PX154 - E4 - the carnot cycle

introduction

introduced by sadi carnot in 1834 as an ideal model of a reversible heat engine cycle
it's theoretical and it can be shown that it's the most efficient heat engine cycle
all real engines have an efficiency lower than that of a carnot engine
work done = area enclosed by the cycle
if this is reversed, a heat pump would be created

$A \to B :$ isothermal expansion at $T_{H}$
$W_{A \to B} = n R T_{H} l n \frac{V_{2}}{V_{1}}$
$Δ U = 0$
$Q_{A \to B} = W_{A \to B}$
$B \to C :$ adiabatic cooling
$Q_{B \to C} = 0$
$W_{B \to C} = - n C_{V} (T_{C} - T_{H})$
$Δ U = - W_{B \to C}$
$C \to D :$ isothermal compression at $T_{C}$
- heat expelled from engine to the cold reservoir
  $W_{C \to D} = n R T_{C} l n \frac{V_{3}}{V_{4}}$
  $Δ U = 0$
  $Q_{C \to D} = W_{C \to D}$
$D \to A :$ adiabatic compression
$Q = 0$
$W_{D \to A} = - n C_{V} (T_{H} - T_{C}) = - W_{B \to C}$
$Δ U = - W_{D \to A}$
for one cycle:
$Δ U = 0$
$W_{t o t a l} = n R T_{H} \ln \frac{V_{2}}{V_{1}} + n R T_{C} \ln \frac{V_{4}}{V_{3}}$
$Q_{t o t a l} = W_{t o t a l}$
can simplify using: $T V^{γ - 1} = c o n s t a n t$ , for adiabatic process
- for $B \to C :$
$T_{H} V_{2}^{γ - 1} = T_{C} V_{3}^{γ - 1}$
- for $D \to A :$
$\begin{aligned} T_{H} V_{1}^{γ - 1} & = T_{C} V_{4}^{γ - 1} \\ ∴ \frac{V_{2}}{V_{1}} & = \frac{V_{3}}{V_{4}} \end{aligned}$
- so:
$W_{t o t a l} = Q_{t o t a l} = n R (T_{H} - T_{C}) \ln \frac{V_{2}}{V_{1}}$

[YF 20.6]

we have: $ϵ = \frac{W}{Q_{H}} = 1 - \frac{| Q_{C} |}{| Q_{H} |}$
- $Q_{H} =$ heat absorbed from $A \to B$
- $Q_{C} =$ heat expelled from $C \to D$

ϵ = 1 - \frac{| Q_{3 \to 4} |}{| Q_{1 \to 2} |}

$Q_{A \to B} = n R T_{H} l n \frac{V_{2}}{V_{1}}$
$Q_{C \to D} = n R T_{C} l n \frac{V_{4}}{V_{3}}$

∴ ϵ = 1 - \frac{T_{C}}{T_{H}}

eg:
- $T_{C} = 273 K$ , $T_{H} = 373 K$ , $ϵ = 0.27 ⟹ 27 %$
- $T_{C} = 300 K$ , $T_{H} = 600 K$ , $ϵ = 0.5 ⟹ 50 %$
- petrol engine $\sim 20.35 %$
- diesel engine $\sim 50 %$
- stirling engine $\sim 15 - 30 %$
- humans $\sim 25 %$
always use kelvin for temperature
all reversible engines operating between the same temperatures have the same efficiency