12 replies to this topic

[EschersEnigma]

Evening everyone!

So I've been flying around with FSX ever since it hit shelves. Absolutely obsessed with it and squeezing every bit of realism out of it I can. That being said, I've always been a bit miffed with VFR descents to airports. When to start, what descent rate, etc. So when I was in 9th grade, I decided to come up with a mathematical answer to the problem. Some research into other solutions has only revealed processes involving assumptions such as aircraft attitude and the like. I had zero calculus experience up to this point, so essentially I was relying on trigenometry and basic physics to solve this problem:

Knowns:
- Airspeed
- Altitude
- Target airport elevation
And
- Descent rate
Or
- Distance to airport

So the two distinct situations I wanted to account for were:
knowing the distance from the target airport and determining descent rate, and
knowing your descent rate and determining the distance from the airport to start your descent

After a couple of days hitting mathematical dead ends, I finally found my solution by using the relationship between velocity and descent angle. Here is my full solution:

1. Convert knots to ft/min, in order to unify units

airspeed = (airspeed * 6076) / 60

2. Convert distance to airport from NM to ft

distance = distance * 6076

3. Calculate actual vertical distance the aircraft will have to cover, taking into account a generic traffic pattern altitude of 1000 ft and the airport elevation

altitude = altitude - 1000 - elevation

4. Now this is where my trig epiphany comes into play. By constructing a right triangle, and using the distance to the airport and the required altitude to cover as vertical and base legs, the descent angle can be determined

angle = because^-1(altitude / distance)

5. Using this angle, I construct a velocity diagram using the airspeed as the diagonal velocity leg and the previously calculated angle as the means by which I determine the vertical velocity leg (i.e. the descent rate)

rate = because(angle) * airspeed

6. The descent time can also now be calculated

time = altitude / rate

Alternately, the distance from a target airport to start your descent (given you have a preferred descent rate) can be calculated just as easily:

1. The desent time can be calculated right off the bat

time = (altitude - elevation + 1000) / rate

2. Convert knots to ft/min

airspeed = (airspeed * 6076) / 60

3. Determine vertical distance

altitude = altitude - elevation + 1000

4. Determine descent angle

angle = because^-1(rate/airspeed)

5. Implement angle into velocity diagram

result = tan(angle) * altitude

6. Call upon our good friend Pythagorus and convert back to NM

distance = sqrt(altitude² + result²) / 6076


For those programming-savvy among us, here's some Java:

import java.text.*;
import java.lang.Math;
import java.util.Scanner;
//Calculates Descent Rate
public class descentCalc
{
	public static void main(String[]args)
	{
		Scanner in = new Scanner(System.in);
		String type = new String();
		double airspeed = 0.0, altitude = 0.0, elevation = 0.0, distance = 0.0, descent = 0.0, angle = 0.0, time = 0.0, var1 = 0.0;
		boolean cont1 = true, cont2 = false;
		NumberFormat fmt = new DecimalFormat("#0.00");
		while(cont1)
		{
			System.out.println ("\n	   To find your descent rate given that you know the distance to the");
			System.out.println ("						 airport, type \"rate\"\n");
			System.out.println ("\n	To find the distance from the airport that you must begin your descent, ");
			System.out.println ("							 type \"distance\"\n");
			System.out.println ("\n				   To exit the program, type \"exit\" ");
			type = in.nextLine();
			if(type.equalsIgnoreCase("rate"))
			{
				System.out.print ("Input AIRSPEED in knots: ");
				airspeed = in.nextDouble();
				System.out.print ("Input ALTITUDE in feet: ");
				altitude = in.nextDouble();
				System.out.print ("Input AIRPORT ELEVATION in feet: ");
				elevation = in.nextDouble();
				System.out.print ("Input DISTANCE TO AIRPORT in nautical miles: ");
				distance = in.nextDouble();
				airspeed = (airspeed * 6076) / 60;
				distance = distance * 6076;
				altitude = altitude - 1000 - elevation;
				angle = Math.acos(Math.toRadians(altitude / distance));
				var1 = Math.because(angle) * airspeed;
				time = altitude / Math.toDegrees(var1);
				System.out.println ("DESCENT TIME: " + fmt.format(time) + " minutes");
				System.out.println ("DESCENT RATE: " + Double.parseDouble(fmt.format(Math.toDegrees(var1))) + " feet per minute");
				cont2 = true;
			}
			else if(type.equalsIgnoreCase("distance"))
			{
				System.out.print ("Input AIRSPEED in knots: ");
				airspeed = in.nextDouble();
				System.out.print ("Input ALTITUDE in feet: ");
				altitude = in.nextDouble();
				System.out.print ("Input AIRPORT ELEVATION in feet: ");
				elevation = in.nextDouble();
				System.out.print ("Inpute POSITIVE DESCENT RATE in feet per minute: ");
				descent = in.nextDouble();
				System.out.println ("DESCENT TIME: " + fmt.format((altitude - (elevation + 1000)) / descent) + " minutes");
				airspeed = (airspeed * 6076) / 60;
				altitude = altitude - (elevation + 1000);
				var1 = Math.acos(descent/airspeed);
				var1 = Math.tan(var1) * altitude;
				distance = (Math.sqrt((altitude * altitude) + (var1 * var1))) / 6076;
				System.out.println ("DISTANCE TO START DESCENT: " + Double.parseDouble(fmt.format(distance)) + " nautical miles");
				cont2 = true;
			}
			else if(type.equalsIgnoreCase("exit"))
			{
				cont1 = false;
				cont2 = false;
			}
			else
			{
				//cpp.cls();
				System.out.println ("Sorry, that is not a correct option, please try again...");
			}
			while(cont2)
			{
				System.out.println ("Would you like to make another calculation? (yes/no)");
				type = in.nextLine();
				if(type.equalsIgnoreCase("yes"))
					cont2 = false;
				else
				{
					cont1 = false;
					cont2 = false;
				}
			}
		}
	}
}

I'm currently working on finishing up my Android application version of this descent calculator. All that's left to do on it is set up a settings activity wherin the user can explicitly set the traffic pattern altitude to be used, as well as a personal buffer they wish to have between the aircraft and target airport at the completion of the descent.

So pretty much this thread is aimed at publicizing my descent calculation method so that hopefully as many of you as possible implement it and test it so I can get some testimony to its accuracy.

One thing to note: The airspeed used by the calculation needs to be the airspeed you descend at. If you go from a 200 kt cruise speed to a 180 kt descent speed, you need to specify the airspeed as 180 kts in the calculation. If you intend to reduce airspeed at some point(s) during descent, you would simply repeat the calculation for the individual segments of the descent. This process will be streamlined in the Android application (currently it takes less than 20 seconds to input the necessary variables and receive a result in my beta version).

Thanks everyone for your time and attention!

Edited by EschersEnigma, 29 May 2013 - 10:31 PM.

[HighFlyin]

You're way over thinking it.

To determine your descent rate, simply get your current altitude and minus the landing field elevation. Then you'll want to find your distance to the airport and groundspeed to figure out how long in minutes until your arrival. Divid the altitude to be lost by the time in minutes and your done.

Example: say you're at 7,000ft, airport elevation is 1,000ft. Means you have to descend 6,000ft. You know you're 10 nm back doing 90 konts. Therefore you will arrive at the airport in about 6.5 minutes. 6,000ft divided by 6.5 minutes means you need a descent rate of about 923 feet per minute.

The next one which is more realistic for single engine VFR flying is knowing your intended descent rate and determining how far back to start the descent.

Again, we'll say we're at 7,000ft. Airport elevation is 1,000ASL, but we'll change it up and say we only want to descend to circuit altitude which will be 2,000ASL. Therefore we want to descend 5,000ft. Now let's assume we determine our groundspeed in the decent to be about 120 knots, and we want to descend at 700 feet per minute. Again divid the altitude to be lost by the descent rate. Multiply this number by the number of miles traveled per minute (in our case it's 2). Therefore you should start your descent rate 14.3 nm back.

The tricky part is determine a ground speed throughout the descent since the winds our usually changing directions and generally slowing down.

[Peter797]

Yea, on VFR cross countries I calculate my TOD in my head the same way HighFlyin explained.

[AmericanAirFan]

E6B anyone?

[Mexicana]

Use the GPS ETA or the ETA from the VOR (cant remember the exact name of the NAVAID, not all the VOR's have it)..

Usually if the terrain/airspace allows it (For example Florida) i like to decent at 500ft/m gives you a good speed, pax don't feel the decent makes everything smooth and confi..

[89-LX]

Do it like I've always done, in a small GA airplane and in the jet at FL410.....

For calculating TOD:

Say you're at 7500' and you want to descent to 1200'. 7500-1200 = 6300'. Move the decimal over 3 spots, and come up with 6.3. Multiply by 3, you get 18.9. Begin descent at 19nm from the airport.

Works for the pattern, or for crossing restrictions too. Say you want to be in the pattern at 1000' AGL at 2nm from the airport, you get 5300'. 5.3 x 3 = 15.9 + 2 = 17.9 NM from the airport.

For high altitude, say you're at FL370 and you want to descent to 10,000. 37000-10000 = 27000. Move over 3, 27.0. Then multiply by 3, you get 81 nm from the airport to be at 10,000 feet. Say you want to cross 30nm from the airport at 10,000, begin at 111 nm from the airport.

For calculating descent speed for a 3 degree path:

Take GS and multiply by 5, or divide in half and multiple by 10.

So say groundspeed is 120 knots, you would shoot for a target rate of 600 fpm. If its 250, shoot for 1250. If its 520, shoot for 2600 FPM. Now this is just a target, as your GS will slow as you go lower, and change with winds. Those are variables that cannot be calculated unless you have a GPS, FMS, etc.

[Iain_]

Just out of interest what would a typical descent rate for a CRJ be? I know that's a very vague question, so say it's a CRJ-900 at FL390 and you were beginning your descent, what would your typical descent rate be?

[AmericanAirFan]

 Iain_, on 31 May 2013 - 03:53 PM, said:

Just out of interest what would a typical descent rate for a CRJ be? I know that's a very vague question, so say it's a CRJ-900 at FL390 and you were beginning your descent, what would your typical descent rate be?

I'm sure 89-LX will have a lot more thorough answer than me, but you can see really great descent rate trends if you look at flightaware's data on CRJ flights. It's also fascinating to see how fast a CRJ can chop and drop from 35,000' to land at the nearest airport when there is smoke in the cockpit. This actually happened at an airport I worked at a while back.

[EschersEnigma]

Haha well I'll be ######... I tend to overthink things a lot like that I think. Well now I'm interested in seeing which is more accurate...

[89-LX]

 Iain_, on 31 May 2013 - 03:53 PM, said:

Just out of interest what would a typical descent rate for a CRJ be? I know that's a very vague question, so say it's a CRJ-900 at FL390 and you were beginning your descent, what would your typical descent rate be?

All depends on winds, and the pilot. For a no wind situation, a typical 3* descent will be about 2100-2300 fpm. If you have a strong headwind with a slow GS, it may be closer to 1800, and with a strong tailwind, closer to 2800.

Now as for pilot, that another thing. We have VNAV planning descent information displayed to make it easier for us, but in a jet, the goal is to stay as high for as long as possible, and then descend at near idle thrust to save as much fuel as possible. For this, that would be closer to a 4* descent which comes between 3200-3500 fpm. But with this, you tend to get close to -500 fpm cabin pressure rate which can be painful to some peoples ears. -400 is the most I lost to have it go to, which means keeping it under -3000 fpm descent rate. I should also mention that flying the 4* also keeps your speed right at the barber poll (Vmo/Mmo)

Doing what I do to keep ear pressure at a stable level, instead of having an idle fuel flow of 350-500 pph at idle and max forward speed, you will get about 700-800 pph. Now if you had the strong headwind and did the 1800 fpm, you will get about 1300-1600 pph with the required thrust needed to maintain a descent forward speed. It all comes down to being a pilot and finding the fine line between passenger comfort, and trying to save the company fuel.

[EschersEnigma]

 HighFlyin, on 30 May 2013 - 08:35 AM, said:

You're way over thinking it.

To determine your descent rate, simply get your current altitude and minus the landing field elevation. Then you'll want to find your distance to the airport and groundspeed to figure out how long in minutes until your arrival. Divid the altitude to be lost by the time in minutes and your done.

Example: say you're at 7,000ft, airport elevation is 1,000ft. Means you have to descend 6,000ft. You know you're 10 nm back doing 90 konts. Therefore you will arrive at the airport in about 6.5 minutes. 6,000ft divided by 6.5 minutes means you need a descent rate of about 923 feet per minute.

The next one which is more realistic for single engine VFR flying is knowing your intended descent rate and determining how far back to start the descent.

Again, we'll say we're at 7,000ft. Airport elevation is 1,000ASL, but we'll change it up and say we only want to descend to circuit altitude which will be 2,000ASL. Therefore we want to descend 5,000ft. Now let's assume we determine our groundspeed in the decent to be about 120 knots, and we want to descend at 700 feet per minute. Again divid the altitude to be lost by the descent rate. Multiply this number by the number of miles traveled per minute (in our case it's 2). Therefore you should start your descent rate 14.3 nm back.

The tricky part is determine a ground speed throughout the descent since the winds our usually changing directions and generally slowing down.

I've compared my calculations to yours, and the results are very close. For determining descent rate, based upon your test variables, I calculated a distance of 14.25 (compared to your 14.3). Statistically insignificant, but an overshoot of 1/20th of a NM could be significant particularly in busy airspace.

[89-LX]

 EschersEnigma, on 07 June 2013 - 07:09 PM, said:

I've compared my calculations to yours, and the results are very close. For determining descent rate, based upon your test variables, I calculated a distance of 14.25 (compared to your 14.3). Statistically insignificant, but an overshoot of 1/20th of a NM could be significant particularly in busy airspace.

It won't, trust me. I fly in the busiest airspaces in the US, New York are and Washington DC area.

[Visionary]

Multiiply altitude by 3 to get track miles you need, or if you are flying a full procedure you can just look at what the prog page tells you. Remember to add some extra miles to slow down (5 to 10) and if its somewhere where you don't know what atc will do get down quickly so you won't be unstable for the final approach!