Fluid FLOW is about the steady flow of fluid.
New concepts
1. Definition of steady-state flow
2. Mass and volume flow rates - continuity equation (what goes in must come out)
3. Head - Pressure, Velocity and Potential Heads
4. Bernoulli equation - conservation of energy of steady flow of fluid
5. Bernoulli with Head Losses  

Lecture Notes Fluid-Flow.pdf

Video Lessons:s

Image Video Lesson Description and Link Duration Date Download
  Fluid Flow 14:08 min 20140731  
  Continuity.htm 10 mins    
  Bernoulli-Intro.htm  9 min    

Fluid Flow (14 mins)



Flow Rates

What goes in must come out

Steady flow means the fluid properties are constant - constant pressure, velocity, temperature etc.

Volume Flow Rate
 = V / t  =   A  
= volume flow rate or volumetric flow rate (m3/s). This may need to be converted from L/s (litres per second) or m3/h etc.
V = volume (m3)
= average velocity of the fluid (m/s)
A = cross-section area of the pipe (m2)


Mass Flow Rate
= m / t = A
 = mass flow rate (kg/s)
m = mass of fluid (kg)
t = time (s)
= average velocity of the fluid (m/s)
 = density of the fluid (kg/m3)



The continuity equation says that mass is conserved. (The mass that goes in) = (The mass that comes out).

in = out = constant

If the density is constant (OK for liquids, sometimes for gases), then

  in =   out = constant

Note that flow rate is AREA related, not volume related.

For example, a car with twin 50mm exhaust pipes has more resistance than a single 75mm exhaust. Twin exhaust = (2*50^2*Pi) = 3927mm2. Single exhaust = (75^2*Pi) = 4417mm2.



In this example, HEAD is the height of water above a turbine or pump, and is a way of measuring pressure.

An example of potential head (as in potential gravity)

However, head means more than just pressure. The velocity of the flow can also raise the fluid to a height (head), and also the  pressure has an equivalent head.

There are 3 types of head:

Pressure Head is the height to which a fluid will rise because of its pressure.

hp = p / g

hp = pressure head (m)
p = pressure (Pa)
= density (kg/m3)
g = gravitational acceleration (9.81 m/s2)  

Velocity Head is the height to which a fluid will rise because of its velocity.

hv2 / 2g

hv = velocity head (m).
= average velocity of the fluid (m/s)
g = gravitational acceleration (9.81 m/s2)  

Potential Head is the height of the fluid. (above a certain datum)
h = (m)

TOTAL Head = sum of pressure, velocity and potential heads;

H = hp + hv + h


More about Velocity Head...

Pressure Head is simply the depth of that fluid equivalent to it's pressure (e.g. 10m of water is 100kPa).

Potential Head is even simpler - that actual height of the fluid (taken at the centreline of the pipe)

Velocity Head needs some explanation.

Example 1: A steam engine scooping water from a trough.

Water Troughs

A locomotive picking up water at speed from a trough laid between the lines. A scoop is lowered from the engine tender when the train reaches the trough, and water rushes up to replenish the tank.

The engine-driver operates a control lowering a scoop, up which the water is forced by the speed at which the train is travelling.


Scoop Design

The scoop was very difficult to raise out of the water (due to dynamic loads).

William F. Kiesel, Jr, obtained a patent in 1894 on a scoop that balanced the force of the water entering the scoop against the water exiting the scoop into the tender tank. This innovation doubled the
efficiency of the system under test conditions: at 70 mph, 3.3 gallons were picked up per linear foot of trough.


Water overflowing the tender while scooping water at speed.

Scoops had their problems, and picking up whatever was in the track pan—debris, dead animals, lumps of coal, or junk tossed in by kids just to see what would happen—was one of them.

Track pans started out about 1200 feet long, but by the 1940s, the typical length was between 1500 and 2500 feet.

A common problem in taking on water at high speed was the rapid buildup of air and water pressure, which would lift the tender hatches open and in extreme cases, spring the tender’s side walls.

In the 1940s, the Central became concerned about window breakage on trains traveling on tracks parallel to other trains scooping water.


Example 2: The Pitot Tube

This is used for measuring the velocity of a fluid.

Above: A vertical tube (piezometer) measures pressure head (hp). A tube pointing upstream (Pitot tube) measures the both pressure head + velocity head. The difference is velocity head, from which we can obtain the actual fluid velocity from hv2 / 2g. This is how air speed is measured in aircraft.

Pitot tube measures air speed from velocity and pressure heads.


The Bernoulli Equation is basically conservation of energy along a pipe. It can be written in different ways by converting energy to head (Kinsky) or pressure, etc.

Bernoulli's equation says in an ideal situation, the TOTAL HEAD = constant.

There are assumptions; An ideal fluid (no friction) flowing steadily...

  • The fluid is incompressible and nonviscous.
  • There is no energy loss due to friction between the fluid and the wall of the pipe.
  • There is no heat energy transferred across the boundaries of the pipe to the fluid as either a heat gain or loss.
  • There are no pumps in the section of pipe under consideration.
  • The fluid flow is laminar and steady state.



Pressure in a fluid is like the energy per unit volume (energy density). From the definition of pressure:

Here is the Bernoulli equation in terms of energy per unit volume;

Note that this correponds exactly with the conservation of energy equation. Where pressure = spring energy, velocity = kinetic energy and height = potential energy.

SE1 + KE1 + PE1= SE2 + KE2 + PE2

Where: PE1 = mgh1 and KE1 = 0.5mv12 and SE1 = 0.5kx12


Total Head (also called Total Dynamic Head) is the sum of the three components; pressure head, velocity head and potential head. From the above equation, divide by g. Head is a simple way to think of each term (pressure, velocity and height) in terms of the number of METRES of that fluid it is equivalent to.

  See Kinsky Eqn 11.4, p241

  • All terms are head (m), which is measured in the working fluid. 
  • You can use gauge or absolute pressure throughout, but gauge is normal
  • Continuity equation is also true (and often needed to solve the question)
  • Ideal fluid is assumed (no friction)

Note: Because the head is in terms of the working fluid, the pressure head of air at 65kPa a much higher number than the pressure head of water at 65kPa. It takes a lot of height to get that much pressure from air depth (5.5km of air = 6.6m of water).


Bernouli's Equation requires velocity. This is usually found using the continuity equation.

 = V / t  =   A

Since most pipes are round, this gives;

v1d2/4 = v2d2/4




Special Cases

Some examples of the application of the Bernoulli Equation.


1. Horizontal Pipe: No potential head change so they cancel each other out.

2. A liquid surface is at atmospheric pressure. Pressure head at surface = 0 (gauge). Assume point 1 is the surface, then...

3. A nozzle discharges to atmosphere. Pressure head at outlet (2) = 0.

4. Pipe is parallel (constant diameter). No change in velocity, so velocity head cancels out.

5. Liquid flow in/out a large tank/lake. The velocity at surface is zero. For example, flow of a tap from a large tank...

Bernoulli with Head Loss

Head loss is the amount of TOTAL HEAD lost between points 1 and 2.

HL = H1 - H2

Interestingly, when losses occur in a pipe they do not affect every term in the Bernoulli equation, but only pressure.
  • Potential: No change because the pipe still has the same start and end heights.
  • Velocity: No change because the pipe still has the same start and end velocities (otherwise continuity is false)
  • Pressure: Head loss will reduce the pressure!

Where HL = Total head loss

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Homework Assignment: Kinksy new edition
Do all questions; Chapter 11: Fluid Flow
11.1 to 11.20 (page 251-253)

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