BERNOULLI’S THEOREM LAB REPORT

DEPARTMENT:CHEMICAL, MATERIALS AND METALLURGICAL

ENGINEERING
COURSE: CHEE 312
Course Coordinator: Dr Mbako Jonas
Technical Team: Mr T. Lekgoba
Laboratory Instructor: Mr Bernard Mosweu and Mr Mompati Bulayani
SUBMISSION DATE: 30 October 2023, Monday
Report by:

Name Student ID Programme

Robin Bernard 21000769 Chemical Engineering
Motheo Keipeile 21000772 Chemical Engineering
Tshegofatso Gotsileng 21000732 Chemical Engineering
Sandile Sibanda 21001330 Chemical Engineering
Govind Shain 21000768 Chemical Engineering
 i. DECLARATION
I declare that the work in this dissertation titled “BERNOULLI’S THEOREM” has been carried
out by me and my colleagues in the department of chemical, materials, and metallurgical
engineering. The information derived from the literature has been duly acknowledged in the text
and a list of references was provided. No part of this dissertation was previously presented for
another work piece of this or any other institution.

Name Student ID Signature

Robin Bernard 21000769

Motheo Keipeile 21000772

Tshegofatso Gotsileng 21000732

Sandile Sibanda 21001330

Govind Shain 21000768

 ii. ACKNOWLEDGEMENTS
We would like to take this moment to thank every personnel who helped us in the writing of this
report. This includes my lab Instructor Mr Lekgoba and lab coordinators Mr Bernard and Mr
Mompati who helped in guiding us on how to write this report.
1. ABSTRACT
TABLE OF CONTENTS
i. DECLARATION___________________________________________________________1
ii. ACKNOWLEDGEMENTS__________________________________________________2
1. ABSTRACT_______________________________________________________________3
 2. INTRODUCTION AND THEORY
Liquid stream is for the most part the movement of a liquid that's subjected to distinctive uneven
strengths. It is basically a portion of liquid mechanics and liquid stream for the most part
bargains with the elements of the liquid. The movement of the liquid proceeds till distinctive
unequal strengths are connected to the liquid. Different parameters influence the stream of a
liquid:
 Density
 Viscosity
 Pressure
Types of Liquids
 Perfect liquid - A liquid is said to be perfect when it cannot be compressed and the
viscosity doesn't drop within the category of a perfect liquid. It is a fanciful liquid which
doesn't exist in reality.
 Genuine liquid - All the liquids are genuine as all the liquid have viscosity.
 Newtonian liquid - When the liquid complies with Newton's law of viscosity, it is known
as a Newtonian liquid.
 Non-Newtonian liquid - When the liquid doesn't comply with Newton's law of
consistency, it is known as Non-Newtonian liquid.
 Perfect plastic liquid - When the shear push is corresponding to the speed slope and shear
stretch is more than the surrender esteem, it is known as perfect plastic liquid.
 Incompressible liquid - When the viscosity of the liquid doesn't alter with the application
of outside constrain, it is known as an incompressible liquid.
 Compressible liquid - When the viscosity of the liquid changes with the application of
outside constrain, it is known as compressible liquid (Bajpai, 2018).

Bernoulli's hypothesis, in liquid flow, connection among the weight, speed, and rise in a moving
liquid (fluid or gas), the compressibility and consistency (inner contact) of which are irrelevant
and the stream of which is steady, or laminar. To begin with inferred (1738) by the Swiss
mathematician Daniel Bernoulli, the hypothesis states, in impact, that the whole mechanical
vitality of the streaming liquid, comprising the vitality related with liquid weight, the
gravitational potential vitality of rise, and the motor vitality of liquid movement, remains
consistent. Bernoulli's hypothesis is the guideline of vitality preservation for perfect liquids in
unfaltering, or streamline, stream and is the premise for numerous designing applications (Ni,
2016).

Bernoulli's guideline states that the entire mechanical vitality of the moving liquid comprising
the gravitational potential vitality of rise, the vitality related with the liquid weight and the active
vitality of the liquid movement remains consistent.
Bernoulli's theorem provides a scientific implies to understanding the mechanics of liquids. It
has numerous real-world applications, extending from understanding the optimal design of a
plane; calculating wind stack on buildings; planning water supply and sewer systems; measuring
stream utilizing gadgets such as weirs, Parshall flumes, and venture-meters; and assessing
leakage through soil, etc. Vitality presents within the frame of weight, speed, and rise in liquids
with no vitality trade due to thick dissemination, warm exchange, or shaft work (pump or a few
other gadget). The relationship among these three shapes of vitality was to begin with expressed
by Daniel Bernoulli (1700-1782), based upon the preservation of energy guideline (Chanson,
2004).

Bernoulli's hypothesis suggests, subsequently, that in the event that the liquid streams evenly so
that no alter in gravitational potential vitality happens, at that point a diminish in liquid weight is
related with an increment in liquid speed. In case the liquid is streaming through a flat pipe of
shifting cross-sectional zone, for illustration, the liquid speeds up in choked regions so that the
weight the liquid applies is slightest where the cross area is littlest. This wonder is some of the
time called the Venturi impact, after the Italian researcher G.B. Venturi (1746–1822), who to
begin with famous the impacts of choked channels on liquid stream. An illustration of Bernoulli's
rule application is to decide the stream speed of a liquid. The devices such as venturi meter or an
orifice plate is used and can be placed into a pipeline to reduce the diameter of the flow. The
reduction in diameter will cause an increase in the fluid flow speed, thus according to Bernoulli’s
there must be decrease in pressure (Hubert Chanson M.E., 1996).

Assumptions:

 Steady-flow
 The fluid is non-viscous
 Incompressible fluid (density is constant).

These assumptions are also based on the laws of conservation of mass and energy. Thus, the
input mass and energy for a given control volumes are equal to the output mass and energy:
Q {in} =Q {out}
E {in} =E {out}
Bernoulli’s principle is essentially a work energy conservation principle which states that an
ideal fluid or for situations where effects of viscosity are neglected, with no work being
performed on the fluid, the total energy is constant. This principle also 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.
The pressure, speed and height (h) at two points in a steady-flowing, non-viscous,
incompressible fluid are related by the equation:

Where:

P1 = static pressure on section 1

𝑣1 = velocity on section 1
h1= height of section 1

P2= static pressure on section 2

𝑣2= velocity on section 2
h2= height of section 2
ρ= density of fluid

𝑔= acceleration of gravity
This equation is the sum of the kinetic energy, potential energy and gravitational energy
at height. If the equation was multiplied through by the volume, the density can be replaced by
mass and the pressure replaced by force times distance. As the replacement, we can see that the
difference in pressure does work which can be used to change the kinetic energy and the
potential energy of the fluid for this equation. The continuity equation of continuity states that
for an incompressible fluid flowing in a tube of varying cross-section, the mass flow rate is the
same everywhere in the tube.
The continuity equation of continuity states that for an incompressible fluid flowing in a tube of
varying cross-section, the mass flow rate is the same everywhere in the tube. The equation of
continuity can be written as:

Where:
ρ = density, A = cross sectional area, v = velocity.
The density stay at the constant and then it is simply the flow rate (Av) that is constant. The
velocity of fluid is greater in the narrow section as we applied the continuity equation.
Bernoulli’s principle also states that if a non-viscous flow along a pipe of varying cross
section. Then, an increment in the speed of the fluid simultaneously with a drop in pressure or a
decrease in the fluid’s potential energy and the pressure increases when the pipe opens out and
the fluid stagnate. Thus, pressure would decreases when the velocity increases.

Venturi meter is arranged horizontally (ℎ1 =ℎ2) in mod. HB5/EV, consequently the previous
equation becomes:

From the point of view of discharge head, the equation can be re-written as indicated here below:

Where:

Consequently:

If fluid is viscous, the frictions inside the fluid and between fluid and pipe walls produce some
Friction heads. Then Bernoulli’s equation becomes:

Where 𝐻𝑓 = friction heads.

In unit mod.HB5/EV, the discharge head measured at a wall-type static-pressure tube represents
the head 𝐻𝑠𝑡𝑎𝑡, whereas the discharge head measured by Pitot tube represents the total head, that
is the sum of static and velocity heads.

Therefore, the head difference between the water column measured by Pitot tube (𝐻𝑡𝑜𝑡) and that
measured on the wall-type pressure tube (𝐻𝑠𝑡𝑎𝑡) represents the velocity head, that is: Therefore,
the head difference between the water column measured by Pitot tube (𝐻𝑡𝑜𝑡) and that measured
on the wall-type pressure tube (𝐻𝑠𝑡𝑎𝑡) represents the velocity head, which is:
For an incompressible fluid (𝜌 = constant) the continuity equation can be rewritten as indicated
here below:

The venturi meter is a tool used to measure the flow through a pipeline. This apparatus consists
of a venturi tube and differential pressure gauge. The venturi tube has a converging portion, a
throat and a diverging portion as shown in the figure below. The converging portion’s function is
to increase the velocity of the fluid and lower its static pressure. A pressure difference between
inlet and throat is thus developed, which pressure difference is correlated with the rate of
discharge. The diverging cone serves to change the area of the stream back to the entrance area
and convert velocity head into pressure head.
3. OBJECTIVES
1. To calculate the volumetric flow rate
2. To calculate dynamic head
3. To determine the experimental value of velocity and theoretical value
4. EXPERIMENTAL EQUIPMENT AND PROCEDURE
4.1. EXPERIMENTAL APPARATUS
Bernoulli’s Demonstration – HB5/ EV

The main technical characteristics of this unit are indicated here below:

 Framework of stainless steel AISI 304

 Venturi tube of transparent Plexiglas with diameter of 20 mm and contraction of 10
mm, converging angle of 14° and diverging angle of 21°
 7-tube differential pressure gauge with range of 0-500 mm
 2 control valves of stainless steel AISI 316

Figure 1: a schematic diagram of the Bernoulli’s theorem demonstrator HB5/EV

Hydraulic Bench – HB-E/EV

Here are the main technical characteristics of this unit:

 Wheeled structure of stainless steel AISI 304

 Tank for the collection of water, with capacity of 120 liters, code D1
 Work top of stainless steel AISI 304 equipped with: - flow channel, dimensions:
170×150×740 mm - measuring tank with double capacity of 10 liters and 40 liters
 Centrifugal pump of stainless steel AISI 304, 80 l/min @ 12 m, 30 l/min @ 20 m,
code G1 Variable-area flow meter; scale of 0.4 – 4 m3 /h, code FI1
 Pressure gauge of stainless steel AISI 304, scale of 0 - 6 bar, code PI1
 Vacuum gauge of stainless steel AISI 304, scale of -1 - 3 bar , code PI2
 4.2. EXPERIMENTAL PROCEDURE
First, the Bernoulli apparatus was linked to the hydraulic bench. Second, the unit was placed on
the mod. HB-E/EV worktop, and the quick connection of the equipment was screwed into the
drain located at the bottom of the flow channel of the bench mod.HB-E/EV the inlet of the
equipment mod. HB5/EV was then linked to the bench mod.HB-E/EV via the pipe with fast
connections of the equipment mod. HB5/EV intake and outflow valves V1 and V2 were opened.
The valve V5 of mod.HB-E/EV was entirely opened, while the valve V1 of mod. HB-E/EV was
totally closed. Following that, the bleed valve (V3) of mod. HB5/EV was opened until the
pressure gauge tubes reached atmospheric pressure and then closed. The Pilot tube tip was then
positioned onto the first wall-type pressure tube. The pump G1 of mod.HB-E/EV was then
started, the valve V1 was opened, and the flow rate was gradually adjusted to approximately
1m3/hr. The closing degree of mod.HB57EV valves V1 and V2 was adjusted so that water
columns are included in the pressure gauge range. After that the flow rate was measured by using
the volumetric tank of 10 liters of mod.HB-E/EV and a chronometer. Thereafter the value of
flow rate and the corresponding values of pressure were recorded. The measurement was
repeated by positioning the point of Pitot tube onto the next wall-type pressure tube, up to the
sixth static-pressure tube. The test was repeated at various flow rates, taking care not to forget to
alter the position of valves V1 and V2 of mod.HB5/EV so that measurements were included in
the differential pressure gauge's range. Finally, at the conclusion of the test, the pump G1 of
mod.HB-E/EV was turned off.
REFERENCES
Bajpai, P., 2018. Hydraulics. Biermann's Handbook of Pulp and Paper, Volume 3, pp. 12-19.

Chanson, H., 2004. Applications of the momentum principle: hydraulic jump, surge and flow
resistance in open channels. Hydraulics of Open Channel Flow, 2(3), pp. 8-12.

Fairclough, C., 2015. Exploring the Venturi Effect.

Hubert Chanson M.E., E. G., 1996. Air Entrainment in High-Velocity Water Jet. Air Bubble
Entrainment in Free-Surface Turbulent Shear Flows, Volume 1, p. 48.

Menon, E. S., 2015. Transmission Pipeline Calculations and Simulations Manual. Meters and
Valves.

Ni, D., 2016. Engine Modeling. Traffic Flow Theory, Volume 2, pp. 9-10.
APPENDICES