Answer
Verified
410.7k+ views
Hint: Reynolds number is a dimensionless value which is applied in fluid mechanics to represent whether the fluid flow in a duct or part of a body is steady or turbulent. Reynolds number is given by ratio of inertial force to viscous force. Use the following formula, $ \left[ 1L={{10}^{-3}}m \right] $
$ R=\dfrac{\rho \nu d}{\eta } $ Here, $ \rho \to $ Fluid density
$ v\to $ Fluid velocity
$ \eta \to $ Fluid viscosity
$ d\to $ Diameter or Length of fluid.
Complete step by step solution
We have given,
( $ \rho $ ) density of water = $ {{10}^{3}}{kg}/{{{m}^{3}}}\; $
( $ \eta $ ) viscosity of water= 10-3 pa.s
We have to find, Reynolds’s number which is given by,
$ R=\dfrac{\rho \nu d}{\eta } $
Here, $ \nu $ is the fluid velocity which can be obtained by following,
$ \nu =\dfrac{4Q}{\pi {{d}^{2}}} $
Q is the volume of water flowing out per second which is given by ,
$ Q=\dfrac{15L}{5\min } $
$ Q=\dfrac{15\times {{10}^{-3}}{{m}^{3}}}{5\times 60s} $ $ \left[ 1L={{10}^{-3}}m \right] $ 1 min =60 sec
$ Q=\dfrac{3\times {{10}^{-3}}{{m}^{3}}}{60s} $
$ Q=5\times {{10}^{-5}}{{m}^{3}} $
Now, d is diameter = $ \dfrac{2}{\sqrt{\pi }} $ cm = $ \dfrac{2}{\sqrt{\pi }} $ ×10-2m
Reynolds’s number is given by,
$ R=\dfrac{\rho \nu d}{\eta } $ = $ \dfrac{\rho }{\eta }\dfrac{4Qd}{\pi {{d}^{2}}} $
$ R=\dfrac{\rho 4Q}{\eta \pi d} $
Put all the values in above equation
$ R=\dfrac{{{10}^{3}}\times 4\times 5\times {{10}^{-5}}}{{{10}^{-3}}\times 3.14\times \dfrac{2}{\sqrt{3.14}}\times {{10}^{-2}}} $ ….. Use $ \left[ \pi =3.14 \right] $
= $ \dfrac{4\times 5}{\sqrt{3.14}\text{ }\times \text{2}}\times \dfrac{{{10}^{-2}}}{{{10}^{-5}}} $
= $ \dfrac{20}{\text{ 2}\times \text{1}\text{.773}}\times {{10}^{3}} $
$ R\approx 5500 $ , This is the approximate value of Reynolds’s number.
Note
Reynolds number formula is used to determine the diameter, velocity and viscosity of the fluid
If Re $ \text{ 2}000 $ , the flow is called Laminar
If Re $ >4000 $ , the flow is called turbulent
If 2000 $ < $ Re $ <4000 $ , the flow is called transition.
Here, Re is Reynolds number,
$ R=\dfrac{\rho \nu d}{\eta } $ Here, $ \rho \to $ Fluid density
$ v\to $ Fluid velocity
$ \eta \to $ Fluid viscosity
$ d\to $ Diameter or Length of fluid.
Complete step by step solution
We have given,
( $ \rho $ ) density of water = $ {{10}^{3}}{kg}/{{{m}^{3}}}\; $
( $ \eta $ ) viscosity of water= 10-3 pa.s
We have to find, Reynolds’s number which is given by,
$ R=\dfrac{\rho \nu d}{\eta } $
Here, $ \nu $ is the fluid velocity which can be obtained by following,
$ \nu =\dfrac{4Q}{\pi {{d}^{2}}} $
Q is the volume of water flowing out per second which is given by ,
$ Q=\dfrac{15L}{5\min } $
$ Q=\dfrac{15\times {{10}^{-3}}{{m}^{3}}}{5\times 60s} $ $ \left[ 1L={{10}^{-3}}m \right] $ 1 min =60 sec
$ Q=\dfrac{3\times {{10}^{-3}}{{m}^{3}}}{60s} $
$ Q=5\times {{10}^{-5}}{{m}^{3}} $
Now, d is diameter = $ \dfrac{2}{\sqrt{\pi }} $ cm = $ \dfrac{2}{\sqrt{\pi }} $ ×10-2m
Reynolds’s number is given by,
$ R=\dfrac{\rho \nu d}{\eta } $ = $ \dfrac{\rho }{\eta }\dfrac{4Qd}{\pi {{d}^{2}}} $
$ R=\dfrac{\rho 4Q}{\eta \pi d} $
Put all the values in above equation
$ R=\dfrac{{{10}^{3}}\times 4\times 5\times {{10}^{-5}}}{{{10}^{-3}}\times 3.14\times \dfrac{2}{\sqrt{3.14}}\times {{10}^{-2}}} $ ….. Use $ \left[ \pi =3.14 \right] $
= $ \dfrac{4\times 5}{\sqrt{3.14}\text{ }\times \text{2}}\times \dfrac{{{10}^{-2}}}{{{10}^{-5}}} $
= $ \dfrac{20}{\text{ 2}\times \text{1}\text{.773}}\times {{10}^{3}} $
$ R\approx 5500 $ , This is the approximate value of Reynolds’s number.
Note
Reynolds number formula is used to determine the diameter, velocity and viscosity of the fluid
If Re $ \text{ 2}000 $ , the flow is called Laminar
If Re $ >4000 $ , the flow is called turbulent
If 2000 $ < $ Re $ <4000 $ , the flow is called transition.
Here, Re is Reynolds number,
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE