What is the maximum mass that the airplane can have and still maintain enough lift to fly?

Question

A commercial airplane travels at a speed which is 85% of the speed of sound. The wings of the airplane are designed such that the bottoms of the wings are flat and the tops of the wings are curved so that air travelling above the wings follows a path which is 15% longer than the straight path of the air below the wings. The wings are approximately rectangular in shape with a length of 35 m and a width of 8 m. The thickness of the wings is negligible. What is the maximum mass that the airplane can have and still maintain enough lift to fly?

Answer 1

Aeroplanes fly by thrusting air downwards and by thus being borne up by the Newton's-Third-Law begotten upwards reaction force of the downthrusted air on the aeroplane. There are many excellent answers to the Physics SE question "What really allows airplanes to fly?" that you should read.

But basically the simplest estimates arise from calculating the ram pressure thrust upwards on the aeroplane given the above principle. The variables you need to know are density of the air at the height, the relative speed of the aeroplane to the air, the angle of attack that the wing makes with the velocity vector of the air relative to a frame comoving with the aeroplane and the scale factor that yields the effective surface area of the wing - which at subsonic speeds is considerably larger than the wing itself because the disturbance to the fluid flow pattern that arises from the wing is felt over a region that is considerably bigger than the wing. The last variable - effective area - can also be expressed as the wing's coefficient of lift.

To illustrate these points, we can do a back of the envelope estimation of ram pressure in this case: see my drawing below of a simple aerofoil with significant angle of attack being held stationary in a wind tunnel. This is the kind of analysis you should do to get an idea of your specific situation. Your air density is going to be rather less than that for the following calculation (commercial jetliners reach their top speed at heights of about 8000m):

Lets suppose the airflow is deflected through some angle $\theta$ radians to model an aeroplane's attitude (not altitude!) on its last approach to landing or as it takes off, flying at $300\mathrm{km\,h^{-1}}$ airspeed or roughly $80\mathrm{m\,s^{-1}}$. I have drawn it with a steep angle of attack. Air near sea-level atmospheric pressure has a density of about $1.25\mathrm{kg\,m^{-3}}$ (molar volume of $0.0224\mathrm{m^{-3}})$. The change in momentum diagram is shown, whence the change in vertical and horizontal momentum components are (assuming the speed of flow stays roughly constant):

$$\Delta p_v = p_b \sin\theta;\quad\quad\Delta p_h = p_b \,(1-\cos\theta)$$

At the same time, the deflecting wing presents an effective blocking area to the fluid of $\alpha\,A\,\sin\theta$ where $A$ is the wing's actual area and $\alpha$ the scale factor to account for the fact that in the steady state not only fluid right next to the wing is distrubed so that the wing's effective area will be bigger than its actual area. Therefore, the mass of air deflected each second is $\rho\,\alpha\,A\,v\,\sin\theta$ and the lift $L$ and drag $D$ (which force the engines must afford on takeoff) must be:

$$L = \rho\,\alpha\,A\,v^2\,(\sin\theta)^2;\quad\quad D = \rho\,\alpha\,A\,v^2\,(1-\cos\theta)\, \sin\theta$$

If we plug in an angle of attack of 30 degrees, assume $\alpha = 1$ and use $A = 1000\mathrm{m^3}$ (roughly the figure for an Airbus A380 wing area), we get a lifting force $L$ for $\rho = 1.25\mathrm{kg\,m^{-3}}$ and $v = 80\mathrm{m\,s^{-1}}$ of 200 tonne weight. This is rather less than the takeoff weight of a fully laden A380 Airbus (which is 592 tonnes, according to the A380 Wikipedia page) but it is an astonishingly high weight just the same and within the right order of magnitude. We see that the wing's effective vertical cross section is bigger than the actual wing by a factor of 2 to 3. This is not surprising at steady state, well below speed of sound flow: the fluid bunches up and the disturbance is much bigger than just around the wing's neighbourhood. So, plugging in an $\alpha = 3$ (given the experimental fact that the A380 can lift off at 592 tonnes gross laden weight), we get a drag $D$ of 54 tonne weight (538kN) - about half of the Airbus's full thrust of 1.2MN, so this ties in well with the Airbus's actual specifications, given there must be a comfortable margin to lift the aeroplane out of difficulty when needed.