Hot and High Deceleration Planning
If you’ve ever flown into Denver, Salt Lake, or other hot and high airports and thought “Hmmm, this airplane really doesn’t want to slow down,” you’re not alone. Wouldn’t it be great if we had a set of rules to adjust for density when planning our slowdowns? Let’s look at the forces at play and define some quick rules to address the challenge so we don’t need to reach for the gear handle as often.
Even if physics wasn’t your favorite class, it’s worth giving it a sporting try to form an appreciation of how the different inputs and outputs relate before skipping to the rules at the end. You might be able to impress a skeptical pilot monitoring someday.
Even if physics wasn’t your favorite class, it’s worth giving it a sporting try to form an appreciation of how the different inputs and outputs relate before skipping to the rules at the end. You might be able to impress a skeptical pilot monitoring someday.
The Challenge
Due to density effects, the typical 10 knots per mile slowdown rule falls apart at hot and high fields. Let’s look at why that happens and then adjust the rules.
The air, by virtue of being less dense, forces us to fly at a faster TAS to maintain the same IAS and lift. The faster TAS also means that any IAS difference, say from 210 down to 200, will represent a larger TAS difference (∆TAS) that we need to bridge.
The air, by virtue of being less dense, forces us to fly at a faster TAS to maintain the same IAS and lift. The faster TAS also means that any IAS difference, say from 210 down to 200, will represent a larger TAS difference (∆TAS) that we need to bridge.
As TAS increases with altitude, you need more distance to slow down.
Pilots should be careful with any quadratic scaling of outputs.
As TAS gets higher, so does the airplane’s kinetic energy, as a quadratic function of TAS. If it’s been a few decades since your high school algebra class, quadratic scaling is something we pilots need to pay attention to, because an increase in the input leads to a much larger output. As an example, a 10% increase in your input gives you 21% higher output (1.1*1.1=1.21).
Not only do we need to bridge a higher ∆TAS at altitude, we also need to dissipate more energy to cover it. This unfriendly stacking of factors means we’ll need substantially more distance to slow down.
Let’s dive in and quantify the scope of the problem. I did the following sections in a spreadsheet because I’m too lazy to spin my whiz wheel. Then I rounded in the presentation to make it look a little bit less ugly. Multiplying the rounded numbers together may give you slight discrepancies.
Not only do we need to bridge a higher ∆TAS at altitude, we also need to dissipate more energy to cover it. This unfriendly stacking of factors means we’ll need substantially more distance to slow down.
Let’s dive in and quantify the scope of the problem. I did the following sections in a spreadsheet because I’m too lazy to spin my whiz wheel. Then I rounded in the presentation to make it look a little bit less ugly. Multiplying the rounded numbers together may give you slight discrepancies.
The Math
In broad terms, we need to proceed in the following manner:
1. The TAS Change
If we want to slow from 210 KIAS to 200 KIAS at 3000 feet, we need to figure out ∆TAS. Recall from Density for Dummies these two equations:
- Find TAS values given IAS at different altitudes before and after decelerating.
- Find kinetic energy values at those altitudes.
- Determine baseline drag.
- Determine the effect of drag on slowdown distances at different altitudes.
- Our base case altitude is 3000 feet and temperatures are ISA.
- Winds are calm.
- Our E175 weighs 72,000 lbs, the base case in many of the manuals.
- We join the arrival area at 210 KIAS and our 10 KIAS/nm rule applies at 210 KIAS/3000 ft.
1. The TAS Change
If we want to slow from 210 KIAS to 200 KIAS at 3000 feet, we need to figure out ∆TAS. Recall from Density for Dummies these two equations:
Putting those into our ISA temp and pressure equations gives us the following numbers:
2. The Energy Change
Now let’s sort out the energy to lose. We know that KE=1/2mv^2. Let’s convert everything to SI units because they’re all inter-defined and require no constants:
- 210 KIAS at 3000 feet is 219.5 KTAS.
- 200 KIAS at 3000 feet is 209.0 KTAS.
- We need to lose 10.5 KTAS.
2. The Energy Change
Now let’s sort out the energy to lose. We know that KE=1/2mv^2. Let’s convert everything to SI units because they’re all inter-defined and require no constants:
Now it’s more straightforward to determine our initial and final kinetic energy states, KE1 and KE2a, by plugging those values into the KE equation:
Subtracting 247.4 MJ – 224.4 MJ gives us 23.0 MJ to lose at 3000 feet to go from 210 KIAS to 200 KIAS.
We can repeat this exercise with a range of altitudes up to 11000 feet. The green row below is the one we just went through. The rest followed the same pattern:
We can repeat this exercise with a range of altitudes up to 11000 feet. The green row below is the one we just went through. The rest followed the same pattern:
Some things to note:
3. Baseline Drag
Next, we need to figure out how much drag we have slowing us. First, we’ll use what we know about deceleration (F=ma) to find the force of drag, then apply the work formula (W=Fd) to find how long we need to apply the force to slow the airplane, referencing the Work-Energy Theorem (W=∆KE) for energy to lose. The Work-Energy Theorem we’re using here is the one out of the physics textbook, not our lack of energy when scheduling calls at 0300 on a reserve day (although that is a good example of inertia).
We know that F=ma. We already know m=38,808 kg, so we can solve for a to find F. We know from the baseline that we lose 10 KIAS in a mile at 3000 feet. Using the equation for acceleration, TAS2 = TAS1 + at, where a is acceleration and t is time, we can input our above numbers to get an answer.
- As expected, ∆KTAS increases with altitude in column Q.
- ∆KE increases in column R.
- In columns S and T, you can see the changes from the 3000-foot baseline. Due to the quadratic scaling of ∆KE, it scales faster than ∆KTAS.
3. Baseline Drag
Next, we need to figure out how much drag we have slowing us. First, we’ll use what we know about deceleration (F=ma) to find the force of drag, then apply the work formula (W=Fd) to find how long we need to apply the force to slow the airplane, referencing the Work-Energy Theorem (W=∆KE) for energy to lose. The Work-Energy Theorem we’re using here is the one out of the physics textbook, not our lack of energy when scheduling calls at 0300 on a reserve day (although that is a good example of inertia).
We know that F=ma. We already know m=38,808 kg, so we can solve for a to find F. We know from the baseline that we lose 10 KIAS in a mile at 3000 feet. Using the equation for acceleration, TAS2 = TAS1 + at, where a is acceleration and t is time, we can input our above numbers to get an answer.
If we assume the drag and deceleration are locally linear, then our average speed is:
At that speed we travel a nm (1852 m) in
Now we have t, and can solve for a:
Plugging a into the force equation gives us the following:
Last, we know that the work (W, measured in joules) done by a force, in this case drag, is W = F*d, where d is distance over which the force is applied. If we plug in our numbers from above, we get:
The Work-Energy theorem tells us that W=∆KE. We already have the ∆KE numbers above in our table. In other words, the drag force of 12.4 kN applied over a mile dissipated the 23.0 MJ of kinetic energy that we needed to lose to slow the aircraft. Now let’s see what happens at other altitudes.
4. Generalized Drag
This part is mercifully simple. Drag follows similar rules as density and lift, scaling as the square of TAS and linearly with density. Because IAS is adjusted for both of those, we can apply the 12.4 kN across all altitudes if we’re calculating drag at the same given IAS, which we are. Now we need to flip our work equation around a bit to give us deceleration distance:
4. Generalized Drag
This part is mercifully simple. Drag follows similar rules as density and lift, scaling as the square of TAS and linearly with density. Because IAS is adjusted for both of those, we can apply the 12.4 kN across all altitudes if we’re calculating drag at the same given IAS, which we are. Now we need to flip our work equation around a bit to give us deceleration distance:
Let’s see what happens at 11000 feet. Due to our faster true airspeed, we need to get rid of 29.4 MJ, not just the 23.0 MJ that we had down at 3000 feet. Applying our constant drag gives us:
That’s 28% additional distance required to slow the airplane down! Plugging those numbers back into the speed equation yields an additional 13% on time, t. No wonder Denver Approach gets antsy when we’re not slowing enough. Putting this into our table gives us the below values:
With a constant drag force, the change in deceleration distance ∆d scales linearly to ∆KE (see columns T and Z). This is an expected outcome given the work equation. Those both increase roughly 3% per thousand feet. The change in time ∆t taken for that deceleration scales linearly to ∆TAS and as the square root of d.
This whole time I’ve been pretty quiet about what happens when we use boards, or other drag devices for that matter. Basically, the scaling works the same. Losing 20 KIAS at 11000 feet takes the same 1.28 nm that losing 10 KIAS did clean. This is because drag scales linearly with the drag coefficient, CD. Here’s the full table. Note the ∆ KE and ∆ KTAS comparisons in S, T, W, and X:
This whole time I’ve been pretty quiet about what happens when we use boards, or other drag devices for that matter. Basically, the scaling works the same. Losing 20 KIAS at 11000 feet takes the same 1.28 nm that losing 10 KIAS did clean. This is because drag scales linearly with the drag coefficient, CD. Here’s the full table. Note the ∆ KE and ∆ KTAS comparisons in S, T, W, and X:
Temperature, as we saw in Density for Dummies, has a 10:1 exchange rate with altitude. In other words, trade 10 degrees C for 1000 feet for the same effect.
The Rules
If you scrolled past the last section, here are the takeaways you should care about:
The physics of this should work for other aircraft as well. Just plug your baseline weights in on #1 and apply the remaining rules to whichever deceleration rules you employ. I’m told some newer types have high idle thrust that may make it a bit trickier. Send me a PM if you want a copy of the spreadsheet to adapt to your own aircraft.
- Deceleration distances scale linearly with mass. For every 720 lbs off 72,000, add or subtract 1%.
- For every thousand feet above 3000, add 3% to your budgeted deceleration distances. If you’re lazy and want a quick rule for Denver, just add 25% at 10000 feet to your distances.
- Add 3% deceleration distance for every 10 C above ISA (Denver on a 35-degree day will require 1.42x from baseline).
- The effectiveness of your drag devices will follow the 3% scaling rule. Move your “Now I gotta drop my gear” spot and other configuration gates back accordingly. Example: “Normally we go gear down flaps 3 a mile out from the FAF, today in Denver we’ll do it at 1.2.”
- Time taken to slow down will increase 1.5% per thousand feet.
The physics of this should work for other aircraft as well. Just plug your baseline weights in on #1 and apply the remaining rules to whichever deceleration rules you employ. I’m told some newer types have high idle thrust that may make it a bit trickier. Send me a PM if you want a copy of the spreadsheet to adapt to your own aircraft.
Conclusions
Hot and high airports require changes to speed planning, flare geometry, and faster thinking because you’re moving at a greater clip. Having some basic rules will help you plan for what’s coming and prompt you to be more aggressive in deploying your drag devices to hit your stability gates.