Why is the thrust/drag increase much lower at a lower FL with weight the only variable?

Question

From and related to: What is the relation between drag and weight?

• Friction drag is not affected by the angle of attack change.
• The [induced] drag will increase with the square of the mass increase.

I have no doubt about that, in fact I was going to post the same answer, the reason I waited is because I could not explain the A320 performance figures using this method.

The drag breakdown for an A320 at FL 370 and M0.78 is:

7,900 lbf of drag is composed of 4,700 lbf of parasitic drag (...) and 3,200 lbf of induced drag.

Let's consider a 32% weight increase -- from 50 to 66 tonnes. Applying 1.32^2 to the induced drag raises the total drag by 30%.

If we used the FF (fuel flow) as a thrust (also drag) measure -- since at a given FL / IAS / TAS the TSFC shouldn't vary that much when the N1 is ± a few percent --

... we find the ΔFF is 7% at FL 290 and 20% at FL 370. 20% is closer to the 30% thrust/drag estimate. Why is the thrust/drag increase much lower at a lower FL with weight the only variable?

Answer 1

At a lower flight level the air density is higher, so a lower lift coefficient can be trimmed and the induced drag is a smaller fraction of total drag. Hence, the mass increase causes a much lower increase in overall drag.

At higher altitude the friction and pressure drag is lower but the induced drag is higher, so increasing the mass will cause a much higher jump in drag.

Now for the numbers: In FL 290 the indicated speed is 155.362 m/s, so the dynamic pressure $q$ is 14,784 N/m². I use surface area $S$ (124 m²) and aspect ratio $AR$ (b²/S = 10.33) from Wikipedia and guess that the Oswald factor $\epsilon$ is 0.8. Now the lift coefficient $c_L$ at FL 290 and 50 tons mass is $$c_L = \frac{m\cdot g}{q\cdot S} = 0.2675$$ This allows us to calculate the induced drag: $$D_i = q\cdot S\cdot\frac{c_L^2}{\pi\cdot AR\cdot\epsilon} = 5052\:\text{N}$$ which is just 1136 pound-force; about a third of what you give for FL370. This fits well with the given thrust increase at FL290 which is about ⅓ of the increase at FL370.