This method is based on the application of the Archimedes' principle: every body immersed in the liquid or gas is exposed to the action of a buoyant force (sometimes called as the Archimedes' force), which acts upwards. This force is equal to the weight of the liquid, gas or vapour displaced by this body. The immersed body is called a buoy, thus giving the name to the method of measurement. Fig. 5.2 presents a schematic of this type of device.

*Figure 5.2. Buoyancy-type level transmitter.*

To measure the level of liquid 1 in a tank 2 a buoy 3 is partly immersed in the liquid. When the level varies so does the resultant force acting on the buoy as follows:

(5.9)

where,

: - gravitational force acted on the buoy, N;

: - the Archimedes’ force acted on the buoy, N ;

m*b *: - mass of the buoy, kg ;

g*loc *: - local gravitational acceleration, m/s2;

:- density of the liquid and the gas (or vapour) above it, respectively, kg/m3;

S*b* : - a horizontal cross-section area of the buoy, m2 ;

:- parts of buoy length in the liquid and in the gas above it, respectively, m;

(5.10)

Since the Archimedes’ force acting from the gas on the buoy is negligible comparing with that from the liquid, and the gravitational force is constant, then the resultant force is proportional to

and, hence, to the level of the liquid.

The displacer element, buoy, is a cylinder of a constant cross-section area, and its density is greater than that of the liquid. The buoy moves up or down, depending on the level variation. The resultant force through a lever 4 is converted by a force-balance or electronic transmitter 5 to a proportional pneumatic (20-100 kPa) or electrical (4-20 mA dc) signal, which is transmitted by the distance. It means that for each value of the level in the tank will correspond the certain value of an output signal. The length, diameter, material of the buoy and transmission ratio can be changed to suit various spans and various liquids. These instruments are used for measurements of liquid level and interface providing the level to vary within the length of the buoy, and for density measurements providing the buoy is fully immersed in liquid in the entire range of measured densities.

It is appropriate now to consider an operational principle of a pneumatic transmitter, which is used for converting level variations into the standard pneumatic signal. To be more precise, these transmitters can be used to convert a mechanical motion (which may be caused by the variation of any process variable) into the standard pneumatic signal. Fig. 5.3 shows a schematic view of the pneumatic transmitter.

When level of the liquid goes up, the Archimedes’ force moves the buoy 1 in the same direction. A membrane 2 separates the measuring part of the pneumatic transmitter from the part with a high process pressure in the tank where the level is to be measured. The motion of the buoy through levers 3 (rotates clockwise) and 4 (rotates clockwise) transmits to the force bar 5 (rotates clockwise). The force bar is connected with the flapper 6, which approaches to the nozzle 7. The pressure supplied to a pneumatic amplifier 8 is equal to 140 kPa. The pressure from this amplifier is fed to the nozzle, and then to atmosphere. When the flapper approaches to the nozzle, the pressure in the nozzle increases. This pressure enters the pneumatic relay in the pneumatic amplifier, where it is amplified, and so the value of the output pressure increases. The output pressure is transmitted to a measuring or controlling instrument, and is applied to the feedback bellows 9, thus increasing the counterclockwise moment of force acting from the lever 10 (rotates clockwise) on the force bar 5. This moment of force is sufficient to restore the force bar to the balance. When the balance has reached the output pressure is linearly related to the value of the measured liquid level. A gain adjustment holder 11 is used for the variation of the measuring range. An additional weight 12 is used for damping the vibration of levers. Zero adjustment can be achieved by the spring 13.

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1.1 Flapper-nozzle system

Fig. 5.3 shows a
flapper-nozzle system and Fig. 5.4 shows a relationship between the
output pneumatic signal and the distance between the flapper and the
nozzle.

The diameter of the supply restriction 1 is 0.2-0.3 mm,
whereas that of the nozzle 2 is 0.8 mm. The distance between the flapper
3 and the nozzle determines the output pressure in the chamber between
them. This pressure is measured by the pressure gauge 4. Small nozzle
diameters increase gain, but also increase the danger of clogging. Large
nozzle diameters increase the air consumption. The variation of the
nozzle clearance by 0.04 mm gives the change in the output pressure from
20 to 100 kPa. Formulars below are taken from(from Bentley J. P.
Principles of Measurement Systems, Longman, 1995, p. 315):

*Figure 5.3. A pneumatic transmitter.*

*Figure 5.4. A flapper-nozzle system.*

1 - restriction, Dor = 0.2 mm; 2 - nozzle, Dn = 0.8 mm; 3 - flapper.* *

*Figure 5.5. Nozzle air pressure vas distance between the flapper and the nozzle.*

For the steady state condition:

and (from Bentley J. P. Principles of Measurement Systems, Longman, 1995, p. 316):.

So,

, and

(5.18)

Finally,

(5.19)

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1.2 . Pneumatic relay amplifier

(5.20)

and

(5.21)

So,

(5,22)

where,

Steady-state sensitivity:

(5.23)

*Figure 5.6. Pneumatic relay amplifier (from Bentley J. P. Principles of Measurement Systems, Longman, 1995, p. 317):.*

1 - flapper;

2 - nozzle;

3 - orifice;

4 - diaphragm;

5 - double vent;

6 - transmission line.

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*1.3*. Simplified model of pneumatic torque-balance transmitter

(from Bentley J. P. Principles of Measurement Systems, Longman, 1995, p. 319-320):

*Figure 5.7. Pneumatic torque-balance transmitter.*

1 – beam;

2 - pivot;

3 - negative feedback bellows;

4 - nozzle;

5 - flapper;

6 -zero adjustment spring;

7 - pneumatic relay amplifier.

Anticlockwise moments:

(5.24)

Clockwise moment:

(5.25)

####
a). Condition of a perfect torque balance:

(5.26)

A simple model for a torque-balance transmitter:

. (5.27)

The sensitivity of the transmitter

. (5.28)

__Example:__

If

then

####
b). Condition of imperfect torque balance:

Anticlockwise moments:

. (5.29)

Clockwise moment:

(5.30)

.

(5.31)

. (5.32)

An accurate model for torque-balance transmitter is as follows:

(5.33)

where,

*Figure 5.8. Block-diagram for a pneumatic torque-balance transmitter.*

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1.4. Simplified model of pneumatic differential pressure transmitter

(from Bentley J. P. Principles of Measurement Systems, Longman, 1995, p. 321-322):

According to Fig. 5.9 the resultant force on the diaphragm is as follows:

(5.34)

Clockwise moment on the force beam due to the action of :

(5.35)

Anticlockwise moment on the force beam due to the action of the span nut:

(5.36)

For the condition of balance:

(5.37) or

(5.38)

Anticlockwise moment produced by the span nut on the feedback beam:

(5.39)

Anticlockwise moment produced by the zero spring force on the feedback beam:

(5.40)

Clockwise moment on the feedback beam produced by

acting on the feedback bellows:

(5.41)

For the condition of balance:

(5.42)

or

(5.43)

*Figure 5.9. Simplified model of pneumatic differential pressure transmitter*

1 - diaphragm capsule; 2 - force beam; 3 - flapper; 4 - nozzle;

5 - span nut; 6 -feedback bellows; 7 - feedback beam;

8 - zero adjustment spring.

Since,

(5.44),

then we can get a simplified model for the differential pressure transmitter.

(5.45)

(5.46)

and

(5.47)

Adjusting the position of the span nut alters the ratio e/d , and the sensitivity.

Adjusting the zero spring force F

*o* gives a zero pressure (when P1 = P2 ) of 20 kPa.

A*bel* - the effective area of the feedback bellows, m2;

A*D* - effective area of the diaphragm, m2.

*Article Source:: Dr. Alexander Badalyan, University of South Australia*