Implicit Newton-Krylov Methods for Modeling Blast Furnace Stoves

In this paper we discuss the use of an implicit Newton-Krylov method to solve a set of partial diierential equations representing a physical model of a blast furnace stove. The blast furnace stove is an integral part of the iron making process in the steel industry. These stoves are used to heat air which is then used in the blast furnace to chemically reduce iron ore to iron metal. The simulation of the stove's behavior is the rst step in a program to reduce the cost of operating these stoves by minimizing the natural gas consumption during the heating cycle, while still maintaining a high enough output air temperature in the cooling cycle to drive the needed chemical reaction in the blast furnace. The formulation and solution of this optimal control problem will also be discussed. The solution technique used to solve the discrete representations of the model and control PDE's must be robust to linear systems with disparate eigenvalues, and must convergence rapidly without using tuning parameters. The disparity in eigenvalues is created by the diierent time scales for convection in the gas, and conduction in the brick; combined with a diierence between the scaling of the model and control PDE's. A preconditioned implicit Newton-Krylov solution technique was employed. The procedure employs Newton's method, where the update to the current solution at each stage is computed by solving a linear system. This linear system is obtained by linearizing the discrete approximation to the PDE's, using a numerical approximation for the Jacobian of the discretized system. This linear system is then solved for the needed update using a preconditioned Krylov subspace projection method.