How is range increased when flying into a HW?
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The general equation for ground range can be simplified into the self-apparent equation:
Rg = (V – Vw)/Q
Where V is aircraft TAS, Vw is wind speed, and Q is fuel flow rate. For a turbofan aircraft Q = cBW
c = a constant or if you want to get fancy it may vary proportionally to speed, something of say c -> V^.45 (for turboprops c is different)
B = Coefficient of Lift over drag
W = weight
As you can see, by maximizing the numerator we increase Rg. Since the Vw is independent of our TAS, if we increase TAS we increase the numerator and also the value of Rg. The problem is we are also increasing the fuel flow, but by a lesser ratio. (The equations can get a lot more fancy, but this should show the relationship fairly well.)
Rg = (V – Vw)/Q
Where V is aircraft TAS, Vw is wind speed, and Q is fuel flow rate. For a turbofan aircraft Q = cBW
c = a constant or if you want to get fancy it may vary proportionally to speed, something of say c -> V^.45 (for turboprops c is different)
B = Coefficient of Lift over drag
W = weight
As you can see, by maximizing the numerator we increase Rg. Since the Vw is independent of our TAS, if we increase TAS we increase the numerator and also the value of Rg. The problem is we are also increasing the fuel flow, but by a lesser ratio. (The equations can get a lot more fancy, but this should show the relationship fairly well.)
Last edited by '%MAC'; 10th Jul 2002 at 07:29.
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Because I am too idle to draw a diagram and post it - here is a word picture. (I could do it on the back of a menu in an instant....)
Graph: Origin bottom left. Vertical is FFlow, Horizontal is TAS (for a given Altitude and Weight).
Plotting useful ranges of FF/TAS you will get the classic smooth "tick" (left to right) of a combined drag curve (well nearly).
The bottom of the tick [Vimd] (or 1.05 - 1.1 of this for steep bases) is the endurance, and the tangent to the bottom of the curve is best range - nominally (classically) 1.32 Vimd (local mileages may vary). In still air this is the best range speed to fly.
But a headwind (component) will shift this graph.
Move the origin of the tangent to the right (on the TAS axis) by the amount of the local headwind. Redraw the tangent. This is now the best range speed for that headwind. Tailwinds are vv.
N.B. This assumes you have a good set of figures for the original curve. It is also constrained by VMO/MMO/Vs etc etc. But you get the idea.
Easier to see than an equation for most.
Hope this helps
MG
Graph: Origin bottom left. Vertical is FFlow, Horizontal is TAS (for a given Altitude and Weight).
Plotting useful ranges of FF/TAS you will get the classic smooth "tick" (left to right) of a combined drag curve (well nearly).
The bottom of the tick [Vimd] (or 1.05 - 1.1 of this for steep bases) is the endurance, and the tangent to the bottom of the curve is best range - nominally (classically) 1.32 Vimd (local mileages may vary). In still air this is the best range speed to fly.
But a headwind (component) will shift this graph.
Move the origin of the tangent to the right (on the TAS axis) by the amount of the local headwind. Redraw the tangent. This is now the best range speed for that headwind. Tailwinds are vv.
N.B. This assumes you have a good set of figures for the original curve. It is also constrained by VMO/MMO/Vs etc etc. But you get the idea.
Easier to see than an equation for most.
Hope this helps
MG
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If fuel flow is proportional to thrust and you are already flying at your best TAS/drag ratio then you will not get more range in a headwind, but less. However, you can minimize the amount you lose by flying slightly faster, as already explained.
Dick W
Dick W
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Let’s assume the following: still air conditions, two airports (A & B) which are 500Nm apart, your aircrafts normal TAS is 500 Kts, so you can nominally fly between A & B in one hour.
Now add a 50Kt headwind component on the sector from A to B.
Accordingly your G/S from A to B is now only 450Kt, so the 500Nm trip takes 66.7 minutes – a 6.7 minute difference from the still-air time, with an effective flight distance now of 555.6Nm (i.e. 55.6 Nm further to fly than in still air)
Of course on the way back (from B to A), with a 50Kt tailwind component, the aircraft has a G/S of 550Kt, so the trip back only takes 54.6 minutes – a 5.4 minute difference from the still-air time, with an effective flight distance of 454.6Nm (i.e. 45.4 Nm less to fly)
The point here is that, on a route between two points, with a consistent TAS and head/tail wind components, a headwind always hurts you more than the tail wind helps you because you spend less time flying along in the tailwind than you do flying into the reciprocal headwind, e.g. in the above example we’re flying into the headwind for ( 6.7 – 5.4 = ) 1.3 minutes more than the we’re exposed to it as a tailwind over the same distance.
So as has been said previously, you would normally try to fly faster into a headwind, thereby lessening your exposure to it, and maybe slow down a bit in a tailwind i.e. let it help you to your destination.
HOWEVER - the above makes no correction for the relationships between fuel cost, aircraft DOC & FOC maintenance costs, fuel tankering policy, obtaining the most efficient flight level, etc, and accordingly is very general.
Hopefully of course, and if you program it correctly, your aircrafts FMC will calculate the best speed to fly at for a particular route ( though it's not aware of your ATC FltPln quoted speed, and or filed FltLvl ) - and albeit that the FMC similarly does not have your airlines financial model programmed into it w.r.t. true aircraft operating costs, i.e. it’s, not surprisingly, very fuel orientated in its calculations.
Now add a 50Kt headwind component on the sector from A to B.
Accordingly your G/S from A to B is now only 450Kt, so the 500Nm trip takes 66.7 minutes – a 6.7 minute difference from the still-air time, with an effective flight distance now of 555.6Nm (i.e. 55.6 Nm further to fly than in still air)
Of course on the way back (from B to A), with a 50Kt tailwind component, the aircraft has a G/S of 550Kt, so the trip back only takes 54.6 minutes – a 5.4 minute difference from the still-air time, with an effective flight distance of 454.6Nm (i.e. 45.4 Nm less to fly)
The point here is that, on a route between two points, with a consistent TAS and head/tail wind components, a headwind always hurts you more than the tail wind helps you because you spend less time flying along in the tailwind than you do flying into the reciprocal headwind, e.g. in the above example we’re flying into the headwind for ( 6.7 – 5.4 = ) 1.3 minutes more than the we’re exposed to it as a tailwind over the same distance.
So as has been said previously, you would normally try to fly faster into a headwind, thereby lessening your exposure to it, and maybe slow down a bit in a tailwind i.e. let it help you to your destination.
HOWEVER - the above makes no correction for the relationships between fuel cost, aircraft DOC & FOC maintenance costs, fuel tankering policy, obtaining the most efficient flight level, etc, and accordingly is very general.
Hopefully of course, and if you program it correctly, your aircrafts FMC will calculate the best speed to fly at for a particular route ( though it's not aware of your ATC FltPln quoted speed, and or filed FltLvl ) - and albeit that the FMC similarly does not have your airlines financial model programmed into it w.r.t. true aircraft operating costs, i.e. it’s, not surprisingly, very fuel orientated in its calculations.
Last edited by Devils Advocate; 11th Jul 2002 at 18:32.
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Doesnt rotation of the earth have a part in this? If you are talking about a west headwind and a long enough flight surely the rotation will increase range if the plane is configured for max range in the headwind.
Is this true?
Is this true?
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I'm sure all the above technical info is absolutely correct. However for us less technically minded it was explained to me in simpler terms. If you look at these problems from the extreme then it often becomes simpler. Imagine you have a 200kt TAS and you were flying into a 200kt headwind. Your SGR would be 0 therefore, by increasing your TAS to 205 kts you will be increasing your SGR. Obviously at higher TAS it is not as significant. On older Boeings i think the rule of thumb was not to worry about it unless the H/W was above 200kt. Now days i just stick the cost index in the computer and pull out the crossword.
