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Homework Help: Finding the density inside a tank as air escapes

  1. Dec 12, 2017 #1
    1. The problem statement, all variables and given/known data

    A tank of constant volume V contains air at an initial density pi. Air is discharged isothermally from the tank at a constant volumetric rate of Q (with SI units of m^3/s). Assuming that the discharged air has the same density as that of the air in the tank, find an expression for the density in the tank, p(t).

    There's also a diagram of the circular control volume V with one outlet which air escapes at Q.

    2. Relevant equations
    Mass conservation equation is integral of (p dv) + integral of (pv*dA) = 0

    3. The attempt at a solution
    I got the equation down to dp/dt = (-pi*Q)/V but that's not right so I'm not sure what to do from here.
     
  2. jcsd
  3. Dec 13, 2017 #2

    stockzahn

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    Gold Member

    There must be something wrong with this equation. The two summands have different dimensions.

    Plus I suggest to use the common symbols. The density's symbol is ##\rho## (rho), whereas ##p## stands for the pressure. Also the symbol ##v## is confusing (I suppose it should be the velocity). However, try to use the common symbols and also explain them in the text, if they could be ambiguously (##v## also could be the specific volume, then ##\int p dv## would be something completely different).
     
  4. Dec 13, 2017 #3

    Okay, I'm new to this but I meant to say ##\int ##\rho## dv## + ##\int ##\rho## V dA##

    I can't quite get the syntax down. But its the integral with respect to the control volume of the density * the dv (volume) + the integral with respect to the surface area of the density * the volume * dA (Area)
     
  5. Dec 13, 2017 #4

    stockzahn

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    Gold Member

    Regarding the syntax: You only have to write two hashtags before and after the entire expression, not for every symbol.

    However, the first summand in your equation ##\int \rho dv## has the unit ##kg/m^3 \cdot m^3 = kg##. The second summand in your equation ##\int \rho v dA## has the unit ##kg/m^3 \cdot m^3 \cdot m^2 = kg\cdot m^2##. So the units are not consistent.

    You start with an initial mass of air ##m_0## in the tank. Then there is a mass flow ##\dot{m}## exiting the tank with time. Now the mass conservation says that the mass in the tank must be the initial mass minus the air flow over the time.

    ##m_0-\dot{m}t=m\left(t\right)##

    Based on this equation, try to find the answer by substituting, re-arranging etc.
     
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