Question

The water flowing through a $\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{}\mathbf{cm}$(inside diameter) pipe flows out through three $\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{}\mathbf{cm}$pipes. (a) If the flow rates in the three smaller pipes are $\mathbf{26}$, $\mathbf{19}$, and localid="1657628147057" $\mathbf{11}\raisebox{1ex}{$\mathbf{}\mathbf{}\mathbf{L}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{min}$}\right.$, what is the flow rate in the$\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{}\mathbf{cm}$pipe? (b) What is the ratio of the speed in the localid="1657628193484" $\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{}\mathbf{cm}$pipe to that in the pipe carrying localid="1657628220414" $\mathbf{26}\mathbf{}\raisebox{1ex}{$\mathbf{L}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{min}$}\right.$?

Short answer

a) The flow rate in the pipe of $1.9\mathrm{cm}$diameter is $\mathrm{R}=56\mathrm{L}/\min$.

b) The ratio of the speed in the $1.9\mathrm{cm}$pipe to that in the pipe carrying $26\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$\min $}\right.$is $\begin{array}{r}{\mathrm{V}}_{56}-1.0\\ {\mathrm{V}}_{26}\end{array}$.

Step-by-step solution

Given information

• The diameter of a large pipe is, ${\mathrm{d}}_{\mathrm{L}}=1.9\mathrm{cm}$.

• The diameter of smaller pipes, ${\mathrm{d}}_{\mathrm{S}}=1.3\mathrm{cm}$.

• The flow rates in the smaller pipes are$26,19\mathrm{and}11\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$\min $}\right.$

Understanding the concept equation of continuity

An ideal fluid is incompressible and lacks viscosity, and its flow is steady and irrotational. A streamline is a path followed by an individual fluid particle. A tube of flow is a bundle of streamlines. The flow within any tube of flow obeys the equation of continuity:

${\mathbf{R}}_{\mathbf{v}}\mathbf{=}\mathbf{V}\mathbf{=}\mathbf{Constant}$

Where ${\mathbf{R}}_{\mathbf{v}}$is the volume flow rate, $\mathbf{A}$is the cross-sectional area of the tube of flow at any point, and $\mathbf{v}$ is the speed of the fluid at that point.

Find the flow rate in a large pipe having diameter $\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{}\mathbf{cm}\mathbf{}$ using the equation of continuity, and to calculate the ratio of speed, assume that both pipes are circular.

Formulae are as follows:

1. The equation of continuity is$\mathrm{R}={\mathrm{R}}_1+{\mathrm{R}}_2+\dots +{\mathrm{R}}_{\mathrm{n}}$

2. The equation for water speed is$\mathrm{V}=\frac{\mathrm{R}}{\mathrm{A}}$

3. The equation for area of cross-section is$\mathrm{A}=\frac{{\mathrm{d}}^2}{4}$

Where, $\mathrm{R}$is rate of flow, $\mathrm{A}$is area,$\mathrm{v}$is velocity and d is diameter of pipe.

Determining the flow rate in the pipe of 1.9 cm diameter

The equation of continuity can be written as,

$\mathrm{R}={\mathrm{R}}_1+{\mathrm{R}}_2+\dots +{\mathrm{R}}_{\mathrm{n}}$

Here, three smaller pipes are given, so $\mathrm{n}=3$.

Hence,

$\mathrm{R}={\mathrm{R}}_1+{\mathrm{R}}_2+{\mathrm{R}}_3$

$=(26+19+11)\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$\min $}\right.$

$=56\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$\min $}\right.$

Hence, the flow rate in the pipe of $1.9\mathrm{cm}$diameter is $56\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$\min $}\right.$.

(b) Determining the ratio of speed of water in 1.9 cm pipe and that in which flow rate is 26 L/min

The equation for water speed can be written as,

$\mathrm{V}=\frac{\mathrm{R}}{\mathrm{A}}$

So, the required ratio can be written as,

$\frac{{\mathrm{V}}_{56}}{{\mathrm{V}}_{26}}=\frac{\frac{{\mathrm{R}}_{56}}{{\mathrm{A}}_{56}}}{{\mathrm{R}}_{26}}=\frac{{\mathrm{R}}_{56}}{{\mathrm{A}}_{56}}\times \frac{{\mathrm{A}}_{26}}{{\mathrm{R}}_{26}}$

$=\frac{56}{{\mathrm{\pi d}}_{\mathrm{L}}^2}\times \frac{\frac{\mathrm{\pi }}{4}{\mathrm{d}}_{\mathrm{s}}^2}{26}$

$=\frac{56}{1{.9}^2}\times \frac{1{.3}^2}{26}$

$\approx 1.0$

Hence, the ratio of the speed in the $1.9\mathrm{cm}$pipe to that in the pipe carrying $26\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$\min $}\right.$is $1.0$.