Page 438 - Bird R.B. Transport phenomena

420 Chapter 13 Temperature Distributions in Turbulent Flow
From this we can calculate the local time-averaged heat flux at the wall:

k(T 0 -T,) \ (13.6-22)
d
p - 1 / 3 D Y=0
r
and the local Nusselt number is then
q D
Nu loc = k(T 0 0 - Т ) = Pr i/3 _ dY (13.6-23)
г
Then the mean Nusselt number over the wall surface for heat transfer, and the analo-
gous quantity for mass transfer, are

= Я,Рг 1/ 3 + Q! 2 Pr 0 (13.6-24)
y=o

*•-*"•((-#) L)--*"* + a Sc° + (13.6-25)
2
In this last equation Sh , Э , and Sc are the mass transfer analogs of Nu , 0, and Pr. We
w
w
л
give the mass transfer expression here (rather than wait until Part III) because electrochem-
ical mass transfer experiments give better precision than heat transfer experiments and the
available range of Schmidt numbers is much greater than that of Prandtl numbers.
If the expansions in Eq. 13.6-24 and 25 are truncated to one term, we are led to
1/3
Nu m 0 е Рг and Sh,, 7 oc Sc . These expressions are essential ingredients in the famous
1/3
5
Chilton-Colburn relations (see Eqs. 14.3-18 and 19, and Eqs. 22.3-22 to 24). The first term
in Eq. 13.6-24 or 25 also corresponds to the high Prandtl (or Schmidt) number asymptote
of Eq. 13.4-20. 6
With the development of electrochemical methods of measuring mass transfer at
surfaces, it has become possible to investigate the second term in Eq. 13.6-25. In
Fig. 13.6-1 are shown the data of Shaw and Hanratty, who measured the diffusion-
limited current to a wall electrode for values of the Schmidt number Sc = /х/рЯЬ from
АВ
693 to 37,200. These data are fitted 3 very well by the expression

v ° / = 0.0575 + 0.1184 Sc- 1/ 3
0.07 о 0037 - Fig. 13.6-1. Turbulent
о J = 0.0889 Sc"
/ mass-transfer data of
D. A. Shaw and T. J. Han-
_ О
о о ratty [AIChE Journal, 28,
23-37,160-169 (1977)]
О О ^ ^ ^ ^ * » i
1
О О *» compared to a curve
0.06 о о based on Eq. 13.6-25 (solid
8 curve). Shown also is a
simple power law function
10 3 10 4 obtained by Shaw and
Sc Hanratty.

5 T. H. Chilton and A. P. Colburn, Ind. Eng. Chem., 26,1183-1187 (1934). Thomas Hamilton Chilton
(1899-1972) had his entire professional career at the E. I. du Pont de Nemours Company, Inc., in
Wilmington, Delaware; he was President of AIChE in 1951. After "retiring" he was a guest professor at a
dozen or so universities.
6 See also О. С Sandall and О. Т. Hanna, AIChE Journal, 25, 290-192 (1979).