Carnot Cycle & the Limits of Heat Engines

1. Heat Engines and the Second Law

A heat engine absorbs heat Q_h from a hot reservoir at temperature T_h, converts some of it into work W, and dumps the remainder Q_c into a cold reservoir at T_c. Energy conservation gives:

Carnot's theorem states that all reversible engines operating between the same two temperatures have identical efficiency, and any irreversible engine has a strictly lower efficiency.

2. The Four Reversible Processes

The adiabatic steps cancel each other in work (W₂₃ + W₄₁ = 0), so the net work of the cycle equals the difference of the two isothermal works.

3. PV Diagram and Work

On a pressure–volume diagram, the enclosed area is the net work output per cycle:

4. Carnot Efficiency

Combining Q_h and Q_c, the logarithm cancels, leaving a remarkably clean result:

This result is profound: efficiency depends only on the ratio T_c/T_h. A perfect engine at room temperature and body temperature (T_h=310 K, T_c=293 K) has a maximum efficiency of merely 5.5%.

5. Entropy and the TS Diagram

Entropy S is defined via reversible heat exchange: dS = δQ_rev / T. The Carnot cycle is a rectangle on a temperature–entropy diagram:

6. Real Engines: Otto, Diesel, Stirling

Real thermodynamic cycles approximate the Carnot ideal in different ways:

7. JavaScript Simulation

// Carnot cycle simulation — ideal gas, n moles
const R = 8.314; // J/(mol·K)

function carnotCycle(n, Th, Tc, V1, gamma = 1.4) {
  // Adiabatic constraint: V2/V1 = (Th/Tc)^(1/(γ-1))
  const adRatio = Math.pow(Th / Tc, 1 / (gamma - 1));
  const V2 = V1 * adRatio;
  // Isothermal expansion gives V3 = V2*(Tc/Th)^(1/(γ-1)) ... simplified:
  const V4 = V1;          // adiabatic compression returns to V1
  const V3 = V2 * adRatio; // symmetric adiabatic

  const Qh = n * R * Th * Math.log(V2 / V1);
  const Qc = n * R * Tc * Math.log(V3 / V4);
  const W  = Qh - Qc;
  const eta = 1 - Tc / Th;

  return { Qh, Qc, W, eta, V1, V2, V3, V4, Th, Tc };
}

// Generate PV curve points for plotting
function pvCurve(type, V_start, V_end, T_or_K, n, gamma, steps = 60) {
  const pts = [];
  for (let i = 0; i <= steps; i++) {
    const V = V_start + (V_end - V_start) * i / steps;
    let P;
    if (type === 'isothermal') {
      P = n * R * T_or_K / V;
    } else { // adiabatic: PVᵞ = K
      P = T_or_K / Math.pow(V, gamma);
    }
    pts.push({V, P});
  }
  return pts;
}

// Draw PV diagram on canvas
function drawPV(ctx, cycle, W, H) {
  const {Qh, Qc, Th, Tc, V1, V2, V3, V4} = cycle;
  const n = 1, gamma = 1.4;
  const P1 = n * R * Th / V1;
  const K_ad12 = P1 * Math.pow(V1, gamma); // adiabatic constant 2→3

  const allV = [V1,V2,V3,V4], allP = [P1];
  allP.push(n*R*Th/V2, n*R*Tc/V3, n*R*Tc/V4);
  const vMin = Math.min(...allV), vMax = Math.max(...allV);
  const pMin = Math.min(...allP) * 0.8, pMax = Math.max(...allP) * 1.2;

  const toX = v => 60 + (v-vMin)/(vMax-vMin) * (W-80);
  const toY = p => H-50 - (p-pMin)/(pMax-pMin) * (H-80);

  ctx.clearRect(0, 0, W, H);
  ctx.strokeStyle = '#f97316'; ctx.lineWidth = 2;
  ctx.beginPath();
  const segments = [
    pvCurve('isothermal', V1, V2, Th, n, gamma),
    pvCurve('adiabatic', V2, V3, K_ad12, n, gamma),
    pvCurve('isothermal', V3, V4, Tc, n, gamma),
    pvCurve('adiabatic', V4, V1, K_ad12*0.5, n, gamma)
  ];
  let first = true;
  for (const seg of segments) {
    for (const {V, P} of seg) {
      first ? ctx.moveTo(toX(V), toY(P)) : ctx.lineTo(toX(V), toY(P));
      first = false;
    }
  }
  ctx.closePath();
  ctx.fillStyle = '#f9731620'; ctx.fill(); ctx.stroke();
}

// Example usage
const cycle = carnotCycle(1, 600, 300, 0.001);
console.log(`η = ${(cycle.eta*100).toFixed(1)}%`); // η = 50.0%

8. Applications and Limits

• Power plants: Modern combined-cycle gas turbines (T_h ≈ 1500 K, T_c ≈ 300 K) reach theoretical η_max ≈ 80%, but achieve ~60% in practice through two-stage conversion.
• Heat pumps and refrigerators: The Carnot cycle run in reverse is a refrigerator (COP = T_c/(T_h−T_c)) or heat pump (COP = T_h/(T_h−T_c)).
• Endoreversible engines: Curzon-Ahlborn efficiency (η_CA = 1 − √(T_c/T_h)) gives realistic predictions for engines optimised for maximum power rather than efficiency.
• Second Law limits: No process can reduce total entropy — Carnot efficiency is the irreducible upper bound set by this law.
• Thermoelectric devices: Peltier coolers and Seebeck generators approach Carnot only at low temperature differences; typical efficiency is 5–10%.