To better illustrate how to use the analytical parts of the multi-scale model, an example calculation will be done that makes use of the formulas in the previous section. All data will be taken from recent work [60]. We will take a simple mortar, with an aggregate volume fraction of 50%. The sieve analysis for the aggregates is given in Table 1.
| Diameter Range (µm) | Fraction of Total Volume (ci ) |
| 1180-600 | 0.0228 |
| 600-300 | 0.749 |
| 300-150 | 0.2193 |
| 150-75 | 0.0089 |
The overall w/c ratio is 0.4. The cement particle size distribution is such that the median diameter is 12 µm, so that the ITZ is taken to have the same thickness [60]. Table 2 shows the model cement particle size distribution, taken from the real cement PSD [60].
| Diameter (µm) | Fraction of Cement Volume |
| 3 | 0.19 |
| 5 | 0.082 |
| 7 | 0.088 |
| 9 | 0.065 |
| 15 | 0.182 |
| 25 | 0.193 |
| 35 | 0.110 |
| 45 | 0.09 |
Using the sieve analysis in Table 1, and the equations given in the previous sections, one can easily calculate the value of VITZ to be 0.0945 (using Assumption 1 and A = 0). A numerical point counting method applied to the same problem gave VITZ = 0.0918, so the analytical result differs from the numerical result by only 3%. Given that the aggregate volume fraction is 0.5, that implies that the ratio of ITZ to bulk cement paste is VITZ/V bulk = 0.233, which is similar to that for Fig. 16.
One can then construct a 220 x 220 x 130 pixel unit cell, with a 2 pixel wide flat aggregate. The scale is one µm per pixel length, similar to that shown in Fig. 16. There are two ITZ regions, each 12 pixels thick, and a bulk region that is 130-2-24 = 102 pixels long. The ratio of ITZ to bulk cement paste volume in this cell is then (220 · 220 · 24) / (220 · 220 · 102) = 0.235. One then uses the model cement PSD in Table 2 and builds up a w/c = 0.4 cement paste in the unit cell. The porosity ratio, and therefore the diffusivity ratio, are determined as a function of degree of hydration.