not that is [Music] f okay you get all right good morning morning everyone sorry I'm a bit late one of the kids got sick you know uh yeah anyway oh thank you for coming in in drw this is the worst morning to come here it's dark it's wet it's cold yikes I mean I'm from tropical country so anywhere below 20° too cold for me um yeah anyways so we're at chapter eight now chapter eight is easy it's not interesting in teaching you chapter 8 well there are problems that I selected that I think are relevant to uh whatever that's has the chance of appearing in the exam and as if you can see uh I don't know if you had time to glance over because I just sent it last night the problems are mostly just applying formulas so take your you know pen and paper and then write these formulas with me there are some explanation as to how those formulas come about I'm going to explain it um in some details but if you take a look at these lecture notes that I put there are some derivations of those formulas uh you're not going to be asked to derive anything in the exam the goal of the module is to allow you to use these principles but not to to derive them well how about the first Formula Reynold's number but before that I want to recap a little bit about what you've studied so far time flies and it uh the first chapter was about viscous flow uh you know everything about sheer stress about viscosity about its effect uh on the flow when solid and fluid are in contact with each other and then you forget about that for a while in Chapter 2 and three in chapter two you deal with how pressure varies according to Heights or according to depths and then chapter three you deal with ideal flow and these are berul equations continuity and all that anyway now all of them come together in this thing viscous flow in pipe now if you have a flow in a pipe so chapter eight is really for chemical engineers and there's a reason why you're not studying chapter 7 or chapter 10 uh some of them deal with uh external flows so those are for aerospace engineers but you deal with internal flow that doesn't mean the flow happens indoor that means that the flow takes place in a pipe it's a in a pipeline in a pipe in a valve anything that is contained uh it that is contained anyway so if you have flow in a pipe that's your pipe right there are many ways by which the velocity profile look like in a pipe if you take a look at the faucet in your kitchen and then you know in the morning you open it and water starts pouring out from the faucet you see that at some point when you don't open it big enough it flows rather uniformly like this for example right like the one you observe in orifice plate when it's steady you notice that you know there are some order in this you can always see it as something that is still right it's almost not moving it's steady state this region as the flow just come out of the orifice plate is what we called laminar flow generally speaking and then somewhere Downstream if you allow it to fall uh far away enough you know things starts to become disorderly right and you know you see something like that you you see that everywhere generally speaking this is an illustration of flow transforming from being laminar to being turbulent now I cannot stress you enough how important this is in terms of looking at these two flows from different lenses as a mathematician or as a scientist or as an engineer when the flow looks like this almost everything here can be described from first principle you can derive for example how the flow profile looks like just by using Newton's Second Law just like what we've been doing in the first three chapters but when the flow becomes turbulent nothing even until now uh people has been able to describe that what I meant is this if you if your flow becomes turbulent and then you want to find out at any depth for example that depth okay at that depth what is the velocity no way to find out precisely no way to find out precisely so what do engineers do well they run experiments they run bunch of experiments and then they average those results and and then take a guess it is like that now when the flow is still laminar it's different provided that it's perfectly laminar yeah laminar any place here there you want to find the flow profile here what's v there V3 V1 V2 you can tell even without going to the laboratory you can tell and the lecture notes will tell you exactly how to do that now in terms of flow in the pipe the most important thing in the exam would be to know when the flow is laminar and when the flow is turbulent well you can tell well that's kind of laminar everything is orderly it's look as if the flow is frozen and this looks disorderly the diameter keep changing fine but when it's in the pipe you cannot tell when the flow is in what regime right so there is one method to do that it's called the Reynolds number now this you have to memorize Reynolds number is a non-dimensional number that means it has no units uh it means um the number isn't followed by anything meter per second nothing it's just a number now that number is a ratio yeah between inertial forces so that's the definition of three lines inertial forces and viscous forces so that's what roughly what that number is inertial forces viscous forces that thing is so so philosophical that I you know even if you are I think and I hope some of you decide to do PhD one day and to di deeper into this these two forces like Good and Evil all the time you know it's like Yin and Yang in Chinese culture folklore whatever I think it manifests in different ways but quite similarly in all cultures this one is chaos this one is order this one you know keeps moving uncontrollably this one dampen things slows things down make it orderly all right now the most beautiful thing in physics is that you can actually put the formula in so you know the number exactly it's not just you know ah this is you know uh you know Enlightenment or anything like that that so row u d That's inertial forces and then the other one is Mu you know about mu mu is viscosity right and row is density U is an average velocity and D is the inner diameter of the pipe so that's your D that's D okay and well I don't draw you the U profile first because I don't know if it's laminina or turbulent okay and row okay now you have to memorize this you have to memorize this um there are many variations of this Reynolds number because you can change absolute viscosity into kinematic viscosity if you want or you can change U into volumetric flow rate if you want what else you can change um you can change it into mass flow rate too if you want but that in general is what you have to memorize so if there's any question in the exam that tells you you know what would be the kind of number or parameter that distinguish two different flow regimes you say Reynold's number and then explain all the terms well you explain all the terms and you tell me you know what these parameters actually mean yeah okay now for example if you want to change it into another form I think the other variation would be u d divided by kinematic viscosity like that it's a n it's a Greek letter knee it's like a V but uh drunk and slanted left so that's kinematic viscosity or you can express this in terms of volumetric flow rate and that will make it Q D over well obviously volumetric flow rate calls for area or you can simplify it into 4 q whichever one is shorter I think D something like that um given that volumetric flow rate and mass flow rate is actually connected don't worry I'll slow down I'll repeat this again hang on oh I could do this what is that no that's what's that oh oh well that doesn't make sense this where's this camera facing you see see that guy it got stuck good thing I bring my own camera I knew it will happen anyway yeah whatever now what I said before because q and mass flow rate m dot is connected well you just multiply it by uh density right given that given m dot equals density mass flow rate F volumetric flow rate then this guy can be 4 m dot divided by this isn't really interesting to explain because it's just formulas after formulas after formulas well I don't like memorizing formulas so I won't at least I won't memorize all of them I think this is the only thing I will memorize and make sure you make sure you memorize it always you can change it into anything right let's say the problem tells you instead of you the problem tells you well there is a flow in a pipe the volumetric flow rate is something thing calculate the Reynold's number what are you going to do well you have to apply this relationship right and then you get the U and then you get the U you multiply it by D divide by this or you can memorize it like this then you don't have to change this into that whichever way is easier for you if you got a photographic memory bam okay that's great I doubt it you might think you have but no anyways so any questions so far let me give you an example a really quick one oh no before that hang on before that um what about if the flow is in a pipe but the pipe isn't circular cross-sectionally speaking that's a sensible sentence in English right yeah like the air ducts no who needs them right anyway in Old buildings you have air ducts and usually uh they're rectangular then what would the diameter be there's no diameter in rectangles so this is what you do you have hydraulic diameter instead hydraulic diameter and that's well DH just to make it different than H is 4 a which is area divided by this is called wet it perimeter so all I'm saying is this if you got a rectangle instead of a circle like that okay well obviously the area you know what actually this not need to be just for rectangle it could be for triangle if you want or hexagonal any shape really uh so long as you can find the area yeah so if the cross-sectional area of the pipe through which your flow take place is nothing is other than Circle then a is the area and P is the wetted perimeter now wetted perimeter depends on what fluid flows through it if if it were gas obviously gas fills the room right and then so everything is wet it because it fills the room like in this room for example all the molecule of the air touch the walls and the ceiling and the bottom but if this room is filled with water for example well it won't touch the ceiling unless the room is full with water right so if it were filled with water instead maybe you know it could be some certain depth yes in that case the weeted parameter is only the length from here to here to here okay if it were filled with gas which fills a room where it's contained then the wetted perimeter would be just the perimeter of the entirety of the cross-sectional area that would what P be okay okay now area is area period so I could change that into this if I want to and that would what your me memorized formula be if you want well you might ask what about if it's Circle well if it's Circle then [Music] If This Were Circle for example yeah you will notice that DH is 4 multiplied by what's the area of a circle Pi yes and what would the perimeter of a circle in terms of diameter yeah your voice is not commensurate with the size of your body yes ccum yes circumference of the exactly great that's all right gentle voice okay I can cancel the whole thing not the whole thing but you know the the pi the four and the D what does that reduces to the reduce to D yes so you know that hydraulic diameter reduce to D for Circle well then I could just replace that with DH if I want to all right so that's that yeah any questions yeah for the a okay yeah is the crosssection yeah yeah yeah the cross-sectional area resulting from D here yeah is it the no just the area just the area okay only this is just the area and P by definition is wet parameter here's the thing and the reason why that might not make sense originally Reynold's number if you're interested in the derivation of it the formula was row U characteristic length divided by mu now you will study flow in Porous media in year two or year three I'm not sure poros media flow is flow that takes place in a pipe but through rocks or through marbles or through complex geometries or beat packs which is ever present in the battery Industries right you know the Eon the ion has to go through in a fuel cell for example the ion has to go through porest media now in that case the characteristic length no longer is the inner diameter the characteristic length becomes the grain diameter this thing can be anything and in fact it is still a subject of a debate what should I put here this is really engineering World engineering world does not talk in certainty unlike math math talks in certainties engineering talks in well does it make me money or not really in the end it's about that all right so I cannot explain to you why this one is not wetted and this one is wetted that's the consensus in the enging engineering community so all right so now we know Reynold's number we know hydraulic diameter we have one equation for you to memorize and then there is this concept about um entrance flow region so let's make this longer now let's say this is a pipe attached to a container like a tank for example right that's the tank opening right that's a tank here right and then there's a valve you open the valve and flow starts rushing in okay now what do you think happen to the velocity proc profile immediately after the valve is opened okay that's the pipe just to make it clear do you think the velocity profile will immediately look like a parable no it doesn't because the viscous forces hasn't had the time to take effect yet so immediately here flaw looks like what newly flow looks like because V viscous versus takes place when fluid is in contact with the solid right and there's not enough body of liquid in contact with the solid yet when the valve just got open so flow looks ideal like this like that that's what flow looks like immediately at that point yeah the point is where the arrow originates the point is not where the arrow ends this is the point okay all the flow here looks like uniform why well because there's only a little area here where viscous forces take place and there's not enough time for viscous forces to creep into the center of the flow now but as flow takes place along the pipe Downstream viscous forces starts to creep in you see these molecules holding hands with the other molecules they start to oh oh no not don't go too far don't go far yet you know and then immediately this guy holds hand with that guy this guy holds hand with that guy and immediately everybody tells everybody not to go too fast that's what happens so if you take a look at somewhere Downstream you know maybe the flow start to look like this okay that means from the molecule here until the molecule here they already know that this guy here is holding hand with the solid the rest don't know yet okay now around here for example finally all the molecules know you know there's viscus going on and they look like that this is what scientists without access to experiment think about ideally it's going to look like that and the flow from being completely nonviscous a viscous or inv visit into fully viscous flow that's viscus flow remember chapter one yeah like that now there is a region region here from here let me put it in red roughly from here to here ah not good one okay something like that yeah that regime here called inv visit core inv visit it means nonviscous core that means in that core if I could I don't need to do this but make it clutter in this core the flow hasn't seen viscosity yet the flow is simply governed by inertial forces okay and finally you know at certain length at certain length that length is called entrance flow region from here until there it's called an entrance flow region L E length entrance uh okay that's an entrance flow region l e that's very important now it's just a defition you might think what the hell am I supposed to do with this information right that's really important let me digress a little bit because your job as an engineer would be to find out roughly speaking energy that I need to transfer fluid from one point to another and the length between one point to that another point is unknown depends on your client and the path depends on the specification of the factory so you need to know Le because it has great implication to the energy required to do that okay now thought that's the entrance flow region now before we move on here let's go back here again now roughly speaking roughly speaking for lamina flow Reynold's number is between 0 to 200 2,000 so 0 to 2,000 or 2,200 some book will say 2,400 some books will say 2,000 okay but let's agree on the average and then for turbulent flow Reynold's number is greater than 4,000 or 4,400 even okay you can memorize this it's recorded and uh I think my handwriting is big enough too now that you have to memorize you might ask me where do these guys get this number and why is it so imprecise they get it from bunch of experiments really carefully in close experiment in close en experiments all right so the way they do that is by flowing fluid through a pipe they can change the diameter right and they can change the U by changing the volumetric flow rate and and so they have many different set of data so they can calculate the Reynolds number the pipe is probably made out of glass so they can tell you know when the flow becomes disorderly and roughly speaking flow starts to become disorderly or turbulent at reyolds number about 4,000 is and for definite laminar flow takes place below 2,000 so what's happening between 2,000 and 4,000 well engineer says sometimes turbulent sometimes laminar the data is inconclusive so your job as an engineer is to not to design anything at ral's number between 2,000 and 4,000 CU nobody has an idea whether the flow will be laminar or turbulent at that time little disturbance or perturbation could switch laminar flow into turbulent any time between that reenal number so just to be safe design something either way above 4,000 or well within 0 to 2,000 all right so that's laminar flow and turbulent flow remember the faucet of water I opened that yeah so you know sometimes it happens near the fet sometimes Downstream right it's like that so so that's really unpredictable all right now that length here the length at which flow becomes fully viscous is very different for laminar and turbulent flow so here's another equation for you to memorize for laminar flow the length over inner diameter is about 0.06 Reynold's number so that's for laminar flow yeah yeah yikes it's like just bam equation eat it anyway the characteristic no not characteristic the entrance flow length it is entrance flow region the length this length what is it length called now okay the length is called entr flow entrance flow region l e divided by diameta divided by diameter is 0.06 renal number for laminar for turbulent sorry I'm going to write it here but so okay I I'll write it here then ah okay for turbulent it is really really ugly 4.4 reyolds number to the power of six for turbulent you have to memorize this there will be no formula given in the exam luckily that's it you memorize it and you apply it well not really there's some thinking going on but not much memorizing is thinking too all right any question so far can you see the yeah so let me repeat again so this is the entrance flow region entrance from what entrance from a container now when flow starts to take place from the entrance point it hasn't yet feel the effect of uh viscosity over time it will as it flows Downstream now there is an entrance flow regime that characterizes the length before flow becomes fully viscous that's viscous flow then right it means every molecule in that flow has felt viscosity effects now that length is different for laminar and turbulent and how do you know laminar from turbulent you know it from the formula Reynold's number so quick exercise take your calculator if you have any if not then use your phone water in 1 in pipe flowing at velocity average 10 ft per second calculate Reynold's number and calculates the entrance flow region go when I got kit cat at 12:00 is the question clear water in 1 in pipe 1 in pipe diameter that 1 in pipe means the inner diameter of that pipe is 1 in okay flowing at average velocity 10 ft per second you have no idea how it looks like laminar turbulent no idea what would the reyolds number be and what's the corresponding entrance flow region yeah what's the visity of uh okay it's about okay okay that's 10 ft per second you can check it online sorry but you know so kinematic viscosity of water is about 1.2 * 10 the^ of minus 5 um feet square per second yeah that's of water let me check it if it's right because it's in nonsi unit I might be wrong water viscosity H this one is with Milli which is exactly okay so in si is 0.001 Pascal second okay in feet okay there 1 point2 okay certainly viscosity depends on temperature now in the exam temperature is at this point is not going to be mentioned if you check Google well it depends on the temperature I think room temperature is about I don't know in Fahrenheit 60 a bit lower than body temperature which is 90 something uh so it's about 1.2 the D Dynamic viscosity is this one okay anybody got the number already what would the Reynolds number be Reynold's number is u d / kinematic viscosity that will be what go that's the point I don't know the renal number will tell you you know it from the Reynolds number so that's why you have to calculate it first anybody got the number already yeah 69444 69444 good okay well in the lecture note is about 68870 fine I guess you use I don't know different rounding okay fine but that's okay 69 people get it the same yeah and then the question is so what regime is the flow in huh laminal flow is between 0 to 2,200 turbulent flow is above 4,000 so this is turbulent flow okay so that flow is turbulent all right now so calculate for me the entrance flow region entrance flow region for that flow what's l e go in Feit please well you need a bit more sophisticated calculator okay you get it uh okay no I don't expect you to convert any any way you feel comfortable now first of all even if you do it in meter it should give you the same number because this one is non-dimensionalized right if you change everything into SI including the uh velocity and the diameter this number will still be the same has no unit anyway the entrance flow region will change because it has feet or meter right so okay so entrance flow region for turbulent follows this yeah you need a bit more sophisticated calculator and that would get you 4.4 where 4.4 come from oh from there okay uh 69 to the power of 16 multiplied by diameter and in feet would be 1 in ided by 12 ft okay that will get you about 2.3 5 ft okay it's about that now anybody confused as to why this formula is used instead of the other one because the flow is turbulent and how do you know the flow is turbulent Reynold's number told you so all right using that rule of thumb now so it's 2.35 ft it's a bit it's almost like 1 M long right so this is almost like a very like for like image it's about that long really well for diameter one inch so it's about okay so if if the diameter is this small it will take that long for water to reach fully viscous flow okay it's going to be like a there I think this is about 2 ft all right okay another one now you got to be quick this time just just to make sure that you know how to calculate it very quickly in the exam SAE 30 oil 1 in diameter u 2 ft per second viscosity kinematic 4.5 10 the^ of minus 3 ft Square per second what's the Reynolds number what's the L go so it's yeah yeah never you don't know so I don't know how to answer that nobody does because the flow is unpredictable it's very engineering right all right now it's oil similar diameter 1 inch twice as fast uh is it twice as fast yeah it is twice as F no no it's smaller okay so only 2 ft per second as opposed to 10 viscosity is considerably higher what's the Reynolds number anyone 37 37 perfect got two Kit cat 37 now very quickly is this flow laminina or turbulent loud really loud good morning laminar flow laminar flow so calculate me the entrance flow region for laminar flow uh what formula will you use and what's the number should start picking on you at some point anybody yes yes exactly you're not looking at the lecture notes right calculate okay so it's about 0.06 multiplied by renold's number multiplied by diameter and that will get you like what you said 0.1 85 was it feet or inches feet so it's about 2.2 in so it's about that yeah so it's about this it takes only that much length for the flow to become fully discus now that length that feeling you know when it comes to that that in your real engineering practice you probably just get it I have um little hobby I play with analog photography so you know in the dark room looking you know like romantic and all but um so in camera you know to get the right exposure you need to know the opening of the lens and how long it opens right that determines how much light get into the film right and then you know the opening of the lens would determine the depth of focus right you know the right bouet they call it you know it depends on how big you open the lens there's a formula to do that and you have to read the light over time though you can just look at the this is about yeah what opening corresponding to what seconds similarly with this you will get that feeling all right there is a room for feeling in chapter eight because if in the exam you get for example instead of 37 you get 45 first of all it depends how you get 45 but there is a lot of margin of error you still get 100 like the actual number in the lecture note was before 68,800 as I calculated it and he came up with 69 69444 that's like over a th well not really it's about almost a thousand different a thousand he'd still get 100 you see so that's why you know that margin of error that's why in different books I don't know even in that book whether it's 2,000 or 200 2,200 some book says 2,100 some book says 4,000 4,100 you don't know there's a lot of margin of error here okay so don't worry if your number is different that's fine okay um good one hour mark 5 minute breaks yeah 5 minute breaks be sorry sorry hey where's my boy [Music] luuka and so talking about yeah Luka I don't know how many there are but yeah Luka the uh you know the the extremely poised guy who seems to get everything no clue no no clue Hey Thomas so you can uh pick uh the dignitaries yeah uh is Leia here do you know how she looks like yeah uh it's Leia here Leia well I I've been slacking if you want I can message oh you know where he is I don't know where he is no I don't need her being here now I just need him to know how she looks like cuz they will pick him up together thanks nice bread nice looking bre [Music] should [Music] Thomas I [Music] [Music] yeah [Music] [Music] okay [Music] [Music] she [Music] [Music] for [Music] I we [Music] [Music] to Mark that we've got is this just to see how the semester's gone the mark the mock that you get the mock test the mock test okay uh yeah it's just going to be so that will be next week by the way I'm going announce it's just uh yeah you don't have to it's not it's not obliged it's not a compulsory thing but it's just to give you a sense of you know this is uh roughly what the exam going to look like and how long it would take you to solve it January yeah it would you will be given feedback all right but it won't affect any of your grades cool cuz I was planning on doing all of my revision for test during the break cuz I've got that's fine that's fine just you know get us off just I'm still going to do it cuz it seems heal yes I think it certainly will yeah all right everyone go back to your seat got a lot of things to cover so yeah go back to the sh slowly you've been you can make this a living so I made that easy for you so that's the distinguishing catch so that's laminar and turbulent uh in terms of Reynolds number and entrance flow region another thing that distinguishes the two is supposedly the pipe is made out of glass and you can see how it looks like hang on it's made out of glass but it's water water is transparent what am I looking at well typically people use Tracer dyed fluid that got injected into a stream of water so that it can it can tell you basically uh what the streamlines look like they do that in the book you you'll find that one anyway but suppose you can see all the vector velocities what would laminar flow profile would look like well like what you've always seen looks like a perfect parabol like that it's going to look like that and nobody can tell what's turbulent look like right it's just going to be like hot mess like that the vector is you know tortuous like that now but if you were to average the X Vector because all of this has many different Vector right this Vector that Vector yeah but if you were to only average the X Vector which you can do pick any of this stream just analyze the X and then you average those X it will look something like that like that if we can see it in the lab that would be great but there isn't right so what's the difference the difference is that um well as you can see you know flow is very well distributed almost every points have very different flow velocities right that one is completely zero that one and this one is completely different in length but this one there is a very steep change of flow near the solid surface but then everyone else is somewhat similar so there is a very steep change of flow velocity for turbulent flow near the interface between fluid and solid like that sharp boom and then after that all the velocity look kind of the same unlike the laminar one and then so another thing that um you can put as a formula here for you to memorize is this this is for laminar flow the maximum U right the one at the center yes the one at the center in a concentric pipe should be the one furthest away from the solid obviously it feels the least of viscous forces that tries to hold it back so obviously it is the biggest velocity now that biggest velocity when compared to the average velocity for laminar becomes two meaning if you were to average all these vectors okay compare it against the maximum velocity that maximum velocity will be twice as big as the average velocity for laminar flow that's why I drew it always like a very sharp bullet rather than like a fat Parable like that because this has to be somewhat geometrically representative yeah it's about twice as high as the average now what about this guy well that's the maximum velocity well put your engineering hat for a while right most of the flows here is the maximum velocity most of it so when compared to the average it's certainly you know near one because they're almost the same so U maximum when compared to U average for turbulent flow is 5 over 4 almost one just slightly bigger okay so that's another thing for you to memorize okay now it's not interesting at all it's just memorizing the thing is how can we predict velocity in a pipe how can we predict velocity in a pipe so the question becomes like this your client comes to you and say I want to transport things from one point to another point it could be 100 m away now now I got a budget for a pump the pump is only that much in power the pipe probably that much in diameter all specified can you tell me roughly how much the volumetric flow rate going to be it's fair question right as an engineer your boss comes to you say here's the pipe um here's the pump here's the diameter how much flow I can get out of that if it were water or oil or whatever is that a fair question it sounds like very easy Right comes to you and then ask that and then so how can we predict velocity now the next part that I'm going to explain to you will tell you exactly how so from the first principle you can make uh force balance of a tiny parcel of a tiny cylindrical parcel of fluid flowing in a pipe unlike before we will take consideration the viscous forces now okay so there will be something like that throughout the envelope of that parcel okay we will take consideration of that all right so if we take force balance in the X direction for that tiny cylindrical parcel of a finite length so that length is known L what would it look like well all right supposedly it is Flowing with a constant velocity that will render the resultant of the forces zero because the acceleration is zero all right now what are the forces acting upon it so pressure certainly placs a roll right there's something pressing from there and there is pressing something there's something pressing from there suddenly there are pressure applied onto the whole envelope but we don't take that into consideration because I'm only caring about X that's X by the way that's my X um good well there is weight but weight is in y direction let's not take that into consideration so that's it that there are only pressure from this side resulting in force pressing from left and there's certainly something from here pressing it the other way around and there is sheer stress resulting in something akin to a friction so this would look like P Pi r² where Pi is the radius of that well force is pressure multiplied by area anyway right well if that's P that can't be P because well you don't need a pipe at all then you know everything will be the same right that would be P minus Delta P I don't know what Delta p is I don't know but it's got to be P minus some difference of P all right multiplied by the same area okay and that one should be because of sheer stress so the balance roughly like this minus there you go to multiplied by area now the area is not the same area as that one the area where tow is applied onto is what you've seen in chapter one it's the envelope of that cylinder right not the cross-sectional area the envelope because that's where T which is actually f t multiplied a right that's F here of f r maybe a friction something like that okay it's the envelope and the envelope is 2 pi r which is the circumference multiplied by the length got it all right right and all of this has to equal zero now long story short keep saying that a lot you can cancel the P's and then maybe the pi because there are pies in every terms and reordering the terms will get you Delta P / L equals 2 to R you will get it like that let me repeat one more thing for you why are we doing this because we want to answer that question how can we predict velocity and how do we answer this by the first principle simply by Newton's Second Law Newton's Second Law applied to a tiny parcel of fluid cylindrically shaped looks like that and that's the relationship that we get okay Delta p a hint for you is what I meant when I say your boss gave you a pump that pump has Delta p in it that pump has to overcome the pressure drop anyways all right now because you know what to is okay don't get surprised to in a cylindrical coordinate for that is minus mu du over Dr R well that looks new oh my God yeah it's not mu du over Dy it's minus mu du over Dr R simply because the coordinate change okay it was before when something like this takes place and that's y then T is Mu du over Dy okay but in that case now T is no longer du over Dy multiplied by mu it's minus du over Dr R because R now originates from here from the center there y originates from the solid there are originates from the the fluid pointing at the solid that's where the minus come from I see all right here y points from the lowest velocity to the highest velocity right there are points from the highest velocity to the lowest velocity that's where the minus come from okay so don't be surpris by that seemingly different formula it is actually the same it's just looking like that because of the now here's the thing too that's why memorizing sucks you change the geometry the formula changes all right be really careful when you have to memorize things like your wife's birthday um so that's that and you can substitute it there it will get you Delta p over L 2 minus mu du over Dr R okay divided by R and then then the equation reduces to D over D Dr R equals minus Delta p l 1 / by you won't be asked to do this in the exam don't worry I keep apologizing as if it is a mistake to give you something difficult I'm not apologizing for that life is tough guys I mean yes you have to know how to do this okay right so that will become that and as you can guess there's some integration action going on because if I were to change this move it to the right hand side and then integrate it I can get many things now I don't want you to spend time solving that all I want is for you to know where this come from okay if you solve this one from zero to R from 0er to R yeah if you solve this integral which you don't have to do in the exam but if you do you will get U equals Delta p over 16 mu l d s multiplied by 1 minus what's the easiest way 2 R over d square solving that integral which the step by step is given in the lecture notes you can tell if if you want to know you can solving that integral will get you U exactly U as a function of diameter Delta P characteristic of the fluid which is the viscosity and the length so you know you as a function of Delta P geometry of the pipe which is what D andl tells you and the fluid which is what me tells you because different fluid gives you different me so knowing that you know you isn't that great you don't have to open the pipe you don't have to make it looks transparent you can tell give me a pump give me what the dimension of the pipe and give me the kind of fluid you want me to flow inside the pipe I'll tell you what the velocity not only that I can tell you the velocity at every point in that pipe every point it could be here which corresponds to R equals z there which corresponds to R equals whatever and there which corresponds to R equals capital D / 2 right I can just substitute those R in the right bracket there now if I do for example for the center of the pipe which is R equals what what is the center of the pipe r equal zero what would that equation reduce to for r equal Z the whole bracket here equals one cuz this guy is gone 1 minus nothing is one so it becomes that so this is the maximum velocity in it right you may as well write it like this U maximum multiplied by 1 / 2 r d square like that you can also write it that way yeah so what's the U maximum that how do you know that well I just substitute little r with zero theoretically the center has got to be the maximum yeah so now do I answer this question that you have to memorize that you have to memorize this bit I also already gave you the conditioning a little bit that might look familiar to you this equation didn't you see something like that in chapter one see it's already in the uh subconscious this bit you have to memorize ah memorizing and then well since you know you you can find U average and you can deduce volumetric flow rate from that how do you find U average U average you find by integrating this which you don't have to do because I've already done it in the lecture notes Let Me erase this it's only if you're interested only if you're interested and I hope you are now that I have you I can find volume flow rate yes well volumetric flow rate is don't be surprised it's not what it looks like before yes sorry no no no that's not maximum velocity maximum velocity is this guy yeah it's not that you know I get maximum velocity when I substitute R equals z which is the center and if I do that the whole thing gone leaving me with that so that has got to be the maximum all right now how do I know volumetric flow rate volumetric flow rate is actually an integral of U da a actually but in chapter three All U is the same all U is U average anyway right the flow looks like like that it's all the same in that case in chapter 3 it reduced to U average a that's what that solution it's not even similar it's exactly that okay those are two are the same only this bit you encountered in chapter 3 this is the actual OG all right that's applicable for any shape any U if you put U which you just derived onto that and you solve the equation which this lecture note will tell you how to if you're interested you will get Q equals make it easy Pi R2 U Max over two okay you will get that by substituting this onto that integral and solve it you will get that here's another thing for you to memorize that's for you to memorize now what I worry about is that you're following this some of you get tired and you start to lose track what am I actually discussing now so let me repeat I'm trying to answer this how can we predict you if I tell you what the pump geometry of the pipe the kind of fluid answer bam how do I know that from Newton's second law in X Direction first principle now once I get you I will want to get volumetric flow rate why because on the actual field nobody's going to give you U like in a formula like that right your boss won't give you that even if it's Chinese right so he will give you Q instead why because it's just one number Q This one is not just one number right okay so now I get that which is a function of still this beautiful right okay so knowing this relationship then I can tell U average then right because this one is U average a so U average is Delta Delta p d Square 32 mu L now from that you actually get that relationship that I mentioned to you before but I erased right think about it this is U Max this is U average which is twice as big yes this this is another one you have to I don't know do you want to memorize this because I can drive this from this easily okay any questions so you see I I gave you the list of problems for chapter 8 I think the first onethird of that problem can be solved simply by knowing all of this equation so far straight to the point applying formulas the question might be well what's Big D is what is this uh fluid is which means what's the viscosity what's the length of the pipe needed for that yeah you know it's just you know once you know is it flow turbulent or laminar uh it's just that you see so this chapter eight is not complex chapter 8 is just complicated there's a big difference between the two it's not complex it's complicated and start speaking like somebody from Liverpool understand now like makes you think oh I need that equation yes okay um only this one on the board yeah not the one before um so what triggers me is this question that's number one the question which has utility is on the field you want to predict velocity given Delta P geometry of the pipe and the kind of fluid you want to do do that and then so that whatever derive in that red box applies for everything so long as this is the question in at hand now the way we get to that point is by simply first principle all of those all of those derive simply from this from that and that alone okay from that you get that from that you get this well you need this as well okay fine you need that and that you get that you solve the integral you get U you solve the integral you get q and with common sense you know what you average is so so that these are all just that so like I said I worry that you lose tracks okay this is bunch of equation when do I have to use what where that's all what chapter eight is to know when to use what equation where which rigorous problem solving will get you there unlike chapter three you know or chapter two that's that's a lot of thinking [Music] uh yeah all right okay now now let's move to the second part of chapter 8 where's my ER eraser okay let's move to second part of chapter 8 now you know that the second part of chapter 8 is to know energy balance or modified beri these two are the same the same again you have a pipe simple straight pipe no fittings no contraction no diffuser nothing it's just a simple horizontally l pipe you've got flowing flow coming in and flow going out okay now you have to first identify your system things can get into the system and get out of the system and what I meant this energy can get into the system and can get out of the system that system can be made as as complicated as you want it need not be a straight pipe it could be a complicated pipeline it need not be just pipe it can have pump in it and it can have friction taking energy out of in terms of friction you know usually it resulted in heat flowing away all right now in addition to energy coming in and out to the system flow gets in and out of the system the flow carries within energy as well all right in chapter three bam that's chap that's point one that's point two what do you do what do you do bruli right you just broli the out of it that's it you put continuity hydrostatic put a tube here you're all set now the game changed slightly cuz you have viscus effects which beri originally cannot account for but now can because you modified the beri so how do we modify the Beres equation well just by adding terms and into it right the bruli will look like P1 row G plus u average square right 2G + Z1 that's U average 1 equals I make it far away for a purpose 2 2 2 G + Z2 now remember this because this is another very important step you will get asked in the exam should equation like this sorry should problem like this occur which it will that's the whole point of flute mechanics to be honest to do this fluid mechanics uh year one okay that's unmodified beri all right how do we modify it and you know what this is this is static head this is kinematic head that's elevation head these threes are heads all right so we're talking about energy getting into and out from the system in terms of heads which is nothing other than energy per weight really okay now in this system there might be friction sometimes it's called Head loss HL in this system you can put a pump so that's number two you call it h pump right in that system you can put fittings which will contribute to loss so I'm not going to write it down because it's going to be embedded here fitting means friction as well it's just a different kind okay now in terms of putting pump you put energy in the system yes in terms of friction energy leak out from the system all right so in terms of pump you add energy into the system add energy into the system right and in terms of friction you lose energy minus all right now that's number one modification there is something into the system usually in the form of a pump and if that's the case you add something into the left hand side and there is losses due to the fittings and due to friction and if that happens you put another term but this time with a negative sign that's one modification to the beri another one is this put an alpha there because in the beri flow looks like that that's U average in beri yes here it doesn't look like that you know it either looks like bullet like that or something like that with very you know anyway it doesn't look like top right that's why you put Alpha here to compensate it so that you can still use U average just like you did in bernly all right those are the two modification that needs to be put in there yes fine Alpha is just a correction constant that you put in the modified bernly so that you can still use U average simply because in the original ber newly flow looks like that which no longer the case so how do I justify using U average while the flow look like that yeah I put an alpha in there now oh too bad I erased this now Alpha for laminar is two for laminar which kind of makes sense right given the ratio between U Max and U average remember that and for turbulent is about one not exactly one but it's about one so in terms of turbulent that Alpha is as good as gone then because it's just one okay let me repeat again now you have a new energy balance now with this equation with this equation you can solve the whole chapter 8 well the complexity from this might be well what's U average there okay and that's where all this equation is for yeah and uh I don't know maybe I want to be fancy and I didn't tell you what P1 is but I stick a tube there uh which I can right then you know what P1 is from the height maybe all right or maybe now the question is all right I know all of this and later you will know how to calculate the losses what's the pump required well you get that from these equations reordered in the way you want it to solve your problems all right now this equation is to be memorized but not just blindly all right it needs to be memorized in terms of energy into the system energy out of the system there might be problems where there is no pump for example you want to flow water from a container up in a mountain Downstream Down Mountain you need a pump no gravity will do it there is no HP there's only losses the question might be well how high should that container be to maintain a specific U velocity I can make so many variation and that's why there's 100 or so problems in chapter 8 just from this any question so far so the modified beri is modified in two ways one by adding these terms for energy included in the system and this term for energy taken away from the system the second one is to put Alpha as a compensator for the fluid velocity profile not looking like chapter 3 Alpha is exactly two for laminar and one for turbulent how do you know things are laminar or turbulent from Reynold's number get it now from this do we have time for one example no we don't okay great now on the 12 you have to come on the 12 o' don't be absent cuz now we will do this and if you can pull it off you're in a really good shape for mock exam all right yes I'm struggling to see how this doesn't go to zero cuz that's diameter over diameter which should be the same so that can SL it becomes zero if the r becomes uh the actual diameter of that but R is a variable you see it can be anything oh which is not the same as the diameter no no it's not the same only when R is at this point which then becomes oh all right right two questions so here that's the radius of the um yeah yeah yeah yeah yeah yeah yeah yeah so this so that's exactly that R and why did you divide by two here that's just coming out of this equation the whole derivation is in the lecture notes okay so that two must have come out from the working of this integral okay so not not this directly then like not no no no you cannot you cannot make sense of this without solving of the interr yeah yeah yeah yeah you have to solve it inter yeah yeah is there a specified date for the mo uh it should be next Monday next Monday yeah the mock is not obligation it's not an you know you it's not compulsory and it's not going to add anything to your mark okay I don't know who first but okay uh I have a question you just this whole lecture you just mentioned L and turbulent why didn't you mention any of the transition stuff yeah okay the transition is the flow in between rynolds 204,000 I I forgot yeah that's called transitional regime yeah I will I will make sure I say it in the 12:00 so we should consider it also home everything that has been mentioned so far no because there is no answer to any of this in transitional regime I don't know what this is for transitional regime I don't know what entrance length I don't know what Alpha for transition there is absolutely no information yeah okay yeah when is the exam like not the main exam but the like test exam oh next week next week it'll be mock one yeah okay yeah okay what does this mean sorry what does this mean um what does what mean this one 16 oh okay I thought it's LV no no 16 yeah yeah 16 okay so uh one moment I need to wrap things up