$4.$ A natural gas compressor station has several large compressors used to boost the pressure in a cross$-$country pipeline. The capacity of the pipeline is $500$ million standard cubic feet $($scf$,$ with standard conditions at $1$ atm and $60\backslash$deg F$)$ per day. The compressors are housed in a building $200$ ft long, $80$ ft wide, and $30$ ft high at the eaves. The building has a sloped roof, and if the volume of the compressors and ancillary equipment is considered, the net free air volume of the building is $510,000$ ft$3.$ The temperature in the building averages about $90\backslash$deg F$,$ and the pressure about $1\mathrm{.}0$ atm. The building has a ventilation system designed to operate at two speeds: the low speed provides a ventilation rate equal to $6$ air changes per hour and the high speed provides $20$ air changes per hour. The ventilation rate is based on entering air, measured at building temperature and pressure. The building is normally ventilated at the low rate, but a gas$-$sensing system monitors the air in the building. If gas is released in the building and reaches a concentration of $25\%$ of the LFL$,$ the ventilation system automatically increases to the high rate.

$($a$)$ Assume one of the compressors develops a seal leak of $10,000$ scf per minute of natural gas $($considered to be pure methane$)$ in the building. How long will it take for an alarm to sound if the alarm sounds when $25\%$ of the LFL concentration is reached $($which will then increase the ventilation system to the higher speed$)?$

$($b$)$ What is the maximum concentration reached if the leak continues indefinitely? Ventilation will be at the higher rate.

$($c$)$ If the gas leak is shut off when the concentration reaches $25\%$ of the LFL$,$ how long will it be until the concentration has dropped to $5\%$ of the LFL if $(1)$ ventilation is at the low rate and $(2)$ ventilation is at the high rate?

$($d$)$ If the leak detection alarm goes off at $25\%$ of the LFL$,$ how long will it be before the concentration reaches the LFL$?$ Ventilation will be at the higher rate as long as the gas concentration is more than $25\%$ of the LFL$.$ Is that sufficient time for the source of gas to be found and shut down in time to prevent the concentration from reaching the LFL$?$

$($e$)$ Calculate the energy released if the natural gas, which has a lower heat of combustion of $1044$ Btu$/$scf$,$ is ignited and completely burned when the concentration of gas is at the LFL$.$ Compare that energy to the equivalent mass of TNT$,$ if the explosive energy of TNT is $2000$ Btu$/$lb$.$

In your calculations $($carry them out to $3$ significant figures$),$ you may assume the air and gas in the building to be completely mixed. Also, assume ideal gas behavior, since the pressure is low and the temperature is moderate.