Adiabatic Adsorption — Thermal Waves

Equilibrium theory of an adiabatic fixed bed: one adsorbable trace component (moisture) in an inert carrier, with a temperature-dependent isotherm. Mass and energy balances form a 2×2 system of conservation laws in (c, T) — the same mathematical object as binary equilibrium chromatography, with temperature as the second “component” (Rhee–Amundson; Pan & Basmadjian). Step changes resolve into two waves: a concentration-like wave and a thermal wave, welded through an intermediate plateau. Drag the I and F markers in the hodograph and the whole solution follows.

Sorbent & isotherm — van ’t Hoff in T

Preset Model K(25 °C) m³/kg −ΔU kJ/mol (isotherm) −ΔH = −ΔU (linked) −ΔH kJ/mol (energy balance) ρ_p kg/m³ c_p,s J/kg/K ε bed voidage

Carrier & column

P kPa c_p,g J/mol/K v m/s interstitial L m

The construction is scale-free in ξ = z/L, τ = tv/L; v and L only set the dimensional clocks shown in the readouts.

Scenario

Step change — initial and feed states

I — initial column state

T K

F — feed (enters at z = 0 from t = 0)

T K

RH is evaluated with the water Antoine equation at the state’s own temperature; both readouts stay live.

Local equilibrium, plug flow, adiabatic wall, gas–solid thermal equilibrium, frozen interstitial velocity (trace limit), no dispersion or axial conduction, constant heat capacities. Two engines run on every problem: the exact hodograph construction (wave curves, Rankine–Hugoniot shocks, admissibility checked) and an independent conservative MUSCL reference; when the construction cannot be certified — strong desorption steps cross a degeneracy where the wave families swap identity — the reference solution is shown and labelled.

Hodograph plane (c, T) drag I and F to set the states

Physical plane — c(ξ, τ)

Physical plane — T(ξ, τ)

Column profiles at τ

Breakthrough at ξ = 1

Engine cross-check