Bourdon and diaphragm gages

1. Bourdon gauge

The most widely used in industry for pressure and vacuum measurements (from 20 kPa to 1000 MPa) is a pressure gauge with sensitive element made of a metallic (various stainless-steel alloys, phosphor bronze, brass, beryllium copper, Monel, etc.) Bourdon tube 1 (see Figure 1). The tube was named after its inventor, E. Bourdon, who patented his invention in 1852. This tube has an elliptical or oval cross-section A-A and has the shape of a bended tube. When the pressure inside the tube 1 increases, its cross-section dimension b₁ also increases by the value of ∆b₁, whereas the cross-section dimension a₁ reduces its length by the value of ∆a₁. Therefore, the tube tends to straighten (if pressure has increased) or twist (if pressure has decreased, for example, during vacuum measurements), and the tip 2 of the tube moves linearly with applied pressure. The movement of the tip is transmitted to the pointer 3 through a mechanism 4. The tube tends to return to its original shape (the pointer returns to the starting position) after pressure is removed. A relationship between the value of the tip movement ∆x and the measured pressure is linear, so the scale of this pressure gauge is uniform.

Figure 1. Bourdon tube pressure gauge.

Some degree of hysteresis still exists during operation of these pressure gages, because metals cannot fully restore their initial elastic properties. If we have two Bourdon tubes made of the same metal, the tube with a bigger radius and a smaller thickness of the wall will have higher sensitivity. An accuracy of a typical Bourdon-tube pressure gauge is equal to ±1%, whereas a specially designed gauge may have better accuracy which varies from ±0.25 to ±0.5%.

2. Diaphragm gauge

Another type of pressure gauge, which utilises elastic-element properties, is a diaphragm pressure gauge. These gages are used when very small pressures (from 125 Pa to 25 kPa) are to be sensed. Fig. 2 shows a sensitive element for this type of pressure gauge.

Figure 2. Sensitive element of a diaphragm pressure gauge.

A flexible disc 1 made of trumpet brass, or phosphor bronze, or beryllium copper, or titanium, or tantalum, etc., is used to convert the measuring pressure to the deflection of the diaphragm. Deflection vs pressure characteristic should be close to linear as much as possible. In reality for a flat diaphragm this characteristic is non-linear. So, flat membranes are not used as sensitive elements. To linearise this relationship special diaphragms with concentric corrugations 6 are designed. Linearisation of a static characteristic of the membrane can be achieved by using a flat spring 2, which is connected, to the diaphragm through the mechanism 3. The movement of the mechanism 3 is transmitted by the link 4 to a pointer of the gauge. The measuring pressure is supplied to the pressure chamber 5 and causes the diaphragm to move upwards until the force developed by this pressure on the diaphragm is balanced by the force acted from the spring. To increase the sensitivity of this type of pressure gauge, we may increase the diameter of the diaphragm, to lengthen the spring, to change the material of the diaphragm and the spring to more elastic, to increase the depth and the number of corrugations of the diaphragm.

When pressure is applied to both sides of the membrane, then the resultant reading is proportional to the differential pressure. The space above the diaphragm is connected to atmosphere, so the diaphragm separates a measured media from the environment. In other words, it serves as a fluid or gas barrier or as a seal assembly, thus preventing contact of corrosive and aggressive fluids with pressure elements.

Accuracy of diaphragm pressure gages varies from ±1.0 to ±1.5% of the span.

3. Bellows pressure gauge

These pressure sensitive elements are usually made of stainless steel; phosphor bronze, brass and are used for pressure measurements for pressures up to 6 MPa. Bellows sensors have large displacement sensitivity. Figure 3 shows this type of sensor.

Figure 3. Bellows pressure sensitive element.

The effective area of a bellows can be calculated using the following formula:

 (36)

,                                              

When pressure is applied to the internal surface of a bellows the force is developed according the formula:

  (37)

            .                                                                     

Let’s now develop a differential equation for bellows pressure sensor.

At steady state condition when          t=0  a displacement of the sealed end of the bellows is equal to:

 (38)

(39) 

(40)

.                            ,                       .                                 

These are initial conditions. The initial force is balanced by the spring force according to:

                                                            

(41)

Let pressure has suddenly increased by the value of P.

Resultant force = mass * acceleration

 (42)

or  ,                                                          

(43)

                                           .                                               

By subtracting (41) from (43) we get equation with variables in deviation form:

 (44)

            ,                                  

or

(45)

            ,                       

or

 (46)

                                                .                                              

Finally, we get a second order linear differential equation with variables in deviation form:

 (47)

                                    .                                              

After introduction of the following parameters:

undamped natural frequency

  (48)

              ,                                             

 damping ratio   

 (49)

                                   ,                                                          

 and a steady-state gain      

 (50)

                                                                            

the equation (47) can be re-written as follows:

(51)

                                                .                                               

Now we apply the Laplace transform:

                        .                                   (52)

Using initial conditions (38) and (39) we can get:

,(53)

or

                                                .                                  

 (54)

Now we can get a transfer function for bellows:

  (55)

                                                ,                                             

where,

A_ef                   - effective area of bellows, m²;

F_ef                    - effective force due to pressure, N;

k                      - bellows stiffness (or spring factor), N/m;

m                     - mass of bellows, kg;

R₁                    - small characteristic radius of bellows,m;

R₂                    - big characteristic radius of bellows, m;

t                       - time, s;

x                      - displacement of the sealed end of bellows, m;

λ                      - constant (friction coefficient) for bellows, (N*s)/m.

1. At z= 1 we have a critically damped response.

  (56)

                        .                                      

2. At z< 1 we have underdamped response.

(57)

.           

3. At z> 1 we have overdamped response.

 (58)

.          

A numerical example will help us to better understand various types of responses.

Let:

 A_ef = 10, cm² = 0.001,m²;

P’ = 6, kPa = 6000, Pa;

k=1000, N/m;

m= 100, g = 0.1, kg;

λ=10, (N*s)/m

Using the above data we can evaluate:

                                   

                                        

;                                   .

Figure 4. shows various types of responses of bellows pressure sensor.

Figure 4. Dynamic responses of bellows pressure sensor.

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