Question Bank • QB_APE_STOR_001 NAT • 2 Marks

Section 6: Agricultural Process Engineering

Storage Systems

Subtopic: Godowns, bins and grain silos

Question Statement:

A deep cylindrical grain silo of diameter $D = 4.0\text{ m}$ (hydraulic radius $R = D/4 = 1.0\text{ m}$) stores wheat. The coefficient of friction between wheat and silo wall is $\mu' = 0.40$, and the ratio of lateral to vertical pressure is $k = 0.45$. According to Janssen's equation, with exponent parameter $m = \frac{k \mu'}{R}$, the percentage of the maximum asymptotic vertical pressure ($L_{\text{max}}$) attained at a depth of $h = 10.0\text{ m}$ is ________ % (round off to two decimal places).

Answer Key:

83.47

Step-by-Step Solution & Explanation

Multi-Approach Solution: **Method 1: Exponent Evaluation** ``` Depth h | Janssen Silo Pressure Curve | /----------------------------- (Asymptotic Lmax) | / | / | / (Non-linear exponential approach: 1 - e^{-m h}) v / ``` 1. Exponent factor $m$: $m = \frac{k \mu'}{R} = \frac{0.45 \times 0.40}{1.0\text{ m}} = 0.18\text{ m}^{-1}$ 2. At depth $h = 10.0\text{ m}$: $m h = 0.18\text{ m}^{-1} \times 10.0\text{ m} = 1.80$ $\frac{L}{L_{\text{max}}} = 1 - e^{-m h} = 1 - e^{-1.80} = 1 - 0.165299 = 0.83470 = 83.47\%$ **Method 2: Differential Slice Integration** Balancing vertical grain slice weight against wall friction: $A dL = \rho g A dh - \mu' k L P dh \implies \frac{dL}{dh} + \frac{\mu' k P}{A} L = \rho g$ Integrating with boundary condition $L(0) = 0$ yields $L(h) = L_{\text{max}}(1 - e^{-m h})$.
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