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MATHEMATICAL MODELING FOR THE CRYSTALLIZATION PROCESS OF HYDROXYAPATITE …
Table 1. The experimental results of the transformation grade of beta-whitlockite
in time and at 20 0 C and 50 0 C for different pH values
pH
Time
[s]
η
0 3600 7200 10800 14400 18000 21600 25200 28800 32400
8.5
20ºC
50ºC
0
0
0.003
0.005
0.009
0.025
0.021
0.052
0.042
0.090
0.063
0.142
0.092
0.190
0.120
0.250
0.158
0.328
0.191
0.390
9.1
20ºC
50ºC
0
0
0.055
0.092
0.215
0.330
0.415
0.590
0.620
0.800
0.775
0.910
0.880
0.971
0.945
0.990
0.981
1.000
0.991
-
9.7
20ºC
50ºC
0
0
0.068
0.102
0.240
0.340
0.470
0.610
0.670
0.815
0.820
0.920
0.920
0.977
0.962
0.990
0.980
1.000
1.000
-
10.2
20ºC
50ºC
0
0
0.101
0.136
0.350
0.450
0.620
0.740
0.820
0.910
0.940
0.978
0.981
0.996
0.996
1.000
1.000
-
-
-
11.3
20ºC
50ºC
0
0
0.110
0.157
0.370
0.490
0.650
0.790
0.840
0.931
0.950
0.982
0.986
0.997
0.996
1.000
1.000
-
-
-
12
20ºC
50ºC
0
0
0.119
0.187
0.400
0.560
0.680
0.840
0.860
0.960
0.960
0.995
0.989
1.000
1.000
-
-
-
-
-
From the slope were determined the numerical values of the K
constants, presented in Table 2.
Tablel 2. The values of K constant
20 0 C 50 0 pH
C
K1 K2 K3 K1 K2 K3
8.5 6.583E-06 2.032E-10 1.920E-19 1.513E-05 4.671E-10 4.412E-19
9.1 1.510E-04 4.649E-09 4.392E-18 2.480E-04 7.655E-09 7.232E-18
9.7 1.740E-04 5.372E-09 5.075E-18 2.620E-04 8.089E-09 7.641E-18
10.2 2.410E-04 8.356E-09 9.915E-18 3.340E-04 1.158E-08 1.374E-17
11.3 2.600E-04 9.036E-09 1.072E-17 3.830E-04 1.329E-08 1.577E-17
12 2.450E-04 9.728E-09 1.494E-17 3.980E-04 1.579E-08 2.425E-17
In order to identify the mathematical model which describes the
process the process simulation was needed. For this purpose the numerical
values of the K1, K2 and K3 from Table 2 were used and different times were
calculated relating to the transformation grade of beta-whitlockite in HAP and
compared with the experimental results. The results obtained at 20 0 C and
50 0 C for pH=9.7 are presented in Figure 3 (the simulations were made for all
pH values, but in this article are presented only the relevant simulation results).
The analysis of the results show a very good fit of the experimental
results with those calculated for equation (16), where the errors for the majority
of the results don’t exceed 2%. In the mathematical models expressed by
equations (14) and (18) the deviations from the experimental values are high.
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