 Good morning in the last class we have stopped at point wherein we were introducing ourselves to the turbojet engines right we said it was developed during the initial phase of the second world war let us look at how a turbojet engine is now this is the sketch of a turbojet engine now this is the compressor this is the combustor this is the turbine this is the afterburner portion this is the nozzle and here you have the intake or diffuser section now if you remember in the last class I had said that while discussing about piston engine propeller combination as we go higher in altitude we needed to do something known as turbo supercharging and therefore we had a compressor connected to a turbine and the exit of the compressor was connected to the piston engine I said the gas turbine engine or the turbojet engine is a evolution from the from this concept to what is seen here here we see a compressor and in between a combustor instead of going into a piston engine and then there is a turbine now the turbine develops enough power to run the compressor and this is the afterburner portion and this is the nozzle now let us look at how pressure and temperature vary across the length of the turbojet engine okay now let us first look at pressure this is the diffuser section in the diffuser what happens is the incoming flow is it has a high velocity its velocities are reduced and therefore you gain in terms of pressure the kinetic energy is converted here and you get higher pressure at this point now this is then further compressed in the compressor and you get much higher pressure here in the combustor it is a constant pressure process again you have the turbine where the pressure drops and then in the afterburner which is nothing but a second combustor pressure is again constant and in the nozzle the pressure drops let us look at how the line goes this is how the pressure varies as a function of the length of the turbojet engine now if we look at what happens to temperature initially in the diffuser portion temperature again slowly increases because it is undergoing compression and in the compressor again the temperature increases but the temperature increase is not similar to the pressure rise pressure rise is much much more and in the combustor you are adding heat you are burning fuel and the fuel plus air is combusted here and therefore the energy is being supplied chemical energy is being converted into heat so the temperature increases tremendously here and then in the turbine again temperatures fall because there is an expansion process and in the afterburner the temperatures let us now look at a case wherein we have not switched on the afterburner so the temperature remains constant and again it expands in the nozzle okay now we will discuss afterburner a little later in the class okay this is how pressure varies this is how temperature varies notice that pressure almost comes back to the ambient pressure at the end of it whereas temperature are still very high at the nozzle exhaust okay now if we were to convert this information into what is known as a TS diagram it looks something similar to what you might already be aware of a Brighton cycle gas turbine engines or turbojet engines follow Brighton cycle if on a TS diagram firstly let us look at ideal processes now I have numbered them 1 to 6 I will do the same here to this is one end of intake or diffuser section is 2 2 to 3 is compressor 3 to 4 is combustor 4 to 5 is turbine 5 I will call this point also 5 because we have not switched on the afterburner so it is a constant pressure constant temperature process right now and lastly this point as 6 that is the nozzle exit as 6 so if you look at this diagram here 1 to 2 is compression in the diffuser and 2 to 3 is compression in the compressor 3 to 4 is temperature rise in the combustor and 4 to 5 is process through the turbine and 5 to 6 to the nozzle this is an ideal cycle for a turbojet engine now what happens if we look at an actual cycle what we have assumed here is isentropic compression and and expansion constant pressure heat addition and rejection this is what constitutes a Brighton cycle now if you see here the x axis is entropy and you see that entropy is constant during the combustion compression process and entropy is also constant during the expansion process this is a constant pressure process and this is also a constant pressure process a notice I have wantonly done this that the slope of this line is much more than the slope of this line this is because if you look at a TS diagram pressure increases much more steep P at higher values than at lower values and therefore the slope of this is much smaller and this is a dotted line because it does not come back to this position now as I said we have assumed constant pressure heat addition and heat rejection and isentropic processes actual processes what kind of processes will be they be will they be compression adiabatic or there is a difference between an isentropic and an adiabatic process isentropic process is reversible adiabatic and adiabatic process is where there is no heat transfer so actual processes are without any heat transfer so it is typically will be a actual cycle will be will have non isentropic processes okay so you will have entropy increasing always so you will have 1 to 2 and then 2 to 3 let me call this new point as 2 dash this new point as 3 dash what happens in the combustor is the process a constant pressure process if you are adding heat in the combustor there is something known as a rally process wherein the pressure drops okay so you will get something like this 4 dash and then expansion is again non isentropic 6 dash this is the typical TS diagram for an actual cycle also okay now let us look at what are the typical efficiencies that we get so 1 to 2 diffuser the typical efficiencies are this is around 0.6 to 0.9 this depends on the mark number at which the vehicle is flying if the vehicle mark number is lower you are going to get somewhere higher efficiencies but as you go to higher mark numbers you will get lower efficiencies and in the compressor the efficiencies range from 0.8 to 0.85 combustor it is very high around 0.97 to 0.99 okay and 4 to 5 you have turbine this is somewhere around 0.87 and lastly 5 to 6 through the nozzle the efficiencies around 0.97 okay you will notice that the efficiencies for these two processors are much lower than efficiencies for these two processes why do you think that is yes if you take a look at the diffuser of the compressor in the diffuser and the compressor the pressure is increasing so you have what is known as an adverse pressure gradient whereas if you look at the turbine and the nozzle there is a favorable pressure gradient okay so therefore the efficiencies with turbine and nozzle are bound to be higher than efficiencies with diffuser and compressor now let us look at some typical engines this is a typical gas turbine engine with axial compressor here you have the intake and this is the axial compressor what we mean by axial compressor is the flow is along the axis and you have the burner or the compressor then you have the turbine and jet pipe or the after burner and then the nozzle okay now this is with a axial compressor as opposed to this you can also have something known as a centrifugal compressor okay wherein the flow comes in axially and goes out radially and again if you want to have another stage it has to come back in and again go in axially and go out radially and you have the combustion chamber here and then the turbine so for a compressor we have two choices that is compressor can be either axial or centrifugal what is the difference between the two wait what do you think is the change from axial to centrifugal notice one thing that if you take a look at this engine there is only one stage of centrifugal compressor whereas in the axial compressor there are many stages here okay so one is in the axial compressor it has large number of stages and this has fewer stages now typically in an axial compressor the pressure rise per stage is around 1.2 whereas for a centrifugal compressor a similar value is somewhere around 8 so you see that if you want a pressure rise in a centrifugal compressor you can achieve the same with fewer number of stages whereas if you go for an axial compressor you need large number of stages to achieve the same okay now is that all the difference or is there something more to it yes typically one would most modern engines would go with axial compressors because it requires a smaller frontal area if you look at the figure here this requires a smaller frontal area compared to this one which has a larger frontal area okay so one would go with a smaller frontal area because it reduces drag so axial compressors compared to centrifugal compressor now if you see here that axial compressor the pressure rise per stage is very small this is because if you go for a larger pressure rise in an axial compressor then flow separation and stall takes place therefore you are forced to look at a very small pressure rise per stage whereas in a centrifugal compressor you do not have such problems we will discuss that in detail as when we look at compressors in a lot in a greater detail little later in the course okay now it is also seen that axial compressors will have higher efficiency compared to centrifugal compressors because the flow distribution is much better in axial compressors if you look at centrifugal compressors if you need to have multiple stages then what you have here is flow coming in axially and then it goes out radially and again if you have to have a second stage it has to come back in axially and then again go out radially so the flow distribution is not very good in centrifugal compressors so therefore centrifugal compressors will have a slightly lower efficiency have higher efficiency compared to centrifugal compressors now we have looked at different kinds of compressors then you go to the combustor here in the compressor there is temperature rise typically this temperature will be of the order of 400 to 750 Kelvin why am I choosing two different values one it depends on compressor pressure ratio and two it depends on altitude at a lower altitude the incoming air temperature is much higher and therefore it will rise to a higher temperature whereas at a higher altitude where the temperatures are much lower it will rise to a lower temperature okay and in the combustor we have temperatures ranging from at the outlet of the compressor to something like 950 to 1600 Kelvin okay again it depends on compressor pressure ratio altitude and also what is known as what is the limit on turbine inlet temperature okay now if you look at the fuel that is used what is the fuel that is used in gas turbine engines or turbojet engines kerosene what is the calorific value of the fuel around 42 mega joules per kg now if you are using kerosene right the maximum temperature that you can attain with kerosene and air is something like 2300 Kelvin it is also known as adiabatic flame temperature so kerosene is used and it has heat of combustion of 42 mega joules per kg and what we know as stoichiometric mixture what do we mean by this what do we mean by stoichiometric mixture is the right amount of fuel and air that is required or it is the amount of air that is required for complete combustion of the fuel if you use this for kerosene and air if you use stoichiometric mixture you get a temperature of around 2300 Kelvin okay the stoichiometric ratio for this is F which is nothing but mass flow rate of fuel divided by mass flow rate of air this is around 0.067 or 1 by 13 for kerosene and air so if you have one part of fuel need 13 parts of air to completely burn it and the temperature that you get at the end of the combustion process is somewhere around 2300 and now compare that with what we are letting this go to this is around 1600 why do you think we are letting not letting it go to such a high temperature turbine inlet temperature turbine inlet temperatures need to be lower why is it to be lower you have nozzle you have compressor combustor takes that kind of temperature why don't you talk about structural integrity there very nearly there if we look at this graph here we see that the strength to weight ratio of any material will decrease rapidly with increase in temperature and in addition turbine blades are spun at very high speeds which contributes to very large centrifugal forces in addition to this there is something known as creep which is nothing but the tendency of material to deform permanently under the influence of mechanical stresses this is caused when you have long term exposure to high levels of stress and is more severe in materials subjected to heating for long periods of time. If you take a look at the compressor sorry the turbine here there are turbine blades and there is an outer casing let me draw that here let us say this is the turbine blade and this is the shroud or the outer casing there is always a small gap between the two of them why do you think it should be there if you see railway tracks there is a small gap between two railway tracks why is it there yes if you during summer the temperature increases and therefore the gap reduces something similar is what you see here firstly the turbine blades are rotated at very high RPMs so there is a enormous amount of centrifugal force acting on them now there is a centrifugal force acting in this direction in addition to that there are thermal stresses that are developed okay now which lead to elongation of the blade now if you do not provide this gap then the blade will rub against the shroud and will deteriorate the performance as well as lead to a shorter life for the turbine blades what happens if we give a larger gap so what is the big fuss about nobody wants to do work just like us the air also does not want to do any work so if you give a larger gap then the most of the flow tends to go through this because this is the path of least resistance okay so you cannot give a very large gap because most of it will anyway pass through that you have to give a small gap such that after elongation it is nearly there right so that is why we see that we are not able to go to very high turbine inlet temperatures although our limit is something like 2300 Kelvin we are forced to something like 1600 or 1800 Kelvin okay this is because of the problem that we just now discussed about okay so if the turbine blades rub on the shroud then it leads to lower performance and reduce life of the so we are forced to compromise and look at a lower temperature okay we are right now at 1600 to 1800 because we use lots of cooling techniques to cool the blades we use bypass air from the compressor to cool the blades so that we are able to go to higher and higher temperatures okay now lastly you have the nozzle here nozzle is the thrust producing device just like in the piston engine plus propeller propeller was the thrust producing device here nozzle is the thrust producing device okay if you notice the pressures after expansion in the turbine are very much lower there will be of the order of around 1.5 to 2.5 atmospheres typically okay this depends again on the altitude and the pressure ratio across the nozzle is very small okay therefore we typically end up using only a convergent nozzle in an aircraft engine that is you only have a convergent portion here and the reason for that is that the pressures at the exit of the turbine are very much lower and you do not have a large pressure ratio across the nozzle so you end up using a convergent nozzle only now the flow at the end of the nozzle here is choked what do we mean by that the Mach number is 1 or the flow attains the speed of flow speed and the acoustic speed are the same so at the exit of nozzle is equal to a which is nothing but speed of sound okay now speed of sound is given by where ? is the ratio of specific heats r u is the universal gas constant m is the molecular weight of gases and T is the temperature at the nozzle exit okay so notice that ue is directly dependent on the temperature at the exit of the nozzle Te so if the temperature at the exit of the nozzle is large then you get a higher velocity okay and therefore higher thrust that is what is the reason for this to produce the thrust you have a higher temperature at the exit and therefore you get higher velocity okay now let us try and look at how to derive the thrust equation now let me call this as the y coordinate and this is my x coordinate and I will assume an engine I will not look at what are all the details in here I will look at an engine something like this well for the time being ignore the details that are inside because all that it does is it changes flow conditions from here to here and if you look at what are the changes that it does we should be able to capture what is the thrust it is producing now let us take a control volume this is the control volume that I take I have taken the control volume sufficiently upstream of the inlet because I do not want things that are happening here two dimensional effects to come into picture we are only looking at what is the thrust in the x direction so what is the thrust that is produced in the x direction is all that we are interested in okay now let me call the parameters at the inlet as rho a v a a a and p a this is rho is density v is velocity a is cross sectional area and p is the pressure now similar quantities at the exit section here let me call this as the intake section I and let me call this as the exit section e at the exit section similar quantities would be rho e v e a e and p e p e is the pressure at the exit of the nozzle the ambient pressure is still p a okay that is the same as this now we want to find out what is the thrust that is produced in the x direction right so what do we need to do in order to do this which law of motion do we need to look at Newton's second law of motion which states that force is equal to rate of change of momentum so sigma f x or the sum of the forces in the x direction must be equal to rate of change of momentum in x direction now what is sigma fx what are the things that constitute sigma fx one is the thrust that it is producing the other is the pressure into this area and pressure into this area okay so sigma fx I can write it as f I have taken p a-p a if you notice I have taken the control volume sufficiently upstream so that the pressure across the intake area does not change much so from here to here there is no change in pressure so this will go to 0 and the other one will be in the negative direction because you are looking at force in the other direction now what is the rate of change of momentum here m. x ue right so you have ve2 ae-rho a va2 aa we know is equal to m. ve okay so we get if you simplify further we get f is equal to I take this on the other side and I can write this as m. a x 1-f ve- va-p e-p a I have taken this term to the right hand side this goes to 0 now m. a is nothing but rho a va okay now the 1 plus f constitutes remember in our earlier earlier in the class we defined f as mass flow rate of fuel divided by mass flow rate of air what happens is inside the engine you are adding fuel in the combustor here you are adding fuel right so therefore there is an increase in mass and that is why you have 1 plus f when it is going out there is an increase in mass and that is why you have 1 plus f so this is the thrust equation for a turbojet we will discuss more about the turbojet in the next class okay we will stop here we will discuss next about the turbojet in the next class.