 The next cycle that we're going to take a look at is actually quite clever from the perspective that it achieves similar performance to what we saw for the cascade cycle that had two different working fluids, yet it does this using only one single working fluid and so this is multi-stage vapor compression. And in a way it's similar to when we looked at multi-stage compression with intercooling for gas compression, but what we have here is a situation where the compressed refrigerant is lower than the atmospheric temperature, so the question would be then how do you cool it? And the way that we'll cool it with this cycle is we mix it with refrigerant from another part of the cycle. It's kind of almost like a regenerative process that we looked at when we looked at other cycles earlier on in the course, both with rank and as well as we looked at the regenerator when we looked at the sterling cycle. Now that was for heat engine, this is for refrigeration. So what we'll do here is in order to cool our refrigerant we will be mixing it with refrigerant from another part of the cycle that's at a low temperature and consequently in a way this is similar to a regenerative heat exchanger, although what we're using is just a mixing chamber, something that we looked at when we studied the rank and power cycle. So let's go ahead and take a look at this. We're going to look at it for a two-stage compression process. So there is our schematic as well as diagram for the two-stage compression refrigeration system and one thing, well what's happening here, the biggest change would be you'll notice we have this flash chamber and basically this is a device that can separate a multi-phase fluid because we have a multi-phase fluid coming out after a condenser with state 6 which on our TS diagram is right here and what we are doing is we're stripping out the vapor and the saturated vapor is what is at state 3 so on our TS diagram that is there and that is then what flows into the mixing chambers, that's the vapor at state 3 there. The percentage of vapor that is going to 3 is equal to the quality of the refrigerant after the throttling process from 5 to 6 so we can say the percentage going to 3 is x6 and if we say that the remaining the liquid is stripped off and it goes to state 7 down to 1 or to 8 and then over to 1 so we can write the percentage of liquid or fluid going there through the condenser, sorry the evaporator at the bottom of the cycle is 1 minus x6 and so we break the fluid in 2 so we have 100% of the fluid coming through here and once we get to 6 we take x6 off and that goes to 3 and then 1 minus x6 goes to 7 and that is one of the different components. The other thing is the mixing chamber, the mixing chamber is occurring right in here and what it does is it takes us to state 9 and so we're taking the fluid from state 3 mixing it with fluid at state 2 and we end up at state 9 which then goes into the high pressure compression process so that is the two stage compression refrigeration cycle what we'll do now is we're going to take a look at the first law applied to the mixing chamber and we'll see what the equations bring by doing that so for the mixing chamber what we can say is that it is adiabatic and that there's no work being done we neglect kinetic and potential and that's what we end up with when we look at the mixing chamber we have fluid at state 3 coming in fluid at state 2 coming in and then fluid at state 9 leaving so that enables us then to figure out what's coming and what's going so the fluid leaving is fluid at state 9 and what's coming in is state 3 plus state 2 now in order to determine m dot 3 and m dot 2 looking back here we have a hundred percent mass flow is going through state 4 5 so up here we have 100 percent mass flow the other streams the mass flow is lower and so we need to acknowledge that when we're looking at this now we're looking at this now for m dot 3 it is x6 which was the quality at the end of the throttling process times m dot 9 and if we say that then m dot 2 is 1 minus that so that enables us to determine the mass flow rate for 3 and 2 we can then plug in for the enthalpy at state 9 and obtain the following and that's an equation that we can use then for our mixing chamber or regenerator and determining the enthalpy at state 9 other things that we would do we would want to be able to get the coefficient of performance so we would look at the low temperature heat transfer coming from our low temperature source whatever we're trying to cool and then expressing that in terms of mass flow rate and we have to do this because we do not have 100 of the mass flow rate going through the evaporator we have 1 minus x6 going through as for work in we will have different mass flow rate going through the low pressure compressor versus the high pressure compressor high pressure compressor will have 100 the low pressure will have 1 minus x6 we need to acknowledge that as well so that's where you have to be a little careful with the work as well as the heat transfer once you've done that you can then determine the coefficient of performance in terms of mass flow rate 9 and and so that would then be a way that you could determine the coefficient of performance for this multi-stage vapor compression cycle