 We are talking about turbine aerodynamics in the course lecture series of turbomationary aerodynamics and we are now on to the area of turbine aerodynamics in which we can now look at certain three dimensional aspects of flow through axial flow turbines. We have done some of the area that is related to the two dimensional flow which normally is also called cascade flow as we have done in case of axial flow compressors. In case of axial flow turbine some of the features as we have seen are somewhat similar to the axial flow compressors, but there are many areas that are quite different from that of axial flow compressors. When we come to the three dimensional flow it is somewhat similar that there are certain areas that have some overlap or similarity with axial flow compressors. On the other hand there are quite a few areas and considerations that are different in case of axial flow turbines. So, today's lecture we will be looking at three dimensional flow in axial flow turbines. Now, in three dimensional flow the axial flow turbine behaves in a manner that is somewhat different from the two dimensional theory you have done in somewhat great detail in the earlier lectures in this lecture series. The three dimensional flow as the name suggests that the flow has acquired a third dimension in its motion and this third dimension is the radial flow. Normally we assume that the flow is two dimensional which means it has axial motion as well as tangential motion which is imparted by the rotating blades and hence it is a two component flow which is what is assumed in two dimensional flow analysis. Here we get into three dimensional flow the third dimension that is the radial component of the flow becomes an important issue. So, many of the theories that have been developed earlier on were developed with the assumption that the flow is essentially two dimensional in nature. So, when the three dimensionality of the flow had to be factored in the assumptions made where the three dimensionality could be factored in a pseudo three dimensional manner that means when the blades or the aerofoils are indeed stacked from root to the tip of a blade the root to the tip stacking is done with the help of firstly getting the two dimensional aerofoils sections at each section at each radial section and then stack them up one on top of another from root to the tip. Now, this two dimensional aerofoils fundamentally the aerofoils are two dimensional entities aerodynamic entities. So, when you stack them up in a three dimensional manner you get a 3 D blade shape which to begin with under with assumption that it follows two dimensional flow aerodynamics. However, the modern designs and the modern analysis have shown that some of the flows do acquire certain amount of three dimensionality. When the flow acquires three dimensionality the problem essentially is that predicting the turbine performance through analysis various kinds of computational analysis becomes quite a lot difficult proposition. In the sense the prediction may not match with the you know actual tests. On the other hand when it was essentially two dimensional flow the it was known that there would be some difference between prediction and test results. However prediction was much easier and much faster. So, many of the laws governing the turbine you know basic turbine analysis and indeed basic turbine design were based on two dimensional understanding. However, they quickly figured out a way to create a blade stacking methodology by which three dimensional turbine blades could be created. And hence a certain three dimensionality of the turbine blade design was brought into the picture or brought into the methodology. In today's lecture we will be talking about some of these methodologies where three dimensionality of the flow has been attempted to be built into the turbine design methodology. And we will say that in what respect certain amount of three dimensionality of the flow could be avoided by design you know large amount of three dimensionality can be indeed avoided by design. And if it is a little bit is there it would have to be found out through a numerical analysis mostly simulation in terms of computational flow dynamics. So, most of the turbine design laws indeed are essentially pseudo three dimensional in the sense they give you a method by which two dimensional blades or aerofoils could be stacked from a root to the tip or quite often from mid section design which is normally done first. And then stacking downwards to the root and upwards to the tip. So, in today's class we will be looking at some of these aspects which are essentially factoring in certain amount of three dimensionality of the flow and then of course telling us how to create a three dimensional axial flow turbine blade for applications in modern gas turbines. So, some of the first assumptions that we would probably like to look at are related to the three dimensionality. One of the first assumptions that we would like to take a look at is the fact that the radial component of the flow is prime of AC very small in relation to the axial component and the whirl component C A and C W. We would like to keep it that way by design. So, the designers attempt to create blades which would promote flow over the blade surfaces and through the blades which have a very small radial component C R. If it is in a rotor we may call it V R. However, the radial component would be rather small compared to the whirl component. Now, if you look at the picture just below it shows that the flow coming into the turbine could have a situation where the flow actually acquires three dimensionality because of the shape of the annular passage. The annular passage that goes through the turbine depending on the turbine pressure drop or pressure ratio would have to open up and in the process of opening up it creates a diverging passage as opposed to a converging passage in axial flow compressor and then this diverging passage you know automatically promotes a certain amount of flow which is non axial. That means, it acquires a certain amount of three dimensionality. For example, flow coming in here into the turbine is most likely to be axial in nature let us say in the first stage of the turbine and then as it goes through the first stage which promotes very high pressure ratio it has to open up. So, it has a large area of flow which has to be given as the density has fallen and then it goes through meridional path which let us say has radius of curvature of R m. So, this meridional path through which it goes then automatically brings in a radial component of the flow. So, if you take the meridional path of the flow at any point on the meridional path the flow would automatically would have an axial component. If it is passing through a rotor blade or just pass through the rotor blade it will have acquired a wheel component as shown here and then because of the nature of this meridional path it will acquire a certain amount of radial component. So, because of the geometry of the flow track sometimes the flow acquires a radial component and that is inevitable because of the aerothermodynamics of the flow as it is going through the axial flow turbine. What people would like to do is to ensure that this radial component is as small as possible as is given out in the first assumption and to do that one may probably invoke this simple radial equilibrium equation that we have done in case of axial flow compressors. So, this radial equilibrium equation then simply equates the pressure static pressure gradient along the radial direction with the centrifugal action that the flow is experiencing due to its wheel component or rotational component and then this balance of static forces on the left hand side and the dynamic forces on the right hand side which is centrifugal force gives us an equilibrium of forces and this equilibrium of forces if it is adhered to or invoked in the design of the axial flow turbines then each of those passages would have a radial balance of forces and then it will be only due to this opening up of the passage which means near the tips for example, in this particular shape it would be quite flat. However, near the hubs it is entirely possible that it would be acquiring certain amount of radial component. So, the designers by design quite often try to balance the forces the statics and the dynamics and as a result of which try to minimize the axial radial flow component to the minimum value. So, that one can say that it is very small compared to axial and the lateral or the wheel component of the flow. Now, this is something which the designers often would like to do by design at the time of designing of the axial flow turbines if you can do that properly when the turbine is actually operating it would more or less stay very close to this design assumption or what has been invoked by design under certain off design operating condition it is possible that the turbine would indeed acquire a certain unbalance of forces and a little more of the radial flow may come out or come up in the actual operation of the flow dynamics or the aerodynamics and in which case those are the things that would need to be found out through more intense analysis which is essentially computational simulation. So, some of those things would have to be found out through intense computational analysis before the turbine is turbine design is finalized. So, some of those things we will be talking about when we talk about computational flow dynamics towards the end of this lecture series. So, having invoked the simple radial equilibrium equation the next thing to do is you know try to find out what are the 3 D models or as I mentioned they are more of pseudo 3 D models flow models which need to be invoked for design and then immediate post design analysis. So, the design and the post design analysis would create the turbine or predict the turbine performance and its fundamental characteristics which would as I mentioned a later on would have to be checked or validated first with intense computational flow dynamics 3 dimensional analysis and later on much later when the design is more or less final through rig testing. So, those are time consuming and indeed rather costly business we will look at the design features of modern actual flow turbines and the same features are used in immediate post design analysis. So, some of the features that we would be invoking or designers do invoke these days first of course, is the free vortex law. Now, free vortex law is something we have done in some detail with reference to actual flow compressors. So, it essentially as you know gives us a very simple handy relation which tells us how the will component should vary from root to the tip of a blade specially of a rotor blade where the flow otherwise is an unknown quantity. If you do not invoke free vortex or any such law what is happening through the rotor essentially would be unknown or untractable. So, a vortex law is absolutely essential in tracking the 3 dimensionality of the flow through the rotor. So, free vortex is the simplest thing that can be done and we have done in case of actual flow compressor in some detail. The one that is used most in modern actual flow turbine is indeed not really the free vortex law. In modern actual flow turbine there are different laws that govern the design and design related immediate analysis and one of them is simply known as the constant nozzle exit angle alpha 2 as we have used in our notations earlier. Coming out of the stator and the flow coming out of the stator would then be coming out at a constant alpha 2 from root to the tip of a blade. So, it is going into the rotor with a constant flow angle from along the length of the blade from root to the tip. Now, this is to be invoked this is not going to happen naturally it does not happen naturally it needs to be invoked or imposed on the design which means the blade shape will be created accordingly. So, you would have a completely different blade shape if you create it with free vortex law and if you create it with constant exit angle feature. So, they are two different design laws and indeed they would create completely two different looking turbine blade shapes both right stator as well as a rotor. The third case that we will be talking about today is a relaxation of the free vortex which is called arbitrary vortex case or arbitrary vortex law which is simply C w into r to the power n. Now, n is a variable and we will be talking about that also in today's lecture. So, we will be talking about these three possibilities as design laws for axial flow turbine and we shall see that three are different from each other and they would indeed give completely different kind of turbine blade shapes. So, depending on what kind of blade you are designing you need to invoke accordingly the design law. The early turbine designs very early turbine designs were indeed made with the help of free vortex design which was known to everybody more than 50 years back and as a result invoking those design laws for turbines as was being done for compressors was a very easy thing to do and as in case of compressors the blades that you get in turbine with using free vortex law indeed would tend to be a rum or twisted. Now, the twisted blade of turbines that came about was a bit of a problem later on because lot of cooling technology needs to be embedded in the turbine blades which is not there in compressor blades and this creating this cooling technology inside the turbine blades which are twisted or heavily twisted is a technological problem it is a huge big technological problem sometimes it is quite impossible to actually do that and of course, it increases the cost of making the turbine blades hugely. So, as it is the cool blade cost is hugely more than an uncool blade. However, the cooling is something we will be talking about very shortly in this lecture series in some detail. So, because of that one reason and the fact that the blades are made of high temperature material the nickel alloys the mnemonic and the inconal and those are high temperature very costly material. So, twisted blade was something that also produces high stress levels. So, turbine is a very heated area the entire blade is highly you know heated up through high temperature gases and then a twisted blade creates lot of stresses due to the temperature gradient along the length of the blade from root to tip as well as from leading edge to trailing edge. So, that kind of an environment for turbines tells us that somewhat simple blade shape is probably a better choice rather than somewhat twisted complicated blade shape that we have seen in case of compressors. So, in compressors they are those complicated blade shapes, but if you try to use them in turbine you get into all kinds of problems which are other than aerodynamic problems and in fact in case of turbine blade design those other people the mechanical designers the heat transfer people indeed often have the veto power. They can overrule the pure aerodynamic design on the basis of other considerations. So, the aerodynamic designer would have to modify his design to accommodate or keep room for blade cooling as well as to ensure that the blades are not unduly stressed during its operation because turbine blades suffer from huge temperature gradient that gives rise to creep and fatigue failure. So, those are the very strong issues based on which turbine design is carried out and that is one of the reasons why after the early era of turbine which were uncooled the blades are not made of normally not made of free vortex law they are made of the other laws and we will be talking about all these laws in today's lecture one after another. So, let us look at some of these laws. First we take up the free vortex law which is of course you know known to all of you and it simply gives us that C w r equal to constant and it is applied to the rotor flow which you know it has a few assumptions behind it. It comes with the radial equilibrium and then it comes with the assumption that the enthalpy gradient along the radius is 0 that means it is a constant enthalpy or iso enthalpy flow through the turbine and then of course one assumes that the flow is going into the rotor that is C w 2 into r equal to constant the rotor entry flow is indeed following the free vortex law that is the important consideration because the rotor flow is the more complex flow and then of course C w is constant from root to tip C a 2 is constant from root to tip that is axial velocity. So, these three are the regular normal assumptions if you remember for free vortex law. So, if you invoke them in the turbine design you end up getting a rotor specific work done which is also normally part of free vortex assumption that rotor specific work done that is h 0 2 minus h 0 3 normally we write it as u into C w 2 plus C w 3 at any station and that is into omega r 2 C w 2 minus r 3 C w 3 and this is constant from root to tip. So, in the third dimension that is in the radial direction now we see that enthalpy at the entry is constant C w 2 into r x constant C a 2 axial velocity is constant and now we find that specific work done is also constant that is work done per unit mass flow is also constant from root to tip that automatically tells us that C w 3 into r is also going to be constant from root to tip and it follows that axial velocity at the exit C a 3 is also constant from root to tip. So, it actually aerodynamically gives us a very simple flow situation where lot of things are very nicely constant from root to tip. However, it does give us a very complex blade shape highly twisted blade shape also in a blade shape that in which the degree of reaction would vary substantially from root to the tip of a blade. Now, that variation of degree of reaction may have certain issues degree of reaction inner turbine can vary again from near 0 to very high degree of reaction which could be 0.5, 0.6 not as high as you get normally in axial flow compressors at the tip, but quite high and the mean or the mid radius degree of reaction for turbines is somewhat less than that of axial flow compressor. So, typically that symmetrical bleeding that we have done in case of axial flow compressor is normally not done at the mean radius of turbine design. However, at the root the design could go to degree of reaction of 0. Now, degree of reaction 0 as you well know actually produces impulse turbine which is acceptable there is no problem with impulse turbine, but once the degree of reaction close to 0 you know that the reaction component from the gas turbine is now very small. Now, in a highly twisted free vortex design this is a possibility that is very strong that some part of the blade would have very low reaction component whereas, the upper part may have reasonably good reaction component. So, the reaction blade that we have talked about in the earlier lectures would then be actually valid for some part of the outer part of the blade and may not really be valid or available for the inboard part of the turbine rotor blade in which case you know that the amount of work done by the turbine would be somewhat lesser because you create reaction blades essentially to get more work done out of a single rotor. So, if the reaction is not available or somewhat non existent in a particular design model you would know that the work done would be somewhat limited by the reaction availability only to the outboard portion of the blade. So, that is one of the limitations among the other limitation that we have talked about with reference to the degree of reaction which indeed varies from root to the tip of the blade. Of course, that brings us to the point that you could have a constant reaction actual turbine blade that is entirely possible normally it is not a done thing people have designed actual flow turbines with constant reaction from root to tip there is nothing fundamentally wrong with that kind of design, but normally in modern actual flow turbines specially with the aero engines it is not a done thing you normally have a variable degree of reaction constant degree of reaction. We are not going to do the details here, but just to mention that it is a possibility that does exist and in the long past long back people have designed actual flow turbines with constant reaction and it works there is no reason why it should not work and the free vortex design as I just mentioned have been used earlier pretty widely, but it is not a used thing in many of the modern designs anymore. So, if you get degree free vortex design this is what you would normally get very similar to what we have received in actual flow compressors. So, the free vortex design essentially gives us the thermodynamic properties are constant from root to tip which means they are constant in the annulus through which the flow is passing in the actual flow turbine remember it is an annular flow and hence it is annular flow passage over which we are now kind of invoking a law which promotes constancy over the entire annulus. Then with the help of all those things it is comparatively easy now to find out that the entry flow angle from the to the rotor tan alpha 2 can be given at any radius can now be related to the one at the mean radius or the reference radius simply by invoking the radius ratio. So, everything kind of gets into the radius ratio in the free vortex design the tan beta 2 which is the relative flow angle going into the rotor which is the flow angle which the rotor indeed feels is also relatable to the radius ratio and then of course, the inverse of the flow coefficient to the turbine that is U m by C A 2 C A 2 by U m of course, would be the average flow coefficient through the particular turbine rotor. In which case as we have seen C W 3 into R is constant from root to tip C A 3 is constant and we can also assume or we have normally used it in free vortex law it is held constant across the rotor that is C A 3 is equal to C A 2. So, free vortex law actually makes the flow variables in a very simple manner and this simplicity of course, is the first attraction of this particular law at the exit tan alpha 3 again is at any radial station is relatable to the mean value tan alpha 3 which is where normally the design is done as I mentioned earlier and we have done that in case of axial flow compressor also. So, that is relatable again through simple radius ratio and then of course, the relative flow angle tan beta 3 which is again relatable through the radius ratio and the inverse of the flow coefficient at the exit. The rigorous designers may find these two values different, but simple design would tell us that C A 2 is equal to C A 3. So, the flow coefficient at the entry and the flow coefficient at the exit indeed are equal to one another. However, as we have seen that the length of the blade at the leading edge and at the trailing edge could be different. So, the values of the variation of this could be different at the exit of the rotor compared to that at the entry to the rotor. If you look at the drawing here it tells you very clearly that at the exit side the radial variation is far more than at the leading edge or the entry of the axial flow turbine rotor. So, those are the simple things that we can get out of free vortex design and as I mentioned it is it does give us a very simple flow feature. However, it does create a complex somewhat twisted and more complex blade shape. The other blade shape which people then resorted to is simply called the constant nozzle exit angle model. Now, this model has been essentially created for the more practical purpose of accommodating cooling. It creates a nozzle or stator blades with zero twist. So, that the flow comes in at alpha 1 goes out at alpha 2 same from root to the tip of the blade. So, this untwisted stator nozzle is the first thing that we wanted and one of the reasons is the stator nozzles phase very high in the temperature coming from the combustion chamber and it normally for a long long time now at least last 40-50 years. The first stator nozzle is embedded with very elaborate cooling mechanism. To have all that cooling mechanism inside the blades requires that blades blade itself does not have complex shape or a twist or a large twist. So, the turbine blade designers very quickly decided to adopt this particular design philosophy to create untwisted stator blade. So, that the cooling can be efficient because for last 50 years the advancement of turbine design has been more through the cooling technique and by increasing the turbine entry temperature to get more work out of turbine rather than creating more and more complex aerodynamic shapes. So, in some sense the aerodynamics of the flow may have been slightly compromised to create elaborate cooling mechanism. So, what we get is alpha 2 is constant from root to the tip of the blade. If alpha 1 is also constant from root to the tip of the blade which is sometimes true especially in the first stage of the HP turbine then you indeed have a untwisted blade. If alpha 1 is some variable in the later stages you have a mildly twisted blade which can still accommodate or have embedded cooling mechanism. So, with this simple invocation of the constant exit angle if we move forward what we get is a simple mathematical model in which now cot alpha 2 which is equal to C a 2 by C w 2 the ratio of axial and wheel component. Now, that is a constant from root to the tip of the blade. Now, as we can see C a 2 would be equal to C w 2 into cot alpha 2. If you take a simple differentiation of that that yields that d C a 2 d r will be equal to d C w 2 d r cot alpha 2. And if we use this in the energy equation if we import the energy equation that we had used in creating the radial equilibrium in case of compressors. If you remember it was and we invoke it here or bring it back here again d h d r that is the enthalpy variation in the radial direction is C a into d C a d r plus C w into d C w d r plus C w square by r and the last term is of course, comes from the radial equilibrium. And then if we invoke d h d r equal to 0 that means, enthalpy variation in the radial direction is 0 that means, enthalpy is constant it is iso enthalpy flow over the entire annulus. If that is so then the right hand side goes to 0 and on the left hand side we have C a d C a d r plus C w d C w d r plus C w square by r and that would then be 0. So, energy equation is brought in here. If we now substitute this in this energy equation we get C w 2 into cot square alpha 2 into d C w 2 d r invoked at the entry to the rotor. We are invoking the energy law at the entry to the rotor as we have done in case of compressors. And then we get with this design law for turbines we get a situation that C w 2 d C w d r and C w 2 square by r that is equal to 0. So, rewriting that we get C w 2 into 1 plus cot square alpha 2 into d C w 2 d r plus C w 2 square by r equal to 0 and this yields 1 plus cot alpha square is sin square alpha 2 and then if we get d C w d r equal to minus sin square alpha 2 d r by r. So, now this on integration this is derived equation from the energy equation by invoking the constant exit angle law turbine design law and having arrived at here if we simply carry out the integration it gives us C w 2 into r to the power sin square alpha 2 equal to constant. Now, this you can see is a different equation very different from the one we had for free vortex law and then of course, we get C w 2 into C w 2 m that is related to the mean radius or the mid radius of a turbine with the radius ratio into sin square alpha 2. So, the variation of the parameters now is beginning to look quite different from that we had got in case of free vortex design. And then of course, the actual velocity variation also falls in line in the same manner and hence we can write C a 2 into r to the power sin square alpha 2 and that would be equal to constant from root to tip. So, the variation in the radial direction is now governed by sin square alpha 2 as the index and then we can get C a 2 variation along the radius similar to C w 2 as C a 2 m into r m by r to the power sin square alpha 2 and in terms of the absolute velocity C 2 we can then write down C 2 would be C 2 into C 2 m into r m by r to the power sin square alpha 2. So, all of them now use this alpha 2 as a parameter which is invoked in case of this present design law. So, at the rotor inlet station we have put alpha 2 equal to constant and hence C w 2 by C w 2 m is equal to C a 2 by C a 2 m is equal to C 2 by C 2 m and all of them are equal to r by r m. So, they are all related to the radius ratio all velocity components are directly related to the radius ratio. Now, this constant exit angle from the stator nozzle has still three possibilities additional three possibilities. The additional possibilities are you can have constant h 0 3 at the rotor outlet that is enthalpy is constant at the rotor outlet as we had indeed got in case of free vortex law or you can have 0 world component at the rotor outlet that means alpha 3 equal to 0 indeed C w 3 would be equal to 0 and then we could have free vortex again brought back continued at the rotor outlet. So, it is possible to look at now what would be the flow at the rotor outlet from three different possibilities. So, let us look at the three possibilities if you have constant total enthalpy at the outlet u is equal to C w 2 plus C w 3 that is equal to delta h 0 that is the work done through the turbines or work being created by the turbine through the hot gas passing and the world component of the velocity at the rotor outlet is indeed found from C w 3 is equal to delta h 0 by u minus C w 2. Now, the first term here can be written in terms of k by r k being delta h 0 by omega omega is the angular velocity of the rotation of the turbine rotor. Now, rotor is the only one which is doing work stator if you remember does not do any work. So, the angular velocity of the work done of the rotor and then that ratio can be taken to be some kind of a constant value for design purposes. Now, you can also find C a 3 which can be computed once you get C w 3 you can sit down and do a little bit of vector analysis very quick velocity diagrams with the help of velocity diagrams and you can find what would be the value of axial velocity at the rotor outlet which could again take a form out of this. Now, both of which can be then computed from root to the tip of the blade. Now, using the kind of variation that we have seen in the earlier slides. So, it is now possible to create C w 3 and C a 3 at the rotor outlet with the help of this particular assumption. If we go for the next assumption that is the world component is 0 at the rotor outlet that is alpha 3 is 0. This means that d h d r would be equal to C a 3 into d c a 3 d r and then one can write down a 0 3 is equal to a 0 2 minus u into C w 2 and C w 2 can be now written down in terms of C w 2 at any station any radial station can be written down in terms of the mean radius station as we have done before substituted with C w 2 m into r m by r equal to the power sin square alpha 2. Now, this produces the enthalpy distribution radially at the exit as d h d r equal to d d r into u into C w 2 m into r m by r to the power sin square alpha 2. So, this is how you get the enthalpy variation radially the earlier one which we just did was that the enthalpy was constant that means the variation would be 0. Now, we find that there is enthalpy variation along the length of the blade. The third condition that we can invoke is the free vortex law which to be applied at the rotor exit now free vortex law is somewhat we are familiar with and if you do that invoking the free vortex laws that we have done in the earlier slides if you bring them over here and simply apply them at the rotor exit you would get a C a variation C a which can be written down or expressed in terms of C a 3 square equal to C a 3 medial square plus twice u m into C w 2 m into 1 plus r by r m to the power cos square alpha 2. Now, this expression would be also valid for this particular case which means the C a variation that you get in case b would be very similar to that you get in case 3. So, the actual velocity variation can also be found by using the laws that we have prescribed. The third possibility that we have talked about is the arbitrary free vortex case or the relaxed free vortex case in which C w is equal to r to the power n instead of writing C w into r equal to constant we are saying that C w has a equality to r to the power n. Now, this value of n is can be now varied. Now, if you put n equal to minus 1 it actually if you remember resolves to free vortex model. So, that is something which we have already done before the other one which if you put n equal to 0 it resolves to constant vortex model or constant it not necessarily constant reaction it is a constant vortex law not constant free vortex, but constant vortex from root to the tip of the blade. On the other hand if you put n equal to 1 it gives what is simply known as solid body rotation that means the flow would be simply directly proportional to r like for example, u is omega r. So, all other parameters would also be directly proportional to r rather than being inversely proportional as in case of free vortex law. So, in some sense this is inverse of free vortex model. The third possibility is where you have n equal to minus 2 in case of compressors we had seen it produces somewhat different kind of blade loading it is same as the same in case of turbine it produces a different kind of blade loading and indeed it produces vortex strength that is different most likely a higher vortex strength at the various sections of the blade if you invoke this law. Now, what the modern designers are doing turbine designers as in case of actual flow compressor designers they invoke these laws in various parts of the blade. So, you do not have the whole blade actually designed as per any one of these laws various length of the blade is designed as per one of these laws. So, you can have one kind of law in one part of the blade another kind of law in other part of the blade and then of course, the blade would have to be blended into a smooth shape through geometrical modeling and aerodynamic analysis. So, modern designers often invoke more than one law in creation of one single turbine blade rotor. So, this is how the design indeed proceeds in the modern actual flow turbine blade creation. So, we have gone through various design models the design models essentially are what I mentioned pseudo three dimensional flow models in the sense they tried very aciduously very consciously and deliberately to avoid creating the radial flow model radial flow component. So, radial component is very deliberately tried to be avoided in the design. So, that radial flow is not created through aerodynamic pressure gradient or aerodynamic laws. However, if the geometry of the flow somehow brings in certain amount of radial flow component and there is nothing much you know you can do about that and we have seen in the earlier lecture that some of the blades indeed do have strong radial component due to the variation of the annular flow track which is curvilinear and diverging flow track. So, there is nothing what you can do there, but the blade designer tries to ensure that radial flow is not created by the operation of the turbine blade rotors. So, this is how the design is normally proceeded with in the modern actual flow turbine and so in today's lecture we have talked about the design laws. In a lecture later on we may have a look at some of the design features or design steps and indeed some of the airfoil sections or the blade sections I do not know whether you can really call them airfoils, but the blade section that are used in actual flow turbine design. In the next lecture we will be actually looking at some of these 3D or pseudo 3D or quasi 3D flow theories that we have done today and try to use them in solving a few very simple standard problems. So, we shall have a quick understanding of the numericals of how this design laws actually come up with numbers in case of actual flow turbines. So, next class will be a problem solving class. So, I will bring a couple of problems for you for you to have a look at solve problems and then I will leave you with a few problems for you to solve by yourselves. So, in the next class we will be doing problems on actual flow turbines using 3 dimensional flow laws that we have done in today's lecture.