The hydrostatic equation PDF

Preview

Summary

hydrostatic equation...

Description

2 Hydrostatics - Manometry Hydrostatics is the name given to the study of non-accelerating fluids. A common case of zero acceleration exists when a fluid is at rest or, in other words, is static. This explains the name "hydrostatics", which means "water at rest". However, the relationship applies to conditions where acceleration is zero, whether the fluid is at rest OR in motion. The Hydrostatic Equation If there is no acceleration, the particles in the fluid must be either at rest, or moving all with the same speed in the same direction. In either case, there will be nowhere in the fluid where velocity gradients exist, so therefore there will be no shearing forces to be taken into account. So we only need to consider Fp (the force on the particle due to the pressures acting around its boundary) and Fg (the weight of the particle). These two forces must balance, since we have said that there is no acceleration. It follows that, since the weight, Fg, acts vertically downwards, the pressure force, Fp, must act vertically upwards, so that all changes of pressure must take place in the vertical direction only. We can describe this mathematically by letting 'h' represent height, measured vertically above some datum level, and then saying that the pressure, 'p', is a function of 'h' only, or (1) Let us consider a fluid particle whose shape is chosen to be cylindrical. (This is only for convenience, what ever shape was chosen the results will be the same).

From the diagram, the force Fp (due to the pressures acting on the fluid particle) must be in the upwards direction and given by

Lecture 2 1st year Fluid Mechanics – The hydrostatic equation and manometry

But we may write so that

(2)

where dp/dh is the rate of change of pressure with height, and is called the PRESSURE GRADIENT. Now the force, Fg is the weight of the fluid particle, and we may write (3) Since we have said that the two forces must balance, (4) This is called the HYDROSTATIC EQUATION. Note the minus sign in equation 4. Since h is measured positively in the UPWARDS direction, the minus sign means that the pressure DECREASES as h INCREASES. Integrating The Hydrostatic Equation In order to integrate equation 4, it is necessary to know whether the density, , of the fluid is constant or varies in some way with height above datum. We shall consider the case of constant density, as would be normal when the fluid is a liquid. Constant Density If the density, , of the fluid can be assumed to remain constant, the hydrostatic equation (4) can be simply integrated to give

Lecture 2 1st year Fluid Mechanics – The hydrostatic equation and manometry

(5) where p1 is the pressure in the fluid at height h1 above datum, and p2 is the pressure at height h2 above the same datum level.

From equation 5 we can deduce the rule:- "For a given body of unaccelerated fluid, pressures are the same at all points that are the same height above a datum". Pressure and Pressure Measurement In practice, pressure is always measured by finding a pressure difference. In the case where that difference is between the pressure of the fluid and that of a vacuum then the answer is called the of the fluid. It is more usual that the difference is that between the pressure of the fluid and the pressure of the surrounding atmosphere. This is normally recorded by pressure gauges and so is known as . If the pressure of the fluid is below that of atmosphere it is called . This should not be confused with the complete vacuum we measure absolute pressure against. From these definitions we can see that absolute pressure is always positive but gauge pressures are positive if they are greater than atmospheric and negative if less than atmospheric. Measurement of Pressure Difference

Lecture 2 1st year Fluid Mechanics – The hydrostatic equation and manometry

The most simple pressure measuring devices are the Piezometer Tube and Liquid Filled Manometers. Although there are a variety of designs, they all work on the principle of using the pressure to be measured to support a column of liquid. The height of this column allows us to calculate the gauge pressure from equation 5, if we also know the local atmospheric/reference pressure we can calculate the absolute pressure. The Piezometer Tube In systems where the 'working fluid' is a liquid, the piezometer tube often provides simple means of measuring pressure in a pipe.

If the pressure at the wall of the pipe is p, the Equation 5 gives

(6)

Lecture 2 1st year Fluid Mechanics – The hydrostatic equation and manometry

The U-tube As above we will use p to represent the pressure to be measured, and Pr to represent the reference pressure.

1

2 Assume the pressure to be measured is applied at A. Then the integrated hydrostatic equation (5) gives

also since B' is on the same level as B, in the same body of fluid. Also, from the hydrostatic equation,

thus or Note, if the working fluid is a gas, 1 is usually negligible, compared to 2, and we obtain

Inclined U-tube

Lecture 2 1st y

1

Mechanics – The hydrostatic equation and manometry

2

The sensitivity of the simple U-tube can be increased by inclining it to the horizontal. In this case, the difference in levels between B' and A becomes Lsin, and hence

(7)

Reservoir Manometer To make the manometer easier to read, it is convenient to replace one limb of the inclined Utube by a reservoir of relatively large cross-sectional area, so that the level in that limb remains effectively constant.

2 1

(8)

Lecture 2 1st year Fluid Mechanics – The hydrostatic equation and manometry...