Changes of state

2

A change of density may be achieved both by a change of pressure and by a Isothermal process

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 Adiabatic process

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 Isentropic process

the process is described as isentropic, and the absolute pressure and dens- ity of a perfect gas are related by the additional expression (developed in Section 11.2):

p/γ = constant (1.6)

whereγ = cp/cvand cpand cv represent the specific heat capacities at con-

stant pressure and constant volume respectively. For air and other diatomic gases in the usual ranges of temperature and pressureγ = 1.4.

20 Fundamental concepts

1.5 COMPRESSIBILITY

All matter is to some extent compressible. That is to say, a change in the pressure applied to a certain amount of a substance always produces some change in its volume. Although the compressibility of different substances varies widely, the proportionate change in volume of a particular material that does not change its phase (e.g. from liquid to solid) during the compres- sion is directly related to the change in the pressure.

The degree of compressibility of a substance is characterized by the bulk

Bulk modulus of

elasticity modulus of elasticity, K, which is defined by the equation K= − δp

δV/V (1.7)

Here_{δp represents a small increase in pressure applied to the material and}

δV the corresponding small increase in the original volume V. Since a rise in

pressure always causes a decrease in volume,δV is always negative, and the minus sign is included in the equation to give a positive value of K. AsδV/V is simply a ratio of two volumes it is dimensionless and thus K has the same dimensional formula as pressure. In the limit, asδp → 0, eqn 1.7 becomes

K= −V(∂p/∂V). As the density  is given by mass/volume = m/V

d = d(m/V) = −m

V2dV = −

dV

V

so K may also be expressed as

K_{= (∂p/∂)} (1.8)

The reciprocal of bulk modulus is sometimes termed the compressibility.

Compressibility

The value of the bulk modulus, K, depends on the relation between pres- sure and density applicable to the conditions under which the compression takes place. Two sets of conditions are especially important. If the com- pression occurs while the temperature is kept constant, the value of K is the isothermal bulk modulus. On the other hand, if no heat is added to or taken from the fluid during the compression, and there is no friction, the corresponding value of K is the isentropic bulk modulus. The ratio of the isentropic to the isothermal bulk modulus is_{γ , the ratio of the specific heat} capacity at constant pressure to that at constant volume. For liquids the value ofγ is practically unity, so the isentropic and isothermal bulk mod- uli are almost identical. Except in work of high accuracy it is not usual to distinguish between the bulk moduli of a liquid.

For liquids the bulk modulus is very high, so the change of density with increase of pressure is very small even for the largest pressure changes encountered. Accordingly, the density of a liquid can normally be regarded as constant, and the analysis of problems involving liquids is thereby simplified. In circumstances where changes of pressure are either very large or very sud- den, however – as in water hammer (see Section 12.3) – the compressibility of liquids must be taken into account.

Viscosity 21

As a liquid is compressed its molecules become closer together, so its resistance to further compression increases, that is, K increases. The bulk modulus of water, for example, roughly doubles as the pressure is raised from 105_{Pa (1 atm) to 3.5× 10}8_{Pa (3500 atm). There is also a decrease of} K with increase of temperature.

Unlike liquids, gases are easily compressible. In considering the flow of gases, rather than using K, it is convenient to work in terms of the Mach number, M , defined by the relation

M_{= u/a}

where u is the local velocity and a is the speed of sound. For gases, compress- ibility effects are important if the magnitude of u approaches or exceeds that of a. On the other hand, compressibility effects may be ignored, if every- where within a flow, the criterion 1₂M2 _{ 1 is satisfied; in practice, this}

is usually taken as M _{< 0.3. For example, in ventilation systems, gases} undergo only small changes of density, and the effects of compressibility may be disregarded.

1.6 VISCOSITY

All fluids offer resistance to any force tending to cause one layer to move over another. Viscosity is the fluid property responsible for this resistance. Since relative motion between layers requires the application of shearing forces, that is, forces parallel to the surfaces over which they act, the resisting forces must be in exactly the opposite direction to the applied shear forces and so they too are parallel to the surfaces.

It is a matter of common experience that, under particular conditions, one fluid offers greater resistance to flow than another. Such liquids as tar, treacle and glycerine cannot be rapidly poured or easily stirred, and are commonly spoken of as thick; on the other hand, so-called thin liquids such as water, petrol and paraffin flow much more readily. (Lubricating oils with small viscosity are sometimes referred to as light, and those with large viscosity as

heavy; but viscosity is not related to density.)

Gases as well as liquids have viscosity, although the viscosity of gases is less evident in everyday life.