Discussion
In the Gaussian Plume Model, the plume size is estimated considering buoyancy of exhaust gases, stack diameter, wind speed and stability of the atmosphere. The plume size \(\Delta h\) depends on inside radius of stack (r) as
Answer: C
The keyed answer is that the plume size depends on the stack radius as r to the power two-thirds, option C. In the standard plume rise formula for a buoyant plume in a neutral atmosphere, the rise varies as the one-third power of the buoyancy flux. The buoyancy flux itself is proportional to the square of the stack radius, so the rise varies as the radius to the power two-thirds.
\[ \Delta h \propto F^{1/3} \]
\[ F \propto r^{2} \Rightarrow \Delta h \propto r^{2/3} \]
Option A, a linear dependence, and option D, a square dependence, would need a different power of the buoyancy flux. Option B, r to the power one-third, forgets that the flux itself goes with the square of the radius.
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