With increasing competition in the steel industry, the survival of a steel plant calls for optimal, efficient operation that generates high-quality products at globally competitive prices. Traditionally, cost-control exercises have focused on individual cost drivers such as productivity, energy, quality, yield and emissions. Because all of these metrics are interrelated, concentrating on any one item often results in sub-optimal operation.

Cost modeling, by comparison, provides a unified approach to analyze and optimize the performance of industrial processes. The broad scope of cost modeling is to correlate the process parameters (product mix, temperature set points, discharge rate and fuel and air flow-rates, for example) to the plant performance metrics (e.g., energy consumption, yield and quality) and, in turn, relate them to the overall process cost. The cost model can then be used to optimize the process parameters to arrive at the lowest operating cost.

In the past, cost modeling has been used as a methodology to quantify the merits of and to choose between various product designs[1] and technology choices[2]. The application of cost modeling for process optimization can be a very powerful tool for cost reduction in a process industry. This article elaborates on the concept of cost modeling and its usefulness for process optimization via a case study of a steel-reheating operation. Specifically, it presents the development of the cost model and how it is used to determine the optimum discharge rate in this industrial application.

## Cost model development

In an industrial steel-reheating operation, important process parameters amenable to control include temperature and air-to-fuel ratio in various control zones, fuel and air inlet temperatures, furnace pressure and discharge rate. Traditionally, fuel consumption was considered to be the major cost driver for this operation and, as a result, most furnace-optimization activities were focused toward minimizing fuel cost.

However, with increasing competition in the industry, every contributor to the process cost must be considered. In a reheating operation, such cost contributors include scale loss and cost of low quality in terms of rejects, rework or product downgrading. In addition, with increasingly stringent environmental regulations and heavy penalties for nonconformance, furnace emissions is becoming a significant cost driver and may become the most important constraint in coming years.

Only the cost drivers amenable to short-term control have been considered in this study. Long-term cost drivers such as capital cost, labor cost and depreciation-factors that are not amenable to short-term control-are assumed to be constant. Also, the overall cost considered here is the sum of the cost contributions of the major cost drivers amenable to short-term control.

This cannot be considered as the real world cost of heating a billet. However, it does provide a work function amenable to optimization because the changes in this cost with respect to the controllable parameters are important rather than the absolute value.

Figure 1 shows the overall process optimization methodology using a cost-modeling approach. The product mix together with the controllable parameters, such as the discharge rate and furnace set points, form the inputs to process models that predict the values of various cost drivers.

The underlying methodology involves:

- Obtaining values of cost drivers from the parameters that affect the process (that is, input parameters and controlled parameters) using a mathematical representation of the real process (process models)

- Converting the values of individual cost drivers into a consolidated cost using cost functions

- Optimizing the controlled parameters using the consolidated cost with cost minimization as the objective

The network of process models required to optimize the steel-reheating operation is illustrated in Fig. 2. The energy model calculates the furnace fuel consumption by conducting an overall heat balance of the furnace based on material throughput, losses from the furnace and flue-gas losses. Using the furnace temperature profile obtained from the furnace temperature model, the slab-temperature model tracks the spatial and temporal temperature of individual slabs traversing through the furnace. This module consists of a transient slab-conduction solver coupled with a three-dimensional (3-D) furnace-radiation solver. The surface temperature profiles of slabs are made available to a coupled oxidation and diffusion kinetics model, which estimates the scale loss and decarburization of the slabs as they traverse the length of the furnace. The scale loss and percentage of decarburization are measures of yield reduction and product quality, respectively. Other relevant quality parameters, such as target temperature and temperature uniformity (the difference between the maximum and minimum temperature within the billet), are obtained from the slab-temperature model.

The combustion model provides the quantity and concentration of the products of combustion through a stoichiometric balance. As shown in Fig. 2, it is possible to quantify all relevant cost drivers for the steel-reheating operation using this approach. Because an object-oriented approach has been used to develop these process models, it is possible to add or upgrade individual models to the desired level of sophistication without changing the overall architecture.

The Cost Functions module converts the cost drivers obtained from the process models to their respective costs via Taguchi-type[3] cost functions, which incorporate the constraints for the optimization problem through cost penalties arising from rework, downgrading, rejections, etc. The total cost is the sum of the cost contributions of individual cost drivers, and forms the objective function for the optimization problem. The optimization problem is solved using an optimizer that calls the process models and the Cost Functions module in a closed loop and finds the values of the controllable parameters that result in the minimum overall cost.

Due to the prevailing market-driven operation, the product mix (grades, dimensions, lot sizes and production sequence) is considered to be an input to the cost model. In a product-driven operation, the same methodology can be applied by assuming the product mix to be variable to arrive at an optimum product mix.

## Cost modeling applications

The cost model, validated and tested using plant data, can be used in a variety of applications aimed toward macroscopic goals such as overall cost reduction, productivity enhancement, energy reduction, quality improvement and reduction in variation. The cost model can effectively be used as an off-line process-optimization tool or as a decision-making component of an on-line reheating control system. In either of these applications, the controllable parameters, such as temperature set points in various zones, air-to-fuel ratio and discharge rates will simultaneously be optimized to achieve minimum process cost.
**Billet reheating-furnace case study**

The reheating-furnace cost model was used to optimize the discharge rate (in metric tons per hour, or tph) of 130-mm square (5-in. square) billets being heated to 1140C (2085F) in a 45-tph capacity furnace, while keeping the temperature set points and the air-to-fuel ratio constant. Decarburization and temperature-uniformity limits for the billet were assumed to be 0.8% and 50C (90F), respectively. Because the set points and the air-to-fuel ratio were kept constant, emission levels also were expected to remain constant and, therefore, were not considered to be a part of the work function.

Figures 3a and 3b show the temperature profile, scale and decarburization of the billets as they traverse the length of the furnace.

The cost drivers plotted in Fig. 4 are converted into process costs via suitable Taguchi-type cost functions. The combined effect of discharge rate on all costs, as well as the overall process cost, is shown in Fig. 5. Fuel and scale costs decrease monotonically with discharge rate. A high discharge temperature or high temperature variation within the billet translates into very high costs as it can result in mill stoppages due to cobbles and roll damage. High decarburization results in a moderate cost due to the subsequent downgrading or reworking of the product. A low discharge temperature results in a higher cost because such a billet is rejected by the mill and has to be reheated again, resulting not only in loss of time but also in further scaling and decarburization. A more uniformly heated billet having low decarburization and discharge temperature being within acceptable limits do not add significantly to the process cost.

An optimum discharge rate is obtained at approximately 44 tph, which results in proper heating of the billets with decarburization and uniformity within the specified limits, and with nominal scale loss and fuel consumption.

## Summary

Optimization of industrial processes requires simultaneous consideration of all cost-driving, plant-performance metrics under a common framework. Cost modeling provides a unified approach to analyze the performance of industrial processes and optimize them. Cost drivers considered in modeling an industrial steel-reheating furnace are fuel consumption, discharge temperature, uniformity, scale loss, and decarburization, while the discharge rate is the controlled variable. Cost drivers are related to process costs via suitable cost functions and the overall cost has been minimized to estimate the optimum discharge rate. In a low discharge rate regime, mill stoppages and rejections due to high discharge temperature and rework due to higher decarburization dominate the overall cost. In contrast, in a high discharge rate regime, mill stoppages and rejections due to poor temperature uniformity and rework due to low discharge temperature play a dominating role. The optimum discharge rate that results in minimum cost provides proper heating of billets within specified quality limits with nominal scale loss and fuel consumption.

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