Water Resources Engineering WEM401 Water Resources Engineering Lecture – 2.4 Pipe Network Analysis; Hydraulic Transients Vishnu Prasad Pandey, PhD Asian Institute of Technology (AIT) Email: vishnu@ait.asia; vishnu.pandey@gmail.com

Hardy Cross Method for pipe network analysis Hydraulic transients Contents Outline Remarks II. Water Distribution Systems Water distribution systems and its components Pipe flow equation Hydraulics of simple networks Pump system analysis Network simulation Hydraulic transients Duration: 7.5 hrs Assignment: 1 Responsibility: Dr. Vishnu Pd. Pandey Introduction Hardy Cross Method for pipe network analysis Hydraulic transients

Q & h must satisfy the continuity & energy Eqns Introduction In municipal water supply system, there are several branches, junctions and circuits, such that the system is complex. Pipe network analysis involves determining pipe “flow rates (Q)” & “pressure heads (h)” at the outflow points of the networks. Q & h must satisfy the continuity & energy Eqns Hardy-Cross (in 1936); Professor of Structural Engineering @ University of Illions Urbana-Champaign, USA)

Hardy-Cross Method Hardy-Cross method is one of the widely used method for analyzing a complex system of pipes forming closed loops This is a method of successive approximation that applies head balance approach Outflows from the system are generally assumed to occur at the nodes/junctions For a given pipe system with known outflows, the Hardy-Cross method is an iterative procedure based on initially iterated flows in the pipe At each junction these flows must satisfy the continuity criteria & energy criteria

Hardy-Cross Method Basic requirements for Hardy-Cross method: Continuity equation: Qin = Qout (for each junction) Energy equation: Algebraic sum of HL around any closed circuit = zero. For each loop or circuit the HL is +ve in clockwise dirn & –ve in anticlockwise dirn Dary-Weisbach Eqn satisfy in each pipe

Hardy-Cross Method: Steps With given inflow & outflow, assume suitable Q & its direction in each pipe that satisfy continuity Eqn at each junction Divide pipe network into a number of loops Include each pipe in at least one of the loops For each loop, compute HL in each pipe & sum them up: Also compute for each circuit; w/o considering sign (i.e., absolute value is taken) Compute the correction:

Hardy-Cross Method: Steps Compute the corrected flow as Qi+1 + ΔQ, where Qi+1 = previous value of flows If ΔQ is +Ve  Add it to Q in clockwise dirN & subtract it from Q in anti-clockwise dirN & vice- versa. For a common pipe of two loops, apply correction from both loops considering appropriate sign for each Use corrected Q for the next trial & repeat steps 3-5 until the correction becomes negligible. Computer solutions: EPANET; KYpipes; WaterCAD; CyberNET

Hydraulic Transients Hydraulic transient developed due to Wave Sudden closure of the valve & corresponding rise in velocity Wave Temporal variation of water surface, which is propagated in the fluid media Celerity (C) The relative velocity of wave w.r.t. velocity of fluid If C is celerity, V is fluid velocity, & VW is wave velocity; C = VW + V; if both move in opposite direction C = VW + V; if both move in the same direction

Hydraulic Transients Celerity (C)   Total pressure: as discussed in Hydropower lecture

Hydraulic Transients: Water Hammer When water flowing in a long pipe is suddenly brought into rest by closing the valve Momentum of flowing water will be destroyed Wave of high pressure will be setup KE of flowing fluid will be converted into the internal pressure energy with rise of pressure The wave of high pressure will be transmitted along the pipe with velocity equal to that of sound (C) May create noise called “knocking” The phenomenon of sudden rise in pressure in the pipe  Water Hammer (of Water Blow) Water hammer is an example of fast hydraulic transient

Hydraulic Transients: Water Hammer Water Hammer is caused by changes in velocity, which are caused by: Valve operation (i.e. closure & opening) Power failures Starting or shutdown of pumps (hydro-turbines) Fluctuation in power demand in turbines Rupture of the line, etc. Mechanical failure of the control devices like valves Effects of water hammer High-pressure fluctuations in pipelines Rupture of pipe or valve if fluctuations beyond safety limit Higher pressure requirements for the design of pipeline & penstocks, etc.

Hydraulic Transients: Water Hammer Magnitude of pressure rise depends on: Time taken to close the valve Velocity of flow Length of pipe Elastic properties of the pipe materials as well as that of the flowing fluid (i.e., water in this case) Critical time of closure = 2L/C; 2L = distance travelled from valve to tank and back; C = velocity of the wave pressure Gradual closure: if T > 2L/C; T = time required to close Sudden closure: if T < 2L/C

Hydraulic Transients: Water Hammer Steady-state condition prior to valve movement Transient conditions at t<L/C Transient conditions at t =L/C Transient conditions at L/C < t < 2L/C Transient conditions at t =2L/C Transient conditions at 2L/C < t < 3L/C Transient conditions at t =3L/C Transient conditions at 3L/C < t < 4L/C. Transient conditions at t = 4L/C. After t =4L/C, the cycle repeats & continues indefinitely if the friction in the pipe is zero. Clockwise & anticlockwise arrows denotes the direction of reflection of the wave front. Fig.: Propagation of Water Hammer Pressure Waves (Neglecting pipe friction)

Hydraulic Transients: Water Hammer Pressure diagram @ different points @ B (Valve)

Hydraulic Transients: Water Hammer @ M (Mid-point) Time (t) Surge tanks are provided to protect the pipes from Water Hammer. @ A (End-point) Time (t)