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?