Thermodynamics: Why Every Engine Pays a Heat Tax

Pump up a bicycle tyre and the barrel of the pump gets warm. Nobody heated it. You only pushed. Leave a phone charging under a pillow and it gets hot, because every bit of electrical energy that is not stored in the battery has to go somewhere, and "somewhere" is heat. Stand next to a power station and you will see a cooling tower pouring white steam into the sky: a machine built to make electricity that spends most of its effort throwing heat away on purpose.

A mechanical or aerospace engineer looks at all three and sees the same two rules. Energy is never lost, only moved around. And when it is moved, some of it always ends up in a form that is hard to use again. Those two rules, written carefully, are thermodynamics. Every engine, every refrigerator and every jet is a way of living inside them.

Draw the box first: systems, properties and the ideal gas

Every thermodynamics problem starts with a drawing: a dashed line around the thing you care about. That is the system. Everything outside is the surroundings. If nothing crosses the line, it is a closed system, like gas sealed in a cylinder. If matter flows through, it is a control volume, like a jet engine: air in at the front, hot gas out at the back. Most of the later subjects, from fluids to propulsion, are control-volume thinking. Draw the box, then count what crosses it.

Inside the box we describe the stuff by properties: pressure, temperature, volume, internal energy. A property depends only on the state the stuff is in, not on how it got there. For air at ordinary conditions, the properties are tied together by the ideal gas law:

A worked number: one litre of air at 100 kPa and 300 K is about 0.040 mole, roughly 1.2 grams. Now squeeze it to a tenth of the volume. What happens depends on how fast. Squeeze slowly and the walls carry the heat away, so the temperature stays 300 K and the pressure rises tenfold. Squeeze fast and the heat has nowhere to go, so the work you put in becomes temperature. Try both.

At 10 to 1 and fast, the air reaches about 754 K, near 480 °C, and about 2.5 MPa. No spark, just squeezing. That is the whole idea of a diesel engine: compress air so hard that it is hot enough to light fuel sprayed into it. Real diesels typically compress about 15 to 20 to 1.

The zeroth and third laws: the bookends

The laws are numbered in the order people needed them, not the order they make sense in. The zeroth law was named last, after the first two already held their numbers. It says: if A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in equilibrium with each other. It sounds trivial. It is what makes a thermometer legitimate. C is the thermometer: it can compare two things that never touch.

At the other end, the third law says you cannot cool anything all the way to absolute zero (0 K, −273.15 °C) in a finite number of steps. Each step removes less heat than the one before. It also says a perfect crystal at 0 K has zero entropy, which gives entropy a true zero to measure from. You will rarely compute with it. Cryogenic engineers live with it.

The first law: energy is a bank balance

Think of the energy inside your box as a bank balance. Two things can change it. Heat (Q) is energy that crosses the line because of a temperature difference. Work (W) is energy that crosses the line because of a force moving something, like a piston. There is no other way in or out of a closed system. So:

Sign conventions differ between textbooks, so watch the first page of yours. Here, heat in is positive and work done by the gas is positive. The bank-balance picture also explains why "heat" and "work" are not things you hold. You cannot own heat; you can only watch it cross the line. Only the balance, the internal energy, is a property of the gas.

Worked number: air has a specific heat at constant volume of about 0.718 kJ per kilogram per kelvin, so putting 50 kJ of heat into 1 kg in a sealed rigid can (no work) raises it by 50 ÷ 0.718 ≈ 70 K. Now let the same heat do something else.

Notice what the third preset shows: 50 kJ of heat goes in, 50 kJ of work comes out, and the temperature does not change. That is the dream of an engine, and it only works for one expansion. An engine has to come back to its start and repeat. That is where the next law bites.

The second law: heat only rolls downhill, and entropy counts the cost

The first law allows a cup of coffee to cool by pushing its heat into the room, and it equally allows the room to push heat back into the coffee and warm it. Both conserve energy. Only one ever happens. The second law is the rule that picks the direction. Clausius's version: heat does not flow by itself from a colder body to a hotter one. Kelvin's version: no cyclic machine can turn heat from a single temperature entirely into work. They are the same law in different clothes.

Engineers make this countable with entropy. When heat Q enters a body at absolute temperature T, its entropy changes by Q ÷ T. The same joule of heat adds more entropy when it enters something cold than when it leaves something hot, because dividing by a smaller T gives a bigger number. So heat that falls from hot to cold creates entropy overall, and heat that climbs from cold to hot would destroy it. The second law says total entropy never decreases.

At 400 K and 300 K, moving 1,000 J from hot to cold gives −2.5 J/K for the hot body and +3.33 J/K for the cold one: a net +0.83 J/K. Run it backwards and the total is −0.83 J/K, which never happens by itself. A refrigerator does move heat uphill, but only by being paid in work, and its exhaust heat creates more entropy than it removed.

Energy is conserved. Entropy is not: it is made every time anything happens, and it can never be un-made.

Carnot: the ceiling over every engine

Now combine the two laws. A heat engine takes heat from a hot source, turns some into work, and must dump the rest into a cold sink. It has to dump some: if it took all its heat from the hot source and rejected none, the entropy of the world would fall. The least heat it can reject is the amount that keeps total entropy from decreasing. Working that out gives the highest efficiency any engine can have, discovered by the French engineer Sadi Carnot in 1824 (before the first law was even clear):

It depends only on the two temperatures, not on the fluid, the pistons or the cleverness. Two consequences stay with every engineer. To get more out, run hotter or reject colder; that is why turbine blades are cast to survive white-hot gas. And no engine reaches 100%, because the cold sink is never at absolute zero.

The steam plant's ceiling is 63.9%. Real steam plants fall well short of it, commonly around a third to 40% of the fuel's heat becoming electricity. The best gas-and-steam combined-cycle plants now reach about 64%: Siemens Energy reported 64.18% for the Keadby 2 plant in the UK, verified in May 2024. The ocean-thermal case shows the other side. A warm-water-over-cold-water engine in the tropics has a ceiling under 7%, which is why it is hard to make economical, even before friction.

Cycles that run the world

An engine must return to where it started, so that it can go round again. That loop of states is a cycle, and the four you will meet in every course have simple names for what the gas does.

• Otto (petrol engine): compress air and fuel, spark, expand, exhaust.
• Diesel: compress air alone until hot, inject fuel which ignites by itself, expand, exhaust.
• Brayton (gas turbines and jets): compress air continuously, burn fuel in a flowing stream, expand through a turbine.
• Rankine (steam and most of the world's electricity): pump water to high pressure, boil it, expand the steam through a turbine, condense it.

For the ideal, air-standard versions of the first three, the efficiency depends on one ratio. For Otto it is 1 − r^−0.4, where r is the compression ratio. At 10 to 1 that is 60%, well above what a real engine delivers, because real engines have friction, heat loss and incomplete combustion. The shape is the lesson: a higher ratio gives a higher efficiency, with diminishing returns. Petrol engines are limited by "knock" (fuel igniting early), which is part of why diesels, which compress only air, can use higher ratios.

The Brayton cycle is the one to remember for later: a jet engine is a Brayton cycle laid out in a tube, with a compressor, a combustor and a turbine in a line. At a pressure ratio of 30 the ideal efficiency is 62%; at 10 it is 48%. That is why modern engines push their pressure ratios so high, and it is exactly the story told in Propulsion (34/21).

Run it backwards: refrigerators, heat pumps and COP

Run an engine in reverse, put work in, and it pumps heat from cold to hot. That is a refrigerator (the useful part is the heat taken from the cold side) or a heat pump (the useful part is the heat delivered to the hot side). Nothing breaks the second law, because you are paying work to climb the hill. For these machines "efficiency" is the wrong word, since the answer can be more than 1. Engineers use the coefficient of performance, COP: useful heat moved per unit of work paid.

A resistance heater turns 1 kWh of electricity into 1 kWh of heat. A heat pump delivering heat at 35 °C from a 5 °C outdoors has an ideal COP of 10.3: it moves ten times more heat than the work it uses, because it is mostly moving heat, not making it. Real machines fall short of the ideal. The slider's 45% is an assumption for illustration, not a measurement; set your own.

Two things to see. A colder day lowers the COP. And so does asking for hotter delivery: a heat pump feeding low-temperature floor heating at 35 °C beats one feeding old radiators at 55 °C, for the same weather. Everything is about the lift between the two temperatures.

What a graduate just sees

• A system is a dashed line: first decide what is inside and count what crosses it.
• Compressing a gas fast heats it; expanding it fast cools it. Work and heat are two ways of moving the same energy.
• "Heat" is energy in transit. Things have internal energy, not heat.
• Energy is always conserved, so the question is never "where did it go" but "what form is it in now, and is it still useful".
• Efficiency has a ceiling set by two temperatures: 1 − T_cold/T_hot. Running hotter is how engines improve.
• Every real process makes entropy. Friction, mixing and heat falling down a gradient all spend usable energy.
• A power plant's cooling tower is not waste by sloppiness. It is the second law's bill.
• A heat pump or a fridge looks like it breaks the rules, but it is moving heat, not creating it, and it is paying for it.

Where it shows up next

• Fluid Mechanics (34/15): the control volume and the steady-flow energy equation become the frame for every pipe, pump and nozzle.
• Heat Transfer (34/16): thermodynamics says heat can flow from hot to cold; heat transfer says how fast through a wall, a fin or a vacuum.
• Propulsion (34/21): a jet is a Brayton cycle in a tube, and a rocket is a heat engine whose "work" is kinetic energy in the exhaust.
• In the world: power stations (Rankine), cars and aircraft (Otto, Diesel, Brayton), air conditioners, freezers, heat pumps in homes, and the energy accounting for any battery or fuel debate.

If you want to go deeper

• 📘 Textbook. Yunus Çengel, Michael Boles and Mehmet Kanoğlu, Thermodynamics: An Engineering Approach (McGraw Hill, 10th edition). Widely assigned and full of worked examples and property tables.
• 📘 Textbook. Moran, Shapiro, Boettner and Bailey, Fundamentals of Engineering Thermodynamics (Wiley). The other standard choice, more careful on the first-principles setup.
• 📘 Gentler. Peter Atkins, Four Laws That Drive the Universe (Oxford, 2007; reissued in 2010 as The Laws of Thermodynamics: A Very Short Introduction). Almost no maths.
• 🌐 Free course. MIT OpenCourseWare 16.050 Thermal Energy (Fall 2002): basic concepts, the second law, propulsion and power cycles, and heat transfer in one aerospace course. Also the thermodynamics half of 16.01 Unified Engineering.