Physics

Physics · thermodynamics

Problem

4. Determine the height of the water column (precipitation level) precipitated by the air stream in 3 hours, assuming a homogeneous rainfall between points

M_{1}

and

M_{2}

. Given constants: Moist air is streaming adiabatically across a mountain range as indicated in the figure. Equal atmospheric pressures of

100 \mathrm{kPa}

are measured at meteorological stations

\mathrm{M}_{0}

and

\mathrm{M}_{3}

and a pressure of

70 \mathrm{kPa}

at station

\mathrm{M}_{2}

. The temperature of the air at

\mathrm{M}_{0}

20^{\circ} \mathrm{C}

. As the air is ascending, cloud formation sets in at

84.5 \mathrm{kPa}

. Consider a quantity of moist air ascending the mountain with a mass of

2000 \mathrm{~kg}

over each square meter. This moist air reaches the mountain ridge (station

\mathrm{M}_{2}

) after 1500 seconds. During that rise an amount of

2.45 \mathrm{~g}

of water per kilogram of air is precipitated as rain. <image_1> Hints and Data The atmosphere is to be dealt with as an ideal gas. Influences of the water vapour on the specific heat capacity and the atmospheric density are to be neglected; the same applies to the temperature dependence of the specific latent heat of vaporisation. The temperatures are to be determined to an accuracy of

1 \mathrm{~K}

, the height of the cloud ceiling to an accuracy of

10 \mathrm{~m}

and the precipitation level to an accuracy of

1 \mathrm{~mm}

. Specific heat capacity of the atmosphere in the pertaining temperature range:

\mathrm{c}_{\mathrm{p}}=1005 \mathrm{~J} \cdot \mathrm{kg}^{-1} \cdot \mathrm{K}^{-1}

Atmospheric density for

\mathrm{p}_{0}

and

\mathrm{T}_{0}

at station

\mathrm{M}_{0}: \quad \rho_{0}=1.189 \mathrm{~kg} \cdot \mathrm{m}^{-3}

Specific latent heat of vaporisation of the water within the volume of the cloud:

\mathrm{L}_{\mathrm{v}}=2500 \mathrm{~kJ} \cdot \mathrm{kg}^{-1}

\frac{\mathrm{c}_{\mathrm{p}}}{\mathrm{c}_{\mathrm{v}}}=\chi=1.4

and

\mathrm{g}=9.81 \mathrm{~m} \cdot \mathrm{s}^{-2}

Context question: 1. Determine temperature

T_{1}

M_{1}

where the cloud ceiling forms. Context answer: 279 Context question: 2. What is the height

h_{1}\left(\right.

\left.M_{1}\right)

above station

M_{0}

of the cloud ceiling assuming a linear decrease of atmospheric density? Context answer: 1410 Context question: 3. What temperature

\mathrm{T}_{2}

is measured at the ridge of the mountain range? Context answer: 271

Related theory

thermodynamics

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