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Show 3 DESCRIPTION OF THE MATHEMATICAL MODEL A physical model of a LIF is shown in Fig. 1. Suffice it to note that the thermal system model of the furnace involves integration of the mathematical models of the furnace enclosure, load, furnace structural components and flat radiant heaters. The model is illustrated schematically in Fig. 2 and consists mainly of three parts: 1) the furnace enclosure model, 2) the load model, and 3) the wall model. The mrnace enclosure model describes the furnace space and the furnace walls which include the FRHs. The model calculates the heat transfer and temperature distribution in the enclosure, the walls and the furnace gas. Once the heat fluxes at the load surface have been determined, the load model calculates the transient temperature distribution in the load. The furnace enclosure model is coupled to the load model through the energy balances at the interface. A discussion of the submodels comprising the furnace enclosure, the load models and their integration into a thermal system simulation model is presented below. Furnace Enclosure A complete thermal systems mathematical model of a batch furnace operation needs to account for all the sources of energy and modes of heat transfer within the furnace. The different modes of heat transfer in a batch furnace are shown in Fig. 1(b). This paper presents a simplified mathematical model wherein the flat radiant heater model has been eliminated from the complete system model by specifying an effective heat flux or temperature on the outer surface of the heater. This specified heat flux is essentially a function of the fuel firing rate within the heater [2]. In industrial furnaces such as LIF, radiation is the dominant mode of heat transfer [4]. Therefore, radiation is treated in greater detail than convection. The total (convective plus radiative) local heat flux at any point P(r) on an internal wall of the furnace, heater or load surface can be expressed as q(r) = qc(r) + qf(r) = h(r)[l (r) - T ] + j"~ q, x(r)dX (1) where h is the local convective heat transfer coefficient, and Tg is the gas temperature inside the furnace. Modeling of convective heat transfer will be discussed later. Presently, furnace thermal performance calculations are being done by assuming that sources, furnace walls, structural elements and load are spectral emitters and reflectors of radiation [3]. The analysis accounts for the spectral nature of thermal radiation and formulates the problem in detail. The need for approximation to calculate total radiant heat fluxes is assessed by making band calculations. The enclosure model and its submodels and the load model are discussed in the following. Furnace Enclosure Model The refractory walls, heaters and load form an enclosure. Since radiation is transported at the velocity of light, radiation exchange can be considered quasi-steady. The gases filling the enclosure are assumed to be radiatively non-participating. For analysis of radiation heat exchange among surfaces, the surface areas are subdivided into an arbitrary number of smaller zones which are assumed to be isothermal. Radiation and total energy balances are applied to each zone to determine the temperatures and heat fluxes at each zone. Radiation Heat Exchange. The zoning procedure and the model simplifying assumptions are described below. The furnace wall (including heaters and load) is divided into a number of arbitrary, finite size zones. The main assumptions and features of the models are: • The outer surface of the flat radiant heater is considered to be isothermal. This idealization is made because presently information is lacking concerning the temperature variation over the |
