CIVIL ENGINEERING AND RESEARCH.

PRESSURE GAUGES AND MANOMETERS

2.1 Introduction. 2.2 Pressure variation in a fluid at rest. 2.3 The hydraulic Jack. 2.4 Systems for Fluid Pressure Measurements. 2.5 Measurement of Atmospheric pressure–Mercury barometer, Aneroid barometer 2.6 Measurement of Fluid Pressure – Mechanical Gauges, Manometers. Typical examples. Highlights. Theoretical problems. Numerical problems.

2.1 Introduction

Pressure is the force exerted by a fluid on the surfaces with which it is in contact or by one part of a fluid on the adjoining part. The intensity of pressure, ‘p’ (or simply pressure) at any point is the force exerted on a unit area at that point and is measured in N/m² (or bar = 10⁵ N/m²). The unit, ‘N/m²’is also known as Pascal (Pa).

In general, p = dp/da, where p is the pressure intensity, dp force acting on a small area da. If the total force P acts uniformly over the entire area A, then,

p = P/A ...(2.1)

2.2 Pressure variation in a fluid at rest

The pressure at any point in a fluid at rest is obtained from hydrostatic law which states: “The rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point”.

The proof of the law is as follows:

Consider a small fluid element as shown in Fig 2.1.

Let

= cross-sectional area of element

= height of fluid element

p= intensity of pressure on face AB

z = distance of fluid element from free surface

The forces acting on fluid element are:

(i) Pressure force on face AB=

(perpendicular to AB and acting downwards)

(ii) Pressure force on face CD=

(perpendicular to CD and acting upwards)

(iii) Weight of fluid element=weight density´volume=

(iv) Pressure forces on surfaces AD and BC, which are equal and opposite (will cancel out).

For equilibrium of the fluid element, we have

(canceling

from both sides) …(a)

Eq. (a) shows that rate of increase of pressure in a vertical direction is equal to weight density of the fluid at that point. This is “hydrostatic law”.

On integrating Eq. (a), we get

…(b)

where p is the pressure above atmospheric pressure.

From Eq. (b), we have

, where z is known as the pressure head.

2.3 The hydraulic Jack

A diagram of a hydraulic jack (hydraulic press is similar in principle) is shown in Fig 2.2.

A force P applied to the piston of the small cylinder forces the liquid out into the large cylinder thus raising the piston supporting load W. The force P acting on area ‘a’ produces a pressure p₁ which is transmitted equally in all directions through the liquid . If the two pistons are at the same level, pressure p₂ acting on the larger piston must equal p₁.

Now

, and

Thus a small force P can raise a large load W, and the jack has a mechanical advantage of A/a.

If the large piston is a distance h below the smaller piston , the pressure p₂ will be greater than p₁,due to the he head h ,by an amount wh ,where ‘w’ is the specific weight of the liquid.

Hence, p₂ = p₁+ wh

Example 2.1: A force of 900N is applied to the smaller cylinder of a hydraulic jack. The diameter of the smaller piston is 5mm while that of the larger piston is 15mm. Determine the load W which can be lifted on the larger piston if , (i) the pistons are at the same level (ii) the larger piston is 0.8m below the smaller piston. The liquid in the jack is water of specific weight 9810N/m³.

Solution

Given: Force applied on smaller cylinder, P=900 N

Diameter of smaller piston, d=5mm=0,005m

Diameter of larger piston, D= 15mm=0.15m

(i) Pistons at the same level:

Area of smaller cylinder,

Area of larger cylinder,

Also

Mass lifted =

(ii) Larger piston at distance h below the smaller piston:

Given: h = 0.8m

p₂ = p₁+ wh

Putting :

and

Mass lifted =

2.4 Systems for Fluid Pressure Measurements

In one system, pressure is measured above the absolute zero (complete vacuum) and is referred to as absolute pressure and in the other system it is measured above the atmospheric and is referred to as gauge pressure. The relationships are as shown in Fig 2.3.

Definitions

· Absolute pressure: This is defined as the pressure which is measured with reference to absolute vacuum pressure as the datum point.

· Gauge pressure: This is defined as the pressure which is measured with reference to the atmospheric pressure as the datum point.

· Vacuum pressure: This is defined as the pressure below atmospheric pressure.

· Atmospheric pressure: This is the pressure due to atmosphere at the surface of the earth, depending upon the head of air above the surface, and at sea level it is about 101.325 kN/m², equivalent to a head of 10.35m of water or 760mm of mercury, and decreases with altitude. Atmospheric pressure may be measured using a mercury barometer or aneroid barometer.

2.5 Measurement of Atmospheric pressure

This is usually measured by means of a mercury barometer or an aneroid barometer.

(i) Mercury barometer: A simple mercury barometer consists of a glass tube about 800mm long and closed at one end (Fig 2.4.It is filled with mercury and then inverted in a small reservoir full of mercury. A vacuum forms in the top portion of the tube.

Atmospheric pressure acting on the surface of the mercury in the reservoir supports a column of mercury in the tube. The height ‘h’ of the column of mercury is about 760mm Hg at the standard temperature. Points A and B, being at the same level in mercury have equal pressures. The pressure at A is equal to the atmospheric pressure, so that pA=ρgh. Thus the height of mercury column is proportional to the atmospheric pressure.

(ii) Aneroid barometer: This consists of a corrugated box which is completely evacuated. The box is prevented from collapse by a strong spring (Fig 2.5). When the box is exposed to atmospheric pressure, the front of the box moves in or out. A spring balances the force due to pressure and these small movements of the face of the box are amplified and cause the movements of a pointer over a calibrated scale.

Pointer

vacuum Thread

Bearing for pointer pivot

p h Spring

mercury Partially evacuated

pA box

Fig 2.4 Mercury barometer

Fig 2.5 Aneroid barometer

2.6 Measurement of Fluid Pressure

This may be carried out using, Mechanical gauges (or pressure gauges), and Manometers. In the case of low pressures, sensitive manometers are used.

2.6.1 Mechanical Gauges

These are used for measuring high fluid pressure and the commonly used ones are: (i) Bourdon tube pressure gauge, (ii) Diaphragm pressure gauge, (iii) Dead weight pressure gauge, and (iv) Bellows pressure gauge.

Advantages of mechanical gauges include portability, direct reading, and a wide operation range. These gauges measure the magnitude of pressure relative to atmospheric pressure i.e. gauge pressures.

(i) Bourdon Gauge (or Tube Pressure Gauge)

This consists of a curved tube (elliptical) which tends to straighten when placed under pressure. The movement of the free end of the tube is used to drive a multiplying mechanism which rotates a pointer over a dial, from which pressure can be read (Fig 2.6).

(ii) Diaphragm Pressure Gauge:

This measures pressure above or below atmospheric pressure. In its simplest form, it consists of a corrugated diaphragm (instead of Bourdons’ tube) as shown in Fig 2.7. When the gauge is connected to a fluid, whose pressure is to be found at C, the fluid under pressure causes some deformation of the diaphragm. With the help of some pinion arrangement, the elastic deformation of the diaphragm rotates the pointer. The pointer moves over a calibrated scale, which directly gives the pressure reading. This type of gauge measures relatively low pressures.

(iii) Dead Weight Gauge:

This is the most accurate pressure gauge and is used for the calibration of other pressure gauges in a laboratory. In its simplest form, it consists of a piston and cylinder of known area, connected to a fluid as shown in Fig 2.8. The pressure on the fluid in the pipe is calculated using the relation; P = weight/area of piston. A pressure gauge to be calibrated is fitted in the other end of the tube. By changing the weight on the piston, the pressure on the fluid is calculated and measured on the gauge at the respective points, indicated by the pointer. A small error due to frictional resistance to the motion of the piston may come into play, but may be avoided by taking adequate precautions.

Fig 2.8 Dead Weight Gauge

Fig 2.7 Diaphragm Pressure Gauge

2.6.2 Manometers. These are defined as devices used for measuring the pressure at a point (or differential pressure) in a fluid by balancing the column of a fluid by the same or another column of fluid. The may be classified as simple manometers or differential manometers.

Merits of manometers include: (i) Easy to fabricate (ii) relatively inexpensive (iii) Good accuracy (iv) High sensitivity (v) Require little maintenance (vi) Not affected by vibration (vii) Suitable for low pressure and low differential pressures (viii) Easy to change the sensitivity by affecting a change in the type manometric fluid in the manometer.

Demerits of manometer include: (i) Usually bulky and large in size (ii) Being fragile, get broken easily (iii) Readings are affected by changes in temperature, altitude and gravity (iv) Capillarity effect is created due to surface tension of the manometric fluid.

2.6.2.1 Simple manometers

A simple manometer consists of a glass tube whose one end is connected to a point where pressure is to be measured and the other end remains open to the atmosphere. Common types of simple manometers include: (i) Piezometers (or pressure tube), (ii) U-tube manometers, and (iii) Single column manometers (or micro manometers).

(i) Piezometer:

This is the simplest form of manometers used for measuring gauge pressures (Fig 2.9). One end is connected to the point where pressure is to

be measured and the other end is open to atmosphere. The

rise of the liquid gives the pressure head at the point in the

form,

P_A = ρgh ...(2.2)

To avoid error due to the velocity of the fluid,

Fig. 2.9 Piezometer

it is essential that the tube should enter the pipe normal to

the direction of flow. Errors due to the formation of eddies

can be avoided by connecting the end of the piezometer

flush with the inner face of the pipe and ensuring that there are no burrs or projections on the connection. To avoid error due to capillary action, the diameter of the piezometer should be greater than 12mm.

Piezometers have limited use due to the following drawbacks: (i) Large pressures require long tubes which cannot be handled conveniently, (ii) If the fluid is a gas, a piezometer cannot be used as the gas would escape through the open end. (iii) Rapid changes in pressure cannot be recorded accurately as the changes in piezometric levels lag behind the changes in pressure. These limitations can be overcome by use of bent tube filled with a heavier liquid.

Example 2.2: Determine the maximum pressure measured by a piezometer tube, 2m high.

Solution

P = ρgh = 1000 × 9.81×2 = 19.62 kN/m²

h₂

h₁

X X

r_m

(ii) U – tube Manometers: These consist of a glass

tube bent in U-shape (Fig 2.10), one end of which is

connected to a point where the pressure is to be

measured, and the other end remains open to the

atmosphere. The tube generally contains mercury or

other immiscible liquid. The pressures at two points at

the same level in a continuous homogeneous liquid are

Fig 2.10 U-tube Manometer

equal. If P_A is the pressure at point A, h₁ depth of the

liquid above datum, h₂ depth of mercury above datum,

ρ_ldensity of fluid in the pipe, ρ_m density of mercury in the manometer, and taking atmospheric pressure as the reference pressure,

Then, P_A+ρ_lgh₁ = ρ_mgh₂

∴ P_A= ρ_mgh₂ - h₁…(2.3)

Example 2.3: The right limb of a simple U-tube manometer contains mercury and is open to the atmosphere, while the left limb is connected to a pipe in which a fluid of specific gravity 0.9 is flowing. The centre of the pipe is 120mm below the level of mercury in the right limb. Find the pressure of fluid in the pipe if the difference of mercury level in the two limbs is 200mm.

120

h₁ h₂=200

X X

Solution:

Given : Sp.gr. fluid , S_l = 0.9

Sp. gr. of mercury S₂ = 13.6

Difference of mercury level, h₂= 0.2m

The connection is as shown in Fig 2.11.

Height of fluid above X-X, h₁= 0.2-0.12 = 0.08m

But, P_A+h₁S₁ = h₂S₂

Fig 2.11 U-tube Manometer

Or P_A = 0.2×13.6-0.08×0.9 = 2.648m of water

= 2.648×10³×9.81

= 25.98 kN/m²

2.5.2.2 Differential manometers

These are used for measuring the difference

in pressures between two points in a pipe, or in two

different pipes. A differential manometer consists of

a U-tube, containing a heavy liquid, whose difference of pressure is to be measured. Common types include: (i) U-tube differential manometer (ii) Inverted U-tube differential manometer.

ρ₁

ρ₂

X h X

ρ_m

(i) U-tube Differential Manometers: These measure

the difference of pressures at two points. The pressure

at one starting point must be known. For a differential

manometer as shown in Fig 2.12, and taking datum at

level X-X,

Pressure head above X-X in the left limb,

= Pressure head above X-X in the right limb

Or (h+x)ρ_lg+P_A = ρ_mgh+ρ₂gy+pB

Hence, P_A- P_B= ρ_mgh+ρ₂gy-(h+x)ρ_lg

If A and B are at the same level, and contain the

Fig 2.12 U-tube Differential Manometer

same liquid, of density ρ then,

P_A- P_B= ρ_mgh –ρgh = gh(ρ_m-ρ)…(2.4)

Example 2.4: A U-tube differential manometer is connected to two pipes A and B as shown in Fig.2.13. Pipe A contains a liquid of sp.gr. 1.5 while pipe B contains a liquid of sp. gr. 0.9. The pressures at A and B are 98.1 kN/m² and 176.58 kN/m². Determine the difference in mercury level in the differential manometer.

X X

Solution:

Given: Sp.gr. of liquid at A, S_l =1.5

Sp.gr. of liquid at B, S₂ = 0.9

Pressure at A, P_A= 98.1kN/m²

Pressure at B, P_B=176.58kN/m²

Taking X-X as datum:

Pressure head above X-X (left limb)

Fig. 2.13

= Pressure head above X-X (right limb) Taking X-X as datum:

(ii) Inverted U-tube differential manometers:

Light fluid

(frequently air)

X X

These consist of an inverted U-tube containing

a light liquid. The two ends of the tube are connected to

the points whose difference of pressures are to be

measured. They are used for measuring difference

of low pressures. For the U-tube differential manometer

shown (Fig 2.14), taking X-X as the datum and assuming

pressure in A is greater than that in B,

Pressure head above X-X in the left limb

= Pressure head above X-X in the right limb

P_A-ρ₁g h₁ = P_B-ρ₂g h_2-ρ_Lgh

where P_A is the pressure head at A, ρ₁is thedensity

of liquid at A, h₁is the height of liquid in the left

Fig 2.14

limb below datum X-X, P_B is the pressure head at B,

ρ₂is thedensity of liquid at B, h₂is the height of liquid

in the right limb below datum X-X and h is the difference in deflection of the light liquid.

\P_A- P_B = ρ₁g h₁-ρ₂g h_2-ρ_Lgh …(2.5)

Example 2.5 An inverted U-tube manometer as shown in Fig 2.15 was used to measure the pressure difference between two points A and B in an inclined pipeline.(a) Derive the equation for the pressure difference between points A and B.(b) Determine the pressure difference between points A and B if the top of the manometer is filled with: (i) air,(ii) oil of specific gravity of 0.8. Given: h = 0.3m, a = 0.25m, b =0. 15m

Solution

(a) For left hand limb, P_x = P_A-ρga-ρ_mgh

For right hand limb, P_x = P_B-ρg(b+h)

Thus, P_B- P_A= ρg(b-a)+gh(ρ-ρ_m)

Light fluid

ρ_m

X X

(b) (i) If top is filled with air, ρ_m is negligible compared with ρ

=10³×9.81(0.15-0.25+0.3)

= 1.962 kN/m²

(ii) If top is filled with oil of specific gravity 0.8,

ρ_m = 0.8×9.81×10³

But P_B-P_A= ρg(b-a)+gh(ρ-ρ_man)

Or P_B-P_A= 10³×9.81(0.15-0.25)+10³×9.81×0.3(1-0.8)

= -392.4 N/m²

Fig.2.15

(iii) Multi-tube manometers

For pressure heads greater than about 30m, a single column

Manometer cannot be used as it requires a long U-tube.

This can be overcome by having a larger number of U-tubes as shown in Fig 2.16, where 2 U-tubes have been used. The manometer is filled with different manometric fluids whose densities are as shown.

Pressure at point B,

Pressure at point C, p_C = p_B

Pressure at point E,

Pressure at point F,

Pressure at point G,

Pressure in the pipe, P=

Note: Result could be written straightaway by looking at the double U-tube configuration.

2.6.2.3 Micro-manometers: These are also referred to as Single Column Manometers and are a modified form of U-tube manometers in which a shallow reservoir having a large cross-sectional area (about 100 times) as compared to the area of the tube is connected to one limb of the manometer. For any variation in pressure, the change in liquid level in the reservoir will be so small that it may be neglected, and the pressure is indicated by the height of liquid in the other limb. The narrow limb may be vertical or inclined. Micro-manometers are used to measure low pressures, where accuracy is of much importance. The following types are to be considered: (i) Vertical single column micro-manometer, (ii) Inclined single column tube micro-manometer, and (iii) Differential micro-manometer.

(i) Vertical Single Column Micro-manometer:

Consider a single column manometer as shown in Fig 2.17. The pressure in the pipe will force the liquid in the basin downward and due to the larger area of the basin, the fall of heavy liquid level will be very small. This downward movement of heavy liquid in the basin will cause a considerable rise of the heavy liquid in the narrow right limb of the column.

For the micro-manometer shown, let:

X-X= level in the basin before connection to the pipe

Z-Z = level in basin after connection to the pipe

h₂

h₁

X X

Z Z

dh = fall of heavy liquid level in the basin

h₁=height of light liquid in the left limb above datum

h₂=height of heavy liquid in the right limb above datum

Px=pressure rise in pipe, (contains light liquid)

A = cross-sectional area of basin

a = cross-sectional area of tube

ρ_l=density of light liquid

ρ_m=density of heavy liquid

Fall of heavy liquid level in the basin causes a

corresponding rise of heavy liquid level in tube.

Fig 2.17 Vertical Single Column Micro-manometer

Therefore,

Taking horizontal surface in the basin at which the heavy and light liquids meet as the datum level Z-Z,

Pressure in the left limb = pressure in the right limb

P_A+ρ_lgh₁+ ρ_lgdh = ρ_mgh₂+ρ_mgdh

or,

Note: If ‘a’ is very small compared to ‘A’, the ratio a/A may be neglected.

so that P_A = ρ_mgh₂-ρ_lgh₁

(ii) Inclined Single Column Micro-manometer:

This type of micro-manometer (Fig 2.18) is more sensitive than the vertical type. Due to the inclination, the distance moved by the heavy liquid in the narrow inclined tube will be comparatively more, giving a higher reading for a given pressure.

Let l = length of heavy liquid moved in the right limb

θ = inclination of right limb with the horizontal

h₁ = height of the centre of the pipe above X-X

h₂ = vertical rise of liquid in the right limb from X-X.

From the geometry of the figure, h₂/l = sin θ, so that h₂ = l sin θ. By substituting the value of h₂in the micro-manometer equation (Eq 2.6), pressure in the pipe can be determined.

Thus, Px = ρ_mglsinθ-ρ_lgh₁ …(2.7)

The movement of the meniscus along the inclined leg read off the scale, is considerately greater than the change in level l. Then the pressure difference, P2-P1= ρgh₂= ρglsinθ.

The manometer can be made as sensitive as may be required by adjusting the angle of inclination of the leg and choosing a liquid with a suitable value of density ρ, to give a scale reading l of the desired size for a given pressure difference. Then the pressure difference may be taken as, P₂-P₁= ρgl sinθ{1+(d/D)²} where d is the diameter of the pipe and D is the diameter of the basin.

(ii) Differential Micro-manometer:

This type of micro-manometer (also referred to as double column enlarged ends manometer or oil gauge with enlarged ends) is a type of a sensitivity gauge which may be obtained by using a U-tube with enlarged ends as shown (Fig 2.19). This type of gauge is used for measuring small differences of pressure in gases. Water (sp. gr. Sw=1) and oil (sp. gr. So=0.95) are placed in the limbs, the free surface of each liquid being in the enlarged ends. For the sensitivity gauge shown, let A= area of enlarged end, usually 50 times the area a= area of tube, So=sp. gr. of oil used. Since water is denser than oil, it will sink to the bottom of the U-tube. When The pressure applied to the two limbs is the same, (P₁=P₂), suppose the separation of the surface between oil and water is at X-X and that the head of oil is h. Pressure at level X-X will be the same in both limbs.