Payload capacity of a quadcopter/helicopter with increasing altitude

Question

I am trying to create a simplified model of a heavy-lift drone payload at different altitude during hover mode. I tried to derive it simply by considering that the thrust, $T$ is equal to the weight of the drone, $G$ and a specified payload, $L$ which is given by the following equation:

$$ n_r \cdot T = G + L $$

From this formula, we know that the thrust at different altitude will vary, and this relationship is captured by the following equation:

$$ T = C_T \cdot \rho \cdot (N)^2 \cdot D_p^4 $$

$$ \rho = (P_a/P) \cdot \rho_0$$

$$ P_a = P \cdot (1 - 0.0065 \cdot \frac{h}{273 + T})^{5.2561}$$

where h is the altitude (m), $T$ is the temperature at sea level, $P_a$ is the local pressure at the given altitude (Pa), $P$ is the pressure at the sea-level, $ \rho $ is the local density at an altitude, $\rho_0$ is the standard air density at sea-level. From this equation, we can simplify it to be the following:

$$T(\rho, N, D_p)$$

$$\rho(P_a)$$

$$P_a(h, T_t)$$

where the relationship can be described by

$$P_a \propto \frac{1}{h}$$

$$ \rho \propto P_a$$

$$T \propto \rho $$

From the relationship above, we can clearly see the connection between the altitude and thrust. In the case of increasing altitude, the local atmospheric pressure decreases, which leads to a decrease in local density. This in turn, leads to a decrease in thrust.

Re-arranging the first equation to give:

$$ L = n_r \cdot T - G $$

Plotting this relationship with a constant propeller speed, constant weight of drone (300kg/3000N) and a constant thrust coefficient, I manage to get the following plot:

Analyzing the plot, from my understanding, as the altitude increases, the payload that the drone can carry decreases. However, what I am confused with is that, in the formula, I only used the weight of the drone without the payload, and the plot tells me the payload capacity of the drone at different altitude. However, if this payload is then placed onto the drone, the plot would have changed as the weight of the drone then changes.

Is there anything that I am missing here?

I then went on to read some literature on helicopter performance and I manage to find this:

The plot looks similar to my plot to some extent, but I've been trying to search for equations that describe this performance for helicopter so that I can compare my derivation to it. Can anyone point me in the right direction?

Answer 1

In helicopter momentum theory, thrust is usually written as $$T = C_T \cdot \rho A (\Omega R)^2$$ And if we keep $C_T, A$ and $\Omega R$ constant, we can see that thrust decreases linearly with decreasing density. Which is basically what hes been depicted in the OP plots.

However, in the hover the thrust must be equal to the weight at a variety of altitudes. Usually this is done by varying $C_T$ - increase collective, and the helicopter rises to a higher equilibrium altitude. Or increase $\Omega$ of a quadcopter.

The altitude limit at a given total weight would be a function of available power, as further detailed in this answer. The IGE hover ceiling graph of the Schweizer 296 depicts the limits imposed by the engine power limit at varying altitudes.