Bernoulli's Principle (with answers)

Bernoulli's Principle

Topic 5: Essential Science for Engineering & Manufacturing

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T-Levels in Engineering & Manufacturing

​Lesson Objectives

​To be able to:

• Recall the 3 forms of energy considered by Bernoulli's Principle

• State Bernoulli's Principle, including the modified version for horizontal pipes

• Use Bernoulli's Principle to solve engineering problems involving the flow of fluid through pipes

​Volumetric Flow Rate & Mass Flow Rate

Last week's lesson

Principle of Volumetric flow rate:

mkm

Volume of fluid entering inlet = Volume of fluid leaving outlet

Principle of Mass flow rate:

mkm

Mass of fluid entering inlet = Mass of fluid leaving outlet

Bernoulli's Principle

Bernoulli's principle is an extension of the principles of volumetric flow rate and mass flow rate, and studies fluid flow through the principle of conservation of energy, i.e:

Energy of fluid system at inlet = Energy of fluid system at outlet

Bernoulli's Principle

Bernoulli's Principle takes into consideration 3 forms of energy of fluids:

1. Flow Energy (F.E)

2. Potential Energy (P.E)

3. Kinetic Energy (K.E)

Bernoulli's Principle

1) Flow Energy (F.E)

F.E = p

Where:

p = pressure (on the fluid) [Pa]

Bernoulli's Principle

2) Potential Energy (P.E)

P.E = ρ ⋅ g ⋅ h

Where:

ρ = density of the fluid [kg/m³]

g = acceleration due to gravity [m/s²] (always 9.81 m/s²)

h = elevation of fluid (from a given base level) [m]

Bernoulli's Principle

3) Kinetic Energy (K.E)

K.E = ½ ⋅ ρ ⋅ ν²

Where:

ρ = density of the fluid [kg/m³]

ν = velocity of the fluid [m/s]

Bernoulli's Principle

Therefore, Bernoulli's Principle becomes:

p₁ + ρ ⋅ g ⋅ h₁ + ½ ⋅ ρ ⋅ ν₁² = p₂ + ρ ⋅g ⋅ h₂ + ½ ⋅ ρ ⋅ ν₂² = C

In actual fact though, there is always an appreciable loss of energy at the outlet due to friction → called Head Loss, H_L

p₁ + ρ ⋅ g ⋅ h₁ + ½ ⋅ ρ ⋅ ν₁² = p₂ + ρ ⋅g ⋅ h₂ + ½ ⋅ ρ ⋅ ν₂² + H_L

But at Level 3 you do not need to worry about this and you can assume that there is no energy loss as the fluid travels through the pipe.

Bernoulli's Principle

For a horizontal pipe (consistent elevation throughout), P.E can be ignored and so Bernoulli's Principle becomes:

p₁ + ½ ⋅ ρ ⋅ ν₁² = p₂ + ½ ⋅ ρ ⋅ ν₂² = C

Multiple Choice

According to Bernoulli's principle as stated on the previous slide, what happens to the pressure of the fluid as the velocity of the fluid increase?

• Increases

• Decreases

• Stays the same

​You will now be using Bernoulli's Principle to solve 10 questions - marks for these questions will go towards your final score for this lesson so please take your time to answer these questions.

p₁ + ρ ⋅ g ⋅ h₁ + ½ ⋅ ρ ⋅ ν₁² = p₂ + ρ ⋅g ⋅ h₂ + ½ ⋅ ρ ⋅ ν₂² = C

p₁ + ½ ⋅ ρ ⋅ ν₁² = p₂ + ½ ⋅ ρ ⋅ ν₂² = C

​Questions

If you haven't already, make sure you make a note of the formulas for Bernoulli's Principle now:

Multiple Choice

Question 1

Pure water flows up a tapering pipe as shown in the diagram. Use Bernoulli's Principle to determine the pressure on the fluid at the outlet of the pipe.

[Keep calculations and answer to 3 d.p]

• 201 Pa

• 314 Pa

• 455 Pa

• 524 Pa

​Answer

Multiple Choice

Question 2

Hydraulic fluid of density 1200 kg/m³ flows up a tapering pipe as shown in the diagram. Use Bernoulli's Principle to determine the pressure on the fluid at the inlet of the pipe.

[Keep calculations and answer to 3 d.p]

• 8.111 kPa

• 14.32 kPa

• 22.22 kPa

• 35.43 kPa

​Answer

Multiple Choice

Question 3

Pure water flows up a tapering pipe as shown in the diagram. Use Bernoulli's Principle to determine the velocity of the fluid at the outlet of the pipe.

[Keep calculations and answer to 3 d.p]

• 4.872 m/s

• 5.766 m/s

• 6.452 m/s

• 7.211 m/s

​Answer

Multiple Choice

Question 4

Hydraulic oil of density 1150 kg/m³ flows up a tapering pipe as shown in the diagram. Use Bernoulli's Principle to determine the velocity of the fluid at the inlet of the pipe.

[Keep calculations and answer to 3 d.p]

• 1.172 m/s

• 2.823 m/s

• 4.566 m/s

• 5.131 m/s

​Answer

Multiple Choice

Question 5

Glycerine flows up a tapering pipe as shown in the diagram. Use Bernoulli's Principle to determine the height of the outlet of the pipe from the ground.

[Keep calculations and answer to 3 d.p]

• 0.45 m

• 0.56 m

• 0.68 m

• 0.78 m

​Answer

Multiple Choice

Question 6

Petrol (vehicle) flows through an industrial pipe as shown in the diagram. Use Bernoulli's Principle to determine the height of the outlet referenced from the height of the inlet.

[Keep calculations and answer to 3 d.p]

• 1.70 m

• 2.52 m

• 2.88 m

• 3.22 m

​Answer

Multiple Choice

Question 7

Pure water flows through a horizontal pipe as shown in the diagram. Use Bernoulli's Principle to determine the pressure on the fluid at the outlet.

[Keep calculations and answer to 3 d.p]

• 2.95 kPa

• 4.52 kPa

• 6.65 kPa

• 8.25 kPa

​Answer

Multiple Choice

Question 8

Acetone flows through a horizontal pipe as shown in the diagram. Use Bernoulli's Principle to determine the pressure on the fluid at the inlet.

[Keep calculations and answer to 3 d.p]

• 1.254 kPa

• 2.776 kPa

• 4.115 kPa

• 5.725 kPa

​Answer

Multiple Choice

Question 9

Crude Oil (Texas) flows through a horizontal pipe as shown in the diagram. Use Bernoulli's Principle to determine the velocity of the fluid at the outlet.

[Keep calculations and answer to 3 d.p]

• 1.0 m/s

• 0.9 m/s

• 0.8 m/s

• 0.7 m/s

​Answer

Multiple Select

Question 10

Pure water flows through three sections of a horizontal, gradually tapering pipe as shown in the diagram. If the velocity of the fluid through Section 1 is 3.0 m/s, use Bernoulli's Principle to determine the velocity of the fluid as it moves through Section 2 and 3 of the pipe. Choose 2 correct answers.

[Keep calculations and answer to 3 d.p]

• v₂ = 2.974 m/s

• v₂ = 2.828 m/s

• v₃ = 2.739 m/s

• v₃ = 2.542 m/s

​Answer

End of Lesson

Please complete the exit poll on the final slide.

Poll

Which lesson objectives do you feel you have achieved from this lesson?

• Recall the 3 forms of energy considered by Bernoulli's Principle

• State Bernoulli's Principle, including the modified version for horizontal pipes

• Use Bernoulli's Principle to solve engineering problems involving the flow of fluid through pipes