Solution:
Given that :
Volume flow is, [tex]$Q_1 = 1000 \ mm^3/s$[/tex]
So, [tex]$Q_2= \frac{1000}{100}=10 \ mm^3/s$[/tex]
Therefore, the equation of a single straight vessel is given by
[tex]$F_{f_1}=\frac{8flQ_1^2}{\pi^2gd_1^5}$[/tex] ......................(i)
So there are 100 similar parallel pipes of the same cross section. Therefore, the equation for the area is
[tex]$\frac{\pi d_1^2}{4}=1000 \times\frac{\pi d_2^2}{4} $[/tex]
or [tex]$d_1=10 \ d_2$[/tex]
Now for parallel pipes
[tex]$H_{f_2}= (H_{f_2})_1= (H_{f_2})_2= .... = = (H_{f_2})_{10}=\frac{8flQ_2^2}{\pi^2 gd_2^5}$[/tex] ...........(ii)
Solving the equations (i) and (ii),
[tex]$\frac{H_{f_1}}{H_{f_2}}=\frac{\frac{8flQ_1^2}{\pi^2 gd_1^5}}{\frac{8flQ_2^2}{\pi^2 gd_2^5}}$[/tex]
[tex]$=\frac{Q_1^2}{Q_2^2}\times \frac{d_2^5}{d_1^5}$[/tex]
[tex]$=\frac{(1000)^2}{(10)^2}\times \frac{d_2^5}{(10d_2)^5}$[/tex]
[tex]$=\frac{10^6}{10^7}$[/tex]
Therefore,
[tex]$\frac{H_{f_1}}{H_{f_2}}=\frac{1}{10}$[/tex]
or [tex]$H_{f_2}=10 \ H_{f_1}$[/tex]
Thus the answer is option A). 10
