Units & Scales of Pressure Measurement | Fluid Mechanics for Mechanical Engineering PDF Download

Pascal (N/m²) is the unit of pressure .

Pressure is usually expressed with reference to either absolute zero pressure (a complete vacuum)or local atmospheric pressure.

• The absolute pressure: It is the difference between the value of the pressure and the absolute zero pressure
  
  P _abs = p - 0 = p
   
• Gauge pressure: It is the diference between the value of the pressure and the local atmospheric pressure
  
  (p_atm) p_gauge = p - p_atm
   
• Vacuum Pressure: If p<p_atm then the gauge pressure (p_gauge) becomes negative and is called the vacuum pressure.But one should always remember that hydrostatic pressure is always compressive in nature 

            
                Fig 4.1 The Scale of Pressure

At sea-level, the international standard atmosphere has been chosen as P_atm = 101.32 kN/m²
 

Piezometer Tube

The direct proportional relation between gauge pressure and the height h for a fluid of constant density enables the pressure to be simply visualized in terms of the vertical height, h = p/p g. 
The height h is termed as pressure head corresponding to pressure p. For a liquid without a free surface in a closed pipe, the pressure head  p/p g at a point corresponds to the vertical height above the point to which a free surface would rise, if a small tube of sufficient length and open to atmosphere is connected to the pipe

Fig 4.2 A piezometer Tube

Such a tube is called a piezometer tube, and the height h is the measure of the gauge pressure of the fluid in the pipe. If such a piezometer tube of sufficient length were closed at the top and the space above the liquid surface were a perfect vacuum, the height of the column would then correspond to the absolute pressure of the liquid at the base. This principle is used in the well known mercury barometer to determine the local atmospheric pressure.


The Barometer
Barometer is used to determine the local atmospheric pressure. Mercury is employed in the barometer because its density is sufficiently high for a relative short column to be obtained. and also because it has very small vapour pressure at normal temperature. High density scales down the pressure head(h) to repesent same magnitude of pressure in a tube of smaller height.

Fig 4.3   A Simple Barometer

Even if the air is completely absent, a perfect vacuum at the top of the tube is never possible. The space would be occupied by the mercury vapour and the pressure would equal to the vapour pressure of mercury at its existing temperature. This almost vacuum condition above the mercury in the barometer is known as Torricellian vacuum.

The pressure at A equal to that at B (Fig. 4.3) which is the atmospheric pressure p_atm since A and B lie on the same horizontal plane. Therefore, we can write

p _B = p_atm = p_v + p g h     (4.1)

The vapour pressure of mercury p_v, can normally be neglected in comparison to p_atm.
At 20⁰C,P_v is only 0.16 p_atm, where p_atm =1.0132 X10⁵ Pa at sea level. Then we get from Eq. (4.1)

For accuracy, small corrections are necessary to allow for the variation of r with temperature, the thermal expansion of the scale (usually made of brass). and surface tension effects. If water was used instead of mercury, the corresponding height of the column would be about 10.4 m provided that a perfect vacuum could be achieved above the water. However, the vapour pressure of water at ordinary temperature is appreciable and so the actual height at, say, 15°C would be about 180 mm less than this value. Moreover. with a tube smaller in diameter than about 15 mm, surface tension effects become significant.
 

Manometers for measuring Gauge and Vacuum Pressure
Manometers are devices in which columns of a suitable liquid are used to measure the difference in pressure between two points or between a certain point and the atmosphere.

Manometer is needed for measuring large gauge pressures. It is basically the modified form of the piezometric tube. A common type manometer is like a transparent "U-tube" as shown in Fig. 4.4.

 
Fig 4.4 A simple manometer    Fig 4.5 A simple manometer to
  to measure gauge pressure         measure vacuum pressure 

One of the ends is connected to a pipe or a container having a fluid (A) whose pressure is to be measured while the other end is open to atmosphere. The lower part of the U-tube contains a liquid immiscible with the fluid A and is of greater density than that of A. This fluid is called the manometric fluid. 

The pressures at two points P and Q (Fig. 4.4) in a horizontal plane within the continuous expanse of same fluid (the liquid B in this case) must be equal. Then equating the pressures at P and Q in terms of the heights of the fluids above those points, with the aid of the fundamental equation of hydrostatics (Eq 3.16), we have



Hence,

               (4.2)

where p₁ is the absolute pressure of the fluid A in the pipe or container at its centre line, and p_atm is the local atmospheric pressure. When the pressure of the fluid in the container is lower than the atmospheric pressure, the liquid levels in the manometer would be adjusted as shown in Fig. 4.5. Hence it becomes,

         (4.2)

 

Manometers to measure Pressure Difference
A manometer is also frequently used to measure the pressure difference, in course of flow, across a restriction in a horizontal pipe.


Fig 4.6 Manometer measuring pressure difference

The axis of each connecting tube at A and B should be perpendicular to the direction of flow and also for the edges of the connections to be smooth. Applying the principle of hydrostatics at P and Q we have,

    (4.3)

where, ρ _m is the density of manometric fluid and ρ_w is the density of the working fluid flowing through the pipe. 
We can express the difference of pressure in terms of the difference of heads (height of the working fluid at equilibrium).

               (4.4)
 

 Inclined Tube Manometer

• For accurate measurement of small pressure differences by an ordinary u-tube manometer, it is essential that the ratio r_m/r_w should be close to unity. This is not possible if the working fluid is a gas; also having a manometric liquid of density very close to that of the working liquid and giving at the same time a well defined meniscus at the interface is not always possible. For this purpose, an inclined tube manometer is used. 
• If the transparent tube of a manometer, instead of being vertical, is set at an angle θ to the horizontal (Fig. 4.7), then a pressure difference corresponding to a vertical difference of levels x gives a movement of the meniscus s = x/sinq along the slope.
  
  
  Fig 4.7   An Inclined Tube Manometer
  
   

• If θ is small, a considerable mangnification of the movement of the meniscus may be achieved.

• Angles less than 5⁰ are not usually satisfactory, because it becomes difficult to determine the exact position of the meniscus.

• One limb is usually made very much greater in cross-section than the other. When a pressure difference is applied across the manometer, the movement of the liquid surface in the wider limb is practically negligible compared to that occurring in the narrower limb. If the level of the surface in the wider limb is assumed constant, the displacement of the meniscus in the narrower limb needs only to be measured, and therefore only this limb is required to be transparent.

Inverted Tube Manometer
For the measurement of small pressure differences in liquids, an inverted U-tube manometer is used.

Fig 4.8 An Inverted Tube Manometer

 

Here  p_m < p_w and the line PQ is taken at the level of the higher meniscus to equate the pressures at P and Q from the principle of hydrostatics. It may be written that

where p^* represents the piezometric pressure, p + pgz (z being the vertical height of the point concerned from any reference datum). In case of a horizontal pipe (z₁ = z₂) the difference in piezometric pressure becomes equal to the difference in the static pressure. If (p_m - p_w) is sufficiently small, a large value of x may be obtained for a small value of   . Air is used as the manometric fluid. Therefore, p_m is negligible compared with p_w and hence,

          (4.5)

Air may be pumped through a valve V at the top of the manometer until the liquid menisci are at a suitable level.


 Micromanometer
When an additional gauge liquid is used in a U-tube manometer, a large difference in meniscus levels may be obtained for a very small pressure difference.


        Fig 4.9     A Micromanometer

 

The equation of hydrostatic equilibrium at PQ can be written as

where p_w , p_G  and p_m are the densities of working fluid, gauge liquid and manometric liquid respectively. From continuity of gauge liquid,

     (4.6)

     (4.7)

If a is very small compared to A

    (4.8)

With a suitable choice for the manometric and gauge liquids so that their densities are close (p_m ≈ p_G)  a reasonable value of y may be achieved for a small pressure difference.