Chapter 1. Properties of fluids and pipes

What you will learn

Understand the basic essentials of fluid properties.

                    Why

It is important to understand and master the characteristic behaviours of fluids,
especially of water and of air. This lays the groundwork for an understanding of
hydrostatics and hydrodynamics - the following chapters of this course.

Duration of chapter 1

1 to 2 hours

1.1. Definition *

A fluid is any substance which flows because its particles are not rigidly attached to one another.
This includes all liquids, all gases and even some materials which are normally considered as
solids, such as glass. Liquids are almost incompressible while gases are readily compressible;
liquids occupy a more or less constant volume and have a surface, whereas gases can expand
and contract to fully occupy an available volume.
In hydraulics, water and air are the two main fluids of interest. Their main properties are described
below.

1.2. Density *

The density of a material is defined as the mass per unit volume. The SI units for density are kg/m3:
ρ: density in kilos per cubic meter [kg/m3]
m: mass in kilos [kg]
V: volume in cubic metres [m3] V

For a gas, the density will vary a lot with pressure and temperature:
P: absolute pressure [N/m2] or [Pa]
R: gas constant, for dry air it is about 29.3 [m/K]
T: temperature in Kelvin [K] (0 °C=273.15 K)
g: earth gravity (9.81 m/s2)


This is a variation of the ideal gas law:
P: is the absolute pressure [Pa]
V: volume in cubic meter [m3]
T: is the temperature in Kelvin [K]

The density of a liquid does not vary significantly with temperature; for example, the density of
water varies by 0.3% between 5°C and 25°C (see annexe). In comparison, the density of a gas will
vary by 7% over the same temperature range.
Relative density is defined as the ratio of the density of a substance to the density of a given
reference material. The reference material is usually water. For example the relative density of
mercury is 13.5 and for ethanol it is 0.789

1.3. Young's and bulk modulus

The Young's modulus for solids and the bulk modulus for fluids (K)
measure the resistance to uniform compression. It is defined as the
pressure increase needed to cause a given relative decrease in
volume. Its base unit is the Pascal.

The following formula can be used to calculate the change in diameter of a pipe
due to an increase in internal pressure ΔP:
D: diameter of pipe [m]
e: thickness of pipe [m]
          K: Bulk modulus of pipe [Pa]
              ΔP: Pressure increase [Pa]


The compressibility of water and pipes can be usually neglected in hydraulics for steady flows; it is
less than 1% for maximum permissible pressures - this effect is overshadowed by the tolerances
used for pipe production.
However, this is not the case for a transitional situation (water hammer), where compressibility
influences the speed and amplitude of a perturbation as it propagates through the pipes.

1.4. Viscosity

The viscosity of a fluid is a measure of its resistance to a tangential force; this resistance is mainly
caused by interactions between the fluid's molecules.
Consider two large parallel plates, close to each other (y is small) and separated by a given fluid.
For the upper plate to move at a constant velocity, a force F should be applied. This force will be
proportional to the surface area of the plates and to the velocity of the upper plate, and is inversely
proportional to the distance between the plates.


F: force in Newton [N]
μ: dynamic viscosity [Pa·s]
A: Surface of the plates [m2]
v: velocity of the upper plate [m/s]
y: distance between the plates [m]

When the dynamic viscosity is independent of the shear stress (F/A), μ is constant (at a given
temperature) and the fluid is called Newtonian, this is the case for water and most gases.

Kinematic viscosity ν, is a useful variable in hydraulics. It is
defined as the ratio of dynamic viscosity to density.
It is expressed in [m2/s] or in Stokes [St].

Fig 1- 1 Water viscosity at different temperatures

As shown in the chart, the viscosity decreases considerably as temperature increases. For
instance, between 0°C and 20°C the viscosity of water decreases by 44%.

1.5. Phase transformations

Phase transformations refer to changes in the physical state of matter. Elements and simple
compounds can generally exist as either solids, liquids or gases.
For water (H2O), these states can be ice, liquid water and/or water vapour.
The state of a material at a given moment depends on its composition and on the temperature and
the pressure exerted.

Fig 1- 2 Phase diagram of water

For instance at sea level (~ 105 Pa), water will be ice below 0°C, liquid between 0°C and 100°C, and water vapour above 100°C. At a pressure of 611 Pa, water will be solid ice up to 0.1 °C and will then sublimate directly to the vapour state. These states are outlined for water in this phase diagram.
NB: For a better understanding, the scales are
not respected.









The triple point: The single combination of pressure and temperature at which liquid water, solid
ice, and water vapour can coexist in a stable equilibrium occurs at exactly 273.16 K (0.01 °C) and
a pressure of 611 Pa. At that point, it is possible to change all of the substance to ice, water, or
vapour by making small changes in pressure or temperature.

The critical point: The vapour-liquid critical point denotes the conditions above which distinct liquid and gas phases do not exist. As the critical temperature is approached, the properties of the gas and liquid phases approach one another, resulting in only one phase at the critical point: a homogeneous supercritical fluid. The heat of vaporization is zero at and beyond this critical point, so there is no distinction between the two phases. Above the critical temperature a liquid cannot be formed by an increase in pressure, but with enough pressure a solid may be formed. The critical pressure is the vapour pressure at the critical temperature.

Note that when there is a mix of elements (like air & water) the situation becomes more complicated (which is why we have water vapour in the air below 100°C). This will be further developed in the next chapter.

Table 1 : Properties of water

1.6. Thermal expansion of pipes 

The change in temperature does not only influence the properties of water (like density and viscosity) but also has impacts on the pipes, which will expand as the temperature rises. For some materials, as for polyethylene, the dilatation effect can be non-negligible. With an increase in temperature, the pipe will increase its length as well as its diameter.

ΔL: total elongation of the pipe [mm]

L: length of the pipe before expansion [m]
ΔT: change in temperature [°K]
αT: thermal expansion coefficient [mm/m°K]


ΔD: total increase in pipe's diameter [mm]
D: (inside or outside) diameter of the pipe
before expansion [m]
ΔT: change in temperature [°K]
αT: thermal expansion coefficient [mm/m°K]

Therefore, the main consequences of thermal expansion from a hydraulic point of view will be an increase of the cross-sectional for plastic pipes, which can influence the flow. The increase in head loss due to the elongation of the pipe is negligible. In practice, the increase in length will mainly matters when we deal with hot fluids, but it is not the purpose of this course.

Basic exercises

1. What is the mass of 1 dm3 of mercury?
2. What is the density of sea water at 5°C knowing that the average salinity is 35 grams per litre?
3. What is the density of air at a pressure of 105 Pa and 30°C?
4. In a closed air tank we have a pressure of 10×10^5 Pa at 0°C. As the maximum pressure
authorised for the vessel is 12×105 Pa, what is the maximum temperature admissible?
5. How does the density of water change with an increase in pressure (30×10^5 Pa) at constant
temperature (5°C)?

Intermediary exercises

6. What corresponding reduction in the volume of a 1m3 steel cube is caused by an increase of
20×105 Pa?
7. In what state does water exist at 5×10^5 Pa and 0°C?
8. What will be the length and diameter of a PE pipe (original length 20m, original diameter
100mm), after thermal expansion due to temperature changing from 5°C to 25°C?
9. A plate 50 x 50 cm is supported by a water layer 1 mm thick. What force must be applied to this
plate so that it reaches a speed of 2m/s at 5°C and at 40°C?

Advanced exercises

10. When a PE pipe with an outside diameter of 200 mm is pressurized to 10×10^5 Pa, what are the
% increases in diameter for pipe wall thicknesses of 9.1 mm, 14.7 mm, and 22.4 mm?
11. On the following three dimensional (P,V,T) phase diagram for water, indicate the following
areas, lines or points:

• critical point (point) & triple point (line)
• liquid water (area); solid ice (area); water vapour (area); supercritical fluid (area)
• liquid water & solid ice (area); liquid water & water vapour (area); solid ice & water vapour (area)

References for this chapter:
WEB:
http://www.thermexcel.com/english/ressourc/pdcgas.htm
http://en.wikipedia.org/wiki/Viscosity
http://en.wikipedia.org/wiki/Bulk_modulus
Books:
Série Schaum, Mécanique des fluides et hydraulique, Ranald V. Giles
ELBS, Fluid Mechanics 2nd edition, Douglas, Gasiorek, Swaffield