What does the ratio of reservoir temperatures have to do with the efficiency of a heat engine?

Question

I am having a conceptual difficulty with heat engine efficiency. I do not understand why having a larger difference in temperature between the hot and cold reservoirs have an effect on the thermal efficiency. In this question I am specifically referring to Carnot heat engines. To clarify, I do not understand the reason the exact same (carnot) heat engine is more efficient when placed between different reservoirs. What is the mathematical or physical reason behind this?

I found this question: Why is it that a Carnot heat engine will reject no heat to a zero temperature sink? which is very similar to what I am asking, but the answers there don't make sense to me and it seems that was the case for the OP too.

I hope the question is clear and cheers in advance!

Answer 1

Even though nobody seems to have come across this yet, I won't delete it just in case someone has a similar lack of understanding I had. This link here: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/CarnotEngine.htm shows very clearly why the assumptions made of the Carnot engine must then mathematically lead to the ratio of temperature differences between the reservoirs (or indeed the ratio of $Q_{in}$ to $Q_{out}$) being part of the equation which defines the efficiency of the engine. On the page, just note that there is an error; instead of $$Q_c = nRT_cln\frac{V_a}{V_b}$$ it should be $$Q_c = nRT_cln\frac{V_b}{V_a}$$

I know this answer isn't very long or in depth, but the article I linked to is very clear, and so hopefully anyone with similar doubts can get use that to improve their understanding!

Answer 2

For a thermodynamic cycle, the internal energy in the system does not change; thus, the first law is $Q$ - $W = 0$ where $Q$ is the net heat added to the system and $W$ is the work done by the system. $Q = Q_{in} - Q_{out}$ where $Q_{in}$ is the heat added from the heat source and $Q_{out}$ is the heat rejected to the heat sink. $W = Q_{in} - Q_{out}$. The thermal efficiently is defined as $W/Q_{in} = (Q_{in} - Q_{out})/Q_{in}$. It can be shown that $Q_{out} /Q_{in} = T_{sink}/T_{source}$ where $T_{sink}$ and $T_{source}$ are the absolute temperature of the sink and source, respectively, so the thermal efficiency can be expressed as $1 - T_{sink}/T_{source}$. Therefore, the thermal efficiency increases as the sink temperature decreases, or as the source temperature increases. See a textbook on thermodynamics, such as one by Sonntag and Van Wylen, for the details relating heat to absolute temperature for the Carnot cycle.

A thermal power plant is more efficient during a cold winter than during a hot summer because the heat sink (river or lake) is colder in the winter.