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.
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 thousandth 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.
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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