of heat conduction and Eq. 13.3-1 a valid one?
5. What is the physical significance of the fact that the turbulent Prandtl number is of the order
of unity?

PROBLEMS
13ВЛ. Wall heat flux for turbulent flow in tubes (ap- in which f = x/B and ](& = \
proximate). Work through Example 13.3-1, and fill in the Jo
missing steps. In particular, verify the integration in going (b) Show how the result in (a) simplifies for laminar flow
from Eq. 13.3-6 to Eq. 13.3-7. of Newtonian fluids, and for "plug flow" (flat velocity pro-
files).
13B.2. Wall heat flux for turbulent flow in tubes. Answer: (b) ff, 3
(a) Summarize the assumptions in §13.4.
for
turbulent
temperature
The
profile
(b) Work through the mathematical details of that section, 13D.1. To calculate the temperature distribution for flow in
turbu-
tubes.
taking particular care with the steps connecting Eq. 13.4-12 lent flow in circular tubes from Eq. 13.4-12, it is necessary
and Eq. 13.4-16. to know C .
2
(c) When is it not necessary to find the constant C in Eq. (a) Show how to get C by applying B.C. 4 as was done in
2
13.4-12? §10.8. The result is 2
13C.1. Wall heat flux for turbulent flow between two 1 2
parallel plates. -j, т&/ш - [ко/кт (13D.1-1)
(a) Work through the development in §13.4, and then per-
form a similar derivation for turbulent flow in a thin slit (b) Verify that Eq. 13D. 1-1 gives C = ^ for a Newtonian
shown in Fig. 2B.3. Show that the analog of Eq. 13.4-19 is fluid. 2
«T - T ) /4
o b dg (13C.1-1)
q B (v ) h [1 + (v /v)(Pr/Fr )]
(t)
{t)
0
z

7
J. S. Newman, Electroanalytical Chemistry, 6,187-352 (1973).