Part I, Chapter 3

Thermodynamic Cycles

Carnot, Otto, Diesel, Rankine, Stirling, and refrigeration cycles

Historical Context

The study of thermodynamic cycles began with Sadi Carnot's 1824 work "Reflections on the Motive Power of Fire," in which he conceived the idealized cycle that bears his name. Carnot sought to understand the maximum possible efficiency of steam engines, a question of immense practical importance during the Industrial Revolution.

Nikolaus Otto patented the four-stroke engine in 1876, while Rudolf Diesel patented his compression-ignition engine in 1893. William Rankine analyzed the steam power cycle in 1859, and Robert Stirling invented his external combustion engine as early as 1816. Each cycle represents a different strategy for converting heat into work, subject to the constraints imposed by the second law.

Key contributors: Sadi Carnot (1824), Robert Stirling (1816), William Rankine (1859), Nikolaus Otto (1876), Rudolf Diesel (1893).

Derivation 1: Carnot Cycle Efficiency

The Carnot cycle consists of four reversible steps: two isothermal and two adiabatic processes. For an ideal gas:

Figure 1. Carnot cycle P-V diagram. The cycle encloses four steps: isothermal expansion at Tₕ (absorbing Qₕ), adiabatic expansion, isothermal compression at Tᶜ (rejecting Qᶜ), and adiabatic compression. The shaded area equals the net work output.

$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = \frac{W_{\text{net}}}{Q_H}$$

The work done in each step for n moles of ideal gas:

Isothermal expansion:$W_{12} = nRT_H \ln(V_2/V_1) > 0$

Adiabatic expansion:$W_{23} = nC_V(T_H - T_C) > 0$

Isothermal compression:$W_{34} = nRT_C \ln(V_4/V_3) < 0$

Adiabatic compression:$W_{41} = nC_V(T_C - T_H) < 0$

The adiabatic works cancel, giving $W_{\text{net}} = nR(T_H - T_C)\ln(V_2/V_1)$.

Derivation 2: Otto Cycle (Gasoline Engine)

The Otto cycle models spark-ignition internal combustion engines. It consists of two adiabatic and two isochoric (constant volume) processes.

1→2: Adiabatic compression.$T_2 = T_1 r^{\gamma - 1}$ where $r = V_1/V_2$ is the compression ratio.

2→3: Constant volume heat addition.$Q_{\text{in}} = nC_V(T_3 - T_2)$

3→4: Adiabatic expansion.$T_4 = T_3 / r^{\gamma - 1}$

4→1: Constant volume heat rejection.$Q_{\text{out}} = nC_V(T_4 - T_1)$

The efficiency depends only on the compression ratio:

$$\eta_{\text{Otto}} = 1 - \frac{1}{r^{\gamma - 1}}$$

Higher compression ratios yield better efficiency, but practical limits include engine knock (pre-ignition) and material strength. Typical gasoline engines use r = 8-12 with $\gamma \approx 1.4$ for air.

Derivation 3: Diesel Cycle

The Diesel cycle differs from Otto by having constant-pressure (isobaric) heat addition instead of constant-volume. This allows higher compression ratios without knock.

1→2: Adiabatic compression (same as Otto).

2→3: Constant pressure heat addition.$Q_{\text{in}} = nC_P(T_3 - T_2)$. Define cutoff ratio $r_c = V_3/V_2$.

3→4: Adiabatic expansion.

4→1: Constant volume heat rejection.

$$\eta_{\text{Diesel}} = 1 - \frac{1}{\gamma \, r^{\gamma-1}} \cdot \frac{r_c^{\gamma} - 1}{r_c - 1}$$

For the same compression ratio, the Diesel cycle is less efficient than Otto (the extra factor is always greater than 1). However, Diesel engines operate at much higher compression ratios (r = 14-25), making them more efficient overall.

Derivation 4: Stirling Cycle

The Stirling cycle uses two isothermal and two isochoric processes. With a perfect regenerator (which stores and returns heat during the isochoric steps), it achieves Carnot efficiency.

1→2: Isothermal expansion at $T_H$.$Q_H = nRT_H \ln(V_2/V_1)$

2→3: Isochoric cooling. Heat $Q_{\text{reg}} = nC_V(T_H - T_C)$ stored in regenerator.

3→4: Isothermal compression at $T_C$.$|Q_C| = nRT_C \ln(V_2/V_1)$

4→1: Isochoric heating. Regenerator returns $Q_{\text{reg}}$.

$$\eta_{\text{Stirling}} = 1 - \frac{T_C}{T_H} = \eta_{\text{Carnot}} \quad \text{(with perfect regenerator)}$$

Derivation 5: Rankine Cycle & Refrigeration COP

Rankine Cycle (Steam Power)

The Rankine cycle is the basis for steam power plants. It involves phase change of the working fluid (water/steam) through four processes:

1→2: Isentropic compression (pump).$W_{\text{pump}} = v(P_2 - P_1) \approx \text{small}$

2→3: Constant pressure heat addition (boiler).$Q_{\text{in}} = h_3 - h_2$

3→4: Isentropic expansion (turbine).$W_{\text{turbine}} = h_3 - h_4$

4→1: Constant pressure heat rejection (condenser).$Q_{\text{out}} = h_4 - h_1$

$$\eta_{\text{Rankine}} = \frac{W_{\text{net}}}{Q_{\text{in}}} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2}$$

Refrigeration Coefficient of Performance

A refrigerator operates as a reverse heat engine, pumping heat from cold to hot:

$$\text{COP}_{\text{ref}} = \frac{Q_C}{W} = \frac{T_C}{T_H - T_C} \quad \text{(Carnot limit)}$$

For a heat pump (heating mode): $\text{COP}_{\text{HP}} = T_H/(T_H - T_C) = \text{COP}_{\text{ref}} + 1$. Note that COP can exceed 1, meaning heat pumps deliver more heating than the electrical energy consumed.

Applications

Automotive Engines

Gasoline engines approximate the Otto cycle (r ~ 10, efficiency ~ 25-30%). Diesel engines approximate the Diesel cycle (r ~ 18, efficiency ~ 35-45%). Turbocharging effectively increases the compression ratio.

Power Plants

Coal and nuclear plants use the Rankine cycle (efficiency ~ 33-40%). Combined-cycle gas turbine plants pair a Brayton (gas) cycle with a Rankine (steam) cycle to reach 60%+ efficiency.

Stirling Engines

Used in solar dish concentrators, submarine propulsion, and cryocoolers. Their external combustion allows any heat source, including solar, biomass, or waste heat.

Heat Pumps

Modern heat pumps achieve COP of 3-5, delivering 3-5 kW of heating per kW of electricity consumed. They are a key technology for decarbonizing building heating.

Simulation: Cycle PV Diagrams & Efficiency

Compare the Carnot, Otto, and Diesel cycles on PV diagrams, and see how efficiency scales with compression ratio. The Carnot limit provides the upper bound for all cycles.

Carnot, Otto, and Diesel Cycle Comparison

Python

script.py165 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Thermodynamic Cycles: Carnot, Otto, Diesel, Stirling
# PV Diagrams and Efficiency Comparison
# ============================================================

R = 8.314
n = 1.0
gamma = 1.4  # diatomic gas (air)
Cv = R / (gamma - 1)
Cp = gamma * R / (gamma - 1)

fig, axes = plt.subplots(2, 2, figsize=(14, 12))

# ---- CARNOT CYCLE ----
ax1 = axes[0, 0]
T_H, T_C = 600.0, 300.0
V1_c = 0.001
V2_c = 0.003
P1_c = n * R * T_H / V1_c
P2_c = n * R * T_H / V2_c
V3_c = V2_c * (T_H / T_C) ** (1.0 / (gamma - 1))
V4_c = V1_c * (T_H / T_C) ** (1.0 / (gamma - 1))
P3_c = n * R * T_C / V3_c
P4_c = n * R * T_C / V4_c

V_12 = np.linspace(V1_c, V2_c, 200)
V_23 = np.linspace(V2_c, V3_c, 200)
V_34 = np.linspace(V3_c, V4_c, 200)
V_41 = np.linspace(V4_c, V1_c, 200)

ax1.plot(V_12*1e3, n*R*T_H/V_12/1e3, 'r-', lw=2.5, label='Isothermal (T_H)')
ax1.plot(V_23*1e3, P2_c*(V2_c/V_23)**gamma/1e3, 'b--', lw=2, label='Adiabatic')
ax1.plot(V_34*1e3, n*R*T_C/V_34/1e3, 'c-', lw=2.5, label='Isothermal (T_C)')
ax1.plot(V_41*1e3, P4_c*(V4_c/V_41)**gamma/1e3, 'm--', lw=2, label='Adiabatic')
ax1.set_title('Carnot Cycle', fontsize=14, fontweight='bold', color='white')
ax1.set_xlabel('V (L)', fontsize=11, color='white')
ax1.set_ylabel('P (kPa)', fontsize=11, color='white')
ax1.legend(fontsize=8, facecolor='black', edgecolor='gray', labelcolor='white')
ax1.set_facecolor('#1a1a2e')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.3)

# ---- OTTO CYCLE ----
ax2 = axes[0, 1]
r = 8.0  # compression ratio
V_bdc = 0.005
V_tdc = V_bdc / r
T1_o = 300.0
P1_o = 1e5

# 1->2: adiabatic compression
T2_o = T1_o * r**(gamma - 1)
P2_o = P1_o * r**gamma

# 2->3: constant volume heat addition
T3_o = 2 * T2_o  # assume Q_in doubles temperature
P3_o = P2_o * T3_o / T2_o

# 3->4: adiabatic expansion
T4_o = T3_o / r**(gamma - 1)
P4_o = P3_o / r**gamma

V_12o = np.linspace(V_bdc, V_tdc, 200)
P_12o = P1_o * (V_bdc / V_12o) ** gamma
V_34o = np.linspace(V_tdc, V_bdc, 200)
P_34o = P3_o * (V_tdc / V_34o) ** gamma

ax2.plot(V_12o*1e3, P_12o/1e3, 'r-', lw=2.5, label='1-2 Adiabatic compression')
ax2.plot([V_tdc*1e3, V_tdc*1e3], [P2_o/1e3, P3_o/1e3], 'y-', lw=2.5, label='2-3 Const-V heat in')
ax2.plot(V_34o*1e3, P_34o/1e3, 'b-', lw=2.5, label='3-4 Adiabatic expansion')
ax2.plot([V_bdc*1e3, V_bdc*1e3], [P4_o/1e3, P1_o/1e3], 'c-', lw=2.5, label='4-1 Const-V heat out')
ax2.set_title('Otto Cycle (r = ' + str(int(r)) + ')', fontsize=14, fontweight='bold', color='white')
ax2.set_xlabel('V (L)', fontsize=11, color='white')
ax2.set_ylabel('P (kPa)', fontsize=11, color='white')
ax2.legend(fontsize=8, facecolor='black', edgecolor='gray', labelcolor='white')
ax2.set_facecolor('#1a1a2e')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.3)

# ---- DIESEL CYCLE ----
ax3 = axes[1, 0]
r_d = 18.0
rc = 2.5  # cutoff ratio
V_bdc_d = 0.005
V_tdc_d = V_bdc_d / r_d
T1_d = 300.0
P1_d = 1e5

T2_d = T1_d * r_d**(gamma - 1)
P2_d = P1_d * r_d**gamma
T3_d = T2_d * rc
V3_d = V_tdc_d * rc
P3_d = P2_d  # constant pressure
T4_d = T3_d * (V3_d / V_bdc_d)**(gamma - 1)
P4_d = P3_d * (V3_d / V_bdc_d)**gamma

V_12d = np.linspace(V_bdc_d, V_tdc_d, 200)
P_12d = P1_d * (V_bdc_d / V_12d)**gamma
V_34d = np.linspace(V3_d, V_bdc_d, 200)
P_34d = P3_d * (V3_d / V_34d)**gamma

ax3.plot(V_12d*1e3, P_12d/1e3, 'r-', lw=2.5, label='1-2 Adiabatic compression')
ax3.plot([V_tdc_d*1e3, V3_d*1e3], [P2_d/1e3, P3_d/1e3], 'y-', lw=2.5, label='2-3 Const-P heat in')
ax3.plot(V_34d*1e3, P_34d/1e3, 'b-', lw=2.5, label='3-4 Adiabatic expansion')
ax3.plot([V_bdc_d*1e3, V_bdc_d*1e3], [P4_d/1e3, P1_d/1e3], 'c-', lw=2.5, label='4-1 Const-V heat out')
ax3.set_title('Diesel Cycle (r = ' + str(int(r_d)) + ', rc = ' + str(rc) + ')', fontsize=14, fontweight='bold', color='white')
ax3.set_xlabel('V (L)', fontsize=11, color='white')
ax3.set_ylabel('P (kPa)', fontsize=11, color='white')
ax3.legend(fontsize=8, facecolor='black', edgecolor='gray', labelcolor='white')
ax3.set_facecolor('#1a1a2e')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.3)

# ---- EFFICIENCY COMPARISON ----
ax4 = axes[1, 1]
r_vals = np.linspace(2, 25, 200)
eta_otto = 1.0 - 1.0 / r_vals**(gamma - 1)

rc_vals = [1.5, 2.0, 2.5, 3.0]
colors_rc = ['#ff6b6b', '#ffa502', '#2ed573', '#1e90ff']
for rc_v, clr in zip(rc_vals, colors_rc):
    eta_diesel = 1.0 - (1.0 / (gamma * r_vals**(gamma - 1))) * (rc_v**gamma - 1) / (rc_v - 1)
    ax4.plot(r_vals, eta_diesel * 100, color=clr, lw=2, linestyle='--',
             label='Diesel rc = ' + str(rc_v))

ax4.plot(r_vals, eta_otto * 100, 'w-', lw=3, label='Otto')
eta_carnot_ref = (1.0 - 300.0/600.0) * 100
ax4.axhline(y=eta_carnot_ref, color='gold', linestyle=':', lw=2, label='Carnot (T_C/T_H=0.5)')

ax4.set_xlabel('Compression ratio r', fontsize=11, color='white')
ax4.set_ylabel('Efficiency (%)', fontsize=11, color='white')
ax4.set_title('Efficiency vs Compression Ratio', fontsize=14, fontweight='bold', color='white')
ax4.legend(fontsize=8, facecolor='black', edgecolor='gray', labelcolor='white')
ax4.set_facecolor('#1a1a2e')
ax4.tick_params(colors='white')
ax4.grid(True, alpha=0.3)
ax4.set_ylim(0, 80)

for ax in axes.flat:
    ax.spines['bottom'].set_color('gray')
    ax.spines['left'].set_color('gray')
    ax.spines['top'].set_color('gray')
    ax.spines['right'].set_color('gray')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=100, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("Thermodynamic Cycles Comparison")
print("=" * 50)
eta_carnot_val = 1 - T_C / T_H
print("Carnot:  eta = 1 - T_C/T_H = " + str(round(eta_carnot_val*100, 1)) + "%")

eta_otto_val = 1 - 1.0/r**(gamma-1)
print("Otto:    eta = 1 - 1/r^(g-1) = " + str(round(eta_otto_val*100, 1)) + "% (r=" + str(int(r)) + ")")

eta_diesel_val = 1 - (1/(gamma*r_d**(gamma-1)))*(rc**gamma - 1)/(rc - 1)
print("Diesel:  eta = " + str(round(eta_diesel_val*100, 1)) + "% (r=" + str(int(r_d)) + ", rc=" + str(rc) + ")")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation: Stirling, Rankine & Refrigeration

Explore the Stirling cycle PV diagram, the Rankine cycle T-s diagram for steam power, and the refrigeration coefficient of performance as a function of cold reservoir temperature.

Stirling Cycle, Rankine Cycle, and Refrigeration COP

Python

script.py133 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Rankine and Stirling Cycles + Refrigeration COP
# ============================================================

fig, axes = plt.subplots(1, 3, figsize=(18, 6))

# ---- STIRLING CYCLE ----
ax1 = axes[0]
R_gas = 8.314
n = 1.0
T_H, T_C = 600.0, 300.0
V1_s = 0.001
V2_s = 0.004

# 1->2: isothermal expansion at T_H
V_12s = np.linspace(V1_s, V2_s, 200)
P_12s = n * R_gas * T_H / V_12s

# 2->3: isochoric cooling (V = V2)
T_23 = np.linspace(T_H, T_C, 200)
P_23 = n * R_gas * T_23 / V2_s

# 3->4: isothermal compression at T_C
V_34s = np.linspace(V2_s, V1_s, 200)
P_34s = n * R_gas * T_C / V_34s

# 4->1: isochoric heating (V = V1)
T_41 = np.linspace(T_C, T_H, 200)
P_41 = n * R_gas * T_41 / V1_s

ax1.plot(V_12s*1e3, P_12s/1e3, 'r-', lw=2.5, label='Isothermal expansion (T_H)')
ax1.plot([V2_s*1e3]*2, [P_12s[-1]/1e3, P_34s[0]/1e3], 'b--', lw=2, label='Isochoric cooling')
ax1.plot(V_34s*1e3, P_34s/1e3, 'c-', lw=2.5, label='Isothermal compression (T_C)')
ax1.plot([V1_s*1e3]*2, [P_34s[-1]/1e3, P_12s[0]/1e3], 'm--', lw=2, label='Isochoric heating')
ax1.set_xlabel('V (L)', fontsize=11, color='white')
ax1.set_ylabel('P (kPa)', fontsize=11, color='white')
ax1.set_title('Stirling Cycle', fontsize=14, fontweight='bold', color='white')
ax1.legend(fontsize=8, facecolor='black', edgecolor='gray', labelcolor='white')
ax1.set_facecolor('#1a1a2e')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.3)

# ---- RANKINE CYCLE (T-s diagram) ----
ax2 = axes[1]
# Simplified ideal Rankine with water properties
# Points: 1(pump inlet) -> 2(pump outlet) -> 3(boiler outlet) -> 4(turbine outlet)
T_cond = 45 + 273.15   # condenser temp K
T_boil = 250 + 273.15  # boiler temp K

# Simplified T-s diagram
s_vals = np.array([1.0, 1.0, 6.5, 6.5, 1.0])
T_vals = np.array([T_cond, T_boil, T_boil, T_cond, T_cond])

# Saturation dome (approximate)
s_dome = np.linspace(0.5, 8.0, 300)
T_dome = 373.15 + 80 * np.sin(np.pi * (s_dome - 0.5) / 7.5)
T_dome = np.minimum(T_dome, 647.0)  # cap at critical point

ax2.fill(s_vals, T_vals - 273.15, alpha=0.2, color='orange')
ax2.plot(s_vals, T_vals - 273.15, 'r-', lw=2.5, label='Ideal Rankine cycle')
ax2.plot(s_dome, T_dome - 273.15, 'w--', lw=1.5, alpha=0.5, label='Sat. dome (approx.)')

ax2.annotate('1 (pump in)', (1.0, T_cond-273.15), fontsize=9, color='yellow',
             xytext=(1.5, T_cond-273.15-20))
ax2.annotate('2 (pump out)', (1.0, T_boil-273.15), fontsize=9, color='yellow',
             xytext=(1.5, T_boil-273.15+5))
ax2.annotate('3 (turbine in)', (6.5, T_boil-273.15), fontsize=9, color='yellow',
             xytext=(5.0, T_boil-273.15+5))
ax2.annotate('4 (turbine out)', (6.5, T_cond-273.15), fontsize=9, color='yellow',
             xytext=(5.0, T_cond-273.15-20))

ax2.set_xlabel('Specific entropy s (kJ/kg K)', fontsize=11, color='white')
ax2.set_ylabel('Temperature (C)', fontsize=11, color='white')
ax2.set_title('Rankine Cycle (T-s diagram)', fontsize=14, fontweight='bold', color='white')
ax2.legend(fontsize=9, facecolor='black', edgecolor='gray', labelcolor='white')
ax2.set_facecolor('#1a1a2e')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.3)

# ---- REFRIGERATION COP ----
ax3 = axes[2]
T_cold = np.linspace(200, 290, 200)
T_hot = 300.0
COP_carnot = T_cold / (T_hot - T_cold)

# Realistic COP (about 40% of Carnot)
COP_real = 0.4 * COP_carnot

ax3.plot(T_cold, COP_carnot, 'r-', lw=2.5, label='Carnot COP (theoretical max)')
ax3.plot(T_cold, COP_real, 'c--', lw=2.5, label='Realistic COP (40% Carnot)')
ax3.axvline(x=255, color='yellow', linestyle=':', alpha=0.7, label='Typical freezer (-18 C)')
ax3.axvline(x=277, color='lime', linestyle=':', alpha=0.7, label='Typical fridge (4 C)')

ax3.set_xlabel('Cold reservoir T_C (K)', fontsize=11, color='white')
ax3.set_ylabel('COP', fontsize=11, color='white')
ax3.set_title('Refrigeration Coefficient of Performance', fontsize=14, fontweight='bold', color='white')
ax3.legend(fontsize=9, facecolor='black', edgecolor='gray', labelcolor='white')
ax3.set_facecolor('#1a1a2e')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.3)
ax3.set_ylim(0, 30)

for ax in axes.flat:
    ax.spines['bottom'].set_color('gray')
    ax.spines['left'].set_color('gray')
    ax.spines['top'].set_color('gray')
    ax.spines['right'].set_color('gray')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=100, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("Additional Thermodynamic Cycles")
print("=" * 50)
eta_stirling = 1 - T_C / T_H
print("Stirling efficiency (ideal with regenerator): " + str(round(eta_stirling*100, 1)) + "%")
print("  (Same as Carnot with perfect regeneration)")
print()
eta_rankine = 1 - T_cond / T_boil
print("Ideal Rankine efficiency: " + str(round(eta_rankine*100, 1)) + "%")
print("  T_boil = " + str(round(T_boil-273.15, 1)) + " C, T_cond = " + str(round(T_cond-273.15, 1)) + " C")
print()
print("Refrigeration COP at T_C = 255 K (-18 C):")
COP_255 = 255.0 / (300.0 - 255.0)
print("  Carnot COP = " + str(round(COP_255, 2)))
print("  Realistic COP = " + str(round(0.4 * COP_255, 2)))

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

• Carnot: $\eta = 1 - T_C/T_H$. Maximum possible efficiency between two reservoirs.

• Otto: $\eta = 1 - 1/r^{\gamma-1}$. Models gasoline engines with constant-V heat addition.

• Diesel: Higher compression ratios possible; constant-P heat addition.

• Stirling: Achieves Carnot efficiency with perfect regenerator; uses isochoric processes.

• Rankine: Steam power cycle with phase change; basis for thermal power plants.

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