A cylindrical grain bin of diameter $4\text{ m}$ is filled with wheat (bulk density $\rho = 800\text{ kg/m}^3$). The coefficient of friction between wheat and bin wall is $\mu' = 0.4$ and the ratio of lateral to vertical pressure is $k = 0.5$. Using Janssen's equation, the hydraulic radius $R = \frac{\text{Area}}{\text{Perimeter}} = \frac{D}{4} = 1.0\text{ m}$. At infinite depth ($h \to \infty$), the maximum theoretical lateral pressure on the bin wall is ________ $\text{kPa}$ (round off to two decimal places).
Answer Key:
9.81
Step-by-Step Solution & Explanation
According to Janssen's theory for deep bins, vertical pressure as $h \to \infty$ is:
$L_{\text{vertical, max}} = \frac{\rho g R}{\mu'}$
where:
- $\rho = 800\text{ kg/m}^3$
- $g = 9.81\text{ m/s}^2$
- $R = \frac{D}{4} = 1.0\text{ m}$
- $\mu' = 0.4$
- $k = 0.5$
$L_{\text{vertical, max}} = \frac{800 \times 9.81 \times 1.0}{0.4} = \frac{7848}{0.4} = 19620\text{ Pa} = 19.62\text{ kPa}$
Lateral pressure at $h \to \infty$:
$P_{\text{lateral, max}} = k \times L_{\text{vertical, max}} = 0.5 \times 19.62\text{ kPa} = 9.81\text{ kPa}$
Wait, let's verify: $k \times \frac{\rho g R}{\mu'} = 0.5 \times 19.62 = 9.81\text{ kPa}$.