 Good morning. We will continue our discussion of non-intrusive techniques, measurement techniques applied to sprays. We will first start by making a quick list of some of the techniques that are commonly used and then we will look at the theory and principle of operation of those instruments. We have already looked at a couple of different non-intrusive measurements which we looked at videography or photography in more specific sense. We looked at particle tracking velocimetry as well as particle imaging velocimetry. The particle imaging velocimetry is a technique that is used to obtain the velocity field given a certain set of particles that are in motion. But for the specific case of a spray, we are also interested in size. So there are techniques, there are algorithms that have been developed to sort of ride upon particle imaging velocity algorithms to give us both size and velocity information. So one very simple particle sizing algorithm in a PIV system is to use the scattered light intensity. So think of a spray and I am looking at a certain region in this spray which is my image and in this image I may have some large drops and some really small drops. Now mind you we are looking at a PIV. So we are not at a level where we are able to see individual drops. So it is sort of a picture that gives us a grayscale intensity and in this grayscale intensity picture if I take a certain region of the spray, I have a certain grayscale distribution of light and this grayscale distribution of light gives me the intensity in an image like this is proportional to d squared where d is the diameter and this comes from what is called me scatter. So if I take this simply just to understand me scatter for take a spherical drop and subject it to a bunch of light rays I have the possibility of reflection at this surface followed by a reflection again sorry I have the possibility of a refraction at that surface followed by refraction again at the second interface or I have the possibility of just reflection at this surface or I have the possibility of refraction at this surface a reflection at the second surface followed following the first refraction and then I could have another refraction at this surface or depending on the situation here I could have like total internal reflection followed by refraction there is a whole combination of physical phenomena that come come into play when I have a set of light rays encountering a spherical transparent object which is capable of both reflecting and refracting at the interface. The combination of all of these effects as well as the gives rise to what is called a me scatter pattern that is if I take if I plot the intensity of the light as a function of the 360 degree angle going around the drop it is known that there is a lobe at the centre along the axis of the light move this to the next page. So, there is an intensity lobe at the centre line let me use another colour there is a second lobe which is smaller in magnitude but slightly off axis there is also a third lobe which is more predominant at for the back scatter level. So, if I plot the intensity versus theta theta being in some sense this angle I have the maximum intensity on the centre line and the as I walk around the centre line away along a clock away from theta the intensity initially decreases reaches what looks like a minimum and then increases. So, this is called the first lobe and this is the second lobe of me scatter this is also another lobe that follows from the back scatter mode. So, where light is for light to reach this play this point approaching from my left hand side the light essentially we are looking at what looks like back scatter scattered light coming back towards the source whereas the first in the second lobe are more forward scatter lobes. Now, this pattern would be symmetric as symmetric as my figure looks and this symmetric distribution about the about the axis has these distinct lobes which are which can be observed. So, if I have a spherical particle suspended let us say in a shaft of light as I walk around I will see the specks of light appear brighter and then dimmer again when I approach one of these minima and then becomes brighter again this is classical me scatter pattern. So, if I now look at this image. So, in this case let us say I have this light sheet approaching from the top this light sheet approaching from the top and if my camera is placed vertically where my where your eye is at this moment then the image formed essentially is as though I am looking at this from a 90 degree scatter point of view. So, this is my camera I am looking at a spherical entity sitting here receiving light from the from the top and scattering light towards your eye and that is in this schematic it looks like a near 90 degree axis. And so, the intensity of the light scatter at the 90 degree position without if I do not change the angle the intensity of the light scatter at that position varies with d squared varies as the diameter of the drop squared. So, it is essentially a phenomena where the intensity is proportional to the surface area of the drop. And if I receive if I receive different amounts of light from different points in this image I can now knowing the intensity of the light coming in and through some calibration process. So, let us say before I came to this experiment I went through a calibration process where I put drops of a known in known diameter into the same light sheet and then measured the intensity of the scattered light coming towards the camera. So, from that if I take this eye proportional to d squared I can now convert this to eye being of the form k times d squared. So, k is a calibration constant that I can calculate or that I can measure from my calibration process where I set up an experiment and in and have drops of a known size in the light sheet and knowing the intensity of the light that the camera sees. I am able to measure I am able to obtain a value of k during the calibration process and I can use that same value of k in my real experiment where I am dealing now with and drops of unknown size, but I know the intensity. This is in some sense the basic principle by which I can add a particle sizing algorithm to a standard PIV setup. Where I take any one of the two images remember PIV deals with pairs of images I can take any one of the two images get the grayscale intensity of the light that the camera sees and knowing this calibration constant k I am able to calculate a distribution of surface area I do not get individual particle sizes. See look at it this way this green image here is a grayscale intensity that is all what you have to deal with. This grayscale intensity at any point is proportional to the surface area the total surface area of the drops present in the per unit volume. So, I if I have mono dispersed drops then I know that I is exactly equal to k times d squared. If I have a whole distribution of drops like in a real spray we will still say that this I is equal to k times the total surface area per unit volume. So, TSA or in some sense instead of saying total you might say this is the mean surface area like an average surface area in this little green rectangle. And that surface area liquid vapor surface area that is present in this green rectangle is responsible for the grayscale intensity like an average grayscale intensity at that point. So, I can look at the entire image look at where the image is bright and assume that there is a higher surface area of the drops. I could get this surface area in one of many ways I could have a few drops that are large in surface area locally concentrated there I could have a large number of very small drops. So, all I know is some sense the total surface area that is available for scattering the light. So, I have now developed a methodology by which we can take the grayscale image and convert it to a surface area distribution spatial distribution of surface area that is essentially what you get from processing a single image. There is a problem with this approach though and it goes back to this mescater pattern. Now, if I have a drop exactly at 90 degrees this dot here indicates the sort of level of intensity I might expect that to be seen by the camera. So, if I take an image like this like I will take this blue image and typically and I will draw this out separately just so we know what we are dealing with. If this is my imaging area and if this is a camera that I have now really speaking the camera is where your eye is. So, I am going to draw a projection of this. This is my imaging area it is a thin light sheet and I have lots of drops inside this light sheet that is the whole idea just like these drops when projected look like this. Now, drops in one part of the drops in this edge of the image subtend a different angle. Now, mind you that we have light coming in from here just like in the image I showed before. So, if I have light coming in from your eye towards the spray in the second in the bottom image the scattered light makes an angle different from different at different points in the spray. So, the angle made by the scattered light coming towards this camera from one of the edges of the spray is different from the angle made by the scattered light coming in from the middle of the spray. So, if my field of view is much larger than the camera aperture which is always the case it is very rare that you are imaging at 1 to 1 or less than 1 to 1 magnification in a typical PIV application. So, you are really looking at a fairly broad region in the spray and so this typical angle may not be as large as the schematic indicates, but it is surely not small it surely not 0 degrees. If the angle is nearly 0 degrees which is the case like in a microscope you are essentially looking at the scatter from all the drops being very close to that 90 degrees. So, light coming in from the top and scattering light towards the observer whereas if the light coming in from the top and the region of interest is much larger than the camera then I have light being scattered towards the observer from different angles from at different angles in different parts of this rectangular image. If you go back to this me scatter pattern the intensity of the light observed is different when observed from different angles. So, the same size drop present in different corners of this image or in the middle of the image would appear brighter or dimmer ok. This is a challenge because this k the calibration constant k now becomes a function of the scatter angle alpha. So, for different points in this image I have to know what the angle over which light is scattered towards the camera. So, I know the angle of the incident ray in some frame of reference and I know the angle subtended by the scattered ray towards the camera I mean I am using ray tracing arguments, but for now this is sufficient at least to get a qualitative picture of what is happening here. So, the angle subtended by the scattered ray coming towards the imaging frame imaging plane the angle between these two is essentially what we saw here as this theta this is the scatter angle. So, if I take an imaging frame like this I need to know the scatter angle for all the points in this imaging plane. So, for example if my camera is right above the imaging plane the light scattered from the top right corner of the image versus the middle of the image versus the bottom right corner of the image would be different and I need this k at different points in order to get a quantitative picture of the total surface area distribution this is a challenge and can only be addressed through proper calibration. So, I populate this light sheet with drops of a given size known size and get the image from different get the entire image over the frame take the entire frame and knowing that I am supposed to get the same grayscale intensity at all points as though I was looking at the entire image from the same scatter angle that is where the calibration constant k has to be derived. Now, some of it is pure geometry where you get the scatter angle and estimate the value of k given i and d squared at that point. There are other sort of more optical effects which are aberrations that have to be accounted for when you are doing light sheet imaging like this, but we would not go into that at the moment because we are only looking we will only focus on sizing and velocity measurements for now. Okay, so give if I was able to get a k as a function of alpha I have a way of taking one of the frames in a PIV imaging system and get an estimate of the total surface area in different points of the image and I will take the surface area as indicative of the size at that point that is the that is a possibility that is a method by which we can get some size related information in a PIV. We will move on and talk about another instrument that is very widely used in particle sizing called phase Doppler particle analyzer. A phase Doppler particle analyzer is an instrument that measures both size and velocity instrument information at a single point. Okay, the basic construction is as follows take a light beam a laser you do require a laser here for PIV really you do not require a laser there have been some studies where people have used reasonably high intensity LED sources to get PIV because all I am doing is imaging I am getting two snapshots of a moving set of particles and I am not taking advantage of coherence in the light source to in a typical PIV system. The one big advantage that a laser has still is its intensity because it is a collimated and coherent light source it remains a very high source it is a very high intensity light source and intensity matters in a PIV because the if I go back for a moment this I is the scattered light intensity really speaking I should strictly write this as I by I naught where I naught is the source light intensity. So for the same I naught if I increase I naught for the same D I am likely to get a higher intensity of the scattered light back towards the camera which also means that if I take a certain source if I take a certain I naught intensity of my source light and if my camera has a threshold of detection that is it is unable to sense intensity levels below a certain lumens or whatever in unit you are comfortable dealing with then that intensity level determines the size of the drop I am able to detect. So if I want to detect very small drops I do require a high value of I naught because I by I naught is the ratio that scales with D squared surface area okay same applies to PDPA here. So I take a laser of a certain intensity there is a mechanism involving some there is a construction involving beam splitters and mirrors inside here which I mean it is not really important but essentially I create two laser beams and in one of the path in the path of one of the beams there is a phase shifter okay this is often called a Bragg cell it is a small it is like a piece of glass if you want to imagine it in very simple terms. If I put a small piece of glass in the path of one of the two beams versus the other I am going to shift the phase because the optical path length is proportional to the refractive index times the physical length if the refractive index of air is one the refractive index of glass is higher in relation to the one. So I am going to create a higher path length for one of for this bottom beam in relation to the top beam. So that is the simplest way to think of these. So if I take the light here if I plot the wave intensity or wave amplitude as a function of time if the top beam has an intensity that looks like that on the same plot the bottom beam may be slightly shifted in intensity as you shifted in phase. So this phase shift you can call it a phase shift in time or a phase shift in phase angle this phase shift is important we will see later. So the moment I take two light beams that are of the same wavelength and potentially even of the same intensity will come to that in just a moment but a one phase shifted in relation to the other okay you see some very interesting things. So the next part of the construction is that there is a convex lens that you use to focus these beams down to a single point this forms my measurement volume. So we said a point but it is really a measurement volume because each beam is approximately about one mm in diameter slightly less than one mm typically one mm. Now the if I take two beams that are approximately one mm in diameter and form an intersection you are essentially looking at the intersection of two cylinders that are one mm in diameter. So this is your measurement volume and it is typically about one mm cubed. So this is a point measurement instrument point that averages information over this scale of length about one mm. Now this part is called the transmission optics. So this is called a transmitter in the phase Doppler particle analyzer construction. So you are essentially consisting of a transmitter, a receiver and a signal conditioning system so those are the three big building blocks. This is the transmitter and the receiver construction involves three detectors that are separated physically in space. So these are three detectors with an appropriate lens in the front. This is called a collection lens. So in relation to this I may place a transmitter over here. Now I am only going to draw one laser beam here but you would have to imagine that the plane of the beam is perpendicular to the plane of this sheet of paper. So I do have two beams that are intersecting in this way and the receiver is placed where my right hand is. So I have two beams coming in this way and the receiver is placed my hand is and so in the plane looking at from the top where I can draw the receiver if the receiver is as you can see slightly off angle and that angle and that in the plane that I can draw this receiver the two beams of light are coming in and intersecting at this measurement point here. Now if I look at the measurement volume let me first draw a zoomed out picture of the measurement volume. I have two beams coming in and these are in a sense plane waves that are shifted in phase slightly. So where the phase there are some parts where the phase add up where the amplitudes add and other parts where the amplitudes are opposite in phase. So you get essentially dark and bright bands. So this is an interference pattern that is created pattern created due to the phase shift. So in fact you can see this if you take a phase Doppler particle analyzer and you just spray mist into the measurement volume where the mist is much much smaller in size than the fringe spacing. Or you just spray mist and look at the measurement volume with a microscope. You can see these dark and bright. So my finger I am imagining to be the bright band followed by a dark bright dark bright dark etc that is at a fixed spacing. The brightness of the bright band in relation to the dark band. So ideally my dark band should be exactly black should be pitch black if the two intensities are exactly equal the two the intensities of the two beams are exactly equal. I will get the dark band to be nearly perfectly dark the bright band to be bright. This is important because this contrast is what I am going to use for particle velocimetry and sizing. So if I come back to the construction this is the part this is my measurement volume and in the measurement volume I have fringes in the fringes in the plane of paper here the fringes are in an orientation perpendicular to my plane of the paper and the spray typically is coming in from the top. So the drops are moving through these fringes and I am only going to focus on the motion of the drops perpendicular to the fringes the component of the velocity perpendicular to these fringes. So we will first look at velocity measurement and then come into the sizing part as far as a velocity measurement is concerned if I have a particle going through these fringes and imagine I have one detector here sensing the light scattered from the motion of this one particle. If I plot the intensity measured by one of these detectors I will call this 1, 2, 3, the I2 the intensity measured by 2 as a function of time goes as I will see I will see the drop or the drop scatters light or the detector sees the drop when it is in the bright part when it is in the dark spot you do not see the drop then again you would see it is like a little speck that appears and disappears that is sort of a simple mental image that you can carry with you and in this process there is another effect that you have to understand that lasers have an intensity distribution. So if I take one of these beams the beam itself has a Gaussian intensity distribution that is the middle of the beam is much brighter than the edges of the beam this is characteristic of any laser source now if I take this Gaussian beam and create these fringes the fringes that are formed at the top and bottom are going to be less bright in comparison to the fringes in the middle as a result the intensity pattern that we see takes on a shape that looks like this. So this is intensity measured by the detector 2 as a function of time and this intensity profile falls in this envelope which is due to the Gaussian intensity distribution. So every time one of these particles passes through the measurement volume I am going to see this I2 as a function of time this is called a Doppler burst I am going to see every time a single particle passes through the measurement volume I will get one of these Doppler bursts and if I know the fringe spacing which is this part in space. So these are perfectly equally spaced interference fringes where the fringe spacing only depends on wavelength of the light and this angle of incidence. So this theta and lambda the wavelength of the light something to that effect. So if I take this fringe spacing and place it if I know the fringe spacing and if I know the time separation delta t between one peak in the Doppler burst and another peak in the Doppler burst we are able to then calculate the velocity. So if I know the fringe spacing d and the spacing between the peaks in time we are able to estimate the velocity of that particle but in reality the way this is done is one of these Doppler bursts is sampled at some fairly high frequency. So I have the intensity measured by this detector to at various instance of time. If I do a fast Fourier transform of this Doppler burst there is a particular frequency in time that shows up as a peak frequency and that peak frequency before we do that in fact the before you do any sort of a fast Fourier transform this intensity I2 versus time is converted into a profile where the mean is subtracted. So you get something that looks like this if I subtract the mean intensity of over this entire data set I will get something that looks like this and now so this is let me write down the different steps. So the first part is where you subtract the mean the second part is where you do an FFT of it and in the FFT you get the peak frequency and from the frequency let us say this I will call this F delta t then is simply 1 over F forget the peak frequency that frequency 1 over that frequency gives me delta t and the velocity of that particle is this d over delta t or d times F d being the fringe spacing. So this is a point wise particle wise measurement system. So I take I get a velocity measurement of every particle passing through the probe volume and if I by chance have more than one particle passing through the probe volume at the same. So if they overlap in time in terms of their residence inside the probe volume I will not get this clean sort of a Doppler burst I will get some image that does not have these clear peaks because I may get a peak from a particle up near the first fringe while the previous particle is still inside the probe volume and that causes a problem to the instrument. So typically it looks for a certain fidelity in the Doppler burst before it does the FFT. So the peak in the frequency should be a very clear peak in relation to everything else. So these are all part of the signal conditioning and signal processing algorithms in the instrument but essentially it can only measure it is hardwired to measure the velocity of one particle at a time. So therein lies a very simple limitation that I cannot take this instrument very close to the spray nozzle where I am increasing the probability of multiple particles residing inside this 1 mm cube volume at the same instant of time. I can only use this instrument sufficiently far away where the particle density is about one particle per mm cube that is the density at level at which particle number density level at which we can safely operate this instrument. So that is the basic method by which one can estimate the particle velocity. Now for the particle size the way this is done is if I now draw out the different fringes I have these bright and dark spots. So this is the bright and this is the dark fringe. If I take a drop let us say a drop that passes through these drop of a certain refractive index pass through these fringes. So I will just quickly superpose the drop on here. Now if I was to look at these bright and dark fringes through a glass sphere. So I am going to just to understand what it looks like. If I look at these static bright and dark fringes through a glass sphere the spacing between the fringes would appear different. That spacing would essentially so if I look at it from through a glass sphere the spacing would increase through the drop. So if I now go back to my transmission optics or receiving optics sorry this is my receiver. If I go back to the receiving optics I have these fringes as the drop goes through these fringes the receiver is seeing scattered light coming in the refraction mode or the forward scatter mode coming towards the receiver and the fringe spacing that the receiver sees increases based on both the volume both the size of the drop and the refractive index. So if I know this if I know the refractive index of the liquid being sprayed I can get the size of the drop. You can do also you can do this increase in fringe spacing also through sort of ray trace optics you know it is just to get a feel for it. But essentially one can understand that the fringe spacing is now higher when viewed through the drop. So the previous argument that we looked at where I took I2 the intensity observed by one of the three detectors and from just that one detector I am able to get the velocity information. But I need to know the fringe spacing. If I now have this transparent drop I do not know the size of the drop but I do know the initial fringe spacing without the drop that is from the geometry of the intersecting beams. So if I take the fringe spacing to be a function of the refractive index and the drop size and I have multiple detectors. So really speaking I only need one more detector. If I know the intensity versus time Doppler burst pattern on detector one and then detector two I can so I have two different Doppler bursts from that I can get both the size which is the fringe spacing as well as the peak in the frequency which is related to the velocity. So really speaking if I have two detectors that gives me enough information to calculate both the fringe spacing and the velocity independently knowing essentially the phase shift in this Doppler burst. So in other words when a particle passes through the probe volume detector one which is spatially separated from detector two inside the receiving optics sees the Doppler burst at a slightly different phase the two detectors see this at a slightly different phase from the phase difference between I1 and I2 as well as the peak in the frequency from any one of the two detectors we can get both the fringe spacing related to the size and the droplet velocity related to the peak frequency. In a real PDPA however there is a third detector that is usually incorporated to get what is called to use what is called a validation procedure. So we look into that at the beginning of next lecture where we look at why the third detector is needed it is essentially needed to make sure that you are seeing light only in the refraction scatter mode and not any part in the reflection scatter mode. So this is important and the third detector is there to ensure that among other things. Now this is the basic principle of operation of a phase Doppler particle analyzer if you remove the sizing part this is also the basic principle of operation of a laser Doppler velocimeter LDV. The basic and of a one component LDV so we are only measuring the component of velocity perpendicular to the fringes and any component any velocity in the in a plane that is not perpendicular to the fringes either this way or this way is ignored. So if I now create fringes in the other two planes we can get three components of velocity independently we look at that and the validation algorithm at the beginning of the next lecture. Thank you.