ChemE 260 The Carnot Cycle Dr. William Baratuci Senior Lecturer Chemical Engineering Department University of Washington TCD 6: E CB 5: 8 & 9 April 29, 2005

Carnot Power Cycle Characteristics Steps Boundary Work Completely Reversible Maximum efficiency between any two thermal reservoirs Steps 1-2: Isothermal expansion 2-3: Adiabatic expansion 3-4: Isothermal compression 4-1: Adiabatic compression Boundary Work W12 > 0 and W23 > 0 W34 < 0 and W41 < 0 Wcycle > 0 Wcycle = enclosed area P V QH QC 1 2 3 4 TH TC We can imagine a wide variety of reversible cycles, but the Carnot Cycle is the standard against which we will compare all other cycles. It is simple and elegant. Steps 1-2: Take some weight off of the back of the piston and let the gas in the cylinder expand. Keep the cylinder submerged in a constant temperature bath at TH. Heat is transferred from the bath to the gas in the cylinder to keep it at a constant temperature. 2-3: Take more weights off the back of the piston, but this time keep the cylinder perfectly insulated so that the process is adiabatic. As the gas expands, the temp[erature drops from TH to TC. 3-4: Put the cylinder into a new constant temperature bath that is at TC. Add weights to the back of the piston to compress the gas in the cylinder. Heat is transferred from the gas in the cylinder into the bath in order to keep the gas in the cylinder at a constant temperature. 4-1: Put more weights onto the back of the piston to raise the pressure back to P1. Keep the cylinder insulated during this compression process so that it occurs adiabatically. As the gas is compressed, its temperature rises from TC to TH to complete the cycle. Cycles that move clockwise on PV Diagrams are power cycles and the enclosed area is the boundary work output ! Baratuci ChemE 260 April 29, 2005

Carnot HP / Ref Cycle If we execute the steps of the Carnot Power Cycle in reverse order, we obtain the Carnot HP / Ref Cycle. Steps 1-2: Adiabatic expansion 2-3: Isothermal expansion 3-4: Adiabatic compression 4-1: Isothermal compression Boundary Work W14 > 0 and W43 > 0 W32 < 0 and W21 < 0 Wcycle < 0 -Wcycle = enclosed area P V QH QC 1 2 3 4 TH TC The Carnot Cycle is REVERSIBLE. This means that you can operate each step in the reverse order and still come back to the original state. That is, we can consider the reverse cycle ! The reverse of the Carnot Power Cycle is the Carnot Heat Pump or Refrigeration Cycle. The Carnot Refrigeration Cycle absorbs heat from the cold reservoir at TC and rejects heat to the hot reservoir at TH. Cycles that move counter-clockwise on PV Diagrams are refrigeration or heat pump cycles and the enclosed area is the boundary work input ! Baratuci ChemE 260 April 29, 2005

Carnot Gas Power Cycle QH P 1 QH 2 TH 4 3 TC QC V QC 1 4 3 2 Isothermal Turbine Compressor Adiabatic 1 4 3 2 P 1 QH 2 TH 4 3 The Carnot Power Cycle can also be executed with continuous flow equipment as shown here. The compression steps take place in compressors and the expansion steps take place in turbines. Notice that the inlet to a turbine is at the narrow end of the trapezoid. I remember this because the gas expands in a turbine, so the trapezoid gets wider as the gas flows across it. The inlet to a compressor is at the thick end of the trapezoid. I remember this because the gas is compressed to a smaller specific volume in a compressor, so the trapezoid gets narrower as the gas flows across it. This is not a very practical cycle because it is difficult to build isothermal turbines and compressors. It just isn’t practical to submerge rotating equipment in constant temperature baths ! Still, in principle, this could be done. Of course, real processes are NOT reversible, or adiabatic for that matter ! QC TC V QC Baratuci ChemE 260 April 29, 2005

Carnot Vapor Power Cycle Pump Turbine Condenser Boiler 4 Hot Reservoir Cold Reservoir QH QC Wturb Wpump 1 3 2 Sat’d Mixture at Plow, low quality Saturated Liquid at PHi Sat’d Vapor at PHi at PLow , high quality QC QH There are some problems with the Carnot Vapor Power Cycle as well. High-efficiency pumps don’t work very well will vapor-liquid mixtures. Pumps are designed to work best on liquids ! Turbines don’t work well if the quality drops below about 90% or even 95%. Liquid droplets collide with the turbine blades at very high velocity and erode them ! Again, in principle, the Carnot Vapor Power Cycle could be built. Of course, real processes are NOT reversible, or adiabatic for that matter ! Baratuci ChemE 260 April 29, 2005

Carnot Vapor HP / Ref Cycle Compressor Expansion Valve Condenser Evaporator 2 Saturated Liquid at PHi 1 Vapor at PHi 3 Sat’d Mixture at Plow, high quality 4 Sat’d Mixture, at Plow , low quality Hot Reservoir Cold Reservoir QH QC WRef QH QC There are also some problems with the Carnot Vapor HP / Ref Cycle. Compressors don’t work very well will vapor-liquid mixtures. Compressors are designed to work best on gases ! I already removed the turbine in this refrigeration cycle because it just isn’t cost effective. In principle, the Carnot Vapor HP / Ref Cycle could be built. Once again, real processes are NOT reversible, or adiabatic for that matter ! Baratuci ChemE 260 April 29, 2005

1st Carnot Principle The efficiency of a reversible cycle will always be greater than the efficiency of an irreversible cycle operating between the same two thermal reservoirs. Hot Reservoir Cold Reservoir HER QC,R HEI QH QC,I WI WR Hot Reservoir QH QH HPR HEI WR WI QC,R QC,I Cold Reservoir A reversible HE can be reversed ! When it is reversed, it becomes a heat pump or a refrigerator. When it is reversed, the directions of all the het and work interactions are also reversed, but they do not change in magnitude. So, the HP and the HE in the diagram at right exchange ZERO net heat with the hot reservoir ! As a result, we can include the reservoir in the system enclosed by the dashed line. This system abosrbs a net amount of heat from the cold reservoir because QC,R > QC,I. It completely converts this heat into a net amount of work equal to WI – WR. This violates the K-P Statement of the 2nd Law ! Therefore, it is not possible for an irreversible HE to have a higher efficiency than a reversible HE. This confirms what you already knew…reversible heat engines have the maximum efficiency. If: Then: The reversible HE could be reversed (right drawing) System in the dashed line violates the K-P Statement of the 2nd Law ! Conclusion: Baratuci ChemE 260 April 29, 2005

2nd Carnot Principle All reversible power cycles operating between the same two thermal reservoirs have the same efficiency Hot Reservoir Cold Reservoir HP1 QC,1 HE2 QH QC,2 W2 W1 In this diagram, I have already reversed the 1st reversible HE and made it into a HP. The QH, QC and W have the same magnitude and opposite direction as they did when it was HE. Now, the heat pump rejects the same amount of heat to the hot reservoir as the HE takes in. So, once again, we can combine the two cycles AND the hot reservoir because there is no NET heat exchange with the hot reservoir. The new system, enclosed by the dashed line absorbs a net amount of heat from the cold reservoir equal to QC,1 – QC,2 The new system completely converts this heat into work equal to W2 – W1. This violates the K-P Statement of the 2nd Law ! Therefore, it is not possible for one reversible HE to have a higher efficiency than any other reversible HE when they operate between the same two thermal reservoirs. All reversible heat engines operating between the same two thermal reservoirs have the same efficiency. If: Then: System in the dashed line violates the K-P Statement of the 2nd Law ! Conclusion: Baratuci ChemE 260 April 29, 2005

Next Class … Ideal Gas Temperature Scales The Carnot Principles (which are consequences of the 2nd law) lead to a natural or Thermodynamic Temperature Scale. The cool part is that we will show that this Thermodynamic Temperature Scale is exactly identical to the Kelvin Scale ! Carnot Cycle Efficiency and COP The new discovery that the Kelvin Scale is a Thermodynamic Temperature Scale will allow us to determine the efficiency or COP of Carnot Cycles by knowing only the temepratures of the two thermal reservoirs with which they interact ! Baratuci ChemE 260 April 29, 2005