Question

Pressure decreases with height y in the atmosphere at a rate given by p =

p0e−αy , where α = 0.116 km−1

. For air, γ = Cp/CV = 1.4.

(a) If the temperature of the air on the ground is 293 K, find what the

temperature at an altitude of 1 km would be if temperature changes at an

adiabatic rate.

(b) If the actual temperature at that altitude is 285 K, would there be upward

convection or would there be an inversion layer?

Answer 1

(a)

Since atmosphere is not isothermal, air temperature decreases with increasing altitude. So, we use the adiabatic law to calculate the temperature

where dT is temperature is the difference is temperature

dz- is height above the ground level (1 KM or 1000m as given)

R-gas law constant (8.314 J/mol.K)

-molecular weight of air ( 29 g/mol or 2.9 x 10-2kg)

g-acceleration due to gravity (9.8 m/s2)

Therfore,

dT = (1000 x 0.4 x 2.9 x 10^-2 x 9.8)/(1.4 x 8.314)

= 113.68/11.63

= 9.77 = 10 (approx)

Temperature is decreasing at a rate of 10 degree celcius per 1km.

Therfore temp at an altitude of 1km = (293-283) = 10 degree celcius

(b) whenever the air is hot, it would try to expand. in this course, it density would decrease. this air would tend to rise and transfer heat upward. This process is called convection.

However, at the altitude of 1KM with 285K, there would be an equilibrium convection as the air would have same density at that elevation.

But earths atmosphere goes upto the height of ~15 km. there is still a possiblity of air to expand above this altitude. so, there would be convection.