Question

A pipe is running full of water. At a certain point A it tapers from 0.6m diameter to 0.2m at B. the pressure difference between A and B is 1m of water column. What is the rate of flow of water through the pipe?

Answer 1

The flow rate is `=0.14m^3s^-1`

Explanation:

Apply Bernouilli's Equation between points `A` and `B`

`p_A+1/2rho v_A^2+rhogh_A=p_B+1/2rho v_B^2+rhogh_B`

The flow rate is constant in the pipe

`Q=v_AxxA=v_Bxx B`

where

`A=pid_A^2/4=pixx0.6^2/4`

and

`B=pid_B^2/4=pixx0.2^2/4`

`v_Axxpixx0.6^2/4=v_Bxxpixx0.2^2/4`

`v_A=v_B*(0.2/0.6)^2=0.11v_B`

`v_B=1/0.11v_A=9v_A`

But

`h_A=h_B`

Therefore,

`p_A+1/2rho v_A^2=p_B+1/2rho v_B^2`

The pressure difference is `Deltap=rhogh=1*1000*9.8=9800Pa`

`Deltap=9800=1/2rho(v_B^2-v_A^2)`

`1/2*1000*(81v_A^2-v_A^2)=9800`

`v_A=sqrt(19.6/80)=0.49ms^-1`

The flow rate is

`q=0.49*pi*0.6^2/4=0.14m^3s^-1`