The Second Law of Thermodynamics
In this chapter we consider a more abstract approach to heat engine, refrigerator and heat pump cycles, in an attempt to determine if they are feasible, and to obtain the limiting maximum performance available for these cycles. The concept of mechanical and thermal reversibility is central to the analysis, leading to the ideal Carnot cycles. (Refer to Wikipedia: Sadi Carnot a French physicist, mathematician and engineer who gave the first successful account of heat engines, the Carnot cycle, and laid the foundations of the second law of thermodynamics).
We represent a heat engine and a heat pump cycle in a minimalist abstract format as in the following diagrams. In both cases there are two temperature reservoirs TH and TL, with TH > TL.
We now present two statements of the Second Law of Thermodynamics, the first regarding a heat engine, and the second regarding a heat pump. Neither of these statements can be proved, however have never been observed to be violated.
The Kelvin-Planck Statement: It is impossible to construct a device which operates on a cycle and produces no other effect than the transfer of heat from a single body in order to produce work.
We prefer a less formal description of this statement in terms of a boat extracting heat from the ocean in order to produce its required propulsion work:
Equivalence of the Clausius and Kelvin-Planck Statements
It is remarkable that the two above statements of the Second Law are in fact equivalent. In order to demonstrate their equivalence consider the following diagram. On the left we see a heat pump which violates the Clausius statement by pumping heat QL from the low temperature reservoir to the high temperature reservoir without any work input. On the right we see a heat engine rejecting heat QL to the low temperature reservoir.
Mechanical and Thermal Reversibility
Notice that the statements on the Second Law are negative statements in that they only describe what is impossible to achieve. In order to determine the maximum performance available from a heat engine or a heat pump we need to introduce the concept of Reversibility, including both mechanical and thermal reversibility. We will attempt to clarify these concepts in terms of the following example of a reversible piston cylinder device in thermal equilibrium with the surroundings at temperature T0, and undergoing a cyclic compression/expansion process.
In summary, there are three conditions required for reversible operation:
- All mechanical processes are frictionless.
- At each incremental step in the process thermal and pressure equilibrium conditions are established.
- All heat transfer processes are isothermal.
Carnot’s theorem, also known as Carnot’s rule, or the Carnot principle, can be stated as follows:
No heat engine operating between two heat reservoirs can be more efficient than a reversible heat engine operating between the same two reservoirs.
The simplest way to prove this theorem is to consider the scenario shown below, in which we have an irreversible engine as well as a reversible engine operating between the reservoirs TH and TLH from the high temperature reservoir, however the irreversible engine produces more work WI than that of the reversible engine WR.
Corollary 1 of Carnot’s Theorem:
The first Corollary of Carnot’s theorem can be stated as follows:
All reversible heat engines operating between the same two heat reservoirs must have the same efficiency.
Thus regardless of the type of heat engine, the working fluid, or any other factor if the heat engine is reversible, then it must have the same maximum efficiency. If this is not the case then we can drive the reversible engine with the lower efficiency as a heat pump and produce a Kelvin-Planck violator as above.
Corollary 2 of Carnot’s Theorem:
The second Corollary of Carnot’s theorem can be stated as follows:
The efficiency of a reversible heat engine is a function only of the respective temperatures of the hot and cold reservoirs. It can be evaluated by replacing the ratio of heat transfers QL and QH by the ratio of temperatures TL and TH of the respective heat reservoirs.
Thus using this corollary we can evaluate the thermal efficiency of a reversible heat engine as follows:
- Draw a diagram representing the heat pump system showing the flow of energy and source and sink temperatures and determine the following:
- The maximum possible Coefficient of Performance of a hot water heater (COPHW) that could be obtained by a reversible heat pump.
- The maximum possible Coefficient of Performance of a space cooling air conditioner (COPAC) that could be obtained by a reversible heat pump.
- Comparing the actual Coefficients of Performance shown above to those of the reversible heat pump determine if the actual heat pump shown above is feasible. State the reasons for your conclusion.
Derive all equations used starting from the basic definition of COPHW and COPAC and the Carnot relations for the ratio of heats of a heat pump.
Solution: For 1), 2), and 3) we need to reduce the system complexity shown above to an energy flow diagram showing only the basic requirements – cool the air to 20°C and heat the water to 50°C. Given the temperature if the heat source (20°C) and the heat sink (50°C) we can evaluate the respective reversible Coefficients of Performance.
Water is found to move through a pipe at a speed of 9m/s where the temperature of the water at the entrance is found to be 50C, and pressure 180 kPa where the pipe diameter is 0.50m. At the exit the pressure was found 160kPa, and temperature at the 100C. Determine the volume flow rates of water at the inlet and exit, the velocity at the exit, and the mass flow rate.
Determine the volume flow rates of water at the inlet and exit, the velocity at the exit, and the mass flow rate.