Degree Name

Doctor of Philosophy


Department of Mathematics


This thesis develops the solution of new time-dependent mathematical models for the operation of furnaces involved in the heat treatment and annealing of steel strip. In a practical context, it is necessary to the variation of the strip temperature throughout each furnace, and most importantly, the strip temperature at the exit of the furnace. Also of practical relevance is the change in temperature (if any) across the width of the the direct-fired furnace, particularly at the edges. Any significant differences between the actual temperature of the steel strip and the desired temperature can have adverse effects on the metallurgical qualities of the steel, and in a final product that does not meet specified requirements.

In each case, the models developed suit the type and geometry of the particular furnace involved. For example, the models for the direct-fired furnace are designed to be run off-line, and the radiant tube furnace model is designed to run in an on-line capacity, so greater liberty is taken with some of the assumptions used. Mathematical equations are developed, or modified and improved, to model the heat transfer between the steel strip, the combusted gas mixture and the highly insulated furnace walls. Methods of solution are derived which are appropriate to each furnace situation, numerical techniques are used in the solution process due to the non-linear form of the equations that are solved. Results show that the off-line direct-fired furnace model works well, although further testing against data is required. The present work extends the existing steady-state model in order to examine the occurrence of transients in the furnace. The on-line radiant tube furnace model uses a different approach to previous models of radiant tube-type furnaces. The present model is extremely rapid and, considering the assumptions that are made, gives very good results when tested against actual data from the furnace. Finally, the new work on the analysis of the temperature variation across the width of the strip in the direct-fired furnace provides some interesting results. In particular, the edges are always hotter than the rest of the strip during steady-state conditions, and the turn-around roll at the base of the furnace has a large influence on the transverse profile, especially when the strip width changes.

The practical aim of the thesis is to install devices that are able to accurately model the strip temperature variation during the heating process, in order to ensure that all of the steel that passes through the furnace is treated to specified requirements and is therefore not wasted due to incorrect treatment. All of the work included in this thesis applies existing mathematical techniques to real industrial problems. The work undertaken here will reduce the amount of improperly treated steel produced, and save costs resulting from this wastage.