SLIDE ON BERNOULLI'S EQUATION

1. BERNOULLI’S EQUATION
AND ITS SIGNIFICANCE
NAME: RUDRASHIS BISWAS
DEPARTMENT: CHEMICAL ENGINEERING
SEMESTER: 3rd
CLASS ROLL NO.: 18/CHE/25
UNIVERSITY ROLL NO.: 10300618025
COURSE: FLUID MECHANICS
COLLEGE: HALDIA INSTITUTE OF TECHNOLOGY

2. Bernoulli’s Principle
THEORY - STATEMENT
 Increase in the speed of the fluid occur simultaneously with a decrease in pressure
or a decrease in the fluid’s potential energy.
 In a horizontal pipe, the highest fluid pressure is in the section where the flow
speed is the lowest, and the lowest pressure is at the section where the flow speed
is the biggest.

3. The Bernoulli’s Equation
 The Bernoulli Equation can be
considered to be a statement of
the conservation of energy principle
appropriate for flowing fluids. The
qualitative behavior that is usually
labeled with the term "Bernoulli
effect" is the lowering of fluid
pressure in regions where the flow
velocity is increased. This lowering of
pressure in a constriction of a flow
path may seem counterintuitive, but
seems less so when you consider
pressure to be energy density. In the
high velocity flow through the
constriction, kinetic energy must
increase at the expense of pressure
energy.

4. Different Forms of Bernoulli’s Equation

5. Deriving Bernoulli’s Equation
Mechanism of fluid flow is a complex process. However, it is possible to get some important
properties with respect to streamline flows by using the concept of conservation of energy. Let us
take an example of any fluid moving inside a pipe. The pipe has different cross-sectional areas in
different parts and is present in different heights. Refer to the diagram below.
 Now we will consider that an incompressible fluid will flow through this pipe in a steady
motion. As per the concept of the equation of continuity, the velocity of the fluid should
change. However, to produce acceleration, it is important to produce a force. This is possible
by the fluid around it but the pressure must vary in different parts.

6. General Expression of Bernoulli’s Equation
 Let us consider two different regions in the above diagram. Let us
name the first region as BC and the second region as DE. Now
consider the fluid was previously present in between B and D.
However, this fluid will move in a minute (infinitesimal) interval of
time (∆t).
 If the speed of fluid at point B is v1 and at point D is v2. Therefore,
if the fluid initially at B moves to C then the distance is v1∆t.
However, v1∆t is very small and we can consider it constant across
the cross-section in the region BC.
 Similarly, during the same interval of time ∆t the fluid which was
previously present in the point D is now at E. Thus, the distance
covered is v2∆t. Pressures, P1 and P2, will act in the two regions,
A1 and A2, thereby binding the two parts. The entire diagram will
look something like the figure given below.

7. Change in Gravitational Potential and
Kinetic Energy
 Now, we have to calculate the change in gravitational potential energy ∆U.
 Similarly, the change in ∆K or kinetic energy can be written as

8. Calculation of Bernoulli’s Equation
 Applying work-energy theorem in the volume of the fluid, the equation will be
 Dividing each term by ∆V, we will obtain the equation
 Rearranging the equation will yield
 The above equation is the Bernoulli’s equation. However, the 1 and 2 of both the
sides of the equation denotes two different points along the pipe. Thus, the
general equation can be written as

9. Bernoulli’s Equation
APPLICATIONS
 Pumps
 Ejectors
 Carburetor
 Siphon
 Pilot Tube

10. Application in Pumps
 Volute in the casing of centrifugal pump converts velocity of fluid into pressure
energy by increasing area of flow.
 The conversion of kinetic energy into pressure is according to Bernoulli’s Equation.

11. Application in Ejectors
 Ejectors are designed to convert the pressure energy of a motivating fluid to
velocity energy to entrain suction fluid and then to recompress the mixed fluids by
converting velocity energy back into pressure energy.
 Ejectors are composed of three basic parts: a nozzle, a mixing chamber and a
diffuser.

12. Application in Carburetor
 The carburettor works on Bernoulli’s Principle: the faster air moves, the lower its
static pressure and the lighter its dynamic pressure.
 The throttle (accelerator) linkage does not directly control the flow of liquid fuel.
Instead, it actuates carburettor mechanism which meter the flow of air being pulled
into the engine. The speed of this flow, and therefore its pressure, determines the
amount of fuel drawn into the air stream.

13. Application in Siphon
 Siphon, a bent tube used to move a liquid over an obstruction to a lower level
without pumping. A siphon is most commonly used to remove a liquid from its
container. The siphon tube is bent over the edge of the container, one end in the
liquid and the other outside end at a lower level than the surface of the liquid in
the container.

14. Application in Pilot Tube
 Pilot Tube is a pressure measurement instrument used to measure fluid flow
velocity.
 Pilot Tubescan be used to indicate fluid flow velocity by measuring the difference
between the static and dynamic pressures in fluids.

15. Limitations of Application of Bernoulli’s
Equation
 One of the restrictions is that some amount of energy will be lost due to internal
friction during fluid flow. This is because fluid has separate layers and each layer of
fluid will flow with different velocities. Thus, each layer will exert some amount of
frictional force on the other layer thereby losing energy in the process.
 The proper term for this property of the fluid is viscosity. Now, what happens to the
kinetic energy lost in the process? The kinetic energy of the fluid lost in the process
will change into heat energy. Therefore, we can easily conclude that Bernoulli’s
principle is applicable to non-viscous fluids (fluids with no viscosity).

16. Conclusion
 From the result obtained, we can conclude that the Bernoulli’s equation is valid for
flow as it obeys the equation. As the area decreases at a section velocity increases
and the pressure decreases.

17. Reference
 Unit Operations of Chemical Engineering by Warren L. McCabe, Julian C. Smith,
Peter Harriott
 Geankoplis, C. J. Transport Processes and Unit Operations
 en.Wikipedia.org
 www.khanacademy.com
 hyperphysics.phy-astr.gsu.edu