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Exponential Temperature Prediction System

Overview

This system predicts cooking times using an exponential approach model based on the physics of heat transfer during cooking.

https://xkcd.com/612/ https://www.youtube.com/watch?v=9gTLDuxmQek

Physics Background

Heat Transfer in Cooking

When you cook meat, the temperature doesn't rise linearly. Instead, it follows an exponential approach toward an equilibrium temperature. This happens because:

  1. Newton's Law of Cooling: The rate of temperature change is proportional to the temperature difference between the meat and its environment
  2. Thermal Mass: Larger pieces of meat heat up more slowly
  3. Heat Capacity: Different materials require different amounts of energy to change temperature
  4. Surface Area: More surface area = faster heat transfer

The Exponential Model

The temperature follows this equation:

T(t) = T_equilibrium - (T_equilibrium - T_start) × e^(-t/τ)

Where:

  • T(t) = temperature at time t
  • T_equilibrium = final equilibrium temperature the meat approaches
  • T_start = starting temperature
  • τ (tau) = time constant (how fast the meat heats up)
  • t = elapsed time

What is Tau (τ)?

Tau is the time it takes to reach 63.2% of the way to equilibrium. Think of it as the "cooking speed":

  • Small tau (fast cooking): Thin chicken breast, high grill temp
  • Large tau (slow cooking): Thick brisket, low-and-slow temperatures

Examples:

  • Tau = 30 minutes: After 30 min → 63% done, after 60 min → 86% done, after 90 min → 95% done
  • Tau = 2 hours: After 2 hours → 63% done, after 4 hours → 86% done, after 6 hours → 95% done

System Architecture

ExponentialPredictor Class

The predictor maintains:

  • Historical data: Recent temperature, grill temp, and time measurements
  • Model parameters: Current tau estimate, start/target temperatures
  • Environmental factors: Grill temperature and setpoint for physics-based corrections

Key Methods

  1. Update(): Processes new temperature readings and re-estimates tau
  2. EstimateTimeToTarget(): Predicts time to reach target temperature
  3. estimateTimeConstant(): Finds the best tau by testing against historical data
  4. calculateEffectiveEquilibrium(): Adjusts target based on grill conditions

Physics-Based Grill Dynamics

The system goes beyond simple exponential curves by considering real grill physics:

Effective Equilibrium Temperature

The meat doesn't just approach the probe target—it's influenced by grill temperature:

Effective Equilibrium = f(grill_temp, grill_setpoint, probe_target)

Examples:

  • Grill at 250°F, target 204°F → meat can reach ~204°F
  • Grill at 180°F, target 204°F → meat might only reach ~185°F (insufficient heat)
  • Grill at 350°F, target 204°F → meat might overshoot to ~210°F (too much heat)

Grill Stability Factor

Unstable grill temperatures reduce prediction accuracy:

stability = 1.0 - |grill_temp - grill_setpoint| / 50.0

Tunable Parameters

Core Model Parameters

Parameter Default Range Purpose
minDataPoints 3 2-10 Minimum measurements before estimating tau
maxHistory 20 10-50 How many recent points to keep
defaultTau 3600s (1hr) 300-28800s Initial tau estimate

Tau Search Parameters

Parameter Default Range Purpose
tauMin 300s (5min) 60-1800s Fastest possible cooking
tauMax 28800s (8hr) 3600-86400s Slowest possible cooking
tauStep 300s (5min) 30-600s Search granularity

Grill Dynamics Parameters

Parameter Default Purpose
Very hot grill factor 0.1 When grill >50°F above target
Moderate hot factor 0.2 When grill 20-50°F above target
Slightly hot factor 0.5 When grill 0-20°F above target
Cool grill factor 0.3 When grill below target
Stability divisor 50.0 How much grill instability matters
Min stability 0.5 Prevents extreme instability penalties

Convergence Parameters

Parameter Default Purpose
Error improvement 0.9 New tau must be 10% better to update
Max prediction time 8 hours Caps unrealistic predictions
Equilibrium threshold 0.1°F When start temp too close to target

Experimental Tuning

For Faster Response (More Aggressive)

  • Reduce minDataPoints to 2
  • Reduce maxHistory to 10
  • Increase tauStep to 60s for coarser search
  • Reduce error improvement threshold to 0.8

For More Stability (Conservative)

  • Increase minDataPoints to 5
  • Increase maxHistory to 30
  • Reduce tauStep to 60s for finer search
  • Increase error improvement threshold to 0.95

For Different Cooking Styles

High-Heat/Fast Cooking:

  • Reduce tauMin to 180s (3 minutes)
  • Reduce tauMax to 7200s (2 hours)
  • Increase grill factors (more aggressive heat transfer)

Low-and-Slow:

  • Increase tauMin to 600s (10 minutes)
  • Increase tauMax to 43200s (12 hours)
  • Reduce grill factors (gentler heat transfer)

Precision Cooking:

  • Reduce tauStep to 60s
  • Increase maxHistory to 50
  • Increase minDataPoints to 8

Code Locations

Primary Implementation

  • exponential.go: Main predictor logic
  • estimateTimeConstant(): Tau calculation (lines ~85-110)
  • calculateEffectiveEquilibrium(): Grill dynamics (lines ~200-240)

Key Constants to Modify

// In NewExponentialPredictor()
minDataPoints: 3,        // Line 36

// In Update()
ep.timeConstant = 3600.0 // Line 55 (default tau)

// In estimateTimeConstant()
for tau := 300.0; tau <= 28800.0; tau += 300.0 // Line 103

// In calculateEffectiveEquilibrium()
effectiveEquilibrium = probeTarget + grillDelta*0.1 // Line 222
effectiveEquilibrium = probeTarget + grillDelta*0.2 // Line 225
effectiveEquilibrium = probeTarget + grillDelta*0.5 // Line 228
effectiveEquilibrium = grillTemp + math.Max(grillDelta*0.3, -20) // Line 232

Testing Your Changes

Using the Forecast Command

Test parameter changes against historical data:

./wifire forecast --input your_cook.json --actual 2025-01-17T20:49:45-05:00

Key Metrics to Watch

  1. Accuracy: How close predictions are to actual finish time
  2. Stability: How much predictions jump between measurements
  3. Convergence: How quickly the model settles on accurate predictions
  4. Tau Values: Are estimated tau values reasonable for your cooking style?

Sample Output Analysis

Time       Delta    Grill    Probe    Target   Velocity   Filtered   Exp ETA      Actual       Accuracy  
19:31:34   61       254      190      204      59.0       190.00     1h17m        1h18m        -0.6%
  • Good accuracy: ±5% error or better
  • Reasonable tau: Visible in debug output (typically 1-6 hours for BBQ)
  • Stable velocity: Not wildly fluctuating between measurements

Advanced Customization

Custom Grill Profiles

You can create cooking-style-specific profiles by modifying the grill dynamics factors:

// Competition BBQ profile (conservative)
grillFactors := []float64{0.05, 0.1, 0.3, 0.2}

// Fast grilling profile (aggressive)  
grillFactors := []float64{0.2, 0.4, 0.7, 0.5}

Environmental Factors

Consider adding parameters for:

  • Ambient temperature effects
  • Wind/weather conditions
  • Meat wrapping (Texas crutch)
  • Probe placement (thickness dependency)

The exponential prediction system provides a solid physics-based foundation that can be tuned for your specific cooking environment and style.