Question

Water enters a house through a pipe with an inside diameter of 2.0cm at an absolute pressure 4.0×10^5 pascal.A 1.0-cm-diameter pipe leads to the second floor bathroom 5.0m above.when the flow speed at the inlet pipe is 1.5m/s.find the flow speed, pressure,and volume flow rate in the bathroom.

Answer 1

Explanation:

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Answer 2

Answer:

The flow speed , pressure and volume flow rate in the bathroom are  6 m/s , $3.3\times10^{5}\ Pa$ and 0.47 L/s.

Explanation:

Given that,

Inside diameter = 2.0 cm

Pressure $P= 4.0\times10^{5}\ pa$

Diameter of pipe = 1.0 cm

Speed at inlet = 1.5 m/s

Height = 5.0 m

(I). We need to calculate the flow speed

Using continuity equation

$A_{1}v_{1}=A_{2}v_{2}$

$v_{2}=(\dfrac{D_{1}}{D_{2}})^2\times v_{1}$

Where, A₁ = area of inside

A₂ = area of

v₁ = speed at inlet

Put the value into the formula

$v_{2}=(\dfrac{2.0\times10^{-2}}{1.0\times10^{-2}})^2\times1.5$

$v_{2}=6 m/s$

(II). We need to calculate the pressure

Using Bernoulli equation

$P_{2}+\dfrac{1}{2}\rho v_{2}^2+\rho g h_{2}=P_{1}+\dfrac{1}{2}\rho v_{1}^2+\rho g h_{1}$

Put the value in the equation

$P_{2}+\dfrac{1}{2}\times1000\times6^2+1000\times9.8\times5.0=4.0\times10^{5}+\dfrac{1}{2}\times1000\times(1.5)^2+1000\times9.8\times0$

$P_{2}+18000+49000=4.0\times10^{5}+1125$

$P_{2}=4.0\times10^{5}+1125-67000$

$P_{2}=3.3\times10^{5}\ Pa$

(III). We need to calculate the volume flow rate

Using formula of the volume flow rate

$Q= vA$

Where, v = final velocity

A = area

Put the value into the formula

$Q=6\times\pi\times(0.005)^2$

$Q=4.7\times10^{-4}\ m^3/s$

Conversation:

$Q=\dfrac{4.7\times10^{-4}\ m^3}{s}\times\dfrac{1000\ L}{1\ m^3}$

$Q=0.47\ L/s$

Hence, The flow speed , pressure and volume flow rate in the bathroom are  6 m/s , $3.3\times10^{5}\ Pa$ and 0.47 L/s.