When condensation occurs in a sample of moist air which is lifted, the heat liberated in the process amounts to - L dxs, where L is the latent heat and xs the saturation-mixing-ratio at the temperature at which the air becomes saturated. This heat is added to the air. The entropy equation, (3.5.3), may, therefore, be written as

where T is the dry-bulb temperature at the level where air becomes saturated, cp the specific heat of dry air at constant pressure, and 9 the potential temperature of the dry air.

Now, the change of xs following motion in a given time normally far exceeds that of L or T, so (4.7.1) may be written as dln 9 ~ -d(L xs/Tcp) (4.7.2)

Integrating (4.7.2) from the initial state (xs, 9) to the state where xs = 0, we get

where 9e is called the equivalent potential temperature.

The expression (4.7.3) may also be used for an unsaturated air provided the temperature T is the temperature at which the unsaturated air becomes saturated while ascending.

Thus, the equivalent potential temperature is conserved during both dry adiabatic and moist pseudo-adiabatic processes.

4.8 Variation of Saturation Vapour Pressure with Temperature 4.8.1 The Clausius-Clapeyron Equation

One of the most important and useful relationships in thermodynamics is what is known as the Clausius-Clapeyron equation which gives the variation of saturation vapour pressure of a liquid in equilibrium with its vapour with temperature. We may derive this equation by considering a reversible cycle of heat exchange between a liquid and its vapour, kept in a closed vessel, and applying the laws of thermodynamics to the system.

As we consider only two phases of a single substance which can exist in three phases, the system, according to Gibbs' phase rule, has only one degree of freedom. We take this to be the temperature. A change of volume at a fixed temperature has little or no effect on the saturation vapour pressure of water, as it would have if we were dealing with an ordinary gas. An increase or decrease of volume would only cause some evaporation or condensation. For the heat exchange, we consider a Carnot cycle, working between the temperatures T + dT and T, as shown in a saturation vapour pressure es - volume v diagram (Fig. 4.2).

In Fig. 4.2, let the initial point in the cycle be A where a quantity of heat equal to the latent heat of vaporization of water L (T + dT) is supplied to vaporize a mol of water at temperature T + dT. Let the saturation vapour pressure at this stage be es + des. The vapour is then allowed to expand at first isothermally upto the point B and then adiabatically to C where the temperature drops to T and vapour pressure to es. From C, the vapour is compressed isothermally to D from where an

water vapour

water vapour adiabatic compression restores the system to its original state. During the process of isothermal compression (from C to D), a quantity of heat equal to the latent heat of condensation of vapour L (T) is released.

If all second-order quantities and the small differences of heat involved in passing from one temperature state to the other are neglected, the work done by the system during the cycle may be approximated to des (vi - v2), where v denotes specific volume and the subscript 2 refers to the liquid and 1 to the vapour. Now, according to the second law of thermodynamics, this work was done by a fraction of the heat supplied to the system, viz., by L (T) dT/T, since L (T+dT) « L (T) andT+dT « T.

Thus, equating the work done to the heat available, we arrive at the relation, des/dT = L(T)/{T(vi - V2)} (4.8.1)

Before proceeding further with the above Eq. (4.8.1), we make the assumptions that v2 is negligible compared to v1 (since v1 is 1674 times larger than v2) and that the ideal gas law is applicable to the case of the saturated water vapour. The simplifications lead to the relation des/dT = es L(T)/Rv T2 (4.8.2)

where Rv denotes the gas constant for saturated water vapour (= R*/Mv).

The Eq. (4.8.2) can be integrated if we know the variation of L (T) with temperature. For this, we treat the specific heats cp' of water vapour and cw of liquid water as constant and consider the liquid to evaporate at temperature T = 0 and pressure es and the resulting vapour to warm up from zero to a temperature T. Alternatively, we may first warm up the water from temperature 0 to T and then evaporate it at T. Under these conditions, we have,

where L(0) is the latent heat of vaporization at T = 0. Substitution of this value of L (T) in (4.8.2) gives des/dT = es{L(0) + (cp' - cw)T}/Rv T2 (4.8.4) The Eq. (4.8.4) may now be integrated to yield ln es = -{L(0)/Rv T} + [{(cp' - cw) ln T}/Rv]+A (4.8.5) where A is a constant.

Fig. 4.3 Saturation vapour pressure over a plain surface of water at different temperatures

The influence of the second factor involving T being negligible, (4.8.6) states that the saturation vapour pressure of water increases rapidly, almost exponentially, with temperature.

In deriving (4.8.5), it was assumed for simplicity that the specific heats of the vapour or the liquid phase were constant. However, this assumption is not quite true. If this restriction is withdrawn, we arrive at a slightly different form of the saturated vapour pressure vs temperature relation. For derivation of this revised form, see Appendix 4.

The variation of saturation vapour pressure with temperature over a plain surface of water is shown in Fig. 4.3.

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