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Carnot Cycle Heat Engine

A simulation of the theoretical limit of thermodynamic efficiency using Causal Monads.

How to Run

cargo run -p physics_examples --example carnot_cycle_engine

Expected Output

=== Carnot Heat Engine Simulation ===

Start (Point A)                | P=415700.0 Pa | V=0.0100 m³ | T=500.0 K | S=+0.00 J/K
Isothermal Expansion (A→B)     | P=207850.0 Pa | V=0.0200 m³ | T=500.0 K | S=+5.76 J/K
Adiabatic Expansion (B→C)      | P= 57960.0 Pa | V=0.0430 m³ | T=300.0 K | S=+5.76 J/K
Isothermal Compression (C→D)   | P=115919.9 Pa | V=0.0215 m³ | T=300.0 K | S=+0.00 J/K
Adiabatic Compression (D→A)    | P=415700.0 Pa | V=0.0100 m³ | T=500.0 K | S=+0.00 J/K

=== Cycle Complete ===
Total Work Done: 1152.57 J
Carnot Efficiency Limit: 40.0%

Physics Overview

The Carnot Cycle represents the theoretical upper limit of efficiency for any heat engine operating between two thermal reservoirs. It consists of 4 reversible processes:

The Four Stages

Stage Process Description
A→B Isothermal Expansion Gas expands at constant $T_H$. Heat $Q_{in}$ absorbed from hot reservoir. Work done by gas.
B→C Adiabatic Expansion Gas expands with no heat exchange ($Q=0$). Temperature drops from $T_H$ to $T_C$. Work done by gas.
C→D Isothermal Compression Gas compressed at constant $T_C$. Heat $Q_{out}$ rejected to cold reservoir. Work done on gas.
D→A Adiabatic Compression Gas compressed with no heat exchange. Temperature rises from $T_C$ to $T_H$. Work done on gas.

Key Equations

Carnot Efficiency: $$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$$

Ideal Gas Law: $$PV = nRT$$

Adiabatic Process (for monatomic ideal gas, $\gamma = 5/3$): $$T V^{\gamma-1} = \text{constant}$$

Isothermal Work: $$W = nRT \ln\left(\frac{V_f}{V_i}\right)$$

Adiabatic Work: $$W = C_V (T_i - T_f) = \frac{3}{2}nR(T_i - T_f)$$

Simulation Configuration

Parameter Value Description
$n$ 1.0 mol Amount of substance
$T_H$ 500 K Hot reservoir temperature
$T_C$ 300 K Cold reservoir temperature
$V_A$ 0.01 m³ Initial volume (10 L)
$P_A$ 415,700 Pa Initial pressure (~4 atm)
Compression Ratio 2.0 $V_B/V_A$

Causal Chain Architecture

The simulation models the Carnot cycle as a causal propagation process using CausalEffectPropagationProcess from deep_causality_core. Each thermodynamic stage is represented as a .bind() operation:

Initial State → Step 1 (A→B) → Step 2 (B→C) → Step 3 (C→D) → Step 4 (D→A) → Final State
     ↓              ↓              ↓              ↓              ↓
  EngineState   EngineState   EngineState   EngineState   EngineState

EngineState Structure

Each state in the cycle tracks:

  • Pressure ($P$): Current gas pressure in Pascals
  • Volume ($V$): Current gas volume in m³
  • Temperature ($T$): Current gas temperature in Kelvin
  • Entropy ($S$): Cumulative entropy change in J/K
  • Work Done: Cumulative work output in Joules
  • Phase: Human-readable description of current stage

Key Design Patterns

  1. Monadic Composition: Each stage transforms the engine state and propagates it to the next stage
  2. Closure Captures: Configuration parameters are captured from the outer scope for use in bind closures
  3. Value Propagation: The EngineState flows through the chain as the monadic value
  4. Physics Validation: Ideal Gas Law is verified at critical points

Key APIs Used

API Purpose
deep_causality_physics::Pressure Type-safe pressure values (Pa)
deep_causality_physics::Volume Type-safe volume values (m³)
deep_causality_physics::Temperature Type-safe temperature values (K)
deep_causality_physics::AmountOfSubstance Type-safe amount (mol)
deep_causality_physics::carnot_efficiency Computes theoretical efficiency limit
deep_causality_physics::ideal_gas_law Verifies P, V, n, T consistency
deep_causality_core::CausalEffectPropagationProcess Monadic effect propagation

Thermodynamic Verification

The simulation demonstrates key thermodynamic principles:

  1. Entropy Conservation: After completing the cycle, entropy returns to its initial value (S=0), confirming reversibility
  2. State Cycle Closure: The system returns exactly to its initial state (P, V, T)
  3. Efficiency Bound: Net work output (~1153 J) is consistent with the 40% Carnot limit
  4. Energy Conservation: Work equals net heat transfer ($W = Q_{in} - Q_{out}$)