General Principles - Climb — Page 175, Lesson 201

Let’s pick this up right where the climb itself left off. We’ve already talked about how an aircraft climbs when Thrust Available exceeds Drag, giving us that Excess Thrust. Now I want to show you how we turn that into a number the performance engineer actually uses — the climb gradient. Here’s the key formula, and I want you to treat it as very significant. The percentage climb gradient equals (T minus D) divided by W, all multiplied by 100. In words: Gradient % = (T – D) / W × 100. T is Thrust Available, D is aerodynamic Drag, and W is the aircraft Weight. So the numerator, T minus D, is that Excess Thrust we talked about — the thrust left over after overcoming drag. The denominator is the full weight. Now, just by looking at that formula, two facts are self-evident. First, for a given weight, the greater the Excess Thrust, the steeper the climb gradient. Less Excess Thrust means a more shallow gradient. Second, for a given Excess Thrust, the greater the weight, the more shallow the climb gradient. Less weight means steeper. So you’ve got two levers: Excess Thrust pushes the gradient up, and Weight pulls it down. Let me walk you through a worked example so you see how the numbers actually flow. Take a twin-engine turbojet. Each engine produces 60,000 Newtons of thrust. The aircraft mass is 50 tonnes, and it has an L/D ratio of 12 to 1. We’re asked for the percentage climb gradient, and we use g = 10 metres per second squared. First, derive the values for the formula. Thrust: 60,000 Newtons per engine, times 2 engines, gives 120,000 Newtons total. Weight: 50 tonnes times 1000 gives 50,000 kilograms, and times g of 10 metres per second squared gives 500,000 Newtons. Now Drag. The L/D ratio is 12 to 1, so Drag equals Weight divided by 12. That’s 500,000 Newtons divided by 12, which is 41,667 Newtons. Now plug into the formula. T minus D is 120,000 minus 41,667, which is 78,333 Newtons. Divide that by the weight of 500,000 Newtons, multiply by 100, and you get 15.7 percent. So with both engines running, this aircraft climbs at a 15.7 percent gradient. Now here’s where it gets really significant. Let’s take the same values but with one engine failed. Now Thrust is just 60,000 Newtons — one engine. Drag stays the same at 41,667 Newtons, and weight is still 500,000 Newtons. So T minus D is 60,000 minus 41,667, which is 18,333 Newtons. Divide by 500,000, multiply by 100, and you get 3.7 percent. Look at what happened. Thrust decreased by 50 percent — from 120,000 down to 60,000. But the climb gradient decreased by approximately 75 percent — from 15.7 percent down to 3.7 percent, roughly one quarter of the original. That’s a hugely important fact. Why did a 50 percent loss of thrust cause a 75 percent loss of gradient? Because the gradient depends on Excess Thrust, not total thrust. When you halve the thrust, you’re not halving the excess — you’re subtracting the same drag from a much smaller thrust, so the leftover shrinks far more than proportionally. Now, one more subtlety, and it’s a good one. In that example, we used the L/D ratio to extract Drag. But in a steady climb, Lift is obviously less than Weight — the aircraft is tilted, so the lift vector isn’t fully opposing weight. Isn’t that inaccurate? Here’s the answer: if the climb angle is less than approximately 20 degrees — and it always will be in normal operations — the difference in magnitude between Lift and Weight in a steady climb is insignificant. So for these and other calculations, we can consider Lift and Weight to be the same. That’s what justifies using the L/D ratio the way we did. So the takeaway: climb gradient is Excess Thrust over Weight, expressed as a percentage, and it’s brutally sensitive to thrust loss because it’s the excess that matters, not the total.