# Speed of air through valve

Right now I'm working on a hot air balloon in Physics formulas. At the moment it's flying up like it should, but once it is at a certain height we want it to go back down.

After some research about hot air balloons we found out that they drop down by opening a valve at the top. So what we want to do is once the balloon gets at a certain height it will open the valve and drop down again. We can get the size of the valve and that it opens at a certain height.

How can we calculate the speed of cold air going into the valve? We know the speed of the balloon, the size, temperature of the air in and outside of the balloon, air pressure etc. Is there some kind of formula we can use to calculate how fast the cold air will flow in and then cool of the air in the balloon making it drop down.

For the flow rate you could use Bernoulli's principle: $$p_1+\frac{1}{2}{\rho}v_1^2+{\rho}gz_1=p_2+\frac{1}{2}{\rho}v_2^2+{\rho}gz_2+\left(f\frac{L}{D}+{\sum}K\right)\frac{1}{2}{\rho}v^2$$ with $K$ a loss factor due to fittings, such as the valve. I am not entirely sure if you could use this formula, since the density, $\rho$, isn't constant, which would also add buoyancy to the system. Or if this only affects the pressure difference at the valve: ${\Delta}p=\left(\rho_{cold}-\rho_{warm}\right)gh$, with $h$ the height of the balloon.