Source: https://guthlabs.ai/docs/instruments/converging-diverging-nozzle-area-ratio-calculator/
instrument / verified-with-binding-errata
# Converging-Diverging Nozzle Area-Ratio Calculator
The current ledger marks this instrument verified with errata that must be applied at binding time.
Status: **VERIFIED_WITH_ERRATA**Lifecycle: **verified-with-binding-errata**
## Inspectable records
[01**Overview**Purpose and lifecycle]()[02**Mechanism**Method and assumptions]()[03**Implementation**Binding and validation]()[04**Qualification**Evidence and errata]()
## Ledger state
The current ledger marks this instrument verified with errata that must be applied at binding time.
**Area**: Thermodynamics, Fluids & Aero
**Tranche**: tranche-6
**Evidence ID**: t6-085
**Ledger rows**: 1
No ledger note is attached.
## Review findings
The core area-Mach formula is correct under the stated ideal 1D assumptions, but the instrument specification is incomplete for its claimed back-pressure-based regime output. Define Pb/P0, enforce Ae/A*>=1 and gamma>1, add the pressure and normal-shock algorithms, and distinguish formal branch roots from the actual unchoked operating Mach. The honest-label warning should explicitly state whether normal shocks and shock losses are modeled; otherwise 'shock-in-nozzle' is only a classification, not a physically computed solution.
### Formula and scope findings
- The area-Mach relation is standard for quasi-one-dimensional, steady, adiabatic, isentropic flow with constant gamma, using A*=the sonic throat area. It is valid only for A/A* >= 1; this input constraint is not stated.
- The relation has two distinct roots only for A/A* > 1. At A/A*=1 the roots coalesce at M=1, so the claimed dual-root output needs an explicit boundary rule.
- The stated formula alone cannot determine the operating regime from back pressure. Shock-in-nozzle classification additionally requires the isentropic pressure relation, the choked-flow criterion, and normal-shock relations, including shock location or an equivalent pressure threshold algorithm.
- The back-pressure ratio is undefined. It must explicitly mean Pb/P0, where P0 is stagnation pressure upstream of the nozzle.
- For a shock-in-nozzle result, the model must include a normal shock and the associated total-pressure loss. Treating the entire nozzle as isentropic while reporting a shock-in-nozzle regime is inconsistent.
- The exit Mach solutions are independent of back pressure only after the nozzle is choked. For sufficiently high back pressure, the nozzle is unchoked and the supplied A/A* relation does not directly provide the actual exit Mach.
### Unit assumptions
- gamma is dimensionless and is the constant ratio of specific heats; test cases use gamma=1.4 for air.
- Area ratio means the geometric exit-area-to-sonic-throat-area ratio Ae/A*.
- Back-pressure ratio means Pb/P0, with Pb static back pressure and P0 upstream stagnation pressure.
- Pressure ratios are dimensionless; no absolute pressure units are required.
- The ideal regime thresholds below assume a converging-diverging nozzle, ideal choking, isentropic regions away from a normal shock, and a normal shock when classified as shock-in-nozzle.
## Boundary hardening
**BREAK_CLAIMED** · SELF_FLAGGED
The specified formula has an undefined gamma=1 input that requires explicit rejection, and the back-pressure ratio convention is not defined well enough to guarantee a unique operating-regime result.
## Errata and corrections
### Tranche errata
**critical_choking_pressure_ratio_Pb_over_P0** · vector 1,2,3
0.528282
0.93716 for AR=2 (p at the subsonic-exit solution; per-AR for others)
ships the throat ratio (2/(g+1))^(g/(g-1)) as the choking BACK pressure — the classic misconception; a C-D nozzle chokes when Pb drops below the subsonic-branch exit pressure, not p*/P0 (v4 with AR=1 coincidentally agrees)
## Inspectable example
4 review vectors are recorded. One source-bound sample is shown below.
````
{
"inputs": {
"area_ratio_A_over_Astar": 2,
"back_pressure_ratio_Pb_over_P0": 0.05,
"gamma": 1.4
},
"expected_outputs": {
"subsonic_M": 0.305904,
"supersonic_M": 2.1972,
"exit_supersonic_pressure_ratio_Pe_over_P0": 0.093934,
"critical_choking_pressure_ratio_Pb_over_P0": 0.528282,
"shock_at_exit_pressure_ratio_Pb_over_P0": 0.5135,
"operating_regime": "supersonic"
},
"working": "For gamma=1.4, A/A*=(1/M)[(5/6)(1+0.2M^2)]^3. Newton solve on g(M)=-ln(M)+3ln[(5/6)(1+0.2M^2)]-ln(2), with g'(M)=-1/M+1.2M/(1+0.2M^2). Subsonic branch: M0=0.300000; g(M0) is approximately +0.01738, g'(M0)=-2.97970, so M1=0.300000-(+0.01738)/(-2.97970)=0.305834. Second iteration gives M2 approximately 0.305904; subsequent iterations give M=0.305904. Supersonic branch: M0=2.00000; g(M0)=-0.169899, g'(M0)=0.833333, so M1=2.00000-(-0.169899)/(0.833333)=2.20388. Second iteration gives M2 approximately 2.19724; subsequent iterations give M=2.19720. Check: for M=2.19720, M^2=4.82768; (5/6)(1+0.2M^2)=(5/6)(1.965536)=1.637947; 1.637947^3/2.19720 approximately 2.00000. Exit pressure: Pe/P0=(1+0.2M^2)^(-gamma/(gamma-1))=1.965536^(-3.5)=0.093934. Choking limit: (2/(gamma+1))^(gamma/(gamma-1))=(5/6)^3.5=0.528282. Normal-shock pressure multiplier at M=2.19720 is 1+[2gamma/(gamma+1)](M^2-1)=1+(7/6)(3.82768)=5.46563; shock-at-exit Pb/P0=0.093934 times 5.46563 approximately 0.5135. Since Pb/P0=0.05 is below the design exit pressure ratio, the ideal flow is supersonic through the diverging section, with any adjustment shock outside the nozzle."
}
````
## Related workflows
No source-defined multi-instrument workflow membership is recorded.
## Sources and evidence
These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.
- `VERIFIED_LEDGER.json · VERIFIED_WITH_ERRATA`
- `evidence-tranche6/tranche6-results.json`
- `BOUNDARY_HAZARD_CATALOG.json`