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Knowledge of the steel strip temperature throughout the length of the annealing line continuous heating furnaces is important to guarantee the physical properties of the metal produced. Given that online temperature measurement is difficult, a stepwise model has been developed that considers the furnace as being divided up into enclosures. This model is easy to implement and has a short run time. The steel strip is considered to be subject to convection and radiation mechanisms, and surface and overall heat balances are conducted in the enclosures. The model has been applied to the conditions of an industrial furnace. The following values have been obtained throughout the length of the furnace: strip and wall temperatures; distribution of convective and radiative heat transfer rates in the strip; and heat transfer rate losses through the walls. The overall heat balances are conveniently fulfilled and the solution is obtained in only a few iterations. List of symbolsA surface area, m 2 c p specific heat at constant pressure, J kg 21 K 21 e strip thickness, m F view factor g gravitational acceleration, m s 22 G Gebhart factors h heat transfer coefficient, W m 22 K 21 L characteristic length, m n number of surfaces in enclosure Nu Nusselt number (5hL/l) Pr Prandtl number (5mc p /l) Q heat transfer rate, W R radiative exchange factors for application of Gebhart method Ra Rayleigh number (5bDTgL 3 r 2 c p /ml) Re Reynolds number (5Lru/m) t material thickness, m T temperature, K u atmosphere velocity, m s 21 v strip velocity, m s 21 w strip width, m X N 2 , X H 2 molar fraction b coefficient of thermal expansion, K 21 d Kronecker delta e emissivity l thermal conductivity, W m 21 K -1 m dynamic viscosity, Pa s r density, kg m 23 s Stefan-Boltzmann constant Y function in equation (18) Subscripts ce ceiling conv convection mechanism hp heating plane hp-l to indicate heat transfer rate from heating plane to enclosure placed on left hp-r to indicate heat transfer rate from heating plane to enclosure placed on right i,j generic surface index in refers to enclosure inlet conditions loss losses through walls lw lateral walls out refers to enclosure outlet conditions rad radiation mechanism s-a to indicate heat transfer rate from strip to enclosure located above s-b to indicate heat transfer rate from strip to enclosure located below s-l to indicate heat transfer rate from strip to enclosure located on left s-r to indicate heat transfer rate from strip to enclosure located on right s1 strip in vertical position s2 strip in horizontal position w walls ws inner surfaces of walls 's thermal conditions in surroundings of furnace 'en thermal conditions in enclosure atmosphere of furnace 1, 2 to indicate material in Table 1, 2

In the present study, a stepwise model based on the enclosure concept has been applied to an annealing line heating furnace. The model has been satisfactorily tested using three industrial manufacturing data sets. As temperature measurement inside the furnace is difficult, the model could be used to improve control and to obtain the outlet temperature of the steel strip, the heat transfer rate loss and the strip heat transfer rate throughout the length of the furnace. Variations in the thermodynamic properties included in the model and in the operational conditions, which cannot be accurately known, have been tested to ascertain their effects on the evolution of the strip temperature. It is found that precise knowledge of the heat capacity and heating power introduced in the furnace are important to obtain good results in application of the model. List of symbolsc p specific heat capacity, J kg 21 K 21 e thickness, m MD manufacturing data set Q heat transfer rate, W T temperature, K v velocity, m s 21 w width, m e emissivity r density, kg m 23 D difference

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