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Let's say we test the coefficient of lift of the same wing at the same AOA and the same flow velocity at the same pressure.

The difference is wind tunnel A will contain helium (molecular weight 4) gas and wind tunnel B will contain air (average molecular weight around 29).

Which test will produce a higher lift coefficient? Why?

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1 Answer 1

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The wing will produce much less lift in helium than in air; all else being equal, lift is proportional to the molar mass of the gas in question.

The lift equation states that lift $L$ is proportional to the air density $\rho$ (rho):

$$L = \frac12 \rho \ v^2 \ S \ C_L,$$

where $v$ is the true airspeed, $S$ is the wing area, and $C_L$ is the coefficient of lift.

Meanwhile, one form of the ideal gas law tells us how the air density $\rho$ is related to the molar mass $M$:

$$\rho = \frac{MP}{RT},$$

where $P$ is the (static) pressure, $R$ is the gas constant, and $T$ is the absolute temperature.

Combining both of these equations, we find that

$$L = \frac{M P \ v^2 \ S \ C_L}{2 R T}.$$

In words, lift is

  • the molar mass of the gas,
  • times the static pressure,
  • times the square of the airspeed,
  • times the wing area,
  • times the coefficient of lift,
  • divided by the absolute temperature,
  • divided by a constant (the gas constant $R$),
  • divided by $2$.
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