Grain Storage: Silos, Bins & Pressure Distribution
Subtopic: Janssen silo pressure asymptote
Question Statement:
According to Janssen's theory for deep grain storage silos, as the depth of grain $h$ becomes very large ($h \to \infty$), the vertical pressure $P_v$ at the bottom:
Options:
(A) Approaches a constant asymptotic maximum limit: $P_{v,\max} = \frac{\gamma R}{\mu' k}$
(B) Increases infinitely and linearly with depth ($P_v = \gamma h$)
(C) Drops to zero due to internal arching
(D) Fluctuates sinusoidally with depth
Answer Key:
A
Step-by-Step Solution & Explanation
In Janssen's equation: $P_v = \frac{\gamma R}{\mu' k} \left[1 - \exp\left(-\frac{\mu' k h}{R}\right)\right]$. As $h \to \infty$, the exponential term vanishes, and $P_v$ asymptotes to $\frac{\gamma R}{\mu' k}$, where grain weight is fully supported by wall friction.