Sampling chamber: virtual flush bench
This is an executed virtual flushing bench. All sensor constants, wall parameters, dimensional tolerances and instrument errors are synthetic reference inputs. No physical chamber, tracer or sensor was operated.
Coupled transport and wall model
The existing finite-volume mass form is extended with two wall states per axial strip. A wall capacity W = K A has units of equivalent liquid volume. Transfer J = (W / tau) (cfluid - cwall) removes tracer from the adjacent fluid cell and adds exactly the same amount to that wall state. Equilibrium is cwall = cfluid. This linear reversible retention model describes sensitivity to extra stored tracer; K and tau are declared parameters, not chemistry fitted to a material. Both walls start at concentration 1. The reported inventory includes fluid and wall mass.
The inlet may pass through a perfectly mixed 20 mm3 volume. Its old-sample fraction decays as exp(-Qt/Vin); each numerical step uses the exact average of that function. The measured probe signal follows tau_sensor dy/dt = cprobe - y. It has an exact exponential decay factor and trapezoidal forcing over each step. The zero-lag case reads the probe average directly. Inlet tracer, outlet tracer and retained inventory are recorded at every time step.
The baseline and response variations use the original 2.5 mm chamber radius to isolate wall and instrument effects. A separate tolerance study evaluates the 2.525 mm nominal-radius variant and exports its CAD. This revised geometry is a dimensional proposal supported by hydraulic calculations, not a fabricated part. Its transport response is not substituted into the original baseline curves.
Hydraulic and dimensional calculation
For a straight concentric annulus, deltaP = 8 mu L Q / [pi (b^4 - a^4 - (b^2-a^2)^2/ln(b/a))]. The 162 cases cover both original and allowance variants, independently varying probe/body radii, flow and viscosity. A new FEM solve checks the worst hydraulic corner of the revised geometry. The 10 Pa budget and minimum 0.5 mm clearance remain unchanged.
The 90 s indicated-response limit is an explicit new virtual-bench criterion, not a requirement from a customer. Failure is reported when a slow sensor or retained wall inventory causes that limit to be exceeded. This prevents a fluid-only result from being presented as sensor performance.
Instrument check
Differential pressure contains hydrostatic head rho g delta_h and instrument zero bias. Twenty readings include the declared 0.2 mm elevation error, 0.2 Pa zero offset and 0.08 Pa independent Gaussian noise. Correcting the known injected errors tests the reduction workflow. It does not establish the calibration of a real low-pressure transducer.
Selected design and operating decision
Adopt the supplied 2.525 mm chamber-radius CAD with the stated radial tolerances. For the declared slow-retention virtual fixture, use a 600 s flush at 4.5 mm3/s before accepting a reading. Four geometry corners and a finer worst-corner solve put fluid, outlet, sensor and total inventory below 1% at that time. Maximum remaining fraction is 0.006958.
The 90 s response objective is rejected for that slow-retention condition. The selected wait does not claim that objective was met. Wall and sensor values remain explicit synthetic inputs; a different measured response requires recalculating the wait.
| probe radius mm | chamber radius mm | flow mm3 s | capacity length mm | wall exchange s | inlet volume mm3 | sensor tau s | check time s | sensor fraction | fluid fraction | inventory fraction | outlet fraction | t99 inventory s | t99 sensor s | maximum mass balance fraction |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.99 | 2.51 | 4.5 | 0.1 | 120 | 20 | 30 | 600 | 0.0019141814356454165 | 0.001383253952207325 | 0.0066926582039379375 | 0.002079523437934192 | 541.4547449814021 | 367.458595003068 | 1.005212343771186e-14 |
| 1.99 | 2.54 | 4.5 | 0.1 | 120 | 20 | 30 | 600 | 0.002115035657780247 | 0.0014683147648261153 | 0.006783047491718452 | 0.0021942154055618742 | 543.001182739786 | 380.18583602416237 | 6.923316804781675e-15 |
| 2.01 | 2.51 | 4.5 | 0.1 | 120 | 20 | 30 | 600 | 0.001803787039346318 | 0.001341889353540995 | 0.006669147176591796 | 0.00203216613442192 | 541.1910334945678 | 359.9417562478399 | 1.34388375921369e-14 |
| 2.01 | 2.54 | 4.5 | 0.1 | 120 | 20 | 30 | 600 | 0.0020002816326127734 | 0.0014252944455530857 | 0.006753627674571143 | 0.002144707236445234 | 542.6021699146293 | 373.0572581969048 | 1.8793824039798466e-14 |
Acceptance results
| requirement | outcome | evidence |
|---|---|---|
| Minimum 0.5 mm physical radial clearance with declared tolerances | ORIGINAL FAIL; REVISED PASS | Original minimum 0.475 mm. Revised 2.525 mm nominal chamber radius gives minimum 0.500 mm. |
| 10 Pa hydraulic budget with revised dimensions and input range | PASS | Worst 8.1736 Pa. |
| 90 s indicated t99 reference criterion | CASE DEPENDENT | Passive probe 40.90 s; 30 s sensor lag 156.31 s; slow wall/inlet/sensor case 345.88 s. |
| Fluid plus adsorbed inventory conserved | PASS | Maximum residual 1.12e-14. |
| Measured pressure equals fluid resistance without corrections | FAIL | Declared 0.2 mm tap-height error alone adds 1.9613 Pa. |
| Selected 600 s flush on revised geometry at the declared slow-retention condition | PASS | Maximum old fraction 0.006958; four tolerance corners and finer worst-corner solve. |
Response cases
| case | capacity length mm | wall exchange s | inlet mixed volume mm3 | sensor tau s | t99 fluid s | t99 probe s | t99 sensor s | t99 inventory s | t99 outlet s | maximum mass balance fraction | minimum fraction | maximum fraction | initial inventory mm3 | initial wall inventory mm3 | elapsed s | nz | nr | dt s | sensor within 90s |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ideal_passive | 0 | 30 | 0 | 0 | 51.9324096829791 | 40.90287971799913 | 40.90287971799913 | 51.9324096829791 | 62.191652826675465 | 1.809382218788134e-15 | 1.8400110146999359e-25 | 1.0000000000000022 | 141.3716694115407 | 0 | 62.75 | 80 | 24 | 0.25 | True |
| sensor_5s | 0 | 30 | 0 | 5 | 51.9324096829791 | 40.90287971799913 | 49.67271812019703 | 51.9324096829791 | 62.191652826675465 | 1.809382218788134e-15 | 1.8400110146999359e-25 | 1.0000000000000022 | 141.3716694115407 | 0 | 62.75 | 80 | 24 | 0.25 | True |
| sensor_30s | 0 | 30 | 0 | 30 | 51.9324096829791 | 40.90287971799913 | 156.30605302163076 | 51.9324096829791 | 62.191652826675465 | 1.809382218788134e-15 | 3.241927943590486e-66 | 1.0000000000000022 | 141.3716694115407 | 0 | 204.75 | 80 | 24 | 0.25 | False |
| inlet_mix_20mm3 | 0 | 30 | 20 | 5 | 57.22342054338996 | 47.160565933615594 | 55.57935394433328 | 57.22342054338996 | 67.57936908611369 | 2.8145945625593194e-15 | 4.0158312651250395e-08 | 1.0000000000000027 | 141.3716694115407 | 0 | 68.5 | 80 | 24 | 0.25 | True |
| wall_fast | 0.02 | 30 | 0 | 5 | 91.7222133707212 | 90.15804176130796 | 95.56349576539468 | 117.42420934857469 | 111.87797034921334 | 2.4199556424121138e-15 | 1.4696509861027557e-11 | 1.0000000000000018 | 152.68140296446393 | 11.309733552923255 | 175.0 | 80 | 24 | 0.25 | False |
| wall_slow | 0.1 | 120 | 20 | 30 | 297.58464437303104 | 311.8635993540652 | 345.88112886072037 | 533.1439862602687 | 346.2949238488806 | 1.1200937544878925e-14 | 1.0100279454565036e-11 | 1.0000000000000013 | 197.92033717615695 | 56.548667764616276 | 764.75 | 80 | 24 | 0.25 | False |
Refinement
| nz | nr | dt s | t99 fluid s | t99 probe s | t99 sensor s | t99 inventory s | maximum mass balance fraction |
|---|---|---|---|---|---|---|---|
| 80 | 24 | 0.25 | 297.58464437303104 | 311.8635993540652 | 345.88112886072037 | 533.1439862602687 | 1.1200937544878925e-14 |
| 160 | 48 | 0.25 | 298.1287903396534 | 312.86343443204106 | 346.84256531091 | 536.4924344285936 | 1.809382218788134e-14 |
| 160 | 48 | 0.125 | 298.0004196465925 | 312.7264367337851 | 346.7075205157091 | 536.2637795198026 | 3.1017980893510866e-14 |
Verification
| id | description | expected | observed | passed | method |
|---|---|---|---|---|---|
| original_clearance_failure | Apply declared manufacturing tolerances to original 0.5 mm nominal gap | minimum gap below 0.5 mm is detected | 0.4750000000000001 | True | Full factorial exact annular hydraulic model |
| allowance_clearance | 2.525 mm nominal radius covers +/-0.015 body and +/-0.010 probe radii | all 81 cases have >= 0.5 mm clearance | 0.5 | True | Worst independent radial tolerance sum, no statistical cancellation |
| allowance_pressure | Unchanged 10 Pa pressure budget for revised nominal radius | all 81 cases below 10 Pa | 8.173571485867228 | True | Exact developed annular Stokes solution |
| regime | Tolerance/flow/viscosity range remains creeping flow | max Re < 0.1 | 0.08693752322834755 | True | Hydraulic diameter 2(b-a) |
| fem_worst_corner | Check revised worst hydraulic case against independent FEM | relative error < 1e-5 | 7.074845043142375e-09 | True | P2/P1 Stokes solve at revised tolerance corner |
| cad_roundtrip | Revised nominal assembly exports and reimports | valid body/probe solids; exact fluid volume | [{'radius_mm': 2.525, 'fluid_volume_mm3': 149.26492095368508, 'body_valid': True, 'probe_valid': True}] | True | CadQuery solid validity and STEP roundtrip |
| mass_ideal_passive | Fluid + reversible wall + inlet/outlet conservation in ideal_passive | max normalized residual < 1e-10 | 1.809382218788134e-15 | True | Sparse mass-form backward Euler; exchange fluxes cancel in coupled capacities |
| bounded_ideal_passive | Concentration bounds in ideal_passive | all fluid/wall concentrations in [0,1] +/- 1e-10 | [1.8400110146999359e-25, 1.0000000000000022] | True | Every solved state checked |
| mass_sensor_5s | Fluid + reversible wall + inlet/outlet conservation in sensor_5s | max normalized residual < 1e-10 | 1.809382218788134e-15 | True | Sparse mass-form backward Euler; exchange fluxes cancel in coupled capacities |
| bounded_sensor_5s | Concentration bounds in sensor_5s | all fluid/wall concentrations in [0,1] +/- 1e-10 | [1.8400110146999359e-25, 1.0000000000000022] | True | Every solved state checked |
| mass_sensor_30s | Fluid + reversible wall + inlet/outlet conservation in sensor_30s | max normalized residual < 1e-10 | 1.809382218788134e-15 | True | Sparse mass-form backward Euler; exchange fluxes cancel in coupled capacities |
| bounded_sensor_30s | Concentration bounds in sensor_30s | all fluid/wall concentrations in [0,1] +/- 1e-10 | [3.241927943590486e-66, 1.0000000000000022] | True | Every solved state checked |
| mass_inlet_mix_20mm3 | Fluid + reversible wall + inlet/outlet conservation in inlet_mix_20mm3 | max normalized residual < 1e-10 | 2.8145945625593194e-15 | True | Sparse mass-form backward Euler; exchange fluxes cancel in coupled capacities |
| bounded_inlet_mix_20mm3 | Concentration bounds in inlet_mix_20mm3 | all fluid/wall concentrations in [0,1] +/- 1e-10 | [4.0158312651250395e-08, 1.0000000000000027] | True | Every solved state checked |
| mass_wall_fast | Fluid + reversible wall + inlet/outlet conservation in wall_fast | max normalized residual < 1e-10 | 2.4199556424121138e-15 | True | Sparse mass-form backward Euler; exchange fluxes cancel in coupled capacities |
| bounded_wall_fast | Concentration bounds in wall_fast | all fluid/wall concentrations in [0,1] +/- 1e-10 | [1.4696509861027557e-11, 1.0000000000000018] | True | Every solved state checked |
| mass_wall_slow | Fluid + reversible wall + inlet/outlet conservation in wall_slow | max normalized residual < 1e-10 | 1.1200937544878925e-14 | True | Sparse mass-form backward Euler; exchange fluxes cancel in coupled capacities |
| bounded_wall_slow | Concentration bounds in wall_slow | all fluid/wall concentrations in [0,1] +/- 1e-10 | [1.0100279454565036e-11, 1.0000000000000013] | True | Every solved state checked |
| zero_retention_limit | New coupled model reduces to existing passive model | relative t99 probe error < 1e-10 | 0.0 | True | Independent public run_transport execution at same mesh and time step |
| refine_grid | Slow-wall case grid sensitivity | max t99 change < 0.03 | 0.006280570079787662 | True | 80x24 to 160x48 and 0.25 s to 0.125 s, original threshold retained |
| refine_timestep | Slow-wall case timestep sensitivity | max t99 change < 0.03 | 0.00043788338034667085 | True | 80x24 to 160x48 and 0.25 s to 0.125 s, original threshold retained |
| exchange_benchmark | Isolated fluid/wall exchange approaches conserved equilibrium | max absolute error < 5e-5 | 2.9697706575071425e-05 | True | Backward Euler compared with independent exact two-capacity exponential |
| exchange_mass | Closed two-capacity benchmark conserves initial tracer | absolute difference < 1e-10 | 1.2594369991347776e-12 | True | Independent dense update and analytical conserved mass |
| pressure_correction | Recover known synthetic hydraulic pressure after elevation/zero correction | absolute mean error < 0.08 Pa | -0.02275547843690351 | True | 20 seeded readings, known imposed elevation and zero bias |
| sensor_lag_challenge | 30 s sensor time constant changes the reference conclusion | sensor t99 exceeds 90 s while passive probe t99 is below 90 s | 156.30605302163076 | True | Causal first-order sensor driven by the spatial probe average |
| retention_challenge | Slow wall inventory is retained in response accounting | inventory t99 exceeds passive-fluid t99 | 533.1439862602687 | True | Reversible exchange with explicitly assumed capacity and time constant |
| invalid_inputs | Reject invalid annular hydraulic conditions | 3 invalid conditions rejected | 3 | True | Zero gap, negative flow, nonfinite viscosity |
| selected_flush_decision | Adopted 600 s flush on tolerance-allowance geometry | all fluid, sensor, inventory and outlet fractions <= 0.01 at four geometry corners | 0.006783047491718452 | True | Four new 80x24 runs at 4.5 mm3/s, 0.1 mm wall capacity, 120 s exchange, 20 mm3 inlet and 30 s sensor |
| selected_flush_refined | Verify the slowest selected corner at finer resolution | all four fractions <= 0.01 at 600 s | 0.006957725391144411 | True | Additional 160x48 run at 0.125 s; full trace retained |
Selected reference dimensions
Model boundary
The model remains axisymmetric and straight. Wall retention is linear, inlet dead volume is perfectly mixed, and the sensor is first order. It does not resolve real side ports, trapped gas geometry, nonlinear binding, seals or leakage. These are completed theoretical sensitivity cases with exposed assumptions, not a replacement for physical characterization.
Method context: COMSOL transport and adsorption example describes coupled fluid and wall species balances. This package uses its own explicitly derived linear two-capacity model, not the COMSOL example chemistry or software.