Executed numerical reference study
Sampling chamber
Completed virtual bench
29 checks; 162 dimensional/operating cases; six coupled response cases; three slow-wall refinement runs; four selected-design corners plus refined worst corner; 20 synthetic instrument readings.
Read the synthetic acceptance results and methods. The supplement records adverse outcomes as well as passing cases.
The original comparison below is retained as baseline evidence. The final selection is the 2.525 mm-radius tolerance-allowance CAD and the 600 s virtual-fixture flush decision in the supplement.
Reference result
The 0.5 mm radial-clearance reference chamber replaces 99% of its initial tracer mass in 50.93 s, with 6.785 Pa modeled pressure loss. It is the smallest declared clearance; no claim is made that a smaller gap is manufacturable or appropriate for a real probe.
Computed comparison
| chamber radius mm | radial clearance mm | volume mm3 | residence time s | pressure Pa | t99 whole s | t99 probe s | t99 outlet s | flush volume whole mm3 | pressure pass |
|---|---|---|---|---|---|---|---|---|---|
| 2.5 | 0.5 | 141.3717 | 28.2743 | 6.785 | 50.9279 | 39.5838 | 60.9967 | 254.6394 | True |
| 3 | 1 | 314.1593 | 62.8319 | 0.7619 | 130.8675 | 106.5055 | 154.7697 | 654.3376 | True |
| 4 | 2 | 753.9822 | 150.7964 | 0.079 | 362.1003 | 294.6016 | 421.0654 | 1810.5017 | True |
Model and measurements
The fixed probe has 2 mm radius and occupies the full 20 mm chamber length. The three chambers have uniform outer radii of 2.5, 3 and 4 mm. An axisymmetric Stokes FEM determines pressure and the axial velocity. The velocity is integrated into annular face fluxes; a tiny normalization enforces the declared flow exactly in the transport mesh. The scalar solver uses mass-form finite volumes, upwind axial advection, centered molecular diffusion and backward Euler. Internal face fluxes cancel; outflow accumulated at each implicit step is compared with mass remaining.
The old-sample fraction starts at one throughout. Incoming fluid has zero old tracer. Diffusive flux is zero at the ends and both walls. The probe metric averages the liquid from z=8 to 12 mm and from r=2 to 2.2 mm, with partial-cell volume weights. Outlet concentration is flow-weighted, while whole-chamber concentration is volume-weighted. t99 is the first interpolated crossing below 0.01.
Refinement and verification
| check | passed | value | limit |
|---|---|---|---|
| 2.5 mm annulus pressure benchmark | True | 1.844085717461752e-08 | relative pressure error < 2e-5 |
| 2.5 mm creeping-flow regime | True | 0.07073553026306459 | Re < 0.1 |
| 2.5 mm spatial refinement | True | 0.02017197635615897 | maximum medium-to-fine t99 change < 4% |
| 2.5 mm time refinement | True | 0.012325079588729526 | maximum t99 change from halving dt < 2% |
| 2.5 mm tracer mass balance | True | 7.237528875152536e-15 | relative initial mass error < 1e-9 |
| 2.5 mm tracer positivity | True | 5.248981356793106e-37 | minimum concentration >= -1e-12 |
| 2.5 mm tracer upper bound | True | 1.0000000000000038 | maximum concentration <= 1+1e-12 |
| 2.5 mm FEM-to-cell flux transfer | True | 5.134959302210973e-07 | flux normalization correction < 1e-5 |
| 3 mm annulus pressure benchmark | True | 5.79637646769271e-08 | relative pressure error < 2e-5 |
| 3 mm creeping-flow regime | True | 0.06366197723675814 | Re < 0.1 |
| 3 mm spatial refinement | True | 0.020103600637870533 | maximum medium-to-fine t99 change < 4% |
| 3 mm time refinement | True | 0.005065248394893196 | maximum t99 change from halving dt < 2% |
| 3 mm tracer mass balance | True | 4.704393768849149e-15 | relative initial mass error < 1e-9 |
| 3 mm tracer positivity | True | 4.6873989073637536e-37 | minimum concentration >= -1e-12 |
| 3 mm tracer upper bound | True | 1.0000000000000062 | maximum concentration <= 1+1e-12 |
| 3 mm FEM-to-cell flux transfer | True | 5.154748532465447e-07 | flux normalization correction < 1e-5 |
| 4 mm annulus pressure benchmark | True | 1.6965606008945144e-07 | relative pressure error < 2e-5 |
| 4 mm creeping-flow regime | True | 0.05305164769729845 | Re < 0.1 |
| 4 mm spatial refinement | True | 0.01808600994572107 | maximum medium-to-fine t99 change < 4% |
| 4 mm time refinement | True | 0.0019243453726162096 | maximum t99 change from halving dt < 2% |
| 4 mm tracer mass balance | True | 6.785183320455503e-15 | relative initial mass error < 1e-9 |
| 4 mm tracer positivity | True | 8.722261628111456e-36 | minimum concentration >= -1e-12 |
| 4 mm tracer upper bound | True | 1.0000000000000113 | maximum concentration <= 1+1e-12 |
| 4 mm FEM-to-cell flux transfer | True | 4.256420818027351e-07 | flux normalization correction < 1e-5 |
| Three nominal CAD chamber solids | True | 3.0 | three valid body/probe solids, volume matched analytically |
| Seven regression tests | True | 0.0 | exit 0 |
Interpretation
First-order upwinding and backward Euler smear the front. The separate space and time studies quantify the remaining t99 sensitivity for this input set; these are not exact analytical washout solutions. The numerical ranking is stable across the tested refinements. The outlet metric clears later than the probe-region metric in these straight annuli. A bypass cavity could reverse that behavior, but no such cavity is included here.
Limits and experiment
Fluid viscosity and diffusivity are reference inputs, not measured properties. The model excludes fittings, expansion chambers, asymmetric probe heads, trapped gas, adsorption and sensor chemistry. The CAD bodies do not include seals, mounting or pressure-vessel qualification. See flush-test.md for an unperformed physical comparison procedure.
Sources
