Head of a liquid and pressure measurement

In this section we will clarify some of the terms used in earlier sections, and look at methods of measuring pressure.

Head of a liquid

Whilst deriving Bernoulli’s equation the total specific energy of a liquid stream was stated as:

Potential energy + kinetic energy + flow energy

$gZ+\frac{C^2}{2}+\frac{p}{\rho }$ 

Dividing these terms by gravitational acceleration, g, each term can then be re-stated as:

$g+\frac{C^2}{g2}+\frac{p}{\rho g}$ 

In other words, Potential head + kinetic head + pressure head.

The energy quantity $\frac{p}{\rho }$ is also referred to as flow work or pressure energy and is the energy which must be continuously expended by a pump (or its equivalent) in forcing a liquid along a pipeline in the presence of hydrostatic pressure.

Although in common use, the term pressure energy tends to imply that a liquid may have its energy increased by pressurization. This is not the case since most liquids are practically incompressible. The term flow work may be considered more appropriate.

Likewise, the three separate types of energy listed above for a liquid stream are often expressed more conveniently as heads of liquid.

Static pressure head and hydrostatic pressure

Pressure measurement

Manometry may be defined as the science of utilising vertical columns (heads) of liquid to measure fluid pressures.

Many different types of pressure measuring devices exist but our studies will be limited to the barometer, piezometer tube and U-tube manometers.

Before considering these devices, however, it is appropriate to recap on definitions of pressure quantities already covered in this unit:

This means that absolute pressure is equal to gauge pressure plus atmospheric pressure

Barometer

For equilibrium and summing forces in the vertical direction:

$F=0=p_B\times A-\rho ghA$ 

In other words

$p_B=\rho gh$ 

We know that

$p_B=p_{atm}$ 

So it follows that:

$p_{atm}=\rho H_{g}gh$ 

As an example, when the height, h, of the column of liquid is 760mm and the density of mercury is 13600 kgm^-3 then:

Atmospheric pressure

$p_{atm}=13600\times 9.81\times 0.76=101.396k{Nm}^{-2}=1.013\text{ bar}$ 

Piezometer tube

Pipelines and vessels carrying liquid under pressure can have their pressure measured by manometers or pressure gauges employing liquid columns.

The simplest type of manometer is the piezometer tube, which is a single vertical transparent open top tube fitted into the pipeline or vessel carrying pressurized liquid whose pressure is to be measured. Two types are shown below.

© A.Henderson, UHI

Due to hydrostatic pressure in the system, the free surface of the liquid in the open tube will stabise at a height, h, above the centre line of the pipeline.

The height, h, is the pressure head and allows gauge pressure to be calculated:

At position/level A:

Gauge pressure, $p_g=\rho gh$ 

For example, a pressure head of 40 mm of water converts to a pressure:

$p_g=\rho gh=1000\times 9.81\times \frac{40}{{10}^3}=392.4{Nm}^{-2}$  

The length of vertical tube which can be conveniently used limits the piezometer to measuring pressures in the lower ranges. For higher liquid pressures, U-tube manometers are often more appropriate.

The U-tube manometer

In order to measure higher pressure levels in pipelines, U-tube manometers employing mercury liquid columns are in widespread use.

© A.Henderson, UHI

In the figure, the level of mercury in the LH limb of the U-tube is at section x-x and distance y below the axis of the pipeline. In the open limb, the mercury column is in equilibrium at a point C, height h above section x-x due to the pressure in the pipeline. B is a point in the LH limb on the same horizontal level as the pipe axis A.

Working on the basis of gauge pressure then, from figure:

Pressure at C, $p_c=0$ (ie atmospheric)

and pressure at x-x in RH limb = ρ Hg g h

likewise pressure at x-x in LH limb = ρ Hg g h  (same horizontal level)

Thus, pB, pressure at B in LH limb = ρ Hg g h - ρ g y

Therefore the pressure in pipeline, pA = pB = g (ρ Hg h - ρ y)

Pressure difference between two pipes

The pressure difference between two levels in a fluid can be measured directly by using a differential manometer.

© A.Henderson, UHI

The U-tube differential manometer shown in figure above is to measure the pressure difference between levels A and B in pipelines carrying a liquid of specific weight w₁. The U-tube contains mercury of specific weight w₂.

Since C and D are at the same level in a liquid at rest:

Pressure at C = pressure at D

$p_c=p_D$ 

For the LH limb

$p_c=$_pA+w1x 

For the RH limb

$p_D=p_B+w_1(Y-h)+w_{2}h$ 

∴  $p_A+w_{1}x=p_B+w_1(Y-h)+w_{2}h$ 

This means the pressure difference can be found using:

$p_A-p_B=w_{1}Y-w_{1}h+w_{2}h-w_{1}X$$=w_1(Y-X)+h(w_2-w_1)$ 

We can work out the pressure difference between the two pipes using this final formula.

Density to specific weight conversion

When working through problems such as in manometry pressure calculations it is often convenient to convert one fluid quantity into another.

Density

This quantity has earlier been defined as mass per unit volume.

$\rho =\frac{m}{V}$ 

Specific weight

This is defined as weight per unit volume.

Since weight is defined as mass times acceleration due to gravity $(w=mg)$ this means that:

Specific weight

$w=\frac{mg}{V}=\rho g$ 

In other words, multiplying density by acceleration due to gravity gives us specific weight, which is also given the symbol  and is measured in Newtons per cubic metre (Nm^-3)

Example

Oil of density 800 kg m^-3 has a specific weight of 800 x 9.81 = 7848 N m^-3

Polytropics