Subtopic: Controlled and modified atmosphere storage
Question Statement:
A circular grain silo with conical bottom (cylindrical top of diameter 4 m, conical section of height 2 m tapering to a 1 m diameter grain outlet), as shown in the figure, is filled with wheat (true density 1200 kg m⁻³) with porosity of 0.6. Five hundred metric tonne of wheat fills 80% of its capacity (by volume). The total height (h) of the silo from its grain outlet end in m is ______
Reconstructed figure (not to exact scale):
Official Answer Key:
104 to 106
Step-by-Step Solution & Explanation
Bulk density of wheat = true density×(1−porosity) = 1200×(1−0.6) = 480 kg/m³. Volume occupied by 500,000 kg of wheat = 500,000/480 = 1041.67 m³. This is 80% of total silo capacity, so total capacity V_total = 1041.67/0.80 = 1302.08 m³. Volume of the conical frustum bottom (top dia 4 m, bottom dia 1 m, height 2 m): V_cone = (πh/3)×(r1²+r1r2+r2²) = (π×2/3)×(2²+2×0.5+0.5²) = (π×2/3)×5.25 = 11.0 m³. Volume of cylindrical portion = 1302.08−11.0 = 1291.1 m³. Cylinder height = V_cyl/(π/4×4²) = 1291.1/12.566 = 102.74 m. Total silo height (outlet to top) = h_cyl + h_cone = 102.74+2.0 = 104.7 m, within the accepted band 104–106.
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