Question

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)

Short answer

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

Step-by-step solution

Given information

(Refer table 5.11 and graph in problem 5.42)

At $25°Cand50\%$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$13.8°C$.

Estimating the height at which clouds will start forming.

From the dry adiabatic lapse rate for unsaturated air mass is

$$\begin{gathered} \frac{dT}{dZ}=9.8°C/km \\ \Rightarrow Z=1.14km \end{gathered}$$

The height at which saturation begins.

The relation between pressure and height is

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

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.

determining the height

$$\begin{gathered} z=\frac{11.8}{9.8}(1.14) \\ z=1.37\cong 1.40km \end{gathered}$$