Numerous times I’ve heard mathematical equations be called “beautiful”. I always took issue with this, as this praise was usually directed at some hideous tangle of symbols intent on making me question my intelligence and confusing the hell out of me. Even if I did finally figure the monstrosity out (and I say “if” deliberately), I’d still struggle to see the appealing side of the headache it caused me. It may sound surprising, seeing as I consider myself something of an engineer (albeit an incomplete one), that I don’t much care for masses of anal mathematics, but have always been a fan of having concepts verbally explained. I find it a lot easier to fully understand something that way; the quantifying maths can come after. If you can’t explain it to me using words and pictures, it’s probably a purely theoretical concept and therefore I don’t like it, and begin to switch off.
Today, however, I found myself longing for an equation. For a mess of numbers, symbols, and single letters. Not to help me calculate anything, just to help me read something. Says a lot about the wording of British Standard 5167-2:2003 regarding the Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full – part 2: Orifice plates.
Guess the clue was in the name with that one.
Virgin Media’s internet service, in the four months since I moved into this house, has been utterly, utterly shite. I stayed with them because for the past two years they were mostly fine, and offered me an upgrade to 30Mb/s from 20Mb/s for a few quid less each month, which seemed ok. But since then, it’s been nothing but bloody hassle.
They have, by their own admission, oversubscribed the area; so they cannot physically supply everyone with anything like the broadband speed they are paying for. I’ve run speed tests most days for the past few months; only once has it exceeded 20Mb. It’s reached the point where a peak of about 12Mb is genuinely exciting, because it means a video might actually start to buffer. Except that 9 times out of 10 the speed has dropped again by the time I’ve run another test. And I’m talking 1Mb/s here, if that. It’s ridiculous. Lowest I’ve seen is a reported 0.03Mb. Thats a thousandth of what I should expect. Thing is, every time you ring up to complain, they say that they’re aware of the problem, they’re working on it at the moment and there’s dick all anyone can do about it (this is after you’ve been on hold for 25 minutes, given all your details, found you’ve spoken to the wrong person, been put on hold for another 10 minutes, then had to give all your details again to someone else). Oh, and you’re in a contract so if you really want to leave, that’ll be 180 quid.
It’s got to the point where I don’t bother following YouTube links posted on Facebook, or even visit YouTube at all, because I know I won’t be able to watch more than the first four seconds. And that’s usually a third of a fucking advert these days. iPlayer? Forget it. 332Mb of a 1.7Gb download has taken five hours so far today, and the estimated time remaining peaked at 22 days and 7 hours before Safari just gave up.
3 weeks to download 1.4Gb?
And they can shove their “peak times” speech; I know for a fact that it’s barely any better at 4 in the morning or 1 in the afternoon.
So there you have it. Virgin Media are denying me the right to watch Postman Pat (scouse overdub) in my own home. What am I actually paying for?
Right, so we have a projected air velocity for full-throttle running, but more important than this is the actual quantity of air moving through the engine. This governs the potential fuel flow, the size of the combustor itself and thus the power output of the engine.
Mass flow rate is calculated using a simple formula:
Mass flow = fluid density * area * fluid velocity
ṁ = ρ*A*V
As the area is known and the velocity has already been calculated, the density of the air is all that remains to be found. However, those with even the most basic grasp of physics will know that density (of a gas at least) is a parameter which is variable. It is greatly affected by two factors in particular which are relevant to this situation: pressure and temperature. An increase in pressure (predictably) causes an increase in density, whilst an increase in temperature causes density to decrease. In an ideal gas:
ρ = MP/RT where M = molar mass, P = pressure, R = gas const. and T = Kelvin temp.
This shows that both of these relationships are linear; doubling pressure alone will double the gas density, whilst doubling temperature alone will halve it. It’s important to remember here that temperature is always in Kelvin, not Celsius; it’s both easy and fatal to make this mistake!
Thanks to these linear relationships, and the fact that both pressure and temperature can be measured fairly easily, we can use the ratios of input to output values for pressure and temperature to calculate the ratio of input to output density:
ρout /ρin = Tin/Tout * CPR where CPR stands for Compressor Pressure Ratio, i.e. Pout /Pin
note that the temperature ratio expression is in reverse order, to account for the fact that it has an inverse relationship with density.
I’m aiming for a maximum CPR of 2.0 for this engine, for no real reason other than that it’s on the safe side! This is still a boost pressure of about 14.6 psi. Working with this number will give me the maximum values I’ll need in order to design the rest of the engine too.
This equation still leaves us with an unknown, however. For these preliminary equations I’m going to use a rough value of air pressure at sea level (which I am pretty much at) of 101 kPa, an ambient room temperature value of 293K (20°C) and thus an ambient air density of 1.205 kgm^-3. Aiming for a CPR of 2 gives an outlet pressure of 202 kPa, but the outlet temperature is hugely dependent on the turbocharger itself, or rather its efficiency. Assuming it were 100% efficient, change in air temperature would be due to pressure increase alone and would follow the relationship:
Tout /Tin = CPR^(γ-1/γ) (see footnote  for an explanation of γ)
and thus, for an inlet temp. of 293K:
Tout = 293K * 2.0^(0.4/1.4) = 357.171K
so ΔT = 64.171K
As stated, this would be the temperature increase if the compressor was 100% efficient, but in the real world nothing ever is. A bit of research indicates that a sensible estimate would be about 70% efficiency, so I’ll factor that in. In this case, poorer efficiency means that the air temperature will increase more than it theoretically should as it is being compressed, so
Actual ΔT = 64.171 * (1/0.7) = 91.673K
This gives us an outlet temperature Tout of 384.673K (or 111.67°C)
(see footnote  for the relevance of this to a car engine)
Now this value is known density can be calculated:
ρout = ρin * CPR * Tin/Tout
= 1.205 * 2.0 * (293/384.673)
therefore ρout = 1.8357 kgm^-3
This is the value we can now plug into the mass flow equation to find out the mass of air that’ll be flowing into the combustor per second:
ṁ = ρ*A*V
ṁ = 1.8357*(3.46*10^-4)*151.287
therefore ṁ = 0.0962 kg/s
So finally we’ve got an estimate for what the sucker is capable of at the intended full throttle. 0.1kg/s of air flowing through at about 111°C and 151m/s (or just under 340mph). Now I have an idea of what the combustor is going to be receiving, I can get to work designing and modelling it.
 γ (Greek letter gamma) is a dimensionless ratio between the specific heat capacities of a fluid, Cp and Cv. Cp is the s.h.c at constant pressure, Cv is the s.h.c at constant volume. γ = Cp/Cv where, for dry air at 300K, Cp = 1.0049 KJ/KgK and Cv = 0.7178 KJ/KgK. Therefore a value of γ = 1.4 was used in the above calculations
 as mentioned, hotter air is less dense air, so for a given volume in the cylinders of a car’s engine the mass of air is lower, meaning less fuel can be mixed & burned and thus a lower power output. This is undesirable, so often an intercooler is employed to cool the air after it’s been compressed (and thus increase its density again). The flip side is that in doing so they restrict the airflow, and so have to be carefully designed in order to balance cooling with pressure loss.
As mentioned, the turbocharger is a good starting point for this project because it comprises two of the key components of a gas turbine engine. However from this point onwards, it’s going to be treated more or less as two separate components, the compressor and the turbine. The performance and parameters of each need to be assessed individually.
The performance of the compressor has huge influence on the design on the combustor, as it determines the amount of air available to it. This governs pretty much every aspect of its operation and thus design; required diameter, length, flame tube design, fuel flow, ignition requirements, everything. Thus, working out a value for the maximum air flow to the combustor was vital if the project was to go anywhere.
As also mentioned, the specs for the KP31 turbocharger proved extremely hard to come by, so to compensate and give myself some numbers to start working with, I needed to approach from another angle with some maths (prior to physical testing). I turned to the engine the turbo was designed to work with, as this at least had some published figures to use:
- 3 cylinders
- 799cc total capacity
- limited to 4500 rpm (as it’s a diesel)
One important figure I couldn’t find for the engine was its volumetric efficiency. This is a measure of how much of the available cylinder capacity is actually filled with air when the engine is running, and is expressed as a ratio. A V.E. of 1 means that the cylinders are filled 100%, e.g. a cylinder with a volume of 200cc receives 200cc of air. For naturally aspirated engines (those without a turbo or supercharger), a V.E. of 1.0 is extremely difficult to achieve, and most modern engines operate in the region of 0.85 – 0.9 (85% – 90%). So a 200cc cylinder will only receive 170-180cc of air each cycle.
Engines with forced induction (a turbo or supercharger), however, can operate at V.E.’s greater than 1. Because the air is compressed before entering the engine, a 200cc cylinder can receive more than 200cc of air. At a V.E. of 2.0, for example, it would contain 400cc of air, compressed to fit inside its 200cc volume. This means it can burn twice as much fuel, making more power without making the engine any bigger.
By how much a turbocharger increases an engine’s V.E. varies from turbo to turbo and with engine speed, so finding a value was difficult. V.E.s for a turbo diesel at full throttle seem to range from around 1.5 – 3.0, which is a massive difference. For my engine, I decided to work with a volumetric efficiency of 1.75.
So what to do with these numbers? Well, if I know how much air the engine can hold, its volumetric efficiency (how much more air it gets fed), and the number of times it gets filled up over a given period of time, I can work out how much air is flowing into it over said period of time. It’s capped at 4500 rpm, so I’ll find out how much air it gets in 1 minute at this speed:
air flow rate = (displacement * volumetric efficiency * engine speed)/2
note I’ve divided the value by 2 here; this is to account for the fact that the engine is a four-stroke, and as such only draws in air once every 2 full rotations.
Q = (V * V.E. * RPM)/2
Q = ((7.99*10^-4) * 1.75 * 4500)/2 I’ve converted the capacity into m^3 here, S.I. units of volume
therefore Q = 3.14606 cubic metres of air per minute, which in S.I. units is:
Q = 0.0524 m^3/s
So this, in theory, is the designed maximum volume of air to flow through the compressor outlet into the engine. This, then, will be the maximum airflow I’ll design my engine to operate at, as I know it’ll be within safe limits.
But the volume of air flowing from the compressor is only part of the overall equation; the mass of air it’s putting out is of much greater importance, as is the velocity of that air.
The air velocity at the compressor outlet Vout can be calculated easily using the volume flow rate equation and the measured diameter of the outlet pipe (21mm):
Q = A * V therefore V = Q/A where A = area of outlet = πr^2
V = 0.0524 / (π * 0.0105^2)
Vout = 151.287m/s
So now I have a value for the velocity of the air as it leaves the compressor. This can be used to help calculate the air mass flow, but to do this the density of the air must also be known. I’ll obtain a value for this next.
This, believe it or not, is a turbocharger. Just about the smallest turbocharger I’ve ever seen, yes, but a turbocharger nonetheless and, most importantly, one in which the bearings aren’t buggered. There were a few other turbos lying about at uni, two of which were enormous truck turbos at least four times the size of this one, but on each of them there was a worrying amount of play in the bearings. By this I’m referring to the bearings supporting the rotating shaft, which connects the turbine to the compressor wheel. Just like with, say, a bicycle wheel, any looseness or wiggle of these wheels is bad news. In this case, however, we’re talking rotational speeds well over 100,000rpm, so any play at all would be absolutely catastrophic. Subsequently (and unfortunately) the big ones were useless.
Still, one must make do with what one can get. Exactly what it was I had I didn’t know, so had to do some research using the numbers stamped into the nameplate. Found out fairly quickly why it’s so small; this is a Borg Warner KP31, the kind found bolted to the 40bhp, 799cc monster lurking beneath the bonnet of a diesel Smart car. Yeesh.
Details beyond this have proven very difficult to obtain. Exact dimensions, operating parameters, compressor maps (a graph detailing the performance of the compressor section), anything of real use is extremely sparse on t’internet. The one important stat I do know is perhaps the only intimidating one this turbo possesses – this thing is designed to run at speeds up to 270,000 rpm. 270,000 rpm. The wheels inside this turbo will be spinning round 4,500 times every second. That’s more revolutions in one second than your car’s engine would ordinarily complete in a minute (assuming you were driving it sensibly). This puts rather a large burden on the lubrication system!
The top image shows clearly the different sections of the turbocharger. The silver section on the left is the compressor; the inlet is the large round opening facing to the left, and the outlet is the tube pointing downwards into the desk. Air rushes out of this tube and into the engine. The compressor wheel is located inside the silver housing, and can be seen in the image below. The black pipe sticking out of the housing is a connector for a pressure gauge.
The turbine section is the thicker, brownish-grey section behind the compressor. Its inlet is the square hole with the triangular flange (at the bottom of the image below). Hot, high-energy exhaust gases flow from the engine into this housing and are directed to the turbine wheel, which is inside. After they have passed over the turbine they flow out through the large opening at the back of the turbo (on the far right in the image below). In a car the gas is expelled through the exhaust pipe as normal; in the case of my jet engine, this is where the thrust comes from.
The hole with the diamond-shaped flange (centre of the image above, between compressor and turbine) is the oil outlet. The shaft assembly connecting the compressor and turbine wheels is just visible inside the hole. This is the part that needs serious lubrication when the turbo is spinning. In most turbos the oil is kept inside the casing at high pressure, and only drops out of this outlet slowly due to gravity. In the case of this one, however, with its ridiculous operating speeds and tiny size, the oil is constantly circulated through the turbo at high pressure. This means that the oil provides cooling as well as lubrication, as there isn’t enough space to include a water circulation cooling system. It also makes the job of designing the oil system that much more complicated. The inlet for the oil is clearly visible in the first image, the round hole immediately aft of the silver compressor housing.
Bear in mind that the oil running through this thing needs to circulate at pressures up to 5 bar (5 times atmospheric pressure, around 70 psi).
Here’s a view in through the turbine outlet. The bladed wheel is the turbine. The disc above & to the left of the turbine wheel is the wastegate. This is a bypass system for the exhaust gases. As mentioned, the turbine and compressor wheels are linked, so the faster the turbine spins the faster the compressor spins. But the faster the compressor spins the more air it will suck in, the more fuel the engine will burn, and thus the more exhaust will flow into the turbine, making it spin faster. Spot the problem? There’s a real risk with these things that they can literally spin out of control, getting faster & faster until something goes bang, so some form of control has to be implemented. The wastegate achieves this by opening when the pressure gets too high, letting the exhaust gases escape without passing through the turbine, allowing it to slow down to an acceptable speed again. The actuator for the gate can be seen to the bottom left of the exhaust flange in the image above, and more clearly on the mid-right of the first image.
To give you an idea of scale, the turbine wheel you can see in the image above has a diameter of about 25mm. The whole outlet flange (big silver ring) has an outer diameter of 75mm. On the compressor side, the inlet pipe has a diameter of 30mm, and the compressor wheel inducer (the bit you can see in the 2nd image) has a diameter of just 22mm. To put that into perspective, the compressor wheel on the turbo one might put on your average 2-3 litre engine has an inducer diameter in the region of 40 – 50mm, so you can see that we’re working on a miniature scale here.
A smaller turbo means a smaller amount of air available to my engine and thus less power, and though this may seem disappointing at first, my sensible side has taken over and made me appreciate (after a couple of days’ moping, admittedly) that for a first attempt at a jet engine, the lower the power output and fuel flow, the better. Or rather, the safer.
This is a project I’ve wanted to undertake for a couple of years now, but never had the time, money, space or equipment to take on. Quite a few people have done it before, and there’re plenty of YouTube videos proving that it can be done. Enter third year university project – now I have an excuse and the wherewithal to finally give it a try.
So what is it I’m attempting? The plan is to make a small gas turbine, or jet engine, using a turbocharger. It’s not going to power a plane, or anything else for that matter I suspect, but it will still be my own jet engine, and being an engineering student the idea of that appeals to me rather a lot. Why do it? If I’m honest; because I can. And because it’ll be cool. I’ll need to come up with a real use for the thing for the purpose of my project report, but just to get the engine up & running and producing some appreciable power will be enough for me personally.
It may seem a bit random to pull bits off a car engine in an attempt to make an aeroplane engine, but the reason for starting here is simple: a turbocharger is, in essence at least, half a jet engine in a small, convenient package. While it may not look anything like the enormous pods hanging beneath the wings of an airliner, a jet engine made from a turbo is largely identical in operating principle. First I’ll briefly explain how your ‘typical’ aircraft jet engine works (in greatly simplified and generic terms, fear not). Though they vary massively in design, all jet engines* follow the same core pattern, shown very clearly here:
*I use the phrase “jet engine” loosely, referring only to turbojets. I do this merely for sake of simplicity; other types of jet engine exist, but I’ve not set out to discuss them here.
image courtesy Wikipedia.org
This is a schematic of a turbojet engine. These vary slightly from the engines on an airliner (turbofans), but as I said the core principles are the same. I won’t go into the different types of gas turbine here. The four main processes taking place are the same as those in the engine in a car: Induction, Compression, Combustion and Exhaust (suck, squeeze, bang and blow). The difference here is that unlike in the cylinder of a car engine, these four processes all take place continuously, in a linear fashion. Air is sucked in through the front of the engine and flows through the rotating compressors. These, as the name suggests, compress the air, squeezing it, heating it and speeding it up. The number of compressor wheels and amount by which they compress the air varies from engine to engine. From the compression section the air moves on the the combustors. This is where fuel is added and burned, giving the air large amounts of energy and providing the thrust required to move the plane. The very hot, high speed air shoots out of the combustors towards the exhaust nozzle, but in doing so passes through the turbines. These act as compressors in reverse, taking some of the energy out of the airflow as it expands. This energy is then used to drive the compressors via a connecting shaft, allowing the whole process to sustain itself. After driving the turbines the hot air is free to escape via the nozzle, which may be shaped in order to maximise the thrust produced.
So that, in very very basic terms, is how a jet engine works. I said that the turbocharger was half of the jet engine rolled into one; that’s true in as much as it provides two of the four main components highlighted above. I’ll give a brief explanation of the turbo as well, simply because I’ll be referring to it a lot more than I’ll be referring to a conventional turbojet and it’s operation is key to the whole project.
Mention the word ‘turbo’ to the uninitiated and they’ll probably conjure up images of fast cars, racing, that sort of thing. While this is rather simplistic, it has to be said that there’s a basis in fact there. A turbocharger’s job when attached to an internal combustion engine is to supply said engine with more air (and thus oxygen) than it would ordinarily draw in. While cars “run on” petrol or diesel, they are in fact combusting nearly 15 times as much air as they are fuel. As you’d expect, the more fuel an engine can burn in each revolution the more power it can produce, so the more air it can get the better.
Rather than simply let the engine suck in air naturally, a turbocharger draws in air, compresses it (making it more dense), then forces it into the engine, thus providing a greater mass of air for the same volume in the cylinders. It does this using a spinning compressor wheel and a specially shaped casing, called a volute. Remember that more air means more fuel can be added, more fuel means a bigger bang, and a bigger bang means more power. How does the turbo forcefully suck air in and compress it? Using a turbine driven by the flow of hot exhaust gases from the engine after combustion of course. The waste gases from the engine are being used to feed it more air than it would usually get. So by its very nature, the turbocharger is supplying me with a compressor and a turbine, connected by a shaft. And seeing as those are the bits that are going to be spinning at high speed, it makes sense to let someone else manufacture them for me.
So this just leaves me with the task of building a combustor and a nozzle, right? Wrong. It doesn’t need a nozzle to work, so I may not make one.
What it does need, however, is the following:
- a combustor
- pipes to connect the combustor to the turbo
- a fuel supply & control system
- a lubrication supply & control system
- an ignition system
- a source of power
All of the above will need to be designed from scratch, even though they’ll incorporate some pre-made components I can get hold of. The specification and design of each depends on parts I can get, the performance of those parts, and the performance/geometry of all the other parts of the engine, as well as the function they’re intended to serve. There’s also a fair amount of maths involved, albeit pretty simple. The processes and problems faced when designing each I intend to post up here as I come to them, along with the solutions I (hopefully) come up with.
To date I’ve already done a fair amount of the research and maths needed, however I’m going to wait until I have some pictures of the fruit it bore (or at least until it’s more immediately relevant) before I post it up.
I’m going to say now that unfortunately, due to financial and time constraints, this cannot be a compact, standalone unit, of the kind one would mount on a bike or kart and take out for a test drive. It will need to be plugged in to various supplies whilst in operation, and though I’ll make it as compact as I can, by nature of the parts employed it won’t be exactly portable.
Still, everything I learn over the course of this project would help enormously if I fancied giving it another go later on…
Riding back from work last night, moved off from a set of traffic lights when some bloke called out of the car next to me: “Well done mate, you’re the first cyclist to know what a red light means! Cheers!”
I appreciated this, because though I harbour an even deeper disdain for both motorists and pedestrians than most cyclists, I’m one of the apparent few to abide by the rules of the road. I don’t understand why most people think it’s ok to breeze (albeit slowly) through red lights, ignore zebra crossings, dart about without looking or signalling, ride about at night without any lights, do all sorts of stupid and dangerous things, and either not give a crap or actually become abusive when a motorist calls them up on it. How are cyclists as a whole ever going to be more respected as road users if they’re always seen to abuse the fact that they don’t have number plates? If you use the road the same way on a bike as you would in a car (within reason obviously), then others will treat you the same as they would another car.
I wish more cyclists would just be more sensible and respectful on the road. Not just because it’s safer, but because then when some arsehole nearly runs you over anyway, you can abuse them as fully as they bloody well deserve – and they can’t say a goddamn thing.