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varies with time as a result of variations in consumption, and in generation.
At a given supersaturation, the consumption rate increases with time since
the total crystal surface area increases with time. The total crystal surface
area increases partly because each crystal increases in size due to growth
and partly because new crystals are generated (nucleation). If the rate of
consumption is low in comparison to the rate of generation, the supersatu-
ration increases, resulting in increasing growth and nucleation rates. This
in turn leads to an increase in total crystal surface area, which together with
the increased linear growth rate causes the consumption of supersaturation
to increase. A mass balance for the substance in solution gives
dc
dt
D
dc
dt
*
=-
+ -
WW
,
(7)
RG
where
∆
c
[
kg/kg inert
]
is the supersaturation driving force,
t
is time
[
s
]
,
represents the
change in concentration of the crystallizing compound due to processes
like evaporation of the solvent or chemical reaction, and
W
G
[
and
c
* is the solubility
[
kg/kg inert
]
.
W
R
[
kg/kg inert, s
]
]
denotes the decrease in concentration because of crystal growth, as can be
described by
kg/kg inert, s
k
k
v
W
= 3
r
AG
,
(8)
G
cT
a
where
G
is the linear crystal growth rate
[
m/s
]
,
A
T
is the total surface area
of crystals in the suspension
[
m
2
/kg inert
]
, and
ρ
c
is the density of the crys-
tals
.
k
v
— and
k
a
— are the volume and area shape factors, respec-
tively, of the crystals. Normally, the consumption of supersaturation
caused by the actual nucleation can be neglected. For a normal batch cool-
ing crystallization,
W
R
is zero, and
[
kg/m
3
]
dc
dt
*
dc
dT
*
dT
dt
(9)
=
,
where
T
is the temperature
. In
batch cooling crystallization
practice,
the cooling is often achieved by pumping coolant water of constant tem-
perature through the cooling jacket:
natural cooling
. The driving force for
[
K
]
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