Theory

The bernoullis principle is generally used in fluid dynamics. Bernoulli’s equation is applicable at

a incompressible and inviscid steady flow fluid. The equation can be denifed as :

Ppg

+ u2

2g+Z = Constant (4.1)

Where (in SI units):      

P = fluid static pressure at the cross section in N/m2.

p = density of the flowing fluid in kg/m3

g = acceleration due to gravity in m/s2 (its value is 9.81 m/s2 )

u = mean velocity of fluid flow at the cross section in m/s

z = elevation head of the center of the cross section with respect to a datum z=0

Based on Giles (1994), the bernoulli’s law indicates that a fluid is flowing along a pipe of

various cross section area. The pressure of the fluid is low at the area where the velocity is high

and the velocity is low when the pressure is high. The pressure is said to be inversely

proportional to the velocity. There are some reccomendations need to be considered in the

bernoullis theorem :

1. The fluid is incompressible

2. The flow is steady along a stream line

3. The flow is frictionless along the streamline

4. The liquid is non-viscous

However, the fluid might also suffer a head loss due to friction during flowing along the

streamline. The head loss due to friction normally encountered for the fluid which is viscid and

only small loss of head due to friction within the fluid and between the fluid and the walls of the

passage if the fluid is not inviscid. To determined the fluid flow, the bernoulli’s equation can be

modified by the inclusion of the frictional head loss :

P1

pg+

u12

2g+z1 = P2

pg+

u22

2 g+z2 + H f (4.2)

Bernoulli’s equation were applied between two different section which are section 1 and section

2. Since the passage is horizontal, the velocity and the elevation head will be equal and the

modified equation will be as below (z1=z2) and (u1=u2¿ :

H f =P1

pg−

P2

pg(4.3)

Most of the pressure loss in the converging part of the duct is recovered in the diverging part of

the duct. Besides that, the pressure and the velocity might differ due to usage of venturi meter:

The venturi meter consists of a venturi tube and differential pressure gauge. The venturi tube has

a converging portion, a throat and a diverging portion which fuction is to increase the velocity of

the fluid and lower its static pressure.

Figure 4.1 : Venturi Meter

Based on the figure, the section of the venturi meter is divided to 1 and 2. The pressure is said to

be high at point 1 and the velocity is inverse of the pressure. As the fluid flow through the stream

line, the pressure started to decrease as the velocity increase. The pressure and velocity were

varies due to the cross sectional area of the pipe.

Next, the continuity equation also must be consider in the bernoulli’s theorem. The continuity

equation is a statement of the conservation of mass. The mass flow rate of fluid at any section

along the streamtube must be constant so that :

M=pAV=Constant (4.4)

For an incompressible fluid flowing in a converging duct it follows that as the area reduces then

the velocity must increase, whilst in a diverging duct as the area increases then the velocity must

decrease. Applying Bernoulli's equation if the velocity increases then the pressure must decrease

whilst as the velocity decreases the pressure must increase.

AV=Constant (4.5)

A1 V 1=A2 V 2 (4.6)

REFERENCES

Coulson, J.M. and Richardson, J.F., (1999). Fluid Flow, Heat Transfer and Mass Transfer, Volume 1, 6th ed, Elsevier.

Bansal, R.K., (2005). Fluid Mechanics and Hydraulic Machines, 9th ed, LAXMI Publications.

Investigation of Bernoulli's theorem (2012,). Retrieved October 28, 2014 from

http://site.iugaza.edu.ps/mymousa/files/Experiment-7-hydraulics-lab-.pdf.

Bernoulli’s Theorem Demonstration (2014,). Retrieved October 28, 2014 from http://fenglab.weebly.com/uploads/2/4/3/3/24334557/c._bernoullis_theorem_demonstration. Pdf.

Bernoulli’s Principle Demonstrator Instruction Manual (n.d.). Retrieved October 28, 2014 from http://www.eng.ucy.ac.cy/EFM/Manual/HM%2015007/HM15007E.pdf.