Development of stepwise thermal model for annealing line heating furnace

Abstract

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

Two-dimensional Temperature Distribution of Strip in Preheating Furnace of Continuous Annealing Line

“…The convective heat transfer coefficients h i are calculated using the Nusselt number. The same models of natural and forced convections as represented in the literature [4] are adopted in the present study. Expression (10), proposed by McAdams [12], is considered for the natural convection in the horizontal heated surfaces facing upwards, as in the ceiling of the chamber, to calculate the heat transfer losses to the surrounding furnace, and also for the cooled horizontal surfaces facing downwards, as in the ceiling of the chamber, to calculate the convective heat transfer rate.…”

“…The temperatures of the atmosphere at each zone are also assumed to be distributed Table 2. Properties of atmosphere [4] Property Function Range q, kg=m 325.38=T 400 < T(K) < 1000 c p , J=kg  K 1.096  10 À4 T 2 þ 5.499  10 À2 T þ 1054.84 400 < T(K) < 1000 k, W=m  K 6.358  10 À5 T þ 1.299  10 À2 400 < T(K) < 1000 m, Pa  s 2.966  10 À8 T þ 1.011  10 À5 400 < T(K) < 1000 uniformly. They generally can be measured by experiment.…”

“…In their models, the effect of convective cooling on the strip was introduced via a constant heat transfer coefficient. Prieto et al [4] have developed a powerful stepwise thermal model to evaluate the one-dimensional temperature distributions for the annealing line. In their model, the furnace was divided into several enclosures in which the overall heat balances were conducted.…”

Temperature Distribution of Taper Rolls in Preheating Furnace of Cold Rolling Continuous Annealing Line

“…In their models, the effect of convective cooling on the strip was introduced via a constant heat transfer coefficient. Prieto et al [4,5] developed a powerful stepwise thermal model for annealing line. In their model the furnace was divided into several enclosures in which overall heat balances were conducted.…”

Overall model of the dynamic behaviour of the steel strip in an annealing heating furnace on a hot-dip galvanizing line

“…In recent times, importance is being given to the modelling of the behaviour of the steel strip in order to improve the control of the annealing process in an HDGL. Thus, in Prieto et al [5] a stepwise mathematical model is reported that allows determining the temperature of the strip based on both its and the furnace's mathematical characterisation, and which considers the phenomena of conduction, convection and radiation existing in the furnace and also present between it and the steel strip.…”