 So we ended the last segment putting together the equation for the control volume formulation for the conservation of energy or the first law of thermodynamics and we saw that we had the work term and we need to deal with the work term. So the work term was shown right here and I said that we needed to have a way to be able to express the different components of the work term. So that's what we're going to do now. We're going to begin by rewriting the work term as the following. So we're going to break work into a number of different components. We're going to say that work could be consisting of what we call shaft works. That could be a shaft crossing the control boundary. We can have work due to normal stresses and we'll call that normal. We can have work due to viscous shear and we can have other forms of work. That could be surface tension work. It could be electromagnetic things like that. But to define these as for shaft work normal stress and shear stress. So what we'll do we will take these one at a time beginning with shaft work. So it's net rate of work transferred in through the control surface. If you think to thermodynamics quite often we'll have our control surface control volume and that would be something like that. Perhaps you have an impeller and you're doing work it's rotating and that is the process by which you are adding work to your control volume or the control surface. So the next one that we have is normal stress. So this is a work associated with a force moving through a distance. Remember we have fluid coming into and leaving our control volume. In thermodynamics we usually call this flow work and we're going to go through the derivation of that. But the work itself remember work is force times a distance. So we have a force applied over some distance. We haven't defined what the force is yet and nor what the distance is. Now we're interested in the time rate of change of that. So we want joules per second or watts. And for that we're going to do it in the limit as delta t goes to zero of delta w divided by delta t. Putting in what we have for delta w. That's f dot ds, the distance over which the force is acting divided by delta t. Now looking at this, that there is just the velocity that the fluid is coming into or leaving and where our normal stresses are being applied. So we can rewrite that then as f dot v. What about force? What we can say with force? We're going to write it generically as some normal stress. We haven't defined that yet. Multiply by some vector for the area. And so if we draw it pictorially and if that's our control volume we have dA here. And that will be a vector. Remember the area always points out from our control volume. And here we will say that we have some stress that we will have it as being the normal stress. We'll leave it kind of generic for now. But with that for the force and with that for the work what we can do is we can rewrite this for normal as being an integral over the control surface. And what I'm going to do here, I'm going to flip the dA and the v around. It's a dot product so it doesn't matter the order of that. And so we can go through and flip that. It'll make sense shortly why I'm doing that flip. I'll just reverse the order of it. So we still have our normal stress multiplied by v dot dA. Okay so that is normal work. The next thing that we need to look at is the shear stress work or work associated due to shear stress. So this is work associated due to tangential shear stress. And just like before for normal stress we're going to define some force and that's going to be the differential of the shear stress vector dA. And so for to look at a little element some surface area we would have tau being the shear stress and I'll call that a vector. And the area that this is being applied to is dA. Now we're applying this over a distance and so just like before we were looking at distance in the way that we were able to figure that out was using velocity. And so we're going to do the same sort of thing here and that enables us to write this as work in terms of joules per second or power. So we have this now what we're going to do we're going to expand this out because there can be different types of viscous shear that occurs on our control volume. Okay so what we have we can have a shear due to shaft work. Remember I drew the schematic showing the control volume and we had some shaft going through with an impeller. Right when you cross there you can have viscous shear on the shaft itself but that's usually already accounted for in the shaft work. So already in W dot shaft so we don't have to worry about that one. The next one is shear on a solid surface and and that would be shear if our control volume is along a solid surface. Now the thing about that at a solid surface if you recall the no slip boundary condition we have V is equal to zero and consequently that term goes away and then finally we have the one for crossing at ports and if you create your control volume in a manner that the control volume is normal to the flow crossing that boundary then we would have tau is perpendicular to V and if tau dot V and they're perpendicular that will equal zero but we don't always necessarily have that condition so we will retain this last term but the other two will go away through careful choice of the control volume as well as the fact that the shaft work is already in the shaft work term. So what we end up with for shear is the following and I write this as area of ports that's wherever mass is crossing our control boundary so in letter exits. So with that what we can do we can expand out and write the work term as follows we had the shaft work plus the normal stress work plus the shear term which we isolated that only to be at the ports and then work other. So with that we can take this now and put it back into our control volume formulation for the first law and what we end up with is this and one thing that I'm going to do here I'm going to take the normal stress term this one right here and instead of putting it on the left hand side of the equation I'm going to move it over to the right hand side of the equation it'll make sense in a second why we're doing that and so here is our normal stress term and you'll notice v.da that we have in the normal stress term is consistent with what we have there and that was the reason why we did that flip around with a dot product. Now that we have this what we can do is we can do some combinations but what is the normal stress term if we think about it sigma nn that is I drew it originally as being the area so that was our area I think I said dA and then we had sigma nn in a fluid any of the fluid systems that we're going to look at the normal stress is really pressure and it does not act outward like I've shown it actually acts in that direction therefore sigma nn is actually equal to minus p and so we can make this substitution here up into this equation and rewrite the formulation for our control volume and what we end up with is this our shaft work shear on the ports other that could be other forms of work surface tension electromagnetic things like that usually things we don't consider but they could be there and you'll notice what I've done here as I pulled in the p term and I put p over row because we have row here and p over row times row is then just p which is what we had originally with the normal stress now there is a thing that I can say about that p over row 1 over row if you recall that is just our specific volume that we look at quite often in thermodynamics so really that term that the p over row is actually equal to p times the specific volume and if you remember from thermodynamics we had internal energy plus pv is equal to enthalpy so that is essentially the enthalpy term and that's the term that's associated with the flow work that you would see in a thermodynamics course which is consistent and it should be because the conservation of energy and fluid mechanics should not be different from the conservation of energy in thermodynamics and so what this is is conservation of energy and it is in the control volume formulation so it enables us to apply it to situations where we have fluid crossing the control boundary so what we're going to do in the next segment we're going to do an example problem of applying the control volume formulation for the conservation of energy to a system and we're going to look and see how to apply it