Page 134 - Separation process principles 2

3.5 Mass Transfer in Turbulent Mow 99

where EM, EH , are ED are momentum, heat, and mass eddy Npr = NSc = 1. Thus, the Reynolds analogy has limited
diffusivities, respectively; v is the momentum diffusivity practical value and is rarely applied in practice. Reynolds
(kinematic viscosity), k/p; and u is the thermal diffusivity, postulated the existence of the analogy in 1874 [53] and
k/pCp. As a first approximation, the three eddy diffusivities derived it in 1883 [50].
may be assumed equal. This assumption is reasonably valid
for EH and ED, but experimental data indicate that Chilton-Colburn Analogy
--
€MI~~ EM/€~ is sometimes less than 1.0 and as low as
=
A widely used extension of the Reynolds analogy to Prandtl
0.5 for turbulence in a free jet.
and Schmidt numbers other than 1 was presented by Colburn
[54] for heat transfer and by Chilton and Colburn [55] for
Reynolds Analogy
mass transfer. They showed that the Reynolds analogy for
If (3-154) to (3-156) are applied at a solid boundary, they can turbulent flow could be corrected for differences in velocity,
be used to determine transport fluxes based on transport temperature, and concentration distributions by incorporat-
coefficients, with driving forces from the wall, i, at z = 0, to ing Npr and Nsc into (3-162) to define the following three
the bulk fluid, designated with an overbar, -: Chilton-Colburn j-factors, included in Table 3.13.

Equation (3-165) is the Chilton-Colburn analogy or the
Colburn analogy for estimating average transport coeffi-
cients for turbulent flow. When NPr = Nsc = 1, (3-165)
We define dimensionless velocity, temperature, and solute reduces to (3- 162).
concentration by In general, j-factors are uniquely determined by the geo-
metric configuration and the Reynolds number. Based on the
analysis, over many years, of experimental data on momen-
tum, heat, and mass transfer, the following representative
If (3-160) is substituted into (3-157) to (3-159), correlations have been developed for turbulent transport to
or from smooth surfaces. Other correlations are presented in
other chapters. In general, these correlations are reasonably
accurate for Npr and Nsc in the range of 0.5 to 10, but should
be used with caution outside this range.
1. Flow through a straight, circular tube of inside
This equation defines the analogies among momentum, heat, diameter D:
and mass transfer. Assuming that the three eddy diffusivities j~ = j~ = jD = o.o~~(N~~)-~'~
(3-166)
are equal and that the molecular diffusivities are either
for 10,000 < NRe= DGlk < 1,000,000
everywhere negligible or equal,
2. Average transport coefficients for flow across a flat
plate of length L:

Equation (3-162) defines the Stanton number for heat
transfer,
h h 3. Average transport coefficients for flow normal to a
-
NStH = - - (3-163) long, circular cylinder of diameter D, where the drag
-
PCPUX GCP
coefficient includes both form drag and skin friction,
where G = mass velocity = iixp , and the Stanton number but only the skin friction contribution applies to the
for mass transfer, analogy:
kc kcp -0.382
N =-=- (3-164) (jM)skin friction = JH = j~ = 0.193(N~e)
st~
ii, G
for 4,000 < NRe < 40,000 (3-168)
both of which are included in Table 3.13.
(j~)skin friction = j~ = j~ = 0.0266(N~,) -0.195
Equation (3-162) is referred to as the Reynolds analogy.
for 40,000 < NRe < 250,000 (3-169)
It can be used to estimate values of heat and mass transfer
coefficients from experimental measurements of the DG
with NRe = -
Fanning friction factor for turbulent flow, but only when k