Bernoulli's Theorem

Experiment No. 1: Bernoulli’s Theorem
Object:
To verify Bernoulli's theorem for a viscous and incompressible fluid.
Theory:
In our daily lives we consume a lot of fluid for various reasons. This fluid is delivered through a network of pipes and fittings of different sizes from an overhead tank. The estimation of losses in these networks can be done with the help of this equation which is essentially principle of conservation of mechanical energy.
Formal Statement:
Bernoulli's Principle is essentially a work energy conservation principle which states that for an ideal fluid or for situations where effects of viscosity are neglected, with no work being performed on the fluid, total energy remains constant. Bernoulli's Principle is named in honour of Daniel Bernoulli. This principle is a simplification of Bernoulli's equation, which states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path.
Mathematical Description:
A+dA
ρ+dρ,V+dV
P+dP

τ =0
A
ds

dz θ ρ,V

CV

P dW Figure 1 Forces and Fluxes for Bernoulli’s Equation for frictionless flow along a streamline.

Applying the conservation of mass and momentum equation yields the following equation BournoulliEquation

1

∂V dP ds +
+ VdV + gdz = 0
(1.1)
∂t ρ Equation (1.1) is the unsteady frictionless flow along a streamline. Integrating, the above equation between any two points 1 and 2 on a streamline.
2
2
∂V
dP 2 1 ds + ∫
+ ∫ dV 2 + ∫ gdz = 0
∫ ∂t
1
1ρ
12
1
2

2
∂V
dP V22 − V12
+
+ g(z 2 − z1 ) = 0 ds + ∫
∫ ∂t
2
1
1ρ
2

(1.2)

For any steady incompressible flow, equation 1.2 reduces to

P2 − P1

ρ

P

or

V22 − V12
+
+ g(z 2 − z1 ) = 0
2
V2

ρ

+

2

(1.3)

+ gz = Cons tan t

(1.4)

2
PV
+
+ z = H (Total Head) ρg 2g

h=

P
+z
ρg

(1.5)
Peizometric
Tubes

Water y x
Centerline
Figure 2 Block Diagram of the set