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CLOUD DETECTION METHOD

Formulae for the Vapour Pressure and Humidity:

Clapeyron (1834) investigated correlations between the pressure, temperature and volume of gases and derived a relationship between the saturation vapour pressure (e ) and temperature, T which is now generally written:

de/dT = Lv / T(vg – vf) (1)

Where

T is the temperature in Kelvin,

Lv is the latent heat of vapourization;

vg is the specific volume of the vapour

and vf the specific volume of the liquid.

This relation later to be proven thermodynamically by Clausius (1850), is now almost universally known as the Clausius - Clapeyron equation. For water, vg >> vf and, assuming that water vapour behaves as an ideal gas (i.e. that evg = RT, where R is the ideal gas constant), Clausius (1850) showed that if the latent heat Lv is assumed to yield, for the saturation vapour pressure of water, ew :

ln(ew/eo) = Lv t / R(273+ t) (2)

where eo is the SVP at 0ºC.

Substituting for water the currently accepted values Lv =2501.5 kJ kg and R = 0.46157 kJ (kg K)-1 into this equation yields the relation for SVP in hPa as a function of temperature t in Celsius:

ew = 6.107 x 10 {8.618 t / ( 273.15 + t )} (3)

which bears a striking resemblance to the formula derived empirically by Magnus.

The simple integration of the Clausius – Clapeyron equation, i.e. equations (2) and (3), has appeared many times in the literature in various forms and with slightly different values for the constants. Hewson and Longley (1944) gave, for the saturation vapour pressure over plane surfaces of pure water, the relation:

ew = 6.11 x 10 {8.573 - 2340 / ( 273 + t )} (4)

Minter (1944) defined this expression in its simplest form as

log10 ew = 6.11 x 10 {8.573 - 2340 / ( 273 + t )} (5)

With all numerical constants concatenated into the minimum possible, while Fleagle and Businger (1963) quote this equation thus:

log10 ew = 9.40 – 2302.7 / T (6)

This latter becomes identical, after rearrangement, to equation (3), assuming that

T = t + 273.155 K.

Hall (1979) cites an equivalent formula, which can be rearranged into a form very similar to equation (6).

ew = 6.11exp{ 19.7 t / (273.15 + t)} (7)

and which is also an exponential version of equation (3) while Henderson-Sellers (1984) expresses this equation in the form

ew =6.1078exp{5417.1{(1/273)– (1/ Td)}} (8)

Where

0ºC is clearly identified with 273.0 K

and Td is the dew point temperature.

Similarly we get the formula of water vapour saturation pressure

esw =6.1078exp{5417.1{(1/273)– (1/ Tt)}} (9)

Where

Tt is the dry bulb temperature.

Now the temperature Tt in ºC, which is converted into a relative inverse temperature variable

q = 300 / (Tt + 273.15) (10) The relative humidity is given by

U = 100(ew / esw) £ 100 %RH (11)

The temperature dependence of water vapour saturation pressure esw (100 %RH) is approximated (error £ 0.2% over 40° C) and, in turn, expressed as vapour concentration,

v = 7.223 ewq = 1.739 x 10 9 U q 5 exp(-22.64q ) g/m3 (12)

Equation (12) allows to correlate absolute (either concentration v or pressure ew) and relative humidity U when the temperature Tt is known.