 What we want to do now is you want to take a look at the first law applied to the regenerative rank end with the open feed water heater. So let's take a look at applying the first law. We can cancel out a few of the terms here. First of all, we're going to assume that our feed water heater is well insulated. The second thing is there are no moving boundaries. It's just a mixing of fluid and agitating with the steam. So really, there's no work being done. And finally, what we will do is we will assume that both kinetic energy and potential energy changes are negligible within the open feed water heater. And so with that, our first law simplifies somewhat. It's basically just the mass flow rate times the enthalpy of the exit streams equating to the mass flow rate of the fluid streams entering multiplied by the enthalpy. So what we can do, we can look back to our diagram of the open feed water heater. We see that we have fluid stream at state 6 coming in, mass fraction y. We have fluid state 2, 1 minus y, and fluid state 3 leaving. So let's take those and we will put them together. However, before we do that, I know what we'll do. We'll look at that first of all. And I'm going to call this equation 1. Another thing that we can say, and I'll go back to looking at our schematic diagram, we can make a comment saying that the fluid at state 3, state 1, and state 5, the mass flow rate there, the fluid going through the boiler is all the same because we're not stripping fluid out or adding it after we leave the open feed water heater and then return to the turbine. So with that, what we can write is that m dot 3 equals m dot 4 equals m dot 5. And in a similar way, we can make comments about the mass flow rate at 7, 1, and 2. So at 7, the flow rate at 7, 1, and 2, that should be a 1 down here. It's kind of hard to see it. We can write the mass flow rate at 7 equals mass flow rate at 1 equals mass flow rate at 2. And that is equal to the mass flow rate at 5 minus the mass flow rate at 6. So basically it's saying that it's what is going, coming out of the boiler, going into the turbine, taking off the fluid stream that we're stripping off, and they would be equal to one another. So we can write m dot 5, which is the flow going into the boiler, minus m dot 6 is what we're stripping off. So those would be equal. And so with this, we can rewrite equation 1. So that's just rewriting equation 1 that we had up here. So that's coming down and making the substitutions for the mass flow rate. Now what we'll do is we're going to divide this equation by the mass flow rate at 5. So let's take a look at what happens when we do that. We get this equation here. So I will box that equation. Now in a similar way, we can come up with other equations for our regenerative ranking with an open feed water heater. So Q in is just the change in enthalpy across the boiler. Q out would be the fluid stream going through our condenser. So we need to take into account the mass fraction, which we're doing with the 1 minus y times h7 minus h1. Now let's go back and take a look at our diagram here, our process schematic. So what we're doing, we're looking at the enthalpy change between state 7 and state 1, but we're acknowledging the fact that the mass fraction or the mass flow rate has been reduced and we have the 1 minus y. And that is why in our term, we have the pre multiplier of the 1 minus y. Now for the turbine, we can write work turbine out. We get the full amount of work for the first part of the expansion, but then what we do, we strip some of the fluid away. So we don't have 100% of the mass flow going through the rest of the turbine and consequently we need to adjust for that. And that's what we're doing in the second part of the equation here, h6 minus h7 times the reduced mass flow rate going through the turbine. Now the work into the pumps, remember we have two of them. And each of them have a different mass flow rates coming through them. And so that would be the work into the pump. And we said that y, our mass fraction, was m.6 divided by m.5. And finally, expressions for pump work, we have our steady flow work equation so we can write work pump in. That would be specific volume at state 1 times p2 minus p1. And for pump 2, let's go back and look at our schematic. Now pump 2 is here and we're going from state 3 up to state 4. And so what we'll need to do is take the specific volume at state 3 in our steady flow work equation and then it's the pressure change. And that would be equal to h4 minus h3. So those are the equations for the regenerative rank and with an open feed water heater. And one point that I would make with this is please do not memorize these equations. Just understand where the equations come from. Every problem can be a little different and consequently to memorize these equations might get messed up. So just understand where the equations come from. And then when you're solving a problem, go ahead and go through kind of the process that we've done here. And you should be in good shape that way and not make any mistakes. So that is regenerative rank and for the open feed water heater. The next thing that we'll do is we will take a look at the closed feed water heater and go through the same process. Look at the schematic and the process diagram. Then we'll look at the first law applied to that for steady flow.