Ind. Eng. Chem. Res. 2010, 49, 5765–5774
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Cyclic Scheduling for Ethylene Cracking Furnace System with Consideration of Secondary Ethane Cracking Chuanyu Zhao, Chaowei Liu, and Qiang Xu* Dan F. Smith Department of Chemical Engineering, Lamar UniVersity, Beaumont, Texas 77710
Cracking furnaces of ethylene plants are capable of processing multiple feeds to produce smaller hydrocarbon molecules, such as ethylene, propylene, and ethane. The best practice for handling the produced ethane is to recycle it as an internal feed and conduct the secondary cracking in a specific furnace. As cracking furnaces have to be periodically shut down for decoking, when multiple furnaces processing different feeds under various product values and manufacturing costs are considered, the operational scheduling for the entire furnace system should be optimized to achieve the best economic performance. In this paper, a new MINLP (mixedinteger nonlinear programming) model has been developed to optimize the operation of cracking furnace systems with the consideration of secondary ethane cracking. This model is more practical than the previous study and can simultaneously identify the allocation of feeds with their quantity, time, and sequence information for each cracking furnace. A case study has demonstrated the efficacy of the developed scheduling model. 1. Introduction Ethylene is the most widely produced organic compound in the world and is essential for daily life. Because of the rapid growth of the worldwide economy and population, global ethylene production in 2007 reached about 115 million metric tons and is expected to increase by 4.4% per year from 2007 to 2012.1 To meet the ever increasing global demand for ethylene, continuous advancements in manufacturing technology are pursued by ethylene plants. Among these advancements, the technological innovation for the cracking furnace operation is one of the most important aspects. The cracking furnace is used to convert hydrocarbon feeds to smaller hydrocarbon molecules through complex pyrolysis reactions, resulting in mostly ethylene and propylene. The cracked gas also contains an amount of ethane. The best practice for handling the produced ethane is to recycle it as an internal feed and conduct the secondary cracking in a specific furnace. Thermal cracking is crucial for ethylene production because the major product yields of an ethylene plant are mainly determined in this operation. The downstream operations after cracking, such as quenching, compressing, chilling, and separating sections, are actually used to recover products from the cracked gas. Thermal cracking is also a semicontinuous dynamic operation. The reason for this is that the pyrolysis byproduct of coke during the operation is gradually generated and deposited on the reaction coils, which increases the heat transfer resistance and the reactor pressure drop and results in the decay of both reaction selectivity and productivity. Thus, a cracking furnace must be periodically shut down for decoking (cleanup). Meanwhile, multiple cracking furnaces are needed in an ethylene plant to sustain the continuity of the cracking operation. Normally, ethylene plants will not allow two or more cracking furnaces to simultaneously shut down for decoking for two major reasons: (i) many ethylene plants have only one decoking facility, which can only clean up one furnace at a time and (ii) simultaneously shutting down two or more cracking furnaces will cause significant disturbances to downstream processes, jeopardizing the plant’s operability and productivity.2 * To whom correspondence should be addressed. Tel.: 409-880-7818. Fax: 409-880-2197. E-mail:
[email protected].
Increasing economic competition and volatile raw-material and product markets have forced ethylene plants to be able to process different types of feeds to enhance their manufacturing flexibility. When multiple cracking furnaces processing different feeds under various product values and manufacturing costs are considered, the scheduling for the entire furnace system implies significant economic opportunities. The scheduling of the cracking furnace system will provide the quantitative answers to the following questions: (i) how are the different feeds allocated to different furnaces for cracking; (ii) what is the processing sequence if two or more feeds are allocated to the same furnace; (iii) how long should each cracking operation take before its cleanup; (iv) what is the best decoking sequence among multiple furnaces if nonsimultaneous cleanups are required? Informed answers to these questions need to optimally coordinate the information elements of feed, furnace, time, quantity, and sequence. Their optimality is extremely important to the ethylene plant profitability, which needs in-depth studies. By far, studies for improving cracking furnace operations mainly focus on the areas of process simulation, control, and operational optimization.3-8 Production scheduling for a cracking furnace system is not well studied because the thermal cracking process involves highly complex reactions. For instance, the well-recognized commercial software package, SPYRO, has over 3000 radical reactions in the kinetic network.9 It is very hard for researchers to consider such complexities in reactions while scheduling the cracking furnace system. Edwin and Balchen have conducted dynamic optimization to determine the optimal batch processing time and time-dependent operation trajectories of a single cracking furnace.10 For multifurnace scheduling, Jain and Grossmann developed an MINLP (mixed-integer nonlinear programming) model for the cyclic scheduling of multiple feeds cracked on parallel ethylene furnaces with exponential decay performance.11 The solving algorithm for global optimality is also exploited and demonstrated. However, the MINLP model does not consider secondary ethane cracking and nonsimultaneous cleanups. Schulz et al. proposed an extension for ethane-fed ethylene plants with the consideration of recycled ethane.12 They estimated the recycling mean value through the addition of a simplified plant model. On the basis of that, a discrete-time-
10.1021/ie1001235 2010 American Chemical Society Published on Web 05/06/2010
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based MINLP model was developed to study cyclic optimal furnace shutdowns and downstream separation systems.13 It employs a time-dependent empirical variable of coil internal roughness as the indicator for furnace shutdown operation. These models capture the decaying performance throughout furnace operations and are proposed to increase overall plant net profit. However, they still allow simultaneous cleanup scenarios and only address cracking of one kind of feed. In 2006, Lim et al. scheduled a neural-network-based cracking simulation, which employed dynamic data on ethylene and propylene yields, coil skin temperature, and pressure drop information to support the scheduling decisions.14 To tackle the computational complexity of the developed large-scale MINLP model, they also developed three alternative solution strategies. This study did not consider cracking multiple feeds. To further consider the unexpected uncertainties, they developed a proactive scheduling for the naphtha cracking furnace system decoking.15 In this MINLP model, scheduling is not in a cyclic scheme but in a dynamic scheme, and the rescheduling is triggered when the gap between the model prediction and the measurement exceeds a chosen threshold value. The model can determine appropriate rescheduling points before actual operational problems arise. Because the profit in an ethylene plant comes from multiple products, Gao et al. addressed the optimization problem for periodic operations of the naphtha pyrolysis.16 Their work highlights the important fact that the evaluation of ethylene furnace performance needs to consider multiple products simultaneously, such as the productivity of propylene and ethylene. Very recently, Liu et al. developed a cyclic scheduling model by considering multiple feeds, multiple cracking furnaces, and nonsimultaneous cleanup constraints.2 However, the model did not consider the issue of secondary ethane cracking, which still has some deficiency in application. In this paper, a new MINLP model has been developed to obtain cyclic scheduling strategies for a cracking furnace system with the consideration of secondary ethane cracking. It can simultaneously identify the allocation of feeds with their quantity, time, and sequence information for each cracking furnace. A case study has demonstrated the efficacy of the developed scheduling model. 2. Problem Clarification Furnace system scheduling addresses multiple cracking furnaces processing multiple feeds. Among the furnaces, there is one furnace specifically designated for cracking ethane feed (either from fresh feedstock or from the recycled ethane product), namely the ethane furnace; while the other furnaces process all the leftover feeds. As shown in Figure 1, the recycled ethane is from down stream separations. Meanwhile, a fresh ethane feedstock is employed to makeup the possible deficiency of the recycled ethane. Note that if the ethane furnace is in decoking operation, the recycled ethane from other furnaces will be accumulated temporarily until the ethane furnace cleanup is completed. For easy understanding of the developed scheduling model, Figure 2 shows the concepts of a batch processing time, cleanup time, total cycle time, and nonsimultaneous cleanup. Batch processing time is the time duration for a furnace starting to crack a feed to the shutdown for decoking. It usually takes 20-90 days, depending on furnace types, feed characteristics, and the cracking severity. Cleanup time is the time duration for decoking a furnace, which is also the down time interval between two adjacent processing batches. Total cycle time is the time span of a schedule for each of the cracking furnaces. Note that a furnace during the cycle time may perform multiple
Figure 1. Illustration for cracking furnace system with secondary ethane cracking.
Figure 2. An illustrative example for a cyclic schedule.
batches with their associated cleanups (e.g., furnaces 1 and 2 have three and two batches, respectively, in Figure 2). For all the cracking furnaces, nonsimultaneous shutdown means every single cleanup cannot be overlapped. Also note that during the batch processing time, the production yields of the products will change dynamically because of the coking and pyrolysis reaction kinetics.8,14 As shown in Figure 2, the ethylene yield during the batch processing time decays with respect to time. It should be emphasized that the developed model has several assumptions: (i) sufficient feedstock supply and steady feed flow rate; (ii) one batch slot in any furnace processing only one type of feed; (iii) ethane contained in the cracked gas will be fully recovered and reused for the secondary cracking; and (iv) product yield models used for the scheduling are reliable. The first three assumptions are generally acceptable in reality. The fourth assumption will be secure if sufficient plant data is regressed to generate the yield models. In this study, all the product yield models are assumed to be already obtained. 3. Cyclic Scheduling Model In this section, a model describing the scheduling of ordinary cracking furnaces and one ethane furnace is introduced, including the objective function, mass balance constraints, recycled flow rate constraint, integrality constraints, feed allocation constraints, timing constraints, nonsimultaneous cleanup con-
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straints, additional logic constraints, and variable bounds. All the parameters are explained both in this section and in the nomenclature section. For illustrations, some auxiliary figures are also presented. 3.1. Objective Function. The objective of the scheduling model is to maximize the average net profit per day over one cycle, which is shown in eq 1. NF NC NB
max J ) [
NP
∑ ∑ ∑{∑P(∫ i)1 j)1 k)1
0
l)1
NF NC NB
∑ ∑ ∑[∫
ti,j,k
0
Di,j(ci,j,NP + ai,j,NPebi,j,NPt) dt] e NB
Di,j(ci,j,l + ai,j,lebi,j,lt) dt) -
DNF,NC
(Cri + Cvi,j)Di,jti,j,k - Csi,jyi,j,k}]/T (1)
where ci,j,l + ai,j,lebi,j,lt describes the dynamic change of product l’s yield with respect to time, when the feed i is cracked in the furnace j during a batch operation. Generally, the dynamic changes of the product yields with respect to time are nonlinear. Thus, using the exponential formulas to describe dynamic product yield will be more acceptable than using linear functions. Di,j is the batch feed flow rate. Pl is the unit price of product l. Then, it can be seen that ti,j,k bi,j,l ∑lNP t) dt) is the total sale income for all ) 1Pl(∫ 0 Di,j(ci,j,l + ai,j,le products during one batch operation. To calculate the net profit of one batch operation, the material and operational costs represented by (Cri + Cvi,j)Di,j ti,j,k and the cleanup cost represented by Csi,j yi,j,k should be accounted. Note that the feed flow rate of ethane to the ethane furnace DNF,NC comes from two parts: ethane produced by all the furnaces and the fresh ethane from the offsite. Only the makeup ethane accounts for the raw material cost. On the other hand, ethane is not a final product for sale. It is just an intermediate product for secondary cracking in the ethane furnace. Thus, eq 1 contains both recycled ethane profit and the equivalent amount of fresh ethane cost. Such a mathematical processing does not influence the optimization results but will make the objective function easy for presentation. Also note that the cleanup cost is controlled by the binary variable of yi,j,k, which is designated as 1 if the feed i is processed in the kth batch in furnace j; otherwise, it is 0. Also note that that the batch processing time, ti,j,k, is subject to yi,j,k (if yi,j,k is 0, ti,j,k will be 0), which will be presented in the model constraints. Overall, the numerator of eq 1 is the total net profit for the furnace system during one cycle operation. It is divided by the total cycle time, suggesting that J represents the average net profit per day of the furnace system. 3.2. Mass Balance Constraints. For each feed, the total amount used by all cracking furnaces should be under the supply capability. Equations 2 and 3 are directly borrowed from Jain and Grossmann.11 In the formulas, Gi is the amount of feed flow rate above the lower bound of Floi. Equation 2 calculates the total average flow rate of the feed i during one cycle. Equation 3 gives the bound of Gi. NC
FloiT + Gi )
during one cycle. Note that the fresh ethane feed will makeup to maintain a steady flow rate to the ethane furnace. Also note that the NPth product represents the recycled ethane, the NFth feed represents the feed of ethane, and the NCth cracking furnace represents the ethane furnace.
i)1 j)1 k)1
ti,j,k
l
∑ j
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∑t
NF,NC,k
(4)
k)1
3.4. Integrality Constraints. Prior to the solution identification, it is unknown how many batches will be needed for each furnace during one cycle time. Thus, the total number of batches, NB, is just a heuristic integer. Some batch slots may not be utilized for cracking. Some logic constraints borrowed from Liu et al. may help reduce the solution searching space, as shown in eqs 5-7.2 Equation 5 suggests all the feeds should be processed during one cycle operation. Equation 6 indicates that, for each furnace, the first-batch slot should always be used for cracking a feed. Equation 7 means one batch slot could only be utilized for cracking one feed at most. NC NB
∑ ∑y
i,j,k
g 1,
∀i ) 1, ..., NF
(5)
j)1 k)1 NF
∑y
i,j,1
) 1,
∀j ) 1, ..., NC
(6)
i)1
NF
∑y
i,j,k
e 1,
∀j ) 1, ..., NC,
∀k ) 2, ..., NB
(7)
i)1
3.5. Feed Allocation Constraints. Because only one cracking furnace is used to process ethane, and all the other furnaces will process all the feeds except ethane, these feed allocation constraints can be modeled as shown in eqs 8 and 9. yi,NC,k ) 0,
∀i ) 1, ..., NF - 1,
∀k ) 1, ..., NB
yNF,j,k ) 0,
∀j ) 1, ..., NC - 1,
∀k ) 1, ..., NB (9)
(8)
3.6. Timing Constraints. Timing constraints restrict the time variables of ti,j,k, Sj,k, and Ej,k. Equation 10 suggests the batch processing time ti,j,k should be within some time window. The time window may be designated by furnace operational characteristics or industrial experience. tloi,jyi,j,k e ti,j,k e tupi,jyi,j,k,
∀i ) 1, ..., NF, ∀j ) 1, ..., NC, ∀k ) 1, ..., NB (10)
NB
(Di,j
∑t
i,j,k),
∀i ) 1, ..., NF
(2)
k
Gi e (Fupi - Floi)T,
∀i ) 1, ..., NF
(3)
3.3. Recycled Flow Rate Constraint. Because the recycled ethane has to be reprocessed, the total amount of ethane produced by all the furnaces during one recycle time should be less than the processing capacity of the ethane furnace; otherwise, the excess ethane would accumulate within the system. As shown in eq 4, the left-hand side represents the total ethane amount produced from all the feeds (including ethane) cracked in all the furnaces (including ethane furnace) during one cycle operation. The right-hand side represents the consumed ethane amount in the ethane furnace
Equations 11 through 14 are used to characterize the startingtime (Sj,k) and ending-time (Ej,k) instants of every batch processing at each furnace. Because the cyclic scheduling problem is a round-table problem, the first batch may start from the previous cycle or from current cycle (Figures 3a and b). Thus, a binary variable zj is introduced. zj is 1 if the starting time of the first batch in the jth furnace is larger than its ending time (as the scenario shown Figure 3a); otherwise, it is 0 (as the scenario shown in Figure 3b). These logic constraints are described by eqs 15 and 16. Equation 17 means the total cycle time at each furnace is the same, which is equal to the summation of all the batch processing time and their cleanup time. Equation 18 suggests the starting time of the first batch should be less than or equal to the total cycle time. Note that
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Figure 3. Illustrative examples for the binary variable of zj.
Figure 5. Illustration for cleanup relations between two furnaces.
Figure 4. Illustrative examples for the binary variable of xj,k,j′,k′.
the starting and ending time of a batch are arranged in such a way that if the kth batch is not actually utilized, its starting and ending time will be exactly the same.
∑τ
Sj,1 ) Ej,NB +
i,jyi,j,NB
- (1 - zj)T,
∀j ) 1, ..., NC
Figure 6. Illustration for the relation between x and y.
i
(11) Sj,k ) Ej,k-1 +
∑τ
i,jyi,j,k-1,
∀j ) 1, ..., NC,
∀k )
i
2, ..., NB (12) Ej,1 ) Sj,1 +
∑t
i,j,k
- Tzj,
∀j ) 1, ..., NC
(13)
i
Ej,k ) Sj,k +
∑t
i,j,k,
∀j ) 1, ..., NC,
∀k ) 2, ..., NB
i
(14) Ej,1 - Sj,1 e (1 - zj)M, Ej,1 - Sj,1 g -zjM,
∀j ) 1, ..., NC ∀j ) 1, ..., NC
(15)
i,j,k
+ τi,jyi,j,k) ) T,
∀j ) 1, ..., NC
(17)
i)1 k)1
Sj,1 e T,
∀j ) 1, ..., NC
(xj,k,j',k' - 1)M e Ej,k - Sj',k'+1 e xj,k,j',kM,
∀j < j' ∀k, k' (19)
-xj,k,j',k'M e Ej',k' - Sj,k+1 e (1 - xj,k,j',k')M,
∀j < j' ∀k, k' (20)
(16)
NF NB
∑ ∑ (t
overlaps, which means the nonsimultaneous cleanup constraints should be considered. Through the study of the nonsimultaneous cleanup scenarios as shown in Figure 2, linear constraints of eqs 19 and 20 can be developed.2 Here, another binary variable, xj,k,j′,k′, is introduced, which is designated as 1 if the kth cleanup in furnace j is no-overlap behind the k′th cleanup in furnace j′; otherwise, if the kth cleanup in furnace j is no-overlap ahead of the k′th cleanup in the furnace j′, it is 0. As shown in eqs 19 and 20, xj,k,j′,k′ equals 0 and 1, resulting in the nonoverlap scenarios of Figure 4 illustrations a and b, respectively.
(18)
3.7. Nonsimultaneous Cleanup Constraints. Cleanup time intervals among different furnaces should have no
3.8. Additional Logic Constraints. Three types of logic constraints can be employed to further reduce the solution searching space. Equation 21 shows that if one batch slot of a furnace is not utilized, the following batches of that furnace should also not be utilized.
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Table 1. Cracking Furnace System Related Parameter Values furnace j parameter
feed i
τi,j (day)
Fa Fb Fc Fpd
2 2 3 0
2 2 2 0
2 3 3 0
2 2 3 0
2 2 2 0
0 0 0 3
Di,j (ton/day)
Fa Fb Fc Fpd
204.4 202.2 201.0 0
214.9 213.7 211.1 0
188.2 187.2 185.8 0
202.3 199.6 197.6 0
212.7 210.6 209.8 0
0 0 0 210.1
tloi,j (day)
Fa Fb Fc Fpd
20 19 18 0
23 21 19 0
18 17 16 0
20 18 17 0
23 22 20 0
0 0 0 59
tupi,j (day)
Fa Fb Fc Fpd
70 53 41 0
72 54 43 0
65 49 39 0
69 52 43 0
73 56 45 0
0 0 0 101
ai,j,Pa
Fa Fb Fc Fpd
-0.0539 -0.0539 -0.0129 0
-0.0500 -0.0560 -0.0140 0
-0.0574 -0.0534 -0.0139 0
-0.0510 -0.0528 -0.0125 0
-0.0585 -0.0584 -0.0134 0
0 0 0 -0.0542
ai,j,Pb
Fa Fb Fc Fpd
-0.0267 -0.0267 -0.0149 0
-0.0243 -0.0287 -0.0149 0
-0.0288 -0.0269 -0.0143 0
-0.0242 -0.0244 -0.0146 0
-0.0278 -0.0257 -0.0148 0
0 0 0 -0.0264
ai,j,Pd
Fa Fb Fc Fpd
-0.2445 -0.0783 -0.0436 0
-0.1675 -0.0855 -0.0485 0
-0.3466 -0.0940 -0.0525 0
-0.2318 -0.0723 -0.0420 0
-0.2901 -0.0671 -0.0394 0
0 0 0 -0.1659
bi,j,Pa (1/day)
Fa Fb Fc Fpd
0.0050 0.0060 0.0070 0.0010
0.0054 0.0062 0.0070 0.0010
0.0053 0.0054 0.0069 0.0010
0.0049 0.0058 0.0073 0.0010
0.0050 0.0057 0.0068 0.0010
0.0010 0.0010 0.0010 0.0051
bi,j,Pb (1/day)
Fa Fb Fc Fpd
-0.0041 -0.0041 -0.0105 -0.0040
-0.0043 -0.0042 -0.0100 -0.0040
-0.0038 -0.0038 -0.0115 -0.0040
-0.0037 -0.0038 -0.0112 -0.0040
-0.0038 -0.0043 -0.0100 -0.0040
-0.0040 -0.0040 -0.0040 -0.0039
bi,j,Pd (1/day)
Fa Fb Fc Fpd
-0.0007 -0.0012 -0.0014 -0.0010
-0.0010 -0.0011 -0.0013 -0.0010
-0.0006 -0.0010 -0.0012 -0.0010
-0.0007 -0.0013 -0.0015 -0.0010
-0.0006 -0.0014 -0.0016 -0.0010
-0.0010 -0.0010 -0.0010 -0.0010
ci,j,Pa
Fa Fb Fc Fpd
0.5900 0.3600 0.3000 0
0.6378 0.3698 0.3109 0
0.5637 0.3548 0.2979 0
0.5987 0.3626 0.3011 0
0.6432 0.3628 0.2985 0
0 0 0 0.6067
ci,j,Pb
Fa Fb Fc Fpd
0.0650 0.1600 0.1600 0
0.0610 0.1581 0.1523 0
0.0747 0.1663 0.1631 0
0.0724 0.1620 0.1572 0
0.0599 0.1524 0.1513 0
0 0 0 0.0666
ci,j,Pd
Fa Fb Fc Fpd
0.5701 0.1136 0.0686 0
0.4430 0.1208 0.0735 0
0.6944 0.1293 0.0775 0
0.5359 0.1076 0.0670 0
0.5733 0.1024 0.0644 0
0 0 0 0.4732
∑y
i,j,k
i
g
∑y
i,j,k'',
1
k < k'' e NB,
2
∀j
(21)
i
Equations 22 and 23 show that if the kth cleanup in the furnace j is after the k′th cleanup in the furnace j′, then the k′′th cleanup in the furnace j should also be after the k′th cleanup in the furnace j′ (k′′ > k), and vice versa. Figure 5 gives an illustration.
3
4
5
6
xj,k,j',k' e xj,k'',j',k',
j < j',
k < k'' e NB
(22)
xj,k,j',k' g xj,k,j',k'',
j < j',
k' < k'' e NB
(23)
Equations 24 and 25 represent logic relations characterized by binary variables of x and y: if two consecutive x’s has the relation of xj,k,j′,k′ - xj,k,j′,k′+1 ) 1, then there must be a batch slot being used between these two cleanups, i.e., ∑iyi,j′,k′+1 ) 1; and vice versa.2 Figure 6 gives the illustration.
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Table 2. Operational Cost and Cleanup Cost Values furnace j parameter
feed i
1
2
3
4
5
6
Fa Fb Fc Fpd Fa Fb Fc Fpd
16.55 16.80 17.20 0 86700.0 87361.0 89154.0 0
17.74 17.61 15.50 0 92758.3 90964.4 87879.6 0
15.55 18.27 16.06 0 81960.6 79695.6 84239.9 0
18.18 17.91 17.29 0 94973.8 80087.9 81375.3 0
16.66 18.25 16.18 0 86733.3 82357.1 92398.7 0
0 0 0 16.94 0 0 0 88625.2
Cvi,j ($/ton)
Csi,j ($)
Table 3. Additional Parameter Values of Feeds and Products Cri Floi Fupi ($/ton) (ton/day) (ton/day)
feed i Fa (gas) Fb (naphtha) Fc (light diesel) Fpd (ethane)
444.13 415.49 411.01 444.13
∑y
i,j',k'+1gxj,k,j',k'
70 420 80 110
210 1260 240 410
product l
Pl ($/ton)
Pa (ethylene) Pb (propylene) Pc (others) Pd (ethane)
866.67 718.53 356.67 444.13
- xj,k,j',k'+1,
j < j',
k' < NB (24)
- xj,k,j',k',
j < j',
k < NB
i
∑y
i,j,k+1gxj,k+1,j',k'
(25)
highest propylene yield. The production yields are also relevant to the design of a furnace. Although furnaces 1-5 are similarly designed; the performance of each is still a little bit different. All the information is shown in Table 1. As mentioned before, the scheduling objective is to maximize the average net profit per day, which comes from product profits subtracting raw material cost, operational cost, and cleanup cost. The related cost data is listed in Tables 2 and 3. The upper time limit of M in the model is set as 365 days. The scheduling model is implemented in GAMS v23.3.17 The developed MINLP model has 1746 binary variables, 2112 continuous variables, and 12247 constraints. The MINLP model is solved by the
i
3.9. Variable Bounds. All the continuous variables have a lower bound of zero, and all the starting times, ending times, batch times, and the total cycle time should be less than the upper bound. Variables x, y, and z are defined as binary variables. Ej,k, Gi, Sj,k, ti,j,k, Ej,k, Sj,k, ti,j,k,
T g 0,
∀i, j, k
TeM
zj, yi,j,k, xi,k,j',k' ∈ {0, 1} ∀i, j, k, j', k'
(26) (27) (28)
The solution scheduling is searched in the pool which obeys the above constraints and variable bounds, and is shown and analyzed in a case study. 4. Case Study A case study with real ethylene plant data is presented in this paper. It is a cyclic scheduling problem for a cracking furnace system with six furnaces processing four types of feeds. The feeds include gas, naphtha, light diesel, and ethane represented by Fa, Fb, Fc, and Fpd, respectively. After cracking, each feed would generate four types of products (NP is 4), which are ethylene, propylene, ethane, and the left products represented by Pa, Pb, Pd, and Pc, respectively. The cracking furnace system includes one specific ethane furnace for processing recycled and fresh ethane and five ordinary furnaces for processing the other feeds. Table 1 shows the product yield data, which comes from the regression of the industrial data. Note that when the product yields of ethylene, propylene, and ethane for a feed cracked at a furnace are obtained, the yield of the remaining products will be identified according to the mass balance. Generally, it can be seen that the ethylene yield is decaying with time, while the propylene and ethane yields are increasing with time. Among the four feeds, ethane feed has the highest ethylene yield, but lowest propylene yield. Conversely, the light diesel has the lowest ethylene yield but
Figure 7. Heuristic scheduling results.
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Table 4. Detailed Solution Results Based on the Optimal Schedule
furnace 1
2
3
4
5 6 total
batch number
feed type
batch starting time
batch processing time (day)
ethylene revenue (M$)
propylene revenue (M$)
ethane revenue (M$)
other product revenue (M$)
feed cost (M$)
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 1 2
Fc Fb Fb Fb Fa Fc Fb Fb Fb Fb Fb Fb Fb Fb Fb Fb Fb Fa Fa Fpd Fpd
203 39 93.7 148.3 183.2 36 81 134 159 3 55 107 205 49 101 153 207 57 132 156 52
41 52.7 52.6 52.7 58.8 43 51 47.2 49 49 49 49 50 50 50 50 56 73 73 101 101
2.04 2.74 2.74 2.74 6.34 2.32 2.87 2.67 2.34 2.34 2.34 2.34 2.61 2.61 2.61 2.61 3.01 7.71 7.71 9.85 9.85 82.39
0.88 1.04 1.04 1.04 0.36 0.91 1.04 0.96 0.93 0.93 0.93 0.93 1.00 1.00 1.00 1.00 1.10 0.40 0.40 0.68 0.68 18.25
0.10 0.18 0.18 0.18 1.57 0.11 0.18 0.17 0.15 0.15 0.15 0.15 0.17 0.17 0.17 0.17 0.20 2.00 2.00 2.97 2.97 14.09
1.59 2.01 2.01 2.01 0.46 1.75 2.05 1.89 1.72 1.72 1.72 1.72 1.86 1.86 1.86 1.86 2.27 0.57 0.57 0.79 0.79 33.08
-3.39 -4.42 -4.42 -4.42 -5.61 -3.73 -4.53 -4.20 -3.81 -3.81 -3.81 -3.81 -4.15 -4.15 -4.15 -4.15 -4.90 -6.90 -6.90 -9.42 -9.42 -104.10
operation cost (M$)
cleanup cost (M$)
batch net profit (M$)
-0.14 -0.18 -0.18 -0.18 -0.22 -0.14 -0.19 -0.18 -0.17 -0.17 -0.17 -0.17 -0.18 -0.18 -0.18 -0.18 -0.22 -0.26 -0.26 -0.36 -0.36 -4.27
-0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.09 -0.09 -0.09 -0.09 -1.80
0.99 1.28 1.28 1.28 2.81 1.13 1.33 1.22 1.08 1.08 1.08 1.08 1.23 1.23 1.23 1.23 1.38 3.43 3.43 4.42 4.42 37.64
solver DICOPT, and the sub solvers CPLEX and CONOPT are adopted to handle the MIP and NLP problems respectively.18-20 For comparison, a heuristic solution based on industrial expertise is shown in Figure 7. This scheduling follows all the
same constraints, and variables are all within their bounds. The idea behind this heuristic solution is that a feed should be allocated to the furnace that could generate higher product yields. For instance, Fb is cracked in furnaces 3 and 4 because it has higher ethylene yield from these two furnaces than that of the
Figure 8. Optimal scheduling results for ethylene production.
Figure 9. Optimal scheduling results for propylene production.
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Figure 10. Optimal scheduling results for ethane production.
other furnaces. Similarly, Fc is only processed in furnace 2 and Fa is processed in furnace 5 for higher ethylene yield consideration. Because Fb has the largest supply rate, furnace 1 can be used to handle Fb. As a result, the heuristic solution gives 150 days of the total cycle time, during which the allocations for different feeds and furnaces are shown in Figure 7. The industrial-experience-based heuristic solution is an effective schedule. The net profit per day for the cracking process is $178,769, which is equivalent to a total profit of $65.25 million per year. Based on the developed scheduling model, the solution gives a total cycle time of 208 days and an average net profit of $180,744/day, about $65.97 million/year. A detailed solution is attached in Table 4. There are four batches in furnaces 1-4, three batches in furnace 5, and two batches in furnace 6 during one cycle. As shown in Figure 8, the first batch of furnace 1 processes Fc (light diesel) followed by three batches of Fb (naphtha); in furnace 2, the first batch processes Fa (gas) followed by an Fc batch and two Fb batches; furnaces 3 and 4 are only cracking the single feed of Fb, because Fb accounts for the majority of the total feed supply and it has higher ethylene yield in these two furnaces; in furnace 5, three batches are arranged as Fb, Fa, and Fa; and for furnace 6 (ethane furnace), it contains only two batches of recycled ethane. Figure 8 illustrations a and b show the ethylene yield curve for each batch and the flow rate fluctuation of the total ethylene product in a cycle, respectively. Similarly, Figures 9and 10 provide yield
and flow rate fluctuation information for products of propylene and ethane, respectively. The overall performance of the cracking furnace system for producing major products is shown in Figure 11, where the decaying and increasing trends of different product yields are consistent with the regressed formulas. Since a furnace decoking will influence the mass throughput of the furnace system, it is valuable to obtain the information of the total flow rate change during the manufacturing. Figure 12 gives the mass flow rate load to the cracking furnace system based on the optimized schedule. It shows the total flow rate drops because of furnace shutdowns for decoking. The fluctuation characterized by the standard deviation of the total flow rate is 86.2, which is better than that of the heuristic solution (90.6 from Table 5). That means the cracking furnace system has inherently unstable features due to the decoking operation. How to mitigate the influence from decoking to downstream manufacturing system is a topic worthy of future study. Note that because simultaneous furnace cleanups are avoided in the developed scheduling model, the inherently unstable situation has been addressed to some extent. The comparison results between the optimal solution and the heuristic solution are summarized in Table 5. It shows that, although the heuristic schedule is acceptable, the solution optimal schedule can obtain nearly $2,000/day more, which is about $0.72 million per year. Since the average product yields are almost the same, the profit increment actually comes from the increase of the average mass flow rate to the cracking furnace system. Meanwhile, the total flow rate fluctuation is also reduced, which is another benefit. Overall, the case study manifests the superiority of the developed furnace scheduling model. Finally, it should be clarified that it would be unnecessary to designate specific ethane cracking furnace(s) if secondary ethane cracking was not considered. Under this situation, the feed allocation constraints (see, eqs 8 and 9) should be removed from the scheduling model, which means the updated optimization model will have less stringent constraints. Thus, the optimization results without considering secondary ethane cracking would supposedly be better than those with considering secondary ethane cracking. However, since the industrial practice is that every ethylene plant has specific ethane cracking furnace(s) to handle the secondary ethane cracking and employs nonsimultaneous furnace cleanups, there would be no practical meaning to study on the difference with/without secondary ethane cracking and nonsimultaneous furnace cleanups. 5. Concluding Remarks For multiple cracking furnaces processing different feeds under various product values and manufacturing costs, the operational scheduling for the entire furnace system should be optimized to achieve the best economic performance. In this paper, a new MINLP model has been developed to obtain cyclic scheduling strategies for cracking furnace systems with the consideration of secondary ethane cracking. It is more practical than previous studies and can simultaneously identify the allocation of feeds with their quantity, time, and sequence information for each cracking furnace. The case study has demonstrated the economic potential of the developed methodology. It also shows that the furnace system scheduling has the capability to mitigate the decoking-induced unstable situations from the very beginning of cracking operations.
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Figure 11. Product yield curves for six furnaces based on the optimized schedule.
Acknowledgment This work was supported in part by Texas Air Research Center (TARC) and the Research Enhancement Grant from Lamar University. Nomenclature
Figure 12. Mass flow rate load to the cracking furnace system based on the optimized schedule. Table 5. Comparison between Optimal and Heuristic Schedule Results
profit ($/day) total cycle time (day) average ethylene yield average ethane yield average propylene yield average flow rate (ton/day) total flow rate fluctuation (standard deviation)
optimal solution
heuristic solution
180,744 208 0.388 0.130 0.104 1,177 86.2
178,769 150 0.388 0.130 0.104 1,167 90.6
Sets and Indices i ) 1, ..., NF ) number of different feeds for cracking j, j′ ) 1, ..., NC ) number of cracking furnaces k, k′, k′′ ) 1, ..., NB ) number of batches assigned for each furnace during one cycle time l ) 1, ..., NP ) number of considered products Parameters ai,j,l ) pre-exponential factor of product l’s yield formula for the feed i cracked in the furnace j bi,j,l ) power factor of product l’s yield formula for the feed i cracked in the furnace j ci,j,l ) constant coefficient of product l’s yield formula for the feed i cracked in the furnace j Cri ) raw material cost for the feed i Csi,j ) one cleanup cost for the feed i cracked in the furnace j Cvi,j ) batch operation cost for the feed i cracked in the furnace j Di,j ) flow rate of the feed i cracked in the furnace j Floi ) lower bound of the total average flow rate for the feed i Fupi ) upper bound of the total average flow rate for the feed i M ) upper bound of total cycle time Pl ) market price for the product l tloi,j ) lower bound of batch processing time for the feed i cracked in the furnace j tupi,j ) upper bound of batch processing time for the feed i cracked in the furnace j τi,j ) cleanup time used after the feed i cracked in the furnace j
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Variables Ej,k ) ending time point of the kth batch in the furnace j Gi ) extra amount of flow rate for feed i that is processed above Floi Sj,k ) starting time point of the kth batch in the furnace j ti,j,k ) processing time for the feed i cracked in the kth batch of the furnace j T ) total cycle time of the scheduling problem xj,k,j′,k′ ) binary variable which is 1 if the kth cleanup in the furnace j is no-overlap behind the k′th cleanup in the furnace j′; otherwise, if the kth cleanup in the furnace j is no-overlap ahead of the k′th cleanup in the furnace j′, it is 0. yi,j,k ) binary variable which is 1 if the feed i is processed in the kth batch of the furnace j; otherwise, it is 0. zj ) binary variable which is 1 if the starting time of the first batch in the furnace j is larger than its ending time; otherwise, it is 0.
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ReceiVed for reView January 19, 2010 ReVised manuscript receiVed April 14, 2010 Accepted April 21, 2010 IE1001235
