To fit the model results to those measured experimentally, a conversion between cycles and time is necessary. The simplest conversion would be to use a linear proportionality ( time=B*cycles). However, it has been previously pointed out to the author that the NIST cement hydration model generates kinetics which closely follow Knudsen's linear dispersion model, but not the parabolic one [41]. With this in mind, an alternative relationship between time and cycles was investigated, mainly
| time(h)= B*cycles2. | (8) |
In this way, the linear kinetics obeyed by the model can be adapted to the parabolic kinetics exhibited by the real cements.
To calibrate the model to the experimental results based on the non-evaporable water content data, the model results for degree of hydration were regressed in Equation 7 using the earlier deduced parameters for Au and k, and a subset of the model degree of hydration data. This subset was generated by selecting single data points at approximately 0.05 degree of hydration intervals for values of degree of hydration between 0.10 and the amount of hydration achieved experimentally at 90 days. In this way, the regression being applied to the model is being weighted in approximately the same manner as that which was applied to the experimental results. This step was deemed necessary due to the fact that the model degree of hydration values are not evenly distributed with number of cycles (i.e., more hydration occurs during the early cycles than during the later ones). The previously determined value of Au was converted to degree of hydration, via normalization by the value for the non-evaporable water content at complete hydration (0.226 or 0.235). The induction time was not included in the regression, as the NIST microstructure model makes no effort to model the induction period, but only the subsequent stages of cement hydration. Thus, this induction time will be included directly in the final equation for conversion between cycles and time, which will take the form:
The determined coefficients for B as a function of cement and w/c ratio are summarized in Table 11. Interestingly, the values for B are relatively constant, suggesting that a constant value of B (such as the average B value of 0.00172) may serve to model all of the results for the two cements and three w/c ratios. From the variability in results in Table 10, an average value of t0 (namely 6.657 hours) may also suffice for these two particular cements. This value is slightly larger than the final times of set measured for the two cements using the Vicat and Gillmore needle techniques which are both on the order of 5 hours [5].
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After using the non-evaporable water contents in a preliminary calibration of the model kinetics, the model chemical shrinkage results were used to determine an appropriate density for the C-S-H gel. According to values in the literature [25], no chemical shrinkage would occur during the hydration of C3S, as the volume of the products is equivalent to that of the reactants. Since the use of the literature density value led to poor predictions of chemical shrinkage for Cements 115 and 116, the molar volume of C-S-H was reduced from 120 to 108 c,3 /mole. For both C3S and C2S hydration, this results in chemical shrinkages of about 6.7 (g H2O/g cement) at complete hydration. For C3S hydration, Powers [33] has directly measured a chemical shrinkage of about 5.3 (g H2O/g C3S) after 28 days hydration, in reasonable agreement with the value being used here. For C2S, Powers measured a value of about 1.2 (g H2O/g C2S) [33], but the C2S would be hydrating at a slower rate than the C3S.
Once a value(s) of B has been determined and the density of C-S-H specified, plots comparing model and experimental results can be generated. Appendix B provides plots of the results for each of the two cements for each of the three w/c ratios for hydration rate, heat release (for w/c= 0.4 and 0.45), and chemical shrinkage. In these figures, the solid lines indicate the model data obtained using the specific values of t0 and B for each w/c ratio and cement as given in Tables 10 and 11, while the dotted lines indicate the results that would be obtained using single average values for these parameters regardless of w/c ratio and cement ID. As can be seen, the agreement between the solid lines and the experimental data is in general excellent. For the dotted lines, the agreement is similar, suggesting that for these two cements, a single relationship can be used to convert model cycles into real time. This suggests that, by capturing the particle size distribution and phase distributions of the cements, much of the hydration kinetics behavior is implicitly included in the hydration model. Knudsen [36] has previously stated that "findings by us have proven the particle size distribution to be a dominant factor in the correct modelling of cement hydration." Pommersheim [42] has also found the particle size distribution to "critically affect the kinetics" of hydration. In the following section, an attempt will be made to indicate that, in addition to particle size characterization, phase distribution quantification is important for the modelling of cement hydration.