Note

This tutorial was generated from a Jupyter notebook that can be accessed here.

Handling source terms

This tutorial can be of interest to researchers working with reactive flows data sets. We present how source terms of the original state variables can be handled using PCAfold software. Specifically, PCAfold functionalities accommodate treatment of sources of principal components (PCs) which can be valuable for implementing PC-transport approaches such as proposed in [SP09].

Theory

The methodology for the standard PC-transport approach was first proposed in [SP09]. As an illustrative example, PC-transport equations adequate to a 0D chemical reactor are presented below. The reader is referred to [BS15], [EM15] for treatment of the full PC-transport equations including diffusion.

We assume that the data set containing original state-space variables is:

\[\mathbf{X} = [T, Y_1, Y_2, \dots, Y_{N_s-1}]\]

where \(T\) is temperature and \(Y_i\) is a mass fraction of species \(i\). \(N_s\) is the total number of chemical species. \(\mathbf{X}\) is also referred to as the state vector, see [HS18] for various definitions of the state vector. The corresponding source terms of the original state-space variables are:

\[\mathbf{S_X} = \Big[-\frac{1}{\rho c_p} \sum_{i=1}^{N_s} ( \omega_i h_i ), \frac{\omega_1}{\rho}, \frac{\omega_2}{\rho}, \dots, \frac{\omega_{N_s-1}}{\rho} \Big]\]

where \(\rho\) is the density of the mixture and \(c_p\) is the specific heat capacity of the mixture, \(\omega_i\) is the net mass production rate of species \(i\) and \(h_i\) is the enthalpy of species \(i\).

For a 0D-system, we can write the evolution equation as:

\[\frac{d \mathbf{X}}{dt} = \mathbf{S_X}\]

This equation can be instead written in the space of principal components by applying a linear operator, \(\mathbf{A}\), identified by PCA. We can also account for centering and scaling the original data set, \(\mathbf{X}\), using centers \(\mathbf{C}\) and scales \(\mathbf{D}\):

\[\frac{d \Big( \frac{\mathbf{X} - \mathbf{C}}{\mathbf{D}} \Big) \mathbf{A}}{dt} = \frac{\mathbf{S_X}}{\mathbf{D}}\mathbf{A}\]

It is worth noting that when the original data set is centered and scaled, the corresponding source terms should only be scaled and not centered, since:

\[\frac{d \frac{\mathbf{C}}{\mathbf{D}} \mathbf{A}}{dt} = 0\]

for constant \(\mathbf{C}\), \(\mathbf{D}\) and \(\mathbf{A}\).

We finally obtain the 0D PC-transport equation where the evolved variables are principal components instead of the original state-space variables:

\[\frac{d \mathbf{Z}}{dt} = \mathbf{S_{Z}}\]

where \(\mathbf{Z} = \Big( \frac{\mathbf{X} - \mathbf{C}}{\mathbf{D}} \Big) \mathbf{A}\) and \(\mathbf{S_{Z}} = \frac{\mathbf{S_X}}{\mathbf{D}}\mathbf{A}\).

Code implementation

We import the necessary modules:

from PCAfold import PCA
import numpy as np

A data set representing combustion of syngas in air generated from steady laminar flamelet model using Spitfire software [HAS+22] and a chemical mechanism by Hawkes et al. [HSSC07] is used as a demo data set.

We begin by importing the data set composed of the original state space variables, \(\mathbf{X}\), and the corresponding source terms, \(\mathbf{S_X}\):

X = np.genfromtxt('data-state-space.csv', delimiter=',')
S_X = np.genfromtxt('data-state-space-sources.csv', delimiter=',')

We perform PCA on the original data:

pca_X = PCA(X, scaling='auto', n_components=2)

We transform the original data set to the newly identified basis and compute the principal components (PCs), \(\mathbf{Z}\):

Z = pca_X.transform(X, nocenter=False)

Transform the source terms to the newly identified basis and compute the sources of principal components, \(\mathbf{S_Z}\):

S_Z = pca_X.transform(S_X, nocenter=True)

Note that we set the flag nocenter=True which is a specific setting that should be applied when transforming source terms. With that setting, only scales \(\mathbf{D}\) will be applied when transforming \(\mathbf{S_X}\) to the new basis defined by \(\mathbf{A}\) and thus the transformation will be consistent with the discussion presented in the previous section.