Question

You just bought a house and the bathroom has a showerhead that is 2m high. There is an issue though where it is not spewing out enough water. In attempt to fix this, the pressure valve on the water system is tweaked. This is located on a pipe at ground level. these pipes have a cross sectional area of 4cm^2, while the slits on the showerheads have a total area of 1 cm^2. What pressure should be set on the valve in order to enable a water flow of 8L per minute or 2.1 gallons a minute from the showerhead?

Answer 1

from equation of continuity

volume flow rate remains same in a fluid flow

at the shower head

V = A1*v1

A1 = 1 cm^2 = 10^-4 m^2

speed at shower head v1 = V/A1 = 1.33*10^-4/(1*10^-4) = 1.33 m/s

in the pipe at ground level

A2 = pi*r2^2

V = A2*v2

speed in the pipe v2 = V/A2 = 1.33*10^-4/(4*10^-4) = 0.33 m/s

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from Bernoullis principle

P1 + (1/2)*rho*v1^2 + rho*g*h1 = P2 + (1/2)*rho*v2^2 + rho*g*h2

P2 = P1 + (1/2)*rho*(v1^2 - v2^2) + rho*g*(h1 - h2)

h1 = 2

h2 = 0 m

rho = density of ethanol = 789 kg/m^3

P1 = 10^5 Pa

v1 = 1.33 m/s

v2 = 0.33 m/s

P2 = 10^5 + (1/2)*1000*(1.33^2-0.33^2) + 1000*9.8*2

P2 = 1.2*10^5 Pa <<<--------ANSWER