Page 132 - Separation process principles 2

3.5 Mass Transfer in Turbulent Flow 97

EXAMPLE 3.15 3.5 MASS TRANSFER IN TURBULENT FLOW
Linton and Sherwood [49] conducted experiments on the dissolu- In the two previous sections, diffusion in stagnant media and
tion of cast tubes of benzoic acid (A) into water (B) flowing through in laminar flow were considered. For both cases, Fick's law
the tubes in laminar flow. They obtained good agreement with pre- can be applied to obtain rates of mass transfer. A more com-
dictions based on the Graetz and Leveque equations. Consider a
mon occurrence in engineering is turbulent flow, which is
5.23-cm-inside-diameter by 32-cm-long tube of benzoic acid, pre- accompanied by much higher transport rates, but for which
ceded by 400 cm of straight metal pipe of the same inside diameter theory is still under development and the estimation of mass-
where a fully developed velocity profile is established. Pure water
enters the system at 25'C at a velocity corresponding to a Reynolds transfer rates relies more on empirical correlations of exper-
number of 100. Based on the following property data at 25°C esti- imental data and analogies with heat and momentum trans-
mate the average concentration of benzoic acid in the water leaving fer. A summary of the dimensionless groups used in these
the cast tube before a significant increase in the inside diameter of correlations and the analogies is given in Table 3.13.
the benzoic acid tube occurs because of dissolution. As shown by the famous dye experiment of Osborne
Solubility of benzoic acid in water = 0.0034 &m3 Reynolds [50] in 1883, a fluid in laminar flow moves paral-
lel to the solid boundaries in streamline patterns. Every par-
Viscosity of water = 0.89 cP = 0.0089 g/cm-s
ticle of fluid moves with the same velocity along a stream-
Diffusivity of benzoic acid in water at infinite dilution
= 9.18 x cmqs line and there are no fluid velocity components normal to
these streamlines. For a Newtonian fluid in laminar flow, the
SOLUTION mornenturn transfer, heat transfer, and mass transfer are by
molecular transport, governed by Newton's law of viscosity,
Fourier's law of heat conduction, and Fick's law of molecu-
lar diffusion, respectively.
In tu~bulent flow, the rates of momentum, heat, and mass
from which transfer are orders of magnitude greater than for molecular
transport. This occurs because streamlines no longer exist
and particles or eddies of fluid, which are large compared to
From (3-149), the mean free path of the molecules in the fluid, mix with
each other by moving from one region to another in fluctuat-
ing motion. This eddy mixing by velocity fluctuations occurs
not only in the direction of flow but also in directions normal
to flow, with the latter being of more interest. Momentum,
heat, and mass transfer now occur by two parallel mecha-
nisms: (1) molecular motion, which is slow; and (2) turbu-
lent or eddy motion, which is rapid except near a solid sur-
From (3-150),
face, where the flow velocity accompanying turbulence
decreases to zero. Mass transfer by bulk flow may also occur
as given by (3-1).
In 1877, Boussinesq [51] modified Newton's law of vis-
cosity to account for eddy motion. Analogous expressions
Using a log-mean driving force,
were subsequently developed for turbulent-flow heat and
mass transfer. For flow in the x-direction and transport in
the z-direction normal to flow, these expressions are written
where S is the cross-sectional area for flow. Simplifying, in the following forms in the absence of bulk flow in the
z-direction:

CA~ = 0 and CA, = 0.0034 g/cm3

and

= 0.01 11 where the double subscript, zx, on the shear stress, 7, stands
0.0034
FA, = 0.0034 - - 0.000038 g/cm3 for x-momentum in the z-direction. The molecular contribu-
=
,0.0111
tions, p, k, and DAB, are molecular properties of the fluid and
Thus, the concentration of benzoic acid in the water leaving the cast depend on chemical composi~on, and pressure;
tube is far from saturation.
the turbulent contributions, p,, kt, and D,, depend on the