Start here: pick the flux condition, then move xA1. Watch the counted flux of A split into diffusion and bulk flow.
Composition and transport
Gas conditions
Probe and rates
Concentration profile
Flux split at the probe z′
Stagnant B over EMCD (xA2 = 0)
Equations, assumptions and references
Governing equations
Steady one-dimensional diffusion of A through B in an ideal binary gas. z is the gas-path length and z′ is measured from the bottom (xA1 at z′ = 0) to the top (xA2 at z′ = z).
NA is the flux relative to fixed coordinates; JA is the diffusive flux relative to the molar-average velocity; xA(NA + NB) is the bulk-flow term.
No bulk flow; the profile is linear.
The bulk-flow factor 1/xB,lm multiplies the EMCD flux. It is the quantity the animation counts as the bulk-flow factor.
P, pA and pBM in Pa. Computed pBM = … Pa; pressure-form NA = … mol/(m²·s). Derived from Eq. 5-198 with pA = xAP; it always describes stagnant B, whatever mode is selected.
Example: r = −0.5 describes a compatible boundary reaction 2A → B. At r = −1, φ is infinite; the simulator then uses the k → 0 limit, which is Eq. 5-197 exactly. The k form is the one used for computation.
P = … Pa; c = … mol/m³; effective r = …; φ = …. For stagnant B (r = 0) the bulk-flow share of NA is xA.
Perry’s 10% holds only for dilute A. Perry’s (p. 5-43) says bulk flow may account for as much as 10% of the flux, is usually much less, and is often ignored. For stagnant B the local bulk-flow share is exactly xA, so it passes 10% wherever xA > 0.1. The integrated flux is within 10% of EMCD for xA1 < 0.176 only when xA2 = 0.
Model validity. Steady 1D ideal binary gas; constant T, P, c and DAB; no distributed reaction, forced axial flow, pressure gradient or natural convection. DAB is entered independently and is not updated with T and P. The default inputs are an illustrative volatile vapor A in air, not a specific dataset. For a literal pure-liquid interface, xA1 must equal the equilibrium vapor-pressure ratio.
Boundary conditions. A liquid Stefan tube represents stagnant B. Counterdiffusion or a reactive flux ratio needs different species boundary conditions, so in those modes the diagram is a gas layer between a source of A and a sink of B, not a tube design.
References
- Perry’s Chemical Engineers’ Handbook, 9th ed., Section 5 (Eqs. 5-189, 5-190, 5-197 to 5-199; p. 5-43).
- R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley (2002), §17.7 and §18.2 (diffusion through a stagnant gas film).
Numerical verification
How the animated schematic is scaled
Stochastic particle model of the gas column (random flight), not molecular dynamics. Red dots are A, blue dots are B. Every molecule flies straight at a kinetic-theory speed and, after an exponentially distributed free path, scatters into a random direction. The free path of each species is set so that both diffuse with the entered DAB:
Sv is one fixed constant: the default vapor (MA = 0.0741 kg/mol at 298.15 K) flies 2 tube lengths per second. B is air (MB = 0.029 kg/mol), so it flies about 1.6 times faster with a longer free path. Raising T or lowering MA speeds every molecule up by √(T/M) and shortens its free path in proportion, so DAB, and therefore the flux, is unchanged: in this model DAB is entered independently (see Model validity).
Bulk flow
On top of the random flight, every molecule of A and B is carried up the tube at the molar-average velocity, drawn as the grey chevrons and the v* arrow:
A crossing made during the random flight counts as diffusion, JA; a crossing made by the drift counts as bulk flow, xAcv*. Their sum is the counted NA:
For stagnant B, B diffuses down at exactly the rate the drift carries it up, so its net count is zero. For EMCD (r = −1) there is no drift and no bulk-flow term.
Ends of the tube
The shaded zones at the bottom (the liquid surface, xA1) and the top (the sweep gas, xA2) are reservoirs: molecules are added or removed there to hold the mole fraction on the analytic profile, extended slightly past each end. The outer edges reflect. In the white gas column nothing is imposed; the profile and fluxes there come from the molecule motion alone.
Length and time
The gas path is always drawn as one tube length, and one diffusion time z2/DAB is shown as 30 s of animation. The animation therefore depends only on xA1, xA2, r and the molar masses; z, DAB, T and P change the SI readouts (the cards above the charts) but not the motion. About 1,200 molecules fill one tube length; sizes, spacings, speeds and free paths are not to scale.
Counted readouts
Net crossings of A and B are counted at 9 planes along the column and averaged over time. NA counted ÷ predicted compares the count with the analytic NA; the band is two standard errors from 3 s batch means, so it shrinks as the count runs longer. Bulk-flow factor is the counted flux relative to the EMCD flux for the same end compositions:
which is the log-mean factor of Eq. 5-198. NB counted ÷ NA should approach r. After a change in xA1, xA2, the mode or r the molecules first relax to the new profile for 12 s (0.4 diffusion times) and the counts restart; other inputs do not reset the counts. Finite molecule numbers bias the counted flux by about 1 to 2% at the defaults and up to about 3.5% for a heavy, cold vapor with long free paths.
Profile and bars
Beside the tube, the red curve is the analytic xA(z′) and the open dots are the counted mole fraction of A in 10 slices, smoothed over 4 s. The bars at the bottom split the counted fluxes of A and B into diffusion and bulk flow; the black tick is the counted net flux and the open chevron is the theory value. Faded bars show theory until counting starts.
Reset run puts the molecules back on the steady profile and clears the counts; the Reset button at the top also restores the default inputs.
Lesson: Diffusion with Bulk Flow.