Lesson 1 - The Reynolds Transport Theorem

hi everybody today is going to be the first of our online lectures that we're going to use to give us a little bit more time in class to work on some example problems so today's topic is the Reynolds transport theorem and this is our introduction to this concept that it ties into thermodynamics and conservation principles so those of you who had thermo with me should see some familiar things here so the big power of the Reynolds transport theorem is that it lets us relate the Lagrangian and oil arian perspectives of fluid mechanics that we discussed last time so the Lagrangian perspective remember this is where we follow one packet of mass as it moves through the flow even if it changes size and shape we always follow that packet of mass and we treat it like a particle that's moving the Alerian perspective this is where we're tied down to some region that we're interested in instead of some sort of fixed packet of mass the oil arian perspective it can still involve a change of size and shape but the big difference is this guy right here that mass might cross the boundary now where mass never crosses the boundary of the Lagrangian approach because that is fundamentally one packet that we're following so we're going to apply names to these and they're the same names that we used in thermo for the Lagrangian approach we're going to call that a system because it has constant mass and for the oil Arian approach we're going to call that a control volume because it's a region that we're interested in not the substance within it now the control volume approach is really what's natural for looking at fluid mechanics if I have a pump I don't want to follow every packet of fluid through that pump I want to look at the pump and look at what comes in and what goes out so that's a control volume unfortunately all of our experience with physics really applies to systems and not control volumes so if I look here at Newton's second law this sum of forces equals MA only applies to a system it doesn't apply to a control volume so it's natural to look at control volumes our physics relates to systems what do we do well what we can do we need to start off with by defining what kind of property we want to track as the fluid moves around so what we're going to do is define B to be any extensive property remember extensive means that it depends on how much of the substance there is so that could be mass momentum volume energy or any other number of things similar to that then what we're going to do is define an intensive version of the same thing that we're going to label lowercase B and that's an intensive version and remembering back to thermo whenever we have an extensive property we can divide by the mass to get an intensive version of that property and that's what we're doing here so by definition little B is equal to big B over m one other thing that we can look at and it gets a the Reynolds transport theorem approach the name integral control volume analysis what we're looking at is a certain volume and we can talk about integrating over that volume to get the B for the whole system so we look at our little B we multiply that by the density which is kind of like the mass and then we integrate that over the entire volume and that gives us the total amount of B in that system or we could do this something similar for a control volume okay so what I want to look at here is how we're going to lay this problem out just looking at an example system and control volume so what we're going to look at is here there's some time t0 we've got a control volume and our system is going to be defined as the mass in that control volume so the system in the control volume coincide initially then what's going to happen is we're going to allow some time DT to go by and the control volume is going to stay stationary but the system will have so see our system has moved out here to the right it expanded a little bit it changed its size and shape but the control volume stayed stationary all right that's a great start and that's going to let us begin this analysis mathematically so the way we're going to look at this is to look at what's going on at time T zero plus DT so we've got our system has now moved out here to the right and if I look I can imagine here in my control volume this region Roman numeral one is mass that had to enter my control volume to replace the system mass that left so that's coming in from the left I also I'm going to notice that my system now has this portion that's outside the control volume and I'm going to call that Roman numeral two and so these regions Roman numeral one which is the mass that entered the control volume that's not part of the system and then Roman numeral two which is mass within the system that has exited the control volume these are going to be important in our accounting so let's look at it mathematically here we go at our initial point time T we can see that the amount of B that's in the system and the amount of B that are in the control volume these are the same because they're on top of each other they coincide we're happy now what I'm looking at though is as a little bit of time has gone by what is my B in the system now and so this is how we're going to write it B in the system remember that's what's in this green thing that's equal to B in the control volume minus what's in region one because that's not part of our system so that's right here and then we're going to add on this part in region two which is not part of the control volume okay so that's what we've got our B in the system is equal to the control volume minus B 1 plus B 2 this is total common sense just looking at our picture well what we want to do is now look at how this changes with respect to time and you can probably see that we're going to get into calculus here so if I look at how B changes with respect to a change in time VT well here's how B changes over these two times here's how my B and the CV changes with respect to these two times and I only have my b1 and b2 happening at the future time because those aren't relevant when my control volume and my system overlap these guys only show up in the future time so what we're going to do here is take a limit when DT goes towards zero that's what we should remember from calculus ties into taking a derivative and what that's going to mean for us is that in that instant when DT is zero the system and the control volume will still coincide so it'll let us be looking at the same volume but we might have different changes in the B that's in the system and the B that is in the control volume because of these boundary terms so when we look at doing this derivative we see that the for the things that occupy a volume which are the system in the control volume when we take the limit as DT approaches zero we get the difference divided by DT that's just the definition of the derivative so I'm going to write that's the derivative the change in the B in the system with respect to time notice I use a capital D because this is the material derivative because the system is a Lagrangian viewpoint thing so when we look at the control volume we see that the terms are the same it's still a definition of the derivative only differences we use the partial derivative notation you don't really need to get caught up in this is the material derivative this is the partial derivative that's kind of my new math but it is helpful to help remind us that this is Lagrangian and this is not if we look at the two terms that are the boundary the b1 and b2 those are going to add in the limit go to look something like this so we're going to have a row a V remember that that's basically the mass flow rate and then times our little B which is the amount of the property per unit mass or the intensive property so we're going to call that B dot in to reflect the fact that that's the rate at which B enters due to mass flow and then below we have the opposite sign Rho AV b2 and that's B dot out so if we put these together in our equation we get the rate of change of B within the system with respect to time equals the rate of change of B within the control volume with respect to time plus the stuff that crosses the boundary B out minus B in in this case so this is similar to our first law of thermo for those of you who had thermo and this is an important fact because what this really represents is conservation of B within the system when we look at the control volume and that's what allows us to relate these two perspectives so it's pretty cool ok so we need to generalize this a little bit because that last equation we just looked at only deals with uniform surfaces and uniform areas for the inlets and outlets and uniform velocities as well so that's a little bit of a simplification we want to be more general so the way we're going to do that is to draw this arbitrary area in the flow that we'll call a control surface and that represents a boundary to a control volume and then we'll look at a little portion of that that we'll call da so da is a differential area within our control surface now at that point we have two vectors one is V the velocity coming through the surface and that could be going in any direction because this is just an imaginary surface in the flow the other vector we see n hat that is a vector that is perpendicular to the surface at the point da it also has a magnitude of 1 so n is defined to be positive of when it's coming outward from the surface and with a magnitude one it makes it a unit normal okay so that's what we call that what we're going to do is write our flow rate of B out of the system as an integral with respect to area so we're going to integrate over this entire surface here with respect to the area da and the thing that we're going to integrate is Rho times B which is the property and then V dot n hat so remember VN hat gives us a dot product so the dot product gives us the component of V that is along the direction and hat an only that component we don't have the component that's normal to n hat so that only gives us V in that direction if our V is flowing along the surface perpendicular or I should say tangent to the surface then we know that V dot n right here goes to zero remember from math class that we can write our V dot n as the magnitude of V times the magnitude of n which is just 1 so it's the magnitude of B times the cosine of theta between these so if they're perpendicular V is tangent and is perpendicular to the surface that gives us a theta of 90 degrees and that would mean that V dot n is 0 the other cool thing is that this works for both inlets and outlets so let's say I had my V going the other way and it was flowing into the system now my angle is going to be something larger than 90 degrees like this so there's my N and here's my V and my theta is now this guy in between them so we can see that that's greater than 90 degrees so that would give us a negative number and I can say that my V dot N is less than zero so this guy covers inlets and outlets because of that dot product that we get this one integral over the entire surface of the control volume gives us all the inlet and outlet floo so this is cool it lets us write our equation the Reynolds transport theorem like this so we're going to say the rate of change of B within our system with respect to time the system is the moving packet of mass that's equal to the rate of change of the stuff in the control volume so this is the integral over the control volume of Rho times B that's the amount of B in the control volume then we have the integral of the control surface of Rho be V dot NDA so this is what we just saw that that is the inlet and outlet stuff so we have the change within the system equals the change within the control volume plus the stuff that crossed the boundary so this should make total sense to us this is right out of thermo what we're describing is the transport and conservation of B due to the flow and it also relates our lagrangian and oil aryan perspectives because remember we have this guy on the left hand side which is a lagrangian perspective and then this control volume guy on the right is an oil aryan perspective so we're relating these two and what we see the difference is is the stuff that flows in and out of our control volume that is where we get the difference between lagrangian and oil aryan so last thing i want to show you really quick is just two problems that we're going to discuss in class and what we're going to focus on is how do we find this V dot N hat da because that's going to be an important skill and we're going to use that later once we start talking about specific properties right now we're really general we're just saying Oh B could be anything but later on we'll apply specific properties like mass or momentum to B and see what happens so for now we want to learn how to calculate V dot n hat da so one way we're going to look at it is for this guy which just has a bunch of nice flat inlets and we can imagine uniform velocities so we can think about that and then the other problem we're going to do in class is to look at ven hat for a cylinder in the flow a cylindrical control volume with my flow coming in sideways through that cylinder so to do this we're going to have to look at n hat as a function of theta so you can start to think about these we're going to do them in class and work as a team and with other classmates to really get a handle on this stuff but I just wanted to show you what we're going to work on so you're thinking about it so that's all we have for this lesson I hope you enjoyed listening to it this way and I will see you in class where we'll work on some of these problems thanks everybody

Channel: Joe Ranalli

Views: 103,374

Keywords: Fluid Mechanics (Field Of Study), Reynolds Transport Theorem

Id: PDq9YQh650g

Length: 16min 12sec (972 seconds)

Published: Tue Jul 15 2014