# Indirect Measurement of Variables in a Heterogeneous Reaction for Biodiesel Production

## TL;DR

This paper develops a method to indirectly measure key variables in biodiesel production using a state observer with a soybean oil and biochar catalyst reaction.

## Contribution

The novelty lies in applying pole placement and LQR methods to estimate reaction variables in a heterogeneous biodiesel process.

## Key findings

- The observer estimates conversion rate and temperature with fast convergence under initial condition variations.
- Indirect measurements of esters, alcohol, and byproducts are achieved using the observer's output.
- Both pole placement and LQR methods show effective performance with error indexes under 30% variation.

## Abstract

This research focuses on the development of a state observer for performing indirect measurements of the main variables involved in the soybean oil transesterification reaction with a guishe biochar-based heterogeneous catalyst; the studied reaction takes place in a batch reactor. The mathematical model required for the observer design includes the triglycerides’ conversion rate, and the reaction temperature. Since these variables are represented by nonlinear differential equations, the model is linearized around an operation point; after that, the pole placement and linear quadratic regulator (LQR) methods are considered for calculating the observer gain vector L(x). Then, the estimation of the conversion rate and the reaction temperature provided by the observer are used to indirectly measure other variables such as esters, alcohol, and byproducts. The observer performance is evaluated with three error indexes considering initial condition variations up to 30%. With both methods, a fast convergence (less than 3 h in the worst case) of the observer is remarked.