From 20f2234b5224d29e02bef8884fdeb546afe09b2a Mon Sep 17 00:00:00 2001 From: Indigo5684 <159226326+Indigo5684@users.noreply.github.com> Date: Tue, 30 Sep 2025 13:19:40 -0500 Subject: [PATCH] Thermal Physics Reading Assignment 11 --- .../physics/thermal/4-engines-refrigerators.md | 15 +++++++++++++++ .../Reading Assignment 11.rnote | Bin 0 -> 423949 bytes 2 files changed, 15 insertions(+) create mode 100644 docs/physics/thermal/4-engines-refrigerators.md create mode 100644 statics/physics/thermal/reading-assignments/Reading Assignment 11.rnote diff --git a/docs/physics/thermal/4-engines-refrigerators.md b/docs/physics/thermal/4-engines-refrigerators.md new file mode 100644 index 0000000..2638f50 --- /dev/null +++ b/docs/physics/thermal/4-engines-refrigerators.md @@ -0,0 +1,15 @@ +# Chapter 4 - Engines and Refrigerators + +## Section 4.1 - Heat Engines + +**Definition**. A *heat engine* is any machine that absorbs heat and converts part of it into work. We model heat engines as accepting heat from a *hot reservoir* with temperature $T_h$ and rejecting heat to a *cold reservoir* with temperature $T_c$, as well as outputting work. + +**Definition**. A *reservoir* is any object so large that its temperature does not change as it accepts or rejects heat + +For a a heat engine, we will denote $Q_h$ and $Q_c$ to represent the heat absorbed from the hot reservoir and heat rejected to the cold reservoir. Then, the net work done by the engine is $W$. In this model, all signs are positive. + +**Definition**. The *efficiency* $e$ is the benefit/cost ratio. Fora heat engine, we see that $e = \frac{W}{Q_h}$. As conservation of energy applies, we know that $Q_h = W + Q_c$, so that $W = Q_h - Q_c$. Then, we can write $e = \frac{W}{Q_h} = \frac{Q_h - Q_c}{Q_h} = 1 - \frac{Q_c}{Q_h}$. We then see that efficiency is always in the range $[0, 1]$. + +By the second law of thermodynamics, $S_h \geq S_c$. We know that $S = \frac{Q}{T}$ for a reservoir, so $\frac{Q_h}{T_h} \geq \frac{Q_c}{T_c}$, which can be rewritten as $\frac{T_c}{T_h} \geq \frac{Q_c}{Q_h}$. Then, substituting into the equation for efficiency, $e \geq 1 - \frac{T_c}{T_h}$. Note that actual efficiency will be less than this limit as entropy will be produced within the engine as well. + +Let us now revisit a classic: the Carnot cycle. This cycle consists of isothermal expansion of a gas at temperature $T_h$, adiabatic expansion of the gas from $T = T_h$ to $T = T_c$, isothermal compression at $T = T_c$, and adiabatic compression from $T = T_c$ to $T = T_h$. By applying the formula of the ideal gas, we see this cycle reaches the maximum efficiency of $e = 1 - \frac{T_c}{T_h}$. 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