Hagen-Poiseuille Equation
By force balance
By Newton’s law of viscosity we have
, where y is the distance from the wall
• This can be combined with the equation above to give,
,
at r = R, u = 0 ( no-slip condition),
• at r = 0, u Vmax, centerline velocity
=
Volume flow rate is given by, Q = vavg A
This is the Hagen-Poiseuille equation for laminar flow through pipe. Therefore, flow rate,
•
Shear Stress,
Wall Shear Stress, ( r=R)
FRICTION LOSS IN TURBULENT FLOW
• Using Darcy equations we can calculate the friction losses in turbulent flow.
• It depends on the surface roughness of the pipe as well as Reynolds number (IN LAMINAR, LOSSES ONLY DEPEND ON THE REYNOLD NUMBER)
FRICTION LOSS IN TURBULENT FLOW
MOODY DIAGRAM FOR TURBULENT FLOW
• One of the most widely used methods for evaluating the friction factor employs the Moody
• An American Engineer called Moody conducted exhaustive experiments and came up with the Moody Chart. The chart is a plot of f vertically against Re horizontally for various values of ε.
• Friction factor is correlated as a function of the Reynolds' Number and the relative pipe roughness. This function is usually presented in the familiar Moody Diagram.
• To understand it, the Moody diagram may be broken down into four zones: Laminar, Transition, Partially turbulent, Fully turbulent.
• The Fully Turbulent zone is the part on the extreme right where the lines flatten-out. In this zone of extremely high flow rate the fluid flows laterally within the pipe in complete turbulence as well as in the primary direction and the friction factor shows no dependency on flow rate.
SIPHON
• A siphon is a device that allows a liquid to be drawn from a storage vessel without the use of a pump.
• A siphon may be as simple as a length of hose. One end of the hose is inserted into the liquid to depth d, the other is positioned below the level of the free surface at a distance H
Water Hammer Phenomenon in pipelines
• Water Hammer is a pressure surge or wave that occurs when there is a sudden
Water Hammer- Cause and Effect
• Water hammer is caused by a change in fluid momentum. The mostcommon cause of this change in momentum is sudden closure of a valve on a pipeline.
• When this occurs, a loud hammer noise can be produced and vibrations can be sent through the pipe (Water Hammer).
• The pressure wave produced from this event can cause significant damage to pipe systems. The large increase in pressure can cause pipes to crack and in some cases burst.
Water Hammer- Cause and Effect
• Another instance that produces a water hammer effect is pump and turbine failure.
• When a pump fails, the sudden halt in flow will produce the momentum change causing the water hammer effect.
• This can also be seen in home plumbing systems when faucets are turned on and off suddenly. A loud hammer noise will be produced, and the plumbing will vibrate in most cases.
• Water hammer can be induced intentionally for various applications.
• A hydraulic ram can be created using a water hammer, and is commonly used in mining practices to break through rock.
• In addition, the water hammer effect creates an increase in pressure within a pipe line and is then used to detected leaks within the pipe line. The increased pressure causes water to shoot out of the pipe at a leak site, which is then easily spotted.
• Despite water hammer being useful at time, it is generally an undesired phenomenon that must be considered when designing pipe lines (Water Hammer).
Analysis of Water Hammer Phenomenon
The pressure rise due to water hammer depends upon:
(a) The velocity of the flow of water in pipe,
(b) The length of pipe,
(c) Time taken to close the valve,
(d) Elastic properties of the material of the pipe.
The following cases of water hammer will be considered:
• Gradual closure of valve,
