Chapter 5: Q. 5.44 (page 177)

Suppose that an unsaturated air mass is rising and cooling at the dry adiabatic lapse rate found in problem 1.40. If the temperature at ground level is 25 C and the relative humidity there is 50%, at what altitude will this air mass become saturated so that condensation begins and a cloud forms (see Figure 5.18)? (Refer to the vapor pressure graph drawn in Problem 5.42)

Expert verified

The altitude at which air mass become saturated so that condensation begins and a cloud forms is 1.37 km

Step by step solution

(Refer table 5.11 and graph in problem 5.42)

At $25 ° C a n d 50 %$ relative humidity, the partial pressure of water is 0.016 bar.

The temperature at which this partial pressure is in equilibrium with vapour pressure is $13.8 ° C$ .

From the dry adiabatic lapse rate for unsaturated air mass is

$\frac{d T}{d Z} = 9.8 ° C / k m \Rightarrow Z = 1.14 k m$

The relation between pressure and height is

$P (z) = P e^{\frac{- z}{8.5}}$

Substituting P=0.016 bar and z=1.14km

P(z) = 0.0014 bar

Using the graph in problem 5.42 we find this pressure is in equilibrium with vapour pressure at T= $11.8 ° C$

For condensation to occur the temperature should reduce to 9.8 $°$ C the air should rise a height.

$z = \frac{11.8}{9.8} (1.14) z = 1.37 ≅ 1.40 k m$

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