Water at `20^(@)C` is flowing in a horizontal pipe that is `20.0m` long.The flow is laminar and the water completely fills the pipe. A pump maintains a gauge pressure of `1400Pa`, at a large tank at one end of the pipe. The other end of the pipe is open to the air, The viscosity of water at `20^(@)C` is `1.005` poise
(a) If the pipe has diameter `8.0 cm`, what is the volume flow rate?
(b) What gauge pressure must the pump provide to achieve the same volume flow rate for a pipe with a diameter of `4.0cm`?
(c) For pipe in part (a) and the same gauge pressure maintained by the pump, what does the volume flow rate become if the water is at a temperature of `60^(@)C` (the viscosity of water at `60^(@)C` is `0.469` poise?

Answer

A, B, C, D

Explanation

(a) `Q=(pi)/(8)((R^4)/(eta))((p_(1)-p_(2))/(L))`
`=((pi)/(8))((0.04)^(4))/(1.005xx10^(-1))xx(1400)/(0.2)`
`=7xx10^(-2)m^(3)//s`.
(b) `Q_(1)=Q_(2)`
`:. [(p_(1)-p_(2))R^(4)]_(i)=[(p_(1)-p_(2))R^(4)]_(f)`
Since, diameter or radius has decreased to half. Therefore gauge pressure should become 16 times
or, `(p_(1)-p_(2))_(f)=16xx1400`
`=2.24xx10^(4)Pa`
(c )`Q prop (1)/(eta)`
`:. (Q_1)/(Q_2)=(eta_2)/(eta_1)`
or, `Q_(2)=((eta_(1))/(eta_(2)))Q_(1)`
`=((1.005)/(0.469))(7xx10^(-2))`
`=0.15m^(3)//s`.