Major & Minor Losses

Under Supervision of:

Prof. Dr. Mahmoud FouadBy students:

Mahmoud Bakr 533 Mohammed Abdullah 511Moaz Emad 619 Mohammed Nabil Abbas 525

Applications

How big does the pipe have to be to carry a flow of x m3/s?

Bernoulli's EquationThe basic approach to all piping systems is to

write the Bernoulli equation between two points, connected by a streamline, where the conditions are known. For example, between

the surface of a reservoir and a pipe outlet .The total head at point 0 must match with the

total head at point 1, adjusted for any increase in head due to pumps, losses due to pipe friction and so-called "minor losses" due to entries, exits, fittings, etc. Pump head developed is generally a function of the flow through the system

Bernoulli's Equation

Friction Losses in PipesFriction losses are a complex function of the

system geometry, the fluid properties and the flow rate in the system. By observation, the head loss is roughly proportional to the square of the flow rate in most engineering flows (fully developed, turbulent pipe flow). This observation leads to the Darcy-Weisbach equation for head loss due to friction

For laminar flow, the head loss is proportional to

velocity rather than velocity squared, thus the friction factor is inversely proportional to velocity

Turbulent flowFor turbulent flow, Colebrook (1939) found

an implicit correlation for the friction factor in round pipes. This correlation converges well in few iterations. Convergence can be optimized by slight under-relaxation.

The familiar Moody Diagram is a log-log plot of the Colebrook correlation on axes of friction factor and Reynolds number, combined with the f=64/Re result from laminar flow. The plot below was produced in an Excel spreadsheet

An explicit approximation

Pipe roughnesspipe materialpipe material pipe roughness pipe roughness (mm) (mm)

glass, drawn brass, copperglass, drawn brass, copper 0.00150.0015

commercial steel or wrought ironcommercial steel or wrought iron 0.0450.045

asphalted cast ironasphalted cast iron 0.120.12

galvanized irongalvanized iron 0.150.15

cast ironcast iron 0.260.26

concreteconcrete 0.18-0.60.18-0.6

rivet steelrivet steel 0.9-9.00.9-9.0

corrugated metalcorrugated metal 4545

PVCPVC 0.120.12

d

d Must be

dimensionless! Must be dimensionless!

Calculating Head Loss for a Known Flow

From Q and piping determine Reynolds Number, relative roughness and thus the friction factor. Substitute into the Darcy-Weisbach equation to obtain head loss for the given flow. Substitute into the Bernoulli equation to find the necessary elevation or pump head

Calculating Flow for a Known HeadObtain the allowable head loss from the Bernoulli equation, then start by guessing a friction factor. (0.02 is a good guess if you have nothing better.) Calculate the velocity from the Darcy-Weisbach equation. From this velocity and the piping characteristics, calculate Reynolds Number, relative

roughness and thus friction factor .Repeat the calculation with the new friction factor until sufficient convergence is obtained. Q = VA

"Minor Losses"Although they often account for a major portion of the head loss, especially in process piping, the additional losses due to entries and exits, fittings and valves are traditionally referred to as minor losses. These losses represent additional energy dissipation in the flow, usually caused by secondary flows induced by curvature or recirculation. The minor losses are any head loss present in addition to the head loss for the same length of straight pipe .

Like pipe friction, these losses are roughly proportional to the square of the flow rate. Defining K, the loss coefficient, by

. K is the sum of all of the loss coefficients in the length of pipe, each contributing to the overall head loss

Although K appears to be a constant coefficient, it varies with different flow conditions

Factors affecting the value of K include: the exact geometry of the component,.the flow Reynolds number , etc.

Some types of minor lossesHead Loss due to Gradual Expansion (Diffuser)

g

VVKh EE

2

221

g

VVKh EE

2

221

diffusor angle ()

00.10.20.30.40.50.60.70.8

0 20 40 60 80

KE

2

1

22

2 12

AA

gV

Kh EE

2

1

22

2 12

AA

gV

Kh EE

Sudden Contraction

losses are reduced with a gradual contraction

g

V

Ch

c

c

21

1 22

2

g

V

Ch

c

c

21

1 22

2

2A

AC cc

2A

AC cc

V1V2

flow separation

Sudden Contraction

0.60.650.7

0.750.8

0.850.9

0.951

0 0.2 0.4 0.6 0.8 1

A2/A1

Cc

Q CA ghorifice orifice 2

g

VKh ee

2

2

g

VKh ee

2

2

0.1eK 0.1eK

5.0eK 5.0eK

04.0eK 04.0eK

Entrance LossesLosses can be reduced by accelerating the flow gradually and eliminating thevena contracta

Head Loss in BendsHead loss is a function

of the ratio of the bend radius to the pipe diameter (R/D)

Velocity distribution returns to normal several pipe diameters downstream

High pressure

Low pressure

Possible separation from wall

D

g

VKh bb

2

2

g

VKh bb

2

2

Kb varies from 0.6 - 0.9

R

Head Loss in ValvesFunction of valve type and

valve positionThe complex flow path

through valves can result in high head loss (of course, one of the purposes of a valve is to create head loss when it is not fully open)

g

VKh vv

2

2

g

VKh vv

2

2

To calculate losses in piping systems with both pipe friction and minor losses use

Solution TechniquesNeglect minor lossesEquivalent pipe lengthsIterative TechniquesSimultaneous EquationsPipe Network Software

Iterative Techniques for D and Q (given total head loss)Assume all head loss is major head loss.Calculate D or Q using Swamee-Jain

equationsCalculate minor lossesFind new major losses by subtracting minor

losses from total head loss

Solution Technique: Head LossCan be solved directly

minorfl hhh minorfl hhh

g

VKhminor

2

2

g

VKhminor

2

2

5

2

2

8

D

LQ

gfh f

5

2

2

8

D

LQ

gfh f

2

9.0Re

74.5

7.3log

25.0

D

f

2

9.0Re

74.5

7.3log

25.0

D

f

42

28

Dg

QKhminor

42

28

Dg

QKhminor

D

Q4Re

D

Q4Re

Solution Technique:Discharge or Pipe DiameterIterative techniqueSet up simultaneous equations in Excel

minorfl hhh minorfl hhh

42

28

Dg

QKhminor

42

28

Dg

QKhminor

5

2

2

8

D

LQ

gfh f

5

2

2

8

D

LQ

gfh f

2

9.0Re

74.5

7.3log

25.0

D

f

2

9.0Re

74.5

7.3log

25.0

D

f

D

Q4Re

D

Q4Re

Use goal seek or Solver to find discharge that makes the calculated head loss equal the given head loss.

Example: Minor and Major LossesFind the maximum dependable flow between the

reservoirs for a water temperature range of 4ºC to 20ºC.

Water

2500 m of 8” PVC pipe

1500 m of 6” PVC pipeGate valve wide open

Standard elbows

Reentrant pipes at reservoirs

25 m elevation difference in reservoir water levels

Sudden contraction

DirectionsAssume fully turbulent (rough pipe law)

find f from Moody (or from von Karman)Find total head lossSolve for Q using symbols (must include

minor losses) (no iteration required)Obtain values for minor losses from notes or

text

Example (Continued)What are the Reynolds number in the two

pipes?Where are we on the Moody Diagram?What value of K would the valve have to

produce to reduce the discharge by 50%?What is the effect of temperature?Why is the effect of temperature so small?

Example (Continued)Were the minor losses negligible?Accuracy of head loss calculations?What happens if the roughness increases by a

factor of 10?If you needed to increase the flow by 30%

what could you do?Suppose I changed 6” pipe, what is minimum

diameter needed?