A plume of heated air leaves the cooling tower of a power plant at 1000mb with a

Question

A plume of heated air leaves the cooling tower of a power plant at 1000mb with a temperature of 30C. If the air may be treated as dry, to what level the plume will ascend if the ambient temperature varies with altitude as a) T(z)=20-8z and b) T(z)=20+z, with z in kilometers. A plume of heated air leaves the cooling tower of a power plant at 1000mb with a temperature of 30C. If the air may be treated as dry, to what level the plume will ascend if the ambient temperature varies with altitude as a) T(z)=20-8z and b) T(z)=20+z, with z in kilometers.

Explanation / Answer

A parcel will rise until its density matches that of its environment. The pressure of the parcel will adjust instantaneously to the environment. Assuming an ideal gas, this means that the temperature of the parcel (adiabatic cooling) will equal the air temperature when the densities are equal. (i.e., it only takes 2 state variables to completely describe an ideal gas, and the problem constrains pressure and density).

(a)T=20-8Z

T'=30-9.8Z

Set them equal and solve for z to find z=5 km.

(b)T=20+z

T'=30-9.8Z

Set them equal and solve for z to find z=0.9 km.

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