# ⚛  L1 Principle — Forced Convection — conjugate heat transfer inversion

**ID:** `L1-197` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Heat Transfer — *Convective heat transfer coefficient estimation*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D heat transfer coeff
> **📡 Carrier:** thermal · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554033

---

## 🧠 1. Introduction

**Forced Convection — conjugate heat transfer inversion** is a **nonlinear inverse problem** whose unknown lives in **2D heat transfer coeff** space, within the **Convective heat transfer coefficient estimation** sub-domain of **Heat Transfer**.

Measurements consist of thermal radiation / temperature measurements via a **ir camera thermocouples** sensing mechanism.

The forward operator applies, in order: L · nav stokes flow operator; L · heat transport operator; L · convect coupling operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Coupled Navier-Stokes + energy; well-posed for moderate Re, unstable at high Re.

## ⚙ 2. Forward Model

Physical chain: **x** → L · nav stokes flow → L · heat transport → L · convect coupling → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.convect_coupling` `L.heat_transport` `L.nav_stokes_flow` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.nav_stokes_flow` | L · nav stokes flow operator |
| `L.heat_transport` | L · heat transport operator |
| `L.convect_coupling` | L · convect coupling operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Heat Transfer |
| Sub domain | Convective heat transfer coefficient estimation |
| Carrier | thermal |
| Problem class | nonlinear_inverse |
| Solution space | 2D_heat_transfer_coeff |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Coupled Navier-Stokes + energy; well-posed for moderate Re, unstable at high Re.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `forced_convection` · **Forward operator:** `forced_convection_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Pr | — | 0.7 |
| Re | — | 50000 |
| Snr db | dB | 30 |
| Pr error | — | 0 |
| Re error | — | 0 |
| Freestream t error | — | 0 |
| Wall emissivity error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 2048 |
| W | px | 64 – 2048 |
| Pr | — | 0.01 – 1000 |
| Re | — | 100 – 10000000 |
| Snr db | dB | 0 – 40 |
| Pr error | — | 0.0 – 0.1 |
| Re error | — | 0.0 – 0.1 |
| Freestream t error | — | 0.0 – 5.0 |
| Wall emissivity error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x16e3321c0919590cd09ecd3e869b8ce2975ee9a06be25d6b4ffc5124ddafbf07`
- **Chain tx hash:** `0x3232cfade9a31082b8966b879ad7ee6a35b8993da9b3907f2e23ee7a78b97d2f`
- **Chain block:** `41554033`

---

## File Mapping

This bundle consists of: `L1-197.md`, `L1-197.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-197.md` | Source of truth — edit this | Human or LLM |
| `L1-197.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-197`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._