1. Chapter 6: Bernoulli and energy equations
University of Palestine
College of Engineering & Urban Planning
Applied Civil Engineering
Chapter 6: Bernoulli and energy equations
Lecturer:
Eng. Eman Al.Swaity
Fall 2009

2. OBJECTIVES Derive the Bernoulli (energy) equation.
Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow.
understand the use of hydraulic and energy grade lines.
Apply Bernoulli Equation to solve fluid mechanics problems (e.g. flow measurement).
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

3. The Bernoulli Equation
Let us first derive the Bernoulli equation, which is one of the most well-known equations of motion in fluid mechanics, and yet is often misused. It is thus important to understand its limitations, and the assumptions made in the derivation.
The assumptions can be summarized as follows:
Inviscid flow (ideal fluid, frictionless)
Steady flow (unsteady Bernoulli equation will not be discussed in this course)
Applied along a streamline
Constant density (incompressible flow)
No shaft work or heat transfer
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

4. The Bernoulli Equation Limitations
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

5. The Bernoulli Equation
The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and is valid in regions of steady, incompressible flow where net frictional forces are negligible.
Equation is useful in flow regions outside of boundary layers and wakes, where the fluid motion is governed by the combined effects of pressure and gravity forces.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

6. Acceleration of a Fluid Particle
Describe the motion of a particle in terms of its distance s along a streamline together with the radius of curvature along the streamline.
The velocity of a particle along a streamline is V = V(s, t) = ds/dt
The acceleration can be decomposed into two components: streamwise acceleration as along the streamline and normal acceleration an in the direction normal to the streamline, which is given as an = V2/R.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

7. Acceleration of a Fluid Particle
Note that streamwise acceleration is due to a change in speed along a streamline, and normal acceleration is due to a change in direction.
The time rate change of velocity is the acceleration
In steady flow, the acceleration in the s direction becomes
(Proof on Blackboard)
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

8. Derivation of the Bernoulli Equation
Applying Newton’s second law in the s-direction on a particle moving along a streamline in a steady flow field gives
The force balance in s direction
gives
where
and
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

9. Derivation of the Bernoulli Equation
Therefore,
Integrating steady flow along a streamline
Steady, Incompressible flow 
This is the famous Bernoulli equation.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

10. Derivation of the Bernoulli Equation
This is the Bernoulli equation, consisting of three energy heads
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

11. Derivation of the Bernoulli Equation
A head corresponds to energy per unit weight of flow and has dimensions of length.
Piezometric head = pressure head + elevation head, which is the level registered by a piezometer connected to that point in a pipeline.
Total head = piezometric head + velocity head.
Applying the Bernoulli equation to any two points on the same streamline, we have
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

12. The Bernoulli Equation
Without the consideration of any losses, two points on the same streamline satisfy
where P/r as flow energy, V2/2 as kinetic energy, and gz as potential energy, all per unit mass.
The Bernoulli equation can be viewed as an expression of mechanical energy balance
Was first stated in words by the Swiss mathematician Daniel Bernoulli (1700–1782) in a text written in 1738.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

13. Example 1 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

14. Example 2 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

15. Example 2 -29.9 kpa EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

16. Example 3 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

17. Example 3 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

18. The Energy Equation
The energy equation is more general than the Bernoulli equation, because it allows for (1) friction, (2) heat transfer, (3) shaft work, and (4) viscous work (another frictional effect).
Where Ws is the shaft work , hL, called the head loss,
In the absence of these two terms, the energy equation is identical to the Bernoulli equation
We must remember however that the Bernoulli equation is a momentum equation applicable to a streamline and the energy equation above is applied between two sections of a flow
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

19. The Energy Equation EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

20. The Energy Equation EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

21. Example 1 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

22. Example 2 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

23. Example 2 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

24. Example 3
Pump draws water from reservoir (A) and lifts it to a higher reservoir (B), as shown below, the head loss from A to the pump = 4v2/2g and the head loss from the pump to B = 7 v2/2g. compute the pressure head the pump must deliver. The pressure head at the inlet of pump = -6m. B A
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

25. Example 3 EGGD3109 Fluid Mechanics2 3
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

26. HGL and EGL
It is often convenient to plot mechanical energy graphically using heights.
P/rg is the pressure head; it represents the height of a fluid column that produces the static pressure P.
V2/2g is the velocity head; it represents the elevation needed for a fluid to reach the velocity V during frictionless free fall.
z is the elevation head; it represents the potential energy of the fluid.
H is the total head.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

27. HGL and EGL Hydraulic Grade Line (HGL)
Energy Grade Line (EGL) (or total head)
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

28. HGL and EGL EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

29. Something to know about HGL and EGL:
For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid, since the velocity is zero and the static pressure (gage) is zero.
The EGL is always a distance V2/2g above the HGL.
In an idealized Bernoulli-type flow, EGL is horizontal and its height remains constant. This would also be the case for HGL when the flow velocity is constant .
For open-channel flow, the HGL coincides with the free surface of the liquid, and the EGL is a distance V2/2g above the free surface.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

30. Something to know about HGL and EGL:
At a pipe exit, the pressure head is zero (atmospheric pressure) and thus the HGL coincides with the pipe outlet.
The mechanical energy loss due to frictional effects (conversion to thermal energy) causes the EGL and HGL to slope downward in the direction of flow.
A steep jump occurs in EGL and HGL whenever mechanical energy is added to the fluid. Likewise, a steep drop occurs in EGL and HGL whenever mechanical energy is removed from the fluid.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

31. Something to know about HGL and EGL:
The pressure (gage) of a fluid is zero at locations where the HGL intersects the fluid. The pressure in a flow section that lies above the HGL is negative, and the pressure in a section that lies below the HGL is positive.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

32. Something to know about HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

33. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

34. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

35. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

36. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

37. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

38. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

39. Examples on HGL and EGL:
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

40. Static, Dynamic, and Stagnation Pressures
The Bernoulli equation
P is the static pressure; it represents the actual thermodynamic pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables.
rV2/2 is the dynamic pressure; it represents the pressure rise when the fluid in motion.
rgz is the hydrostatic pressure, depends on the reference level selected.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

41. Static, Dynamic, and Stagnation Pressures
The sum of the static, dynamic, and hydrostatic pressures is called the total pressure (a constant along a streamline).
The sum of the static and dynamic pressures is called the stagnation pressure,
The fluid velocity at that location can be calculated from
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

42. APPLICATIONS OF BERNOULLI & MOMENTUM EQUATION
Pitot tube.
Changes of pressure in a tapering pipe:
Orifice and vena contracta.
Venturi, nozzle and orifice meters.
Force on a pipe nozzle.
Force due to a two-dimensional jet hitting an inclined plane.
Flow past a pipe bend.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

43. PITOT TUBE EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

44. PITOT TUBE EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

45. PITOT TUBE IN THE PIPE-(method1)
Two piezometers
for ideal flow
To account for real fluid effects, the equation can be modified into
where is the coefficient of velocity to be determined empirically.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

46. PITOT TUBE IN THE PIPE-(method2)
Using a static pressure taping in the pipe
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

47. PITOT TUBE IN THE PIPE-(method2)
Example
Sol.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

48. PITOT TUBE IN THE PIPE-(method3)
Using combined Pitot static tube
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

49. PITOT TUBE IN THE PIPE-(method3)
Using combined Pitot static tube. In which the inner tube is
used to measure the impact pressure while the outer sheath
has holes in its surface to measure the static pressure
The total pressure is know as the stagnation pressure (or total pressure)
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

50. PITOT TUBE IN THE PIPE-(method3)
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

51. CHANGES OF PRESSURE IN ATAPERING PIPE.
Changes of velocity in a tapering pipe were determined by using the continuity equation.
Changes of velocity will accompanied by a changed in pressure, modified by any changed in elevation or energy loss, which can be determined by the use of Bernoulli's equation.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

52. CHANGES OF PRESSURE IN ATAPERING PIPE-Example
Find:
¢ the pressure difference across the 2m length ignoring any losses of
energy.
¢ the difference in level that would be shown on a mercury manometer connected across this length.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

53. CHANGES OF PRESSURE IN ATAPERING PIPE-Example
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

54. CHANGES OF PRESSURE IN ATAPERING PIPE-Example
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

55. Orifice and vena contracta
We are to consider the flow from a tank through a hole in the side close to the base
Looking at the streamlines you can see how they contract after the orifice to a minimum cross section where they all become parallel, at this point, the velocity and pressure are uniform across the jet. This convergence is called the vena contracta (from the Latin 'contracted vein'). It is necessary to know the amount of contraction to allow us to calculate the flow.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

56. Orifice and vena contracta
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

57. Orifice and vena contracta
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

58. Venturi , nozzle and orifice meters
The Venturi, nozzle and orifice-meters are three similar types of devices for measuring discharge in a pipe.
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

59. A Venturi meter in laboratory
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

60. Venturi Meter EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

61. Venturi Meter EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

62. Venturi Meter EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

63. Venturi Meter This is also the theoretical discharge in terms of
manometer readings
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

64. Venturi Meter Horizontal Venturimeter EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

65. Venturi Meter Example 1 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

66. Venturi Meter Example 2 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

67. Venturi Meter EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

68. Venturi Meter Example 3 EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

69. Venturi Meter EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations

70. PIPE ORIFICES
¢ A similar effect as the venturi meter can be achieved by inserting
an orifice plate
¢ The orifice plate has an opening in it smaller than the internal pipe
diameter
EGGD3109 Fluid Mechanics
Chapter 6: Bernoulli and energy equations