Jangam, Mujumdar - Basic Concepts and Definitionty is regarded as a lumped property that does not really distinguish between the trans-
port of water by liquid or vapor diffusion, capillary or hydrodynamic flow due to pres-
sure gradients set up in the material during drying. Further, the diffusivity values will
show marked variations if the material undergoes glass transition during the drying
process. Keey (1978) has provided analytical expressions for liquid diffusion and capil-
larity models of falling rate drying. Table 1.7 gives solution of the one-dimensional
transient partial differential equations for cartesian, cylindrical and spherical coordinate
systems. These results can be utilized to estimate the diffusivity from the falling rate
drying data or to estimate the drying rate and drying time if the diffusivity value is
known.
It is noteworthy that the diffusivity, DL, is a strong function of Xf as well as tempera-
ture and must be determined experimentally. Thus, the liquid diffusion model should be
regarded purely as an empirical representation drying in the falling rate period. More
advanced models are, of course, available but their widespread use in the design of
dryers is hampered by the need for extensive empirical information required to solve
the governing equations. Turner and Mujumdar (1997) provide a wide assortment of
mathematical models of drying and dryers, and also discuss the application of various
techniques for the numerical solution of the complex governing equations.
One simple approach for interpolating a given falling rate curve over a relatively
narrow range of operating conditions is that first proposed by van Meel (1958). It is
found that the plot of normalized drying rate ν = N/Nc versus normalized free moisture
content η = (X - X)/(Xc - X) was nearly independent of the drying conditions. This plot,
called the characteristic drying rate curve, is illustrated in Figure 1.8. Thus, if the con-
stant rate-drying rate, Nc, can be estimated and the equilibrium moisture content data
are available, then the falling rate curve can be estimated using this highly simplified
approach. Extrapolation over wide ranges is not recommended, however.
Table 1.7. Solution to Fick's second law for some simple geometries
(Pakowski and Mujumdar, 1995)Geometry Boundary conditions Dimensionless average free M.C.Flat plate of thickness 2b (^) t= − < < 0 ; b z b X; =X 0
t> 0 ;z= ±b X; =X*
X
n
n
b
Dt
b
L
n=
−
− −
=∞∑
8
1
2 1
2 1
4
2221π
π
( )
exp ( )
Infinitely long cylinder of
radius R
t=0 0; < <r R;X=X 0
t> 0 ;r=R;X=X*
X
R
D t
nL n
n= −
=∞(^4) ∑
1
2 221 α
exp( α )
where n are positive roots of the
equation J 0 (Rn) = 0Sphere of radius R (^) t=0 0; < <r R;X=X 0
t>0;r=R;X=X*
X
n
n
R
Dt
R
L
n=
−
=∞∑
6 1
2 22 2π 1
π
exp^
Waananen et al. (1993) have provided an extensive bibliography of over 200 refer-
ences dealing with models for drying of porous solids. Such models are useful to de-