T-Level Unit 5 Session 17

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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5. Essential science for engineering and
manufacturing

5.7 Fluid dynamics in engineering

PowerPoint 11: Bernoulli's Principle

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Objectives

By the end of this session, learners should
be able to:

•State Bernoulli’s principle and apply it
to carry out calculations

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Definitions

One aspect of fluid dynamics is the movement of fluids, for example
flowing in pipes. Some useful definitions:

•

Flow is a measure of the amount of fluid in terms of volume per unit
of time. This can be measured in terms of, for example, litres per
minute.

•

Mass flow rate is the mass of fluid passing a point in a system per
unit time. This can be measured in, for example, kg s-1.

•

Flow velocity refers to how quickly the fluid is moving. This can be
measured in, for example, m s-1.

•

Pressure is the measure of force applied on an area. This can be
measured in, for example, N mm-2or Pa.

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Viscosity

•

Most materials (and gases) occupy a smaller volume when they are
squeezed – which means they become denser.

•

The viscosity of a fluid is a measure of its resistance to
deformation at a given rate - different fluids have different levels of
viscosity.

•

For example, oil, glycerin and tar are highly viscous.

Volume at lower

pressure

Volume at higher

pressure

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T Level Technical Qualification in
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300 Engineering Common Core Content

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‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Incompressible fluids

•

An ideal hydraulic system uses
incompressible fluid.

•

This is a fluid whose density does not
change when the pressure changes. This
means that with an increase in pressure its
volume stays the same.

Volume at lower

pressure

Volume at higher

pressure

Incompressible

fluid=

volume stays
same under

pressure

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

© 2022 City and Guilds of London Institute. All rights reserved.

‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Bernoulli's Principle

In fluid dynamics, Bernoulli's principle states that an increase in the
speed of a fluid occurs simultaneously with a decrease in pressure or a
decrease in the fluid's potential energy.

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T Level Technical Qualification in
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300 Engineering Common Core Content

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‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

•The Bernoulli Equation can be considered to be a statement of the
conservation of energy principle appropriate for flowing fluids.

•The qualitative behaviour that is usually labelled 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.

Understanding Bernoulli's Principle

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T Level Technical Qualification in
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300 Engineering Common Core Content

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Pressure and Bernoulli Equation
•Pressure, P = force x area = F A

•For a hydraulic system where two points (1 and 2) lie on a streamline,
Bernoulli’s equation states:

•p + ½ ρ v2 + pgh = constant, where:

•p = pressure

•ρ = density

•v = flow velocity

•h = elevation

•g = gravity

This assumes that:

•the fluid has constant density/is incompressible

•the flow is steady

•there is no friction

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T Level Technical Qualification in
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300 Engineering Common Core Content

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‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Mass flow rate

•The volume flowing through a hydraulic system in a certain time can
be found by multiplying the cross sectional area by the flow velocity.

•V = A x v.

•Note that V is the volume and v is the flow velocity.

•Mass = density x volume = ρ V = ρ v A

•Mass flow rate

= mass / time

= (density x volume) / time

= (ρ V) / t

(which, for t=1, = ρ v A)

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T Level Technical Qualification in
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300 Engineering Common Core Content

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‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Mass flow rate example

In a hydraulic system, a pipe has a cross
section area of 120 mm2.

A fluid of density 1000 kg m3flows through the
pipe at a velocity of 1.5 m s-1.

Calculate the mass flow rate.

Answer:

Taking t = 1:

mass flow rate = ρ v A = 1000 x 1.5 x 1.2 x 10-4

mass flow rate = 0.18 kg s-1

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Changes in pipe diameter

•If the diameter of a pipe changes (such as in a tapered pipe), its
cross sectional area changes.

•When an incompressible fluid flows at a constant rate in a pipe, the
mass flow rate must be the same at all points along the pipe.

•As the mass flow rate and pressure are constant a change in the
cross sectional area changes the flow velocity.

•I.e. v1A1 = v2A2

•Using Bernoulli’s equation, the change in pressure between two
points in the system where there is a difference in diameter (such as
in a tapered pipe) can be expressed as:

•p1 – p2 = ½ ρ (v2

2 – v1

2)

•Note: this assumes that the difference due to gravity is negligible.

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T Level Technical Qualification in
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300 Engineering Common Core Content

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‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Example calculation
In a hydraulic system, an incompressible fluid is pumped at a
constant mass flow rate through a tapered pipe.

The fluid is pumped into a pipe of area 1.6 x 10-3m2, where the flow
velocity is measured at 0.2 m s-1and the pressure is 80 bar. The
density of the fluid is 900 kg m-3.

Calculate the pressure where the area of the pipe has reduced to 0.8
x 10-3m2.

Answer:

Rearranging v1A1 = v2A2, v2 = v1A1 / A2
v2 = 0.2 x 1.6 x 10-3/ 0.8 x 10-3= 0.4 m s-1

Rearranging p1 – p2 = ½ ρ (v2

2 – v1

2), p2 = p1 – ½ ρ (v2

2 – v1

2)

p2 = 80 – ½ 900 (0.42– 0.22) = 36 bar

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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‘T-LEVELS’ is a registered trade mark of the Department for Education.
‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

Bernoulli's principle can also be used to calculate the lift force on an
aerofoil, if the behaviour of the fluid flow in the vicinity of the foil is known.

For example, if the air flowing past the top surface of an aircraft wing is
moving faster than the air flowing past the bottom surface, then Bernoulli's
principle implies that the pressure on the surfaces of the wing will be lower
above than below.

Bernoulli’s principle in flight

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T Level Technical Qualification in
Engineering and Manufacturing (Level 3)
300 Engineering Common Core Content

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‘T Level’ is a registered trade mark of the Institute for Apprenticeships and Technical Education

This pressure difference results in an upwards lifting force (lift).

Whenever the distribution of speed past the top and bottom surfaces of a
wing is known, the lift forces can be calculated (to a good approximation)
using Bernoulli's equations.

Bernoulli’s principle in flight

Open Ended

Worksheet 11: Bernoulli’s Principle

1.    Water is flowing in a fire hose with a velocity of 1.0 m/s and a pressure of 200000 Pa. At the nozzle the pressure decreases to atmospheric pressure (101300 Pa), there is no change in height.

 

Use the Bernoulli equation to calculate the velocity of the water exiting the nozzle. (The density of water is 1000 kg/m3 and gravity g is 9.8 m/s²)

Open Ended

Worksheet 11: Bernoulli’s Principle

Through a refinery, fuel ethanol is flowing in a pipe at a velocity of 1 m/s and a pressure of 101300 Pa. The refinery needs the ethanol to be at a pressure of 2 Atm (202600 Pa) on a lower level.

 

How far must the pipe drop in height in order to achieve this pressure?

 

Assume the velocity does not change. 

(The density of ethanol is 789 kg/m3 and gravity g is 9.8 m/s²)