Question

Water from a pipe is coming at a rate of 100 litres per minute. If the radius of the pipe is 5 cm, Reynolds number for the flow is of the order: (density of water = 1000 kg/$m^3$, coefficient of viscosity of water is 1 mPa.s)
A. ${{10}^{6}}$
B. ${{10}^{3}}$
C. ${{10}^{4}}$
D. ${{10}^{2}}$

Answer 1

In this question we have been asked to calculate the Reynolds number, more specifically the order of Reynolds number. Now we know that Reynolds number is given as the ratio of inertial forces to viscous forces within a fluid which arises due to the different fluid velocities. Therefore, to solve this question we shall be using the equation of Reynolds number.
Formula used:
$R=\dfrac{\rho vd}{n}$
Where,
R is the Reynolds number
$\rho$ is the density of fluid kg/cubic meter
v is the velocity of the fluid in m/s
d is the diameter of the pipe in m
n is the dynamic viscosity of fluid in Pa.s

Complete answer:
We know that Reynolds number is calculated by taking the product of fluid velocity and pipe diameter (internal) and then dividing it by dynamic viscosity.
The equation of Reynolds number is given as,
$R=\dfrac{\rho vd}{n}$
Substituting the values after appropriate conversion
We get,
$R=\dfrac{{{10}^{3}}\times v\times 10\times {{10}^{-2}}}{{{10}^{-3}}}$ …………………………. (1)
We have been given that the volume flow rate says ${{V}_{R}}$ of water is 100 litres per minute.
Therefore, for velocity of fluid
We know that,
${{V}_{R}}=v\times A$ …………. (A is the internal area pipe through which water is flowing)
Solving for v,
$v=\dfrac{{{V}_{R}}}{A}$
Substituting the values after making appropriate conversion of unit
We get,
$v=\dfrac{100\times {{10}^{-3}}}{60}\times \dfrac{1}{\pi \times 25\times {{10}^{-4}}}$
On solving
We get,
$v=0.212m/s$
After substituting the above value in (1)
We get,
$R=\dfrac{{{10}^{3}}\times 0.212\times 10\times {{10}^{-2}}}{{{10}^{-3}}}$
Therefore,
$R=2\times {{10}^{4}}$

So, the correct answer is “Option C”.

Note:
Reynold number is a dimensionless number which helps to predict the flow pattern of a fluid in different situations. At low Reynolds numbers such as 2100 the flow of fluid can be found to be laminar and if the Reynolds number is greater than 4000 the flow is found to be turbulent. There is transitional flow between these limits. The area where the transition of flow takes place is called boundary layer. The laminar flow is ordered flow however, turbulent flow is chaotic.