# Determining pressure as a function of time for water filled balloon

Looking to determine pressure as a function of time for a water filled "balloon" at extremely high pressures (~90MPa) with a fixed flow rate out. This is to determine the time it takes to hit a certain pressure $$P_f$$ from an initial pressure $$P_0$$. I have recognized the importance of creating an equation of state.

$$\frac{\delta(P)}{\delta(t)} = \frac{\delta(P)}{\delta(\rho)} * \frac{\delta(\rho)}{\delta(t)}$$

$$\frac{\delta(P)}{\delta(\rho)}$$ may be found with a compressed water table.

$$\frac{\delta(\rho)}{\delta(t)} = \frac{\delta(\rho)}{\delta(m)} * \frac{\delta(m)}{\delta(t)}$$

Where $$\frac{\delta(m)}{\delta(t)}$$ is simply the fixed mass flow rate.

Now $$\frac{\delta(\rho)}{\delta(m)}$$ is where I begin to get difficulties. Volume will change as a function of pressure, ie. $$V=f(P)$$ and mass will change as a function of density and volume, ie. $$m=f(\rho,V)$$.

Simply, it should functionally look like something along these lines: $$\Delta(\rho (\frac{kg}{m^3}) ) = (\rho (\frac{kg}{m^3}) )_0 - \Delta(m (\frac{kg}{s}) ) + \Delta(V (\frac{m^3}{s}) )$$

The change in volume with pressure is a problem I have saved for later, so for now, assume that equation can be readily expressed.

Assume all initial conditions are known and the system is isothermal.

Is this a correct way to express this problem?