Finding the density inside a tank as air escapes

[Lee Cousins]

Homework Statement

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.

Homework Equations

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

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.

 

[stockzahn]

Homework Equations

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

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).

 

[Lee Cousins]

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).

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)

 

[stockzahn]

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)

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.