 In our previous lecture, we were discussing about the venturimeter and we will continue with that as an example. So if you recall, what was the purpose of the venturimeter? The purpose was to measure the flow rate, volume flow rate through a pipe. Now for that we utilize this type of an arrangement where you have a converging section which is sometimes also called as a converging cone. Then you have a throat where the area of cross section of the entire arrangement is a minimum and then a diverging section. And the purpose is obvious that this part has to fit with the pipe. So if you have reduced the cross section area, somehow you have to increase it so that it again matches with the pipe. So this black colored portion is like a fitting which is fitted with the pipe to measure the flow which is occurring through it. The question is why you have such a converging section? We have seen that it gives rise to an accelerated flow. So it gives rise to a high velocity and therefore a change in pressure. And we have seen that it is not just the pressure that is important but the sum of the pressure and the elevation head together that is p by rho g plus z. That term together is something which is changing because of the change in the kinetic energy head and we gave it a name called as piezometric head. So the difference in piezometric head is reflected by the difference in reading in the heights of the liquid columns in the 2 limbs and that delta H therefore will be an indicator of the rate of flow which we found out by utilizing the Bernoulli's equation. The question is that is it a very reliable way of finding out the flow rate? The answer is straightforward in some sense that it is not that reliable. The reason is that we have utilized an ideal type of equation. This all the assumptions which are there in built with the Bernoulli's equation in steady form are inbuilt with this and therefore all the idealities which are also inbuilt with this form of equation those are assumed to hold. At the same time we understand that in practice such idealities do not hold true. What is the biggest deviation? In practice it is never a frictionless flow. So you have viscous effects and because of the viscous effects something happens or quite a few things happen. Now one of the important things is because of the viscous effects if you have these as the total head or expressed in terms of different units total energy represented it is a representative of the total energy at section 1 and this is a representative of the total energy when we say energy here we only mean the mechanical form of energy at section 2. We assume these 2 are equal because there is no loss because it is a frictionless flow. In practice there is a loss. So you expect that if you call this as say E1 and if you call this as E2 you expect that there is which one is more E1 is more or E2 is more? E1 is more because you expect that there may be a loss of energy and the loss of energy will be because of the travel of the fluid from the initial point to the final point. So when it is travelling from 1 to 2 it will have a loss. So that means E1 will be more so that when you come to the section 2 it is E2 plus some losses the losses which have taken place between 1 and 2. In one of our later chapters we will try to characterize these losses in a more formal way but we will just keep in mind that there are certain losses because of viscous effects and these losses will give you a guideline of like what is the direction of flow. So say you know nothing you are given some E1 you are given some E2, if E2 is greater than E1 you must be assured that flow is taking place from 2 to 1 not from 1 to 2. So it is basically taking place from a high head to a low head. It cannot be the other way because where will the head come from. Now here because of this loss what will happen? See this eventually will boil down to a large larger drop in the piezometric head. So if it is flowing from 1 to 2 the piezometric head which is coming into the picture the piezometric head drop that is the difference between the 2 terms present there in the bracket this you expect to be more because you expect a more severe drop in pressure because of the overcoming the frictional resistance effects. So that means whatever delta H you read here it is not the ideal delta H. See in this formula what delta H you put to get the value of the ideal Q. This delta H is what you experimentally observe. We have to keep one thing in mind what is the basic principle of measurements that we use in experiments that we have an expression in which we have certain measurable quantities certain easily measurable quantities and we express some more difficult to measure quantities in terms of the easily measurable quantity. So here delta H is something which you measure easily flow rate you do not measure directly but use this formula to write flow rate express flow rate in terms of the measure delta H. So here maybe to get Q ideal it would have been better if you could put delta H ideal but that you cannot do because delta H is what you are reading from the practical thing. So it is giving the delta H which has got manifested because of by considering all practicalities. So this delta H also considers the practicality that there is some loss of energy to overcome the fluid friction effects. That means this delta H is higher than what delta H would have been if it were a frictionless flow but you cannot help this is what you read experimentally and it is what you put here. That means even in terms of theoretical flow rate it is not giving the correct theoretical flow rate it is overestimating that because you are putting an overestimated delta H. The reason is straight forward because experimentally you cannot reveal an idealistic picture experimentally you reveal the real picture where the delta H is much more severe than what delta H you could perhaps get in a frictionless condition. So one important thing we realize that if you consider no other effect this particular effect alone should give you that Q actual should be less than Q theoretical which you calculate by using this formula but there are other important reasons also. What is the other important reason? See when you have written v1 square by 2 actually we were bothered about the velocity at the point 1 but the velocity of the point 1 how did we evaluate? We evaluate by using this Q equal to a1 v1 equal to a2 v2. So by that we implicitly presume that v1 and v2 are like same as the average velocities over the section that is possible only when the velocity profiles are uniform over the sections 1 and 2 but because of the viscous effects we have seen that those are not uniform. So there is a non-ideality because of viscous effects not only in terms of the frictional resistance but also in terms of putting the v square by 2 term. So there is also some error in that. So these 2 errors are very very significant one error is the frictional resistance another error is the like miscalculating the velocity expression or misinterpreting the velocity expression. So when you put the velocity here ideally you should have put a velocity here in such a way that this would have represented the kinetic energy across the section 1. Again in our later chapter we will see that how to exactly put that but here we will just appreciate that we have not put it correctly. So whenever we make a mistake in writing something the first and foremost thing is to appreciate that we have made a mistake. So let us appreciate that this is not correct there is some error in it. Now incidentally engineers are such classes of people who are happy to get the final result disregarding maybe some mistakes which have already been done and then to adjust that mistake let us say that some adjustment factor is put. Let us say we call a new coefficient CD as Q actual by Q theoretical. So if somehow this coefficient is known to us whatever by magic we will see what magic will tell us this number but if somehow you get this value of CD then you straight away multiply that with the Q theoretical that you get from here to get what is the Q actual if it is very close to the final result that you want in many practical engineering applications people are happy. So we have to see that what is there in the CD which will try to make us more and more happy and this coefficient is known as coefficient of discharge. So this takes into account that we have realized that if it is an ideal case all together this would have been equal to 1. So deviation of this from 1 represents the extent of non-ideality in the flow and not only non-ideality in the flow in general but more specifically how that non-ideality has got manifested in the prediction of flow rate. So that means what is the total influence of this friction in terms of the delta H and what is the influence of the inaccuracy in the velocity distribution that has already got inbuilt in the corresponding expression for energy. To look into that we will not go into all sorts of details but we will just consider one thing that see at the section 1 say we have sort of uniform velocity profile somehow we have maintained. Again it is we will later on see that it is not easy to maintain that but fortunately we will be easily maintaining such a situation when the flow is turbulent and when we will be discussing about turbulent flows we will see that turbulence is a kind of situation which will create almost a uniform velocity profile over the section. So that will in some way take care of some of our acts of ignorance in writing or describing the correct velocity profile but even then let us try to see that at which section the error will be more severe at section 1 or section 2. To do that let us say that we consider 2 stream lines which are very close to each other say you have a stream line like this and another stream line which is very close to each other both stream lines are connecting the sections contained by 1 and 2. So here 1 and 2 are points but let us say that these are sections which contain the points 1 and 2 so we have one stream line and we have another stream line. Now these 2 stream lines are very close so close that let us say that here the velocity is u1 here the velocity is u2 here the velocity is u1 plus delta u1 and here it is u2 plus u2 where delta is a small change in comparison to the other value. Now if these stream lines are very close to each other what will happen? There will be negligible difference in pressure between these 2 stream lines always remember that there is a difference in pressure between that stream lines because of the curvature effects of the stream lines but stream if they are very close that effect is negligible. So if these 2 are very close there is negligible difference in the pressure head between these 2 stream lines at 1 and at 2 and if they are very close there is negligible difference in the height also that is the z coordinate. So what we can say is that the difference in kinetic energy heads between the 2 points it remains same for the stream line above and the stream line below because the other ones they do not change. So how we may reflect that in our analysis? Let us try to do that. So what we want intend to write is u1 square minus u2 square is equal to u1 plus delta u2 square. So let us try to simplify it keeping in mind that delta u1 and delta u2 are much smaller as compared to u1 and u2 respectively. So you are clear that why such an equation has come because other terms like p by rho g and g those are the same. So they have cancelled out when you consider the Bernoulli's equation for the stream line above and stream line below and when you have subtracted that effect has gone. So only these terms remain basically u square by 2 and all those things but that division by 2 gets cancelled out that is how these terms come up. So if you now write it like this so what you have here? So you can write this as I mean u1 plus delta u1 square that you can bring in one side u1 plus delta u1 square minus u1 square is equal to u2 plus delta u2 square minus u2 square. So you can write it as 2 u1 into delta u1. So when you write this this like a square minus b square formula when you write a plus b it is 2 u1 plus delta u1. Delta u1 is much much smaller as compared to 2 u1 so only 2 u1 and the difference only delta u1 is there so that is equal to 2 u2 delta u2. So we are interested to express delta u1 by u1 in terms of delta u2 by u2 to see that what is the relative error in or relative change between the velocities into adjacent stream lines. So you will have delta u1 by u1 as if we are dividing both sides by u1 square so you write as u2 by u1 square delta u2. So here you can write this as u2 by u1 whole square into delta u2 by u2 that means delta u2 by u2 is equal to u1 by u2 whole square into delta u1 by u1. Now which velocity you expect to be more u1 or u2? Look at the sections 1 and 2 here the area is large so the velocity will be less. So u1 by u2 more is this reduction in area u1 by u2 will be lesser and lesser square of that will be small. So from this our conclusion is that this delta u2 by u2 is expected to be much much less than delta u1 by u1 provided there is a great reduction in section. That means if it is approximately uniform at section 1 it will be even better at section 2 because the non-uniformity is much less what this represents a non-uniformity when you go to a different stream line along the same section you expect the velocity to be different and that difference give rise to a non-uniformity. So when you have this non-uniformity but again see this is an estimation because for estimating the non-uniformity we have again utilized the ideal equation which is like the Bernoulli's equation. But what we have considered that even for a non-ideal case this is not very very invalid because whatever is the frictional effect that also has got cancelled out when you have subtracted the 2 equations. Assuming that the frictional effects are also same as the fluid flows from 1 to 2 along the 2 streamlines above and below. So even if frictional effects are considered and they these the Bernoulli's equations for the 2 or the modified Bernoulli's equations considering the frictional effects they are cancelled or they are subtracted one from the other that effect will be cancelled. So this is not a bad estimation. So this estimation shows that if the velocity is such that you are going towards a cross section of reduced size if at the bigger cross section the velocity was more or less uniform the smaller cross section it is expected to be even more uniform. The reason is quite clear that if there were streamlines like these streamlines will more converge to each other because they are now confined to be there within a very small space as compared to how they were earlier. So if the streamlines were quite a large distance apart so if the streamlines were like this now when their streamlines are confined so what will happen all the streamlines will try to converge. So when the streamlines try to converge you see the distance between the streamlines corresponding streamlines becomes smaller and smaller and eventually different streamlines represent the sort of like different states of flow. So if you have them quite close to each other and almost parallel to each other that non-uniformity in the velocity is almost like it is not totally nullified but it becomes a better situation. So by having a section 2 like this which is like a convergent section it is not bad it sort of eliminates one non-ideality. The other non-ideality because of negligible friction that may be reduced to some extent by what? By minimizing the length travelled between 1 and 2 because the frictional resistance will be related to how much length the fluid has travelled against the viscous effects. So how do you reduce that? One of the ways is like you have this angle this cone angle quite large okay so that it converges quickly to a small section. In practice this angle is like typically kept as like 20 degree or so. These are like design considerations of this device it is not that it is 20 degree is a magic number and it is always kept like that I am just giving you a rough idea of what is the range in which it is kept in practice. Now there are different issues like you cannot make it as a large angle as you like there are issues of manufacturing the device and so on. So it is not that whatever angle you want and you propose one has to also fabricate it and put it in practice. One particular aspect on which one may not make a compromise it is like by putting by locating this section 1 where you are having this manometer limb it should be preferably somewhat away from the place where the reduction has started. So that this disturbance is not influencing the velocity at this point significantly and that is why it is kept a little bit away from this one and roughly it may be if the diameter of the pipe is d it is roughly like a distance d away I mean it is again a rough estimate there are more accurate estimates for each device. So the connection of the pressure tapings are also very important that is where are there to be put. So one is here then this is roughly like 20 degree and this creates a good accelerated flow if you achieve it in a very small or a short distance it is good you have less frictional resistance and smaller the cross section you expect that more will be your resolution in terms of this delta H. So the experimental objective is the delta H is if the delta H is more it is better because that is your reading if it is very small your error in resolution will be affecting your result significantly. But if the readability of this is good then the error corresponding error is less and that is why you are trying desperately to reduce the cross section area so that there is a change in the kinetic energy head very severely which is manifested in terms of this delta H. Now after this section has come and then what you have to do then you have to divert back to the pipe diameter again so you have a diffuser which is like a diverging section. So you have a converging cone you have the throat where you have the minimum area and then you again have a diffuser. The question is what should be the angle of this diffuser I mean do not get confused with the sketch that I have drawn in the board it is just because of lack of space that I have drawn it not to scale. So this angle does not represent what is there in reality it just represents the shape but not really the sense of the angle so what should be this angle. Now again there are two conflicting requirements engineering is such an area where when you want to design something there are two aspects that you have to keep in mind one is it should satisfy the fundamental scientific requirements so that the device is based on a thorough scientific principle. The other important thing is that it must optimally satisfy the performance requirements so what are the corresponding influencing parameters always you will see that there will be opposing parameters so opposing parameters means if you increase this angle then something good will happen and something bad will also happen. So let us see that if we increase this angle of the diverging section first let us see that what good thing will happen that is very obvious so if you increase the angle of the diverging section what good thing will happen yes if you increase the angle of this what What is the good effect of that? That portion will decrease in length. So, in a relatively short length, this device will merge with the pipe. So, the loss due to frictional resistances will be less. So, just like I mean what would have been a good effect of making this angle large, the same logic holds there also. But one of the logic that does not hold is that there is a great difference between an accelerating and decelerating section. This is an accelerating section, but this is a decelerating section. Why this is a decelerating section? So, if you see the area of cross section is increasing. So, you expect the kinetic energy head to reduce. That means if you expect the kinetic energy head to reduce, that means p by rho g plus g, the piezometric head will increase to compensate for that. So, if you say, let us say that you are having a horizontal venturimeter. So, if you are having a horizontal venturimeter, z1 and z2 are the same or maybe 2 and 3. Here you consider another 0.3, z2 and z3 are the same. So, then if you go from a 0.2, say to a 0.3, you expect what? If these 2 are located at the same height, then what you expect? You expect that pressure will increase or decrease, pressure will increase. So, when pressure increases, that means, so if you consider the direction of the flow as x. So, you can write the dp dx as the rate of change of pressure with respect to x. In the converging section, dp dx was what? Less than 0. But here in this particular section, dp dx will be greater than 0 because pressure is increasing with x. What is the consequence? The consequence is, see you expect that if the pressure is decreasing with x, that is fine because then a higher pressure is creating a drive for you. That is if pressure is decreasing with x, that means p1 is greater than p2 and that is in some way it is trying to create a driving force for you. On the other hand, where from 2 to 3, the pressure is opposing you because as you are moving from 2 to 3, you are experiencing higher and higher pressure. So, that means it is a sort of effect that tries to inhibit the motion of the flow. So, that is why this type of pressure gradient is called as adverse pressure gradient. So, this type of pressure gradient is called as a favourable pressure gradient. So, favourable and adverse, the English names are quite clear. Favourable means which favours the flow, adverse means which is not good for the flow. So, when you have an adverse pressure gradient which is like this dpdx greater than 0, what happens? The flow has a tendency to be decelerated because of that kind of a pressure gradient. So, if you try to sketch that what happens to the streamlines in such a case, so the flow tries to move like this but because of the deceleration effects and the deceleration effects are more severe close to the wall, why? Because viscous effects propagate from the wall. So, at those locations what will happen is the flow may not be capable enough of being dragged with the main or the core flow because it is slowed down so severely that it just creates a local rotation but it does not contribute to the main flow. So, that type of thing is called as a flow separation. That means you have a main flow like this. Now the flow, the fluid particles, these poor guys close to the wall, they are so severely disturbed because of the adverse pressure gradient which are acting on them that they really cannot maintain the flow and they might even reverse their direction of flow. So, local vortices are created close to the wall. How do these vortices contribute? They contribute in a sort of negative way. See, these vortices by virtue of the rotation have some energy but that energy is not contributed to the main flow. The main flow is like this which is moving. Now here this energy which is there because of the rotation of these vortices because of flow separation that does not contribute. So effectively as if some energy is taken away from the main flow to sustain the rotation of these vortices. So effectively there is a kind of loss of energy of the main flow and that loss has been created because of this flow separation and this flow separation effect is stronger. More is this angle of diffuser. The reason is more is this, more severe will be the adverse pressure gradient because more severe will be the pressure increase over a given length. The length becomes smaller. So, this is a conflict with the requirement of the frictional resistance. So, we have seen that if you increase the length of this one or maybe reduce this angle then this effect will be less. So, the adverse pressure gradient effect will be less if you make this angle quite small. So, that this length is large but if this length is large the direct frictional resistance will be more. So, these 2 are 2 conflicting parameters in the design that is where you have to come to an optimal design where you cannot keep this angle maybe as large as this and the common optimization is that this is typically like 5 degrees, 6 degrees like that much less than the angle of the converging section. So, this is something we have to understand very carefully that why in the practical design the diffuser angle is much much less than the converging section angle. When you have the converging section you do not have such a case of flow separation. So, only the frictional resistance is because of the length is the only important resistance because flow separation will be there when the flow is decelerated but in the converging section the flow is accelerated. So, it does not suffer from a resistance because of adverse pressure gradient. In fact, the pressure gradient here is favorable which makes it move in a much more convenient manner. So, the design aspects are quite clear that why you should have different angles for converging and diverging sections and what are the parameters which should decide the range of these angles. So, and keeping these things in mind one may if one designs this device quite well by minimizing the losses then the coefficient of discharge which is the ratio between the actual flow rate and the ideal flow rate it is actually very close to 1.98, 0.97 like that. So, somehow the device is very cleverly designed some of the non-idealities are taken care of in some way not that it becomes ideal but our ignorance about non-ideality does not get manifested so much the reason is that one is you are using like a continuously converging section in this way and the diffusing section is also say properly well designed. Now, this venturimeter is therefore a very common device which may be used in a pipeline to measure the flow rate at the same time this is not a very inexpensive device it is not very highly expensive but at the same time for very routine applications one might look for some cheaper devices which are broadly following the similar principles and let us see one such device. So, that device we call as orifice meter orifice meter is another application of the Bernoulli's equation. So, orifice meter is something like in the purpose is the same that you have a pipeline you want to measure the flow rate through the pipeline. So, what you are doing here you are putting an obstruction in the form of a plate. So, this is like a circular plate if it is a circular pipe it may be a circular plate with a hole at the centre which is called as orifice. So, here also what happens here if you consider the stream lines the stream lines were originally say parallel to each other not that they always have to be but just as an example. Now, you know that because of this constriction the stream lines have to be forced to flow through this small section. So, stream lines will convert like this and then when the stream lines pass through this. So, there is also a stream line at the centre. So, when the stream lines pass through this constriction after that what happens that is very interesting. So, after the stream lines pass not that they become parallel because of the inertial effects the stream lines go on tending to converge. So, there is not that after coming out of this they become parallel. So, they go on converging till the stream lines come to a condition where the distance between the extreme stream lines is a minimum and then the stream lines tend to diverge again from that and the divergence is again to match with the pipe contour. So, that type of stream line behaviour is there qualitatively it is important to first appreciate this because from this you will get an apparent similarity with the venturimeter. What is that? In the venturimeter you try to have a reduction in the area available for flow. Here also you are having the same thing but what is the difference? Difference is in the venturimeter you had a gradual transition from the bigger area to a smaller area and here you are trying to have a more abrupt transition. An abrupt transition is something which is not so good because the flow does not get enough opportunity to be adjusted to that abrupt change and that may create additional losses. Not only that there are more uncertainties in the measurement why there are more uncertainties in the measurement let us try to see again our policy will be that we will try to measure the pressure difference or to be more fundamental the piezometric head difference between 2 points. What are the points that we should choose? See when we are choosing a particular point we are making a tapping in the wall of the pipe right. So, as if we are making a hole and fitting a manometer that is the arrangement. The arrangement does not change here the philosophy also does not change but implementation becomes more difficult why? See here you have taken it at a distance substantial enough from here so that this effect in the curvature of the stream lines is not important. You are interested about the pressure at this point actually not actually you are interested about the pressure at a point which is at the central line but at the same section. There may be a difference in this pressures if the stream lines are curved but if the stream lines are parallel that will be not. So, the pressure red here and the pressure here will obviously mean almost the same effect of stream line curvature will not change anything. Here also if you want to utilize the same principle you should come to a location where there is negligible stream line curvature and that is there only at the place when this has come to a minimum. So, if you consider a curve which has come to a minimum the tangent is parallel to the axis. So at this location where the distance between the consecutive stream lines or the extreme stream lines is a minimum here almost stream lines are parallel to each other. So, there is negligible error because of neglecting the curvature of the stream lines at that location and this location where the distance between extreme stream lines is a minimum is known as a Venna contractor. So, that is the name Venna contractor and that is located somewhat away from the orifice. It is not exactly located at the orifice. So, if you connect this limb of the manometer at the position of the Venna contractor then your analysis is quite good. Question is how will you know where the Venna contractor is located? One has to do a lot of experiment to figure it out and it depends on the flow condition. So, it is not like a universal location where it will always be located. So, it is not that trivial to put the manometer location correctly that is one of the big errors because we are assuming that the manometer limb is being put at the section of the Venna contractor and we are writing our equations accordingly but actually it may not be. But let us say that this is put in the section of the Venna contractor. Let us say that area of cross section of this is AC and the velocity of flow through this section entire section is uniform and is equal to VC. Again we are assuming uniform velocity profiles which is a deviation from the reality and with such a kind of abrupt change the deviation of from the reality is more severe. Now here also let us say we consider this as section 1 or maybe a 0.1 on the section 1 but if it is a uniform velocity profile we consider V1 to be same throughout the section. Let us consider A1 as the area of cross section which is basically if capital D is the diameter of the pipe then A1 is pi capital D square by 4. Let us say that small d is the diameter of the orifice and let us utilize the subscript o to indicate the orifice. So, let us say A o is the area of cross section of the orifice which is pi small d square by 4 where small d is the diameter of this orifice. And let us say that V o is the velocity through the orifice again we consider it is a uniform otherwise there is no meaning of the term velocity through the orifice it will vary across the section. So, if you write the Bernoulli's equation between say 2 points let us mark 2 points let us say we have a 0.1 and a 0.c. Point c is located on the same streamline as that of 1 but in the Vienna contractor section. So, we are writing the Bernoulli's equation between points 1 and c along the streamline which is identified by this black colour. So, what is the equation P1 by rho g plus V1 square by 2g plus Z1 is equal to Pc by rho g plus Vc square by 2g plus Z2. So, again the question comes that how will you find out the difference between P1 by rho g plus Z1 and P2 Pc by rho g plus Zc that is by using the manometer principle. So, let us say that you have the depth of the limb as marked in the figure and let us say that delta h is the difference in the reading of the 2 limbs of the manometer. So, utilizing the principle of manometry you can write that if you have a and b as these 2 points you have pressure at a is equal to pressure at b. So, pressure at a is nothing but pressure at 1 plus if rho is the density of the fluid that is flowing through the pipe plus rho g and let us say that this is the datum also with respect to which we measured the height. So, rho g z1 is equal to pressure at b is equal to pressure at c plus z1 minus delta h rho g plus delta h into rho of the manometric fluid into g where rho m is the density of the manometric fluid. This is say like the same equation what we had for the venturi meter there is no difference. So, from here you will be getting a difference between P1 by rho g plus z1 minus Pc by rho g plus sorry this is gc right plus gc what is that? That is equal to delta h into rho m by rho minus 1 into g. So, g is already there. So, only this so that you can substitute in this expression and you can write a1 v1 is equal to a c vc right. So, you can eliminate v1 by expressing it in terms of vc. So, from this expression what you will eventually get you will get vc by combining this manometric equation and a1 v1 equal to a c vc. But when you get this vc let us call it say vc theoretical because again we have used the theoretical equation. This assumes that you know the idea of cross section of the Vienna contractor which you actually do not know. Now the actual velocity vc actual by vc theoretical this is not equal to 1 because of certain non-idealities which have not been considered in this equation. Just like the volume flow rates are not also same the velocity is calculated the actual and this theoretical consideration they are not going to be the same. So, this is again considered to be a coefficient c v this is called as coefficient of velocity. Coefficient of velocity you have to keep in mind it is also a coefficient of ignorance used by the engineers. Because actually we do not know what is the velocity we can only estimate from reading the some kind of theoretical velocity. But there is a difference between these 2 and because of losses the actual will be less than theoretical. So, this will be less than 1. Now if you want to find out the flow rate q, q is the actual q it is vc actual into the area of cross section of the Vienna contractor. This is q actual vc actual you can express in terms of vc theoretical. So, vc theoretical into c v into now see area of cross section of the Vienna contractor you cannot really measure when you are doing experiments. What area of cross section you know with more confidence you know area of orifice because that is like it is usually given the geometrical construction and everything the manufacturer knows exactly what it is. So, you can change the basis from Ac by writing this as Ac by area of cross section of the orifice into area of cross section of the orifice by changing the basis from Ac to AO. So, this is again another coefficient which is the coefficient of ignorance. We do not know but we expect that the manufacturer has done a lot of experiment to figure it out and this is again not a constant it depends on many things that what is the ratio of the big diameter to the small diameter what is the velocity of flow. So, it depends on many things but if the manufacturer has done lots of experiments and has calibrated the device against something more standard then the manufacturer can give a data on that. So, this we call Ac by AO as another coefficient. So, Ac by AO this we call as Cc which is called as contraction coefficient. So, we can say that the final expression is you have Q is equal to A naught into vc theoretical both of which you have determined A naught you know area of the orifice you know the vc theoretical from this simple analysis multiplied by Cc into Cv and this we call as the Cd here coefficient of discharge because this is the sort of ideal flow rate. But the thing is this is a different this is bit different from the previous case because in this case the areas and velocities are referred to 2 different sections area is referred to orifice but velocity is referred to Vienna contractor that is a basic difference but otherwise notionally it is like a sort of ideal velocity and this is an actual velocity. So by the definition of the coefficient of discharge it is like Q actual by Q ideal. So you can say that Cd is equal to Cc into Cv because these 2 combined non idealities are there in the calculation the Cd is much less than what you get in a venturimeter. So here the Cd in such a device may be say 0.7, 0.65, 0.7 like that it is not as close to 1 as that for a venturimeter that makes it a more inaccurate device than the venturimeter but the advantage is that is much cheaper than the venturimeter. You just have to put a plate with a hole in the pipe and put the manometer tapping properly and classically if capital D is the diameter of the pipe this manometer tapping is kept at a distance of roughly say capital D from the plate and this is roughly like capital D by 2. This is one of the standard engineering practices of putting these tapings. It is not necessary always that one has to put that but with a lot of experiment that has been found that then these 2 represent the proper sections with the kind of considerations that we are looking for. So this type of device is known as orifice meter and this plate with a hole is called as orifice plate. The whole objective is to reduce the cross section area so that the velocity is increased and the piezometric head is reduced and the reduction in piezometric head is measured through a manometer. So same principle as that of a venturimeter but a much less accurate one. In reality there is a there is a there is some device which is in between that is called as a flow nozzle. So what is a flow nozzle? We will not go into the detail construction of a flow nozzle we will just try to go through the philosophy because it is something in between the venturimeter and orifice meter. It is the cost is in between the accuracy is also in between. So what it does is instead of putting a sharp orifice with an abrupt change puts a kind of a nozzle at the wall to have a more gradual change of cross section of the area. It does not make it as good as the venturimeter but sort of compromise between the venturimeter and the orifice meter. So that is a flow nozzle its performance is also a compromise. So with this kind of flow through the orifice let us consider a very simple example to illustrate it that in what other conditions these types of concepts of Vena contractor also come into the picture. And one such example is something which you have encountered many times that if you have a tank and if you have a hole through the tank there is a water jet that goes out. So example of flow through an orifice let us say you have a tank like this and there is a hole through this hole some water jet comes out okay. Let us try to draw a streamline say let us say you are coming from the free surface the streamline gets bent or curved to accommodate this one. All the streamlines which are there they are getting bent or curved and just like the previous case the streamlines come to or converge to a location of minimum distance of separation between these 2 before they diverged and then maybe the water is falling like this. So the location where the extreme streamlines come to a minimum distance of separation is somewhere here which is the Vena contractor here but not at the orifice okay. So this is the place what we are looking for and let us say we identify a streamline from going like this and we want to apply a Bernoulli's equation from in between the points 1 and 2 along the streamline assuming it to be ideal and let us make certain approximations so that it matches with the high school thing that you have learnt. So what are the approximations we will make we will make that we will assume that it is a steady flow that is number 1, number 2 we will assume that it is frictionless flow and then we will also assume that the area of the thickness of the orifice is such that it is much less than the area of cross section of the main tank. So if you have that then you neglect V1 as compared to V2 so if you write say P1 by rho g plus V1 square by 2g plus G1 is equal to P2 by rho g plus V2 square by 2g plus G2 okay. So once you have that what happens see 1 and 2 we are assuming both are at atmospheric pressure so you cancel the 2 pressures V1 you neglect as compared to V2 and G1-G2 let us say that that is equal to h which is a function of time maybe but at a particular instant therefore you can write V2 is equal to root 2gh a very famous formula known as Torricelli's formula because Torricelli first derived this you know from high school physics. Question is other than the approximation that we made one very important thing deviation that we have made from the high school physics what we have not considered the area 2 to be at the tank orifice why because we have considered the pressure at 2 to be P atmospheric if there is a serious streamline curvature then there is no guarantee that throughout 2 pressures P atmospheric because of the streamline curvature there will be a difference in pressure as you go across it only where it is a Vena contractor that is true because streamlines are parallel so there is no normal gradient of pressure across the streamline. So whenever you have called it that same as P atmosphere you have to take this section 2 at Vena contractor so that is that means this is not the velocity at the exit so if you want to find out the flow rate if you write the flow rate it should be a0 into the V0 where o is 0 or o is the orifice but this is actually the velocity at the Vena contractor so you must compensate for this you can write this also in terms of the coefficient Cc Cv like that so your V0 is not same as root 2 gh and you can therefore write this q in terms of the coefficient some coefficient C times root 2 gh where this coefficient C takes into account this deviation that is not actually at this section that you are considering but at a section which is located at the Vena contractor so that you have to keep in mind so that is how so this is like a coefficient of a coefficient of velocity times the area of cross section times root 2 gh where this is like a coefficient that takes care of that non-ideality. Now finally we will come into one example where we show the use of an unsteady Bernoulli's equation for a practical device so let us consider that we have similar arrangement like a tank with a pipeline in this pipeline there is a valve so you have a valve this valve when it is fully closed it does not allow the water to be discharged through this pipeline okay. Now suddenly this valve is made open and water is allowed to flow so you have to find out that how the velocity changes with time assuming the flow velocities to be uniform over each section so then if we consider a streamline between the say points 1 and 2 and if you write the Bernoulli's equation here the velocity is clearly a function of time so you have to write p1 by rho g plus b1 square by 2g plus g1 is equal to p2 by rho g plus v2 square by 2g plus g2 plus integral of 1 by g that is the extra term that you get because of the unsteadyness p1 is like p atmosphere and when the valve is open it is also released to atmosphere so this is when the valve is suddenly made open that is what we are trying to analyze so then these 2 pressures are the same because then this is atmosphere when this valve is totally open that is exposed to atmosphere let us say L is this length of this pipe so you can and you may neglect v1 as compared to v2 if the area of cross section of the tank is much larger than that of the pipe so let us say that you neglect that g1-g2 is like h let us say so you have h is equal to v2 square by 2g plus 1 by g now we have to approximate this term so this clearly has 2 parts one part is like you may consider the part within the pipeline and another part within the tank so what is this you have you are at each and every point you are locally finding this time derivative of velocity and integrating this over this entire length so how you are doing it you are doing it by considering maybe this part and this part clearly for the part within the tank the velocity is much less than the part within the pipe so this may be approximated to be as good as the part within this length L and because the area of cross section is not changing here like v is not changing with the length so this is approximately same as like dv2 dt into L okay so this is like 1 by g into dv2 dt into L so from this consideration you can you can integrate this by considering at time equal to 0 v2 equal to 0 because time equal to 0 is the time at which the valve is suddenly kept open and then like you can separate variables and integrate to find out how v2 varies with time it is a very simple integration so the whole idea is that how you utilize this unsteady term properly to find out an estimate so I will end up the discussion of this class by giving you a very simple exercise again like a junior class level problem so you have a tank like this you have 3 holes in the tank if this height is h the central hole is at h by 2 and the others are symmetrically located one at the top one at the bottom so when water jets are ejected like this which one will traverse the greatest distance this is like your entrance examination problem you have solved it many times now you try to figure it out can you tell from your maybe memory or whatever which one should be the most yes not the upper one not the lower one but the middle one and now with all these background that we have developed your objective will be to find out yes the middle one will be like that second is what are the approximations or assumptions under which that analysis will hold true so I hope that you will complete that exercise so with that we stop our discussion today in the next class we will start with a new chapter the conservation equations for control volumes thank you.