Speed of air through valve

Question

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.

Answer 1

The valve will, if not to big, not let cold air in, but will let warm air out, which will be replenished by cold air from the bottom (I am assuming that the balloon is open at the bottom). If the valve would be very big, then cold air would also flow in through the valve, but I don't know how you could calculate this.

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.

However your question gets even more complicated since you would have to solve a partial differential equation, since the cold air from below will cool the air in the balloon and therefore change its density. However this will happen both due to convection and conduction (thermal radiation will be neglected).

I think you will be better of to do some experiments, since this model will give you some insight into the physics involved, but you would have to make to much assumptions to be able to make an accurate calculation.