Executed numerical reference study
Dispensing nozzle
Completed virtual bench
14 verification checks; 243 tolerance/operating cases; 30 synthetic observations; four dosing cases. Original maximum-flow condition fails added system effects; computed bound 26.4 mm3/s.
Read the synthetic acceptance results and methods. The supplement records adverse outcomes as well as passing cases.
Reference case
Three 20 mm passages have the same 1 mm outlet diameter and 6 mm body diameter. The fluid properties and pump budget are explicit design assumptions in inputs.json. The chosen wide inlet reduces nominal modeled pressure by 42.4%. Only that candidate meets both reference limits at the upper flow and viscosity.
Computed comparison
| candidate | nominal pressure Pa | worst pressure Pa | retained volume mm3 | pressure pass | volume pass |
|---|---|---|---|---|---|
| baseline | 5329.459 | 10658.918 | 38.2227 | False | True |
| long-taper | 3874.8577 | 7749.7154 | 39.7935 | False | True |
| wide-inlet | 3071.3094 | 6142.6188 | 56.9414 | True | True |
Method
A body-fitted triangular meridian mesh solves the incompressible axisymmetric Stokes equations. Quadratic velocity and linear pressure include radial, axial and circumferential strain. Walls are no-slip, the axis has radial symmetry, and developed pipe profiles prescribe equal flow at inlet and outlet. One pressure value fixes the arbitrary gauge. The reported difference is area-averaged static pressure at the modeled inlet and outlet. No external supply line, free jet or fittings are included. The flow/viscosity sweep follows the linear Stokes system; the separate regression test re-solves two scaled cases.
Verification
| check | passed | value | limit |
|---|---|---|---|
| Straight-pipe pressure benchmark | True | 1.9246826354901714e-12 | relative error < 1e-8 |
| baseline pressure mesh refinement | True | 2.275942063256764e-06 | medium-to-fine relative change < 0.005 |
| baseline section flow check | True | 0.0016555006285932095 | maximum sampled sectional deviation < 0.003; includes quadrature error |
| baseline inlet/outlet balance | True | 1.000088900582341e-12 | relative imbalance < 1e-8 |
| baseline free equation residual | True | 5.161267705717814e-11 | relative residual < 1e-8 |
| long-taper pressure mesh refinement | True | 1.031203919810153e-06 | medium-to-fine relative change < 0.005 |
| long-taper section flow check | True | 0.001005742201407811 | maximum sampled sectional deviation < 0.003; includes quadrature error |
| long-taper inlet/outlet balance | True | 1.000088900582341e-12 | relative imbalance < 1e-8 |
| long-taper free equation residual | True | 3.61676409459339e-11 | relative residual < 1e-8 |
| wide-inlet pressure mesh refinement | True | 2.8778594611811315e-06 | medium-to-fine relative change < 0.005 |
| wide-inlet section flow check | True | 0.0014377521215825917 | maximum sampled sectional deviation < 0.003; includes quadrature error |
| wide-inlet inlet/outlet balance | True | 1.2001066806988093e-12 | relative imbalance < 1e-8 |
| wide-inlet free equation residual | True | 3.258543912591118e-11 | relative residual < 1e-8 |
| Developed boundary extension | True | 2.7001853184493996e-06 | extra straight-pipe resistance removed; pressure change < 0.01 |
| Creeping-flow regime | True | 0.07639437268410976 | outlet Reynolds number < 0.1 |
| Pressure budget at maximum flow and viscosity | True | 6142.618759363431 | <= 6500 Pa |
| Retained volume limit | True | 56.941366846315 | <= 60 mm3 |
| baseline CAD solid validity | True | 1.0 | one valid solid; bore volume checked against frustum equation |
| long-taper CAD solid validity | True | 1.0 | one valid solid; bore volume checked against frustum equation |
| wide-inlet CAD solid validity | True | 1.0 | one valid solid; bore volume checked against frustum equation |
| Six regression tests | True | 0.0 | exit 0 |
Boundary placement
Adding five inlet radii upstream and five outlet radii downstream, then subtracting their exact developed pipe resistance, changed the selected pressure by 0.0003%. This checks the imposed profiles for the selected geometry; it does not account for real fittings.
Limits and physical check
All operating cases have outlet Re below 0.1. Results assume fixed viscosity and single-phase Newtonian liquid. No droplet, shutoff, compressibility, surface-tension, contamination, cleaning or physical accuracy claim is made. The STEP files are nominal reference solids, not a released manufacturing drawing. Read bench-test.md before adapting the geometry.
Sources
