Gel/space ratio and strength (600 gm cement)

Powers' model gives the strength of concrete in terms of the gel/space ratio $x$:

$$x = \frac{\text{volume of gel}}{\text{volume of gel} + \text{capillary pores}} = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

where $\alpha$ is the degree of hydration and $w/c$ the water-cement ratio by mass. The numbers come from cement of specific gravity about 3.14: 1 g of cement has 0.319 cm$^3$ volume and produces gel of $2.06\times0.319 = 0.657$ cm$^3$ (including gel pores). Compressive strength:

$$f_c = 240\,x^3\ \text{MPa}$$

(240 MPa is the intrinsic strength of the gel; it varies from about 234 to 255 MPa for different cements.) The ratio $x$ does not depend on the mass of cement; the mass is only needed to give volumes.

Given

Cement = 600 g, $w/c = 0.45$, so water = $0.45\times600 = 270$ g. Volume of cement $= 600\times0.319 = 191.4$ cm$^3$; gel at full hydration $= 600\times0.657 = 394.2$ cm$^3$.

Full hydration ($\alpha = 1$)

$$\begin{aligned} x &= \frac{394.2}{191.4+270} = \frac{394.2}{461.4} = 0.854 \\ f_c &= 240\times0.854^3 = 149.7\ \text{MPa} \end{aligned}$$

60% hydration ($\alpha = 0.6$)

$$\begin{aligned} x &= \frac{0.6\times394.2}{0.6\times191.4+270} = \frac{236.5}{384.8} = 0.615 \\ f_c &= 240\times0.615^3 = 55.7\ \text{MPa} \end{aligned}$$

Answer: gel/space ratio 0.854 and strength 149.7 MPa at full hydration; 0.615 and 55.7 MPa at 60% hydration. The result is independent of the 600 g mass because only the ratio $w/c$ matters.