after we take off we settle into the climb up to our cruising altitude but what sort of performance characteristics do we need to achieve on the way up there let's find out [Music] hi I'm Grant and welcome to the sixth class in the performance Series today we're going to be taking a look at the climb procedure there's various external environmental factors that will affect how steep our climb is so we're best off understanding them so we know if we'll be able to safely climb above any obstacles that are in our way the claim phase of flight starts after the takeoff has been completed and we pass the screen height basically up to the altitude we're going to cruise at but we can also climb on route or after I go around for example so it's not exclusive to just after takeoff when we're climbing the aircraft we usually break it down either into an angle of client that we need to achieve say there's a mountain off the end of the runway that we need to get over or a certain rate of climb imagine there's an aircraft closing in on us at the same level we need to be able to climb fast enough to clear it in time we're going to start off by taking a look at the angle of climb in a steady climb this is what the forces look like we have lift equal to weight times cosine Theta and we have thrust equal to drag plus wait times sine Theta we have thrust which is larger than drag and weight which is larger than lift if we didn't know the angle we could work it out if we knew all the other factors by rearranging one of the equations to look like this basically it equates to any excess thrust that we have so we can say that sine theta equals the excess thrust overweight if we have much more thrust and a lot less drag making our excess thrust larger then we would get a larger number over here as a result sorry over here as a result and a larger value for sine Theta which equates to a larger steeper angle so if we plug in some numbers let's say we have a thousand Newtons of extra thrust over our aircraft which weighs 10 000 Newtons then our sine Theta value is going to be equal to 0.1 because that's a tenth of that then if we take the inverse sine of that that means that we can find out the actual angle then we will get the actual angle equal to 5.7 degrees if we had 2 000 Newtons of extra thrust that'd be 0.2 and that equals 11.5 just to give you a flavor for how this works at small angles that are normally experienced in aviation basically the reason because it's only small angles is because there's only ever a little bit of extra thrust available there's never that much extra thrust available so this number rarely gets it rarely gets that high but basically when we're considering small angles sine Theta roughly equals the gradient of a climb the vertical change over the horizontal in the example four we get an angle of 5.7 degrees which we if we convert into a gradient using trigonometry we'll see why these two angles are similar so let's see using trigonometry that tan five point seven equals the if we had our triangle we have the angle in here we've got the vertical which is the opposite over the adjacent which is the horizontal vertical over horizontal is gradient So Tan 5.7 equals vertical over horizontal and let's say we've covered a hundred meters horizontally that means that if we rearrange we're trying to find the vertical if we do the vertical equal to tan 5.7 multiplied by 100 then we can see the vertical change if you calculate that out will be 9.98 meters which makes our gradient vertical over the horizontal or tan value would be 9.98 over 100 or we could say it's roughly 10 over 100 which is roughly well a bit of rounding our gradient is 0.1 so by using that trigonometry we can see that the value for sine Theta is basically the same as the gradient so what we can do is we can re-jig this equation and say that sine Theta is equal to the climb gradient so client gradient equals sine Theta which also equals thrustmaster drag over weight so the client gradient equals excess thrust overweight VX is one of many V speeds and this one stands for the speed that we fly to achieve the best angle of climb or the best climb gradient the best angle we know will occur where excess thrust is maximized this happens where we have the largest difference between thrust and drag so we look at the total drag curve with the thrust available overlaid we can actually figure out where this happens for a turbojet aircraft it occurs here at the bottom of the drag curve because this is where we have the largest gap the largest amount of excess thrust and this is our speed for minimum drag this is vmd so in a jet aircraft vmd equals the same as VX our speed for best client in a propeller driven aircraft it's slightly different we have vmd over here but that's not actually the largest gap the largest gap occurs slower than vmd and that's actually fairly close to the speed for minimum power VMP and the rvx speed again just without the gaps the biggest nice and easy so there's quite a few things that will influence the angle of our climb or the climb gradient but the thing that's key is excess thrust so anything which increases drag will be bad anything which reduces thrust will be bad and anything that increases thrust and reduces drag will be good for that angle allow us to apply a steeper angle basically Mass we'll start with mass mass increasing means in our equation for climb gradient we're dividing by a larger number which makes the climb gradient smaller simple mathematics would get a lower climb angle aircraft configuration basically flaps and the landing gear will mean more drag that means a lower excess amount of thrust that excess thrust if you think about it in terms of lines you would have uh uh larger my drag that line for the total drag curve would move up smaller Gap again Lower climb gradient as a result altitude increasing as we know from class 4 on thrust means we have lower air density and less thrust is produced as a result that means there's less excess rust and that means that we can't climb um as steep as we'd like so as thrust goes down with increasing altitude our climb gradient would also go down temperature increasing has a very similar effect to altitude as the altitude increase sorry as the temperature increases that means we're in less tense air which produces less thrust output so as the temperature goes up less excess thrust we can't climb steep we have a smaller climb gradient on multi-engine aircraft if one of our engines fail then we lose a large amount of thrust therefore we lose a large amount of excess thrust and if you combine this with the increase in drag caused by maybe a wind Milling propeller or a deflected Rudder being used to keep us straight then we get a large reduction in excess thrust and we get very reduced climb performance in terms of the angle wind will not affect the climb angle but will reflect something called the flight path angle the flight path angle is the aircraft position relative to the ground and the climb angle is the aircraft's position relative to the air mass not the ground which does sound a bit confusing but I like to think of it as a train just bear with me for a second hopefully it'll make sense stay where say we are in a train station not moving that's this first example here then if I walked up a slope ladder resting at 45 degrees against the front of the train Carriage carrying a toy airplane the toy airplanes climb angle would be 45 degrees and the overall position relative to the ground the flight path angle would also be 45 degrees it's climbed there the ladder with me um in the stationary train so then we climbed down the ladder and the train starts to move it starts to reverse a little bit we climb up the ladder again with a climb angle of 45 degrees we're going straight up the ladder but the position of the toy airplane and US relative to the train station or relative to the ground has moved because as a whole the train has been moving backwards so the angle would be a lot steeper in this case it's actually backwards just the way I've drawn it then the train or the air mass starts to move forward imagine the trains being pushed by a Tailwind for example we climb down and then climb up again the 45 degree ladder with the toy plane The Climb angle relative to the air mass or the train Carriage remains 45 degrees but relative to the ground it's going to be a lot shallower because the whole thing has moved across it's pushed us along um relative to the ground so whilst The Climb angle might be 45 degrees the flight path angle will be a lot shallower so the claim angle isn't affected by wind it's only the flight path angle in cases where there's no wind such as the starting condition the flight path angle and the flight angle will be the same but when there's wind to consider the flight path angle and the climb angle will be different even I'm getting confused seeing it but yeah hopefully that train analogy works the rate of climb or the rate of vertical change will be larger for larger angles of climb and at low angles that we normally climb at the relationship is fairly linear if we double the angle we would double the rate of climb also if we fly at a faster speed up the slope of the climb then we would increase the rate of climb we're covering more vertical distance per second for example that basically means that the rate of climb depends on both the angle and the true air speed that we fly up the slope so if we take our equation that we know for angle of climb and multiply it by the true air speed we can extend it out and see some interesting things so that's the midpoint and then we multiply the thrust by the trigger speed and the drag by the trigger speed divide that all by the weight and what we can do is we can use our knowledge of thrust and power to say that the thrust times the true air speed is the power available and the drag times the trigger speed is the power required so what's the difference between power required and power available well that would be Excess power so the rate of climb equals Excess power over weight very similar to climb but we're doing the angle over time and power is essentially um thrust over time so that's the way would be easy to remember it VY is another V speed and this one stands for the speed for the best rate of climb the best rate we know will occur where Excess power is maximized but we have the largest difference between power available and power required so if we look at a power available versus Power required graph for a jet we can speak see that the speed for v y occurs around about here that'd be where the largest gap is between the two lines and that is both faster than vmd which equals VX in the case of a jet and also the speed for minimum power when we look at the graph for a propeller driven aircraft we can see that the largest difference between the power available and the power required occurs in this region here in between these two lines it's actually usually a bit closer weighted to vmd but it will be faster always than VMP and VX as well so just like with the angle there's various things that influence the rate of climb various environmental factors but the key thing is Excess power for angle it's excess thrust for rate it's Excess power so simply put anything that reduces thrust will be bad anything that increases drag will be bad just as it was for angle so Mass increasing will basically mean that we're dividing by a larger number we got a lower rate of climb nice and easy to understand but if we think about it on the power available versus Power required graph we still have the same amount of power available thrust times to your speed but because we're heavier we need more lift this causes an increase in induced drag and therefore our total drag curve would move up and to the right if we multiply all the points on that new Total drag curve by the Trier speed we will get a new power required curve which also moves up and to the right I've drawn these lines a bit weird but basically it means that we have a smaller gap between the heavy aircraft and the power available when compared to the light aircraft and the power available graph and it also means because everything is heavier and because the lines have moved up and to the right it means that our speeds would be faster so all speeds would increase so VY is going to be faster for the heavier aircraft when compared to this one it's the same with VX and all the speeds in general if you're heavier you need more lifts you need to fly faster no matter what you're doing aircraft configuration flaps and gear will increase Dragon aircraft if we have more drag on the aircraft if we multiply it by the true air speed we get a larger amount of power required Shifting the overall line upwards therefore reducing the gap between power available and power required and we get a lower rate of client as a result as we increase in altitude we know that the thrust output of the engines is reduced due to that reducing density meaning that as we multiply it by the true air speed we get a much shallower line for power available in a jet and this means a lower smaller gap between the and power available and power required and as we get higher and higher the power available lighting will continue to shallow off and when we get to the point where the two lines cross over something like that then we now require more thrust than we have available and we've hit the theoretical maximum altitude of the aircraft which is known as the absolute ceiling or in other words it's where the rate of climb is zero feet per minute we use feet per minute when we're talking about rate of climates the units we also um have an altitude which we call the service ceiling this is slightly lower than the absolute ceiling basically to give a bit of a buffer uh in uh jet this is when it's 500 feet per minute that's the maximum amount of climb that we can achieve when we're in a propeller it's a hundred feet per minute temperature increasing has the same effect as altitude increasing less dense air less stress to help us climb at a fast rate basically imagine the line shallowing off just as it has for altitude increasing on multi-engine aircraft if one of our engines fail then we lose a large amount of thrust or um Power available and we also increase the drag due to the windmill and propeller or the deflected Rudder and we get a large increase in the power required as a result that means less XX Excess power overall and a lower rate of climb as a result wind doesn't affect the rate of climb basically the rate of climb is the vertical change that we're concerned about and wind only really acts in the horizontal plane so it's not affecting the vertical plane simple as that so in terms of claim speeds we have VX for angle and v y for rate simple as that well not really hence all the other stuff on the page but if we fly using an indicator speed which is what we normally do in front of us using the dials and displays that actually show us our indicator speed then if we climb the true air speed will increase above the indicator speed if we climb with a constant indicator speed sorry the true air speed will start to increase above the indicator speed as we get higher and higher basically because of this equation if we are flying with a constant indicate air speed this stops reducing then it means that the Tas has to rise as a result to keep our indicator speed constant this will result in the Mach number also increasing so as the task is increasing Mach number is going to go up as well and to be climb it gets colder local speed design goes down leading to the Mach number and increasing quite a lot this could lead to over speeding the aircraft above structural limits which are normally given as a Mach number this MMO Mach number maximum operating in Jets at least so what we do if we're going to climb at the constant indicator speed we'd run into this problem so what we do is we climb using both a combination of indicated AirSpeed calibrator speed or equivalent AirSpeed whatever you want to call it and Mach number and we do a change over at a certain change over altitude where we stop climbing at a constant indicate AirSpeed and then start climbing at a constant Mac speed you can see I've made a little mistake here obviously the lines line up with the equivalent speeds that they're supposed to be lined up with so by climbing up this constant Mach number we keep ourselves safe from this maximum uh operating Mach number and you might see some questions in the exam asking you about the different climbing speeds and certain types of speed the easy way to remember it is basically with these three fingers on your left hand I make sense for me because I'm right-handed so the spear hand is the one I've got look inside your hand the middle finger is the Mach number some numbers that indicate air speed and the index finger is the true air speed so you've got it you know it and Mach number for the middle whichever is constant goes straight up and you can see what happens to others so for example if we climb a constant indicate air speed the Taz and the Mach number both go up then we switch over to the Mach number we can see the Taz and the indicated AirSpeed start to reduce simple to summarize climbing then climb gradient is given by the equation excess thrust overweight and we got there by rearranging the forces in a climb so that we had sine theta equals thrust minus drag over weight and if we were to find the angle we would take the the inverse sine of the result on this side but for small angles sine Theta roughly equals tan Theta and tantia is the same as the gradient the vertical change over the horizontal change so we can say that tan Theta or the gradient is equal to excess thrust overweight to maximize the gradient and the angle we want to maximize this value over here which means either maximizing excess thrust or minimizing weight so if we have more mass we would have a smaller angle if we have less Mass we have a larger angle if we have more thrust and or less thrust that would mean less excess thrust things that cause less thrust would be an engine failing for example they usually get a lot less thrust increasing altitude causes a reduction in thrust and increasing temperature also causes a reduction in thrust causing a reduction in excess thrust which means that climb angle is smaller client gradient is smaller anything which increases drag will also reduce the excess thrust think about flaps think about gear think about things like contamination on the aircraft like ice Frost that kind of stuff that'll increase drag which will reduce excess thrust reduce excess thrust and reduce the angle so whereas climb angle is all about excess thrust rate of climb is all about Excess power and we need to maximize Excess power if we want to maximize our rate of climb it's basically the same things as we do for angle we want more thrust less drag and less weight to achieve a faster rate of climb and we do this by reducing the weight that's a simple one and if we are at low altitudes with uh more thrust available then we'd climb faster and if we've got low drag then again we'd climb faster because we have more Excess power available um things like wind don't affect the rate of climb because rate of climb is to do with vertical change not horizontal change where wind is concerned and with climb actually never talked about wind but think of that train analogy the wind doesn't actually change the angle um of the climb just the flight path angle the angle relative to the ground so as we climb if we climb with a constant indicated air speed the trigger speed and the Mach number increase up to the limiting speeds of the aircraft so we have to climb with a combination of both indicated air speed and Mach number and basically we do that the change over altitude we switch from indicator speed to Mach number to make sure the trigger speed and the indicator speeds remain in safe sort of ranges and the Mach number more importantly stays in the safe range