Centre of Pressure for Vertical and Inclined Plane Surface

Centre of Pressure for Vertical Plane Surface

The point of application of the total pressure on a plane surface is known as centre of pressure. For a plane surface immersed horizontally since the pressure intensity is uniform the total pressure would pass through the centroid of the area i.e., in this case the centroid of the area and the centre of pressure coincide with each other. For a plane surface immersed vertically the centre of pressure does not coincide with the centroid of the area. Since the pressure intensity increases with the increase in the depth of liquid, the centre of pressure for a vertically immersed plane surface lies below the centroid of the surface area. The position of the centre of pressure for a vertically immersed plane surface can be determined as shown below.

Fig. 1 Total Pressure on a Vertical Plane Surface 

As shown in Fig. 1 let ‘ħ’ be the vertical depth of the centre of pressure for the plane surface immersed vertically. Then the moment of the total pressure ‘P’ about axis OO is equal to (P ℏ).

The total pressure on the strip

dP = pdA = wx(bdx)

and its moment about axis OO is

(dP) x = wx² (bdx)

Likewise, by considering a number of small strips and summing the moments of the total pressure on these strips about axis OO, the sum of the moments of the total pressures on all the strips becomes

∫ (dP) x = w ∫ x²(bdx)

By using the “Principle of Moments”, which states that the moment of the resultant of a system of forces about an axis is equal to the sum of the moments of the components about the same axis, the moment of the total pressure about axis OO is

P ℏ = w ∫ x² (bdx)                  (1)

In Eq. (1), ∫ x²(bdx) represents the sum of the second moment of the areas of the strip about axis OO, which is equal to the moment of inertia I0 of the plane surface about axis OO. That is

I₀ = ∫ x² (bdx)                      (2)

Eq. (1) in Eq. (2) and solving for ℏ,

Substituting for the total pressure from Equation P = wAx̅, we obtain

From the “parallel axes theorem” for the moment of inertia, 

I₀ = I_G + A x̅                         (4)

where I_G is the moment of inertia of the area about an axis passing through the centroid of the area and parallel to axis OO.

Introducing Eq. (4) in Eq. (5), it becomes 

Equation (5) gives the position of the centre of pressure on a plane surface immersed vertically in a static mass of liquid. Since for any plane surface the factor (I_G/Ax̅) is always positive, Eq. (5) indicates that ℏ > x̅, i.e., the centre of pressure is always below the centroid of the area. 

Further it is seen that deeper the surface is submerged, i.e., the greater is the value of x̅, the factor (I_G/Ax̅) becomes smaller and the centre of pressure comes closer to the centroid of the plane surface. This is because, as the pressure becomes greater with increasing depth, its variation over a given area becomes smaller in proportion, thereby making the distribution of pressure more uniform. Thus where the variation of pressure is negligible the centre of pressure may be taken as approximately at the centroid. This is justifiable in liquids, only if the depth is very large and the area is small, and in gases because in them the pressure changes very little with depth.

The lateral location of the centre of pressure can also be readily determined by taking moments about any convenient axis in the vertical direction. Thus if ‘OX’ is the reference axis (as shown in Fig.1) in the vertical direction, lying in the same vertical plane in which the plane surface is lying, from which ȳ is the distance of the centre of pressure of the plane surface and ȳ is the distance of the centre of pressure of the small strip on the plane surface, then the distance ȳ may be determined by taking the moments about axis OX. The moment of dP about axis OX is (dP) y =wx (bdx) y and the sum of the moments of the total pressure on all such strips considered on the plane surface is

∫(dP)y = w ∫ xy (bdx) 

which by the principle of moments is equal to the moment of P about axis OX, i.e., (P ȳ).

Thus

The centre of pressure of the plane surface immersed vertically in a static mass of liquid is therefore, at a vertical depth ℏ below the free surface of the liquid and at a distance ȳ from an assumed vertical reference axis OX. If the plane surface has a vertical axis of symmetry passing through its centroid, then this axis may be taken as the reference axis OX, in which case ∫ xy (bdx) = 0, and the centre of pressure lies on the axis of symmetry at a vertical depth ℏ below the free surface of the liquid.

Centre of Pressure for Inclined Plane Surface

Fig. 2 Total pressure on inclined plane surface

As shown in Fig. 2, let ℏ be the vertical depth of the centre of pressure for the inclined plane surface below the free surface of the liquid and its inclined distance from the axis OO be y_p.

Total pressure on the strip shown in Fig. 2 is [w(y sin θ) dA ] and its moment about axis OO is

(dP) y = (w sinθ y²  dA)

By summing the moments of the total pressures on such small strips about axis OO and using the “Principle of Moments”, we get

Py_p = w sin θ ∫ y²  dA                   (6)

Again in Eq. (6), ∫ y²dA represents the sum of the second moments of the areas of the strips about axis OO, which is equal to the moment of inertia I0 of the plane surface about axis OO. That is,

I₀ = ∫ y² dA                       (7)

Introducing Eq. (7) in Eq. (6) and solving for y_p, we obtain

Further from the “parallel axes theorem” for the moments of inertia, 

I₀ = I_G +A ȳ ²                     (9)

where I_G is the moment of inertia of the area about an axis passing through the centroid of the area and parallel to axis OO. Introducing Eq. (9) in Eq. (8) and substituting for the total pressure from P = wA (ȳ sin θ), we get

        (11)

Eq. (10) gives the vertical depth of centre of pressure below free surface of liquid, for an inclined plane surface, wholly immersed in a static mass of liquid. The lateral location of the centre of pressure in this case may also be determined in the same manner as in the case of vertically immersed plane surface, by considering a reference axis OY perpendicular to the axis OO (or OZ) (as shown in Fig. 2) and lying in the same plane in which the inclined plane surface is lying. From the axis OY let z_p be the distance of the centre of pressure of the plane surface and z be the distance of the centre of pressure of the small strip on the plane surface. The moment of the total pressure dP on the small strip about axis OY is (dP) y = wx(dA) z and the sum of the moments of the total pressure on all such strips considered on the plane surface is

∫ (dP) z = w ∫ xz (dA)

which by the principle of moments is equal to the moment of P about axis OY, i.e., (Pz_p). Thus

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Total Pressure and Centre of Pressure

When a static mass of fluid comes in contact with a surface, either plane or curved, a force is exerted by the fluid on the surface. This force is known as total pressure. Since for a fluid at rest no tangential force exists, the total pressure acts in the direction normal to the surface. The point of application of total pressure on the surface is known as centre of pressure. As indicated later an engineer is often required to compute the magnitude of total pressure and to locate its point of application in the design of several hydraulic structures.

Total Pressure on a Horizontal Plane Surface

Consider a plane surface immersed in a static mass of liquid of specific weight w, such that it is held in a horizontal position at a depth ‘h’ below the free surface of the liquid, as shown in Fig. 1. Since every point on the surface is at the same depth below the free surface of the liquid, the pressure intensity is constant over the entire plane surface, being equal to p = wh. Thus if ‘A’ is the total area of the surface then the total pressure on the horizontal surface is

                                                                       P = pA = (wh) A = wAh                                                (1)

Fig. 1 Total pressure on a Horizontal Plane Surface

The direction of this force is normal to the surface, as such it is acting towards the surface in the vertical downward direction at the centroid of the surface.

Total Pressure on a Vertical Plane Surface

Fig. 2 shows a plane surface of arbitrary shape and total area ‘A’, wholly submerged in a static mass of liquid of specific weight ‘w’. The surface is held in a vertical position, such that the centroid of the surface is at a vertical depth of ‘x’ below the free surface of the liquid. It is required to determine the total pressure exerted by the liquid on one face of the plane surface.

Fig. 2 Total Pressure on a Vertical Plane Surface

In this case since the depth of liquid varies from point to point on the surface, the pressure intensity is not constant over the entire surface. As such the total pressure on the surface may be determined by dividing the entire surface into a number of small parallel strips and computing the total pressures on each of these strips. A summation of these total pressures on the small strips will give the total pressure on the entire plane surface.

Consider on the plane surface a horizontal strip of thickness ‘dx’ and width ‘b’ lying at a vertical depth ‘x’ below the free surface of the liquid. Since the thickness of the strip is very small, for this strip the pressure intensity may be assumed to be constant equal to p = wx. The area of the strip being dA = (b × dx), the total pressure on the strip becomes

                                                                            dP = pdA = wx(bdx)                                                  (2)

Total pressure on the entire plane surface is

P = ∫ dP = w∫ x(bdx)

But ∫ x (bdx) represents the sum of the first moments of the areas of the strips about an axis OO, (Which is obtained by the intersection of the free surface of the liquid with the vertical plane in which the plane surface is lying) which from the basic principle of mechanics is equal to the product of the area A and the distance x of the centroid of the surface area from the same axis OO. That is

∫ x (bdx) = Ax̄

                                                                              P = wA x̄                                                        (3)

Equation (3) thus represents a general expression for total pressure exerted by a liquid on a plane surface. Since w x̄ is the intensity of pressure at the centroid of the surface area, it can be stated that the total pressure on a plane surface is equal to the product of the area of the surface and the intensity of pressure at the centroid of the area.

Total pressure on a horizontal plane surface can also be determined by Eq. (3), since in this case

x̄ = h.

Total Pressure on Inclined Plane Surface

Consider a plane surface of arbitrary shape and total area ‘A’, wholly submerged in a static mass of liquid of specific weight ‘w’. The surface is held inclined such that the plane of the surface makes an angle ‘θ’ with the horizontal as shown in Fig. 3. The intersection of this plane with the free surface of the liquid is represented by axis OO, which is normal to the plane of the paper.

Let x̄ be the vertical depth of the centroid of the plane surface below the free surface of the liquid and the inclined distance of the centroid from axis OO measured along the inclined plane is ȳ.

Consider on the plane surface, a small strip of area ‘dA’ lying at a vertical depth of ‘x’ and its distance from axis OO being ‘y’. For this strip the pressure intensity may be assumed to be constant equal to p = wx.

Fig. 3 Total Pressure on Inclined Plane Surface

Total pressure on the strip is 

dP = wx (dA)

Since x = y sin θ

dP = w (y sin θ) (dA)

By integrating the above expression the total pressure on the entire surface is obtained as

P = (w sin θ) ∫ y (dA)

Again ∫ y dA represents the sum of the first moments of the areas of the strips about axis OO, which is equal to the product of the area A and the inclined distance of the centroid of the surface area from axis OO. That is

∫ y dA = A ȳ

                                                                                       ∴ P = wA (ȳ sin θ)                                      (4)

But x̄ = ȳ sin θ

                                                                                           P = wA x̄                                                (5)

Eq. 5 is same as Eq. 3, thereby indicating that for a plane surface wholly submerged in a static mass of liquid and held either vertical or inclined, the total pressure is equal to the product of the pressure intensity at the centroid of the area and the area of the plane surface.

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Mechanical Gauges

Mechanical gauges are those pressure measuring devices, which embody an elastic element, which deflects under the action of the applied pressure and this movement mechanically magnified, operates a pointer moving against a graduated circumferential scale. Generally these gauges are used for measuring high pressures and where high precision is not required. Some of the mechanical pressure gauges which are commonly used are as noted below.

i) Bourdon Tube Pressure Gauge

It is the most common type of pressure gage which was invented by E. Bourdon (1808–84). The pressure responsive element in this gauge is a tube of steel or bronze which is of elliptic cross-section and is curved into a circular arc. The tube is closed at its outer end and this end of the tube is free to move. The other end of the tube, through which the fluid enters, is rigidly fixed to the frame as shown in Fig. 1. When the gauge is connected to the gauge point, under pressure enters the tube. Due to increase in internal pressure, the elliptical cross-section of the tube tends to become circular, thus causing the tube to straighten out slightly. The small outward movement of the free end of the tube is transmitted, through a link, quadrant and pinion, to a pointer which by moving clockwise on the graduated circular dial indicates the pressure intensity of the fluid.

The dial of the gage is so calibrated that it reads zero when the pressure inside the tube equals the local atmospheric pressure and the elastic deformation of the tube causes the pointer to be displaced on the dial in proportion to the pressure intensity of the fluid. By using tubes of appropriate stiffness, gauges for wide range of pressures may be made. Further by suitably modifying the graduations of the dial and adjusting the pointer Bourdon tube vacuum gauges can also be made.

When a vacuum gauge is connected to a partial vacuum, the tube tends to close, thereby moving the pointer in anti-clockwise direction, indicating the negative or vacuum pressure. The gauge dials are usually calibrated to read Newton per square metre (N/m²),or pascal (Pa), or kilogram (f) per square centimetre [kg(f)/cm²]. However other units of pressure, such as metres of water or centimetres of mercury, are also frequently used.

Fig. 1 Bourdon Tube Pressure Gauge

ii) Diaphragm Pressure Gauge

The pressure responsive element in this gage is an elastic steel corrugated diaphragm. The elastic deformation of the diaphragm under pressure is transmitted to a pointer by a similar arrangement as in the case of Bourdon tube pressure gauge. However, this gauge is used to measure relatively low pressure intensities. The Aneroid barometer operates on a similar principle.

Fig. 2 Diaphragm Pressure Gauge

iii) Bellows Pressure Gauge

In this gauge, the pressure responsive element is made up of a thin metallic tube having deep circumferential corrugations. In response to the pressure changes this elastic element expands or contracts, thereby moving the pointer on a graduated circular dial.

Fig.3 Bellows Pressure Gauge

iv) Dead -Weight Pressure Gauge

A simple form of a dead-weight pressure gauge consists of a plunger of diameter d, which can slide within a vertical cylinder, as shown in Fig. 4. The fluid under pressure, entering the cylinder, exerts a force on the plunger, which is balanced by the weights loaded on the top of the plunger. If the weight required to balance the fluid under pressure is W, then the pressure intensity ‘p’ of the fluid may be determined as, 

The only error that may be involved is due to frictional resistance offered to motion of the plunger in the cylinder. But this error can be avoided if the plunger is carefully ground, so as to fit with the least permissible clearance in the cylinder. Moreover, the whole mass can be rotated by hand before final readings are taken.

Fig. 4 Dead-Weight Pressure Gauge

Dead-weight gauges are generally not used so much to measure the pressure intensity at a particular point as to serve as standards of comparison. Hence as shown in Fig. 4, a pressure gauge which is to be checked or calibrated is set in parallel with the dead-weight gauge. Oil under pressure is pumped into the gauges, thereby lifting the plunger and balancing it against the oil pressure by loading it with known weights. The pressure intensity of the oil being thus known, the attached pressure gauge can either be tested for its accuracy or it can be calibrated.

A dead-weight gauge which can be used for measuring pressure at a point with more convenience. In this gauge a lever, same as in some of the weighing machines, is provided to magnify the pull of the weights. The load required to balance the force due to fluid pressure is first roughly adjusted by hanging weights from the end of the main beam. Then a smaller jockey weight is slide along to give precise balance. In more precise type of gauge the sliding motion may be contrived automatically by an electric motor.

The following points should be kept in view while making connections for the various pressure measuring devices.

• At the gauge point the hole should be drilled normal to the surface and it should flush with the inner surface.
• The diameters of the holes at the gauge points should be about 3 to 6 mm.
• The holes should not disturb the internal surface and no burrs or irregularities must be left.
• There should be no air pockets left over in the connecting tubes, which should be completely filled with the liquid. The presence of air bubbles can easily be detected if the connecting tubes are made of polythene or similar transparent material.

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Measurement of Pressure

In practice, pressure is always measured by the determination of a pressure difference. If the difference is that between the pressure of the fluid in question and that of a vacuum then the result is known as the absolute pressure of the fluid. More usually, the difference determined is that between the pressure of the fluid concerned and the pressure of the surrounding atmosphere. This is the difference normally recorded by pressure gauges and so is known as gauge pressure. If the pressure of the fluid is below that of the atmosphere it is termed vacuum or suction. (The term high vacuum refers to a low value of the absolute pressure.) The absolute pressure is always positive but gauge pressures are positive if they are greater than atmospheric and negative if less than atmospheric.

Most of the properties of a gas are functions of its absolute pressure and consequently values of the absolute pressure are usually required in problems concerning gases. Frequently it is the gauge pressure that is measured and the atmospheric pressure must be added to this to give the value of the absolute pressure. The properties of liquids are little affected by pressure and the pressure of a liquid is therefore usually expressed as a gauge value. The absolute pressure of a liquid may be of concern when the liquid is on the point of vaporizing.

Barometer

If the pressure of a liquid is only slightly greater than that of the atmosphere, a simple way of measuring it is to determine the height of the free surface in a piezometer tube. (The diameter of the tube must be large enough for the effect of surface tension to be negligible.) 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 mercury barometer. Mercury is employed because its density is sufficiently high for a fairly short column to be obtained and also because it has, at normal temperatures, a very small vapour pressure. A perfect vacuum at the top of the tube is not in practice possible; even when no air is present the space is occupied by vapour given off from the surface of the liquid. The mercury barometer was invented in 1643 by the Italian Evangelista Torricelli and the near vacuum above the mercury is often known as the Torricellian vacuum. All air and other foreign matter is removed from the mercury and a glass tube full of it is then inverted with its open end submerged in pure mercury.

The various devices adopted for measuring fluid pressure may be broadly classified under the following two heads.

   1) Manometers

   2) Mechanical Gauges

Manometers

Manometers are those pressure measuring devices which are based on the principle of balancing the column of liquid (whose pressure is to be found) by the same or another column of liquid. The manometers may be classified as

   a) Simple Manometers

   b) Differential Manometers

Simple Manometers are those which measure pressure at a point in a fluid contained in a pipe or a vessel. On the other hand differential manometers measure the difference of pressure between any two points in a fluid contained in a pipe or a vessel. In general a simple manometer consists of a glass tube having one of its ends connected to the gauge point where the pressure is to be measured and the other remains open to atmosphere. Some of the common types of simple manometers are given below.

   i) Piezometer

   ii) U-tube Manometer

   iii) Single Column Manometer

i) Piezometer

Piezometer is the simplest form of manometer which can be used for measuring moderate pressures of liquids. It consists of a glass tube (Fig. 1) inserted in the wall of a pipe or a vessel, containing a liquid whose pressure is to be measured. The tube extends vertically upward to such a height that liquid can freely rise in it without overflowing. The pressure at any point in the liquid is indicated by the height of the liquid in the tube above that point, which can be read on the scale attached to it. Thus, if ‘w’ is the specific weight of the liquid, then the pressure at point m in Fig. 1 is pm = whm. In other words, ‘hm’ is the pressure head at ‘m’. Piezometers measure gauge pressure only, since the surface of the liquid in the tube is subjected to atmospheric pressure.

Fig. 1 Piezometer

From the foregoing principles of pressure in homogeneous liquid at rest, it is obvious that the location of the point of insertion of a piezometer makes no difference. Hence piezometers may be inserted either in the top or the side or the bottom of the container, but the liquid will rise to the same level in the three tubes.

Negative gauge pressures (or pressures less than atmospheric) can be measured by means of the piezometer. It is evident that if the pressure in the container is less than the atmosphere no column of liquid will rise in the ordinary piezometer. But if the top of the tube is bent downward and its lower end dipped into a vessel containing water (or some other suitable liquid), the atmospheric pressure will cause a column of the liquid to rise to a height ‘h’ in the tube, from which the magnitude of the pressure of the liquid in the container can be obtained.

Neglecting the weight of the air caught in the portion of the tube, the pressure on the free surface in the container is the same as that at free surface in the tube and the equation may be expressed as

p = –wh,

where ‘w’ is the specific weight of the liquid used in the vessel. Conversely ‘–h’ is the pressure head at the free surface in the container.

Piezometers are also used to measure pressure heads in pipes where the liquid is in motion. Such tubes should enter the pipe in a direction at right angles to the direction of flow and the connecting end should be flush with the inner surface of the pipe. All burrs and surface roughness near the hole must be removed and it is better to round the edge of the hole slightly. Also, the hole should be small, preferably not larger than 3 mm. In order to prevent the capillary action from affecting the height of the column of liquid in a piezometer, the glass tube having an internal diameter less than 12 mm should not be used. Moreover, for precise work at low heads the tubes having an internal diameter of 25 mm may be used.

ii) U-tube Manometer

Piezometers cannot be used when large pressures in the lighter liquids are to be measured, since this would require very long tubes, which cannot be handled conveniently. Furthermore gas pressures cannot be measured by means of piezometers because a gas forms no free atmospheric surface. These limitations imposed on the use of piezometers may be overcome by the use of U-tube manometers. A U-tube manometer consists of a glass tube bent in U-shape, one end of which is connected to the gauge point and the other end remains open to the atmosphere (Fig. 2). The tube contains a liquid of specific gravity greater than that of the fluid of which the pressure is to be measured.

Fig. 2 U-tube Manometer

Sometimes more than one liquid may also be used in the manometer. The liquids used in the manometers should be such that they do not get mixed with the fluids of which the pressures are to be measured. Some of the liquids that are frequently used in the manometers are mercury, oil, salt solution, carbon disulphide, carbon tetrachloride, bromoform and alcohol. Water may also be used as a manometric liquid when the pressures of gases or certain coloured liquids (which are immiscible with water) are to be measured. The choice of the manometric liquid, depends on the range of pressure to be measured. For low pressure range, liquids of lower specific gravities are used and for high pressure range, generally mercury is employed.

When one of the limbs of the U-tube manometer is connected to the gauge point, the fluid from the container or pipe A will enter the connected limb of the manometer, thereby causing the manometric liquid to raise in the open limb as shown in Fig. 2. An air relief valve V is usually provided at the top of the connecting tube which permits the expulsion of all air from the portion A’B and its place taken by the fluid in A. This is essential because the presence of even a small air bubble in the portion A’B would result in an inaccurate pressure measurement.

iii) Single Column Manometer

The U-tube manometers described above usually require readings of fluid levels at two or more points, since a change in pressure causes a rise of liquid in one limb of the manometer and a drop in the other. This difficulty may however be overcome by using single column manometers. A single column manometer is a modified form of a U-tube manometer in which a shallow reservoir having a large cross-sectional area (about 100 times) as compared to the area of the tube is introduced into one limb of the manometer, as shown in Fig. 3. For any variation in pressure, the change in the liquid level in the reservoir will be so small that it may be neglected and the pressure is indicated approximately by the height of the liquid in the other limb. As such only one reading in the narrow limb of the manometer need be taken for all pressure measurements.

Fig. 3 Single Column Manometer

The narrow limb of the manometer may be vertical or it may be inclined. The inclined type is useful for the measurement of small pressures. Since no reading is required to be taken for the level of liquid in the reservoir, it need not be made of transparent material. If the pressure at A in the container is negative, the manometric liquid surface in the reservoir will be raised by a certain distance and consequently there will be drop in the liquid surface in the tube. Again by adopting the same procedure the gage equations for the negative pressure measurement can also be obtained.

Differential Manometers

For measuring the difference of pressure between any two points in a pipeline or in two pipes or containers, a differential manometer is employed. In general a differential manometer consists of a bent glass tube, the two ends of which are connected to each of the two gauge points between which the pressure difference is required to be measured. Some of the common types of differential manometers are as noted below.

   i) Two–Piezometer Manometer

   ii) Inverted U-Tube Manometer

   iii) U- Tube Differential Manometer

   iv) Micromanometer

i) Two-Piezometer Manometer

As the name suggests this manometer consists of two separate piezometers which are inserted at the two gauge points between which the difference of pressure is required to be measured. The difference in the levels of the liquid raised in the two tubes will denote the pressure difference between the two points. Evidently this method is useful only if the pressure at each of the two points is small. Moreover it cannot be used to measure the pressure difference in gases, for which the other types of differential manometers described below may be employed.

ii) Inverted U-tube Manometer

It consists of a glass tube bent in U-shape and held inverted as shown in Fig. 4. Thus it is as if two piezometers described above are connected with each other at top. When the two ends of the manometer are connected to the points between which the pressure difference is required to be measured, the liquid under pressure will enter the two limbs of the manometer, thereby causing the air within the manometer to get compressed. The presence of the compressed air results in restricting the heights of the columns of liquids raised in the two limbs of the manometer. An air cock as shown in Fig. 4, is usually provided at the top of the inverted U tube which facilitates the raising of the liquid columns to suitable level in both the limbs by driving out a portion of the compressed air. It also permits the expulsion of air bubbles which might have been entrapped somewhere in the pipeline. If pA and pB are the pressure intensities at points A and B between which the inverted U-tube manometer is connected, then corresponding to these pressure intensities the liquid will rise above points A and B upto C and D in the two limbs of the manometer as shown in Fig. 4.

Fig. 4 Inverted U-tube Manometer

iii) U-Tube Differential Manometer

It consists of glass tube bent in U-shape, the two ends of which are connected to the two gauge points between which the pressure difference is required to be measured. Fig. 5 shows such an arrangement for measuring the pressure difference between any two points A and B. The lower part of the manometer contains a manometric liquid which is heavier than the liquid for which the pressure difference is to be measured and is immiscible with it.

Fig. 5 U-Tube Differential Manometer

iv) Micromanometers

For the measurement of very small pressure differences or for the measurement of pressure differences with very high precision, special forms of manometers called micromanometers are used. A wide variety of micromanometers have been developed, which either magnify the readings or permit the readings to be observed with greater accuracy. One simple type of micromanometer consists of a glass U-tube, provided with two transparent basins of wider sections at the top of the two limbs, as shown in Fig. 6. The manometer contains two manometric liquids of different specific gravities and immiscible with each other and with the fluid for which the pressure difference is to be measured.

Fig. 6 Micromanometer

Before the manometer is connected to the pressure points A and B, both the limbs are subjected to the same pressure. As such the heavier manometric liquid of specific gravity S1 will occupy the level DD’ and the lighter manometric liquid of specific gravity S2 will occupy the level CC’. When the manometer is connected to the pressure points A and B where the pressure intensities are pA and pB respectively, such that pA > pB then the level of the lighter manometric liquid will fall in the left basin and rise in the right basin by the same amount Δy. Similarly the level of the heavier manometric liquid will fall in the left limb to point E and rise in the right limb to point F.

Bourdon Gauge

Where high precision is not required a pressure difference may be indicated by the deformation of an elastic solid. For example, in an engine indicator, the pressure to be measured acts at one side of a small piston, the other side being subject to atmospheric pressure. The difference between these pressures is then indicated by the movement of the piston against the resistance of a calibrated spring. The principle of the aneroid barometer may also be adapted for the measurement of pressures other than atmospheric. A curved tube of elliptical cross-section is closed at one end; this end is free to move, but the other end – through which the fluid enters – is rigidly fixed to the frame as shown in Fig. 5. When the pressure inside the tube exceeds that outside (which is usually atmospheric) the cross-section tends to become circular, thus causing the tube to uncurl slightly.

The movement of the free end of the tube is transmitted by a suitable mechanical linkage to a pointer moving over a scale. Zero reading is obtained when the pressure inside the tube equals the local atmospheric pressure. By using tubes of appropriate stiffness, gauges for a wide range of pressures may be made. If, a pressure higher than the intended maximum is applied to the tube, even only momentarily, the tube may be strained beyond its elastic limit and the calibration invalidated. All gauges depending on the elastic properties of a solid require calibration. For small pressures this may be done by using a column of mercury; for higher pressures the standard, calibrating, pressure is produced by weights of known magnitude exerting a downward force on a piston of known area.

Fig. 6 Bourdon Gauge

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Variation of Pressure in a Fluid

Consider a small fluid element of size δx × δy × δz at any point in a static mass of fluid as shown in Fig.1. Since the fluid is at rest, the element is in equilibrium under the various forces acting on it. The forces acting on the element are the pressure forces on its faces and the self-weight of the element.

Let ‘p’ be the pressure intensity at the midpoint O of the element. Then the pressure intensity on the left hand face of the element is

The pressure intensity on the right hand face of the element is 

The corresponding pressure forces on the left hand and the right hand faces of the element are

 and

 respectively.

Fig. 1 Fluid Element with Forces Acting on it in a Static Mass of Fluid

Likewise the pressure intensities and the corresponding pressure forces on the other faces of the element may be obtained as shown in Fig. 1. Further if ‘w’ is the specific weight of the fluid then the weight of the element acting vertically downwards is (w δx δy δz). Since the element is in equilibrium under these forces, the algebraic sum of the forces acting on it in any direction must be zero. Thus considering the forces acting on the element along x,y and z axes the following equations are obtained

                                 ΣFx = 0

                        or                                                     ΣFy = 0

                               or                                               ΣFz = 0

Equations 1, 2 and 3 indicate that the pressure intensity p at any point in a static mass of fluid does not vary in x and y directions and it varies only in z direction. Hence the partial derivative in eq. 3 may be reduced to total (or exact) derivative as follows.

In vector notation Eq. 4 may be expressed as

– grad p = wk = ρgk

where ‘k’ is unit vector parallel to z axis.

The minus sign (–) in the above equation signifies that the pressure decreases in the direction in which z increases i.e., in the upward direction.

Equation 4 is the basic differential equation representing the variation of pressure in a fluid at rest, which holds for both compressible and incompressible fluids. Equation 4 indicates that within a body of fluid at rest the pressure increases in the downward direction at the rate equivalent to the specific weight ‘w’ of the liquid. Further if dz = 0, then dp is also equal to zero; which means that the pressure remains constant over any horizontal plane in a fluid.

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Equation of State: The Perfect Gas

The assumed properties of a perfect gas are closely matched by those of actual gases in many circumstances, although no actual gas is perfect. The molecules of a perfect gas would behave like tiny, perfectly elastic spheres in random motion and would influence one another only when they collided. Their total volume would be negligible in comparison with the space in which they moved. From these hypotheses the kinetic theory of gases indicates that, for equilibrium conditions, the absolute pressure p, density ‘ρ’, the volume V occupied by mass m and the absolute temperature T are related by the expression

p = ρRT

                                                                              or           pV = mRT                      (1)

in which ‘R’ is a constant called the gas constant, the value of which is constant for the gas concerned and ‘V’ is the volume occupied by the mass m of the gas. The absolute pressure is the pressure measured above absolute zero (or complete vacuum) and is given by

p_abs = p_gage + p_atm 

The absolute temperature is expressed in ‘kelvin’ i.e., K, when the temperature is measured in °C and it is given by

T°(abs) = T K = 273.15 + t°C

No actual gas is perfect. However, most gases (if at temperatures and pressures well away both from the liquid phase and from dissociation) obey this relation closely and hence their pressure, density and (absolute) temperature may to a good approximation, be related by Eq.1.

Similarly, air at normal temperature and pressure behaves closely in accordance with the equation of state. It may be noted that the gas constant R is defined by Eq. 1 as p/ρT and therefore, its dimensional expression is (FL/Mθ). Thus in SI units the gas constant R is expressed in Newton-metre per kilogram per kelvin i.e., (N.m/kg. K). Further, since 1 joule = 1 newton × 1 metre, the unit for R also becomes joule per kilogram per kelvin i.e., (J/kg. K). Again, since 1 N = 1 kg × 1 m/s², the unit for R becomes (m²/s² K).

In metric gravitational and absolute systems of units, the gas constant R is expressed in kilogram (f)-metre per metric slug per degree C absolute i.e., [kg(f)-m/msl deg. C abs.] and dyne-centimetre per gram (m) per degree C absolute i.e., [dyne-cm/gm(m) deg. C abs.] respectively. For air the value of R is 287 N-m/kg K, or 287 J/kg K, or 287 m²/s² K.

In metric gravitational system of units, the value of R for air is 287 kg(f)-m/msl deg. C abs. Further, since 1 msl = 9.81 kg (m), the value of R for air becomes (287/9.81) or 29.27 kg(f)-m/kg(m) deg. C abs.

Since specific volume may be defined as reciprocal of mass density, the equation of state may also be expressed in terms of specific volume of the gas as

pv = RT

in which v is specific volume.

The equation of state may also be expressed as

p = wRT

in which w is the specific weight of the gas. The unit for the gas constant R then becomes (m/K) or (m/deg. C abs). It may be shown that for air the value of R is 29.27 m/K. For a given temperature and pressure, Eq. 1 indicates that ρR = constant. By Avogadro’s hypothesis, all pure gases at the same temperature and pressure have the same number of molecules per unit volume. The density is proportional to the mass of an individual molecule and so the product of R and the ‘molecular weight’ M is constant for all perfect gases. This product MR is known as the universal gas constant. For real gases it is not strictly constant but for monatomic and diatomic gases its variation is slight. If M is the ratio of the mass of the molecule to the mass of a hydrogen atom, MR = 8310 J/kg K.

Any equation that relates p, ρ and T is known as an equation of state and equation of state is therefore termed the equation of state of a perfect gas. Most gases, if at temperatures and pressures well away both from the liquid phase and from dissociation, obey this relation closely and so their pressure, density and (absolute) temperature may, to a good approximation, be related by Eqn. 1. For example, air at normal temperatures and pressures behaves closely in accordance with the equation. But gases near to liquefaction – which are then usually termed vapours – depart markedly from the behaviour of a perfect gas. Equation 1 therefore does not apply to substances such as non-superheated steam and the vapours used in refrigerating plants. For such substances, corresponding values of pressure, temperature and density must be obtained from tables or charts.

It is usually assumed that the equation of state is valid not only when the fluid is in mechanical equilibrium and neither giving nor receiving heat, but also when it is not in mechanical or thermal equilibrium. This assumption seems justified because deductions based on it have been found to agree with experimental results.

Calorically Perfect Gas

A gas for which the specific heat capacity at constant volume, cv, is a constant is said to be calorically perfect. The term perfect gas, used without qualification, generally refers to a gas that is both thermally and calorically perfect.

Changes of State

A change of density may be achieved both by a change of pressure and by change of temperature. If the process is one in which the temperature is held constant, it is known as isothermal. On the other hand, the pressure may be held constant while the temperature is changed. In either of these two cases there must be a transfer of heat to or from the gas so as to maintain the prescribed conditions. If the density change occurs with no heat transfer to or from the gas, the process is said to be adiabatic.

If, in addition, no heat is generated within the gas (e.g. by friction) then the process is described as isentropic, and the absolute pressure and density of a perfect gas are related by the additional expression p/e^γ  = constant, where γ = cp/cv and cp and cv represent the specific heat capacities at constant pressure and constant volume respectively. For air and other diatomic gases in the usual ranges of temperature and pressure γ = 1.4.

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Pressure

When a fluid (either liquid or gas) is at rest, it exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. While the fluid as a whole is at rest, the molecules that makes up the fluid are in motion, the force exerted by the fluid is due to molecules colliding with their surroundings. A fluid always has pressure as a result of innumerable molecular collisions. Pressure at any part of the fluid must experience forces exerted on it by adjoining fluid or by adjoining solid boundaries. If, therefore, part of the fluid is arbitrarily divided from the rest by an imaginary plane, there will be forces that may be considered as acting at that plane. As shown in Fig. 1, pressure occurs when a force is applied to an area. Fluid pressure is the force exerted by the fluid per unit area. Fluid pressure is transmitted with equal intensity in all directions and acts normal to any plane. In the same horizontal plane, the pressure intensities in a liquid are equal.

Fig.1 Force Producing a Pressure

The relationship between force, pressure and area is

Using the base units of Newton (N) for force and square metres (m²) for area, the unit of pressure is N/m², which for convenience is called a pascal (Pa). Because the pascal is an extremely small unit of pressure (car tyre pressure are around 200 000 Pa) the more commonly used unit is the kilopascal (kPa) or the megapascal (MPa). The pascal unit is used for the low pressures that occur in fans or in ventilation ducts. Kilopascals are used for normal gas and liquid pressures. The pressure in an oil hydraulic system would be measured in megapascals. The other multiple of the pascal that may b;e used on the weather forecast is the hectopascal (hPa), being used for the barometric pressure. A hectopascal is 10² pascals. A typical barometric pressure is 1013 hPa.

Pressures of large magnitude are often expressed in atmospheres (abbreviated to atm). For precise definition, one atmosphere is taken as 1.01325 × 10⁵ Pa. A pressure of 10⁵ Pa is called 1 bar. The thousand^th part of this unit, called a millibar (abbreviated to mbar), is commonly used by meteorologists. It should be noted that, although they are widely used, neither the atmosphere nor the bar are accepted for use with SI units. For pressures less than that of the atmosphere the units normally used are millimetres of mercury vacuum. These units refer to the difference between the height of a vertical column of mercury supported by the pressure considered and the height of one supported by the atmosphere. In the absence of shear forces, the direction of the plane over which the force due to the pressure acts has no effect on the magnitude of the pressure at a point. The fluid may even be accelerating in a particular direction provided that shear forces are absent – a condition that requires no relative motion between different particles of fluid. Many other pressure units are commonly encountered and their conversions are detailed below.

            1 bar =10⁵ N/m²

            1 atmosphere = 101325 N/m²

1 psi (1bf/in² - not SI unit) = 6895 N/m²

1 Torr = 133.3 N/m²

Pressure is determined from a calculation of the form (force divided by area), and so has the dimensions [F]/[L²] = [MLT⁻²]/[L²] = [ML⁻¹T⁻²]. Now although the force has direction, the pressure has not. The direction of the force also specifies the direction of the imaginary plane surface, since the latter is defined by the direction of a line perpendicular to, or normal to, the surface. Here, then, the force and the surface have the same direction and so in the equation Force = Pressure × Area of plane surface pressure must be a scalar quantity. Pressure is a property of the fluid at the point in question. Similarly, temperature and density are properties of the fluid and it is just as illogical to speak of ‘downward pressure’, for example, as of ‘downward temperature’ or ‘downward density’. To say that pressure acts in any direction, or even in all directions, is meaningless; pressure is a scalar quantity.

Terms commonly used in static pressure analysis include the following.

Pressure Head

The pressure intensity at the base of a column of homogenous fluid of a given height in metres.

Vacuum

A perfect vacuum is a completely empty space in which, therefore the pressure is zero.

Atmospheric Pressure

It is the pressure of earth's atmosphere. This changes with weather and elevation. The pressure at the surface of the earth due to the head of air above the surface is called atmospheric pressure. At sea level the atmospheric pressure is about 101.325 kN/m² (i.e. one atmosphere = 101.325 kN/m² is used as units of pressure).

Gauge Pressure

The pressure measured above or below atmospheric pressure is called Gauge pressure. Pressure cannot be measured directly; all instruments said to measure it in fact indicate a difference of pressure. This difference is frequently that between the pressure of the fluid under consideration and the pressure of the surrounding atmosphere. The pressure of the atmosphere is therefore commonly used as the reference or datum pressure that is the starting point of the scale of measurement. The difference in pressure recorded by the measuring instrument is then termed the gauge pressure.

Gauge pressure = Absolute pressure – Atmospheric pressure

Absolute Pressure

The pressure measured above absolute zero or vacuum is called Absolute pressure. The absolute pressure, that is the pressure considered relative to that of a perfect vacuum, is then given by

                        Absolute Pressure = Gauge Pressure + Atmospheric Pressure

pabs = pgauge +patm

The pressure of the atmosphere is not constant. For many engineering purposes the variation of atmospheric pressure (and therefore the variation of absolute pressure for a given gauge pressure, or vice versa) is of no consequence. In other cases, especially for the flow of gases – it is necessary to consider absolute pressures rather than gauge pressures and a knowledge of the pressure of the atmosphere is then required.

Positive and Negative Pressures

Because we are subjected to an atmospheric pressure, the pressure indicated on a gauge can be either positive (pressure) or negative (vacuum). Above atmospheric pressure is positive and called a gauge pressure for clarity. A typical pressure gauge would be calibrated in kPa. Below atmospheric pressure is negative and called a vacuum or a negative pressure. Fig.2 indicates the relationship between the pressure and vacuum ranges and introduces the concept of one pressure range starting from absolute zero pressure and called the absolute pressure range.

Fig. 2 Pressure/Vacuum Relationships

We normally express pressures in terms of gauge pressure and before these values may be used in calculations regarding the change of state of a gas, the gauge pressure must be changed into an absolute pressure. Changing to absolute values is done by adding the accepted value for atmospheric pressure, nominally 101.3 kPa.

Absolute Pressure (kPa) = Gauge Pressure (kPa) + 101.3 kPa

It is important, when specifying pressures or using pressures in a calculation, to determine if the values given are in terms of gauge pressure (sometimes written `kPa g’ or `kPa gauge’) or absolute pressure (sometimes written `kPa abs’).

Compressibility

A parameter describing the relationship between pressure and change in volume for a fluid. A compressible fluid is one which changes its volume appreciably under the application of pressure. Therefore, liquids are virtually incompressible whereas gases are easily compressed. The compressibility of a fluid is expressed by the bulk modulus of elasticity (K), which is the ratio of the change in unit pressure to the corresponding volume change per unit volume.

Vapour Pressure

When evaporation of a liquid having a free surface takes place within an enclosed space, the partial pressure created by the vapour molecules is called the vapour pressure. Vapour pressure of a liquid is the partial pressure of the vapour in contact with the saturated liquid at a given temperature. Vapour pressure increases with temperature.

All liquids possess a tendency to evaporate or vaporize i.e., to change from the liquid to the gaseous state. Such vaporization occurs because of continuous escaping of the molecules through the free liquid surface. When the liquid is confined in a closed vessel, the ejected vapour molecules get accumulated in the space between the free liquid surface and the top of the vessel. This accumulated vapour of the liquid exerts a partial pressure on the liquid surface which is known as vapour pressure of the liquid. As molecular activity increases with temperature, vapour pressure of the liquid also increases with temperature. If the external absolute pressure imposed on the liquid is reduced by some means to such an extent that it becomes equal to or less than the vapour pressure of the liquid, the boiling of the liquid starts, whatever be the temperature. Thus a liquid may boil even at ordinary temperature if the pressure above the liquid surface is reduced so as to be equal to or less than the vapour pressure of the liquid at that temperature.

If in any flow system the pressure at any point in the liquid approaches the vapour pressure, vaporization of liquid starts, resulting in the pockets of dissolved gases and vapours. The bubbles of vapour thus formed are carried by the flowing liquid into a region of high pressure where they collapse, giving rise to high impact pressure. The pressure developed by the collapsing bubbles is so high that the material from the adjoining boundaries gets eroded and cavities are formed on them. This phenomenon is known as cavitation. When the liquid pressure is dropped below the vapour pressure due to the flow phenomenon, we call the process cavitation. Mercury has a very low vapour pressure and hence it is an excellent fluid to be used in a barometer. On the contrary various volatile liquids like benzene etc., have high vapour pressure. Cavitation can cause serious problems, since the flow of liquid can sweep this cloud of bubbles on into an area of higher pressure where the bubbles will collapse suddenly. If this should occur in contact with a solid surface, very serious damage can result due to the very large force with which the liquid hits the surface. Cavitation can affect the performance of hydraulic machinery such as pumps, turbines and propellers, and the impact of collapsing bubbles can cause local erosion of metal surfaces.

Variation in Pressure with Depth

If the weight of the fluid can be neglected, the pressure in a fluid is the same throughout its volume. But often the fluid's weight is not negligible and under such condition pressure increases with increasing depth below the surface.

Let us now derive a general relation between the pressure ‘P’ at any point in a fluid at rest and the elevation ‘y’ of that point. We will assume that the density ′ρ′ and the acceleration due to gravity ‘g’ are same throughout the fluid. If the fluid is in equilibrium, every volume element is in equilibrium.

Consider a thin element of fluid with height ‘dy’. The bottom and top surfaces each have area ‘A’ and they are at elevations y and (y + dy) above some reference level where y = 0. The weight of the fluid element is

                                     dW = (volume) (density) (g)

                                            = (A dy) (ρ) (g)

             or                  dW = ρ g A dy

The pressure at the bottom surface P, the total y component of upward force is PA. The pressure at the top surface is P + dP and the total y-component of downward force on the top surface is (P + dP)A. The fluid element is in equilibrium, so the total y component of force including the weight and the forces at the bottom and top surfaces must be zero.

               Σ Fy = 0

                                                            PA – (P + dP)A – ρ g A dy = 0

This equation shows that when y increases, P decreases, i.e., as we move upward in the fluid pressure decreases.

If P1 and P2 be the pressures at elevations y1 and y2 and if ρ and g are constant, then integration of Equation (1), we get

                                   or                                     P2 – P1 = – ρ g (y2 – y1) (2)

It's often convenient to express equation (2) in terms of the depth below the surface of a fluid. Take point 1 at depth h below the surface of fluid and let P represents pressure at this point. Take point 2 at the surface of the fluid, where the pressure is P0 (for zero depth). The depth of point 1 below the surface is,

                                   h = y2 – y1

    and equation (2) becomes

                                           P0 – P = – ρ g (y2 – y1) = – ρgh

                                                      P = P0 + ρ gh                     (3)

Thus pressure increases linearly with depth, if ρ and g are uniform. A graph between P and h is shown below.

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