Any Progress on inverse pyrolysis modelling

with ensemble learning methods?

Patrick Lauer
University of Wuppertal

Content

What is pyrolysis?

Burning of solids can be separated in two phases:

  • Thermochemical decomposition of solid material and phase change from solid to gas phase (Pyrolysis)
  • Chemical reaction in the gas phase (Combustion)

To predict fire spread, we need to model burning of solids, hence pyrolysis.

Pyrolysis

pyro


Figure 1. [Schematic of pyrolysis [10]]

How do we model pyrolysis?

Boundary condition

(1)
\[-k_{s,1} \frac{T_{s,1}^{n+1}-T_{s,0}^{n+1}}{\delta x_{\frac{1}{2}}}=\dot{q}_c'' + \dot{q}_r'' \]

Heat conduction

(2)
\[\rho_s c_s \frac{\partial T_s}{\partial t} = \frac{\partial}{\partial x} (k_s \frac{\partial T_s}{\partial x})+\dot{q}_s''' \]

Reaction rate:

(3)
\[r = A Y^n \cdot e^{-\frac{E_a}{RT}} \]

Parameter overview

Parameter
Activation energie $E_a$
Pre-exponential factor $A$
Reaction order $n$
Density $\rho$
Conduction coefficient $k$
Heat capacity $c$
Emissivity $\epsilon$
Heat of reaction $\Delta H$

How do we get these parameters?

Find parameters with small scale experiments and mathematical fitting, scale up to parts and devices

Usual experiments:

  • Thermogravimetrical analysis
  • Cone calorimeter
  • Micro combustion calorimeter

Exkurs: Thermogravimetrical analysis (TGA)

A small specimen (mg scale) gets heated in a furnace with a constant or transient heating rate. Mass loss of the specimen is captured. It allows to estimate reaction kinetics of this sample.

tga


Figure 2. TGA

Approaches

  • Forwad fitting
    • Basic graphical fitting [4, 5, 8]
    • Advanced automated fitting [3]
  • Inverse modeling [9]
    • Optimization algorithms [1, 7]
    • Machine learning

Forests of randomized trees

Common Methods:

  • Random Forests [2]
  • Extremly Randomized Trees [6]

Concept

  • Ensemble learning methods

Advantages

  • Efficient for big data sets
  • Fast to train
  • Easy to parallelize

Method

  • Train a model to predict reaction kinetic parameters with given reaction rate
  • Case study: mockup TGA experiment with constant heating rate
  • All data used is randomly generated with the pyrolysis model

Method II

invers


Figure 3. Invers modelling then

Method III

invers2


Figure 4. Invers modelling now

Process I

  1. Generating sample data set with the regarding model
    • Samples for 1, 2 and 3 reactions, with 3 heating rates applied each
    • Up to 1M samples generated, with $r(T)$ and $A_n$, $E_n$

Exkurs: Sampling I

Problem:

  • Arrhenius equation is an exponential function
  • $A_n$ might be $10^{10}...10^{40}$
  • $E_n$ might be $10^{1}...10^{20}$
  • Hence we can't sample from a uniform distribution

Challenge:

  • Find a distribution to sample from

Exkurs: Sampling II

Solution:

  • Introducing $T_p$ (reference temperature), $\Delta T_p$ (reference range) and $r_p$ (reference rate), characterizing a triangle of temperature at maximum mass loss rate and width of the peak

peak


Figure 5. $T_p$, $\Delta T_p$, $r_p$

Exkurs: Sampling III

  • $T_p$ and $r_p$ are sampled from a uniform distribution and then mapped to $A_n$ and $E_n$
  • Mapping between $A_n$, $E_n$ and $T_p$, $\Delta T_p$, $r_p$ is done with these equations:
(4)
\[E_{i,1} = {\frac {er_{p,i}}{Y_{s,i}(0)}}{\frac{R{T}_{p,i}^2}{\dot T}} \]
(5)
\[A_{i,1} = {\frac {er_{p,i}}{Y_{s,i}(0)}} e^{\frac {E}{RT_{p,i}}} \]
(6)
\[\frac {r_{p,i}}{Y_{s,i}(0)}={\frac{2 \dot{T}}{\Delta T}}(1-\nu_{s,i}) \]

Process II

  1. Splitting data set in two independent sets (75 % training data set and 25 % validation data set)
  2. Train model with training data set
    • Input: $r(T)_{train}$
    • Output: $A_{n, train}$, $E_{n, train}$
    • Model adopts to transform input to output
  3. Validate trained model by feeding $r(T)_{prescribed}$ of validation data set and check for expected outcome

Process III

  1. Recalculate $r(T)_{predicted}$ with $A_n$, $E_n$, calculate RMSE between $r(T)_{validation}$ and $r(T)_{predicted}$

Process IV

  1. Evaluate
  2. Repeat with different algorithms and different hyperparameter settings

Results

for inverse replacement models with

  • 1 reaction, 3 heating rates,
  • 2 reactions, 3 heating rates and
  • 3 reactions, 3 heating rates

built with Extremly Randomized Trees algorithm

1 Reaction, 3 Heating rates

1r


Figure 6. Histogram of RMSE Test vs. Predicted Data (12.5k cases)

2 Reactions, 3 Heating rates

2r


Figure 7. Histogram of RMSE Test vs. Predicted Data (250k cases)

2 Reactions, 3 Heating rates II