Gel/space ratio and strength (550 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 = 550 g; $w/c$ = 0.45, so water = $0.45\times550 = 247.5$ g.

Full hydration ($\alpha = 1$)

Volume of cement $= 550\times0.319 = 175.5$ cm$^3$. Gel volume $= 550\times0.657 = 361.4$ cm$^3$. Space available $= 175.5 + 247.5 = 423.0$ cm$^3$.

$$\begin{aligned} x &= \frac{0.657\times1}{0.319\times1+0.45} = \frac{0.657}{0.769} = 0.854 \\ f_c &= 240\times(0.854)^3 = 149.7\ \text{MPa} \end{aligned}$$

60% hydration ($\alpha = 0.6$)

Gel $= 361.4\times0.6 = 216.8$ cm$^3$; space $= 175.5\times0.6 + 247.5 = 352.8$ cm$^3$.

$$\begin{aligned} x &= \frac{0.657\times0.6}{0.319\times0.6+0.45} = \frac{0.3942}{0.6414} = 0.615 \\ f_c &= 240\times(0.615)^3 = 55.7\ \text{MPa} \end{aligned}$$

Answer: gel/space ratio 0.854 and strength 149.7 MPa at full hydration; gel/space ratio 0.615 and strength 55.7 MPa at 60% hydration. (Powers' theoretical strengths are for the paste; actual concrete strength is lower.)