 The last form of control volume analysis that we are going to do is going to be applied to the conservation of energy and so what we'll be looking at will be the first law of thermodynamics and in fluid mechanics we sometimes call the first law of thermodynamics just the energy equation we have continuity momentum and energy and this will be the last one that we'll be looking at for the control volume formulation so what we'll begin with is the system equation remember these are equations that apply to a fixed mass and so what we will need to do is to be able to rewrite that equation in the control volume formulation that applies to mass moving through our control volume so the first law of thermodynamics we have heat transfer we have work and that is equal to the time rate of change of energy within our system and so that's a system with fixed mass and we can write out the energy in the system very much like we did earlier when we looked at continuity as well as momentum and in this equation you notice we have little e that is our intensive property and that is the intensive property for energy and that is represented by the internal energy plus the kinetic energy plus the potential energy that we would have in our fluid and that is total energy per unit mass so like before what we need to do is come up with a control volume formulation for this before we do that let's take a look at our heat transfer term and our work term and we'll have to spend a little bit of time on work it's going to take time to solve that and relate it to the fluid mechanic properties heat transfers easy work is a little bit more laborious now the convention that we are using here if you look back at thermodynamics any class you take in thermodynamics quite often we say when you add heat to the system it is viewed as being positive and we'll do the same thing here in fluid mechanics however in thermodynamics typically if the system is doing work that is considered positive when the system does work on the surroundings we're going to use an opposite convention in fluid mechanics and that is it will assume that work is positive when it is done on the system so that will be the way that we do our sign convention for fluid mechanics okay so heat transfer positive when heat is added to the system and work is positive when work is done on the system so with that what we're going to do we have our basic equation the conservation of energy what we now need to do we need to go through the process of relating that between a system and a control volume and and so we'll use the equation that enables us to do that relation okay so that's the formulation that we have now in this equation what we're going to do we will let capital N equal energy and eta being the intensive property that will be energy per unit mass so with that what we can do we can rewrite it with the energy a term embedded for eta and N okay so we have that and the other thing that we are going to need to make an assumption about is the relationship with our work and heat transfer term so let's look back here this is our basic equation so on the left hand side we have this when we looked at momentum equation that was the force and it was equal to mass times acceleration here we have heat transfer and work and so we need to make a relationship between fixed mass and the control volume formulation and and for that if you recall back to the derivation that we did to come up with the equation that relates the system to the control volume we assume that at one point in time the control volume and the system were co-located and and so with that we can then make an assumption that relates heat transfer and work so at time t0 through that derivation that was uh reynolds analogy we had a situation where the control volume and the system were coincident or co-located and consequently what we can say is that heat transfer plus work applied to the system of fixed mass will be equal to heat transfer plus work applied to our control volume and therefore we can then translate the equation into the control volume and with that what we have is this for the energy equation so we have that we're dealing with the scalar energy as a scalar when we looked at linear momentum remember that was a vector so that this is a little simpler uh the place where we're now going to need to spend a little bit of time however is figuring out how to handle the work term here uh this is going to require us to go through a little bit of analysis heat transfer is pretty straightforward it's just heat transfer across the boundary but work will require a little bit more work because we have fluid coming into and across the boundary and so we need a way to be able to account for that uh in thermodynamics that is called flow work but essentially we're going to go through the same sort of analysis so in the next segment what we're going to do we're going to spend time looking at the work term and be able to come up with a formulation and then once we have that we'll be able to bring it all together and we will have the control volume formulation for the energy equation so in the next segment we will look closer at work