Primary Drying: The Sublimation of Ice - Freeze-drying of Pharmaceuticals and Biopharmaceuticals - page 100

simple kinetic theory of gases. Thus, for sublimation from a free ice
crystal surface:

dw

dt ¼ K v

fp
ð 2pRT s Þ 1 = 2

DH ð T c T s Þ¼

ð 1 Þ

where
DH is the latent heat of ice sublimation;
w is the mass of ice sublimed in time t;
T c is the temperature of the condenser;
T s is the temperature at the subliming ice front;
K v is the heat transfer coefficient;
p is the SVP of ice at T s ;
f is the ''drying factor'' to account for the probability of removal of
water molecules before they can recondense; 0 r f r 1.
For sublimation of ice, grown in an aqueous solution, Equation (1)
may need to be substantially modified; for details see below.
In practice, the magnitudes of the various mechanisms to the total
heat transfer, and thus also K v , depend on the quality of the contact
between solid phases, e.g. vial bottom and shelf or plate surface, and the
chamber pressure, which is itself a function of the total number of gas
molecules and the temperature. Figure 2 shows these relative magnitu-
des for three types of glass vials, standing on a polished stainless steel
shelf, at two pressures, 14 and 52 Pa, respectively. The cross-hatched
area represents heat transfer by conduction through and between solid
phases (e.g. shelf, glass vial and the frozen plug), the shaded area
represents the contribution from radiation, and the clear area is due
to gas conduction (convection). An inspection of the two figures shows
that in every case the major contribution to heat transfer is due to

Figure 2 Contributions of radiation (left), conduction (centre) and convection (right) to
the total heat transfer for three vial types at two pressures. Redrawn with
changes from Pikal