Modeling of heat transfer processes for controls applications requires that the modeler make a set of assumptions that strikes a balance between accuracy and simulation run time. Common points of consideration include choosing the appropriate dimension of the model, the importance of transient effects, numerical and temporal discretization, etc. The results of these decisions directly impact the speed and accuracy of the model. It can be useful for the modeler to build a framework containing a set of models for a given system where each model contains a different set assumptions. This allows for easy comparison of different classes of models and aids in the decision process for selecting the model that best fits the needs of the user.
This work takes the modeling of an open-flame hearth furnace as a case study and examines the implications of different modeling simplifications. This scenario is similar to plug-flow reactors (PFRs) and continuous stirred-tank reactors (CSTRs) in that reactions are occurring in a chamber where mass is flowing in to and out of the system. Modeling of these systems focuses on the concentration of the reactants, whereas the hearth furnace model seeks to capture the heat transfer to the work-piece and walls. Numerous models of PRFs and CSTRs exist in literature as well as several open-fire furnace models using Hottel’s zone method. This work builds off of these modeling techniques by testing several types of models for speed and accuracy.
Both steady-state and transient operation are considered and each of the different heat transfer phenomena are modeled in varying dimensions. The open-flame environment inside the furnace produces gases that participate in radiative heat transfer for which there are a number of different models. The P1 approximation to the method of spherical harmonics is used, both in three dimensions and a simplification to one dimensional radiation. The flow of gases inside the furnace is considered to be plug flow with a series of burners injecting hot gases along the length of the furnace. Conduction to the walls and work-piece occurs at much longer time scales than convection, i.e. the combustion gases are moving through the furnace much faster than the characteristic thermal response time of the walls and work-pieces (τgas ≪ τwall). This brings the modeler to the question of how to model transient effects at such disparate time scales. A model is posited that is a combination of quasi-steady radiative heat transfer, transient fluid evolution, and lumped transient conduction. This formulation takes advantage of the disparity in time scales to take large steps in time when solving the radiation equation while solving the energy equation at each time step for the internal gas control volumes. This model is then compared on the grounds of speed and accuracy to the more traditional explicit advancement of a consistent time step in both the gas energy balance and radiative transfer solution.
