Engine Efficiencies

Heat engines convert heat to energy, specifically mechanical energy, which can be used to move stuff around $(W=F\cdot\Delta x)$. It does this by using a substance to move things, while cooling the substance. A heat source initially heats the substance, the substance moves whatever it needs to move while losing heat. This substance is anything that has a heat capacity. Some heat is lost to the surroundings, and there are practical issues of friction and drag.

Note that heat pumps (refrigerators) do the exact opposite by performing work on a substance. They usually compress (do work) on a refrigerant (the working substance) and draw out heat. This is a diversion I do not wish to explore.

Carnot engines operate on the carnot cycle. The Carnot cycle has in order 1) isothermal expansion, (where it expands in volume by pulling in heat) 2) adiabatic expansion, (where it loses temperature because all the heat has to be dispersed among the expanding molecules) 3) isothermal compression (where it compresses by expelling heat) and 4) adiabatic compression where it gains temperature.

Figure 1. Pressure vs. Volume curve of the Carnot cycle.

Figure 2. Entropy vs. Temperature curve of the Carnot cycle

Lets see what happens if $Q_C=0$ (which means our engine doesn't lose any waste heat). Then we'll have negative entropy because $\Delta S=\displaystyle\frac{-Q_H}{T_H}<0$ which violates the second law of thermodynamics. Which means we have to consider waste heat. So the maximum work we can get is $\Delta W=Q_H-Q_C$. Now, considering both sources: $\Delta S=\displaystyle\frac{-Q_H}{T_H}+\displaystyle\frac{Q_C}{T_C}$ and we know that $\Delta S>0$. For the entropy to be positive, the minimum value of $Q_C$ will be $\displaystyle\frac{Q_H}{T_H}T_C$ and this means that $$\Delta W=Q_H\left(1-\displaystyle\frac{T_C}{T_H}\right)$$.

This means we can't have 100% efficiency because we can't exhaust the heat sink to 0 K, and we will definitely lose waste heat to the universe due to the second law of thermodynamics. Also, this is a purely theoretical concept, taking a Carnot cycle through its steps without losing any additional energy will require us to do it infinitely slowly, and it is impractical to build an engine that takes forever to do its job. Another way to think of the whole entropy argument before is to look at the S-T diagram and realize that for the cycle to complete we need to gain back all the entropy we lost.

Bonus: TIL Diesel is apparently the name of a dude called Rudolf Diesel who invented the Diesel Engine that runs on Diesel.