Correct Answer - Option 3 : 553
Concept:
The average velocity is given by
\({{\rm{V}}_{{\rm{avg}}}} = \frac{1}{{\rm{t}}}\mathop \smallint \limits_0^{\rm{t}} {\rm{udt}}\)
Let, At = Total area , Ab = Area of each bag
\({{\rm{A}}_{\rm{t}}} = \frac{{{{\rm{Q}}_{\rm{g}}}}}{{{{\rm{V}}_{{\rm{avg}}}}}}\)
Ab = πdl
Where, d = diameter of the bag, l = length of the bag
N = Number of bags = \({\rm{N}} = \frac{{{{\rm{A}}_{\rm{t}}}}}{{{{\rm{A}}_{\rm{b}}}}}\)
Calculation:
Given:
The ratio of flow rate air to cloth is (u) = \(\frac{1}{{0.267 + 0.08{\rm{t}}}}\)(m3/m2 min of cloth)
Time, t = 30 min, diameter of bag, d = 0.2, Length of bag, l = 3 m
Flow rate, Qg = 1000 m3/min.
Put the values in the equation, we get the average velocity
\({{\rm{V}}_{{\rm{avg}}}} = \frac{1}{{\rm{t}}}\mathop \smallint \limits_0^{\rm{t}} {\rm{udt}} = \frac{1}{{\rm{t}}}\mathop \smallint \limits_0^{\rm{t}} \frac{{{\rm{dt}}}}{{0.267 + 0.08{\rm{t}}}}\)
\({{\rm{V}}_{{\rm{avg}}}} = \frac{1}{{30}}\mathop \smallint \limits_0^{30} \frac{{{\rm{dt}}}}{{0.267 + 0.08{\rm{t}}}} = \frac{{28.78}}{{30}} = 0.959\;{\rm{m}}/{\rm{min}}\)
\({{\rm{A}}_{\rm{t}}} = \frac{{1000}}{{0.959}} = 1042.390\;{{\rm{m}}^2}\)
Ab = π × 0.2 × 3 = 1.8849 m2
\({\rm{N}} = \frac{{1042.390}}{{1.884}} = 553.005\) ≈ 553