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Ind. Eng. Chem. Res. 2009, 48, 2598–2604
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 with long transition times and possible ream defects. A reduction in the product turnover time implies a higher product yield and a reduced energy consumption. This turnover time relies to a large extent on the understanding of furnace dynamics. In this work, the disadvantages of the current transition practice (based on the concepts of minimum residence time and perfect mixing) are discussed. An optimization approach that is based on the residence time distribution of the glass furnace is tailored to accommodate practical implementation issues in the product transition control. Comparisons of the current transition practice and the proposed approach are made through computer simulations, which demonstrate that significant savings in the product turnover time are expected if the latter is used. 1. Introduction The physical properties of a glass product are significantly influenced by its composition. One example is the Saint-Gobain ultraclear glass used in the Grand Canyon Skywalk,1 whose exceptional clarity owes to its much lower concentration of iron oxide as compared to the regular glass. Because of this unique property, the ultraclear glass also finds applications in furniture and photovoltaic substrates. As another example, the Solargreen and Solextra automotive glass manufactured by PPG Industries differ in the percentage of iron oxide, and therefore, they have different aesthetic appearances and solar performances.2 Besides iron oxide, other colorants such as nickel oxide and cobalt oxide may also be introduced to the glass melt as body tints to adjust the properties of the glass product.3 When these glass products with different compositions are manufactured in the same furnace, the transition from one product to another is critical because of its continuous operation nature. Because of the long nominal holding time (typically 50 h or longer) and several flow recirculations in a glass melting furnace, the product change process might take several days or even weeks (see Figure 1 for a schematic of the complicated flow characteristics in a Siemens glass furnace). The long transition time is a disadvantage of the body tints as compared to the solar control glazings, which can be made by online chemical vapor deposition with very flexible operation times.4,5 A reduction in the turnover time implies more saleable glass product and less energy consumed to remelt the glass with unacceptable composition and color. Besides the long transition time, another potential issue in the product change is the socalled ream defect that might accompany the new product for a long period of time. One well-known mechanism of the ream defect is the Rayleigh instability.6 It occurs in the refiner section where the glass melt loses heat to the crown and overturns near the surface of the vitreous space as a result of density difference. The current transition practice is an open-loop control strategy based on perfect mixer theory.7 Because of the complexity of the raw material feeding process, a real-time adjustment of the batch composition in the transition process is currently not practical. If the furnace behaves like a continuous-stirred tank reactor (CSTR), a single overdose in the batch is applied and * To whom correspondence should be addressed. E-mail: minghengli@ csupomona.edu. Tel.: +1-909-869-3668. Fax: +1-909-869-6920.
its duration time is calculated following the dynamics of a CSTR.7 Several operating parameters such as gas firing rates, cooler duties, and bubbling flow rates are adjusted accordingly to enhance homogeneity of the glass melt during the transition process.8 As will be shown later, the current transition practice uses very limited information of the glass furnace to predict the transition behavior, which leads to unreconcilable discrepancies between model predictions and experimental observations. A further improvement in the transition practice requires a better understanding of the furnace dynamics. Recent efforts have been made in the computational fluid dynamic (CFD) modeling of glass furnaces to provide detailed information about the flow, concentration, and thermal fields in the combustion space and the glass melt as well.9-11 Transient CFD simulations have also been developed,8 which shed more insight into the dynamics of the product transition process. Because of the long solution time of a CFD simulation which makes it difficult to be directly used for dynamic optimization purposes, efforts have been made in the reduction of CFD models for glass furnaces.12 More work on identification and control of a glass furnace itself is also available in literature.13-17 To the best knowledge of the author, a previous work focusing on the product concentration control beyond the perfect mixer theory is not yet available. Our recent research efforts have led to the development of various optimal control approaches to a class of concentration transition problems using input/output models18 or reduced-order state-space models.19 These optimization algorithms to the optimal transition control problem relied on results developed previously.20-22 In this work, the input/output approach is tailored to accommodate several practical considerations in the glass product transition process. Compared to the perfect mixer
Figure 1. A schematic of complicated flow patterns in a Siemens glass melting tank.
10.1021/ie801134b CCC: $40.75 2009 American Chemical Society Published on Web 02/05/2009
Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2599
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 overdose ratios.
model, the proposed approach would potentially result in a better evolution of the colorant concentration in the glass product. It is also able to explain several observations that are not predicted by the perfect mixer model. 2. Discussions on the Current Transition Practice In general, under the assumption that the mixing behavior of a glass furnace follows the one of a perfect CSTR, the evolution of colorant concentration at the outlet of the glass furnace during the transition process is described by the following dynamic equation:7 FV
dC(t) ) F[Cb(t) - C(t)], dt
C(0) ) C0
y(0) ) y0
u(t) )
(2)
where τ ) FV/F (the characteristic time of the glass furnace) and y0 ) 0. The currently used practice in industry is to first apply an overdose in the batch for a certain period of time. Once the colorant concentration in the product reaches its desired value at a finite time, the batch formula is switched to the one of the new product which can be manufactured thereafter.7 This approach is somewhat similar to the so-called “bang-bang control” in the control community.23 Several time scales in the transition process are defined as follows: (i) tf, time elapsed from
{
N, 0 e t < to 1, t g to
(3)
where N is sometimes referred to as the overdose ratio in the literature.8 With such a batch sequence, it can be derived that
(1)
where F is the density of the glass, V is the volume of the furnace, and F is the throughput. Cb(t) and C(t) are the colorant concentrations in the batch and in the glass product at time t, respectively. Let C0 and Cf be the colorant concentrations in the old and the new products, eq 1 can be converted to a dimensionless form by introducing two variables u(t) ) (Cb(t) -C0)/(Cf - C0) (dimensionless colorant concentration in the batch), and y(t) ) (C(t) - C0)/(Cf - C0) (dimensionless colorant concentration in the product): dy(t) + y(t) ) u(t), τ dt
the inducing of the overdose to the saving of the new product; (ii) tt, transition time calculated from the discarding of the old product to the saving of the new product; (iii) to, overdose duration time calculated from the inducing of the overdose to the inducing of the new product batch. Mathematically, the batch sequence used in the current transition pratices has the following form:
y(t) )
{
( τt )],
[
N 1 - exp -
0 e t < to
1,
t g to
(4)
provided that the overdose duration time (to) satisfies the following relationship: to 1 ) -ln 1 N τ
(
)
(5)
The dynamic evolution of the dimensionless outlet concentration under different overdose ratios and different overdose duration times are shown in Figure 2. It can be readily verified that tf ) to ) tt if the glass furnace is a CSTR without any time delay. Moreover, eq 5 implies that the overdose ratio N should be greater than 1 for a successful transition. In fact, the larger the overdose ratio, the shorter the transition time (see Figure 3). However, it is a well-known fact that the glass furnace exhibits a long time delay (typically no less than 10 h in order to make bubble-free, completely melted glass)16 during the product transition. In the glass community, the term minimum residence time is used to describe such a time delay.16,24 Because of this behavior, the old glass product can be saved for a certain
2600 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009
the outlet concentration after which the concentration gradually reduces to approach the target.8 The reason to these discrepancies is that the glass furnace exhibits high-order dynamics which cannot be adequately described by a CSTR or a PFR and a CSTR in series. For such a system, even when y(t) reaches the target at a finite time, it will continue to change due to its nonzero derivatives. For example, a second-order reactor system consisting of two CSTRs in series (for simplicity, both reactors are 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: 2 2
+ y(t) ( 2τ ) ddty(t) + τ dy(t) dt 2
) u(t),
y(0) ) 0,
dy(t) dt
|
)0 t)0
(6)
Figure 3. Product turnover time as a function of the overdose ratio in a perfect mixer.
With an overdose ratio of N and an overdose duration time of to followed by a dose ratio of 1, the trajectory of y(t) can be derived as y(t) )
Figure 4. Profiles of the dimensionless concentrations at the outlet of a system consisting of two CSTRs in series using an overdose ratio of 4 and different overdose duration times.
period of time even after a batch with a new composition is introduced into the furnace. To take this effect into account, an improved reactor model might consist of a plug flow reactor (PFR) and a CSTR in series. From reactor engineering fundamentals it is known that switching the order of these two reactors does not change the residence time distribution of the entire system.25 If the glass furnace is considered as a CSTR followed by a PFR, it can be derived that with the same trajectories of the input u(t) shown in Figure 2, the profile of the output y(t) will just shift to the right when the time delay (td) is taken into account, which is shown in the same figure for a comparison. Alternatively, one could solve y(t) analytically and the result will be the same. Apparently, the transition time is still the same as the one without time delay. A difference between these two cases is that tf ) to + td ) tt + td if the time delay is present. Even when it is combined with the concept of minimum residence time, the perfect mixer model is not able to explain several phenomena observed in practice. For example, both industrial practices and transient CFD simulations indicate that the transition time is typically much longer than the overdose duration time, and it does not necessarily reduce as the overdose ratio N increases. Moreover, there is usually an overshoot in
{
t < to N · P(t), N · P(t) - (N - 1) · P(t - to), t g to
(7)
where P(t) ) [1 - (2t/τ +1) exp(-2t/τ)] is the response of the output concentration under a unit step change in the inlet concentration, which is a non-negative, monotonically increasing function defined for t g 0. As will be shown later, this P(t) is the cumulative residence time distribution (RTD) function of a reactor system from a viewpoint of reactor engineering.25 Its first-order derivative p(t) ) dP(t)/dt is the so-called RTD function, which has been widely used in the glass community to evaluate the mixing behavior of the glass melt.7,26-29 For the system described by eq 6, a concentration evolution similar to Figure 2 may not be achieved because one can readily verify that a finite tf cannot be found such that dy(t)/dt ≡ 0 for any t g tf. If N ) 4, the profiles of y(t) under several overdose duration times are shown in Figure 4. Overshoots occur in three cases where to g 0.2τ, after which the dimensionless outlet concentration gradually decreases to its target. In the case where to ) 0.1τ, it will take a long (theoretically infinite) time for y(t) to reach its target. Therefore, the transition strategy derived from a first-order system cannot be directly applied to a second-order or high-order system. If more CSTRs are connected in series (the characteristic times of each reactor are assumed to be the same for simplicity), the time delay becomes more obvious and the reactor network behaves similar to an actual glass furnace,7 as shown in Figure 5. The RTD and cumulative RTD functions of the CSTRs in series are available in reactor engineering textbooks,25 and those of the glass furnace are obtained through CFD simulations. It is worth pointing out that the above analysis does not necessary imply that a successful transition cannot be made in a glass furnace. This is primarily because the glass product might be aesthetically acceptable if the colorant concentration in the glass product falls into a tolerant region of its desired value. Given the fact that the furnace is a stable system in terms of colorant concentration evolution, a successful turnover can always be made as long as a dose ratio of N ) 1 is applied at the end of the batch sequence. However, one can still differentiate the performance of different transition approaches. For example, among the four cases shown in Figure 4, the one with an 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
Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2601
function of the glass throughput. The transition time should be less with a larger throughput during the transition and this is outside of the scope of this work. In the following paragraphs, the previously developed input-output optimization approach18 will be tailored to accommodate certain practical considerations in the glass product transition processes. Let p(t) be the RTD of a glass furnace, the amount of the colorant that enters the furnace at time t - θ and spends time θ 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 furnace is the convolution of u(θ) and p(θ), or: y(t) )
∫
∞
0
u(t - θ) p(θ) dθ )
∫
∞
0
u(t - θ) dP(θ)
(8)
The optimization problem under consideration is as follows: Figure 5. RTD and cumulative RTD functions of CSTRs in series25 and of a glass furnace obtained using CFD simulation.
∫
∞
min J ) u(t)
0
(y(t) - 1)2 dt + 2
∫
∞
0
(u(t) - 1)2 dt (9)
subject to
∫
∞
y(t) ) 0 u(t - θ) dP(θ) umin e u(t) e umax
Figure 6. Profiles of the dimensionless concentration at the outlet of glass furnace under different overdose ratios (overdose duration time based on τ ) FV/F).
value of the integral implies a closer match of the outlet concentration and its desired value, and thus, a better transition. 3. Optimal Transition Strategy Based on the Concept of RTD As discussed earlier, the limitation of the conventional transition strategy is that the RTD (impulse response of the glass furnace) or the cumulative RTD (step response of the glass furnace) is forced to follow the one of a single CSTR or a CSTR plus a PFR. In fact, these functions in a glass furnace can be measured experimentally using a tracer element that does not affect the glass quality, e.g., lithia.28 If the RTD varies little during the transition process, a previously developed optimization approach18 can be applied to calculate the optimal trajectories of the feed concentration in continuous time. This assumption is reasonable because the colorant concentration is typically less than 1 wt%. As a result, the thermal and flow fields in the furnace do not vary significantly. Transient CFD simulations using different bubbling flow rates, positions of orifices’ tips and rotation speeds of the stirrers indicate that these parameters do not have a significant effect on the dynamic evolution behavior in a glass furnace if the batch sequence is the same.8 However, it should be noted that the RTD is a strong
where represents the weight on the colorant concentration in the batch. A small implies “cheap control” which is true for glass product transition. This term is included to avoid ill-defined control problems where the control action is infinite. The lower and upper bounds of the u(t) are set to ensure that the colorant concentration in the batch is in a desirable and attainable range. For example, a very high colorant concentration is not desirable for uniform mixing with the glass melt. Too low a concentration might not be achievable in practice (e.g., if iron oxide is the colorant, the lowest concentration cannot be zero because the feedstock usually contains a certain amount of it). In solving the optimization problem, the upper bound of the integrand can be changed to a prespecified finite time as long as a steady state is approached at this time, which can be judged after the optimization problem is solved.18 Alternatively, a final time constraint 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 equivalent least-squares minimization problem as follows by discretization: min J ) ||Cu - d|| 2 u
(10)
subject to umin e e u
e umin e
where u ) [u(t1) u(t2) ... u(tn)]T P C) ∆ EI e d) Ee
[ ] []
[
e ) [11...1]T P(t1) P(t2) - P(t1) P∆ )
l l P(tn) - P(tn-1)
0 ... P(t1) ··· ·
··
·
··
·
0 l
·· l P(t1) ·· 0 ... P(t2) - P(t1) P(t1) ·
...
...
]
2602 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009
and I is the identity matrix; n∆t is the horizon length for optimization which is chosen such that a steady-state is approached at this time. Solving the above optimization problem will result in an optimal feed trajectory changing with a discretized time ∆t.18 This is technically challenging to implement in a manufacturing environment due to the complexity of the batch feeding system. To avoid unintentional mistakes, only a couple of batches should be used and each batch should last for a reasonable period of time (i.e., greater than ∆t). Whereas the transition time might be slightly longer, it is easier for the operators to follow. To take this factor into account, the optimization problem should be modified such that the input is constant over a longer time interval (∆T ) k∆t). For example, in the first batch, the following equality constraints should be satisfied: A1u1 ) 0 where
[
1 -1 1 -1 · ·· ··· A1 ) 1 -1 0 0 ... 0 0
(11)
]
Figure 7. Profiles of the dimensionless concentration in the batch material using optimal batch formula with a large time interval.
u1 ) [u(t1) u(t2) ... u(tk-1) u(tk)]T A similar equation is written for the second batch, the third one, and so on. When these equations are combined, the following equation can be obtained: Aequ ) 0 where
(12)
[ ] A1
Aeq )
A2
·
··
Am
and n ) k · m. As a result, the following least-squares minimization problem can be obtained: min J ) ||Cu - d|| 2 u
(13)
subject to Aequ ) 0 umine e u e umaxe Equation 13 is in the standard form of constrained linear leastsquares. Note that they are m consistent equality constraints, which can be recognized automatically by Matlab if lsqlin is used. Given the fact that C is a sparse matrix, an acceleration method proposed in a previous work18 can be used to accelerate the solution process, which is based on singular value decomposition. To illustrate the proposed method, a glass melting furnace similar 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 the characteristic time of the furnace (τ ) FV/F), the process is approximated using a CSTR and a PFR in series. If the overdose duration times are calculated based on eq 5 using different overdose ratios (N ) 1, 2, 4, and 6), the profiles of y(t) are solved using eq 8 and the results are shown in Figure 6. It is
Figure 8. Profiles of the dimensionless concentration at the outlet of the glass furnace using optimal batch formula with a large time interval.
seen that with different overdose ratios, there is always an overshoot in the outlet concentration. Moreover, there is not much difference in the transition time when different overdose ratios are used, which indicates that the currently used transition practice does not lead to significant improvements. To investigate whether a better evolution of the colorant concentration could be obtained, the proposed optimization method is used. The solved profiles of the dimensionless inlet and outlet concentrations are shown as dash-dot plots in Figures 7 and 8. In this case, the constraints on the inlet concentration are chosen to be 0 e u(t) e 6 and the minimum duration time of each batch is 0.16τ. The concentration profiles based on the current transition practice are shown in the same figures for a comparison. It is seen that the turnover time is much shorter when the proposed method is used. It is also interesting to notice that for a transition from a low colorant concentration in the old product to a higher concentration in the new product, the colorant concentration in the batch might be lower than the new product during a certain period of time. In this way the overshoot in the outlet concentration is suppressed. Even though further investigations are necessary, a suppression in concentration overshoot is potentially helpful for the reduction of ream defect. A scenario can be imagined in which the density of the new
Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2603
A simple economic analysis is conducted to demonstrate the benefit of the implementation of the developed approach. For a Siemens float furnace with a throughput of 600 ton/day, a reduction of 20 h per product transition is equivalent to 500 ton savable glass. If an average sale price of $300/ton glass is used, the profit gain is $150,000 per transition. Given the fact that the glass industry is conservative by nature, a major challenge of this approach would be whether the same product quality can be achieved. To the knowledge of the author, an overdose ratio as high as 8 (based on the conventional transition practice) has been tried in the glass industry without any mixing problem. In the approach developed in this work, the largest overdose ratio is only 6. Some training for the plant engineers might be necessary since the number of batch formulas is more and the overall batch duration time is longer in the developed approach than in the conventional transition practice. 4. Summary Figure 9. Profiles of the dimensionless concentration at the outlet of the glass furnace if the RTD function varies (15%.
glass product is lower. After the overshoot, the low density glass is gradually replaced by the high density glass in the refiner section, and overturns might occur near the surface of the vitreous phase in the refiner. It is noticed that there are several batches with u(t) very close to 1 in the optimal solution. To reduce the number of batches used in the product transition, it is desired to fix the composition at u(t) ) 1 for these batches. On the basis of the optimal solution already determined, a suboptimal solution might consist of five batches with different compositions. This can be achieved by solving eq 13 with a few changes. For example, A5 to An should be replaced by identity matrices, and the according elements in beq should be replaced by 1. With such a modification, the solution is shown in Figures 7 and 8 with the dash plots. It is seen that a batch sequence with five different compositions is able to achieve a very similar performance in terms of transition time. It is worth noting that the developed approach does not provide a detailed guidance to dynamically operate bubblers, coolers, and burners during the glass product transition. The objective of this work is to capture the most dominant dynamics (i.e., the evolution of colorant concentration) with a reasonable computational effort. A comprehensive description of process dynamics for optimization purposes would require reduced CFD models12,30 that provide coupled flow, thermal, and mass transport phenomena but might involve a significant amount of time. Another simplification made in this work is that the RTD does not change during the product transition process. To test the robustness of the proposed method in this work, it is assumed in the following simulation that the RTD function p(t)(t) during the 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 random variable uniformly distributed between [-0.15 0.15] (however, to guarantee p(t)(t) is nonnegative, R ) 0 if p(t) ) 0). It is shown in Figure 9 that with the same input profile, the output profile is still close to its predicted trajectory. Moreover, given the fact that the furnace is a stable system in terms of colorant concentration evolution and a dose ratio of 1 is applied at the end of the batch sequence, it is guaranteed that the outlet concentration will eventually reach its desired value. This simulation demonstrates that a slight change in the RTD would not affect the performance of the proposed approach.
It has been shown in this work that current practice of glass product transition, which is based on the concepts of minimum residence time and perfect mixing, does not adequately characterize the high-order dynamics exhibited in a glass furnace. The cumulative RTD of a glass furnace, which can be measured using lithia as a tracer, better describes the dynamic evolution of the colorant concentration. On the basis of the RTD function obtained through CFD simulations, an optimal batch sequence which contains five different compositions is shown to obtain a significant reduction in the transition time and a suppressed overshoot in the outlet concentration. Acknowledgment Start-up funding from the Dean’s Office and support from the Faculty Center for Professional Development at California State Polytechnic University, Pomona are gratefully acknowledged. The author would also like to thank Jiri Brada from Glass Service, Czech Republic, for providing data of the modified Ford glass furnace.8 Nomenclature C ) colorant concentration, kg/ton F ) through put, ton/h I ) identity matrix N ) overdose ratio p(t) ) residence time distribution function P(t) ) cumulative residence time distribution function ) ∫0t p(θ) dθ t ) time, h td ) time delay, h tf ) final time counted from inducing the overdose, h to ) duration time of the overdose, h tt ) transition time counted from discarding the old product to the saving of the new product, h u ) dimensionless colorant concentration in the batch V ) volume of the furnace, m3 y ) dimensionless colorant concentration in the glass product ∆t ) time interval, h ∆T ) time interval between different batches, h F ) density of the glass, ton/m3 τ ) characteristic time, h
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ReceiVed for reView July 23, 2008 ReVised manuscript receiVed December 8, 2008 Accepted January 5, 2009 IE801134B
