Pseudo-Steady Evaporation Explorer (embed)

Pseudo-Steady Evaporation

Start here: pick a liquid, drag T and read Time for z₀ → z_t.

Geometry

Liquid, gas and geometry

Liquid A, into air

Temperature T, K

Total pressure P, atm

ρ_L, kg/m³

M_A, kg/mol

D_AB, 10⁻⁶ m²/s

p_A*, kPa

Named liquids: cited data, 290 to 320 K (Property data tab).

Tube geometry and top boundary

Initial gas path z₀, mm

Target gas path z_t, mm

Top vapor pressure p_A2, kPa

Tube diameter d, mm

Level clock and graph

0 h

Playback speed

Elapsed time t, h

Plot z² (or r²)

Time for z₀ → z_t, h…

Elapsed t, h…

Gas path z now, mm…

Level fall rate dz/dt, mm/h…

N_A at the surface, mol/(m²·s)…

Mass evaporated, g…

…

Animated schematic

Physical view

Path z versus time

Property data and sources

Property data used for the named liquids

Liquid A (in air)	`M`_A, g/mol	ρ_L at 298.15 K, kg/m³	`D`_AB at 298 K, 1 atm, cm²/s	Antoine `A`, `B`, `C` (range, K)
Water, H₂O	18.0153	997.0	0.260 [1]	5.40221, 1838.675, −31.737 (273 to 303); 5.20389, 1733.926, −39.485 (304 to 333) [3]
Methanol, CH₃OH	32.0419	786.3	0.166 [4]	5.20409, 1581.341, −33.50 (288.1 to 356.8) [3]
Ethanol, C₂H₅OH	46.0684	785.1	0.129 [4]	5.37229, 1670.409, −40.191 (273 to 351.7) [3]
Acetone, C₃H₆O	58.0791	784.7	0.107 [4]	4.42448, 1312.253, −32.445 (259.2 to 507.6) [3]

Named liquids are limited to 290 to 320 K. Densities are held at their 298.15 K values [5]; over 290 to 320 K the reference equations of state change them by at most 3.2% (acetone). The Antoine values agree with those equations of state within 0.8% over the same range. D_AB is scaled with T¹·⁷⁵/P [6]. Independent checks: Fuller estimates are within 5% of every tabulated D_AB, and a second tabulation (NIST TN 2279 for water, WWWR Table J.1 for the alcohols) is within 2.4%. Choose Custom to enter any other property values.

References

Welty, Wicks, Wilson and Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer, 5th ed., Wiley (2008): §26.1 (Arnold diffusion cell, pseudo-steady state) and Appendix Table J.1, p. 691 (water in air).
Bird, Stewart and Lightfoot, Transport Phenomena, 2nd ed., Wiley (2002), §18.2: diffusion through a stagnant gas film.
NIST Chemistry WebBook, SRD 69: Antoine parameters for water (Bridgeman and Aldrich, 1964), methanol (Ambrose and Sprake, 1970), ethanol (Kretschmer and Wiebe, 1949) and acetone (Ambrose, Sprake and Townsend, 1974); molar masses.
Burgess, NIST Technical Note 2279 (2024), doi:10.6028/NIST.TN.2279, Table 1c, compiling Tang et al., Atmos. Chem. Phys. 15, 5585 (2015).
Reference equations of state: water, Wagner and Pruss, J. Phys. Chem. Ref. Data 31, 387 (2002); methanol, de Reuck and Craven (IUPAC, 1993); ethanol, Schroeder et al., J. Phys. Chem. Ref. Data 43, 043102 (2014); acetone, Lemmon and Span, J. Chem. Eng. Data 51, 785 (2006); evaluated with CoolProp, Bell et al., Ind. Eng. Chem. Res. 53, 2498 (2014).
Fuller, Schettler and Giddings, Ind. Eng. Chem. 58(5), 18 (1966).
Perry’s Chemical Engineers’ Handbook, 9th ed., Eq. 5-198 (stagnant B, plane film).

Equations, assumptions and references

Stefan tube: A diffuses through stagnant B

Flux through the gas column, N_B = 0

N_{A} = \frac{c D_{AB}}{z} \ln [\frac{1 - y_{A2}}{1 - y_{A1}}] = \frac{c D_{AB} Λ}{z}

WWWR §26.1, BSL §18.2. z is the gas path from the liquid surface to the tube top.

Quasi-steady balance on the liquid

\frac{ρ_{L}}{M_{A}} \frac{d z}{d t} = N_{A} \Rightarrow z \frac{d z}{d t} = K = \frac{M_{A} c D_{AB} Λ}{ρ_{L}}

Level position and time to drop

z (t) = \sqrt{{z_{0}}^{2} + 2 K t} t = \frac{ρ_{L} ({z_{t}}^{2} - {z_{0}}^{2})}{2 M_{A} c D_{AB} Λ}

Equivalent log-mean pressure form

p_{BM} = \frac{p_{A1} - p_{A2}}{Λ} t = \frac{ρ_{L} R T p_{BM} ({z_{t}}^{2} - {z_{0}}^{2})}{2 M_{A} D_{AB} P (p_{A1} - p_{A2})}

Gas profile and species velocity drawn in the schematic

\frac{1 - y_{A}}{1 - y_{A1}} = {[\frac{1 - y_{A2}}{1 - y_{A1}}]}^{s / z} v_{A} = \frac{N_{A}}{c y_{A}}

s is the height above the liquid surface. The vapor flux cy_Av_A is the same at every height.

Common relations

Ideal-gas concentration and boundary mole fractions

c = \frac{P}{R T} y_{A1} = \frac{p_{A}^{*}}{P} y_{A2} = \frac{p_{A2}}{P}

R = 8.314462618 J/(mol·K). For the sphere y_As = p_A*/P.

Pseudo-steady validity

\frac{τ_{gas}}{τ_{motion}} = \frac{ℓ^{2} / D_{AB}}{ℓ^{2} / K} = \frac{K}{D_{AB}} < 0.01

ℓ = z (tube) or r (sphere). A time-scale estimate; numerical prefactors are neglected.

Property relations for the named liquids

\log_{10} (\frac{p_{A}^{*}}{bar}) = A - \frac{B}{T / K + C}

D_{AB} = D_{AB, ref} {(\frac{T}{298.15 K})}^{1.75} \frac{1 atm}{P}

Antoine constants from the NIST WebBook; temperature and pressure scaling of D_AB after Fuller, Schettler and Giddings (1966). See Property data.

Reading guide. z² grows linearly in time, so the level falls ever more slowly. Doubling both z₀ and z_t gives 4× the time; doubling only the end point does not: with z₀ = 20 mm, moving z_t from 30 to 40 mm gives (40² − 20²)/(30² − 20²) = 2.4×.

Assumptions. Isothermal ideal binary gas (evaporative cooling of the surface is ignored), constant D_AB, P, T and liquid density; B insoluble in the liquid (N_B = 0); no natural or forced convection inside the tube; the gas profile relaxes quickly compared with the level motion. The sphere sits in an infinite stagnant gas with no curvature effect on vapor pressure.

How the animated schematic is scaled

What the dots are

Maroon dots are parcels of vapor A. Each moves at the species velocity v_A = N_A/(cy_A) computed from the exact stagnant-B profile, so dots crowd near the liquid where y_A is high and thin out and speed up toward the open top. Because cy_Av_A = N_A at every height, the same number of parcels crosses each level per second: a constant flux through a falling concentration. New parcels leave the surface at a rate proportional to N_A, which falls as 1/z. Blue rings are air (B); they have no net motion because N_B = 0, and their density follows 1 − y_A.

Two clocks

The level moves on a compressed clock: one full run from z₀ to z_t takes 16 s on screen at 1× playback, and the schematic states how many real minutes or hours each second represents. Playback speed (½×, 1×, 2×, 4×; Reset returns to 1×) scales both clocks together, so a run takes 32, 16, 8 or 4 s; the stated factors update with it, and the physical time t and every readout are unchanged. The gas moves on its own clock, at real speed or sped up by a stated power of ten so a parcel crosses the tube in a few seconds. That the gas is far faster than the level (K/D_AB ≪ 1) is exactly the pseudo-steady assumption. Changing an input restarts the level clock at t = 0.

Scale

Heights in the tube are to scale (the tube is drawn 12% taller than z_t); its width is not. The y_A curve beside the tube is the exact profile at the current z. For the sphere the view is a slice through the center with radii to scale; the gas is cut off at 3.2 r₀, so the outer region is not to scale. In the slice, parcels are removed with probability dr/r as they move out, which keeps the drawn density proportional to y_A while the real flux spreads over 4πr².

Numbers

Every readout (time to target, level, rate, flux, mass) comes from the analytic solution, not from counting dots.