Page 119 - Separation process principles 2

84 Chapter 3 Mass Transfer and Diffusion

The structure of wood, which often consists of (1) highly Typical results are given by Sherwood [25] and Stamm [24].
elongated hexagonal or rectangular cells, called tracheids in For example, for beech with a swollen specific gravity of
softwood (coniferous species, e.g., spruce, pine, and fir) and 0.4, the diffusivity increases from a value of about
fibers in hardwood (deciduous or broad-leaf species, e.g., 1 x cm2/s at 10°C to 10 x lop6 cm2/s at 60°C.
oak, birch, and walnut); (2) radial arrays of rectangular-like
cells, called rays, which are narrow and short in softwoods
3.3 ONE-DIMENSIONAL, STEADY-STATE
but wide and long in hardwoods; and (3) enlarged cells with
AND UNSTEADY-STATE, MOLECULAR
large pore spaces and thin walls, called sap channels because
they conduct fluids up the tree. The sap channels are less DIFFUSION THROUGH STATIONARY MEDIA
than 3 vol% of softwood, but as much as 55 vol% of For conductive heat transfer in stationary media, Fourier's
hardwood. law is applied to derive equations for the rate of heat transfer
Because the structure of wood is directional, many of its for steady-state and unsteady-state conditions in shapes such
properties are anisotropic. For example, stiffness and as slabs, cylinders, and spheres. Many of the results are plot-
strength are 2 to 20 times greater in the axial direction of the ted in generalized charts. Analogous equations can be de-
tracheids or fibers than in the radial and tangential directions rived for mass transfer, using simplifying assumptions.
of the trunk from which the wood is cut. This anisotropy ex- In one dimension, the molar rate of mass transfer of A in
tends to permeability and diffusivity of wood penetrants, a binary mixture with B is given by a modification of (3-12),
such as moisture and preservatives. According to Stamm which includes bulk flow and diffusion:
[24], the permeability of wood to liquids in the axial direc-
tion can be up to 10 times greater than in the transverse
direction.
Movement of liquids and gases through wood and wood If A is a dissolved solute undergoing mass transfer, but B is
products takes time during drying and treatment with preser- stationary, ng = 0. It is common to assume that c is a constant
vatives, fire retardants, and other chemicals. This movement and XA is small. The bulk-flow term is then eliminated and
takes place by capillarity, pressure permeability, and (3-54) accounts for diffusion only, becoming Fick's first law:
diffusion. Nevertheless, wood is not highly permeable be-
cause the cell voids are largely discrete and lack direct
interconnections. Instead, communication among cells is
through circular openings spanned by thin membranes with Alternatively, (3-55) can be written in terms of concentration
submicrometer-sized pores, called pits, and to a smaller ex- gradient:
tent, across the cell walls. Rays give wood some permeabil-
ity in the radial direction. Sap channels do not contribute to
permeability. All three mechanisms of movement of gases
and liquids in wood are considered by Stamm [24]. Only dif- This equation is analogous to Fourier's law for the rate of
fusion is discussed here. heat conduction, Q:
The simplest form of diffusion is that of a water-soluble
solute through wood saturated with water, such that no di-
mensional changes occur. For the diffusion of urea, glycer-
ine, and lactic acid into hardwood, Stamm [24] lists diffu- Steady State
sivities in the axial direction that are about 50% of ordinary
liquid diffusivities. In the radial direction, diffusivities are For steady-state, one-dimensional diffusion, with constant
about 10% of the values in the axial direction. For example, DAB, (3-56) can be integrated for various geometries, the
at 26.7"C the diffusivity of zinc sulfate in water is most common results being analogous to heat conduction.
5 x lop6 cm2/s. If loblolly pine sapwood is impregnated 1. Plane wall with a thickness, 22 - zl:
with zinc sulfate in the radial direction, the diffusivity is
found to be 0.18 x cm2/s [24].
The diffusion of water in wood is more complex. Mois-
ture content determines the degree of swelling or shrinkage.
Water is held in the wood in different ways: It may be 2. Hollow cylinder of inner radius rl and outer radius r2,
physically adsorbed on cell walls in monomolecular layers, with diffusion in the radial direction outward:
condensed in preexisting or transient cell capillaries, or
absorbed in cell walls to form a solid solution.
Because of the practical importance of lumber drying
rates, most diffusion coefficients are measured under drying
conditions in the radial direction across the fibers. Results
depend on temperature and swollen-volume specific gravity.