Cold weather climb performance: fact or fiction?

[Keith.Williams.]

The only set of charts to which I have regular access are those in the CAP 698 which is published by the CAA for use in JAR ATPL exams. These are reputedly based on a version of the 737, but I know not which.

The engines used in these charts are flat rated up to ISA +15 degrees C. The climb performance charts show a steady decline in climb performance at any given altitude up to the flat rating temperature limit. This is followed by an abrupt increase in performance degradation at higher temperatures. The overall implication of this being that flat rating reduces the detrimental effects of increasing temperatures, but does not entirely eliminate these effects.

If we ignore flat rating to simplify the arguments we will see that increasing temperature has two effects. These are a decrease in power available and an increase in power required. The overall result of increasing ambient temperature is therefore a reduction in excess power and hence a reduction in best possible ROC.

The principal cause of the reduction in power availabe is the reduction in thrust. In a non-flat rated engine the thrust reduction is caused by a reduction in mass flow due to reduced air density. In a flat rated engine it is caused by the reduced acceleration of the air, which is in turn caused by the increased TAS to CAS ratio.

The principal cause of the increase in power required is the increase in TAS at any given CAS. Power required is equal to drag x TAS. If we assume that drag at any given CAS remains constant, then the power required is proportional to any temperature-induced change in TAS.

The drag equation includes TAS squared, so multiplying drag by TAS to get power required, gives something that is proportional to TAS cubed. So any increase in TAS at any given CAS will cause an even bigger increase in power required.

To visualise the effects of temperature we need to sketch power required and power available curves. Power required looks like a drag curve which has been rotated in an anti-clockwise direction..A bit like a NIKE tick.

For a jet aircraft if we simplify the situation by assuming that thrust is constant at all values of TAS, the power available is a straight line starting at the origin (0,0) and moving up towards the top right hand corner of the chart. At low altitude these two lines will cross at two points. At these crossing points the power available is equal to the power required. At all higher and lower speeds the power available is less than the power required. So these two points are the minimum and maximum speeds for which sufficient power is available.

At all speeds between the two crossing points the power available is greater than the power required. This excess power can be used to provide a rate of climb. The excess power and hence ROC are proportional to the vertical distance between the two power curves. ROC will be greatest at the speed at which the vertical distance between the two lines is greatest.

If we draw a tangent from the origin to touch the under surface of the power available curve we can use this to predict the effects of changes in altitude or temperature. All subsequent power available curves are similar to the first and all curves touch the same tangent. Increasing altitude or increasing temperature cause the power required curve to slide up the tangent towards the right hand end of the chart.

The power available curve always starts at the origin because at this point it is equal to thrust multiplied by zero TAS. Its angle is dependent upon altitude and temperature. Increasing altitude and increasing temperarture both cause this curve to rotate clockwise about the origin, thereby reducing power available at any given TAS.

The overall effect of these changes is that increasing altitude or increasing temperature will increase power required, reduce power available and reduce ROC.

The use of flat rated engines reduces the rate of reduction in power available but has no effect on the rate of increase in the power required.

[mbga9pgf]

Turbofan/Turbojet thrust increases with reduced Compressor inlet temperature, as more work can be done with greater heat input into the compressed airflow. the limiting factor of course being Tmax experienced by the turbine blades (Turbine inlet Temp), which is set by the material they are made of. More thrust=Greater climb performance.

[Black Baron]

Thankyou Dorset , excellent post.
Not sure about the climb gradient questions hawk, surely the figures for distance to top of climb represent the gradient?

[bookworm]

One point I don't follow Keith:

The drag equation includes TAS squared, so multiplying drag by TAS to get power required, gives something that is proportional to TAS cubed. So any increase in TAS at any given CAS will cause an even bigger increase in power required.

Surely the drag depends on the dynamic pressure and therefore the CAS squared?

[HotDog]

Hawk,

"Do either of you two have a V final segment climb GRADIENT chart, and are you capable of providing
any figures on how much the gradient changes based on temperature? "

No, I do not have a V final segment climb gradient chart. All I can tell you from personal experience of around 20,000 hours of operating a variety of jet aircraft with RR, GE and P&W engines, mostly in the tropics, is that climb performance deteriorates with a rise in temperature.

[hawk37]

Wow, Keith, quite the post. You took a tremendous amount of time to write that, an amount of time that will only be surpassed by the number of times I re-read it. I appreciate your efforts.

What still bothers me is that I can make an argument, based on THRUST, and not power, that is still academically unclear to me.

Consider thrust changes with tas and temp as being small enough to not be a factor, perhaps as with early non bypass jet engines on a cool day.

Regardless of temperature, climbing at V min drag gives a constant amount of excess thrust. This constant amount of excess thrust gives a constant climb angle. This constant climb angle gives a higher rate of climb on cool days than very cold days.

I’m still assuming this ACADEMIC argument is 100% logical I’m also assuming that it is not what happens in the REAL world for ONE simple reason. As you pointed out, attaining constant thrust with changes in tas and temperature do not happen.

I'm also assuming that if engine manufacturers came up with engine controls that could maintain a constant thrust throughout some defined envelope, that my argument of higher rate on a warmer day, at V min drag would be correct

thanks

Edited for formatting only

[Tinstaafl]

Even if the excess thrust relationship is unchanged AND the climb rate is unchanged**, the a/c will be at a faster TAS as temp. rises. Faster TAS vs same RoC = shallower angle.

**Of course the RoC will be reduced, making the gradient worse, but for the sake of argument...


Hawk, I get the impression you're trying to use a Thrust &/or Power &/or Drag graph for a particular circumstance (WAT) and possibly with unsuitable axis (IAS/CAS?) then applying it in slightly different conditions eg different TAS? Just a thought...

[bookworm]

Even if the excess thrust relationship is unchanged AND the climb rate is unchanged**, the a/c will be at a faster TAS as temp. rises. Faster TAS vs same RoC = shallower angle.

But the climb rate wouldn't be unchanged if the excess thrust is unchanged. It would be increased. The rate of climb depends on excess power, which is excess thrust times TAS. With TAS as a factor in both power and rate of climb, when dealing with approximately constant thrust it makes sense to divide through by TAS and deal with thrust and angle of climb.

[hawk37]

Tinstaffl, I didn’t say the climb rate is unchanged. In my scenario, which is academic, the excess thrust is unchanged, which implies the climb gradient is unchanged, which implies the ROC is increased for a temp increase. And actually, I wasn’t using any graph, just a 3 step hypothesis that I thought is clear to follow, but difficult to explain. See my previous post.
Oh, I see bookworm posted a reply so yours also.
Here's the 3 steps again

1. Regardless of temperature, climbing at V min drag gives a constant amount of excess thrust.
2. This constant amount of excess thrust gives a constant climb angle.
3. This constant climb angle gives a higher rate of climb on cool days than very cold days.

Caveat: dealing with a case of near CONSTANT thrust output from the engine, eg non bypass jets.
(See Keith's earlier posts about why this is not entirely possible).I use "cool" and "very cold"
to avoid the flat rated questions.

Bookworm, your question about Keith's post was

"Surely the drag depends on the dynamic pressure and therefore the CAS squared?"

Not sure what you mean. I get dynamic pressure is .5 * V ^ 2 * p

and therefor to tas squared. Drag = thrust required, and P = T x V, so Power is proportional to tas cubed.

this help?

[Keith.Williams.]

Hawk,

I think that the fault in your argument lies in the fact that the thrust does not remain unchanged as you climb. Thrust is proportional to mass airflow multiplied by the acceleration the engine gives to that air. The acceleration is equal to exhaust velocity minus TAS. As we climb the air density and hence mass airflow at any given engine RPM reduce, causing thrust to reduce.

In addition to this, the increasing TAS to CAS ratio causes the acceleration we give the air to reduce. This again reduces thrust. So even if we keep the same CAS and hence the same drag, excess thrust and climb gradient will decrease. This process continues until we hit the absolute ceiling at which maximum thrust equals drag, and excess thrust and best climb gradient are zero.

We can visualise this effect by drawing a drag and thrust available curve on a single chart. The drag curve will be the typical bucket shape. If we simplify the situation by assuming constant thrust at all airspeeds, thrust will be represented by a horizontal straight line. (In reality it would be a bucket a bit like the drag curve, but much shallower).

At low altitude the thrust line will be some distance above the bottom of the drag curve and will cross it at two points. These are the minimum and maximum speeds for which sufficient thrust is available. Excess thrust is the vertical distance between the two lines and this will be greatest at Vmd. Excess thrust is proportional to best climb gradient so this will occur at Vmd.

If we have marked our speed range in CAS we will find that the drag curve does not change with changing altitude. But as explained above, the thrust available will decrease as altitude increases. This can be represented by repeatedly drawing further thrust lines, each lower than its predecessor. When the thrust line just touches the bottom of the drag curve we are at our absolute ceiling.

It is tempting to think that we can treat excess power and excess thrust as if they were unrelated. This is not the case. As we climb, the power required increases and the power available decreases. This causes excess power and best ROC to fall to zero at the absolute ceiling. So at the absolute ceiling we have no excess thrust, no excess power, and the best climb gradient and best ROC are zero. Worse than this, there is only one speed at which we have enough thrust and power to fly. For a jet this is speed Vmd.

If we repeat the process described above, but this time with power available and power required, we can see how excess power and ROC change. As altitude increases the power required curve keeps being repeated ever higher. At the same time the power available line rotates clockwise about the origin, becoming shallower with each increase in altitude. When the power available line just touches the power required curve we are at the absolute ceiling. We have just enough power to fly straight and level at a single airspeed. If we superimpose the power and drag curves on the same chart we will see that the single speed at which we have enough thrust to fly is also the single speed at which we have enough power. For a jet this speed is Vmd. At this altitude it should really be called Vonly, but for some strange reason it is not.


Bookworm,

Regarding your comment "Surely the drag depends on the dynamic pressure and therefore the CAS squared?"

We need to be a bit careful here. If we change the CAS at any given altitude and temperature it is true that drag is (in part at least) proportional to CAS (or more correctly EAS) squared. But it is more complicated than that, because total drag is made up of induced drag and profile drag. Induced is proportrional to one over EAS squared and profile is proportional to EAS squared. Worse still the coefficient of induced is proportional to one over the fourth power of EAS and the coefficient of profile is more or less constant at most angles of attack. The overall relationship is therefore a bit complicated.

But the situation we were considering above was one of increasing temperature. In this case if we keep the altitude constant, we see TAS increases while CAS remains constant.

Looking at the drag equation Drag = Cd 1/2 Rho V squared S, the 1/2RhoVsquared is the dynamic pressure and the V squared is TAS squared. The ASI captures dynamic pressure and gives us a CAS indication proportional to it. So if we fly at constant CAS we are flying at constant dynamic pressure. But if temperature increases, the reducing air density Rho must be matched by a corresponding increase in TAS squared, in order to keep the dynamic pressure and CAS constant. This is the cause of the increasing ratio of TAS to CAS as temperature increases.

So as temperature increases at any given CAS, the drag stays constant but the TAS increases. Power required is equal to Drag times TAS. As temperature increases at any given CAS, the drag remains constant but is multpilied by an increased TAS. Because TAS appears as TAS squared in the drag, when this is multiplied by TAS to give power required we get something that is proportional to TAS cubed. The same thing happens as altitude increases.

[hawk37]

Keith, reference your quote "I think that the fault in your argument lies in the fact
that the thrust does not remain unchanged as you climb"

I’m talking instantaneous climb gradient, instantaneous climb rate. I know that normally
thrust decreases as you climb. I’m not trying to suggest this scenario can occur at any other speed
other than at, or very close to V md. And certainly, I understand best ROC is at max excess power etc.

My scenario has been at ONE altitude, say 1000 ft msl., and only at V md. And, most importantly,
with CONSTANT THRUST AS TAS CHANGES WITH TEMPERATURE.

I’m still don’t see why my argument isn’t sound.

IE
1. Regardless of temperature, climbing at V min drag gives a constant amount of excess thrust.
2. This constant amount of excess thrust gives a constant climb angle.
3. This constant climb angle gives a higher rate of climb on cool days than very cold days

What I SUSPECT is the problem, is that the caveat that I’ve put in, that thrust be considered
constant with a change in temperature (note, I’m not also adding altitude, since I’m looking at
instantaneous gradient/rate) IS TOO FAR OFF from the real world, even in pure jets, to have the
effect I’m expecting on gradient, and ROC.

[Black Baron]

You are correct that if thrust is constant, and TAS increases through temperature then angle remains constant, and rate increases.

If you were talking rocket engined aircraft this would be usual.

To the gas turbine engine a rise in temperature causes a rise in power required to produce the same thrust;
T=mv, but kinetic energy is 1/2mv2.
A drop in air density with temperature increase, requires an increase in V to maintain thrust. But that is squared into power required, whilst the drop in mass is halved.

Then the TAS is increasing so the prop/fan/jet speed must again be increased, again requiring more power yet.

An increase in temperature causes an increase in power required, a reduction in power available, requires higher compressor speeds, brings you closer to your EGT limits.

So as far as a gas turbine goes you are comparing a reduced power climb on a cool day to a max power climb on a warm day.

[bookworm]

Keith

I think this TAS cubed thing is just a lack of clarity about what we're holding constant and what we're considering as a variable. We're probably saying the same thing in different ways.

Neglecting compressibility, and using a slight variant on your terminology:

dynamic pressure = 1/2Rho(TAS)squared = 1/2Rho_standard(CAS)squared

(where Rho_standard is a reference constant, density at ISA SL)

or TAS = d * CAS where d is the square root of the density ratio.

So

Drag = Cd * dynamic pressure * area

and

Power required = Drag * TAS

So you can write that as:

Cd * 1/2Rho_standard(CAS)squared * TAS * area

in other words proportional to (retaining density as a variable)

CAS squared * TAS

You can, if you want, write this as

Power required is proportional to CAS cubed * d

or you can write it as

Power required is proportional to TAS cubed / d squared

But for a constant CAS climb, the key is that the d comes in once not three times.

I think the complex shape of the drag curve is a bit of red herring, isn't it? Since lift depends on speed through the dynamic pressure too, the a point at constant CAS on the curves represents a particular drag-to-lift ratio, and therefore, for an aircraft of a particular weight a particular drag-to-weight ratio, regardless of density ratio. Turning that into a power required involves multiplying by the TAS or d only once.

hawk

I think Keith answered this the first time round, but let me share my understanding of his explanation by putting it another way. The thrust available depends explicitly on both temperature and TAS. Since TAS also depends on temperature for a constant CAS, there is also an indirect dependence on temperature.

If thrust is flat-rated, it means that the explicit dependence on temperature is removed. The static thrust (TAS = CAS = 0) will not depend on temperature. But for non-zero CAS, there is still that indirect temperature dependence. Thus the thrust available at climb CAS does vary with temperature even for a flat-rated engine.

[Keith.Williams.]

Bookworm,

If we hold pressure altitude and temperature constant then the relationship between CAS and TAS is also constant. Under these circumstances it is true to say that power required is proportional to CAS cubed just as much as it is proportional to TAS cubed.

But if we wish to consider real world situations where pressure altitude and temperature change, the ratio of CAS to TAS changes. Under these circumstances it is more useful to say that power required is proportional to TAS cubed.

To illustrate this let's use msl and 40000 ft in the ISA. At msl the CAS equals the TAS. At 40000 ft the CAS is about half of the TAS. If we climb from msl to 40000 ft at any given CAS the CAS cubed will be unchanged but the TAS cubed will be 8 times its original value.

The power required will in fact only have increased to twice it original value because the density will be about 1/4 of its msl value. Under these circumstances it would be reasonable to say that power required is proportional to Cd x density x TAS cubed. But it would not be at all accurate to say that power required is proportional to Cd x density x CAS cubed.

In summary if we keep pressure altitude and temperature constant, then power required is proportional to CAS cubed in exactly the same way that it is proportional to TAS cubed. But if we want to allow for changes in altitude and density we must use TAS cubed.


Hawk,

I think that I can see what you are saying, but again you are starting with a false assumption. Thrust will not remain constant as temperature increases. It will decrease, causing excess thrust and climb performance to decrease. I suspect that you are using the fact that very high exhaust speeds do not suffer much of an acceleration reduction for each knot of TAS increase. But they still suffer this to some extent. Although we might imagine that the old pure jets would benefit from this effect, they did not do so. This is partly because they did not enjoy the benefits of flat rated engines and partly because increasing temperature reduces thrust in all aircraft propulsion systems (jet or prop).

[bookworm]

The power required will in fact only have increased to twice it original value because the density will be about 1/4 of its msl value. Under these circumstances it would be reasonable to say that power required is proportional to Cd x density x TAS cubed. But it would not be at all accurate to say that power required is proportional to Cd x density x CAS cubed.

Looks like we agree on the equations, Keith. We'll just have to agree to differ as to how we describe those in words.

[hawk37]

Keith, thanks, and your quote "I think that I can see what you are saying, but again you are
starting with a false assumption. Thrust will not remain constant as temperature increases."

I was considering flat rating here. Based on all comments though, I'll assume my scenario would be
true if thrust was somehow constant with TAS. But it isn't, and the effect is significant
enough that it's effect on my academic scenario is that the reverse is true. I'm done. thanks
Bookworm too and all

[Black Baron]

Boeing 767 -300 cf6-80
brw 150 tonnes
climb to FL 370

ISA +10> clb / 18 minutes over 115 miles

ISA +10> clb 2/ (reduced climb) 21 minutes over 132 miles

ISA +15 clb / (fulll climb ) 20 minutes over 130 miles

ISA +15 clb 2 / 24 minutes over 151 miles

ISA +20 clb / 23 minutes over 155 miles

ISA +20 clb 2 / 27 minutes over 179 miles

clb2 represents a 20% thrust deduction to 10,000 feet decreasing linearly to Max climb thrust at FL 300

I think it is then fair to say that a 5'C rise in temperaure equals a 20% or greater loss of thrust available, and visa versa.

[hawk37]

Baron, I don't think this figures are any where near V min drag. Also, way out of the flat rating
range. And 20% loss in thrust available with a 5 deg C rise sound WAAAAAAY too much. Was that a published number?

For Keith, I have found a thrust vs mach graph for the Garret 731 at sea level, 59 deg F. Its a 2.79
bypass fan. The graph has thrust in lbs on the y axis, and mach on the x axis. The curve starts at
3700 lbs at zero mach, and slopes down to the right. Other values are 2800 lbs at mach .3, and 2200 lbs at mach .6.
Concerning thrust decrease with tas near V min drag, say mach .3, about 198 ktas, if I draw a
tangent, the slope of the line works out to be 3.4 lbs of thrust per knot.

So with a 10 ktas increase near V min drag, the decrease in thrust from the engine is 34 lbs.
Based on the 2800 lbs it is producing, this is a 1.2 percent decrease in thrust available. Regarding the excess thrust available, this percent would be considerably higher, of course.
Now it you had a model to calculate drag, the decrease in excess thrust due to tas increase could give you the decrease in climb gradient. I’ll leave that up to Baron to calculate

[timzsta]

Hawk37 - when Black Baron talks about a 20% reduction in thrust he is correct. What he has done in his "reduced climb" is derated his engines through the FMC. This allows us to lengthen engine life, reduce wear and tear etc etc. I dont have the figures for the 767 but I do for the aircraft I am familar with, 737-400, from doing my ATPL exams.

The 737-400 comes usually with engines that are flat rated at 23,500lbs static thrust. However we can de-rate the engines to 22k or 20k through the N1 page on the FMC. We do this when we can still safely achieve our performance requirements with the use of less thrust, in order to reduce wear and tear on engine, extend engine life and reduce noise. At London Stansted (where I dispatch) over the last few days daytime temp has not got much over 5 deg celcius. So its pretty much ISA - 10 deg which is great for performance (the fog however isnt great for flying!!!). With a long runway and light pax load we dont need 23.5k, so you can de-rate to 22k. If we went for 20k you can see that we have reduced thrust by approx 15%.

On a cold day the air is more dense so the engine produces more thrust. The wing produces more lift because the air is more dense, so the climb gradient is higher.

The other thing you need to consider is that performance charts for GRADIENT of climb are only really applicable at low altitude when you want to clear an obstacle in the take off flight patch - ie as at London City, all those big buildings nearby.

With a jet airliner what you are really interested in is getting to cruise altitude in the shortest TIME as that make the aircraft more economical. Hence you aim to fly a jet at best rate of climb speed for all but the initial part of the climb.

Remeber max angle of climb (GRADIENT) - max altitude gain over shortest distance over the ground. (Vx)
Max rate of climb - largest altitude gain for shortest TIME in the air (Vy).
On the 737 FMC you can select from the climb page max angle or max rate climb.

I think the problem, Hawk37 , is that you are trying to make a hypothetical "text book/labatory" situation fit the real world. Interesting discussion though and useful performance refresher as it was some months ago I sat (and passed) the exam.

[bookworm]

The 737-400 comes usually with engines that are flat rated at 23,500lbs static thrust. However we can de-rate the engines to 22k or 20k through the N1 page on the FMC.
....
On a cold day the air is more dense so the engine produces more thrust.

So, what does flat rated mean then?