I'm trying to understand the air flow within a melodica.
A melodica is a wind instrument that has a piano-like keyboard. Pressing on a key opens an airway so that air entering the melodica's air chamber can flow past a brass reed and exit the system. Low note reeds are bigger than high note reeds and the opening through which the air flows is just slightly bigger than the reed.
Here is a simplified diagram in which two keys are pressed simultaneously: At the moment, I am just looking at the mass flow, the total mass of air flowing past any point in a tube- or pipe-like system per given unit of time (at melodica air speeds, air can be treated as incompressible, so the mass flow can be specified in terms of volume per unit of time).
I am simplifying my analysis by considering only horizontal systems and only constant air flows.
In a system open at two ends, the mass flow at any point in the system is a constant. Therefore, if I play a single note, the mass flow through any reed is equal to the mass flow entering the system. This is not true if I play two notes.
I could apply the rule for parallel pipes; in the diagram above, the mass flow exiting A + B is equal to the mass flow entering the system. The mass flow for A would be twice that of B or 2/3 of the entry flow. The mass flow for B would be 1/2 that of A or 1/3 of the entry flow.
However, note that the path from the entry point to A is shorter (much shorter in real life than in this diagram) than the path to B.
My question is whether this introduces some additional factor that I have to account for; in other words, will A's mass flow be greater than 2/3 of the entry flow?
I don't have an answer, but I have done further research and realize my question needs work.
First, the "rule" for parallel pipes is $A_1v_1 = A_2v_2 + A_3v_3$, which does not mean the mass flow rates would be 2/3 and 1/3. There are multiple solutions for $v_2$ and $v_3$ even when we know all the areas and the initial velocity. For instance, if I block one branch, its velocity becomes 0 and the other branch's velocity will increase to compensate.
Originally, I thought I could apply rules for pipes in parallel and pipes in series. In other words, once a flow branched, I thought each branch could be treated independently. Blocking one branch shows that the flows are not independent. This makes me think that, for example, if one branch is constricted even after it splits, it might still have an effect on the other branch.
In the case of the melodica, there is a large air chamber with the input flow closer to one outlet than the other. At this time, I don't know if there are any general rules that can be applied; perhaps a simulation (or experiment) is the only way to answer the question. I suspect that reed A gets more than it's share of the air.
As a test, I connected two melodicas using equal diameter, equal length tubes. If I play a low note on one and a high note on the other, the high note does not drop in volume the way it does when I play both notes on the same melodica. Topologically, it looks like the same situation, but it's clearly not.
Some potential changes to a melodica's design might be to direct the airflow first to the higher, smaller reeds. Given their smaller area, they might balance the mass flow distribution more equally. Another thought would be to have a more even distance to each reed. This could be done by having the airflow enter in the middle, separating the reeds into individual chambers and ensuring that the distances from the entry to each reed is the same.
I'm leaving the question open, but I'm going down the simulation path. If I get an answer, I will post it. If anyone knows a way to answer the question (even in a general way) without simulation, please do.