Page 416 - Modelling in Transport Phenomena A Conceptual Approach

396 CHAPTER 9. STEADY MICROSCOPIC BALANCES WTH GEN.

9.5.1.3 Sherwood number for a fully developed velocity profile

For water flowing in a circular pipe of diameter D at a Reynolds number of 100
and at a temperature of 20°C, Skelland (1974) calculated the length of the tube,
L, required for the velocity, temperature and concentration distributions to reach
a fully developed profile as

fully developed velocity profile
L= 350 fully developed temperature profile (9.572)
("" D fully developed concentration profile
6000
Therefore, a fully developed concentration profile is generally not attained for fluids
with high Schmidt number and the use of Eqs. (9.554) and (9.571) may lead to
erroneous results.
When the velocity profile is fully developed, it is recommended to use the fol-
lowing semi-empirical correlations suggested by Hausen (1943):

Sh = 3.66 + 0.668 [(DIL) ReSc] CAW = constant (9.573)
1 + 0.04 [(D/L) R~SC]~'~

Sh = 4.36 + 0.023 [(D/L) Re Sc] NA, = constant (9.574)
1 f 0.0012 [(DIL) Re Sc]
In the calculation of the mass transfer rates by the use of Eqs. (9.5-73) and (9.5-74),
the appropriate driving force is the log-mean concentration difference.

Example 9.12 Pure water at 25°C flows through a smooth metal pipe of 6cm
internal diameter with an average velocity of 1.5 x m/ s. Once the filly de-
veloped velocity profile is established, the metal pipe is replaced by a pipe, cast from
benzoic acid, of the same inside diameter. If the length of the pipe made of a ben-
zoic acid is 2m, calculate the concentration of benzoic acid in water at the exit of
the pipe.

Solution
Physical properties

From Example 4.8:
p = 1000 kg/ m3
For water (B) at 25 "C (298 K) : p = 892 x kg/ m. s
{ DAB = 1.21 x m2/s
sc = 737
Saturntion solubility of benzoic acid (A) in water = 3.412 kg/ m3