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Thread: Air flow in a room. Differential equations

  1. #1
    Newbie leebatt's Avatar
    Jul 2008

    Air flow in a room. Differential equations

    The air in a small 12 ft by 8 ft by 8 ft room is 3% carbon monoxide. Starting at t = 0, fresh air containing no carbon dioxide is blown into the room at a rate of 100 cubic feet per minute. If air in the room flows out through a vent at the same rate, when will be the air in the room be 0.01% carbon monoxide?

    Thank you
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  2. #2
    MHF Contributor
    skeeter's Avatar
    Jun 2008
    North Texas
    volume of room ... $\displaystyle 12 \cdot 8^2 = 768 \, ft^3$

    let r = percentage of room air changed in 1 minute ...

    $\displaystyle r = \frac{100}{768} \cdot 100 \approx 13.02$%

    let p = percentage volume of CO in the room at any time t

    $\displaystyle \frac{dp}{dt} = -rp$

    $\displaystyle p = p_o e^{-rt}$

    $\displaystyle p = 3e^{-rt}$

    $\displaystyle .01 = 3e^{-rt}$

    $\displaystyle t = \frac{\ln{300}}{r} \approx 43.8 \, min$
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  3. #3
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA
    Here's another way using the integrating factor thing. More roundabout than the skeets though.

    $\displaystyle \frac{dy}{dt}=\text{rate in-rate out}$

    The rate in of the air is $\displaystyle \left(\frac{100 \;\ ft^{3}}{min}\right)$

    $\displaystyle \text{rate out}=\left(\frac{y(t)}{768} \frac{ft^{3}}{ft^{3}}\right)\left(\frac{100 \;\ ft^{3}}{min}\right)=\frac{25y}{192}$

    The equation is then:

    $\displaystyle \frac{dy}{dt}-\frac{25y}{192}=100$

    The integrating factor is $\displaystyle e^{\int \frac{25}{192}dt}=e^{\frac{25t}{192}}$

    Use the IC to solve for C. The amount of air at t=0 is


    $\displaystyle ye^{\frac{25t}{192}}=768e^{\frac{25t}{192}}+C$

    Using y(0)=744.96, we find that C=-23.04

    We have $\displaystyle \boxed{y=768-23.04e^{\frac{-25t}{192}}}$

    .01% CO means 99.99% air. .9999(768)=767.9232

    $\displaystyle 767.9232=768-23.04e^{\frac{25t}{192}}$

    Solving or t we find $\displaystyle \boxed{t=43.805 \;\ min.}$

    You can also do this in terms of the CO instead of air. Either way should give the same solution.
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