Page 102 - Separation process principles 2

3.1 Steady-State, Ordinary Molecular Diffusion 67

plane or stationary coordinate system. When a net flux the direction selected as positive. When the molecular and
occurs, it carries all species present. Thus, the molar flux of eddy-diffusion fluxes are in one direction and N is in the
an individual species is the sum of all three mechanisms. If opposite direction, even though a concentration difference
Ni is the molar flux of species i with mole fraction xi, and N or gradient of i exists, the net mass-transfer flux, Ni, of i can
is the total molar flux, with both fluxes in moles per unit time be zero.
per unit area in a direction perpendicular to a stationary In this chapter, the subject of mass transfer and diffusion
plane across which mass transfer occurs, then is divided into seven areas: (1) steady-state diffusion in
stagnant media, (2) estimation of diffusion coefficients,
Ni = xi N + molecular diffusion flux of i
(3) unsteady-state diffusion in stagnant media, (4) mass
+ eddy diffusion flux of i (3-1)
transfer in laminar flow, (5) mass transfer in turbulent flow,
where xiN is the bulk-flow flux. Each term in (3-1) is positive (6) mass transfer at fluid-fluid interfaces, and (7) mass
or negative depending on the direction of the flux relative to transfer across fluid-fluid interfaces.

3.0 INSTRUCTIONAL OBJECTIVES
After completing this chapter, you should be able to:
Explain the relationship between mass transfer and phase equilibrium.
Explain why separation models for mass transfer and phase equilibrium are useful.
Discuss mechanisms of mass transfer, including the effect of bulk flow.
State, in detail, Fick's law of diffusion for a binary mixture and discuss its analogy to Fourier's law of heat
conduction in one dimension.
Modify Fick's law of diffusion to include the bulk flow effect.
Calculate mass-transfer rates and composition gradients under conditions of equimolar, countercurrent diffusion
and unimolecular diffusion.
Estimate, in the absence of data, diffusivities (diffusion coefficients) in gas and liquid mixtures, and know of some
sources of data for diffusion in solids.
Calculate multidimensional, unsteady-state, molecular diffusion by analogy to heat conduction.
Calculate rates of mass transfer by molecular diffusion in laminar flow for three common cases: (1) falling liquid
film, (2) boundary-layer flow past a flat plate, and (3) fully developed flow in a straight, circular tube.
Define a mass-transfer coefficient and explain its analogy to the heat-transfer coefficient and its usefulness, as an
alternative to Fick's law, in solving mass-transfer problems.
Understand the common dimensionless groups (Reynolds, Sherwood, Schmidt, and Peclet number for mass
transfer) used in correlations of mass-transfer coefficients.
Use analogies, particularly that of Chilton and Colburn, and more theoretically based equations, such as those of
Churchill et al., to calculate rates of mass transfer in turbulent flow.
Calculate rates of mass transfer across fluid-fluid interfaces using the two-film theory and the penetration
theory.

3.1 STEADY-STATE, ORDINA
direction and then in another, with no one direction pre-
MOLECULAR DIFFUSION
ferred. This type of motion is sometimes referred to as a
Suppose a cylindrical glass vessel is partly filled with water random-walk process, which yields a mean-square distance
containing a soluble red dye. Clear water is carefully added of travel for a given interval of lime, but not a direction of
on top so that the dyed solution on the bottom is undisturbed. travel. Thus, at a given horizontal plane through the solution
At first, a sharp boundary exists between the two layers, but in the cylinder, it is not possible to determine whether, in a
after a time the upper layer becomes colored, while the layer given time interval, a given molecule will cross the plane or
below becomes less colored. The upper layer is more col- not. However, on the average, a fraction of all molecules in
ored near the original interface between the two layers and the solution below the plane will cross over into the region
less colored in the region near the top of the upper layer. above and the same fraction will cross over in the opposite
During this color change, the motion of each dye molecule is direction. Therefore, if the concentration of dye molecules in
random, undergoing collisions mainly with water molecules the lower region is greater than in the upper region, a net rate
and sometimes with other dye molecules, moving first in one of mass transfer of dye molecules will take place from the