Question

Instead of running blood through a single straight vessel for a distance of 2 mm, one mammalian species uses an array of 100 tiny parallel pipes of the same total cross-sectional area, 4.0 mm2. Volume flow is 1000 mm3/s. The pressure drop for fluid passing through the single pipe is lower than that through the 100 vessel array by a factor of:_______.

A. 10
B. 100
C. 1000

Answer 1

Given that :

Volume flow is,
`$Q_1 = 1000 \ mm^3/s$`

So,
`$Q_2= (1000)/(100)=10 \ mm^3/s$`

Therefore, the equation of a single straight vessel is given by

`$F_(f_1)=(8flQ_1^2)/(\pi^2gd_1^5)$` ......................(i)

So there are 100 similar parallel pipes of the same cross section. Therefore, the equation for the area is

`$(\pi d_1^2)/(4)=1000 *(\pi d_2^2)/(4) $`

or
`$d_1=10 \ d_2$`

Now for parallel pipes

`$H_(f_2)= (H_(f_2))_1= (H_(f_2))_2= .... = = (H_(f_2))_(10)=(8flQ_2^2)/(\pi^2 gd_2^5)$` ...........(ii)

Solving the equations (i) and (ii),

`$(H_(f_1))/(H_(f_2))=((8flQ_1^2)/(\pi^2 gd_1^5))/((8flQ_2^2)/(\pi^2 gd_2^5))$`

`$=(Q_1^2)/(Q_2^2)* (d_2^5)/(d_1^5)$`

`$=((1000)^2)/((10)^2)* (d_2^5)/((10d_2)^5)$`

`$=(10^6)/(10^7)$`

Therefore,

`$(H_(f_1))/(H_(f_2))=(1)/(10)$`

or
`$H_(f_2)=10 \ H_(f_1)$`

Thus the answer is option A). 10