The Optimal Product Transition In Glass Furnaces

Mingheng Li*

Department of Chemical and Materials Engineering, California State Polytechnic UniVersity, Pomona,California 91768

Product change in glass furnaces is a common and challenging industrial problem generally associated withlong transition times and possible ream defects. A reduction in the product turnover time implies a higherproduct yield and a reduced energy consumption. This turnover time relies to a large extent on the understandingof furnace dynamics. In this work, the disadvantages of the current transition practice (based on the conceptsof minimum residence time and perfect mixing) are discussed. An optimization approach that is based on theresidence time distribution of the glass furnace is tailored to accommodate practical implementation issues inthe product transition control. Comparisons of the current transition practice and the proposed approach aremade through computer simulations, which demonstrate that significant savings in the product turnover timeare expected if the latter is used.

1. Introduction

The physical properties of a glass product are significantlyinfluenced by its composition. One example is the Saint-Gobainultraclear glass used in the Grand Canyon Skywalk,1 whoseexceptional clarity owes to its much lower concentration of ironoxide as compared to the regular glass. Because of this uniqueproperty, the ultraclear glass also finds applications in furnitureand photovoltaic substrates. As another example, the Solargreenand Solextra automotive glass manufactured by PPG Industriesdiffer in the percentage of iron oxide, and therefore, they havedifferent aesthetic appearances and solar performances.2 Besidesiron oxide, other colorants such as nickel oxide and cobalt oxidemay also be introduced to the glass melt as body tints to adjustthe properties of the glass product.3

When these glass products with different compositions aremanufactured in the same furnace, the transition from oneproduct to another is critical because of its continuous operationnature. Because of the long nominal holding time (typically 50 hor longer) and several flow recirculations in a glass meltingfurnace, the product change process might take several days oreven weeks (see Figure 1 for a schematic of the complicatedflow characteristics in a Siemens glass furnace). The longtransition time is a disadvantage of the body tints as comparedto the solar control glazings, which can be made by onlinechemical vapor deposition with very flexible operation times.4,5

A reduction in the turnover time implies more saleable glassproduct and less energy consumed to remelt the glass withunacceptable composition and color. Besides the long transitiontime, another potential issue in the product change is the so-called ream defect that might accompany the new product fora long period of time. One well-known mechanism of the reamdefect is the Rayleigh instability.6 It occurs in the refiner sectionwhere the glass melt loses heat to the crown and overturns nearthe surface of the vitreous space as a result of density difference.

The current transition practice is an open-loop control strategybased on perfect mixer theory.7 Because of the complexity ofthe raw material feeding process, a real-time adjustment of thebatch composition in the transition process is currently notpractical. If the furnace behaves like a continuous-stirred tankreactor (CSTR), a single overdose in the batch is applied and

its duration time is calculated following the dynamics of aCSTR.7 Several operating parameters such as gas firing rates,cooler duties, and bubbling flow rates are adjusted accordinglyto enhance homogeneity of the glass melt during the transitionprocess.8 As will be shown later, the current transition practiceuses very limited information of the glass furnace to predictthe transition behavior, which leads to unreconcilable discrep-ancies between model predictions and experimental observa-tions. A further improvement in the transition practice requiresa better understanding of the furnace dynamics. Recent effortshave been made in the computational fluid dynamic (CFD)modeling of glass furnaces to provide detailed information aboutthe flow, concentration, and thermal fields in the combustionspace and the glass melt as well.9-11 Transient CFD simulationshave also been developed,8 which shed more insight into thedynamics of the product transition process. Because of the longsolution time of a CFD simulation which makes it difficult tobe directly used for dynamic optimization purposes, efforts havebeen made in the reduction of CFD models for glass furnaces.12

More work on identification and control of a glass furnace itselfis also available in literature.13-17 To the best knowledge ofthe author, a previous work focusing on the product concentra-tion control beyond the perfect mixer theory is not yet available.

Our recent research efforts have led to the development ofvarious optimal control approaches to a class of concentrationtransition problems using input/output models18 or reduced-orderstate-space models.19 These optimization algorithms to theoptimal transition control problem relied on results developedpreviously.20-22 In this work, the input/output approach istailored to accommodate several practical considerations in theglass product transition process. Compared to the perfect mixer

* To whom correspondence should be addressed. E-mail: minghengli@csupomona.edu. Tel.: +1-909-869-3668. Fax: +1-909-869-6920.

Figure 1. A schematic of complicated flow patterns in a Siemens glassmelting tank.

Ind. Eng. Chem. Res. 2009, 48, 2598–26042598

10.1021/ie801134b CCC: $40.75 2009 American Chemical SocietyPublished on Web 02/05/2009

model, the proposed approach would potentially result in a betterevolution of the colorant concentration in the glass product. Itis also able to explain several observations that are not predictedby the perfect mixer model.

2. Discussions on the Current Transition Practice

In general, under the assumption that the mixing behavior ofa glass furnace follows the one of a perfect CSTR, the evolutionof colorant concentration at the outlet of the glass furnace duringthe transition process is described by the following dynamicequation:7

where F is the density of the glass, V is the volume of thefurnace, and F is the throughput. Cb(t) and C(t) are the colorantconcentrations in the batch and in the glass product at time t,respectively. Let C0 and Cf be the colorant concentrations inthe old and the new products, eq 1 can be converted to adimensionless form by introducing two variables u(t) ) (Cb(t)-C0)/(Cf - C0) (dimensionless colorant concentration in thebatch), and y(t) ) (C(t) - C0)/(Cf - C0) (dimensionless colorantconcentration in the product):

where τ ) FV/F (the characteristic time of the glass furnace)and y0 ) 0.

The currently used practice in industry is to first apply anoverdose in the batch for a certain period of time. Once thecolorant concentration in the product reaches its desired valueat a finite time, the batch formula is switched to the one of thenew product which can be manufactured thereafter.7 Thisapproach is somewhat similar to the so-called “bang-bangcontrol” in the control community.23 Several time scales in thetransition process are defined as follows: (i) tf, time elapsed from

the inducing of the overdose to the saving of the new product;(ii) tt, transition time calculated from the discarding of the oldproduct to the saving of the new product; (iii) to, overdoseduration time calculated from the inducing of the overdose tothe inducing of the new product batch. Mathematically, the batchsequence used in the current transition pratices has the followingform:

where N is sometimes referred to as the overdose ratio in theliterature.8 With such a batch sequence, it can be derived that

provided that the overdose duration time (to) satisfies thefollowing relationship:

The dynamic evolution of the dimensionless outlet concentra-tion under different overdose ratios and different overdoseduration times are shown in Figure 2. It can be readily verifiedthat tf ) to ) tt if the glass furnace is a CSTR without any timedelay. Moreover, eq 5 implies that the overdose ratio N shouldbe greater than 1 for a successful transition. In fact, the largerthe overdose ratio, the shorter the transition time (see Figure3).

However, it is a well-known fact that the glass furnaceexhibits a long time delay (typically no less than 10 h in orderto make bubble-free, completely melted glass)16 during theproduct transition. In the glass community, the term minimumresidence time is used to describe such a time delay.16,24 Becauseof this behavior, the old glass product can be saved for a certain

Figure 2. Profiles of the dimensionless concentrations at the inlet and the outlet of a CSTR with/without time delay (td ) 0.2τ) using different overdoseratios.

FVdC(t)

dt) F[Cb(t) - C(t)], C(0) ) C0 (1)

τdy(t)dt

+ y(t) ) u(t), y(0) ) y0 (2)

u(t) ) { N, 0 e t < to

1, t g to(3)

y(t) ) { N[1 - exp(- tτ)], 0 e t < to

1, t g to

(4)

to

τ) -ln(1 - 1

N) (5)

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2599

period of time even after a batch with a new composition isintroduced into the furnace. To take this effect into account, animproved reactor model might consist of a plug flow reactor(PFR) and a CSTR in series. From reactor engineeringfundamentals it is known that switching the order of these tworeactors does not change the residence time distribution of theentire system.25 If the glass furnace is considered as a CSTRfollowed by a PFR, it can be derived that with the sametrajectories of the input u(t) shown in Figure 2, the profile ofthe output y(t) will just shift to the right when the time delay(td) is taken into account, which is shown in the same figurefor a comparison. Alternatively, one could solve y(t) analyticallyand the result will be the same. Apparently, the transition timeis still the same as the one without time delay. A differencebetween these two cases is that tf ) to + td ) tt + td if the timedelay is present.

Even when it is combined with the concept of minimumresidence time, the perfect mixer model is not able to explainseveral phenomena observed in practice. For example, bothindustrial practices and transient CFD simulations indicate thatthe transition time is typically much longer than the overdoseduration time, and it does not necessarily reduce as the overdoseratio N increases. Moreover, there is usually an overshoot in

the outlet concentration after which the concentration graduallyreduces to approach the target.8 The reason to these discrep-ancies is that the glass furnace exhibits high-order dynamicswhich cannot be adequately described by a CSTR or a PFRand a CSTR in series. For such a system, even when y(t) reachesthe target at a finite time, it will continue to change due to itsnonzero derivatives. For example, a second-order reactor systemconsisting of two CSTRs in series (for simplicity, both reactorsare assumed to have the same characteristic time τ1 ) τ2 )τ/2, where τ is the nominal holding time of the entire system)is described by the following differential equation:

With an overdose ratio of N and an overdose duration timeof to followed by a dose ratio of 1, the trajectory of y(t) can bederived as

where P(t) ) [1 - (2t/τ +1) exp(-2t/τ)] is the response of theoutput concentration under a unit step change in the inletconcentration, which is a non-negative, monotonically increasingfunction defined for t g 0. As will be shown later, this P(t) isthe cumulative residence time distribution (RTD) function of areactor system from a viewpoint of reactor engineering.25 Itsfirst-order derivative p(t) ) dP(t)/dt is the so-called RTDfunction, which has been widely used in the glass communityto evaluate the mixing behavior of the glass melt.7,26-29

For the system described by eq 6, a concentration evolutionsimilar to Figure 2 may not be achieved because one can readilyverify that a finite tf cannot be found such that dy(t)/dt ≡ 0 forany t g tf. If N ) 4, the profiles of y(t) under several overdoseduration times are shown in Figure 4. Overshoots occur in threecases where to g 0.2τ, after which the dimensionless outletconcentration gradually decreases to its target. In the case whereto ) 0.1τ, it will take a long (theoretically infinite) time for y(t)to reach its target. Therefore, the transition strategy derived froma first-order system cannot be directly applied to a second-orderor high-order system. If more CSTRs are connected in series(the characteristic times of each reactor are assumed to be thesame for simplicity), the time delay becomes more obvious andthe reactor network behaves similar to an actual glass furnace,7

as shown in Figure 5. The RTD and cumulative RTD functionsof the CSTRs in series are available in reactor engineeringtextbooks,25 and those of the glass furnace are obtained throughCFD simulations.

It is worth pointing out that the above analysis does notnecessary imply that a successful transition cannot be made ina glass furnace. This is primarily because the glass product mightbe aesthetically acceptable if the colorant concentration in theglass product falls into a tolerant region of its desired value.Given the fact that the furnace is a stable system in terms ofcolorant concentration evolution, a successful turnover canalways be made as long as a dose ratio of N ) 1 is applied atthe end of the batch sequence. However, one can still differenti-ate the performance of different transition approaches. Forexample, among the four cases shown in Figure 4, the one withan overdose duration time of to ) 0.2τ is better than the others.To evaluate how close the outlet concentration is to its target,one may consider the following functional ∫0

∞(y(t) - 1)2 dt,which is similar to the H2 norm in optimal control. A smaller

Figure 3. Product turnover time as a function of the overdose ratio in aperfect mixer.

Figure 4. Profiles of the dimensionless concentrations at the outlet of asystem consisting of two CSTRs in series using an overdose ratio of 4 anddifferent overdose duration times.

(τ2)2d2y(t)

dt2+ τdy(t)

dt+ y(t) ) u(t), y(0) ) 0,

dy(t)dt |

t)0

) 0

(6)

y(t) ) { N · P(t), t < to

N · P(t) - (N - 1) · P(t - to), t g to(7)

2600 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009

value of the integral implies a closer match of the outletconcentration and its desired value, and thus, a better transition.

3. Optimal Transition Strategy Based on the Concept ofRTD

As discussed earlier, the limitation of the conventionaltransition strategy is that the RTD (impulse response of the glassfurnace) or the cumulative RTD (step response of the glassfurnace) is forced to follow the one of a single CSTR or a CSTRplus a PFR. In fact, these functions in a glass furnace can bemeasured experimentally using a tracer element that does notaffect the glass quality, e.g., lithia.28 If the RTD varies littleduring the transition process, a previously developed optimiza-tion approach18 can be applied to calculate the optimaltrajectories of the feed concentration in continuous time. Thisassumption is reasonable because the colorant concentration istypically less than 1 wt%. As a result, the thermal and flowfields in the furnace do not vary significantly. Transient CFDsimulations using different bubbling flow rates, positions oforifices’ tips and rotation speeds of the stirrers indicate that theseparameters do not have a significant effect on the dynamicevolution behavior in a glass furnace if the batch sequence isthe same.8 However, it should be noted that the RTD is a strong

function of the glass throughput. The transition time should beless with a larger throughput during the transition and this isoutside of the scope of this work. In the following paragraphs,the previously developed input-output optimization approach18

will be tailored to accommodate certain practical considerationsin the glass product transition processes.

Let p(t) be the RTD of a glass furnace, the amount of thecolorant that enters the furnace at time t - θ and spends timeθ in the furnace before flowing out is given by u(t - θ) p(θ)dθ. Therefore, the concentration y(t) at the outlet of the furnaceis the convolution of u(θ) and p(θ), or:

The optimization problem under consideration is as follows:

subject to

where ε represents the weight on the colorant concentration inthe batch. A small ε implies “cheap control” which is true forglass product transition. This term is included to avoid ill-definedcontrol problems where the control action is infinite. The lowerand upper bounds of the u(t) are set to ensure that the colorantconcentration in the batch is in a desirable and attainable range.For example, a very high colorant concentration is not desirablefor uniform mixing with the glass melt. Too low a concentrationmight not be achievable in practice (e.g., if iron oxide is thecolorant, the lowest concentration cannot be zero because thefeedstock usually contains a certain amount of it). In solvingthe optimization problem, the upper bound of the integrand canbe changed to a prespecified finite time as long as a steady stateis approached at this time, which can be judged after theoptimization problem is solved.18 Alternatively, a final timeconstraint can be added to the optimization problem in eq 9.

It has been proven that if the time interval (∆t ) ti+1 - ti, i )1, 2,...) is small enough, eq 9 can be converted to an equivalentleast-squares minimization problem as follows by discretization:

subject to

where

u ) [u(t1) u(t2) ... u(tn)]T

C ) [P∆

EI ]d ) [ e

Ee ]e ) [11...1]T

P∆ ) [ P(t1) 0 ... ... 0P(t2) - P(t1) P(t1)

··· ll ···

······ l

l ··· P(t1) 0P(tn) - P(tn-1) ... ... P(t2) - P(t1) P(t1)

]

Figure 5. RTD and cumulative RTD functions of CSTRs in series25 and ofa glass furnace obtained using CFD simulation.

Figure 6. Profiles of the dimensionless concentration at the outlet of glassfurnace under different overdose ratios (overdose duration time based on τ) FV/F).

y(t) ) ∫0

∞u(t - θ) p(θ) dθ ) ∫0

∞u(t - θ) dP(θ) (8)

minu(t)

J ) ∫0

∞(y(t) - 1)2 dt + ε

2∫0

∞(u(t) - 1)2 dt (9)

y(t) ) ∫0

∞u(t - θ) dP(θ)

umin e u(t) e umax

minu

J ) | |Cu - d| |2 (10)

umin e e u e umin e

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2601

and I is the identity matrix; n∆t is the horizon length foroptimization which is chosen such that a steady-state is approachedat this time.

Solving the above optimization problem will result in anoptimal feed trajectory changing with a discretized time ∆t.18

This is technically challenging to implement in a manufacturingenvironment due to the complexity of the batch feeding system.To avoid unintentional mistakes, only a couple of batches shouldbe used and each batch should last for a reasonable period oftime (i.e., greater than ∆t). Whereas the transition time mightbe slightly longer, it is easier for the operators to follow.

To take this factor into account, the optimization problemshould be modified such that the input is constant over a longertime interval (∆T ) k∆t). For example, in the first batch, thefollowing equality constraints should be satisfied:

where

A1 ) [1 -11 -1

······1 -1

0 0 ... 0 0]

u1 ) [u(t1) u(t2) ... u(tk-1) u(tk)]T

A similar equation is written for the second batch, the thirdone, and so on. When these equations are combined, thefollowing equation can be obtained:

where

Aeq ) [A1

A2···

Am ]and n ) k ·m. As a result, the following least-squares minimiza-tion problem can be obtained:

subject to

Equation 13 is in the standard form of constrained linear least-squares. Note that they are m consistent equality constraints,which can be recognized automatically by Matlab if lsqlin isused. Given the fact that C is a sparse matrix, an accelerationmethod proposed in a previous work18 can be used to acceleratethe solution process, which is based on singular valuedecomposition.

To illustrate the proposed method, a glass melting furnacesimilar to the one discussed in a published work8 is considered.Its RTD and cumulative RTD functions are shown in Figure 5.Based on the concepts of minimum residence time and thecharacteristic time of the furnace (τ ) FV/F), the process isapproximated using a CSTR and a PFR in series. If the overdoseduration times are calculated based on eq 5 using differentoverdose ratios (N ) 1, 2, 4, and 6), the profiles of y(t) aresolved using eq 8 and the results are shown in Figure 6. It is

seen that with different overdose ratios, there is always anovershoot in the outlet concentration. Moreover, there is notmuch difference in the transition time when different overdoseratios are used, which indicates that the currently used transitionpractice does not lead to significant improvements.

To investigate whether a better evolution of the colorantconcentration could be obtained, the proposed optimizationmethod is used. The solved profiles of the dimensionless inletand outlet concentrations are shown as dash-dot plots in Figures7 and 8. In this case, the constraints on the inlet concentrationare chosen to be 0 e u(t) e 6 and the minimum duration timeof each batch is 0.16τ. The concentration profiles based on thecurrent transition practice are shown in the same figures for acomparison. It is seen that the turnover time is much shorterwhen the proposed method is used. It is also interesting to noticethat for a transition from a low colorant concentration in theold product to a higher concentration in the new product, thecolorant concentration in the batch might be lower than the newproduct during a certain period of time. In this way the overshootin the outlet concentration is suppressed. Even though furtherinvestigations are necessary, a suppression in concentrationovershoot is potentially helpful for the reduction of ream defect.A scenario can be imagined in which the density of the new

A1u1 ) 0 (11)

Aequ ) 0 (12)

minu

J ) | |Cu - d| |2 (13)

Aequ ) 0umine e u e umaxe

Figure 7. Profiles of the dimensionless concentration in the batch materialusing optimal batch formula with a large time interval.

Figure 8. Profiles of the dimensionless concentration at the outlet of theglass furnace using optimal batch formula with a large time interval.

2602 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009

glass product is lower. After the overshoot, the low density glassis gradually replaced by the high density glass in the refinersection, and overturns might occur near the surface of thevitreous phase in the refiner.

It is noticed that there are several batches with u(t) very closeto 1 in the optimal solution. To reduce the number of batchesused in the product transition, it is desired to fix the compositionat u(t) ) 1 for these batches. On the basis of the optimal solutionalready determined, a suboptimal solution might consist of fivebatches with different compositions. This can be achieved bysolving eq 13 with a few changes. For example, A5 to An shouldbe replaced by identity matrices, and the according elements inbeq should be replaced by 1. With such a modification, thesolution is shown in Figures 7 and 8 with the dash plots. It isseen that a batch sequence with five different compositions isable to achieve a very similar performance in terms of transitiontime.

It is worth noting that the developed approach does notprovide a detailed guidance to dynamically operate bubblers,coolers, and burners during the glass product transition. Theobjective of this work is to capture the most dominant dynamics(i.e., the evolution of colorant concentration) with a reasonablecomputational effort. A comprehensive description of processdynamics for optimization purposes would require reduced CFDmodels12,30 that provide coupled flow, thermal, and masstransport phenomena but might involve a significant amount oftime. Another simplification made in this work is that the RTDdoes not change during the product transition process. To testthe robustness of the proposed method in this work, it is assumedin the following simulation that the RTD function p(t)(t) duringthe transition process is different than its original profile (i.e.,the profile before the transition occurs) used for optimization,and the relationship is p(t)(t) ) p(t)(1 + R), where R is a randomvariable uniformly distributed between [-0.15 0.15] (however,to guarantee p(t)(t) is nonnegative, R ) 0 if p(t) ) 0). It is shownin Figure 9 that with the same input profile, the output profileis still close to its predicted trajectory. Moreover, given the factthat the furnace is a stable system in terms of colorantconcentration evolution and a dose ratio of 1 is applied at theend of the batch sequence, it is guaranteed that the outletconcentration will eventually reach its desired value. Thissimulation demonstrates that a slight change in the RTD wouldnot affect the performance of the proposed approach.

A simple economic analysis is conducted to demonstrate thebenefit of the implementation of the developed approach. For aSiemens float furnace with a throughput of 600 ton/day, areduction of 20 h per product transition is equivalent to 500ton savable glass. If an average sale price of $300/ton glass isused, the profit gain is $150,000 per transition. Given the factthat the glass industry is conservative by nature, a majorchallenge of this approach would be whether the same productquality can be achieved. To the knowledge of the author, anoverdose ratio as high as 8 (based on the conventional transitionpractice) has been tried in the glass industry without any mixingproblem. In the approach developed in this work, the largestoverdose ratio is only 6. Some training for the plant engineersmight be necessary since the number of batch formulas is moreand the overall batch duration time is longer in the developedapproach than in the conventional transition practice.

4. Summary

It has been shown in this work that current practice of glassproduct transition, which is based on the concepts of minimumresidence time and perfect mixing, does not adequately char-acterize the high-order dynamics exhibited in a glass furnace.The cumulative RTD of a glass furnace, which can be measuredusing lithia as a tracer, better describes the dynamic evolutionof the colorant concentration. On the basis of the RTD functionobtained through CFD simulations, an optimal batch sequencewhich contains five different compositions is shown to obtaina significant reduction in the transition time and a suppressedovershoot in the outlet concentration.

Acknowledgment

Start-up funding from the Dean’s Office and support fromthe Faculty Center for Professional Development at CaliforniaState Polytechnic University, Pomona are gratefully acknowl-edged. The author would also like to thank Jiri Brada from GlassService, Czech Republic, for providing data of the modifiedFord glass furnace.8

Nomenclature

C ) colorant concentration, kg/tonF ) through put, ton/hI ) identity matrixN ) overdose ratiop(t) ) residence time distribution functionP(t) ) cumulative residence time distribution function ) ∫0

t p(θ)dθ

t ) time, htd ) time delay, htf ) final time counted from inducing the overdose, hto ) duration time of the overdose, htt ) transition time counted from discarding the old product to the

saving of the new product, hu ) dimensionless colorant concentration in the batchV ) volume of the furnace, m3

y ) dimensionless colorant concentration in the glass product∆t ) time interval, h∆T ) time interval between different batches, hF ) density of the glass, ton/m3

τ ) characteristic time, h

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Figure 9. Profiles of the dimensionless concentration at the outlet of theglass furnace if the RTD function varies (15%.

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ReceiVed for reView July 23, 2008ReVised manuscript receiVed December 8, 2008

Accepted January 5, 2009

IE801134B

2604 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009