Question $1[25$ marks in total$].$

A manufacturing plant emits $20m^3{s}^{-1}$ of gas from a chimney stack that is $50m$ high with

an exit diameter of $1m.$ This gas contains hydrogen sulphide $(H_2(S))$ at a concentration of

$0\mathrm{.}3g{m}^{-3}.$ The effluent is being released at a temperature of $450K$ thus the plume initially

rises vertically above the chimney exit before being carried horizontally by a wind speed

of $12m{s}^{-1}.$ The atmospheric temperature can be taken to be $293K$ with neutral

atmospheric conditions, in an urban environment.

Assume that the molecular weight of the gas leaving the chimney stack is $29gmo{l}^{-1},$ the

specific heat capacity is $1kJk{g}^{-1}{K}^{-1},$ and the molecular weight of $H_{2}S$ is $34gmo{l}^{-1}.$

a$.$ What is the plume rise hight from the stack?