Pressure measurement

Languages for engineering communication - 6

Pressure measurement

6.1 Introduction

Pressure is represented as a force per unit area and has the same units as parameter discussed previously in lecture No.4 as stress,

The new aspect of the same parameter which will be discussed in this lecture is the force exerted by a fluid per unit area on a containing wall. The pressure measurement devices for fluid are usually measure pressure in three different forms: absolute pressure, vacuum and gage pressure.Absolute pressure is the absolute value of the force per unit area exerted on the containing wall by a fluid.

Gage pressure is the positive difference between the absolute pressure and the local atmospheric pressure.

Vacuum is the negative difference between the local atmospheric pressure and the absolute pressure.

Fig. 1 Relationship between pressure terms

From these definitions in Fig. 1 we see that:

1. The absolute pressure is usually positive.

2. The gage pressure may be positive or negative (vacuum)

3. The vacuum is usually smaller than the local atmospheric pressure.

Some common units of pressure

Because a fluid pressure is result of a momentum exchange between the molecules of the fluid and containing wall, it depends on the total number of molecules striking the wall per unit time and on the average velocity of the molecules.For an ideal gas the pressure is:

(1)

where n is molecular density, molecules/unit volume; m is molecular mass and velocity is root mean square velocity:

(2)

where T is the absolute temperature of the gas, k =1.3803 is Boltzmanns constant, molecular mass, velocity is:

Since the pressure depends on collisions between the molecules, it should be dependent on the average distance passed by a molecule of radius r between collisions. The mean free path in an ideal gas is

[]

(3)

More n smaller is the mean free path. For air the Eq. (3) reduced to

(4)

Where and pressure p must be in Pascal [].The mean free path decreases with increase in the gas pressure p (or the density n) and with decrease in temperature T. The typical examples of mean free path in air at 20 C were calculated using Eq.(4):

1atm = 1.0132x

1torr = 133.32 Pa

1 = 0.13332 Pa

0.01 = 1.332

At a standard conditions (p=1 atm) is quit small (about 0.1A), while in vacuum it reaches several meters length. 6.2 Vacuum devices with electrical outputLow absolute pressure or vacuum is used in many practical operations: the manufacture of computer chips and laser crystals, for example, are completed in vacuum. Vacuum is measured in torr which is defined as 1/760 of the standard atmosphere. Since the standard atmosphere is 760 mm Hg, 1 torr is 1 mmHg. Usual definitions for the ranges of vacuum are:

Low vacuum

760 to 25 torr

Medium vacuum25 to torr

High vacuum

to torr

Very high vacuum

to torr

Ultrahigh vacuum below torr

Since the length in vacuum is large this is used in a series of gages for measuring vacuum which have electrical output. The vacuum gages contain all components of the measuring device shown in Fig. 1*. ) (2) (3

Fig. 1* The main components of the measuring device.6.2.1 Pirani gage

Fig.2 Pirani thermal conductivity vacuum gage arrangement

In gage shown in Fig. 2 a measured parameter (1) is vacuum. A sensing element (2) is heated filament ( ) which is located at the center of a chamber connected to the vacuum source. The variation in filament temperature (electrical resistance) is mechanical reaction (3). Temperature depends on heat loss to the wall. Heat loss in torn, dependent on the thermal conductivity ( ) of the gas. When the pressure within a chamber is low the length is large and the heat transfer from a filament to the walls is strongly sensitive to the variations in pressure:

where is the filament temperature, is the chamber wall temperature. Coefficient C depends on the gas in the chamber, the wall temperature, the geometry of the chamber, and the filament surface area. The lower is the pressure, the lower the thermal conductivity and, consequently, the higher the filament temperature (resistance) for a given electric-energy input. Since the filament is part of electrical bridge circuit (4), the variations in temperature of the filament causes variation in the filament resistance (tangsten) which affects the resulted output ( ) (5). The procedure

The reference chamber is sealed and evacuated. Since both gages are connected in series and exposed to the same environment, variations in the ambient conditions are compensated. A balance condition initially adjusted through resistance and then the test gage is exposed to the vacuum. Since the change in environment temperature is compensated, all deflection of the bridge from the null position shows the vacuum.

Advantages1. An overall range of this gage is about 0.1 to 100 Pa (from 1 to about 1 torr) medium vacuum.

2. Electric output allows automation.

Disadvantages 1. The establishment of thermal equilibrium is rather slow and the transient response is several minutes.2. Preliminary calibration of each pressure gage is required. 6.2.2 The ionization gage

The arrangement shown in Fig. 3 is similar to the ordinary triode vacuum tube when the heated cathode emits electrons, which are accelerated by the positively charged grid.

Fig.3 Schematic of ionization gageThe electrons move toward the grid and produce ionization of the gas molecules through collisions. The plate is maintained at a negative potential, so that the positive ions are collected there, producing the plate current.The electrons and negative ions are collected by the grid, producing the grid current The number and effectiveness of the collisions effect the ratio of plate current to grid current. This ratio is sensitive part of the gage which is proportional to the pressure of the gas (or vacuum):

where the constant S is called the gage sensitivity which should be determined by calibration.

Advantages

1. An overall range is between 1.0 to Hg - high vacuum, while special types of ionization gages can be used for very high and ultra high vacuum up to .2. Due to linear output current these gage can be used for automation purpose.

Disadvantage

Danger of burning out of the cathodes due to a high sensitivity to a pressure change.6.3 Devices with a visual control

The several types of traditional pressure measuring devises that have no electrical output still deserve mention. The U-manometers and the bourdon gages enable a pressure to be read quickly and reliably while the MacLeod gage and the dead-weight tester are used for calibration. None are suitable for dynamic measurements and automation and composed of only three first components (1) (3) of the schematics shown in Fig.1*. The fluid U-tube manometer shown in Fig. 4 is used for measurements of fluid pressure and vacuum under laboratory conditions. It consists of a U-shaped transparent glass or plastic tube partially filled with a liquid. The sensing lines of the pressures to be sensed are connected to the tubes as shown in Fig. 4. The U-tube manometer is used readily for measurement of differential pressure (A) (the difference in pressure at two points in a system) or gage pressure (B). In the last case one sensing lines is opened to atmospheric pressure. Hence the input parameter for this type of the measuring device is pressure, ( ) - (2) the measuring liquid and mechanical reply (3). The U-tube manometer is used either for gases or for liquids. When the fluid being sensed is a liquid, then the manometer fluid must be both denser than the sensed fluid and be immiscible with it. The fluids should also have different colors so the interface (meniscus) is readily visible. The pressure difference at the top of the manometer can be computed from:

where is the difference in levels of the two interfaces (given the symbol R), is the manometer liquid density, is the density of the sensed fluid and g is the acceleration of gravity.

Fig. 4 U-tube manometer for measurement of differential pressure (a) the difference in pressure at two points A and B or gage pressure (b) in point A.

For gases, since , and the result of , known as reading R has the unit of a fluid column height:

If we use the density of water, the pressure can be expressed as feet of water or inches of water, atmospheric pressure is usually expressed as 30 in Hg or 760 mm Hg. The accuracy of fluid U-tube manometer depends on the scale and on the manometer fluid density. A new term: specific gravity, S ( ) specifying the ratio of the fluid density to the density of water at a specified temperature (usually 4C).Example 1* GivenU-tube manometer uses the mercury which is 13.6 times denser then water. g=9.8

The density of water at 4 C is 1000

Objectives1. Define specific gravity, S of the mercury.2. Determine the reading of the manometer or the difference in levels of the two interfaces (given the symbol R) when it measures pressure difference 4 atm.

3. How long this device should be?

Solution1. Specific gravity of the mercury is S=13.62.1atm=

EMBED Equation.3

EMBED Equation.3 More simple solution is: 4atm x 760 torr = 3040mm

Example 2

We have to measure absolute pressure 0.4atm by U-tube manometer which scale is 1mm/div. Manometer is filled by oil with specific gravity, S =0.9. To define the reading and to compare a reading accuracy for the two U-tube manometers: one filled by Hg and another filled by oil/?

Given.U-tube manometer with scale 1mm/div is filled by oil.

Oil specific gravity is S =0.9.

Hg specific gravity is 13.6

An absolute pressure which has to be measured is 0.4atm.

Solution.

A. Reading For Hg manometer

0.4atm=0.4 x 760torr=304torr For oil manometer

x 304torr=4.59m

B. Reading accuracy: For oil manometer reference error is:

For Hg manometer reference error is:

0.16/0.01=16

Accuracy of the oil reading is 16 times of this for Hg.

Simple estimate was already obtained previously as ration of Hg to Oil specific gravities:

.6.3.1 An inclined manometer

The minimum pressure measured by U manometer (of 0.5 in. of water ) can be reduced to 0.1 in. (0.19mmHg) by the use an inclined manometer.

Fig.5 Inclined manometer

A small change in a fluid height causes larger displacement in the transparent tube when it is inclined to the angle and filled by oil. A reading formula for gases is:

Example An inclined manometer with a resolution of the sloping scale of 0.05 in has to measure a gas pressure 3 in. of water with a resolution of 0.01 in. What is the angle and how long is the inclined tube of manometer which is filled with water?

Solution

In order to choose the angle one should remember that the minimum pressure level to be measured has to be equal to requested value of the resolution 0.01in. At the same time this resolution must be equal to reading R of single division on the inclined scale of the manometer. Hence

= Hence 11.5.

Since the total rise of a water column is 3 in. one can obtain:

, L=15in38.1cm.

6.3.2 Well-type manometerTo read the locations of the two interfaces of U-manometer is rather inconvenient. To overcome this problem, on can use the well with the cross-sectional area larger then the area of the transparent tube. When a pressure is applied, the change in surface elevation of the well is extremely small and only one reading is required:

Since for gases- ,

Fig.6 Well-type manometer

ExampleA gas pressure applied to the well port of a well-type manometer results in a reading R= 47.5 cm. To define the applied pressure if the manometer fluid has a specific gravity of 2.0. Solution

==9310

Pa=N/=

Finally we have .A well-type manometer is also used for measure absolute atmospheric pressure. Fig.7 Well barometer

For this purpose one leg is evacuated and the well is exposed to atmospheric pressure. The top of the column contains saturated mercury vapor at the local temperature. However in comparison to atmospheric pressure this saturated pressure is negligible

The height R is thus a measure of the absolute atmospheric pressure read directly in inches (or mm of mercury).A small temperature correction has to account for the vapor pressure of mercury and differential thermal expansion of the mercury and the measuring scale.Correction to altitude for absolute pressure

When pressure measuring devices indicates gage pressure, the absolute pressure is:

P(absolute)=p (gage)+

A common mistake is to take as the local atmospheric pressure the value given by the local weather bureau.The value stated by the weather bureau is usually corrected to the see level, using the altitude at the weather station Z:

where standard atmospheric pressure at sea level

Z= altitude, m or ft

B=0.003566=0.00650K/m

When the problem is to specify P(absolute), the local atmospheric pressure must be measured near the place where the gage pressure is measured.

Example

Calculate the standard atmospheric pressure in mmHg at the altitude 7000ft (2133.6m).

The relative error in atmospheric pressure is:

Advantage of U-manometers

1. Simple, inexpensive and based on clear principles.

2. With light oils differential pressures as low as 0.5 - 0.1 in. of water can be measured with a resolution of the order of 0.02 in (0.5mm).

Disadvantage

Limited pressure range (due to the length restriction).

The densest fluid normally available in manometers of this type is mercury, with a density 13.6 times that of water. To measure a pressure difference about 4 atm, for example, a device height is about 10 ft3m and a ladder is required to read it.

6.3.3 Bourdon gage.

This is another type of a simple device which is used for readings of fluid pressure with visual control.In this device a sensing element is curved, flattened tube (bourdon tube) which attempts to straighten out when subject to internal pressure. The end of the tube is linked to a rotary dial indicator.

Advantages

Relatively inexpensive and used for a wide range of pressures, from low vacuum up to 20,000psi (1350bar), error of 5% of full scale.The most expensive gages with a mirror scale (about 2000-3000$) have an accuracy about 0.5% of full scale.

Fig.8 Bourdon gage

Disadvantages

1. If not protected, easily destroyed by high overpressure.

2. Are mostly suited for a steady state measurements.

3. After the year of application can change its characteristics due to repair or other reasons.

6.4 Static calibration of the pressure gagesAfter the some period of time since production (usually several years) the manufacturers' specifications can not be further taken at face value of the measuring accuracy. To be sure of the validity of the measurements in this case we have to make a calibration before accepting the reading of an instrument. The term calibration means to check the instrument against a standard and thus, to reduce the measurement errors. The term static indicates that the pressure change during calibration is slow and the device is allowed to come to equilibrium prior to taking a reading. Usual calibration procedure involves a comparison of the particular instrument with one of the following:

(1) primary standard;

(2) known input source;

(3) secondary standard which accuracy is higher than that of the instrument to be calibrated,

For static calibration of pressure gage three possibilities can be realized:(1) comparing it with a standard pressure-measurement facility of the National Bureau of Standards (is very expensive).(2) direct calibration with a primary source of a known force applied normally over a full cross section area of the pressure gage (rather complex).

(3) comparing it with a secondary standard (another gage of known accuracy). The term "known accuracy" means here a gage which accuracy was specified by a reputable source.Since first two possibilities are too expensive or complex, many instrument manufacturers use only third option shown schematically in Fig.9. Calibrated gage and gage of known accuracy (secondary standard) connected to the same pressure chamber. Since both

Fig.9 Schematics of the pressure gage calibration

Gages are subject to equal pressure, the output of the secondary standard serves as measure of applied pressure, which value is referred to the calibrated gage reply. A standardized procedure regulated by the document ANSI/ISA (1979) note that the values of the applied pressure must cover the intended range of use of the pressure gage to be calibrated. In the first step, using a fine pressure valve the pressure values are incrementally increased from the low end of the range to the top end of the range. The pressure values are then incrementally reduced to the low end of the range and this process is repeated several times (cycles). A final result is a set of the data which include calibrated gage outputs as a function of the output of the secondary standard. A graphical presentation of these data is known as the calibration curve which in most cases is correlated using a mathematical function using the process of curve fitting. When correlating function is a straight line this results in a single value of the gage sensitivity "s", for example [mv/Pa]. When correlating function is a parabola or a higher-order polynomial, the obtained correlating function is used to convert the calibrated gage output to the pressure in real experiment. In this procedure, the accuracy statement effectively combines system's contribution to the overall errors due to nonlinearity, hysteresis and nonrepeatability. 6.4.1 Secondary standardsCalibration philosophy prefers the secondary standard (gage or measuring system) which are based on simple and clear physical principles. This ensures the experimenter that calibration accuracy is really maintained.Such simple and clear physical principle is base of McLeod gage, which used as a secondary standard for calibrating vacuum measuring devices. In this gage a large volume of the low-pressure gas is firstly compressed into a much smaller volume and then its pressure is measured as shown in Fig.10.

Fig.10. The McLeod gage

Initially, when the gage is connected to the vacuum source, the mercury is below point A, as shown on Fig.9a. Thus, the gage is filled entirely with gas under a low pressure p. Next, the plunger is pushed downward until the mercury rises to the level in capillary tube 2. If the volume of the capillary per unit length is a, then, the volume of the gas which contained within the length y of the capillary1 is

(1)

The volume of the capillary, bulb, and tube down to the point A is V. If we assume that the gas initially occupies a volume V is further compressed isothermally in the capillary 1 to volume, we have since

hence

(2)

In the mode of operation in Fig. 9b, the level of is the same as the top of capillary tube 1. Now, the pressure indicated by the capillary is

(3)

where the pressure in terms of the height of the mercury column is .

Using (2) one can obtain

(4)

Since usually , Eq.(3) becomes which result in

Note that parameter is characteristics of the manometer and the scale of capillary reads directly the units of torr.

Advantages1. Wide range of the measured pressure: from 10 to torr.

2. The capillaries are calibrated directly in micrometers and any additional calibration of the gage is requested.Disadvantages1. Available for dry gases only that not condense when compressed into the capillary tube.

2. Experiments with large capacity of Hg are dangerous.

ExampleA McLeod gage has V=100 and a capillary diameter 1mm. Calculate the pressure indicated by a reading of 3.00cm. Calculate the error resulted by assumption that

Solution.

1mm Hg = 133.3Pa

The fraction error resulted by assumption that is negligible small value.

6.4.2 Dead-weight tester The dead-weight tester shown in Fig. 11 is another example of mechanical device which based on a simple and clear principle and used for calibration purposes. A sensitive element of this device is a fluid pressure within the chamber. When balanced with a known weight this pressure used as a secondary standard for static calibration of pressure gages.

Fig.11 Dead-weight tester

Calibrated pressure gage G is connected to the chamber when the valve V is closed. To fill the tester with a clean oils the plunger first moving to its most forward position. Withdraw the plunger slowly the oil is poured in through the opening for the piston.

When the valve V is open the known pressure exerted on the fluid by the piston is now transmitted to the calibrated gage. This pressure may be varied or by adding weights to the piston or by using different piston-cylinder combinations of varying areas.

Advantages

1. Wide range of the application: from1 to 1000 bar

2. Direct calibration by a known standard fluid pressure

Disadvantages

1. The accuracy of dead-weight testers is limited by two factors:

(a) The friction between the cylinder and the piston;

This factor can be reduced by rotation of the piston and use of long enough surfaces to ensure negligible flow of oil through the annular space between the piston and the cylinder.

(b) The uncertainty in the area upon which the weight force acts.

This is neither the area of the piston nor the area of the cylinder; while it is some effective area which depends on the clearance spacing and the viscosity of the oil. The smaller the clearance, the closer the effective area will approximate the cross-sectional area of the piston. The percentage error due to this clearance varies according to formula:Percent error =0.02 1%where = oil density; = pressure differential on the cylinder; b = clearance spacing; = viscosity; D = piston diameter and L = piston length. It is seen from this formula that at high pressures when an elastic deformation of the cylinder increases the clearance spacing, b rises and thereby the error of the tester also increases. To overcome this disadvantage a pressure supplied by the tester acts simultaneously on calibrated gage and on the secondary standard gage of known sensitivity as shown in Fig.9. Typical characteristics of several pressure measuring devisesDeviceRange, torrRelative error, %

Dead-weight tester 760 -

0.02 -1.0

Bourdon-tube1.0 -

0.5 5(% of scale)

Pirani gages - 1.0 2 - 7

Ionization gage - 1.05 - 50

U-manometers0.2 - 7600.02 -1.0

McLeod gage - 101 - 3

Questions to exam1. Distinguish among gage pressure, absolute pressure and vacuum.

2. What are the advantages of the manometer pressure-measurement device?3. What is the advantage of a well-type manometer?4. What are some advantages of the bourdon-tube, diaphragm, and bellows gages?5. Describe the principle of operation of a McLeod gage.6. Describe the Pirani gage.7. Describe the ionization gage.

8. How does ionization gage differ from the Pirani gage? What disadvantages does it have?9. Describe the principle of operation and main disadvantage of Dead-weight tester.

10. The static calibration what is it means?

11. How many possibilities exist for static calibration of pressure gage?

12 Describe s standardized procedure of the static calibration of pressure gage.

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