Question Bank • QB_APE_EXP_048 NAT • 2 Marks

Section 6: Agricultural Process Engineering

Storage Systems

Subtopic: Godowns, bins and grain silos

Question Statement:

In a cylindrical grain silo of diameter $D = 4.0\text{ m}$, grain having bulk density $\rho = 800\text{ kg/m}^3$ is stored. The hydraulic radius $R = D / 4 = 1.0\text{ m}$. The ratio of horizontal to vertical grain pressure is $k = 0.40$ and the coefficient of wall friction is $\mu' = 0.40$. Acceleration due to gravity $g = 9.81\text{ m/s}^2$. Using Janssen's equation, the maximum asymptotic vertical pressure ($L_{max} = \frac{\rho g R}{k \mu'}$) at infinite depth in $\text{kPa}$ is ________ (round off to 1 decimal place).

Answer Key:

49.1

Step-by-Step Solution & Explanation

Janssen's equation for vertical pressure at depth $y$: $V = \frac{\rho g R}{k \mu'} \left[1 - \exp\left(-\frac{k \mu' y}{R}\right)\right]$ As $y \to \infty$, the term $\exp(-\dots) \to 0$, giving asymptotic maximum vertical pressure: $V_{max} = \frac{\rho g R}{k \mu'}$ Given: • $\rho = 800\text{ kg/m}^3$ • $g = 9.81\text{ m/s}^2$ • $R = 1.0\text{ m}$ • $k = 0.40$ • $\mu' = 0.40$ $k \mu' = 0.40 \times 0.40 = 0.16$ $V_{max} = \frac{800 \times 9.81 \times 1.0}{0.16} = \frac{7848}{0.16} = 49,050\text{ Pa} = 49.05\text{ kPa} \approx 49.1\text{ kPa}$
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