Control of the ledge thickness in high-temperature metallurgical reactors using a virtual sensor

Abstract

A non-intrusive inverse heat transfer procedure for predicting the time-varying thickness of the phase change ledge on the inside surface of the walls of a high-temperature metallurgical reactor is presented. A Kalman filter, based on a state-space representation of the reactor, is coupled with a recursive least-square estimator in order to estimate online the position of the phase front. The data are collected by a heat flux sensor located inside or outside of the reactor wall. The inverse method, used here as a virtual sensor, is coupled to a classic proportional–integral controller in order to control the ledge thickness by regulating the air cooling applied on the outside surface of the reactor wall. The virtual sensor and the control strategy are thoroughly tested for typical phase change conditions that prevail inside industrial facilities. Results show that a virtual sensor that relies on a heat flux sensor embedded inside the reactor wall provides more accurate and stable information, but at a price of a more complicated installation. In that case, it is shown that the discrepancy between the exact and the estimated ledge thicknesses remains smaller than 3% at all times, and that the control strategy ensures a null steady-state tracking error, a maximum tracking error less than 10%, no overshoot and no significant time lag.

Keywords:

• ledge
• metallurgical reactor
• virtual sensor
• soft sensor
• inverse method
• control
• PI
• Kalman filter
• recursive least-square estimator

Nomenclature

A	=	FDM matrix
	=	state-space matrix
	=	state-space matrix
	=	state-space matrix
	=	specific heat (J kg−1 K−1)
	=	time step (s)
	=	space step (m)
	=	width of the PCM layer (m)
	=	tracking error (m)
	=	liquid fraction
	=	optimal Kalman gain
	=	RLSE correction factor
	=	heat transfer coefficient (W m−2 K−1)
	=	identity matrix
	=	thermal conductivity (W m−1 K−1) or sampling instant
	=	a priori estimate
	=	a posteriori estimate
	=	controller proportional gain
	=	width of the brick wall (m)
	=	maximum value
	=	RLSE sensitivity matrix
	=	initial RLSE sensitivity matrix
	=	total number of input/output data
	=	heat flux (W m−2)
	=	input heat flux imposed at the boundary x = L + D (W m−2)
	=	estimated input heat flux (W m−2)
	=	initial estimated input heat flux (W m−2)
	=	heat flux measurements (W m−2)
	=	covariance matrix of the estimated state
	=	initial covariance matrix of the estimated state
	=	covariance matrix of the estimated heat flux
	=	initial covariance matrix of the estimated heat flux
	=	covariance matrix of the process noise
	=	ledge thickness set point (m)
	=	covariance matrix of the measurement noise
	=	ledge thickness (m)
	=	estimated ledge thickness (m)
	=	residual covariance
	=	time (s)
	=	temperature (K)
	=	integral time constant (s)
	=	input signal vector
	=	estimated input signal vector
	=	measurement noise (W m−2)
	=	PEM algorithm cost function
	=	process noise
	=	RLSE sensitivity matrix
	=	state vector or space direction
	=	estimated state vector
	=	initial estimated state vector
	=	measurement vector (W m−2, m)
	=	measurement residual (W m−2)
	=	predicted output of the state-space model (W m−2, m)

	=	parameter related to RRMSE
	=	parameter related to RRMSE
	=	volumetric enthalpy change (J m−3)
	=	difference between the outputs of the FDM and the state-space model
	=	RLSE forgetting factor
	=	heat of fusion (J kg−1)
	=	density (kg m−3)
	=	process noise standard deviation (W m−2, m)
	=	measurement noise standard deviation (W m−2)

	=	initial or constant
	=	ambient
	=	brick wall
	=	liquidus
	=	liquid (PCM)
	=	solidus
	=	solid (PCM)

=

transposed matrix

1. Introduction

An interesting melting/solidification phenomenon that arises in high-temperature metallurgical reactors such as electric arc furnaces 1, blast furnaces 2 and aluminium electrolysis cells 3 is the formation of solid layers, sometimes called ledges or banks, on the inside surface of the reactor (Figure 1). This ledge is formed as the molten material comes into contact with the cooled surface of the wall and solidifies. The presence of the ledge is highly desirable. It serves as a protective barrier against the corrosive molten material, thereby maintaining the integrity of the cell and prolonging its active life 4,5. On the other hand, too thick a ledge is detrimental to the reactor as the active cell volume is reduced, and in the case of electrolysis cells, horizontal currents may be generated thereby disturbing the magneto-hydrodynamics stability of the cell and, as a result, decreasing the current efficiency 5,6. Maintaining a ledge of optimal size at all times is therefore crucial for the safe and profitable operation of the metallurgical reactor.

Figure 1. Cross view of a typical high-temperature metallurgical reactor; an aluminium electrolysis cell with molten material (electrolyte) and protective ledge.

Unfortunately, due to the hostile conditions that prevail inside the reactor, probing the time-varying thickness and shape of the ledge with sensors submerged into the molten electrolyte is a very difficult task, not to say impractical. The standard method for measuring the thickness of the ledge is to probe it manually 7. This method is time consuming and requires qualified personnel to perform the task. Also, the measurements are usually taken days apart and they require the opening of the cell which results in significant heat losses to the surroundings and the release of undesirable chemicals into the atmosphere 8.

The alternative is to attempt to predict the ledge profile with inverse heat transfer methods that rely on thermal measurements taken from sensors located in the reactor walls. This non-intrusive method can be seen as a virtual sensor, or soft sensor, defined as the combination of an estimation algorithm with measurements that allows to estimate on-line ‘hard to measure’ process variables 9.

The prediction of banks or ledges on reactor walls with inverse heat transfer methods has been the subject of few investigations in the past 1,10–14. Figure 2 shows such an inverse one-dimensional (1-D) problem where the ledge is represented by the solid phase of the phase change material (PCM). In the direct problem (Figure 2a), the time-varying heat flux at the right boundary is known. In the inverse problem (Figure 2b), the time-varying heat flux at the right boundary is unknown. It can be estimated, however, with an inverse method that uses heat flux sensors located inside or on the external surface of the brick wall. With the estimated heat flux , the ledge thickness can therefore be evaluated.

Figure 2. Schematic of a 1-D phase change problem inside a high-temperature metallurgical reactor. (a) Direct problem: and are known; , and are predicted with the finite difference numerical model. (b) Inverse problem: is unknown. It is determined from heat fluxes measurements taken inside or outside the brick wall. With this information, the ledge thickness is estimated.

Most of the proposed inverse calculation strategies rely on the conjugate gradient method with an adjoint equation. These strategies are referred as the Alifanov's iterative regularization method. In spite of its success, this method is CPU time consuming, because in most cases, it uses complex direct models composed of finite differences or finite elements. As the simulation time and the number of nodes grow, the conjugate gradient method with an adjoint equation become computationally inefficient 15,16. Other inverse strategies employ the virtual area method. In spite of the fact that the least-square algorithm is CPU time efficient, this method is applied to problems for which heat conduction is the sole thermal process, i.e. there is no phase change. Also, this inverse technique is only employed with a temperature sensor (not a heat flux sensor) and the simulations are conducted in steady-state. In the iron smelting industry, inverse methods have also been applied in order to estimate the lining erosion inside blast furnaces. Such methods include Alifanov's iterative regularization method 17–19 and the Fourier series combined with an elastic membrane concept 20. Even if these inverse techniques are formulated for two-dimensional (2-D) heat transfer, the main drawback is that they are also limited to the steady-state regime, which is not interesting for online inverse predictions or for control applications. Finally, the vast majority of inverse methods rely on temperature measurements 15,16,21,22 due to the fact that it is one of the easiest measured physical quantities in thermal sciences. However, this study focuses on heat flux measurements only. As reported in references 23,24, using heat flux measurements in the inverse calculation enables to recover the unknown variables much more efficiently. This can be explained by the fact that a heat flux corresponds to a vector (modulus, direction), hence providing a richer information than the temperature which is only a scalar quantity (modulus) 24.

Moreover, in an industrial context, the interest (and the challenge) is to control at all times the thickness of the ledge. A classic feedback loop used for control process is shown in Figure 3 25. The measurement given by a sensor is first subtracted to the desired set point. From this error, the controller calculates the control action needed for the process that will ensure the desired set point even if the process is subjected to disturbances.

Figure 3. Classic feedback loop 25.

This feedback loop implies that the measurement from a sensor is available. In the case of metallurgical reactors, it means that a ledge thickness sensor exists. This is obviously not the case since the hostile conditions that prevail inside the molten bath forbid it. Moreover, manual measurement of the ledge thickness is not only impractical but also barely useful as the frequency of the data collection is bound to remain insufficient.

The objective of this study is to combine a virtual sensor, which provides an estimation of the ledge thickness from heat flux measurements, to a control strategy that will ensure the desired ledge thickness even when the process is subjected to disturbances (Figure 4).

Figure 4. The control of the ledge thickness in high-temperature metallurgical reactors using a virtual sensor.

This article is organized as follows. First, a finite difference model (FDM) is developed for predicting the nonlinear melting/solidification process inside the reactor. Second, a state-space model of the reactor, more computationally efficient than the FDM, is presented. Third, the inverse method (the virtual sensor), combining a Kalman filter, a recursive least-square estimator (RLSE) and the state-space model, is shown. The virtual sensor enables to estimate the ledge thickness from heat flux measurements taken inside or outside of the reactor walls. And finally, the control strategy, a classic proportional–integral (PI) controller, is coupled with the virtual sensor in order to control the ledge thickness inside the metallurgical reactor. The control strategy is thoroughly tested for typical operating conditions that prevail inside an industrial facility.

2. FDM of the metallurgical reactor

The 1-D phase change problem under investigation is depicted in Figure 2a). A time-varying ledge of thickness , which corresponds to the solidification front location, is built against the inner surface of a brick wall of thickness . The outer surface of the brick wall, i.e. the left boundary condition, is cooled by forced convection heat transfer with an impinging air jet. The air temperature remains constant and equal to . The time-varying heat transfer coefficient is found using an empirical correlation that employs the non-dimensional Reynolds and Nusselt numbers of the flow 26. The heat transfer coefficient may easily be controlled by varying the air mass flow rate of the impinging jet. For the present cooling system, the heat transfer coefficient ranges from 60 to 120 W m⁻² K⁻¹ 26. A time-varying heat flux , which represents the heat load supplied to the reactor, is imposed on the right boundary condition at . The heat load may be affected by different operating conditions: ore feeding, current shutdown and anode replacement 1,6,27. These operations can be thought as thermal disturbances to the smelting process. They can be correlated to the variation of the input heat flux from energy balances 6.

The mathematical model for the phase change problem rests on the following assumptions:

•	The FDM represents faithfully the thermal behaviour of a real high-temperature metallurgical reactor 1,3,4.
•	The temperature gradients across the wall (x direction) are much larger than the ones in the vertical direction so that a 1-D analysis can be applied 1,3,4.
•	As a result, a single portion of the metallurgical reactor is examined (at the metal/bath interface where the ledge thickness is minimal).
•	The phase change problem is non-isothermal. It is characterized by a mushy zone between the solidus temperature and the liquidus temperature .
•	The 1-D cooling of the reactor external wall by an impinging jet corresponds to forced convection heat transfer only. The cooling process is thermally decoupled from the phase change problem inside the reactor. Therefore, the ambient temperature and the heat transfer coefficient are well known at all times.
•	The thermal contact resistance between the brick wall and the ledge is neglected.
•	The thermal properties of the PCM are temperature independent; they may, however, be different for the liquid and the solid phases.
•	The heat transfer across the liquid layer of the PCM is conduction dominated. The effect of the flow circulation in the liquid layer could, however, be accounted for in the diffusion model by means of an effective thermal conductivity. The reader is referred to references 1,28 for further details on this matter.

With the foregoing assumptions, the governing heat diffusion equation for the wall and the PCM may be stated as: (1)

The boundary conditions are given as follows: (2) (3) and the initial condition is (4)

The second term on the right-hand side of Equation (1) accounts for the solid/liquid phase change. Of course, this term vanishes inside the brick wall. The enthalpy change is defined as . The liquid fraction varies linearly between the solidus temperature and the liquidus temperature in the following manner: (5)

Equations (1–5) were discretized using second-order finite differences in space and first-order differences in time with an implicit scheme. The resulting set of algebraic equations of the form is then solved using a tri-diagonal matrix algorithm. For each time step, the liquid fraction in Equation (1) is determined according to the enthalpy method, an iterative procedure developed by Voller and Swaminathan 29. Thus, with the known heating and cooling h(t) boundary conditions, the FDM provides the evolution of the temperature field , the heat flux distribution and the ledge thickness .

The above FDM was thoroughly tested and validated using analytical solutions and results available in the open literature. Further details concerning the validation of the numerical model are reported in reference 23.

The physical properties of the brick wall and of the PCM for a typical high-temperature metallurgical reactor are provided in Table 1 3,30. From these references, the thickness of the brick wall was fixed at  = 0.1 m and the width of the PCM layer (solid, mushy and liquid) was set equal to  = 0.1 m (Figure 2a). The outside temperature was kept constant at  = 300 K. As for the time step and the mesh size , they were set to 100 s and 0.5 × 10⁻³ m respectively, which enables to obtain a good compromise between numerical accuracy and CPU time requirement. Moreover, the mesh size was chosen so as to yield the required resolution for the predicted ledge thickness and smoothen the thermal conductivity jump inside the control volumes that undergo phase change.

Table 1. Physical properties of the brick wall and of the PCM for a typical high-temperature metallurgical reactor 3,30.

Parameter	Value
(J kg^−1 K^−1)	875
(W m^−1 K^−1)	16.8
(kg m^−3)	2600
(J kg^−1 K^−1)	1800
(W m^−1 K^−1)	1
(kg m^−3)	2100
(J kg^−1 K^−1)	1800
(W m^−1 K^−1)	10
(J kg^−1)	5.1 × 10^5
(K)	1213
(K)	1233

3. State-space identification of the reactor

With the above mesh size, the resulting size of the matrix A is 400 × 400 which can become a problem due to the fact that it needs to be inversed for calculating the temperature vector T. This problem is accentuated for multidimensional problem (2-D, 3-D) which brings the question: is there a different way to model the thermal behaviour of the reactor with a simpler and faster model?

In the last decades, the popularity of model reduction techniques have increased significantly in various fields such as heat transfer, fluid mechanics, solid mechanics and process control. The main objective of these methods is to extract, from knowledge of the physical process (simulations or experimental data), a low-dimensional system (reduced model) that has nearly the same response characteristics as the process under study. The idea behind model reduction is the need, in many instances, for a simplified model that captures the main features of the original complex model. The simplified model, much more computationally efficient, may then be used in place of the original complex model, either for simulations or for control applications 31. A variety of different model reduction techniques exist, such as the modal identification method 32, the Litz methods 33, proper orthogonal decomposition 34, artificial neural networks 35 and state-space models 16,21,22. Since the inverse method used in this study is composed of a Kalman filter and a RLSE (Section 4), state-space models were a more convenient choice for building the reduced model.

The state-space identification consists in casting the predictions of the FDM, the exact values, into the following linear discrete state-space form: (6) (7) Where is the measurement vector. It contains the information extracted from the FDM, that is the heat flux measurement and the ledge thickness . is the state vector. Due to the fact that canonical form of the state-space model is used, the state vector is equal to the measurement vector . is the input vector which is equivalent to the boundary conditions (Equations (2) and (3)). Finally, the state-space matrices are defined by the symbols , and , respectively, and corresponds to the sampling instant.

Given the input vector and the corresponding measurement vector , the objective of the state-space identification is to compute the matrices , and that minimize the cost function defined as: (8) where is the difference between the measurement vector (extracted from the FDM) and the predicted output of the state-space model and the total number of sampling instants. In order to compute the matrices , and , the state-space identification algorithm relies on the iterative prediction-error minimization (PEM) method reported in reference 36.

To characterize the accuracy of the state-space identification, the relative root-mean-square error (RRMSE) is adopted: (9) where corresponds to the measurement vector and is equal to the predicted output of the state-space model .

The usefulness of the present state-space identification procedure is exemplified for the two following cases: (1) using a heat flux sensor located at the brick/ledge interface, i.e. and (2) using a heat flux sensor at the external surface of the brick wall, i.e. . The input vector corresponds to typical values for the heat load inside high-temperature reactors 37 and the heat transfer coefficient for forced convection in air 26 (Figure 5). The corresponding predictions of the state-space models are compared in Figures 6 and 7 where the identified state-space matrices are presented in Table 2. For the first case, it is seen that the agreement between the predictions of the state-space model and those of the FDM is excellent, i.e. the RRMSE values remain lower than 3% for the heat flux and ledge thickness predictions. As for the second case, even if the RRMSE takes somewhat larger values, especially for the heat flux prediction, the agreement between the predictions of the state-space model and those of the FDM remains reliable (RRMSE < 5%).

Figure 5. Input heat flux and heat transfer coefficient , i.e. the input vector u, used for computing the FDM predictions and the state-space model outputs.

Figure 6. Predictions of the FDM versus that of the state-space models when using the heat flux sensor : (a) heat flux at the brick/ledge interface and (b) ledge thickness.

Figure 7. Predictions of the FDM versus that of the state-space models when using the heat flux sensor : (a) heat flux at the external surface of the brick wall and (b) ledge thickness.

Table 2. Identified state-space matrices.

State-space matrices	Using the heat flux sensor	Using the heat flux sensor

Table 3. Parameters for the inverse method 23.

Parameter	Value
(s)	100
(W m^−2)	50
for (W m^−2)	10
for (m)	0.0001
	[0 0]^T
	Zero matrix
(W m^−2)	0
for	0.90
for	0.97

4. Inverse method for estimating the ledge thickness

In the inverse problem, the heat flux on the right boundary is unknown (Figure 2b). It may be estimated however with an inverse method which employs a heat flux sensor embedded into the brick wall or placed on its outside surface. Once the heat flux has been estimated based on these measurements , the solidification front location, or ledge thickness, , is determined from the state-space model of Section 3. A schematic of the overall inverse methodology based on the Kalman filter, RLSE and state-space model is provided in Figure 8.

Figure 8. The inverse method; a Kalman filter coupled with a RLSE and a state-space model. It acts as a virtual sensor for estimating the ledge thickness.

The inverse method is thus equivalent to a virtual sensor 9 allowing on-line measurements of heat fluxes . The virtual sensor relies on an estimation algorithm (Kalman filter, RLSE, state-space model) so as to provide on-line estimates of the non-measurable ledge thickness .

In spite of the fact that various estimation methods are available for implementing computationally efficient inverse calculations, such as the function specification method 38,39, the filter coefficient method 38,40], the Kalman filters with augmented state 21,22, the Kalman filters with RLSE 41,42, the partial least-square regression methods 43,44 or the adaptive iterative filters 45,46, this study will focus on the procedure already reported in reference 23: a Kalman filter coupled with a RLSE and a state-space model (Figure 8). The main features of the proposed linear method are as follows: (1) the algorithm is simple; (2) it handles complex inverse phase change processes; (3) it does not rely on a statistical database; and (4) the inverse methodology can be carried out indiscriminately with temperature and/or heat flux sensors. The proposed inverse method may be divided into four steps.

The first step is concerned with the thermal behaviour of the reactor. Bear in mind that it is assumed that the FDM represents faithfully the thermal behaviour of a real high- temperature metallurgical reactor 1,3,4. While the industrial process is running, the PCM inside the reactor may undergo melting/solidification cycles due to the time-varying heating and cooling conditions, i.e. and . This will in turn generate data for the heat flux sensor located inside or outside the brick wall. These data contain noise which is considered as a zero mean Gaussian white noise with covariance defined as the covariance matrix of the measurement noise (Equation (10)). (10)

The second step consists in using a Kalman filter which is a set of mathematical equations that provide an efficient recursive mean to estimate the state of a process (the phase change process) 47. The Kalman filter estimates a process using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of noisy measurement. Referring to Figure 8, the Kalman filter estimates the process state of the reactor with the sensor measurement . It also uses simultaneously only one of the input signals, i.e. the heat transfer coefficient . In order to estimate correctly the process state, the Kalman filter proceeds through two steps: (1) the time update step (prediction) followed by (2) the measurement update step (correction) 48.

The time update equations project the current state estimate and the error covariance matrix estimate forward in time in order to obtain the a priori estimate for the next time step, i.e. and (Equations (11) and (12)). (11) (12)

The covariance matrices and characterize the accuracy of the computed state estimate and , respectively. is the covariance matrix of the process noise which accounts for the disturbances to the metallurgical reactor dynamics. The process noise is assumed once again to be zero mean Gaussian white noise with covariance defined as the covariance matrix of the process noise (Equation 13). (13)

The feedback is then performed by the measurement update equations that incorporate a new measurement into the a priori estimates and in order to obtain an improved a posteriori estimate and (Equations (14) and (15)). Due to the fact that only the measurement of heat flux is available in the measurement vector (Equation (7)), the matrix will be used with the current algorithm. (14) (15) where is the identity matrix and the optimal Kalman gain (Equation (16)) that minimizes the a posteriori error covariance . (16)

One important result of this process, , corresponds to the measurement residual which reflects the discrepancy between the predicted measurement and the actual measurement . Finally, stands for the residual covariance which is the covariance matrix of the measurement residual .

In the third step of the inverse method, the RLSE is applied to the present heat transfer problem. The RLSE is a well-known recursive technique used in the field of adaptive signal processing. The reader is referred to reference 49 for a full discussion on this technique. The RLSE estimates the unknown heat flux using the information generated from the Kalman filter, that is the measurement residual , the optimal Kalman gain and the residual covariance . The full RLSE algorithm applied to heat transfer is reported in references 41,42. The overall results can be stated in terms of the following set of five equations: (17) (18) (19) (20) (21) where corresponds to the estimated heat flux, the covariance error of the estimated heat flux, a correction gain and and the sensitivity matrices. The magnitude of the forgetting factor ranges from 0 to 1. It is employed to prevent to shrink to zero, thereby maintaining the RLSE updating ability when the time-varying heat fluxes need to be estimated. This is a consequence of the RLSE algorithm where older data are gradually discarded in favour of more recent measurements. As reported in reference 49, measurements older than samples are forgotten. The magnitude of is usually chosen so as to ensure good tracking ability without the loss of accuracy.

In the fourth and last step of the inverse method, the solidification front location is calculated with the state-space model of Section 3 (Equations (6) and (7)) and with the estimated heat flux given by the RLSE, that is, (22) (23) where is a new input vector and the estimated ledge thickness.

The above inverse heat transfer procedure was thoroughly tested for estimating the time-varying input heat flux on the right boundary and therefore for predicting the time-varying thickness of the ledge inside a reactor. For this reason, numerical simulations were carried out using once again typical operating conditions that prevail in such facilities (Figure 5) 26,37. Moreover, the main parameters for the Kalman filter and of the RLSE are provided in Table 3. The selection of these parameters is documented in reference 23.

Table 4. Proportional gain and integral time constant values.

Case		(s)
Virtual sensor using heat flux	7500	5000
Virtual sensor using heat flux	5000	7500

One can note that forgetting factor is also larger when the heat flux sensor is employed. The purpose of the forgetting factor is to reach a compromise between stability and accuracy of the inverse prediction. Due to the fact that the heat flux sensor is farther away from the ledge compared to , the damping and lagging effect caused by heat conduction diffusive nature are more significant in that case 38. Therefore, the inverse solution is more sensible to the measurement noise, and in order to achieve the desired compromise on stability and accuracy, the value of the forgetting factor needs to be increased.

The corresponding time-varying exact (FDM) and estimated (inverse method) ledge thicknesses are compared in Figures 9 and 10. In the first case, a heat flux sensor at the brick/wall interface is used with the inverse method, i.e. , and in the second case, the inverse method employs a heat flux sensor at the external surface of the brick wall, i.e. . To characterize the accuracy of the inverse method, the RRMSE (Equation 9) is used once again; corresponds to the exact ledge thickness and the estimated ledge thickness .

Figure 9. Predicted (inverse method) and exact (FDM) ledge thickness using a heat flux sensor at the brick/ledge interface, i.e. .

Figure 10. Predicted (inverse method) and exact (FDM) ledge thickness using a heat flux sensor at the external surface of the brick wall, i.e. .

Figure 9 shows that the discrepancy between the exact and the estimated ledge thicknesses remains smaller than 3% and gives a maximum error of 4 × 10⁻³ m. The main drawback is that the heat flux sensor needs to be embedded into the brick wall, and thus its installation will be more cumbersome. However, even if it is simpler to install a heat flux sensor at the external surface of the brick wall, by looking at Figure 10, one can note that the inverse method is less accurate; the RRMSE value is three times larger (9.3%) and the maximum error is now 10 × 10⁻³ m. Figure 10 also shows that the time lag between the exact (FDM) and estimated (inverse) is higher using this heat flux sensor. This can be explained by the fact that the magnitude of the forgetting factor is increased in order to maintain the stability of the inverse solution.

5. Control of the ledge thickness using the virtual sensor

As presented earlier, no ledge thickness sensor is available for high-temperature metallurgical reactors because of the hostile conditions that prevail inside the reactor, and the use of the standard manual method for measuring the ledge thickness is not conceivable for control application because the measurement frequency is too low. Therefore, in order to control the ledge thickness inside metallurgical reactors, the virtual sensor described in Section 4 is merged to the following control strategy (Figure 11). As before, it is assumed that the thermal behaviour of the reactor may be described by the FDM of Section 2 1,3,4.

Figure 11. Control of the ledge thickness in metallurgical reactors using a virtual sensor.

From the heat flux measurements contaminated with noise , the virtual sensor estimates the ledge thickness . Using this prediction, the tracking error , defined as the difference between the set point (desired ledge thickness) and the actual ledge thickness , can be computed. Its value is then fed to the controller which adjusts the cooling (the heat transfer coefficient ) of the reactor wall in order to minimize the tracking error when subjected to unknown external disturbances .

The control strategy used in this study rests on a standard PI controller. Incidentally, more than 90% of the controllers employed in the process industry are PI–PID controllers 50,51. It is noted that more complex control strategies such as the internal model control 52, the model predictive control 53, the artificial neural networks 54 and the optimal control 55 are available for controlling different thermal processes. But the simplicity and the availability of PI controllers as well as their easiness of implementation make them a more attractive choice. Moreover, electrolysis cells are usually cooled by natural convection 56, that is without any controlled forced convection cooling devices and the implementation of a simple PI is considered to be a bold new application for the smelting industry.

The implementation of the control strategy is done in discrete time where a sampling time dt = 100 s is employed. The choice of this sampling time is a compromise between stability and accuracy 57. For a PI controller, the control action, defined as the cooling () of the reactor wall, is computed in the following manner (Equation (24)) 25. (24) where is the proportional gain and the integral time constant. Based on common guidelines typically used to manually tune a PI–PID controller for open-loop stable systems 58, the proportional gain K_p and the integral time constant T_i were determined in order to obtain a maximum tracking error of 3 × 10⁻³ m, a null steady-state tracking error and no overshoots (Table 4). These specifications are required in order to minimize the ledge thickness variation, thus ensuring a continuous protective barrier against the corrosive molten material 4–6.

The control scenario of the metallurgical reactor is the following. The initial operating conditions, i.e. the input heat flux and the heat transfer coefficient , are set at constant values of 15,000 W m⁻² and 60 W m⁻² K⁻¹, respectively 26,37. These values imply an initial ledge thickness of 3.78 × 10⁻² m. At time t = 1 × 10⁴s, the ledge is suddenly upset with an input heat flux of 17,000 W m⁻² until time t = 3 × 10⁴s where it lowers to 16,000 W m⁻² up to the end of the simulation. The sudden change in the heat load by the square wave signal can be related to different operating conditions that act as thermal disturbances of the smelting process 1,6,27. Finally, the ledge thickness set point is kept at its initial value of 3.78 × 10⁻² m.

The control of the ledge thickness by combining a PI controller with a virtual sensor is depicted in Figures 12 and 13. Once again, the comparison of using a heat flux sensor at the brick/ledge interface and at the external surface of the brick wall is done. In the first case, it is shown that the controller provides a maximum tracking error of only 2.8 × 10⁻³ m compared to 8 × 10⁻³ m for the case without control. Moreover, no overshoots are present and a null steady-state tracking error is ensured. The corresponding RRMSE value is equal to 1.6% where corresponds to the ledge thickness set point and the estimated ledge thickness (Equation (9)).

Figure 12. Control of the ledge thickness by combining a PI controller with a virtual sensor using a heat flux sensor at the brick/ledge interface .

Figure 13. Control of the ledge thickness by combining a PI controller with a virtual sensor using a heat flux sensor at the external surface of the brick wall .

In the second case, the tracking error is equivalent with a value of 2.85 × 10⁻³ m. However, one can note the significant difference in the time lag which is caused by the larger forgetting factor (Table 3). As stated earlier, because the heat flux measurements are farther away from the ledge, the accuracy of the inverse solution needs to be decreased (larger time lag) in order to stabilize it (less oscillation). Figure 13 also shows that the controller response is less aggressive than that in the previous case, that is the set point is attained less rapidly and with a more oscillatory response. This can also be observed in Figure 14 in which the controller output signal, i.e. the heat transfer coefficient h, is depicted. This behaviour can be explained by looking at the proportional gain K_p and integral time constant T_i values from Table 4. These PI settings were chosen in order to ensure the stability of the control loop even if the virtual sensor using this heat flux sensor provides a larger time lag 59.

Figure 14. The PI controller output signal, i.e. the heat transfer coefficient h, when the virtual sensor uses heat flux (a) and (b) .

Finally, the control response of the two cases presented above are compared with a PI controller (K_p = 20,000, T_i = 10,000 s) that employs the exact ledge thickness, which is the ledge thickness provided by the FDM of Section 2 (Figure 15). This is shown for comparison purposes only. Figure 15 shows once again the time lag difference between the two virtual sensors. One can note also that using the exact ledge thickness provided by the FDM, the results are equivalent to the case employing the virtual sensor with heat flux sensor . However, in the latter case, the measurements are non-intrusive, i.e. they are not taken inside the metallurgical reactor.

Figure 15. Control of the ledge thickness using a PI controller combined to a virtual sensor with heat flux (a) , (b) and (c) the exact ledge thickness provided by the FDM.

6. Concluding remarks

A non-intrusive inverse heat transfer procedure for predicting the time-varying thickness of the phase change ledge on the inside surface of the walls of high-temperature metallurgical reactors was presented. A Kalman filter, based on a state-space representation of the reactor, was coupled with a RLSE in order to estimate on-line the position of the phase front. The data were collected by a heat flux sensor embedded into the reactor wall or placed on the outside surface of the wall. The inverse method, used here as a virtual sensor, was then combined to a traditional PI controller in order to control the ledge thickness, by regulating the air cooling applied on the outside surface of the reactor wall. The virtual sensor and the control strategy were thoroughly tested for typical phase change conditions that prevail inside industrial facilities. Results have shown that using a virtual sensor that employs a heat flux sensor inside the reactor wall provided more accuracy and stability in the inverse calculation and in the control application, but at the cost of a more complicated installation. In that case, it was shown that the discrepancy between the exact and the estimated ledge thicknesses remained smaller than 3% at all times, and that the control strategy ensured a null steady-state tracking error, a maximum tracking error less than 10%, no overshoot and no significant time lag.

Acknowledgements

The authors are very grateful to the Fonds québécois de la recherche sur la nature et les technologies (FQRNT), to the Centre québécois de recherche et de développement de l’aluminium (CQRDA) and to the Natural Sciences and Engineering Council of Canada (NSERC) for their financial support.