A cylindrical storage bin with an internal diameter of 4 m and a height of 16 m is completely filled with paddy having bulk density of 640 kg·m⁻³. The angle of internal friction between grain and bin wall is 30° and the ratio of horizontal to vertical pressures is 0.4. When the grain filling rises from 4 m to 16 m in height, the lateral pressure increases by a multiple of ______. [round off to 2 decimal places]
Official Answer Key:
1.60 to 1.70
Step-by-Step Solution & Explanation
By Janssen's equation for a cylindrical bin, vertical pressure Pv(h) = (γR/μk)[1 − exp(−μkh/R)], and lateral pressure Ph = k·Pv, where μ = tan(wall friction angle) = tan(30°) = 0.5774, k = 0.4, and R = hydraulic radius = D/4 = 1.0 m for a circular bin. Exponent coefficient = μk/R = 0.5774×0.4/1.0 = 0.2310/m. At h=16 m: [1−exp(−0.2310×16)] = 1−exp(−3.695) = 1−0.0248 = 0.9752. At h=4 m: [1−exp(−0.2310×4)] = 1−exp(−0.924) = 1−0.3971 = 0.6029. Ratio = 0.9752/0.6029 = 1.62, matching the key.
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