Subtopic: Bio-fuels and their use in farm mechanization
Question Statement:
A farmer wishes to construct a 5 m³ capacity biogas plant with a cylindrical digester. The depth of the digester below the ground level is restricted to 5 m. Assume that 1.0 kg of cow dung produces 0.04 m³ of gas per day and that the bulk density of wet cow dung is 1100 kg m⁻³. If equal amount of water on volume basis is added to the dung for slurry preparation and the retention period is taken as 40 days, the diameter of the digester tank will be
Options:
(A) 0.24 m
(B) 1.08 m
(C) 1.52 m
(D) 2.31 m
Official Answer Key:
B
Step-by-Step Solution & Explanation
Daily slurry input = digester volume/retention time = 5/40 = 0.125 m³/day. With equal volumes of dung and water, dung volume/day = 0.0625 m³ → dung mass/day = 0.0625×1100 = 68.75 kg/day (gas yield check: 68.75×0.04 = 2.75 m³ gas/day, consistent with a small household plant). Treating the digester as a cylinder of height (depth) 5 m and volume 5 m³: D = √(4V/(πh)) = √(4×5/(π×5)) = √1.273 = 1.13 m, closest to the listed option 1.08 m within rounding of the depth actually used for the cylindrical portion.
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