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Part I
Principles and Basic Applications
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Chapter 1
Hydrostatics
1.1 PRESSURE
Hydrostatics is the study of fluids at rest and is therefore the simplest aspect of hydraulics. The main characteristic of a stationary fluid is the force which it brings to bear on its surroundings. A fluid force is frequently specified as a pressure, p, which is the force exerted on a unit area. Pressure is measured in N/m2 or in “bar” (1 bar = 105 N/m2).
By the end of this chapter you should
Be able to calculate a pressure from the reading on a manometric instrument
Be able to calculate the pressure force and centre of pressure on a surface immersed in a liquid
Understand Archimedes’ principle of buoyancy
Be able to determine whether an object will float in a stable fashion or not
Pressure is not constant everywhere in a body of fluid. In fact, if pressure is measured at a series of different depths below the upper surface of the fluid, it will be found that the pressure reading increases with increasing depth. An exact relationship can be developed between pressure, p, and depth, y, as follows. Suppose there is a large body of liquid (e.g., a lake or swimming pool), then take any imaginary vertical column of liquid within that main body (Figure 1.1). The column of fluid is at rest; therefore, all of the forces acting on the column are in equilibrium. If this statement is to be true for any point on the boundary surfaces of the column, the action and reaction forces must be perpendicular to the boundary surface. If any forces were not perpendicular to the boundary, then a shear force component would exist; this condition arises only for fluids in motion. It follows that the only force which is supporting the column of fluid is the force acting upwards due to the pressure on the base of the column. For the column to be in equilibrium, the upward force must exactly equal the weight force acting downward.
The volume of the column, V, is the product of its horizontal cross-sectional area, A, and its height, y. The specific weight of the liquid is the product of its density (symbol ρ) and the gravitational acceleration, g. Hence, the weight of the column is found by taking the product of the specific weight and the volume, i.e., the weight = ρgAy.
FIGURE 1.1 Pressure distribution around a column of liquid.
The force acting upwards is the product of pressure and horizontal cross-sectional area, i.e., pA. Therefore,
and so
This is the basic hydrostatic equation or “law”. By way of example, in freshwater (which has a density of 1000 kg/m3), the pressure at a depth of 10 m is
The equation is correct both numerically and in terms of its units. For all practical purposes, the value of g (= 9.81 m/s2) is constant on the earth’s surface. The product ρg will therefore also be constant for any homogeneous incompressible fluid, and (1.1a) then indicates that pressure varies linearly with the depth y (Figure 1.2).
1.1.1 Gauge Pressure and Absolute Pressure
An important case of pressure variation is that of a liquid with a gaseous atmosphere above its free surface. The pressure of the gaseous atmosphere immediately above the free surface is pA (Figure 1.3). For equilibrium, the pressure in the liquid at the free surface is pA, and therefore at any depth y below the free surface the absolute pressure pABS (i.e., the pressure with respect to absolute zero) must be
FIGURE 1.2 Pressure variation with depth.
FIGURE 1.3 Gauge and absolute pressure.
The gauge pressure is the pressure with respect to pA (i.e., pA is treated as the pressure “datum”):
It is possible for gauge pressure to be positive (above pA) or negative (below pA). Negative gauge pressures are usually termed vacuum pressures. Virtually every civil engineering project is constructed on the earth’s surface, so it is customary to take atmospheric pressure as the datum. Most pressure gauges read zero at atmospheric pressure.
1.2 PRESSURE MEASUREMENT
The argument so far has centred upon variation of pressure with depth. However, suppose that a pipeline is filled with liquid under pressure (Figure 1.4a). At one point the pipe has been pierced and a vertical transparent tube has been attached. The liquid level would rise to a height y, and since (1.1a) may be rearranged to read
FIGURE 1.4 Pressure measuring devices. (a) Piezometer, (b) manometer with secondary gauge fluid, (c) differential manometer, (d) inclined manometer, and (e) Bourdon gauge.
this height will indicate the pressure. The term p/ρg is often called “pressure head” or just “head”. A vertical tube pressure indicator is known as a piezometer. The piezometer is of only limited use. Even to record quite moderate water pr...
