Computers and Chemical Engineering 33 (2009) 1727–1734 Contents lists available at ScienceDirect Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng The solution of very large non-linear algebraic systems Davide Manca ∗ , Guido Buzzi-Ferraris, Alberto Cuoci, Alessio Frassoldati Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Piazza Leonardo da Vinci 32, 20133 Milano, Italy article info Article history: Received 15 October 2008 Received in revised form 13 March 2009 Accepted 28 April 2009 Available online 5 May 2009 Keywords: Applied numerical analysis Very large non-linear algebraic systems Mixed CFD and detailed kinetics abstract The manuscript discusses the feasibility and the methods for solving systems of non-linear algebraic equations, the main numerical subjects being convergence tests, stop criteria, and expedients for large and sparse systems. After a detailed discussion on the features of large non-linear systems, the paper focuses on the numerical simulation of complex combustion devices and on the formation of macro- and micro-pollutants. This quantification is not possible by simply introducing a detailed kinetic scheme into a fluid dynamics (CFD) code, especially when considering turbulent flows. Actually, the resulting problem would reach a so huge dimension that is still in orders of magnitude larger than the feasible one (by means of modern computing devices). To overcome this obstacle it is possible to implement a separate and dedicated kinetic post-processor (KPP) that, starting from the CFD output data, allows simulating numerically the turbulent reactive systems by means of a detailed kinetic scheme. The resulting numerical problem consists of a very large, non-linear algebraic system comprising a few millions of unknowns and equations. The manuscript describes the KPP organization and structure as well as the numerical challenges and difficulties that one has to overcome to get the final numerical solution. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Nowadays, there is a big interest in the solution of non-linear algebraic systems of large dimensions (i.e. millions of equations with millions of unknowns), because of the complexity of simu- lation problems in several branches of science. This interest is also supported by the computational power of available computers and, more specifically, of common processors and personal worksta- tions. This manuscript does not focus on parallel programming and does not try to exploit the multiple-core features of present pro- cessors. However, the suggestions proposed in this paper can be implemented directly on both multi-core processors and parallel computers, in order to improve the performance of the discussed numerical procedures. The manuscript begins with the discussion of some open-issues on the numerical solution of non-linear algebraic systems. Even- tually, it focuses on a rather complex and quite large problem concerning the numerical modeling of a turbulent jet flame. The research interest is addressed to the evaluation of both macro- and micro-pollutants, by coupling CFD and detailed kinetics. Modern CFD codes can work with a reduced number of chemical species when simulating complex combustion units such as burn- ers, furnaces and kilns. A reduced number of species means that only a rather simplified kinetics can be accounted for. Conversely, if ∗ Corresponding author. Tel.: +39 02 23993271; fax: +39 02 70638173. E-mail address: davide.manca@polimi.it (D. Manca). one wants to quantify the formation of macro- and micro-pollutants (such as CO, NOx, SOx, PAHs and soot) in a combustion cham- ber, then a detailed kinetic scheme is mandatory. By working with detailed kinetics, it is possible to optimize the burner geometry while assessing both its impact on the environment and its ther- modynamic efficiency. Unfortunately, a detailed kinetic scheme usually comprises hun- dreds of chemical species (molecular and radical species) and thousands of elementary direct and inverse reactions. Conse- quently, the direct coupling of a CFD code with a detailed kinetic scheme is still unfeasible within any program and any available computing device being either a personal computer or a worksta- tion. This is due to the overall dimensions of the resulting numerical problem and, as a consequence, to the CPU time required. It is quite simple to get an idea of the overall dimensions of the numerical problem since they are proportional to the product between the number of computational cells (N C ), which discretize the equipment, and the number of chemical species (N S ), along with a few additional variables (e.g. temperature, pressure, and momen- tum). Consequently, it would be quite common to have some tens of millions of variables describing the coupled CFD and kinetic problem. Nowadays, commercial CFD codes cannot solve such large problems. To overcome this obstacle, the manuscript describes a kinetic post-processor (KPP) that, starting from the CFD output data produced by a well-known commercial program, simulates the combustion in a burner by means of a detailed kinetic scheme. The numerical case study is based on the combustion of syngas, which is a mixture of hydrogen and carbon monoxide that can be 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.04.010