 So what we're going to do in this lecture is we're going to come up with an equation that provides us with system derivatives for a control volume formulation. If you recall, we talked about in the last segment how the basic laws, so the law of conservation of mass, linear and angular momentum and energy, are derived for a system of fixed mass. But in fluid mechanics we always deal with fluid in motion and consequently it's difficult to deal with the fixed mass things. So we're going to come up with a way to be able to take the time rate of change or the derivative of those four different properties and be able to express them in a manner that we can apply it to fluid mechanic analysis. So before doing that I'm going to define both an extensive and intensive properties and those will be for mass, momentum and energy and then we'll move ahead and come up with that derivative equation. So the extensive properties that we're going to be dealing with, the mentioned mass, momentum, momentum to momentum and energy and we can also define an intensive property which is what we will put into our equations. So the intensive property is just the extensive property per unit mass and what we're going to do, we're going to express this property as an integral across the system that we're looking at. And so we can integrate the intensive property either across the mass or across the system and the properties that we're looking at are for the four governing equations or basic laws. Okay, so those are the intensive and extensive properties for our four basic laws, conservation of mass, linear momentum, angular momentum and energy and what we're going to be working on is coming up with an equation that enables us to express the time rate of change of any of those properties with respect to time rate of change, with respect to time but expressing it in terms of the control volume formulation. Okay, so we're after a way to be able to express this time rate of change for any of our system properties in terms of a control volume. And so in order to do this, what we're going to begin with is a schematic and what the schematic will show is a control volume and a system that coincide in time at t0 but then later in time what will happen is the system will continue to move because there's fluid flow in the system that we're looking at but the control volume will remain fixed in space and we're going to use that as the basis for our derivation of getting that d by dt of whatever the property we're looking at in terms of control volume variables. So let's begin by looking at our system and control volume at time t0. So what we have here is we have time t0 where we have a system and a control volume that coincide and occupy the same chunk of mass but what we're going to do, we're going to have it where the system is moving and the control volume is fixed in space and so it is not moving and now what we're going to do, we're going to redraw this at a later point in time so this will be time t0 plus delta t and we want to look and see what has happened to the system and the control volume we know control volume is going to be fixed but the system will have moved off into the right so let's re-sketch that and take a look at what it looks like ok so what we show here is the control volume as I said was fixed it's not moving but the system is the system is occupying the fixed mass and what we're going to do with this is we're going to assume that there are three regions the first region is region 1 which is defined by the boundary between the control volume and the system then we'll have region 2 and this will be again bound by the system on the left and the control volume on the right and then finally we're going to have region 3 and that will be defined by the control volume on the left and the system on the right and what I'm going to do is I'm going to write two little differential volumes or elements one in region 1 and one in region 3 and that's where we're going to be focusing some of our analysis so let's begin with the differential element in region 1 and the second differential element we're going to look at is over here in region 3 and I'll call that sub-region 3 so that's sub-region 3 in region 3 and we're going to be looking at those in further detail so there you can see that this is a schematic that we're going to use when we come up with the formulation for the time rate of change of our properties and be able to convert it between intensive and extensive and you know what I'm going to do let me just clean that up a little bit these lines should really be parallel to the stream lines so I apologize for that my artistic abilities are not great that's a little better but not great so anyways there's region 3 and region 1 okay so with that what we're now going to do recall what we're after is a way to be able to express the time rate of change of whatever intensive property we're looking at from a system into a control volume but let's begin by defining that as a system and what I'm going to do is I'm going to say that intensive property whatever it might be if it's mass, momentum, or energy I'm going to define it as N capital N with a subscript capital S and so we can define the time rate of change in differential form so this is a definition of the time rate of change and what I'm going to do is I'm going to call this equation 1 and this is going to be the equation that we're going to build from as we go through this derivation and what I'm going to begin with is looking at what is happening to our intensive property or extensive property at time t0 plus delta t so and I'm going to refer to our schematic in coming up with this expression but we want to know what is contained within our system at t0 plus delta t and so referring back to our schematic this would be the figure on the right hand side that we're looking at and at t0 plus delta t what we see is our system is defined by the black region around that encompasses region 2 and region 3 and consequently we can write that as being our intensive or extensive property in this case it's extensive because it's capital N so it's what is in region 2 plus what is in region 3 at time t0 plus delta t now what I'm going to do I want to write this in terms of the control volume and so I'm going to make a substitution here for what we have for region 2 and looking back we can see that region 2 consists of its part control volume so we can take what is within the control volume minus what is in region 1 and so let's do that and this here is equal to the extensive property in region 2 and then adding that to what is in region 3 and that is all evaluated at time t0 plus delta t so we have that now another thing that I'm going to do this is where I'm going to start bringing in the intensive property so if you recall earlier we said that the extensive property could be evaluated by an integral over the system of the intensive property times rho dV so I'm going to make that substitution for each of the terms in this equation here and what we get is the following so this gives us an expression for this is that is what we're showing here so that gives us the value of the extensive property at t0 plus delta t for our system and we have a little bit of the control volume and then regions 1 and 2 so that's what's happening at t0 plus delta t let's take a look at what is happening at time t0 so at time t0 the control volume and the system were overlapping with one another so we can say that our system value our extensive property at t0 is also equal to what was happening in the control volume at t0 and so this we can represent as an integral across the control volume of eta rho dV at t0 so if you're wondering where we're going with this what we're working on is we're trying to evaluate the different terms in equation 1 and we're going to make some substitutions as we work along so there we have what is happening at time t0 and now what I'm going to do is I'm going to take these and I'm going to plug them back into equation 1 and this is all divided by delta t because we're plugging this into equation 1 so what you can see we have a little bit of a mix of things at t0 plus delta t we have some things over the control volume and some things of region 1 and region 3 I'm now going to rewrite this equation and I'm going to break it into things that are either control volume region 1 or region 3 so let's go ahead and rewrite it ok so our equation is getting cleaner and cleaner that's a joke now what we're going to do I'm going to take this and I'm going to call it a and that's the thing that had only control volume integrals then I'm going to take this here and I'm going to call it region or equation B and then I'm going to take this one here and we're going to call it equation C and so what we're going to do we're going to try to simplify or re-express each of these one at a time but what you'll notice right off the bat this one A involves only control volume B involves only region 3 and C involves only region 1 and so if you remember we had those little differential elements in region 3 and region 1 we're going to be using them for understanding what's going on in B and C control volume, we'll use what's going on in the control volume to get the first part so let's work our way through A, B and C and then we'll try to simplify this equation, make it look a little neater and more presentable so in the limit as delta T goes to 0 what you'll notice that we have in the expression for A this here and this here that's just the value of N evaluated for the control volume at either T0 plus delta T or T0 so that's just the time rate of change of the property N in the control volume which we can also express in the following manner as an integral over the control volume of our intensive property times rho dB in order to get the mass so that takes care of A so that is good we have A figured out so that's A now let's move on to B so what we have here is this is the change in region 3 let's go back to our schematic where was it? so region 3 was this region here so what we're expressing is the time rate of change of our extensive property because we're talking about capital N here with respect to time so what we're going to do we're going to take advantage of this little sub-region to be able to come up with an expression for the time rate of change of our extensive property with respect to time so that will be the approach that we will take in order to evaluate this so what I'm going to do I'm going to zoom in on that sub-region that we had in our schematic and we call that sub-region 3 in region 3 one of them has roman numerals ok so what I'm going to sketch out here remember this was bounded by we had streamlines so first of all what color was our control system our control volume was red so I'll try to be consistent with that so the control volume was red and then our system boundary we drew that as being black and we had streamlines and I think the streamlines were in blue so I'll draw this blue to be consistent and this here is little region 3 so it's a differential element and this here is a streamline and then we have our control surface on the left hand side so if we have streamlines on the upper and the lower what we can assume is that the velocity vector is going to be parallel to those streamlines between the streamlines so the velocity vector is something like that the other thing that I'm going to assume is the control surface the area of that it could be some arbitrary area and what I'm going to do I'll draw a projection of that area over here we're not exactly sure what that shape is going to be but I'll do something arbitrary like that and I'll call that DA so that's the projection of the area on the control surface and it may turn out that the unit vector for that area could be a little bit different from the streamline and the velocity vector I'll do that as area vector DA and the angle between the area unit vector and the velocity vector we'll call that alpha and here alpha is less than 90 degrees for the way that we've drawn it and that means that this is for mass outflow or mass leaving our control volume because this was on the right hand side of the system that we're looking at and the final thing I'm going to draw here is the length of this of this element that we're looking at and we'll call that delta L okay, so I think that describes everything now what we're after is we're trying to determine let's go back here we're after this so we want to know the time rate of change of the extensive property in region 3 so this is a differential element out of region 3 so what I can write is the differential element at time t0 plus delta t so this is the differential element that we're looking at here is equal to our intensive property rho times the differential element of volume in this little differential element that we're looking at and we can express this then as the intensive property times the density multiplied by the volume of this is just going to be the area times the length so the length is delta L now remember the area and the vector itself might be a little bit off with respect to one another and so consequently what I'm going to do is I'm going to put the values of the angle because this volume is parallel to the streamlines but the area itself might not be and we need to be able to preserve that or take that into account so that's what we're doing by introducing that cos alpha we get that expression so out of that we can then write that extensive property in region 3 at t0 plus delta t is just the integral of that over the control surface at 3 and what we're integrating is eta rho delta L cos alpha dA and this would be a t0 plus delta t okay so we're getting closer to understanding B what we're going to do we now want to find the time rate of change of that because if we go back here we're interested in the time rate of change of that extensive property so that's what we're now going to do and what we'll notice here is we have a delta L and a delta t well if we look back at our diagram delta L over a period of delta t all that is that's the velocity the magnitude of the velocity and so we can make that substitution as well and we can rewrite it as the magnitude of velocity and then the magnitude of the area vector so that will take us part way to understanding what's happening in B that was term B and then the final thing that we need to look at is C in our expression and C going way way backwards our diagram C is describing what's happening in this region on the inlet and so that's what we're going to work with now so that was the expression that we had and so what this translates into is basically what we need to do is come up with an expression for our extensive property at time t0 plus delta t in region 1 and just like before what we're going to do we're going to go back and we're going to take a look at the streamlines and at this little chunk or region 1 in order to figure out what is going on here so again we have our control surface we have our system boundary we have our streamlines now the area here is going to be a little different from the way that we drew it before because if you recall area we always have it projecting outward from our control system and so this is the inlet to the control system so the area is projected outwards and consequently alpha the angle between the velocity vector and the area alpha is greater than 90 degrees so this represents mass inflow so mass coming into our control volume and we have a projected area some arbitrary area we don't know exactly what it is but I'll draw it as being dA ok so what we need to do is follow the steps as we did for sub region 3 come up with a differential element so eta, rho are intensive property density but the volume is going to be the volume of this little element or differential element that we have here so we'll just follow like we did before you might be wondering why did I put in a minus cos alpha that's because we want the volume to be positive and we know with the way the sign convention is that it would come out negative if we didn't do that so we put in a negative cos alpha in order to make it positive and what do we end with we end with an expression for the extensive property in region 1 at t0 plus delta t we get that but remember we're after the time rate of change of that which is what c was so let's re-express that ok so there we have an expression for c the last thing I'm going to do here notice that we have magnitude of velocity v and dA and cos alpha well if you remember back from looking at mathematical operations this is just the dot product and we talked about that before or the direction cosine so wherever we have this we can replace it with the dot product between the velocity vector and the area vector doing that we take all of this back to equation 1 and so if you remember this is a b and c from equation 1 we're going to plug them all back in now you might be wondering how I got rid of the negative for control surface 1 the way that we do that the dot product takes care of it automatically we don't need to worry about it so we can simplify this a little bit by saying that control surface 1 plus control surface 3 if you look back to our original diagram that represents the entire control surface around the control volume so we can couple those together and we can rewrite the time derivative of our extensive property which goes into our basic laws so this term here is a system formulation and on the right hand side of the equation it has been recast into a control volume formulation so we can use it then for control volume analysis by applying our basic laws the laws of conservation of mass momentum be it linear or angular and energy okay so that's a lot of work to get one little equation but this equation will be very very useful as we'll see coming up this is sometimes called the Reynolds transport theorem it links a system approach to control volume so we can express the basic equations in control volume so that concludes the derivation in the next segment what we're going to do we're going to start applying this to mass momentum and energy