Temperature gradient → Useful heat

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.01 × C.33 · Temperature gradient × Thermodiffusivefamily: Thermodiffusive

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, filtered out by its applicability
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c
• Onsager–Casimir reciprocity · constitutive-relation · listed, never decisive
  L_ij(B) = ε_i ε_j L_ji(−B): the cross-coefficients of coupled linear transport are equal up to the time-reversal parities ε of the variables and reversal of the magnetic field; L_ij = L_ji is the zero-field, even-parity case

Steps

1. 01
   Temperature gradient —drives→ Soret effectestablishedclaim record

   thermal → chemical · loss to thermal

   • a liquid mixture or solution under a steady temperature gradient

   liquid · temperature gradient required

   refs [1] [2]
2. 02
   Soret effect —produces→ Concentration gradientestablishedclaim record

   chemical → chemical

   refs [2]
3. 03
   Concentration gradient —drives→ Dufour effectestablishedclaim record

   chemical → thermal

   • a composition gradient in a liquid or gas mixture; the heat flux is the Onsager-reciprocal partner of Soret thermodiffusion

   refs [3] [2]
4. 04
   Dufour effect —produces→ Heat flowestablishedclaim record

   thermal → thermal

   refs [3] [2]
5. 05
   Heat flow —converts into→ Useful heatestablishedclaim record

   thermal → thermal

   • heat delivered at a temperature above the point of use

   refs [4]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 4

1. [1]
   C. Soret. Sur l'état d'équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homogène dont deux parties sont portées à des températures différentes (1879)

   Archives des Sciences Physiques et Naturelles 2, 48–61 · paper, no DOI · record
2. [2]
   S. R. de Groot, P. Mazur. Non-Equilibrium Thermodynamics (1962)

   North-Holland · book, no DOI · record
3. [3]
   R. L. Rowley, F. H. Horne. The Dufour effect. II. Experimental confirmation of the Onsager heat–mass reciprocal relation for a binary liquid mixture (1978)

   The Journal of Chemical Physics 68, 325-326 · doi:10.1063/1.435458 · crossref ✓ 2026-09-20 · record
4. [4]
   A. Bejan. Advanced Engineering Thermodynamics (4th ed.) (2016)

   Wiley · doi:10.1002/9781119245964 · crossref ✓ 2026-09-19 · record