Carburetors are supposedly volumetric (i.e. they deliver a constant air:fuel ratio by volume), so as the air density decreases we need to decrease the amount of fuel delivered in order to maintain the same mass flow ratio. This is how nearly every pilot is taught. But that explanation is incorrect. Carburetors do not maintain a constant volumetric ratio of fuel and air. The amount of fuel delivered actually declines as the air density declines. The problem is that it declines slower than the air density, and this is what makes the mixture richer as we climb. Hence we need to restrict the flow of fuel to maintain the proper air:fuel ratio.

As air flows through the carburetor’s constriction, the drop in pressure can be written as

\delta P = \frac{1}{2}\rho_a {v_a}^2where \delta P is the pressure difference (compared to ambient), \rho_a is the density of air, and v_a is the velocity in the constriction. This pressure difference \delta P is what forces the fuel from the bowl into the venturi.

Next, we need to relate \delta P to the fuel draw rate. For this, we need to use the orifice plate equation, which states

Q = K\sqrt{2\delta P \rho_f}where Q is the fuel flow rate, \rho_f is the fuel density and K is a function of the geometry of the orifice.

Combining the above two equations, we can get

Q = K\sqrt{\rho_a \rho_f}v_a.Now we can explain why mixture has to be leaned as we climb.

- If the aircraft climbs to 8000 ft, where the density is 75% that of sea level, assuming you leave the mixture at full-rich, the square root function makes the fuel flow rate to decline to \sqrt{0.75}=0.87 or 87%. In other words, there will be 12% more fuel in the mixture than at sea level. One way to compensate for this is by making the orifice smaller (which would make K smaller). This is exactly what the mixture knob does.
- If the throttle is opened more, the pressure in the intake manifold will rise, resulting in an increase in the total air intake. This will increase the air velocity v_a. The fuel flow Q will increase in direct proportion to v_a, and this will maintain the same mixture ratio as before. In other words, increasing the throttle should not, under normal circumstance, require a change in mixture setting.
- All of this assumes that we want to maintain a fixed air:fuel ratio. The stoichiometric mass flow ratio for complete combustion is 14.7:1. That is, we need 14.7 grams of air for each gram of fuel. However, due to other factors such as cooling and combustion efficiency, the ideal mass flow ratio is not always 14.7:1. At very high power outputs, it is better to use something like 12:1 (which is known as rich of peak) and at low power settings it is more efficiency to use something like 16:1 (known as lean of peak). As a result, even if there is no change in air density, the mixture should be adjusted whenever there is a change in power. On takeoff, the mixture should be slightly enriched. After takeoff, the mixture should be leaned immediately after the first power reduction. We don’t need to wait for a higher altitude.