Laminar flow through circular conduits, flow through pipes , Boundary Layer theory

What is the difference between a laminar flow and turbulent flow?
In the laminar flow the fluid particles move along smooth paths in laminas (or) layers with one layer gliding over the adjacent layer .
In turbulent flow the fluid particles move in a very irregular path causing an exchange of momentum from one portion of the fluid to the other. The turbulence setup greater shear stress throughout the fluid and causes more irreversibility and losses.
In laminar flow through a pipe the maximum velocity at the pipe axis is 0.2m/s. Find the average velocity.
The average velocity Uavg = Umax /2 = 0.2/2 = 0.1m/s
Write the relationship between shear stress and pressure gradient in a laminar flow through pipe.
The shear stress at any point t = - (dp/dx) (r/2) Where (dp/dx) is the pressure gradient and -ve sign shows that pressure decreases in the direction of flow.
What is a boundary layer?
For fluids having relatively small viscosity, the effect of internal friction in a fluid is appreciable only in a harrow region surrounding the boundaries, where the velocity gradients are large and also larger shear stress. This region is known as boundary layer.
Why does the boundary layer increase with distance from the upstream edge ?
At the upstream edge, the free stream velocity is retarded by the solid surface causing a steep velocity gradient form the boundary to the flow. The velocity gradient sets up boundary shear forces that reduce the flow relative to the boundary. As the layer moves along the body the continual action of shear stress tends to slow down causing the thickness of boundary layer to increase.
Define boundary layer thickness.
The boundary layer thickness is defined as the distance from the boundary in which the velocity reaches 99% of main stream velocity and usually denoted by d.
Define displacement thickness.
The distance perpendicular to the boundary , by which the stream velocity is displaced due to the formation of boundary layer. It is denoted by d*.
Define momentum thickness.
Momentum thickness is defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on account of boundary layer formation. It is denoted by q.
Define Energy thickness .
Energy thickness is defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the fluid flowing on account of boundary layer formation.
What is the purpose of moody diagram ?
The Moody diagram shows the variation of friction factor with the governing parameters namely Reynolds number and relative roughness. This diagram is employed for predicting the values of friction factor in turbulent flow.
What do you understand by energy losses in a pipe ?
When a fluid flowing through a pipe the fluid experiences some resistance due to which some of the energy of the fluid is lost. This loss of energy is classified as
Major losses ---- This is due to friction
Minor losses ---- This due to enlargement , contractions, bents, valve fittings, entrance & exit present in the pipe system.
Give the Darcy-Weisbach equation for frictional head loss in pipe flow.
The frictional head loss hf = f L U2 / (2 g D) D - diameter of the pipe
f - Friction factor L – length of the pipe U – mean velocity of the pipe
The diameter of water pipe is suddenly enlarges from 350mm to 700mm. The rate of flow through it is 0.25m3 /s. Calculate the loss of head in enlargement.
The diameter of the smaller pipe D1 = 350mm = 0.35 m
The diameter of the larger pipe D2 = 700mm = 0.7 m
The rate of flow Q = 0.25 m3/s
The velocity through the pipe U1 = Q/A1 = (0.25 x 4 ) / ( p x 352) = 2.60 m/s
The loss of head in enlargement henl = (U12/2g) ( 1 - (A1/A2) )2 = (U12/2g) ( 1 - (D1/D2)2 )2
= (2.62/2x 9,81) ( 1 - (0.35/0.7)2 )2 = 0.1938 m of H2O
A horizontal pipe carries water at the rate of 0.04 m3/s. Its diameter reduces abruptly from 300mm to 150mm. Calculate the head loss across the contraction. Take the coefficient of contraction as 0.62.
Diameter of the pipe D1 = 0.3m Diameter of the contraction D2 = 0.3m Discharge Q = 0.04 m3/s
The loss of head in enlargement hcont = (U12/2g) ( (1/Cc) - 1 )2
= (2.262/2x 9,81) ( (1/0.62) - 1 )2 = 0.0978 m of H2O
The velocity at the contraction U2 = Q/A2 = (0.04 x 4 ) / ( p x 0.152) = 2.26 m/s
What are energy lines ?
The energy line is a longitudinal display of the total head at all salient sections of the pipe. The energy line therefore represents the degradation of energy along the flow due to friction, minor losses etc as well as additional input or output by means of pumps and turbines.
What are hydraulic gradient lines ?
A hydraulic gradient line is a longitudinal display of the pressure and datum head ie., piezometric head at all salient sections of the pipe line.
What is a compound pipe ?
A compound pipe is one in which a number of pipes of different diameters, different lengths and different friction factors are connected in series
What is an equivalent pipe ?
The equivalent pipe is the pipe of uniform diameter having loss of head and discharge equal to the loss of head and discharge of a compound pipe consisting of several pipes of different lengths and diameters.
Two pipes of lengths 800mm and 400mm of diameters 50cm and 30cm respectively are connected in series. These pipes are replaced by a single pipe of length 1000m. Find the diameter of single pipe.
Length of pipe –1 L1 = 800m Length of pipe –2 L2 = 400m
Length of equivalent pipe L = 1000m Diameter of pipe-1 D1 = 0.5m
Diameter of pipe-2 D2 = 0.3m
(L/D5) = (L1 /D15 ) +(L1 /D15 ) = (800/0.555 ) +(400 /0.35 )
` = 25600 + 164609 = 190209
D5 = (1000 / 190209) D = 0.416m
Explain the term pipes in parallel.
If two pipes are connected between two given points of a flow system it is called a parallel pipe system.