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Entropy, the Second Law and Power Plant Efficiency (with no math)

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Charles Bayless's picture
Retired CEO, Tucson Electric Power

Mr. Bayless is a retired Utility Executive and a lecturer on Energy Policy, Climate Change and Ocean Acidification. Until June 30, 2008 Mr. Bayless was President and Provost of the West Virginia...

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  • Sep 24, 2020

Entropy,  Efficiency and Mr. Spock


“... there is an art to flying", said Ford, "or rather a knack. The knack lies in learning how to throw yourself at the ground and miss.”

Douglas Adams’ “A Hitchhiker’s Guide to the Galaxy.”


Many point out that power plants are inefficient, converting only 40-60 percent of the input energy to electricity. Why can’t we do better?  Easy answer - the Second Law of Thermodynamics. The Second Law puts improving power plant efficiency to much more than 65% - 70% in the same category as   “throw yourself at the ground and miss”,  easy to say – but impossible to do.  

The First law of Thermodynamics is rather simple, “Energy can neither be created nor destroyed.” The Second Law, “In any adiabatic transaction entropy must increase” is, while largely unintelligible, critical to understanding of thermodynamic efficiency and performance. The term adiabatic means no energy is added or subtracted from external sources. Entropy is a measure of the quality of energy. Further as entropy always increases it dictates the direction of the flow of energy in any thermodynamic transaction. There are many transactions that could proceed in reverse without violating other Laws of Physics but the fact that entropy must always increase dictates that they are irreversible, they can only proceed in one direction.

Spock, Temperature Scales, and Efficiency

To understand entropy, we will turn to a world galactic class science teacher, Mr. Spock. Spock, warping past earth one day, was stunned to learn that entropy calculations were not taught in the second grade and incredulous that most earthlings did not understand the temperature scales. Spock thus beams us aboard for a five-hour mission to explore Temperature, Entropy, Efficiency, and the Food Replicator.

Spock starts by pointing out that based on a survey of 348 Class M Planets the Fahrenheit Temperature Scale still holds the galactic record for most disorganized, ill-conceived scale ever. First, the thermometer was made, temperatures were labeled, and it was “discovered” that water froze at 320F and boiled at 2120F.

In 1742 Andres Celsius decided there must be a better way. He built a thermometer and put it in freezing water and marked the that point as 00C and put it in boiling water and marked that point as 1000C and put 100 degrees in between. This is the Centigrade or Celsius Scale

Both scales however miss an important point they don’t start at zero temperature. Which then raises the question, “Is there such a thing as zero temperature?”

Figure 1. Temperature Scales

The answer is yes, it is the point where all molecular motion ceases. To confuse  High School Students even further we have two scales starting at  Absolute Zero, The Rankine Scale which uses degrees the same size at the Fahrenheit scale and where 00R is minus 459.670F and the Kelvin Scale which uses degrees which have the same size as centigrade degrees and where 00K is minus 273.150C. Figure 1 shows a comparative look at Temperature scales. To simplify matters let’s assume that absolute zero is -4600F and that the earths average temperature is +600F putting earth at about 5200R. 

Once the scales are understood Spock starts  his entropy discussion with two soft drink cans as shown in Figure 2, one can at 1000F and another at 00F( the soft drink, from Vogon, is so concentrated that it does not freeze even at 00F ). He puts them in a perfectly insulated box. Later, after a break, half the class[1] returns and both cans are at 500F as shown in Figure 3. Spock then asks the question, “Will the cans in Figure 3. ever go back to their state in Figure 2? Clearly not.

Figure 2. Soft Drink Cans

Figure 3. Soft Drink Cans


But why not? If the cans went back to the 1000F - 00F state the First Law would not be violated, energy would neither been created nor destroyed. Common sense however tells us that the cans will not reverse their trip. The Second Law tells us they cannot reverse their trip.

As energy transfers from the hot can to the cold can the warmer, faster molecules slow down and the cooler, slower molecules speed up. Once the water molecules are all at 500F there is no way they can collide so that one molecule speeds another back up to the 1000F energy, the transition is irreversible. For a parallel example consider a demolition derby. Can you think of a way that two 50 mph cars could collide so that one of them ended up at 100 mph?

her example is a box with fifty White Balls lined up on one side and 50 Black Balls on the other side. Once you start shaking the box what is the probability that the Balls will ever reach their initial State. There is nothing to prevent this, but the probability of this outcome is exceedingly low. Next consider the probability that the several zillion molecules in the soft drink cans will return to their initial state and you can see why Thermodynamic Transactions increase entropy and flow in one direction.

In the 1000F/00F the 1000F molecules were in one can and the 00F molecules were in the other can, a state of order prevailed.  In the 500F/500F scenario disorder prevailed, everything was the same. 

This use of the terms “Order” and “Disorder” is confusing to many as they believe that the final state where everything is the same should be a state of order while the initial state where temperatures are different should be a state of disorder. But think about two Marching Bands. Order is when they everyone is lined up, disorder prevails when they are all wandering around on the field. It is sometimes easier to view the Second Law in Terms of concentrated Energy and Dispersed Energy. In any Adiabatic Transaction Concentrated Energy will be dispersed, disorder will increase. In Thermodynamic transactions disorder always increases. Entropy is a measure of the degree of order and disorder.

Some may point out that it is easy to concentrate energy, simply put a pot of water on a stove and boil it. But that is not an Adiabatic Transaction, you have added heat. If you include the Heat input from the Stove disorder and Entropy increased.

Now let’s consider the Soft Drink Can Experiment from a work point of view. A thermocouple is a device that can generate electricity from temperature differences. In Figure 1 a thermocouple could extract energy from the cans due to the temperature difference.  

But if we let the cans reach 500F/500F state we have forever given up the opportunity to extract energy from this 1000F – 00F temperature differential. The same amount of energy is still present in the 500F/500F state, energy can neither be created nor destroyed.  But by allowing the concentrated energy in the 1000F can to  combine with the lower grade energy in the 00F the concentrated energy has dispersed, disorder has increased and we have  forever  lost the ability to  extract  useful work from the initial  temperature difference.

As well as having a quantity such as BTU’s or Joules energy has quality. Entropy is not a “thing” It is a measure, a measure of quality.  As we disperse the energy from a high temperature to a low temperature, we destroy that quality and create entropy or already used up energy. The energy is still there, it has not been destroyed, it is just of lower quality. In any thermodynamic transaction the total energy input is converted into either useful work or already used up energy or entropy. The trick for designing for maximum efficiency is to minimize entropy generation and maximize exergy output or Exergy.

Here is the simple equation for work in a process where gas at a constant pressure is used to drive a piston to do work.



In this equation the first “d” is just the mathematical symbol for differential or difference and H is Enthalpy, the total energy content. So, dH means the change in total energy content in the system, in this case the input Energy. T represents temperature while dS means the change in Entropy. PdV represents Pressure(P) times change in volume (dV). Note that in units PdV is simply pressure(lbs/ft2) * volume (ft3) which when multiplied together give ft-lb which is work.

Thus, the input energy can either go into work (PdV) or entropy (TdS). The goal of system design is thus to minimize entropy generation and maximize work.

Consider two weights tied to two ropes wound in different directions around a pipe. As the weight heavier weight falls it turns the pipe which then lifts the other lighter weight. In this example most of the energy remains concentrated, it is now in the lighter weight and not much entropy has been created. Now consider the same pipe and weight but this time the pipe drives a fan. In this case all the energy has gone to mix up air molecules, it has gone from concentrated to dispersed and cannot be used to do useful work. All the input work has created entropy, already used up energy.

Why can’t we just make Heat Engines that don’t create Entropy?


Figure 4. Simple Piston Engine 

To answer this question Spock takes you to the Enterprise’s Museum of Ancient Technology where he shows you the simple (but thermodynamically perfect) piston engine shown in Figure 4. On the left-hand we have a piston (blue) with a purple weight on the top resting on 700F air. In the next diagram a 7000F heat is introduced either by igniting fuel or through a heat exchanger. This heat heats up the 700F air to 5000F lifting the piston and weight as the air expands. Note that by mixing 7000F heat with 700F air to create 5000F air we have created entropy just as we did with the soft drink cans. We have used some of energy to heat air, not to do work. As the air does work by expanding and lifting the piston and weight it cools to 2000F.

The next question is, “What now?” As our machine is perfect and lossless the 2000F air will remain at 2000F forever and the piston will not go back to its original position to begin another cycle. Thus an exhaust valve must open letting the 2000F high pressure air out of the cylinder to mix with the 700F ambient air creating even more entropy due to the pressure and temperature differentials between the 2000F pressurized piston air and the 700F outside-unpressurized  air. Note that both temperature and pressure differentials can create entropy.

Spock then points out we had to let the air out of the piston to return to the original state on order to run in a cycle. Another statement of the Second Law uses this observation: “ It is impossible to construct an engine, operating in a cycle, which takes heat from a high temperature reservoir and does work without exhausting heat into a low temperature reservoir.” Note that in both the intake cycle and exhaust cycle we created entropy. Your next question is why we can’t just design better machines that don’t create as much entropy.


Figure 5. Vulcan Second Grade Entropy Kit

To answer that Spock gives you the Vulcan Second Grade Entropy Kit; four perfectly insulated Containers filled with a liquid at 1,0000 R (~5400F), and a perfect thermocouple as shown in Figure 5. He explains that he is using the Rankine Temperature Scale as ALL Thermodynamic Calculations must be done using a scale that starts at Absolute Zero.

Next you visit a 7000R world. Here your thermocouple begins to work and extracts energy from the container.  But as energy is extracted the containers cools. When the container reaches 7000R no more energy can be extracted. You were thus able to extract 30%, of the containers total  energy.He first beams you (well insulated) onto a planet with a temperature of  1,0000R where  you discover that the thermocouple  cannot do any work with the heat in container as the temperature on both ends of the thermocouple is 1,0000R.

Next you warp to a somewhat cooler but earth-like 400F world (5000R). On this world you will be able to extract energy until the energy of the container reaches 5000R or 50% of the energy.

Finally, you get to the 00R world where, well insulated, you discover that the thermocouple can extract 100% of the energy from the container


Regardless of which world which Mr. Spock beams you to or what initial conditions existed in the vials or what process was used to heat them,  you will quickly discover one of the fundamentals laws of thermodynamics and entropy: The maximum efficiency of any heat engine is;

Efficiency (Max) = (Tin - Tout) / Tin.

Where Tin is the input temperature and Tout is the exhaust temperature (using a scale that starts at Absolute zero.)  On the 7000R world our 1,0000R container would result in a maximum theoretical efficiency of (1,000-700)/1000 or 0.3 = 30%. Thus, efficiency is a direct function of the difference between input and output temperature, the larger the difference the higher the efficiency.

You may think this isn’t right. You just, heated a substance from 700F (5300R) to 1000F (5600R) and you should be able to get back as much heat as you put in. Sorry, your situation is no different than if you had been beamed to earth with a vial of liquid at 1000F (5600R). The maximum efficiency, the ratio of energy in-to energy-out is (560-530)/530. It makes no difference what method you used to heat the container, what method you use to extract the heat, what the initial conditions were etc., the Formula gives the Maximum Theoretical Efficiency of any Heat Engine.

Efficiency (MAX) =(Tin-Tout)/Tin   

This points out that we do not have a World Energy Crisis, we have a World Entropy Crisis. In a 600F (5200R) world we have plenty of energy. What we do have is an entropy Crisis as most of this energy is unavailable for use as it is already at or near the ambient temperature.

Another way of looking at thermodynamic efficiency is to consider a person on a diving board fifteen feet above the bottom of the pool. If the pool has ten feet of water the person can only change one-third of their potential energy into kinetic energy as when they hit the water, they can go no further.  If the pool has five feet of water, then can change two-thirds and if the pool has no water, they can change all their potential energy into kinetic energy with the attendant unpleasant consequences. Molecules having a heat content are the same, once they cool down to the ambient temperature they can fall no further, no more energy can be extracted, the energy is still there it just can’t be extracted to do useful work unless you decrease the outside ambient temperature.

For power plants if the inlet temperature of a turbine is  10400F (about 15000R) and the outlet (ambient air) temperature is 400F ( about  5000F Rankine) the maximum theoretical efficiency it could obtain is (1,500-500)/1500 or 1,000/1500 or about 33%. Since it is impossible to construct a plant operating at the maximum theoretical efficiency (there are losses) power plants operate at lower efficiencies than the Maximum Theoretical Efficiency. Due to low temperature such as these power plants in the 1940’s operated at about 33% efficiency. Today plant temperatures are higher resulting in higher efficiencies.

As outlet temperature is not controllable the main driver of modern turbine design is thus the quest for higher inlet temperatures using single crystal metal blades with thermal and ceramic coatings. These blades are cooled by lower temperature steam circulating inside the blade and also escaping from tiny holes near the Leading Edge forming a protective cooler layer on the surface of the blades.

An interesting fact is that many Electric Markets plants must list Summer and Winter Qualified capacity (Peak Output) as in the winter plants have lower Outlet Temperatures and hence higher efficiency and therefore higher output.

The ability to handle high temperatures is one of the main reasons why gas turbines are more efficient than steam turbines such as used in coal plants. For instance, the turbine inlet temperature of a gas fired Mitsubishi 501J is about 2,9000F while in a supercritical coal plant it is about 1,3000F. Temperature however is not the whole story. Coal plants however have much higher inlet pressures which can also perform work. And, as explained below, coal plants must also contend with the Heat of Vaporization. The amount of energy needed to convert 1000C water to steam.

Nucear plants run at low temperatures (about 5700F but, again, at higher pressures than gas turbines) and hence low efficiencies as the zirconium cladding on the fuel rods would melt at higher temperatures. Next generation nuclear plants having ceramic fuel or other improvements allowing them to run at far higher temperatures and efficiencies.


Figure 6. Space Suit Refill

Spock next explains that work can be done, and entropy can also be created, by differences in pressure as he walks you to the Enterprise Space Suit Oxygen Refill Tanks.  When you arrive Spock takes three tanks, one empty and the other two at 3,000 PSI. First, he hooks the tanks together (“B” in Figure 6) and the resulting pressure in the tanks is 2,000 PSI.

Next Spock takes three other tanks having the same initial conditions and hooks one 3,000 PSI tank to the empty tank resulting in the two tanks each having 1,500 PSI (“C” in Figure 6). He then hooks the 1,500PSI tank to the remaining 3,000 PSI tank (“D” in Figure 6) and both end up at 2,250 PSI which is 250 PSI higher than just hooking all three tanks together.

Why? By holding one 3,000 PSI tank in reserve you did not expose it to the lower pressure of the empty tank creating entropy or already used up energy. Every SCUBA Diver is familiar with this process as this is the way Dive Shops fill SCUBA Tanks. You will see this process again, using temperature rather than pressure, when we discuss Feedwater Heaters in Power Plants.


Adding Heat at the Highest Possible Temperature.

Figure 7. Adding Heat at the Highest Possible Temperature

Spock then decides to show you another experiment which has major implications for powerplant design. Although Spock hates pounds as they, like the Fahrenheit Scale, are irrational he decides as he is in geostationary orbit over the U.S it will be okay. But he points out that pounds and Fahrenheit together lead to an even more irrational unit, the British Thermal Unit or BTU. A BTU is the amount of heat required to increase the temperature of one pound of water one degree of Fahrenheit.

Spock then gives you four one-pound containers of water, “A”, “B”, “C”, and “D”. as shown in the Bottom Box of figure 7. Next, he gives you two BTU’s which you use to raise the temperature of Containers “B” and “D” 10F as shown in the middle box.  “B” and “D” are now one 10F degree hotter than “A” and “C”.

Now Spock gives you another BTU and a choice as shown in the top Box. You can use your BTU to raise Container ”A” one degree so that it is now equal in temperature to Container  “B” as shown in the left portion of the top box. Or, on the right in the top box you can leave Container  “C” at its’ original temperature and raise Container  “D” another one degree. Spock then asks you which option is best.

Having been through the Spock demonstration with soft drink cans you quickly recognize that the raise Container “D” option is best.  With a very good thermocouple we could get energy from the “C-D” configuration and it would decay to the “A-B’ State as energy was removed. While in the “A-B” state we have forever lost the energy that could be obtained from the “C-D” temperature differential.  This experiment and Space Suit Oxygen Refill experiment show another fundamental consequence of the Second Law: it is always more efficient to add additional pressure at the highest pressure and additional heat at the highest temperature as less entropy is created. 

An interesting story about entropy is how one industries quest to decrease entropy radically transformed another Industry. Early jet engines such as those used on the Boeing 707 and 727 aircraft compressed air, injected fuel into a combustion chamber and then expelled the high temperature and high-pressure air out the back providing thrust. As much of the high temperature and high pressure dissipated into the atmosphere 707 engines were among the most efficient machines ever invented for converting jet fuel into noise and entropy.

The solution was to place turbine blades in the hot gas pathway converting the previously wasted energy into shaft horsepower. The shaft ran to the front of the Engine and turned a large “Fan” (propeller) forcing air backwards outside the engine but inside a shroud. The “High Bypass” Engine was born. This configuration is the reason modern aircraft engines are so much wider. Today when you fly a 777 or 787 you are flying a propeller driven aircraft, over 90% of the thrust cones from the Fan vastly improving energy efficiency and aircraft range.

How did this change the Electric Utility Industry? Before the 1970’s the plant with the highest efficiency was usually the largest plant. These plants cost over one billion dollars to build and only large well-financed companies could enter the industry.  With the advent of the High Bypass Engine engineers quickly figured out that one could remove the fan, insert a generator, and use the engine for power generation.  The cheapest, most efficient plant to build was now an Aeroderivative Gas Turbine. Further construction time for a coal plant was measured in years. For an LM-6000 Aeroderivative it was months. The Gar Turbine Unit was standardized, and factory built; build the foundation, the gas and electric infrastructure, hook it up and you were in the electric business. Competition arrived and the industry changed forever. All of this because the airlines were getting tired of having to make a fuel stop in Bangor, Maine on the way back from Europe


Power Plants, The Brayton and Rankine Cycles

Figure 8. Typical Coal Plant (Rankine Cycle)

Figure 8 shows a one-line schematic of a Rankine Cycle Coal Plant which shows several instances of “adding heat at the highest possible temperature.” Let ‘start with the turbine exhaust, the steam coming out of the Low-Pressure Turbines (the Yellow Triangle LP Turbines in the diagrams).

From the LP Turbines the steam exhausts into a condenser vacuum where it condenses into water. The “vacuum” condition is so that the steam condenses at a lower temperature decreasing exhaust temperature and pressure hence increasing efficiency and plant output. Once the condensate water leaves the Condenser (flowing through the blue dashed line) it enters the feedwater heaters, devices designed specifically to add heat at the highest possible temperature.  

The feedwater heaters, (the green cylinders, four in this diagram) take extraction steam from the turbines and use it to heat the feedwater. The first stage feedwater heater extracts low temperature steam from the LP Turbine and increases the water temperature through a heat exchanger mechanism. After passage through the Boiler Feed Pump, which increases the pressure up to about 2,000PSI, the feedwater comes to the next stage of feedwater heating which takes even higher temperature steam raising the water temperature even further. The remaining feedwater heaters continue the process, taking the increasingly hotter feedwater and adding increasingly higher temperature. steam.

The water then pass through the economizer where even hotter exhaust gas raises the temperature even further, the water then proceeds to the steam drum where it descends to the bottom of the boiler and then passes upward through the  “Waterwall”, hundreds of vertical pipes, welded together to from the boiler walls where it picks up even more heat. From the waterwall the water returns to the steam drum where it either turns to steam or makes another circuit through the waterwall.

As the water changes to steam a huge amount of energy is required. A Calorie is defined as the amount of heat necessary to raise one gram of water one degree centigrade. As we heat the feedwater it’s One Calorie per Degree Centigrade; then we hit 1000 Centigrade. To change that one gram of 1000C water to 1000C steam takes another 540 Calories. So, to go from 700Farenheit (210C) to 2120F (1000C) took (100-21) or 79 Calories/gram. To change the water to steam took another 540 Calories per gram, over 7 times as much. Although this seems like a huge waste of energy when viewed from a temperature point of view we have gained much, we have changed water to steam and the that steam has the ability to do work as it moves through the turbines.

After leaving the Steam Drum the steam goes through the First Stage Superheater and finally to the Last Stage Superheater which is just above the flame front where the hottest stem picks up the hottest heat. All of this to “Add heat at the highest possible temperature.

After leaving the Superheater the steam travels to the first stage or High Pressure (HP) Turbine. After passing through the HP turbine it returns to the boiler to be reheated (adding heat at the highest possible temperature) and then to the Intermediate Pressure Turbine. As the steam cools it expands. To constantly reduce the pressure so as to keep the steam as a vapor each row of turbines blades is larger than the one before it increasing the volume as the pressure drops. Finally, after leaving the IP Turbine the steam has expanded so much that it usually requires two low pressure turbines.

After the steam it is exhausted from the LP Turbine it goes into the condenser and it is here that a major heat loss occurs. Remember the 540 calories per gram it took to turn the water to steam? When we turn that steam back into water, we get an instant replay but in reverse. To turn 700F (about 200C) steam back into 700F Water we emit 540 calories per gram. The condenser brings in colder water from a lake, river, or ocean. This water travels through the inside of thousands of condenser tubes to extract the 540calorie per gram heat generated from condensing the water and in doing so the cooling water rises about 400Fcreating large amounts of entropy.

This water can be returned to the source but is usually to hot so it must go through a cooling tower where the water falls through the air and vaporizes it again gives up 540 calories. Thus, as the steam system is a closed circuit, the 540 calories per gram used to vaporize the water is all rejected to the condenser and then to the cooling tower.


Figure 9. Combined Cycle Cogeneration Plant (Brayton and Rankine Cycle)


Everything we have learned about entropy is utilized in a Combined Cycle Plant. In a Combined cycle plant as shown in Figure 9 we utilize two different Thermodynamic Cycles. The first (the “Brayton” Cycle)  

 Uses a gas turbine to power a generator. The gas turbine exhaust however is about 10000F and still contains much useable energy.

Note that in the Brayton Cycle there is no reason to turn water into steam with the attendant losses as there is in a Rankine Cycle Unit. The hot gases run directly through the turbine producing work. Thus, a pure Brayton Cycle Unit such as a gas turbine does not have a need for cooling water.  The Brayton cycle could not be run in a conventional coal plant as the molten ash and particulate matter would soon destroy the turbine blades. Thus, the push for coal gasification so that a higher efficiency Brayton Cycle can be utilized for coal.

In a Combined Cycle Unit, the hot turbine exhaust is then run through a Heat Recovery Steam Generator or HRSG where the exhaust energy is used to turn water to steam which then runs a turbine using a conventional Rankine Cycle. By using the Rankine Cycle, we avoid discharging 10000F air into the atmosphere creating massive entropy. instead we use it to do work. The fact that these units use the Brayton Cycle on the front end and the Rankine Cycle on the back end is where the name “Combined cycle comes from. As the Brayton Cycle portion of the plant does not need cooling water a combined cycle unit only uses about 40% of the cooling water as an equivalent coal plant.

Due to the differences in efficiencies  modern combined cycle plants can generate one kw-hr with just less than 7,000 BTU’s/kw-hr while a Coal plant normally requires about 10,000 BTUS/kw-hr (this number is referred to as the plants “Heat Rate.) This lower fuel usage also translates into lower CO2 emissions.

Many large combined cycle plants stop there but others, located near industrial facilities or areas needing central heat, go even further in utilizing waste heat for other purposes. These units are known as Cogeneration or “Cogen” facilities. In figure 9 the HRSG has two additional outputs. Fist some of the exhaust is sent to another facility to power a dryer and separate heat exchanger section is placed in the HSRG to provide hot water to the customer.

Why does this make sense? Entropy. Methane burns at about 3,5000F.  Going back to the soft drink cans think about the energy forever lost by burning methane at 3,5000F and mixing it with 700F air to get 2900F air for the dryer when the power plant exhaust is available for no added energy. On a per unit basis one of the greatest entropy generators around is in your house, burning methane at 3,5000F to heat 700F air to 1200F or so to heat your home. Many cities such as Boston and New York thus have Cogen plants and use their waste heat to supply central heat.


TS Diagrams

Figure 10. TS Diagram Brayton Cycle

A good way to visualize Entropy is through a TS Diagram (Temperature – Entropy Diagram, S stands for Entropy) as in Figure 10. which shows a TS Diagram for the Brayton Cycle used by a gas turbine. Thermodynamics majors may think this is an unusual looking TS Diagram and it is. I have started T at absolute zero. Normally, as in Figure 11, below showing a superheat Rankine Cycle we start T just below the lowest temperature in the cycle as everyone knows there is a lot of space below the diagram (in addition to saving energy engineers like to save paper space.) But by running the curve to Absolute Zero all of the energy involved in the transaction is shown.  The area labeled W is the work done in the transaction the Area labeled OE is the other energy in the transaction.

Point 1 in Figure 10. indicates the temperature and entropy of the inlet air. The path between Points 1 and 2 is where the air is compressed and heated in the turbine compressor. The path from 2-3 is the combustor while 3-4 is the gas expanding through the turbine doing work. Path 4-1 is the exhaust expanding and cooling back to atmospheric pressure and temperature.

An interesting aspect of TS Diagrams is that the area bounded by 1-2-3-4 Curve is the work being done in the transaction. You night wonder why a TS Diagram would show work but remember the equation given above for work and entropy.




It is clear from this equation that all the energy in the transaction goes into work or entropy. If you view the plane in the TS Diagram as the total temperature – entropy possibilities

the area outside area 1-2-3-4 equals TdS and the area inside 1-2-3-4 is equal to the work done.

These diagrams also give us an easy way to visualize Efficiency - W/(W+OE). From Figure 9 one can see that the area 1-2-3-4 or work is determined by inlet and outlet temperatures and entropy generated.   

Some may point out that generating more entropy (Going further along the X Axis will increase the area under the curve and thus more work. That is just a matter of scale. I have run the curve down to Absolute Zero to show all the energy of the substance involved in the transaction including its latent energy before it was heated. If you extend the area further along the entropy axis you also increase the 1-2-3-4 area but also increase the “OE” Area proportionally and the ratio of work to total energy remains constant.

The greater the difference between Tin and Tout the greater the area under the TS Curve in relation to the area under the line 1-4 line and the greater the efficiency.

The only way to increase the work -to-total energy or efficiency is to raise the 2-3 line by running at a higher inlet temperature or lower the 1-4 line by running at a lower outlet temperature. Lowering line 1-4 is the reason we produce a vacuum in the condenser so that the steam will condense at a lower temperature thus gaining efficiency.


Figure 11. TS Diagram Rankine Cycle

Figure 11 shows the TS Diagram for a Superheat Rankine Cycle as used in a Coal Plant. Superheat simply means that heat is applied to a gas above a temperature where all the liquid has converted to vapor. In Figure 11 There is a upside “U” shaped curve in the background. That curve is a steam curve and shows the state of water as either Water or Steam.  Simplistically, everything to the Left of the Peak of the Curve is water (1-2-3) while everything to the Right(5-6-7-8) is  steam. The Area in the Middle of the Curve is the area where we are transitioning from Steam to Water or Vice Versa.

Point 1 is the temperature inside the condenser. which represents the outlet and inlet temperature of the cycle as the cycle is a closed loop.  From Point 1-2 the water passes thorugh one stage of Feedwater Heating and the Boiler Feed Pump. From Points 2-3 the water passes through additional Feedwater Heaters and the Economizer to the Steam Drum. From 3-4 the water circulates through the Steam Drum / Waterwall System until it converts to steam. During this segment all the added energy, 540 Calories per gram, goes into the Heat of Vaporization and hence entropy, none of it is raising temperature. But there is a good reason for doing this as we are converting water to stem allowing us to utilize the steam in a turbine to convert the high pressure to work. At point 4 all the water has converted to steam and we now add heat in the superheater from points 4-5. It is clear from the slope of the 4-5 line that temperature is increasing raising the plant efficiency. Thus is the whole point of the Superheater; to add heat at the highest possible temperature after all of the water has been converted to steam so that the  heat goes into raising stem temperature and not converting water to steam thereby creating Entropy.

From 5-6 the steam is expanding through the High-Pressure Turbine giving up energy and doing work. It is important for the temperature at point 6, the exhaust from the HP Turbine be above the point where the steam would begin condensing to water. From 6-7 the steam passes through the reheater adding heat at the highest possible temperature above the temperature where some of the heat would go into converting water to steam. From 7-8 the water passes through the IP and LP Turbines. When the steam reaches the condenser as it is in a closed system it is lower than atmospheric pressure, usually at about 2 PSI enabling it to remain as steam at a lower Temperature.

Finally, from 8-1 the steam converts to water in the condenser giving up its’ hard earned 540 Calories per gram Heat of Vaporization to the Cooling Water. As before the total work done is the area inside the curve which also shows the increase in area (work) due to Superheat and reheat.

We live in a universe of entropy. In addition to thermodynamics the mathematics of entropy. have applications in quantum mechanics, computer science, information theory and a host of other fields. And, in the end, most believe the Universe will die an entropy death, everything will cool to a temperature near absolute zero. No work will be possible. However even here there will be some order left in the Universe as molecules can have different states. 

Although we cannot reach  absolute zero if we were able to continue the journey from the “Entropy Death” Temperature down to absolute Zero we would find that a perfect crystal has only one state at that temperature and hence has exactly  zero entropy.  This is the Third Law of Thermodynamics.

We do not know the ultimate temperature of our universe. Some believe it is around 40K while others have different theories.

Spock believes that believe that due to interactions between the quantum vacuum fluctuations, Dark Energy  and the High Level of Energy found in the Higgs Field in Empty Space (246 GEV) and cosmological constant considerations the correct answer to the Heat Death Question  is 420K finally providing the ultimate question to go with the already known answer to the Ultimate Question – 42.[2]

Spock, happy that you have achieved the Vulcan Second Grade level, now prepares to beam you back to earth. Knowing you are newly knowledgeable in entropy and energy conversion; Spock modifies his customary Goodbye saying, “Live Long and Prosper in the energy conversion business by always adding heat at the highest possible temperature.”




Authors note. This paper is open source. The paper is intended to help beginning Thermodynamics Students understand entropy before they get into the math and help people in the energy field to develop a better understanding of energy conversion and  power plant operation and efficeincy. Hopefully it will help fulfill those needs.

I am an Electrical Engineer (who has hung around power plants for most of my life) and fully realize that Mechanical Engineering Instructors  can improve on this paper and that I have taken a couple of shortcuts. So feel free to use excerpts, edit the paper for your own needs, etc. with or without attribution.  

Charles E. Bayless

­­­Retired President, West Virginia University institute of Technology

Retired CEO of Tucson Electric Power and Illinois Power

Co-Chair of The Climate Institute.


[1] During the break, the other half used the Food Replicator, and each consumed a Pan Galactic Gargle Blaster. According to the “Hitchhikers Guide to the Galaxy” the effects of a Pan Galactic Gargle Blaster are like "having your brains smashed out by a slice of lemon wrapped round a large gold brick." Other authoritative sources point out that consuming two Pan Galactic Gargle Blasters will cause your Brains to be expelled through your ears. Ford Perfect however is known to have consumed three at a single setting which probably explains a lot of Ford Perfect’s behavior.

[2] In “The Hitchhikers Guide to the Galaxy “a Giant Computer “Deep Thought” was constructed and told to search for the answer to the Ultimate Question of Life the Universe and Everything. Deep thought calculated for 7.5 million years and then announced before a Grand Galactic Gathering the Answer – 42. After several moments of confusion, blank looks and stunned silence, Deep Thought was asked, “Err, exactly what is the Ultimate Question.” Whereupon Deep Thought replied, “I don’t know, I wasn’t told to find the question.” A new, much larger computer, earth, was then built to discover what was the ultimate question. Unfortunately, earth was destroyed by Vogons to make a clearing for a hyperspace expressway ten minutes before it arrived at the answer. If Spock is correct that the ultimate temperature of the universe is 420K we may finally have discovered the long-lost question to go with ultimate answer of 42.

Matt Chester's picture
Matt Chester on Sep 24, 2020

Really valuable post, thanks for the detailed and thorough review of a complex but critical topic (though I must admit the TS Diagram Rankine Cycle gave me immediate horror flash backs to my engineering classes from college!)

Michael Keller's picture
Michael Keller on Sep 28, 2020

Not only are combined-cycle plants exceptionally efficient, they are also easy to operate with very low build costs and low priced energy. By contrast, coal and nuclear plants are much more expensive to build and operate. Land needs are insignificant, unlike renewable energy which requires vast acreage..

Nathan Wilson's picture
Nathan Wilson on Mar 22, 2021

Yep, combined cycle plants are the perfect choice for situations where fossil gas is abundant & domestic, and no one cares about emission of air pollution or CO2.

When that isn't the case, we have to ask ourselves how we can make nuclear affordable.  Nuclear costs have soared, even though for the energy delivered, nuclear uses much less concrete and steel than solar/wind, and has much lower environmental impact.

Matt Chester's picture
Matt Chester on Mar 22, 2021

Yep, combined cycle plants are the perfect choice for situations where fossil gas is abundant & domestic, and no one cares about emission of air pollution or CO2.

When that isn't the case..

Genuine question, Nathan-- when isn't that the case? The nature of emissions is that it's a global impact, so it's not that we care about emissions in one country or region but not others, but rather emissions being spilled into the atmosphere anywhere is a problem for us everywhere. Perhaps if we're talking about a successful carbon capture technology being attached then we don't care about the immediate emissions from the plants since they won't make it to the atmosphere, but we're definitely not at that point 

Nathan Wilson's picture
Nathan Wilson on Mar 23, 2021

Well, globally, there are lots of places where fossil gas is not abundant & domestic: China, India, Germany, and many others.  The US is a leader in frac'ing technology so we don't face the same political risk that others do when buying gas that comes from Russia or Iran.  Thus, those listed countries place a great value on their coal industries (even if they publicly claim to love renewables above all others).  

Fair point about people everywhere caring about emissions.  But we engineers always use number for decision making.  If the customer isn't willing to pay a premium for a feature (e.g. low emissions) or the premium isn't enough to make a difference, then to me, that's the same as "the customer doesn't care about the feature".

Low emissions are like good health care.  Not every country can afford it, and even among those that can, not all will make the choice.  That's why making clean energy cheaper than coal is so important.  The US fossil gas industry has done that successfully; if the nuclear industry could do it as well, then every nation could have clean energy, since nuclear fuel is the most portable and easily stockpiled fuel of all.

Jim Stack's picture
Jim Stack on Mar 10, 2021

Charles, I really like your examples and formulas as well as you knowledge of inventors and systems. I am a huge fan of efficiency. QUOTE=Efficiency (MAX) =(Tin-Tout)/Tin   you always include great formulas for each example.

  We you show how poor the efficiency is on a piston engine IR =E max 30% you don't then compare it to an electric motor at 80-110% efficiency.  They also in the reliability at 10 times more in the electric since it only has 2 vs 2,000 moving parts and the total system energy with REGENERATIVE braking giving my 110% efficient at times AC electric motor. 

   In the entropy example I would think it is worth mentioning Geo-Thermal since it trades heat and cold with the earth to make a very comfortable temperature for people. It's a great example in many areas of the world.

   Then when we also look at the poor efficiency of COAL we should also consider how it uses and pollutes a lot of water to make power. It also creates tons of air pollution. It can't be ramped  up and down very fast so it has to run at full output each day to be able to cover the max need the next day. Overall it is a very poor system that would embarrass us if it was invented today. 

    So I love your examples but would like to see some quick comparisons of other systems with all the outputs including pollution and water use. Your data is right on the mark but I feel some people may not be able to put it all together. 

Bob Meinetz's picture
Bob Meinetz on Mar 19, 2021

Wonderful article, Charles. I look forward to one where you debunk the current trend of considering efficiency as equivalent to energy, as if energy efficiency alone might one day deliver a carbon-free world (a pet peeve of mine).

Nathan Wilson's picture
Nathan Wilson on Mar 22, 2021

Thanks for the thermodynamics refreshers.  It helped me understand an issue that had been troubling me: why efficiency of thermo-chemical cycles for producing hydrogen are limited by the same Carnot efficiency that applies to thermal power plants (answer: otherwise heat could not drive the reaction, same as in Spock's demonstrations above).

The natural follow-up question: since fuel cells and reverse fuel cells are not heat engines, what limits their efficiency?

Charles Bayless's picture
Thank Charles for the Post!
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