Unravelling the mystery surrounding Bernoullis Theorem

This presentation has been prepared to assist those sitting the science paper of the Membership Examination.  We will dispel the mystery surrounding Bernoulli’s Theorem in an easily understood manner and then to show how to apply this to previous examination questions by giving model answers.

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Daniel Bernoulli was a Dutch born Swiss Scientist, who discovered basic principles of fluids.  The Bernoulli principle is that a fluid (liquid or gas) in motion can have three types of energy

• Potential energy
• Kinetic energy
• Pressure energy

These can be interchanged but unless energy is taken out (e.g. turbulence or friction) or energy is put in (e.g. pump) then the total energy must be constant.

The frictional loss is neglected in calculations due to being small compared to the total energy, however, you must consider frictional loss in certain circumstances e.g. sprinkler calculations

To use Bernoulli’s theorem in calculations it is important to have all three forms of energy in the same units. The Systems International (SI) unit for energy is the Joule (kg.m2/s2) however; when using Bernoulli the energy is expressed per unit mass or per unit volume.

Therefore, there are different forms of the Bernoulli Equation depending on whether we are working with either joules per kilogram (j/kg) or joules per meter cubed (j/m3). In order to simplify the matter I am only going to use the Bernoulli Equation that expresses the energy in the form of joules per metre3 (j/m3) which I believe is easier to apply to IFE examination questions.

Potential energy

This is the energy due to the potential above the datum line from which all the energies are measured.  The potential energy per m3 of fluid can be considered as

ρgH    (Joules/m3)

Where

p = density (kg/m3)

g = acceleration due to gravity (9.81 m/sec2)

H = height (m)

Kinetic energy

The kinetic energy is due to the fluid being in motion.  The Kinetic energy can be considered as

½ ρv2       (Joules/m3)

Where             p = density (kg/m3)

V = Velocity (m/sec)

Pressure energy

The pressure energy is due to being under pressure.  The SI unit of pressure is Pascal but in the Fire Service, the Bar and metres head are still used.  Therefore, you must remember to use the correct formula

To convert from Bar to Pascal’s you use the following

P  X 100,000      (Joules/m3)

Where      P = Pressure (Bar)

To convert metres head pressure to Pascal’s, you use the following

Where             ρ     = Density (kg/m3)

g = acceleration due to gravity 9.81 m/sec2

z   = height (m)

Bernoulli – Pascal’s

Where the pressure energy is Pascal’s

PA + ρgHA + ½ ρvA2     = PB + ρgHB + ½ ρvB2

Where             PA is pressure energy at point A (joules)

ρgHA is the potential energy at point A ( joules)

ρ = Density of fluid (kg/m3)

g = Acceleration due to gravity 9.81 m/sec

H = height of column of water

½ ρvA  is the kinetic energy at point A (joules)

V = velocity m/sec

Where the pressure energy is Bar

PA x 100,000+ ρgHA + ½ ρvA2 = PB x100, 000+ ρgHB + ½ ρvB2

Where the pressure energy is metres head

ρgzA + ρgHA + ½ ρvA2     = ρgzB + ρgHB + ½ ρvB2

Continuity equation

When considering Bernoulli it is also very important to understand the continuity equation.  This is due to the fact that in a closed system the rate of flow Q (m3 /sec) can be considered as constant.

Q = VA

Where            Q = Rate of flow (m3/sec)

V = Velocity (m/sec)

A = Area (m2)

If the flow is constant then

Q = VAAA  = VBAB

This is shown here in this diagram showing a pipe As the water flows down the pipe and it tapers out what you will find is that the waters velocity will reduce. In other words as the area increases the velocity falls. This is a very important relationship when attempting Bernoulli calculations as will be shown later.    Before we attempt questions involving the use of these equations, I would like to give you a few tips to ensure mistakes are not made.

Tip one

Produce a sketch and enter all the details given in the question first.  This will make understanding the problem much clearer. Tip two

Always convert ALL units to SI units before attempting to answer the question.  Many candidates make mistakes because they don’t convert the units and simply place the number in the formula.

Here is a list of the most common units.

Tip three

You have to place a datum line which is where you are measuring the energies from.  Now if this is a horizontal pipe you always put the datum in the centre of the pipe because in this way you have zero potential at both points.  This is because the potential energy above and below the datum cancels out.  Now if the situation is not in a horizontal pipe for example like this example.

What you do is always place your datum line at the lowest point in the system.  In this way only one of the points will have potential energy and it makes it easier to answer the question. Click to download exercises to attempt Module 1

Attempt the question below and when completed you can view the model answer.

Question 1

A pump is pumping 2m3/min of water the surface of which is 5m below the pump inlet.  At the outlet the pump has a diameter of 100mm and at this point the pressure is 8 Bar.  From the nozzle (which is at the same level as the pump outlet) the jet rises 35m.

A)     Calculate the energy/kg of the water

(1)   At the outlet of the pump

(2)   At the top of the throw of the jet

B)     Explain why (1) and (2) are not equal

(acceleration due to gravity is 9.81m/sec2) ..

Module 2

Attempt the question below and when completed you can view the model answer.

Water is flowing horizontally through a 250mm diameter pipe and into a constriction of 100mm diameter. The pressure difference is measured as 23.5mm of mercury.  Using Bernoulli’s theorem, calculate the rate of flow.

(Density of mercury = 13,600 kg/m3)

(Density of water = 1000 kg/m3)

( g = 9.81) ..

Module 3

Attempt the question below and when completed you can view the model answer.

A foam generator consists of a horizontal tube of circular cross section which tapers from an input of 80mm internal diameter to 20mm diameter.  750 lts/min of concentrate (Density 1200 kg/m) is flowing through the generator and the pressure inlet is 12 Bar.

What is the pressure at the point where the diameter is 20mm? ..

Module 4

Attempt the question below and when completed you can view the model answer.

Water is flowing in a vertical tapering pipe 2 metres in length.  The top of the pipe is 100mm diameter and the bottom is 50mm diameter.  The quantity of water flowing is 1300 litres/minute.

Calculate the pressure difference between the top and the bottom of the pipe? ..

Module 5

Attempt the question below and when completed you can view the model answer.

A pump supplies 4kw of energy to the water flowing through a 45mm hose.  The water flows 15m vertically and through a 25mm branch at a rate of 500 litres/minute.  Use Bernoulli’s theorem and find the pressure at the branch. ..

Module 6

Attempt the question below and when completed you can view the model answer.

If the manometer readings are 800mm and 200mm, what is the flow ?

(Density of water = 1000kg/m3) ..