The restrictions of gas temperature uniformity imposed in the stirred-reactor model and of a one-dimensional temperature field and absence of axial radiation in the plug flow model severely limit the usefulness of these models when applied to furnaces where there are significant gradients in temperature. Allowance for gradients in temperature, and other properties, may be made using the multizone model to be described in this section. Furthermore, the model may be used to examine the effect on heat flux distribution of complex factors such as recirculatory flow and combustion pattern.

A. Basis of model

The volume of the furnace chamber is subdivided (conceptually) into small zones in which the temperature, composition, and other physical properties can be assumed to be uniform. Similarly, the surfaces within the furnace chamber are subdivided into small surface zones that are assumed to have uniform temperature and emissivity and over which the incident and leaving fluxes can be considered to be uniform. The surfaces are assumed to be gray and to emit and reflect radiation diffusely. The geometric arrangement of zones is determined by the furnace shape and the ease with which radiation transfer factors between zones can be evaluated.

The stirred-reactor furnace model is a special case of the multizone model with one gas zone and two surface zones. This can be extended to a longitudinal series of well-stirred zones forming the so-called long furnace zone model. In this model combustion gases and process material can flow between the zones. There is a stepwise variation in gas, refractory and heat sink temperatures and heat fluxes. In its simplest form, axial radiation transfer between well-stirred zones is ignored. In some models, axial radiation is allowed for by considering just the interchange across the boundaries between adjacent gas zones. To include axial radiation, it is more usual however, to allow for the complete interchange of radiation between all surface and gas zones representing the furnace enclosure. When temperature gradients across the furnace need to be modelled, then the full multizone furnace model needs to be implemented and various models have been developed including two-dimensional cylindrical zone models to three dimensional models with rectangular zoning.

B. Model formulation