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we obtain:
64
V.R. DEJEU, R. BARABÁS, AL. POP, E.S. BOGYA, P.-Ş. AGACHI
V
T
V
=
1−v []m
[]n
dv[]n dNp dvp
= (1−v []n) ⋅v p ⋅ + (1−v []n) ⋅Np⋅ dτ V ⋅dτ V ⋅d τ
(9)
[]m []m
Equation (9) represents the nuclei forming and growing combined
macrokinetic model, where the first right term expresses the rate of the
nuclei formation and the second one the growing rate.
When the process is developing after this macrokinetic model and
because the reaction rate is higher than the germs formation rate, solutions
with high supersaturation will be formed.
MATHEMATICAL DESCRIPTION OF THE PROCESS
For the mathematical description of the process according to the
macrokinetic model presented in equation (9) we must refer to the geometrical
shape of the macro particles and to the τ signification ( τ - the time to reach
the final value of the volume fraction v []n in case of the new phase).
Considering that the macroparticles are spherical, equation (9) can
be written as follows [9]:
dv[]n 4 dNp N
3 p 2 dr
= πr ⋅ + ⋅4πr ⋅
(1−v ) ⋅dτ 3 V ⋅dτ V d τ
(10)
[]n []n []m
dNp
where: w3
=
V ⋅d τ
and = dr
w2
d τ
.
[]n
Because the process is developing at constant supersaturating, it is
proposed that not only the rate of the germs formation but also the rate of
transformation-growing are constant. Based on this hypothesis results that:
and equation (10) becomes:
r
Np = w3 ⋅V []n ⋅τ;
( w 2 = ) (11)
τ
dv 16
= π⋅w2⋅w3 ⋅τ
(1−v ) ⋅dτ 3
[]n
[]n 3 3
(8)
(12)