General Principles - Climb — Page 175, Lesson 201

Let’s get into the climb gradient calculation. This is one of the most important formulas in performance work, so I want to walk you through it carefully. We start with a simple idea. If we know the Thrust Available and the aerodynamic Drag, we can find the maximum backward component of Weight. That backward component is what actually pulls the aircraft down the slope of the climb. We get it by subtracting Drag from Thrust Available. That difference is called the Excess Thrust. Now, for most purposes we use climb gradient rather than climb angle. Climb angle is the actual angle of the flight path above the horizontal. Climb gradient is the percentage of the backward component of Weight relative to the aircraft Weight. So it’s a ratio, expressed as a percentage. Here is the formula, and I want you to remember it because it is very significant: Gradient percent equals (T minus D) divided by W, all multiplied by 100. So Gradient % = (T – D) / W × 100. Let me define each symbol. T is Thrust Available. D is aerodynamic Drag. W is the aircraft Weight. The numerator, T minus D, is the Excess Thrust. So the formula says: Excess Thrust divided by Weight, times 100, gives you the climb gradient as a percentage. Now, just by looking at this formula, two facts are self-evident. First, for a given weight, the greater the Excess Thrust, the steeper the climb gradient. The less the Excess Thrust, the more shallow the climb gradient. Second, for a given Excess Thrust, the greater the weight, the more shallow the climb gradient. The less the weight, the steeper the climb gradient. So Excess Thrust helps you climb, and Weight hurts you. Let me show you a worked example. We have a twin-engine turbojet aircraft. Each engine produces 60,000 Newtons of thrust. Its mass is 50 tonnes. It has a Lift-to-Drag ratio, L/D, of 12 to 1. We use g equal to 10 metres per second squared. The question is, what is the percentage climb gradient? First, we derive the values for the formula. Thrust is 60,000 Newtons times 2 engines, which gives 120,000 Newtons. Drag equals Weight divided by 12, because the L/D ratio is 12 to 1. Weight is 50 tonnes times 1000, which is 50,000 kilograms, times 10 metres per second squared, which gives 500,000 Newtons. So Drag is 500,000 divided by 12, which is 41,667 Newtons. Now we plug into the formula. 120,000 minus 41,667, divided by 500,000, times 100. That gives 78,333 divided by 500,000, times 100, which equals 15.7 percent. So the climb gradient is 15.7 percent. Now let’s consider the same aircraft, but with one engine failed. Thrust is now only 60,000 Newtons, because one engine is out. Drag stays the same at 41,667 Newtons. Weight stays the same at 500,000 Newtons. So we have 60,000 minus 41,667, divided by 500,000, times 100. That gives 3.7 percent. Here is the significant fact. Thrust has decreased by 50 percent, but the climb gradient has decreased by approximately 75 percent, or to one quarter of the gradient possible with all engines operating. That is a very significant fact. So why did a 50 percent loss of Thrust Available cause a 75 percent decrease in gradient? Think about it. The climb gradient depends on Excess Thrust, which is Thrust minus Drag. When you lose an engine, you lose half your thrust, but Drag stays the same. So the Excess Thrust shrinks much more than proportionally. The fixed Drag eats into the reduced thrust, leaving a much smaller Excess Thrust. That is why the gradient drops so dramatically. Now, there is a note about accuracy. The aircraft data gave us the L/D ratio, from which we extracted Drag. But in a steady climb, Lift is obviously less than Weight. So isn’t it inaccurate to use Weight divided by L/D to get Drag? The answer is no, for practical purposes. If the climb angle is less than approximately 20 degrees, and it always will be, the difference in the magnitude of 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. So the key takeaways are the formula, Gradient % = (T – D) / W × 100, the two self-evident relationships about Excess Thrust and Weight, and the dramatic effect of engine failure on gradient. That’s the core of climb gradient calculation.