How Long to Ventilate Linked Rooms to Safe Nitrogen Levels?

[SamRS]

Homework Statement

There are two storage rooms, (A) and (B) connected by a single doorway. Only room (B)
has an outward vent, so all air exiting the rooms does so through this vent in room (B). The
ventilator you will install drives outside air (78% nitrogen) at a certain flow rate through
a duct that splits into two and distributes the total flow evenly between an inward vent in
room (A) an an inward vent in room (B). All air exiting room (A) does so through the door
and then mixes with the air in room (B) before leaving through the vent in room (B). No air
ever flows from (B) to (A). You may assume that the air entering each room (either through
the vent or in the case of (B) also through the door from (A)) mixes instantaneously with the
rest of the air in the room. The dimensions of room (A) are 20 feet × 15 feet × 12 feet
(3600 cu ft) and the dimensions or room (B) are 20 feet × 18 feet × 12 feet (4320 cu ft).

The laboratory knows that if there is a leak in room (A) then by the time the atmosphere
in (A) reaches 85%, the atmosphere in (B) will have reached 81%. This is the situation you
are hired to deal with: (A) starting at 85%, (B) at 81% and you need to reduce both to 79%.
Support your conclusions with graphs.

Homework Equations

In the first situation, the ventilator forces air through the main duct (before it splits
into two) at 400 cu ft/min on high. How long does it take for this ventilator to reduce
the atmosphere in both rooms to below 79%?

The Attempt at a Solution

Okay, so I understand that the initial conditions are A(0)=.85 and B(0)=.81 and that A(t) and B(t) both =.79

Saying x is the nitrogen level, I need to form an equation something like dx/dt = 200(.79)-200(.85t) but that's where I get confused. I feel like this should be fairly simple but I keep getting stumped. Any help you can give is greatly appreciated!

-Sam

 

[Filip Larsen]

It may help you get started if you try to relate the different volumes involved. For instance, if you select (as you already seem to have done) the volumetric ratio of nitrogen in room A as a one state variable for your differential equation then try to write down how big a volume of nitrogen in room A this corresponds. Likewise you can use ratios to describe the change of volume of nitrogen entering (using atmospheric nitrogen ratio) and leaving room A (using room nitrogen ratio).

Having a volume and the time rate change of it, you can now write this up as a differential equation for room A. Repeat this process for room B and you should end up with two first order ODE's that you can solve for the given initial conditions and use in your further analysis. Alternatively, you can combine the two first order ODE's into a single second order ODE if you like to solve that better.

 

[SamRS]

Thanks for your reply, I've finally had time to get back to this problem and have made little progress.

I took your advice and turned the ratios into volumes of nitrogen;

V_{room A} = 3600, .85(3600)=3060, and .79(3600)=2844

so dx/dt = 200(2844/3600) - 200(3060t/3600) is this right?

would that lead to an equation like 3060 = 2844e^-200t/3600 ?