Relative Concentration-Based Mathematical Optimization for the

Article pubs.acs.org/IECR

Relative Concentration-Based Mathematical Optimization for the Fluctuant Analysis of Multi-Impurity Hydrogen Networks Qiao Zhang, Huachao Song, Guilian Liu,* and Xiao Feng School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an, 710049, China ABSTRACT: Hydrogen is a crucial and expensive resource for refineries. Hydrogen network integration is an effective way to conserve fresh hydrogen. On the basis of relative concentration analysis, a mixed integer linear programming (MILP) mathematical model is established to synthesize refinery multi-impurity hydrogen networks. By correlating the hydrogen consumption of a sink with its oil processing throughputs, this model reveals the fresh hydrogen variation tendency caused by fluctuation of each capacity, and determines the key sink(s) of a hydrogen network with multiple impurities on the premise of minimum fresh hydrogen consumption. With a structure matrix variable, practical hydrogen networks can be analyzed through the proposed model. The results show that the novel method in this paper is superior to traditional absolute concentration-based methods in fresh hydrogen conservation under cases of static, fluctuant throughputs, and fluctuant throughputs with restricted network structure. Two cases are employed to demonstrate its application.

1. INTRODUCTION Hydrogen is a more and more important and expensive utility for chemical industries, especially in oil refineries. Stricter environmental protection regulations and continuous increase in cleaner product oil demand have resulted in a huge deficiency of fresh hydrogen supply. Process integration plays an important role in resource conservation of refinery utilities, including energy, water, hydrogen, etc. Overall, both pinchbased conceptual and mathematical programming methods are widely applied in hydrogen network integration, including both single-impurity and multi-impurity hydrogen networks. Alves and Towler1 employed hydrogen purity profile and hydrogen surplus diagram for the analysis of hydrogen distribution system as a pinch-based conceptual methods. ElHalwagi et al.2 put forward a graphical targeting method to conserve pure fresh resources in impurity load versus flow rate pinch diagram, and later Almutlaq et al.3 extended this method to the case of impure fresh resources. Subsequently, Zhao et al.4 performed the minimum fresh hydrogen consumption targeting in hydrogen load versus flow rate diagram. Foo et al.5 put forward a new method based on pinch analysis named property cascade analysis to determine the maximum resource recovery network and soon generalized it to gas cascade analysis for gas integration.6 Afterward, Zhang et al.7 and Yang et al.8 proposed respective methods to deal with the case considering purified reuse, so that more fresh hydrogen can be conserved. However, the literature stated above can only solve single impurity network problems, and although a few works9−11 are suitable to the case of multiple impurities, their targeting processes are not easy to implement and the fresh hydrogen targets are conservative. Pinch analysis includes the advantages of clear © XXXX American Chemical Society

concepts and visual solution. However, although it can cope with a static network, indicating settled operating conditions, it is not able to handle the fluctuant case. Mathematical programming is usually based on a superstructure. Hallale and Liu12 achieve the objectives of saving operating cost and debottlenecking while accounting for pressure constraints and existing equipment. Van den Heever and Grossmann13 developed a multiperiod mixed integer nonlinear programming (MINLP) optimization model taking both reactive scheduling and production planning into consideration. Liu and Zhang14 selected appropriate purifiers through an understanding of trade-offs of hydrogen consumption, compression cost, and capital investment among pressure swing adsorption processes, membranes, and their combinations. Kumar et al.15 set the total annual cost as the target, proposed a systematic approach to maximize the amount of hydrogen recovered by fully accounting for the pressure constraint as well as the existing equipment. Liao et al.16,17 set up mixed integer nonlinear programming (MINLP) optimization models to maximize hydrogen reuse with a consideration of compressors, purifiers, and threshold problems. Wu et al.18 established an optimization model by introducing a hydrogento-oil ratio to minimize the total exergy consumption, and further generalized it to minimize the total cost and the number of compressors.19 Jia and Zhang20 developed an improved nonlinear programming model (NLP) to optimize hydrogen Received: May 31, 2016 Revised: August 1, 2016 Accepted: September 7, 2016

A

DOI: 10.1021/acs.iecr.6b02098 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 1. Composition configuration of traditional hydrogen network superstructure12

not considered, the method can give the same fresh hydrogen target and network design as existing methods. Even more important, when purification is introduced, it gives a better solution than its counterparts because of the flow rate relaxation. Thus, the superiority of a relative concentrationbased method over absolute ones1−5 has been demonstrated in a single impurity case. At present, static and fluctuant single impurity, static multiimpurity hydrogen networks integration methods on the basis of absolute concentration, and relative concentration-based static single impurity network on the basis of relative concentration have been investigated. To conserve fresh hydrogen and make a network more flexible, it is of great importance to develop a relative concentration-based multiimpurity hydrogen network integration method with fluctuant consideration, which is the target of this paper. This work first presents relative concentration analysis for multi-impurity hydrogen networks on the basis of an existing one for the single impurity case and formulates the corresponding static mathematical model. Afterward, by correlating the hydrogen demand of units to their raw oil throughputs, we generalize the static model to the fluctuant case and establish a fluctuant multi-impurity hydrogen networks optimization model. Meanwhile, the network structure matrix is introduced to account for a practical case. This method can not only improve hydrogen utilization efficiency by constraints relaxation but also perform better network flexibility in throughput fluctuation.

networks with light hydrocarbon production and integrated flash calculation. Zhou et al.21 proposed a model that accounts for both economic and environmental aspects of hydrogen networks to perform sustainable hydrogen network optimization. Although those mathematical programming methods are able to handle either single or multiple impurities hydrogen networks, they are all based on absolute concentration and rarely account for fluctuant networks. In fact, hydrogen network is not static but fluctuant, because throughput of each unit, properties of streams, availability of fresh resources and utilities, catalysts deactivation, facilities maintenance, as well as economic considerations, etc., all have uncertainties. For hydrogen networks, Amad et al.22 suggested that multiperiod operation is more practical than constant benchmark to describe hydrogen networks with varying operating conditions. A MINLP model is established and solved by linear relaxation to minimize the total annual cost. Jiao et al.23 further developed a MINLP model to address uncertainties of hydrogen demand fluctuation, pipeline level difference, and hydrogen units shutdowns under minimum annual cost, so as to get more flexible network design. Wang et al.24 considered the fluctuant hydrogen network integration by linearly correlating hydrogen sink demand to its throughput, and minimized the fresh hydrogen consumption using a linear programming model. Although those recent works focus on the fluctuant operating conditions, they are only suitable to single impurity hydrogen networks. The hydrogen source and sink properties of both pinchbased graphical and mathematical programming methods stated above are performed by total flow rate (sum of hydrogen and impurities flow rates) and absolute hydrogen/impurity concentrations that lie in [0,1]. Practically, either hydrogen source or sink encompasses two parts: hydrogen and impurity, and they are considered separately. Therefore, it is unreasonable to unify them by the total flow rate and concentration normalization, which obviously constrain the flexible allocation between sources and sinks and consequently go against effective use of a fresh resource. To overcome such a drawback, Wu et al.18 employed a hydrogen-to-oil ratio, instead of total flow rate, to measure the net pure hydrogen over feed oil in volume benchmark. Using this constraint, they built mathematical models to optimize hydrogen network performance on minimization exergy18 and number of compressors,19 respectively. On the basis of such a concept, Zhang et al.25 put forward relative concentration-based pinch analysis to synthesize single impurity hydrogen networks. When purification is

2. THEORY 2.1. Relative Concentration Description of Hydrogen Sources and Sinks. Figure 1 shows the traditional absolute concentration-based source−sink superstructure12 of multiimpurity hydrogen networks with m sources, n sinks, and s impurities. In Figure 1, each blank frame filled by F represents the total flow rate of a source or sink, The divided gray frames denote, for each source or sink, the composition, including hydrogen and impurities. Specifically, H is hydrogen flow rate, while I is flow rate for each impurity; yH2 and C are absolute hydrogen and impurities concentrations, respectively. Subscripts sr and sk are employed to distinguish source and sink, and i, j, and k are the number of source, sink, and impurity among 1, 2, 3, ···, m, 1, 2, 3, ···, n and 1, 2, 3, ···, s, respectively. Within each source or sink, the length sum of gray frames is exactly equivalent to that of the blank one, namely the total flow rate encompasses B


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Figure 2. Relative concentration-based superstructure of hydrogen networks with multiple impurities.

and lowering the hydrogen partial pressure.18 In other words, when the hydrogen flow rate is satisfied, the flow rate of each impurity can be varied within a feasible range instead of a decided level, which is, however, the result given by absolute concentration basis in Figure 1. Therefore, the total flow rate constraint, eq 1, and the concentration normalization, eq 4, should both be released. Consequently, similar to the relative concentration-based pinch analysis for a single impurity hydrogen network proposed by Zhang et al.,25 this paper performs the constraints by substituting relative concentration for the absolute basis, and Figure 2 shows such a novel superstructure. Similarly, the subscripts and quantifications in Figure 2 are the same as that in Figure 1, with the exception of blank frames representing total flow rate (H) and divided gray frames (I) representing the relative contents of sources and sinks, respectively. After elimination of total flow rate and concentration normalization, the constraints of eqs1−4 are transformed to the following three:

hydrogen and impurities, and such balances can be expressed by the following four equations: s

F=H+

∑ Ik ,

k ∈ {1, 2, 3, ···, s} (1)

k=1

yH = 2

H , s H + ∑k = 1 Ik

Ck =

Ik , s H + ∑k = 1 Ik

yH +

∑ Ck = 1,

k ∈ {1, 2, 3, ···, s} (2)

k ∈ {1, 2, 3, ···, s} (3)

s 2

k=1

k ∈ {1, 2, 3, ···, s} (4)

where eq 1 is the flow rate mass balance, eqs 2 and 3 are definitions of absolute concentrations of hydrogen and impurities, and eq 4 is the absolute concentration normalization, respectively. Such a superstructure indicates all hydrogen sources and sinks with multiple impurities are performed by absolute concentrations to implement the network integration. Specifically, sr1 is the fresh hydrogen and the others are internal sources; each source can supply hydrogen for any sink while each sink can receive hydrogen from any source; any sources excepting fresh hydrogen can be fully or partially sent to the fuel gas system. The parametric difference between sources and sinks is that flow rate and concentrations of the former are practical properties of streams, whereas that of the latter are lower (for flow rate and hydrogen concentration) and upper (for impurity concentrations) bounds. The target of this superstructure is to make full use of internal sources and minimize fresh hydrogen consumption, as well as the waste discharge, under the precondition that fulfills the hydrogen demand of sinks. Although Figure 1 can reflect composition and the mapping road of source and sink streams in refinery hydrogen networks, its composition analytical criterion is not consistent with practical concerns. Refinery hydroprocessing operations do not inspect hydrogen and impurities integrally, but separately, namely by hydrogen-to-oil ratio and impurity contents. Hydrogen is the desired component used to satisfy hydroprocessing, while impurities are undesired and should be controlled under certain levels so as to avoid poisoning catalysts

H = F ·yH

Ik = F ·Ck , RCk =

(5)

2

k ∈ {1, 2, 3, ···, s}

Ik F · Ck C = = k, H F ·yH yH 2

2

(6)

k ∈ {1, 2, 3, ···, s} (7)

where eq 5 is hydrogen flow rate balance, eq 6 is impurity flow rate balance for each impurity, and RC is relative concentration for all sources and sinks. Equations 5−7 are relative concentrations of impurities to describe sources and sinks. Since hydrogen network integration is performed by flow rate and concentration, which are stream properties, property description is crucial. Property parameters should be consistent with practical concerns, so that a reasonable target can be determined by integration methods. Relative concentration hydrogen networks are the practical case and thus the properties eqs 5−7 are superior to eqs 1−4 in performing constraints. Since the advantages have been demonstrated in the single impurity case, relative concentration-based integration has even better performance in multi-impurity hydrogen networks as it has more complex constraints. 2.2. Problem Statement. C


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Fluctuant Networks. The hydrogen demand, Hj, in eq 11 is assumed to be constant; however, it is fluctuant. Wang et al.24 proposed a linear correlation of total flow rate and raw oil feed to quantify the hydrogen demand as throughput fluctuation:

Model Assumptions 1. The uncertain condition considered in this paper are the throughput of sinks and availability of fresh hydrogen; 2. For any sink, the hydrogen consumption is linearly related to its feed flow rate; 3. Only one sink among all has throughput fluctuation. Parameters 1. Total flow rates (Fsri), hydrogen, and impurities concentrations (yH2,sri,Ck,sri) of all sources. 2. For each sink, the current feed throughput(VO,j), the correlation between throughput and hydrogen demand (eq 14), upper bound (RCk,skj) of relative concentration for each impurity. Variables and Objective εj is the fluctuant variable to quantify the sink throughput range, f jis the control variable used to select the fluctuant sink, and Vj is the dependent variable of εj. Network structure matrix x(i,j). Hi,j is the decision variable. The target is to identify the minimum FFH as selected εj changes, so as to investigate the influence of the relative concentration property on fresh hydrogen minimization. 2.3. Mathematical Model. Hydrogen Source Availability. For each source, the sum of hydrogen flow rates supplied to all sinks and the fuel gas system is consistent with its current hydrogen flow rate.

FH = C1Ff + C2

where FH is the total flow rate and Ff is the raw oil feed flow rate, C1 and C2 are two fitting constants. Similarly, this work also linearly correlates hydrogen demand (Hj) and raw oil throughput (Vj) of sinks: Hj = KjVj + bj ,

Vj = (1 + εj)VO , j ,

εj ≤ f j ·εUP ,

The total hydrogen flow rate(Hfg) of fuel gas system: m

(10)

Hydrogen Sink Hydrogen Flow Rate Requirement. For each sink, the sum of hydrogen flow rates supplied from all sources should not be less than its hydrogen flow rate demand, Hj. m

∑ Hi ,j ≥ Hj ,

j ∈ {1, 2, 3, ···, n}

Hydrogen Sink Impurity Relative Concentration Constraint. For each impurity in each sink, the concentration (RCi,k) average of all supplied sources should not be more than its upper limit (RCj,k). m

(17)

(18)

where Hi,jUP is the upper bound of Hi,j. Besides, in order to avoid Hi,j being too small, a lower bound should be set, as eq 19:

m

Hi , j ≥ HiLO ,j ,

i=1

j ∈ {1, 2, 3···n}, k ∈ {1, 2, 3···s}

=1

i ∈ {1, 2, 3, ···, m}, j ∈ {1, 2, 3, ···, n}

∑ Hi ,j RCi ,k /∑ Hi ,j ≤ RCj ,k , i=1

(16)

Hi , j ≤ x(i , j) ·HiUP ,j ,

(11)

i=1

(15)

Equation 16 is used to define the upper bound of a fluctuant factor, and eq 17 allows only one sink to fluctuate. After incorporating eqs(14−17, the static model is developed for the fluctuant case, which is a mixed integer linear programming (MILP) model. Network Structure. For nominal hydrogen networks, its structure is unrestricted. Practically, the hydrogen network structure is not random as Figure 1 or 2 but optional among several scenarios, because the connections of sources and sinks depend on the availability of the infrastructure, such as pipelines, valves, compressors, and so on. Thus, network synthesis should be performed within feasible structure designs, which is also a constraint that can be controlled by a binary variable x(i,j). When x(i,j) is 0, there is no hydrogen supply or receipt between corresponding source and sink, and thus not any aforementioned infrastructure placed between those pairs of source and sink in this given refinery; when x(i,j) is 1, hydrogen supply and receipt happens and practically the aforementioned infrastructure is placed. Such a structure matrix can judge allocations and Hi,j among sources and sinks as follows:

(9)

i=1

j ∈ {1, 2, 3, ···, n}

j=1

where Hi,j is hydrogen flow rate supplied from source i to sink j and Hi,fg is that from source i to fuel gas system. The first source is the fresh hydrogen and it should not be sent to fuel gas system.

∑ Hi ,fg = Hfg

j ∈ {1, 2, 3, ···, n}

n

∑ fj

(8)

H1,fg = 0

(14)

where VO,j is the current feed oil throughput of sink j and εj is its fluctuant factor.

i ∈ {1, 2, 3, ···, m}

j=1

j ∈ {1, 2, 3, ···, n}

where V is the raw oil feed flow rate, and K and b are two hydrogen flow rate related correlation coefficients. Besides, to estimate the throughput flexibility, the raw oil feed flow rate is stated as the linear function of current throughput:

n

∑ Hi ,j + Hi ,fg = Hi ,

(13)

i ∈ {1, 2, 3, ···, m}, j ∈ {1, 2, 3, ···, n} (19)

(12)

After incorporating eqs 18 and 19, the fluctuant model is restricted by limited connections among sources and sinks, and then the fluctuant model with network structure is a mixed integer linear programming (MILP) model.

Equations 5−12 constitute the constraints of static multiimpurity hydrogen networks, and it is a linear programming (LP) model. D


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Industrial & Engineering Chemistry Research Objective Function. Relative concentration in this work is the improvement of property benchmark and there is not any newly added units or equipment, so the target is to minimize the fresh hydrogen consumption: n

min FFH =

∑ H1,j/yH j=1

2

(20)

In summary, eqs 5−12 and 14−19 are constraints, and eq 20 describe the objective function. Since there are binary variable x(i,j) and not nonlinear terms, this model is mixed integer linear programming (MILP). Totally, three mathematical models for three cases are proposed above: LP model static network, MILP model fluctuant network, and MILP model for fluctuant network with given structure. In each case, a multi-impurity hydrogen network will be optimized under both absolute concentration model, which is performed by eq 1−4, 8−12, and 14−19, and relative concentration basis, which is performed by eq 5−7, 8−12, and 14−19, so as to demonstrate the advantage of the proposed superstructure and models. In three cases, either basis is performed in the same constraints excepting absolute and relative concentrations, and thus the advantage can be clearly verified.

3. CASE STUDIES In this section, two cases are employed to demonstrate the application of the proposed method. The first case is from the Table 1. Source and Sink Data of a Hydrogen Network11 concn of impurities (%) streams

total flow rate (mol/s)

Hydrogen Sources 500 sr1(FH) sr2 176 sr3 428 sr4 576 sr5 295 sr6 104 sr7 271 Hydrogen Sinks sk1 219 sk2 127 sk3 191 sk4 275 sk5 315 sk6 391

absolute hydrogen concn (%)

H2S

N

C

99.99 87.2 95.4 97.2 93.22 85.4 89.4

0 1 1.3 1.7 0.6 2.26 4.5

0.01 2.3 1.5 0.5 5.14 7.23 2.6

0 9.5 1.8 0.6 1.04 5.11 3.5

82.3 91.33 87.1 98.2 84.7 96.5

0 1.2 3.5 0.5 3 1.5

7.8 5.4 3.9 0.6 1.2 0.7

9.9 2.07 5.5 0.7 11.1 1

Figure 3. Static hydrogen network for case 1.

Table 2. Source and Sink Data of a Practical Refinery Hydrogen Network in China concn of impurities (%) streams Hydrogen sr1(FH) sr2 sr3 sr4 sr5 sr6 Hydrogen sk1 sk2 sk3 sk4 sk5

literature,11 and the second case data is from a large practical refinery in China. 3.1. Case 1static Network. The first hydrogen network has three impurities, H2S, N, and C, seven sources and six sinks, and sr1 is the fresh hydrogen with maximum total flow rate 500 mol/s and absolute hydrogen concentration of 99.99%. The detailed data are shown in Table 1. For case 1, on the basis of static network, when the static model is employed and solved by GAMS 22.3, Baron solver, the minimum fresh hydrogen consumption is 373.5 mol/s. Meanwhile, the absolute concentration-based model gives a minimum fresh hydrogen 413.2 mol/s. Therefore, the relative concentration method can further reduce 39.7 mol/s fresh

total flow rate (Nm3/h) Sources 70000 56000 58000 32000 7500 10000 Sinks 55200 (76t/h) 90860 (130t/h) 40630 (65t/h) 8441 (22 t/h) 11388 (18t/h)

absolute hydrogen concn (%)

H2S

N

C

99.5 91.5 89.9 88 90 90

0.09 0.7 2.16 3.7 5.6 0.6

0.3 1.6 6.34 4.1 2.2 8.3

0.11 6.2 1.6 4.2 2.2 1.1

92 90.86 89.8 91.8 88.3

3.2 1.2 2.1 3.2 2.2

1.7 5.54 2.3 2.1 6.5

3.1 2.4 5.8 2.9 3

hydrogen. Figure 3 shows the corresponding hydrogen network. 3.2. Case 2Static Network. Table 2 shows the second case which encompasses same impurities as case 1, six sources and five sinks, and sr1 is also the fresh hydrogen with maximum total flow rate 70000 N m3/h and absolute hydrogen concentration 99.5%. Furthermore, Table 3 gives detailed data about sinks, including current throughputs of all sinks, E


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Industrial & Engineering Chemistry Research Table 3. Current Crude Oil Data and Their Correlation Coefficients hydrogenation unit

K

b

minimum hydrogen to oil rate (V/V)

current throughput (t/h)

fluctuant range of throughput (t/h)

high pressure hydrogenation (sk1) petrol and diesel hydrogenation (sk2) hydrorefining (sk3) removal acid (sk4) fission (sk5)

725 700 525 380 635

100 −140 5 81 −42

639 623 473 335 553

76 130 65 22 18

60.8−91.2 104−156 52−78 17.6−26.4 14.4−21.6

Figure 6. Fresh hydrogen variation of sink 2.

Figure 4. Static hydrogen network for case 2.


For case 2, the hydrogen demand of each sink is estimated by eq 14, and the throughputs are current values in Table 3. The targeting procedure is the same as case 1, and the corresponding minimum fresh hydrogen consumption determined by relative concentration is 54117.8 N m3/h, while that by absolute basis is 57815.5 N m3/h. Similarly, the relative concentration method can have another margin of 3697.7 N m3/h. The resulting hydrogen network is illustrated by Figure 4. 3.3. Case 2Fluctuant Network. The results of those two static networks are sufficient to demonstrate the advantage of relative concentration over absolute concentration methods in conservation of fresh hydrogen. Furthermore, there are throughputs data in case 2, and thus the method can be used to perform fluctuation analysis. In fluctuation analysis, the fluctuant MILP model is used and only one sink is chosen every time. Since the operation


their correlation coefficients, hydrogenation unit names, minimum hydrogen-to-oil ratio, as well as fluctuant throughputs upper and lower bounds. All those are given by refinery technicians. F


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Figure 10. Fresh hydrogen variation of sink 2 with certain network structure.


Figure 11. Fresh hydrogen variation of sink 5 with certain network structure.

Table 4. Connection of Fixed Network Structure Mode x(i,j)

sk1

sk2

sk3

sk4

sk5

sr1 sr2 sr3 sr4 sr5 sr6

1 1 0 1 1 0

1 1 1 0 0 1

1 1 0 1 1 0

1 0 0 0 0 0

0 0 1 0 0 0

fresh hydrogen consumption variation while nonkey sink means only a certain level of fluctuation can cause such variation. Specifically, in Figures 5 to 8, sinks 1−4 are key sinks, while in Figure 9, sink 5 in nonkey. The reason for such phenomena is their hydrogen demand qualities, that is, sink 5 has the lowest inlet absolute hydrogen concentration; hence, internal sources, that is, any one between sr2 and sr6 that is higher than or nearly the same as the quality of sink 5, can be used to fulfill sink 5. In such a way, fresh hydrogen will only increase when the fluctuation goes to a certain level. Oppositely, qualities of key sinks are near to that of fresh hydrogen, and thus the fresh hydrogen increase starts relatively earlier. 3.4. Case 2Fluctuant Network with Network Structure. In the synthesis of the fluctuant MILP model with network structure, it is sufficient for us to only select two sinks for two corresponding categories, respectively. Sink 5 is the only nonkey sink and thus should be selected. Note that the plus 20% fluctuation of sink 2 will cause a deficiency (71283.7 N m3/h of relative basis, 76630 N m3/h of absolute basis) of the current fresh hydrogen supply (70000 N m3/h), so sink 2 is selected as a key sink. The network structure matrix x(i,j) is shown in Table 4. Such a matrix extracted by refinery technicians gives the practical network.

resilience of this refinery is generally within minus 20% and plus 20%, this is selected as the fluctuation range, with an increment of 4% as well. By using GAMS 22.3, Baron solver, we obtained the graphs in Figures 5−9 indicating the minimum fresh hydrogen consumption variation as the fluctuation range change of each sink, respectively. What can be seen from those five graphs is that for each sink, whether throughput increases or decreases, the relative concentration basis is always better in fresh hydrogen conservation. Such results reveal that this method is superior not only in a static network but also in the fluctuant case. In addition, from the tendency of all the curves, all sinks can be divided into two categories: one is the key sink and the other is the nonkey sink. Key sink means any fluctuation can cause G


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Figure 14. Comparison between individual fluctuation of sinks 1, 2, and 3 and simultaneous fluctuation of the three sinks.

sr2−sr6 are internal sources with respective total flow rates and sr1 is fresh hydrogen with maximum total flow rate (78319 N m3/h) determined by 20% fluctuation of sink 2 in Figure 10 because sink 2 always has the largest fresh hydrogen consumption. Such data indicate that the fresh hydrogen total flow rate and the network structure in Figure 12 can fulfill fluctuation of any sink within minus 20% and plus 20% under minimum fresh hydrogen consumption. In addition, all these mathematical models are coded in GAMS 22.3 and solved by Lenovo T4900c, with Intel Pentium CPU G2030, 3.00 GHz. All of them have 10 constraints, 7 continuous variables, and 2 binary variable at most. Finally, they are solved by three solvers: DICOPT, CPLEX, and BARON, and all in less than 1 s, whichever solver is used. Therefore, the solver hardly has influence on solution of those models. In summary, the relative concentration-based, the relaxed constraint in other words, multi-impurity hydrogen network synthesis can conserve more fresh hydrogen than the traditional absolute concentration-based ones under the same conditions. The reason is that relative concentration released the total flow rate and concentration normalization. Such a property benchmark reveals that the hydrogen demand of sinks must be satisfied by sources while impurities content can be flexible within a feasible range. Therefore, compared to the traditional absolute concentration basis which results in a decided impurities content upon certain hydrogen demand, the relative concentration constraints are relaxed. Therefore, the allocation of sources and sinks is more flexible, the direct reuse of sources is enhanced, and thus better performance is gained in fresh hydrogen conservation under static, fluctuant, and fluctuant with given network structure circumstances. Besides, the advantage of the relative concentration-based method in fresh hydrogen conservation for single impurity networks can only be demonstrated by considering purified reuse. However, such an advantage in multi-impurity hydrogen networks does not resort to purified reuse. Therefore, the superiority of the relative concentration basis in multi-impurity hydrogen networks is even more remarkable than that in the single case. Although the fluctuation in this work is performed for a single sink every time, note that, the variation of fresh hydrogen as throughput fluctuation is partly linear, which can be seen from all seven figures. Practically, multiple sinks fluctuation is

Figure 12. Fluctuant hydrogen network for case 2 (The flow rate unit is Nm3/h, numbers in blankets are absolute impurity concentrations in unit of mol %).

Figure 13. Comparison between individual fluctuation of sinks 1 and 3 and simultaneous fluctuation of the two sinks.

For those two sinks, the results of fluctuation analysis are illustrated in Figures 10 and 11, where similar fresh hydrogen variation tendency as that in Figures 6 and 9 can be seen. However, the minimum fresh hydrogen consumption in these two graphs is wholly larger than their counterparts. This is occurs because the introduction of network structure constraints makes the network constraints more rigorous, and therefore worse results are obtained. Figure 12 shows the hydrogen network specified by matrix x(i,j). In this figure, all sinks are quantified by their linear correlations with throughputs, and all upper (20% margin) and lower (−20% margin) bounds of throughputs are also listed. H


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Table 5. Fluctuation Factors and Fresh Hydrogen Consumption of Sinks 1 and 3 in Both Individual and Simultaneous Casesa

a

fluctuation scenarios a b c d e f g h i j k

sink 1 fluctuant only


FR

min FH (Nm3/h)

FH change (Nm3/h)

FR

min FH (Nm3/h)

FH change (Nm3/h)

sinks1 and 3 fluctuant simultaneously min FH (Nm3/h)

min FH (Nm3/h)

0.20 0.16 0.12 0.08 0.04 0.00 −0.04 −0.08 −0.12 −0.16 −0.20

65137 63661 62232 60803 59374 57945 56516 55087 53658 52228 50799

7192 5716 4287 2858 1429 0 −1429 −2858 −4287 −5717 −7146

−0.20 −0.16 −0.12 −0.08 −0.04 0.00 0.04 0.08 0.12 0.16 0.20

54642 55303 55963 56624 57284 57945 58605 59266 59926 60586 61249

−3303 −2642 −1982 −1321 −661 0 660 1321 1981 2641 3304

61788 61019 60250 59482 58713 57945 57176 56407 55639 54870 54102

3843 3074 2305 1537 768 0 −769 −1538 −2306 −3075 −3843

Notation: FR, fluctuation range; min FH, minimum fresh hydrogen consumption; FH change, fresh hydrogen change.

Table 6. Fluctuation Factors and Fresh Hydrogen Consumption of Sinks 1, 2, and 3 in Both Individual and Simultaneous Casesa sink 1 fluctuant only

a


sink 1, 2, 3 fluctuant simultaneously


flunctuation scenarios

FR

min FH (Nm3/h)

FH change (Nm3/h)

FR

min FH (Nm3/h)

FH change (Nm3/h)

FR

min FH (Nm3/h)

FH change (Nm3/h)

min FH (Nm3/h)

FH change (Nm3/h)

a b c d e f g h

−0.12 −0.12 −0.12 −0.12 −0.12 −0.12 −0.12 −0.12

53658 53658 53658 53658 53658 53658 53658 53658

−4287 −4287 −4287 −4287 −4287 −4287 −4287 −4287

0.20 0.16 0.12 0.08 0.04 0.00 −0.04 −0.08

72635 69438 66241 63044 59847 57945 56272 54599

14690 11493 8296 5099 1902 0 −1673 −3346

−0.08 −0.04 0.00 0.04 0.08 0.12 0.16 0.20

56624 57284 57945 58605 59266 59926 60586 61249

−1321 −661 0 660 1321 1981 2641 3304

67027 64490 61954 59417 56880 55639 54627 53615

9082 6545 4009 1472 −1065 −2306 −3318 −4330

Notation: FR, fluctuation range; min FH, minimum fresh hydrogen consumption; FH change, fresh hydrogen change.

throughput, especially the linearity of individual fluctuation. Although the linear combination is only verified in this case, it can be a reference for future work. Consequently, we can deduce that when network structure is given and within linear region of individual fluctuation, the variation of fresh hydrogen as a fluctuation of those throughputs can be their individual linear combination. Although the fluctuation range is selected as 20% minus and plus, the relative concentration method can be expected to be always better than the absolute basis method in even a larger range, as the constraints of the relative concentration method are mathematically relaxed. In addition, for sinks 2 and 5, from their fresh hydrogen consumption, as seen in the difference between cases with and without network structure constraint, the practical connections of sources and sinks have a significant influence on fresh hydrogen minimization. The more flexible a hydrogen network is, the more fresh hydrogen will be conserved.

probably more common, and hence we further analyzed such a case. However, multifluctuation relies so much on mathematical models and algorithms and has so many options. Therefore, as a compromise, multifluctuation under the given network structure and the rigorous linear region in Figures 5−9 for all involved fluctuant sinks is investigated, and the results are shown in Figures 13 and 14. In Figure 13, the fluctuation of sinks 1 and 3 are considered simultaneously and separately, illustrated by blue, black, and red lines, respectively. The horizontal points a−k indicate the fluctuant factor εj selection, and the vertical points are the fresh hydrogen consumption, which both correspond to the values in Table 5. From the rigorous linear curves and data summary in Table 5, the fresh hydrogen change caused by the sum of individual fluctuations equals that by the simultaneous fluctuation. That is, point b in Figure 13 shows that when sink 1 increases by 20% capacity, fresh hydrogen increases by 5716 N m3/h and when sink 3 decreases by 20% capacity, fresh hydrogen decreases by 2642 N m3/h. By comparison, when sink 1 increases by 20% and sink 3 decreases by 20% capacity simultaneously, the fresh hydrogen increases by 3074 N m3/h, which is almost the sum of 5716 and −2642 N m3/h. Although some points have a little gap, the linear relationship can be applied to other points in this Figure. Therefore, the simultaneous fluctuation of sinks 1 and 3 are rigorously the linear combination of the two individual fluctuations. The same case can be applied to Figure 14 and Table 6. The reason for such a linear combination may be the linear model, the linear correlation of hydrogen demand and

4. CONCLUSIONS This paper puts forward a relative concentration-based mixed integer linear programming (MILP) mathematical model for multi-impurity hydrogen networks synthesis. First, a relative concentration-based property benchmark is generalized from the single impurity case to the multi-impurity case, so as to construct a new superstructure for source−sink allocation. Afterward, three situation mathematical models, static, fluctuant, and fluctuant with network structure, are established. I


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The results of the case studies fully demonstrate the superiority of this proposed method in fresh hydrogen conservation under the three situations. Relative concentration released the total flow rate and concentration normalization, making allocation of sources and sinks more flexible and enhancing direct reuse of sources, thus creating better performance. Because of partial linearity of the model, the linear section of the single fluctuation results can be generalized to multifluctuation in the same region case by linear combination. The uncertainty considered in this work is only the throughput, other uncertain conditions, such as temperatures, pressures, facilities maintenance, and economic considerations, as well as others, should be included together with throughput fluctuation to generate more practical models in the future.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Postdoctoral Science Foundation of China under Grant 2015M582666 and the National Natural Science Foundation of China under Grant 21506169 is gratefully acknowledged.

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NOMENCLATURE

Symbols

C = contaminant concentration, mol % F = total flow rate of a stream, mol/s or Nm3/h f = fluctuation control binary variable FH = fresh hydrogen ε = throughput flexibility factor H = hydrogen flow rate, mol/s or Nm3/h I = flow rate of impurities, mol/s or Nm3/h m = number of hydrogen source n = number of hydrogen sink RC = relative concentration of impurities, mol % s = number of impurities, mol % sr = hydrogen source sk = hydrogen sink V = oil feed flow rate x = network structure matrix y = absolute hydrogen concentration of sources, mol % Subscripts

i = hydrogen source i i,j = hydrogen source i to sink j j = hydrogen sink j fg = fuel gas k = impurity k min = minimum Superscripts

LO = lower bound UP = upper bound

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REFERENCES

(1) Alves, J. J.; Towler, G. P. Analysis of Refinery Hydrogen Distribution Systems. Ind. Eng. Chem. Res. 2002, 41 (23), 5759−5769. J


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Relative Concentration-Based Mathematical Optimization for the

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