Laminar Flow through Pipe MCQ Quiz in मल्याळम - Objective Question with Answer for Laminar Flow through Pipe - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Explanation:

Poiseuille's Law:

• Real fluids have viscosity, So a pressure difference is required between the ends of the tube or pipe for the flow to happen.
• The relation between pressure difference and discharge in the capillary tube is given by Poiseuille's law.

\(Q = \frac{\pi R^{4}\left ( P_{1}-P_{2} \right )}{8η L}\)

Where, Q = Discharge in the pipe/tube, R = Radius of pipe, P₁, P₂ are the pressure at point 1 and 2 respectively, η = Dynamic viscosity of the fluid, L = Length of pipe between point 1 and 2

Assumptions for the use of Poiseuille's law

1. The fluid must be incompressible.
2. It should be laminar between the given boundaries

Explanation:

For a fully developed laminar viscous flow through a circular pipe, the maximum velocity is equal to twice the average velocity.

i.e. \({U_{max}} = 2 \times {U_{avg}}\)

\({U_{max}} = \frac{1}{{4\mu }}\left( { - \frac{{\partial P}}{{\partial x}}} \right).{R^2}\)

\(U_{avg}=\bar U = \frac{1}{{8\mu }}\left( { - \frac{{\partial P}}{{\partial x}}} \right).{R^2}\)

Hence \(\frac{{{U_{max}}}}{{\bar U}} = 2\)

For the fully developed laminar flow through the parallel plates, the maximum velocity is equal to the (3/2) times of the average velocity.

i.e. \({U_{max}} = \frac{3}{2} \times {U_{avg}}\)

Explanation:

Reynolds Number

Reynold number is a dimensionless number that helps to predict flow patterns in different fluid flow situations.

\({\rm{Re}} = \frac{{{\rm{Inertia\;force}}}}{{{\rm{Viscous\;force}}}} = {\rm{\;}}\frac{{{\rm{\rho V}}{{\rm{L}}_{\rm{c}}}}}{{\rm{\mu }}} = \frac{{{\rm{V}}{{\rm{L}}_{\rm{c}}}}}{{\rm{\nu }}}\)

Where,

Re = Reynolds number, ρ = density, V = velocity of flow

μ = dynamic viscosity ν = kinematic viscosity, LC = characteristic linear dimension

For pipe flow:

LC = diameter of pipe = D

Reynold number = \(Re = \frac{{{\rm{\rho VD}}}}{\mu } = \frac{{{\rm{VD}}}}{\nu }\)

• Laminar flow  Re ≤ 2000
• Transition flow 2000 ≤ Re ≤ 4000
• Turbulent flow Re ≥ 4000

Additional Information

The Reynolds number value below which flow can be definitely considered to be Laminar flow is called Critical Reynolds number Rc.

The values of the critical Reynolds number are listed below:

• For Pipe flow Rc = 2300
• For Flow between plate Rc = 1000
• For open channel flow Rc = 500
• For flow over plate Rc = 5 × 105
• For flow over sphere Rc = 1

Concept:

For pipe:

The loss of pressure head in case of laminar flow is given by:

\({h_{L}} = \frac{{32{μ \bar UL}}}{{ ρ gd^2}}\)

Parallel plate:

\({h_L} = \frac{{12\mu \bar UL}}{{ρ g{t^2}}}\)

where μ = viscosity, L = length of pipe/plate, U̅ = average velocity, d = diameter of the pipe.

Explanation:

From the above head loss expression, it is clear that 

Head loss is directly proportional to the length of the pipe.

∴ Pressure loss (ρgh_L) is also directly proportional to the length of the pipe.

Concept:

The head loss in laminar flow through a pipe is given by:

\({h_{L}} = \frac{{32{μ ̅ UL}}}{{ \rho gd^2}}\)

where μ = viscosity, L = length of pipe/plate, U̅ = average velocity, d = diameter of the pipe.

Calculation:

Given:

L₂ = L₁, μ₂ = μ₁, U̅₂ = 2U̅₁, d₂ = 0.5d₁

The head loss in laminar flow through a pipe is given by:

\({h_{L}} = \frac{{32{μ ̅ UL}}}{{ \rho gd^2}}\)

\(\frac{h_2}{h_1}= \frac{{{\bar{U_2}}}}{{\bar{U_1}}}\times\frac{{{{d_1^2}}}}{{d_2^2}}\)

\(\frac{h_2}{h_1}= \frac{{{2\bar{U_1}}}}{{\bar{U_1}}}\times\frac{{{{4d_1^2}}}}{{d_1^2}}=8\)

Additional InformationUsing Darcy-Weisbach equation:

\(h_L=\frac{fLV^2}{2gd}\)

where f = friction factor = 64/Re

Reynold's number is given by:

\(Re=\frac{\rho V d}{\mu}\)

Putting the value of friction factor:

\(h_L=\frac{64}{Re}\times\frac{LV^2}{2gd}=\frac{64\mu}{\rho V d}\times\frac{LV^2}{2gd}\)

\(h_L=\frac{64\mu LV}{2 \rho gd^2}\)

When length and viscosity remain same then:

\(h_L\propto\frac{V}{d^2}\) which is the same as above.

Explanation:

Kinetic energy correction factor(α): 

• It is defined as the ratio of kinetic energy/second based on actual velocity to the kinetic energy/second based on average velocity.
• \(α = \frac{1}{A}\mathop \smallint \limits_A^{} {\left( {\frac{u}{V}} \right)^3}dA\)
• where A = area, V= average velocity, u= local velocity at distance r.
• α = 1 for uniform velocity distribution and tends to become greater than I as the distribution of velocity becomes less and less uniform.
• α = 1.02 to 1.15 for turbulent flows.
• α = 2 for laminar flow.

The marked answer is most appropriate according to the given options.

Additional Information

Momentum correction factor: 

The momentum correction factor is defined as the ratio of momentum of the flow per second based on actual velocity to the momentum of the flow per second based on average velocity across a section.

 \(β = \;\frac{{Momentum\;per\;second\;based\;on\;actual\;velocity}}{{Momentum\;per\;second\;based\;on\;average\;velocity}}\)

 \(β = \frac{1}{{A{V^2}}}\smallint {u^2}.dA\)

β = 1 for uniform flow,

β = 1.01 to 1.07 for turbulent flow in pipes, and

β = 3 1.33 for laminar flow in pipes.

Explanation:

Surge Tank:

• A surge tank is a water storage device used as a pressure neutralizer in hydropower water conveyance systems in order to dampen excess pressure variance.
• A surge tank (or surge drum or surge pool) is a standpipe or storage reservoir at the downstream end of a closed aqueduct.
• A feeder dam, barrage, or pipe to absorb sudden rises of pressure, as well as to quickly provide extra water during a brief drop in pressure.
• In mining technology, ore pulp pumps use a relatively small surge tank to maintain a steady on the pump.
• For hydroelectric power uses, a surge tank is additional storage space or reservoir fitted.
• Surge tanks are usually provided in high or medium-head plants when there is a considerable distance between the water source and the power unit,

The main function of surge tanks are as follows:

• When the load decreases, the water moves backward and gets stored in it.
• when the load increases, an additional supply of water will be provided by the surge tank.
• In short, the surge tank mitigates pressure variations due to rapid changes in the velocity of the water and protects the pipeline against water hammers.

Water Hammer

• When a liquid flowing through a long pipe is suddenly brought to rest by closing the valve at the end of a pipe, then a pressure wave of high intensity is produced behind the valve. This pressure wave of high intensity has the effect of hammering action on the walls of the pipe. This phenomenon is known as a water hammer or hammer blow.

Explanation:

In the case of a pipe, 

Velocity distribution for laminar flow is given by:

\(u = - \frac{1}{{4\mu }}\frac{{\partial p}}{{\partial x}}\left[ {{R^2} - {r^2}} \right]\)

∴ Velocity follows a parabolic law in case of laminar flow.

Important Points

Velocity distribution for turbulent flow in smooth pipe:

\(u = \frac{{{u_\infty }}}{k}\ln y + c\)

In the case of turbulent flow, the velocity distribution is logarithmic in nature.

In the pipe, flow is laminar up to Reynolds number 2000.

So taking the Reynolds number as 2000,

\(\begin{array}{l} Re = \frac{{\bar v D}}{\nu }\\ 2000 = \;\frac{{v ̅ \times 0.08}}{{0.4 \times {{10}^{ - 4}}}} \end{array}\)

V̅ = 1 m/s

For a pipe flow: V_max = 2V̅ = 2 m/s

Maximum velocity up to which flow is laminar is 2 m/s.

Important Point:

The pipe flow is laminar when the Reynolds number is less than or equal to 2000.

The pipe flow is turbulent when the Reynolds number is more than 4000. In between 2000 and 4000 Reynolds number flow is in a transition zone.

Explanation:

The pressure drop per unit length in a fully developed laminar flow through a pipe is given as,

\(\Delta P = \frac{{32\mu {V_{mean}}}}{{{D^2}}}\)

We know Q = A × V

so V = \(\frac{Q}{A}\)

∴ \(\Delta P = \frac{{32\mu {Q}}}{{{AD^2}}}\)

\(\Delta P\propto\frac{1}{A}\)

∴ For laminar flow through a long pipe, the pressure drop per unit length is in inverse proportion to the cross-sectional area.