Local Head Losses Local head losses are the “loss” of energy at point where the pipe changes dimension (and/or direction). Pipe Expansion Pipe Contraction Entry to a pipe from a reservoir Exit from a pipe to a reservoir Valve (may change with time) Orifice plate Tight bends They are “velocity head losses” and are represented by

Value of kL For junctions and bends we need experimental measurements kL may be calculated analytically for Expansion Contraction By considering continuity and momentum exchange and Bernoulli

Losses at an Expansion As the velocity reduces (continuity) Then the pressure must increase (Bernoulli) So turbulence is induced and head losses occur Turbulence and losses

Value of kL for Expansion 1 2 Apply the momentum equation from 1 to 2 Using the continuity equation we can eliminate Q From Bernoulli

Value of kL for Expansion Combine and - Borda-Carnot Equation Using the continuity equation again

Losses at Contraction Flow converges as the pipe contracts Convergence is narrower than the pipe Due to vena contractor Experiments show for common pipes Can ignore losses between 1 and 1’ As Convergent flow is very stable

Losses at Contraction Apply the general local head loss equation between 1’ and 2 And Continuity The value of k depends on In general

Losses: Junctions

Losses: Sharp bends

kL values Bell mouth Entry T-branch kL= 1.5 kL= 0.1 Sharp Entry/Exit T-inline kL Values Bellmouth entry 0.1 Sharp entry 0.5 Sharp exit 90o Bend 0.4 90o Tees flow in line to line Branch 1.5 Gate value (open) 0.25

Pipeline Analysis Bernoulli Equation equal to a constant: Total Head, H Applied from one point to another (A to B) With head losses

Bernoulli Graphically Reservoir Pipe of Constant diameter No Flow A

Bernoulli Graphically Constant Flow Constant Velocity No Friction A

Change of Pipe Diameter Bernoulli Graphically Constant Flow Constant Velocity No Friction Change of Pipe Diameter A

Bernoulli Graphically Constant Flow Constant Velocity With Friction A

Reservoir Feeding Pipe Example

Reservoir Feeding Pipe Example Apply Bernoulli with head losses pA=pC = Atmospheric uA = negligible

Find pressure at B: Apply Bernoulli A-B pA= Atmospheric u = uB = 2.41 m/s Negative i.e. less than Atmospheric pressure

Pipes in series Consider the situation when the pipes joining two reservoirs are connected in series 1 2 Total loss of head for the system is given as A1U1=A2U2=AnUn=Q Q1=Q2=Qn=Q

Pipes in Series Example Two reservoirs, height difference 9 m, joined by a pipe that changes diameter. For, 15m d=0.2 m then for 45 m, d=0.25 m. f = 0.01 for both lengths. Use kL,entry= 0.5, kL,exit = 1.0. Treat the joining of the pipes as a sudden expansion. Find the flow between the reservoirs.

Apply Bernoulli A-B

Pipes in parallel The head loss across the pipes is equal Diameter, f, length L, Q, andu may differ Total flow is sum in each pipe

Knowing the loss of head, the discharges in each pipe can be obtained The values of U1, U2 and U3 may be obtained from the above equations and hence the discharges Q1, Q2 and Q3 obtained.

The discharge Q is given, the distribution of discharge in different branches is required The discharges Q1, Q2, Q3 can be expressed in terms of H Where

Similarly Therefore: As the discharge is known and K1,K2 and K3 are constants, the value of H is obtained and the discharge in individual pipes obtained

Pipes in Parallel Example Two pipes connect two reservoirs which have a height difference of 10 m. Pipe 1 has diameter 50 mm and length 100 m. Pipe 2 has diameter 100 mm and length 100 m. Both have entry loss kL,entry = 0.5 and exit loss kL,exit =1.0 and Darcy f of 0.008. Find Q in each pipe Diameter D of a pipe 100 m long and same f that could replace the two pipes and provide the same flow.

Apply Bernoulli for each pipe separately

Pipe 2:

Flow required in new pipe = Replace u using continuity Must solve iteratively

Must solve iteratively for D Get approximate answer by leaving the 2nd term Increase deq a little, say to 0.106 … a little more … to 0.107 So

Flow through a by-pass A by-pass is a small diameter pipe connected in parallel to the main pipe Meter By-pass Main pipe The ratio of q in the by-pass, to the total discharge is known as the by-pass coefficient. by-pass coefficient

Head loss in main pipe = head loss in by-pass Minor losses in the by-pass due to bend and the meter etc Divide by Implies

Using continuity equation Adding one on both sides

By-pass coefficient Knowing the by-pass coefficient, the total discharge in the main pipe can be determined