1. C.k.pithawalla college of enginnering & technology
Prepared by:
Sr.no
Name
Enrollment no. 1
Nayani paril 2
Patel Nikunj 3
Parmar Mihir 4
Patel Adarsh 5
Patel Bhavin

2. Topics:
Euler’s Euqation
Bernolli’s Equation
Energy Correction Factor

3. Euler’s Equation Define a function along a trajectory.
y(a,x) = y(0,x) + ah(x)
Parametric function
Variation h(x) is C1 function.
End points h(x1) = h(x2) = 0
Find the integral J
If y is varied J must increase x2
y(a, x)
y(x) x1

4. Parametrized Integral
Write the integral in parametrized form.
Condition for extremum
Expand with the chain rule
Term a only appears with h
Apply integration by parts …
for all h(x)

5. Euler’s Equation It must vanish for all h(x) This is Euler’s equation
h(x1) = h(x1) = 0
It must vanish for all h(x)
This is Euler’s equation

6. Geodesic
A straight line is the shortest distance between two points in Euclidean space.
Curves of minimum length are geodesics.
Tangents remain tangent as they move on the geodesic
Example: great circles on the sphere
Euler’s equation can find the minimum path.

7. Soap Film Find a surface of revolution. Find the area y
Minimize the function y
(x2, y2)
(x1, y1)

8. Action next Motion involves a trajectory in configuration space Q.
Tangent space TQ for full description.
The integral of the Lagrangian is the action.
Find the extremum of action
Euler’s equation can be applied to the action
Euler-Lagrange equations Q q’ q
next

9. Energy Conservation (Bernoulli’s Equation)
Integration of Euler’s equation
Bernoulli’s equation
Flow work + kinetic energy + potential energy = constant Dx
Under the action of the pressure, the fluid element
moves a distance Dx within time Dt
The work done per unit time DW/Dt (flow power) is p A

10. Energy Conservation (cont.)
It is valid for incompressible fluids, steady flow along a streamline,
no energy loss due to friction, no heat transfer.
Determine the velocity and mass flow rate of efflux
from the circular hole (0.1 m dia.) at the bottom of the
water tank (at this instant). The tank is open to the
atmosphere
and H=4 m
Examples: 1
p1 = p2, V1=0 H 2

11. Energy Equation(cont.)
Example: If the tank has a cross-sectional area of 1 m2, estimate
the time required to drain the tank to level 2. 1
First, choose the control volume as enclosed
by the dotted line. Specify h=h(t) as the water
level as a function of time.
h(t) 2 20 40 60 80
100 1 2 3 4
time (sec.)
water height (m)
2.5e-007 h ( ) t

12. Energy conservation (cont.)
Energy added, hA
(ex. pump, compressor)
Generalized energy concept:
Energy loss, hL
(ex. friction, valve, expansion)
Energy extracted, hE
(ex. turbine, windmill) hL
loss through
valves
heat exchanger hA hE
pump
turbine
hL, friction loss
through pipes hL
loss through
elbows
condenser

13. Energy conservation(cont.)
Examples: Determine the efficiency of the pump if the power input of the motor
is measured to be 1.5 hp. It is known that the pump delivers 300 gal/min of water.
6-in dia. pipe
4-in dia.pipe
hE=hL=0, z1=z2 1
pump 2
Q=300 gal/min=0.667 ft3/s=AV
V1=Q/A1=3.33 ft/s
V2=Q/A2=7.54 ft/s zo
Z=15 in
Mercury (m=844.9 lb/ft3)
water (w=62.4 lb/ft3)
1 hp=550 lb-ft/s

14. Energy conservation (cont.)
Example (cont.)

15. Frictional losses in piping system
R: radius, D: diameter
L: pipe length
w: wall shear stress P2
Consider a laminar, fully developed circular pipe flow w p
P+dp
Darcy’s Equation:

17. Energy Conservation (cont.)
Energy: E=U(internal thermal energy)+Emech (mechanical energy)
=U+KE(kinetic energy)+PE(potential energy)
Work: W=Wext(external work)+Wflow(flow work)
Heat: Q heat transfer via conduction, convection & radiation
dE=dQ-dW, dQ>0 net heat transfer in dE>0 energy increase and vice versa
dW>0, does positive work at the expense of decreasing energy, dE<0
U=mu, u(internal energy per unit mass), KE=(1/2)mV2, PE=mgz
Wflow=m(p/)
Their difference is due to external heat transfer and work done on flow

18. Energy Conservation (cont.)
Heat in q=dQ/dt
Work out dW/dt

19. Energy Conservation(cont.)
Example: Superheated water vapor is entering the steam turbine with a mass
flow rate of 1 kg/s and exhausting as saturated steam as shown. Heat loss from
the turbine is 10 kW under the following operating condition. Determine the
power output of the turbine.
From superheated vapor table:
hin= kJ/kg
P=1.4 Mpa
T=350 C
V=80 m/s
z=10 m
10 kw
P=0.5 Mpa
100% saturated steam
V=50 m/s
z=5 m
From saturated steam table: hout= kJ/kg

20. Energy correction factor
Kinetic energy per unit mass
Potential energy per unit mass

21. Energy Equation
Reynolds Transport Theorem
b = e ; Bsys = E

22. Work
Rate of Work
Shaft work work done by a mechanical device which crosses the CS
Flow work work done by pressure forces on the CS
Viscous work work done by viscous stresses at the CS

23. Flow Work
Work occurs at the CS when a force associated with the normal stress of the fluid acts over a distance. The normal stress equals the negative of the fluid pressure.

24. Energy Equation

25. Kinetic Energy Correction Factor
For nonuniform flows, this term requires special attention
We can modify this kinetic energy term by a dimensionless factor, a, so that the integral is proportional to the square of the average velocity
where
and
Kinetic energy correction factor so

26. Energy Equation for Pipe Flow

27. Thank You