Macroscopic Heat Conduction Formulation

Estimation of kinetic coefficients in micro-reactors for biodiesel synthesis: Bayesian inference with reduced mass transfer model

“…The three-dimensional partial differential model in Eq. (3) can be reformulated as a lumped-differential system through the Coupled Integral Equations Approach -CIEA (Aparecido and Cotta, 1989;Cotta and Mikhailov, 1997;Naveira et al, 2009;Sphaier et al, 2017;Kakaç et al, 2018). The purpose of the CIEA is to reduce the original mathematical model by eliminating one or more spatial variables through lumping operations in the corresponding coordinate directions.…”

“…Following the CIEA approach, firstly the lumping was performed in the transverse direction (Z coordinate), followed by the vertical direction (Y coordinate), considering the integrals that define the dimensionless average concentration and average mass flux of each species. The selected Hermite formulae for each pair of integrals are the approximations H 1,1 and H 0,0 , which correspond to the corrected trapezoidal and trapezoidal integration rules, respectively (Cotta and Mikhailov, 1997;Sphaier et al, 2017).…”

Hybrid Approach in Microscale Transport Phenomena: Application to Biodiesel Synthesis in Micro-reactors

“…Higher order approximations that provide a more informative relation between boundary and average potentials are thus quite useful in model reduction efforts. The Coupled Integral Equations Approach (CIEA) [14][15][16][17][18][19][20] is a systematic methodology for constructing improved lumped-differential formulations, based on approximating the integral expressions for the average potential and its associated flux through Hermite formulae for integration, which employ only the function and its spatial derivatives at the boundary points [21]. Then, more informative expressions relating average and boundary potentials are achieved, which are then incorporated in the averaged differential formulation.…”

“…The Coupled Integral Equations Approach (CIEA) [14][15][16][17][18][19][20] aims at reducing the number of independent variables in partial differential equations, via averaging (lumping) operations in the coordinates selected to be eliminated, but partially preserving the local information in these space variables through the corresponding boundary conditions. The use of Hermite formulae to approximate the averaging integrals [21] provides improved formulations when compared to the classical lumped system analysis [20], which is the simplest possible lumped approximation. The partial differential three-dimensional model shown in eqs.…”

“…The partial differential three-dimensional model shown in eqs. ( 1) is here reformulated using the CIEA technique, yielding a nonlinear coupled mathematical model composed only of first order ODEs that enables simulations with reduced computational costs [14][15][16][17][18][19][20].…”