Physics, 30.03.2020 21:45 yiikes4012

Air flows through a pipe at a rate of 124 L/s. The pipe consists of two sections of diameters 22 cm and 10 cm with a smooth reducing section that connects them. The pressure difference between the two pipe sections is measured by a water manometer.
Neglecting frictional effects, determine the differential height of water between the two pipe sections. Take the air density to be 1.20 kg/m^3.

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Answer from: mmblair14

Complete Question

The diagram for this question is shown on the first uploaded image

The differential height is h= 1.458cm

Explanation:

The schematic diagram of the venturimeter is shown on the second uploaded image

The continuity equation is mathematically given as

$\r V = Av$

Where A is the area

$\r V$ is the flow rate

v_1 is the velocity

At the inlet making v the subject to obtain the inlet velocity we have

$v_1 = \frac{\r V}{A_1}$

Substituting 124 L/s = 0.124 m^3 /s for $\r V$ and $A_1 = \frac{\pi}{4 } *d^2 = \frac{\pi}{4} * 0.22^2$ given that

$d =22cm= \frac{22}{100} = 0.22m$

So $v_1 =\frac{0.124}{\frac{\pi}{4} * 0.22^2 }$

=3.26m/s

At the exit point the velocity is

$v_2 = \frac{\r V}{A_2}$

Where $A_ 2 = \frac{\pi}{4}d^2 = \frac{\pi}{4} * 0.10^2$ given that $d =10cm= \frac{10}{100} = 0.10m$

So $v_2 = \frac{0.124}{\frac{\pi}{4} *(0.10)^2}$

= 15.78m/s

The Bernoulli's flow equation between the inlet and exist is mathematically given as

$\frac{P_1}{\rho_o } + \frac{v_1^2}{2} = \frac{P_2}{\rho_0} +\frac{v_2^2}{2} +gz_2 ---(1)$

And

$P_1 - P_2 = \rho_o [\frac{v_2^2}{2} + \frac{v_1^2}{2} ] = \rho_{water} gh ---(2)$

Where P_1 is the pressure at inlet

P_2 is the pressure at exist

$\rho_{water}$ is the density of water with value of 1000kg/m^3

g is acceleration due to gravity

h is the height of the water column

making h the subject in the equation 2

$h = \rho _o \frac{v_2^2 - v_1^2}{2g\rho_{water}}$

where $\rho _o$ is the density of air given in the question

Substituting value

$h = (1.20) \frac{15.78^2 - 3.26^2}{2 (9.81 )(1000)}$

h= 1.458cm

Air flows through a pipe at a rate of 124 L/s. The pipe consists of two sections of diameters 22 cm

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Air flows through a pipe at a rate of 124 L/s. The pipe consists of two sections of diameters 22 cm...

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