 Okay, we're now going to begin the process of trying to determine what the pressure drop is within pipe flow, and this is going to be kind of a long process. The actual lecture segments will be required, and what we're going to do, we're going to come up with two different expressions, one for the case of laminar flow, which it turns out is relatively straightforward, and you can do things analytically, and then the other case we'll look at is that of turbulent pipe flow, which we need to go to experiments and do dimensional analysis in order to figure out how to put everything together. But what we're going to do, we're going to begin by looking at the energy equation applied to pipe flow. So what we're going to consider is the case of a pipe section that goes through an angle, an elbow, and it is reducing the diameter as we go through that elbow. So we'll begin by sketching out our pipe. Okay, so there we have our pipe, and what we've done is we've also sketched a control volume in the inside of it, and so what we're going to do, we're going to apply the energy equation for a control volume, and we're going to go through and take a look at some analysis with that. Okay, so there is our energy equation. Now to begin with, there is no shaft work, so that term disappears. Now in terms of shear, what we've done, although I didn't quite draw it perfectly, the control volume is supposed to be in the pipe walls, and by doing that, what we can do, we can then set the shear equal to zero, and that is based on the fact that control volume is in walls. Other that disappears, it is steady flows, so the time rate of change disappears. Energy, that is how we've defined energy, and so what we can do is we can plug that into the equation that results, and all we have on the left hand side is heat transfer, and on the right hand side we have the energy plus P over rho, so let's take a look at what we get. Okay, so that's what we get for our energy equation. Now you'll notice that with the terms involving the kinetic energy flowing into and out of the control volume, I've left those in integral form, and there's good reason for that, and those are the two terms there and there, and the reason for that is when we're dealing with viscous flows, we don't have the nice characteristic uniform velocity profile that we had when we did a lot of our other control volume analysis. When we're dealing with viscous flows, you're going to have some sort of velocity profile, and consequently you need to take that into account and integrate across the profile, and that's why I've left it in that form. So when we're dealing with viscous flows, the velocity is not uniform, and so the way that we handle that with the energy equation is we're going to introduce a kinetic energy coefficient, and so let's take a look at that now. So what we have here is I've introduced this kinetic energy coefficient, and if you look at what is within both integrals right now they're identical, now what I'm going to do is I am going to recast this as being the average velocity across the channel, and consequently that value of alpha would now come into effect, where the thing on this integral here will not equal that integral, so those two are no longer equal, and we need to have this correction factor alpha, and I can further express what's on the right hand side with alpha, and then we have mass flow rate, that is the rho VDA is m dot, and it's multiplied by the average velocity squared divided by 2, and so with this we can then define what alpha would be, so we get that as the definition for this kinetic energy coefficient alpha, and we will see that in the energy equation when we're dealing with pipe flow, but for laminar flow it turns out that this kinetic energy coefficient alpha is 2.0, and for turbulent flow alpha is approximately equal to 1.0, and consequently for turbulent flow, with alpha being 1, we'll see that it doesn't really modify the energy equation that much, or it doesn't modify it as at all because we assume it to be 1, so let's rewrite the energy equation with this new kinetic energy coefficient embedded within it, and so we get that expression there, what I'm going to do is divide by the mass flow rate and rearrange, okay, so that's what we get. Looking at this term by term, what we can do is we can compare, so those first two terms that we have there, the one on the left and then the right hand side of the equation, that is mechanical energy per unit mass at location 1 and location 2, and if we think about it, if we have a pipe where we have turbulent flow, or it could be laminar flow, but it's viscous flow within it, as the fluid moves along, what happens is the pressure is dropping, and the reason why the pressure is dropping is because pressure is being used to overcome the viscous shear within the fluid flow, and consequently the mechanical energy is dropping, and where is that energy going? If we look at pipe flow, it's moving along pressure, so a delta P is driving the flow, and that energy is going into viscous shear, and that viscous shear is then represented in these next two terms on the right hand side here, okay, so we know the mechanical energy is dropping, so the dropping of mechanical energy is then being realized by something else, and the place where it is going is it's actually going into increasing the temperature of the fluid, and that is an irreversible process, once you get the thermal energy being formed, kinetic energy is converting into thermal energy through viscous shear, it's pretty much impossible to get back unless you put a heat engine in there, and you're dealing with such low temperature differentials that, and that's something of science fiction, so what we're dealing with here, the first term on the right, the U2 minus U1, that's actually a realization of mechanical energy going into internal energy, if you remember from thermodynamics we have DU is CVDT, so essentially you can integrate that if you assume constant specific heats, even variable you can have it in table form, but you get U2 minus U1 would then be CV times T2 minus T1, so what we see happening here is the change in internal energy is being realized by an increase in temperature of the fluid, and when you increase the temperature you can have heat transfer going on, that's what the last term on the right hand side is telling us, and so what happens here is the fluid heats up, you have convection, so I guess you would have Q convection here, and then you would have Q conduction going through the walls, so that would be the heat loss, that's where the energy goes, it goes into the surrounding environment, air, soil, whatever your pipe might be flowing or sitting in, and that is the energy loss mechanism, and so that's the conversion of kinetic energy, kinetic energy, or pressure energy actually, because pressure is what drives the flow, so pressure goes into kinetic energy which goes into internal energy, and that then goes out through heating up the soil, so heating up the internal energy of the soil around or the air, and that is through conduction convection, so that is the process by which the energy loss in pipe flow and where the energy is going, and if you have a turbulent flow it would be a higher loss mechanism, and we have a lot of theories talking about that, because all turbulent energy goes from kinetic and eventually goes to the inner scale, and then it goes into viscous shear, we won't be getting into a lot of that in this course, but with that the reason why that is important is what we're going to do, we're going to take these last two terms over here, and we're going to encapsulate those into a loss term, and we'll call it a head loss term, and that's how we handle this loss mechanism, so what we do is we lump those irreversible terms into this head loss, and we'll call it a head loss parameter hf, okay, so that's the head loss, so what we're now going to do, we're going to spend a number of segments trying to figure out how to quantify that head loss term, because that's very very important for pipe flow, if you can figure out what that head loss is, you can then figure out your pressure drop, that size is pumps, you can then relate it to flow rate, diameters, all kinds of things, so it's a very very important aspect of the pipe flow, and this is what we call a major head loss associated with pressure drop due to distance, but that is the energy equation, and we'll continue with that into the next segment.