Numerical Reconstruction of the Catalyst Bed Temperature Distribution in a Multitubular Fixed-Bed Reactor by Karhunen–Loève Expansion

A Karhunen–Loève (K–L) expansion has been utilized to numerically reconstruct the bed temperatures in different tubes of a cross-flow multitubular reactor for the partial oxidation of propylene to acrolein. The K–L expansion was performed in a two-step mode on a data set consisting of the catalyst bed temperature distribution under different operating conditions. Eigenvectors, as well as the corresponding eigenvalues and coefficients, were obtained in each step. Statistical analyses were performed on the coefficients, and the cumulative variance was used as an indication of accuracy of the reconstruction. The two-step K–L expansion, with three terms in the first step and one term in the second step, can well reconstruct the catalyst bed temperature distribution. Thus, the catalyst bed temperature distribution can be represented by three coefficients of the K–L expansion, which can be further correlated with operating variables to establish simple models for process modeling, monitoring, and control.