Pinch Sliding Approach for Targeting Hydrogen and Water Networks

Article pubs.acs.org/IECR

Pinch Sliding Approach for Targeting Hydrogen and Water Networks with Different Types of Purifier Junyi Lou, Zuwei Liao,* Binbo Jiang, Jingdai Wang, and Yongrong Yang State Key Laboratory of Chemical Engineering, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, Peoples’ Republic of China ABSTRACT: Purifiers are widely used in refinery hydrogen networks. The placement of purifiers is critical in reducing the utility consumption. Various conceptual methods have been employed to find the optimal placement of the purifier inside the hydrogen network. However, previous research only adopts the performance based purifier model, leaving the mechanism based purifier model behind. More practical and simplified approaches are expected to deal with the complicated mechanism models. This paper improves the existing graphical approach in two aspects: incorporate algebraic equations into graphical methods to find the initial location of purifiers with complicated models; develop a novel pinch sliding approach to simplify the shifting procedure of composite curves. In this method, both performance based and mechanism based models of purifier can be solved. Another advantage of this targeting approach is that the flow rate of streams to the purifier also can be optimized simultaneously. Case studies indicate that this method is effective for both hydrogen and water networks.

1. INTRODUCTION Hydrogen and water are important resources for refineries and other process industries. The rising costs of hydrogen utility and fresh water along with stricter environmental regulations have motivated the judicious use and management of these precious resources. To manage the hydrogen and water resources efficiently, process integration methodologies have been widely applied and recognized as an effective tool. Process integration methodologies stand on the perspective of the system, which treats the research object as hydrogen networks and water networks. For hydrogen networks and water networks, typical process integration methodologies involve mathematical programming approaches1−15 and pinch based conceptual techniques.16 This paper presents a pinch based graphical approach to the hydrogen and water management. The management of hydrogen and water networks involves two alternatives: reuse and purification (regeneration). The pinch based method in hydrogen networks was pioneered by Alves and Towler,17,18 and many targeting methods have been developed since then for the hydrogen reuse cases. They are essentially graphical methods: a hydrogen surplus diagram,18 material recovery pinch diagram,19−21 and limiting composite curves22 and algebraic algorithmic methods on concentration intervals23 and mass load intervals.24 Compared to the fruitful achievement of hydrogen reuse targeting, the targeting of hydrogen purification still has a long way to go. Alves17 suggested a qualitative conclusion that it would be best to place the purifier across the pinch. For quantitative conclusions, early research23,22 fixed the feed purity of the purifier. Liu et al.25,26 developed systematical targeting methods for such situations. Later, research shifted to variable feed purities. Zhang et al.27 introduced an effective tool, the triangle rule, which presents the purifier on the graphical composites. Liao et al.28,29 proposed the sufficient and necessary conditions for the optimal placement of the purifier and provided an algebraic targeting method. These recent achievements have driven the pinch targeting of hydrogen © 2013 American Chemical Society

purification to a real quantitative stage, and this paper is inspired by the new approaches. To the best of our knowledge, the difficulty of purifier targeting mainly lies in the model of the purifier. There are various purifier models while the methods proposed by Zhang et al.30 and Liao et al.29 only work for the performance based purifier model. In addition, the targeting of purifier capacity (its feed flow rate) has not been considered when the purifier feed purity is not fixed. In the present work, an improved graphical targeting approach, i.e., the pinch sliding approach, for hydrogen and water networks with different types of purifier is presented. The triangle rule and the optimal condition theorem are integrated into our method to target the minimum flow rates of utility and purifier feed simultaneously. The proposed approach can be further applied to more complicated purifier models such as membrane separation with a solution-diffusion model.

2. THE OPTIMAL CONDITIONS AND TRIANGLE RULE FOR PURIFIER Purification processes such as pressure swing adsorption (PSA) and membrane and cryogenic separation processes are widely used in refineries to further reduce hydrogen utility consumption. Although the purifiers rely on different separation theories and have different operating characteristics, they can be described by the same equations as follows: Fin = Freg + Fr

(1)

FinC in = FregCreg + FrCr

(2)

Received: Revised: Accepted: Published: 8538

February 25, 2013 April 14, 2013 May 31, 2013 May 31, 2013 dx.doi.org/10.1021/ie4006172 | Ind. Eng. Chem. Res. 2013, 52, 8538−8549

Industrial & Engineering Chemistry Research R=

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FregCreg FinC in

(3)

where F and C represent the flow rate and hydrogen concentration, respectively. Subscripts in, reg, and r represent the feed, purified product, and tail gas of purifier, respectively, and R represents the hydrogen recovery ratio. Equations 1 and 2 describe the material balance of the purifier, while eq 3 defines the hydrogen recovery ratio. 2.1. Necessary Conditions for Optimal Purifier Placement. In the above equations, Fin and Cin are determined by the mixing of hydrogen sources in the hydrogen network. Consequently, Fin and Cin are critical parameters in the consumption of hydrogen utility. To obtain the minimum hydrogen consumption, the optimal Fin and Cin have to be calculated. This is not easy for pinch based methods, because the optimizing variables are more than one. Liao et al.28,29 found that the upper bound of Fin can be expressed as the sum of a series of concentration adjacent hydrogen sources, which is determined by Cin. Therefore, if Fin is fixed to its upper bound, then the optimizing variable is reduced to one. Furthermore, if Cin is larger than the concentration where maximum mixing of hydrogen source streams occurs, the series of concentration adjacent sources starts at the lowest hydrogen source. On the basis of these conditions, Liao et al.28,29 proved the necessary optimal condition for given R and Creg: If a solution of the hydrogen network with purification is an overall optimal, then the following conditions 1, 2, and 3 hold simultaneously or an equivalent solution with the same hydrogen utility consumption exists which satisfies these conditions. (1) Cin > Cin,mm, where Cin,mm is the concentration at which maximum mixing of hydrogen source streams occurs (2) Fin = Fin,Cin,max, where Fin,Cin,max is the maximum flow rate under given Cin (3.1) When Cin = Fin,max, the pinch is located in Cr, where Cin,min is the minimum value of Cin or (3.2) When Cin,mm < Cin < Cin,max, the system has two pinches, which are located in Cr and the concentration interval between Creg and Cin or (3.3) When Cin = Cin,mm, the pinch is in the concentration interval between Creg and Cin or (3.4) The pinch belongs to interval between Creg and Cutility. Although the optimal condition is under the assumption that Fin is fixed to its upper bound, it still can provide useful guidelines to the targeting of utility consumption. Followed by this theorem, a systematic targeting algorithm which involves solving a set of seven quadratic equations has been proposed.29 2.2. Triangle Rule Based Targeting Method. At the same time, Zhang et al.30 developed a visualized tool for purifiers. They employed a triangle rule to describe eqs 1 and 2 in the hydrogen load vs flow rate coordinate. As shown in Figure 1, the feed, product, and tail gas of a purifier can be represented by the three sides of ΔDEF, which shows the material balance of the purifier. Next, they added this triangle rule to the graphical method which was initiated by El-Halwagi et al.19 and further developed by Kazantzi and El-Halwagi31 and Zhao et al.20 The targeting procedure is described as follows: Step 1. Construction of the Composite Curves. Figure 2a shows the source composite curve AHPD and sink composite curve APC constructed by Zhao et al.20 In this figure, line AH

Figure 1. Material balance of purifier.

Figure 2. (a) The sink and source composite curves without purification. (b) Result of hydrogen network with purification (Creg > Cutility).

represents the hydrogen utility while point P is the pinch point where the source and sink composite curves intersect each other. Step 2. Fixing the Initial Purifier Triangle. The source streams which will be sent to the purifier are suggested to be from the lower part of the source composite curve, and they are actually a continuous section of the source composite curve that starts from the lower end point D. This source stream selection agrees with the method proposed by Liao et al.;29 in other 8539

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The two steps can be described by Henry’s law and Fick’s law, respectively. Therefore the gas stream passed through a membrane can be modeled as

words, it meets the specification of necessary condition. As shown in the left part of Figure 2b, the purifier triangle ΔDEF is presented on the composite curves. Note that point F is on the source composite curve, and the purifier feed line DF is a mixture of sources. The angles of ΔDEF are fixed by specified Creg and Cr. The side lengths of ΔDEF are fixed by intersecting line EF with the sink composite curve at point P′. Step 3. Addition of the Purified Product Line DE to the Source Composite Curve. The addition is implemented by inserting an assistant product line which is parallel to the purified product line. When the purified product has higher hydrogen concentration than utility, the assistant purified product line should be drawn from the top point of the source composite curve, as line AI shown in Figure 2b. Then, the source composite curve together with the purified product line DE is shifted until the length of the assistant product line equals that of the product line. The shifting direction is not fixed; any directions that lead to an increase in the assistant product line are attemptable. The right part of Figure 2b shows the shifted result, where line AK is the obtained assistant purified product line while line E′F′ represents the tail gas of the purifier. It should be noted that the shifting procedure might be tedious, because it has two targets: reducing utility consumption and increasing the assistant product line. This method provides an effective visualized tool to target the minimum utility consumption. In this paper, we will extend the above method in two aspects: incorporate algebraic equations into the graphical procedure to make it suitable for complicated purifier models; develop a novel pinch sliding approach to simplify the shifting procedure. The improved method will be suitable for both performance-based and mechanism-based purifier models.

FregCreg = Lgi A mem (PinC in − PregCreg)

(4)

Freg(1 − Creg) = Lgj A mem [Pin(1 − C in) − Preg(1 − Creg)] (5)

Lg n =

Sn·Dn z

(6)

where Lgn represents the permeability of component n, Amem denotes the membrane area, Sn is the solubility, and Dn and z stand for the diffusivity and membrane thickness, respectively. Combining eqs 1−3 with 4−6, we get the solution-diffusion model of the membrane unit. This model is represented by purifier model type II for short. Although hydrogen streams in refineries are mixtures of many different components, we assume all components other than hydrogen in the streams to be CH4 during the membrane separation process. Then we can construct the initial triangle of membrane purifiers with this solution-diffusion membrane model.

4. GENERATION OF INITIAL TRIANGLE FOR PURIFIER On the basis of the above models, an attempt is made to fix the initial triangle. In the method of Zhang et al.,30 the initial triangle is generated by translating the tail gas line upward until it intersects the hydrogen sink composite curve. However, the concentration of the tail gas line is not specified in our purifier models; hence the tail gas line cannot be drawn directly. Instead, the initial triangle is fixed by solving a set of algebraic equations which relate the coordinates of the triangle by our model specifications. Without a loss of generality, let us suppose line DM and line ST in Figure 3 represent the purified product line and the tail

3. MODEL OF PURIFIER As described earlier, hydrogen purifiers like cryogenic units, membrane units, and PSA units can be modeled as one sink (feed) and two sources (purified product and tail gas). The mass balance of these units can be described by eqs 1, 2, and 3. These three equations include five variables if Fin and Cin are given; therefore additional equations or assumptions are required to complete the model. The purifiers can be described as two kinds of models: the performance-based model and the mechanism-based model. In this article, we employ both kinds of models, and for each model a typical case is presented to elaborate the proposed approach. 3.1. Type I: Performance-Based Model. For cryogenic units and PSA units, we chose assumptions rather than the complicated mass transfer equations. The performances of cryogenic units and PSA units are mainly assessed by the purified product concentration Creg and hydrogen recovery ratio R. In industrial operation, both Creg and R can be maintained by adjusting the operating parameters such as the reflux ratio of the cryogenic distillation process and the pressure of the PSA process. Taking this into consideration, we assume that Creg and R are specified for cryogenic units and PSA units. It should be noted that eqs 1−3 can be solved by this assumption, and let us denote this performance based model as purifier model type I. 3.2. Type 2: Solution-Diffusion Model. For the membrane separation unit, a mechanism-based model has been adopted, the solution-diffusion model, to support the additional equations. As presented by Liu and Zhang,13 the gas penetrates the membrane in two steps: solution and diffusion.

Figure 3. Construction of the initial triangle of purifier and targeting for scenario a.

gas line, respectively. According to the triangle rule, the tail gas line will intersect the hydrogen sink composite curve, hydrogen source composite curve, and the purified product line at points P′, F, and E, respectively, in Figure 3. Consequently, ΔDEF is the initial triangle. Let (xP′,yP′), (xE,yE), and (xF,yF) denote the coordinates of points P′, E, and F, respectively. Since points P′, E, and F are on the tail gas line, we get 8540



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Figure 4. New graphical targeting method for scenario b.

yF − yP ′ xF − xP′

=

yE − yP ′ xE − xP′

⎛ y − yD ⎞ (x E − yD )⎜1 − E ⎟ x E−x D ⎠ ⎝ ⎡ ⎛ y − yD ⎞ = Lg CH AMEM ⎢Pin⎜1 − F ⎟ 4 ⎢⎣ ⎝ xF − xD ⎠

(7)

Because point F is on the segment GN (shown in Figure 3) of the hydrogen source curve, its coordinate satisfies the following equation:

yF − yG yN − yG

=

x F − xG x N − xG

⎛ y − yD ⎞⎤ − Preg ⎜1 − E ⎟⎥ x E − x D ⎠⎥⎦ ⎝

(8)

In order to fix ΔDEF, eqs 7−10 are used for the case of purifier model type I, while eqs 7, 8, 11, and 12 are used for the case of purifier model type II. For the sake of convenience in representation, these two equation sets are denoted by equation set I and equation set II, respectively. We can see that both of the equation sets involve the coordinates of six points: E, F, P′, D, G, and N. Among these points, the coordinate of D is certain for its position is fixed at the end of the source composite curve. The coordinates of G, N, and P′ are tied to the kink points of the source and sink composite curves; hence they have several possible locations. The coordinates of E and F are completely uncertain while they are the vertex of the triangle which we are trying to fix. We will assume the position of G, N, and P′ first, then substituting the coordinates of P′, D, G, and N into either of the two equation sets. If the solution of these equations does not exist, the assumed positions are refreshed and the solving process is repeated. If the solution does exist, then we obtain the coordinates of E and F. Note that although there are many alternatives of G, N, and P′, the alternatives are tried in the following sequence: points P′, G, and N are gradually moved upward from the lower segment of the composite curves. There is also one special case: when point F slides on the last segment of source composite curve, both Cin and Cr will not change.

In the next step, equations for purifier model types I and II are developed separately. For purifier model I, the slope (which represents the concentration of the stream) of line DE is fixed to Creg. Therefore, substituting the coordinate of point E into the equation of line DE yields yE − yD = Creg(x E − x D)

(9)

The hydrogen recovery ratio R is also given for purifier model I. From the definition of R in eq 3, we obtain R=

yE − yD yF − yD

(10)

where the numerator and denominator of the right-hand side represent the hydrogen loads of the product and feed of the purifier, respectively. Substituting the coordinates of P′, D, G, and N into eqs 7−10, we can obtain the coordinates of E and F for purifier model I. For purifier model II, if we substitute the coordinates of ΔDEF into eqs 4 and 5, then they can be rearranged as follows: ⎛ y −y y − yD ⎞ D ⎟ − Preg E yE − yD = Lg H AMEM ⎜⎜Pin F 2 x E − yD ⎟⎠ ⎝ xF − xD

(12)

5. IMPROVED TARGETING PROCEDURE After the construction of the initial triangle, the targeting procedure originally proposed by Zhang et al.30 will be improved to determine the minimum utility consumption and

(11) 8541



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Figure 5. New graphical targeting method for scenario c.

minimum feed flow rate to the purifier. The improvement mainly lies in the shifting procedure. As is well-known, the pinch-based graphical methods obtain targets by shifting the composite curves in the utility reducing direction until pinch occurs. The process is easy to follow, but in the case of purification, it is not easy. It is because in the case with purification the shifting procedure has two targets: reducing utility consumption while increasing the assistant product line. We will simplify the shifting procedure with a new concept: pinch sliding. Now, let us consider three scenarios for the targeting procedure, i.e. (a) Creg = Cutility, (b) Creg > Cutility, (c) Creg < Cutility. For scenario a, Creg = Cutility, since the purified product and utility concentration are identical, there is no need to shift the sink composite curve. As shown in Figure 3, the purified product DE can be used to substitute part of the utility; therefore the projection difference on the x axis between the purified product DE and the utility consumption without purifier AH is the minimum utility consumption with purifiers. This is the simplest scenario. For scenario b, Creg > Cutility, the original composite curve has to be shifted to take in the new segments of purified product and tail gas. First, DE needs to be shifted to the position where points E and A coincide; the purified product line DE turns into AI, as shown in Figure 4a. Since the connection of assistant purified product line AI and the sink composite curve is fixed at point A, the sink composite curve together with AI (the red line shown in Figure 4) is shifted to find the position where point I is on the utility line HB (the green line shown in Figure 4). Point P is the original system pinch point. The red line in Figure 4a will be shifted by sliding point P upward or downward along the intersected hydrogen source line GN to make point I approach line BH. When point I stands on line BH, the line section HI represents the new hydrogen utility consumption. During this procedure, the sink composite curve may intersect the source composite curve at another point. If the new intersection point is formed, the sliding line will be changed. The red line will be slid along the new intersected hydrogen source line.

When the shifting procedure is completed, it may be found that the sink composite curve crosses the purifier tail gas line EF as shown in Figure 4b. Therefore, another triangle of the purifier has to be constructed with the new position of point P′ as illustrated in Figure 4c. The procedure described above is repeated until the new coordinate of point P′ after the shift of the hydrogen sink composite curve is the same (or within a certain tolerance) as that of point P′ before the movement. Then, in Figure 4d, we get the optimal solution of the hydrogen network with purifier. As we can see in Figure 4d, the minimum utility consumption is the projection of HI′ on the x axis; the projection of DF′ on the x axis is the purifier feed flow rate. It should be noted that P′ does not necessarily stand on E′F′ in the final result. This happens when condition 3.3 or 3.4 in the necessary condition holds. From the shifting procedure, we can conclude that there will always be at least one pinch except the pinch on the tail gas line in the result. In addition, we can prove that the resulting purifier feed flow rate is also the minimized one under the minimum utility consumption. The proof is made by contradiction. It is assumed that the minimized purifier feed flow rate is less than our result. Consequently, the length of line AI in Figure 4d will be reduced while the other lines are unchanged. However, since the source and sink composite curves are just intersected at points P and A, reducing the length of AI must lead to some overlaps between these two curves. This contradicts the satisfaction of hydrogen demands. Therefore, the length of AI cannot be reduced any more. The conclusion has been proved. For scenario c, Creg < Cutility, the targeting procedure is similar to scenario b. In the first step, the purified product line DE is shifted to the position where points D and H coincide. Hence, the purified product line DE turns into HI (the green line shown in Figure 5a). Then the hydrogen sink composite curve will be moved together with AH (the red line shown in Figure 5a) by sliding point P upward or downward along the intersected hydrogen source line GN to make point I approach line AH′. As shown in Figure 5b, the coordinate of point P′ will change after the shift. Then another initial triangle is constructed with the new coordinate of point P′ in Figure 5c. 8542



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Figure 6. The whole graphical targeting procedure.

The steps described above are repeated until the new coordinate of point P′ after the shift of hydrogen sink composite curve is the same (or within a certain tolerance) with that of the point P′ before the movement. Then we get the optimal solution of the hydrogen network with a purifier. As we can see in Figure 5d, the minimum utility consumption is the projection of AI on the x axis; the projection of DF′ on the x axis is the minimum feed flow rate to the purifier. For the same reason as that of scenario b, the resulting feed flow rate of the purifier is also the minimized one. To summarize, the entire graphical targeting procedure is presented as a flowchart in Figure 6.

this example for both purifier model type I and purifier model type II. Table 1. Process Data of Example 1 flow (mol/s)

sinks 1 2 3 4

HCU NHT CNHT DHT sources

1 2 3 4 5 6 utility

6. CASE STUDY The applicability of the proposed new graphical method is now demonstrated through the following hydrogen and water network examples. Example 1. This example is taken from Alves and Towler.18 The system consists of four hydrogen consuming processes: hydrogen cracker unit (HCU), naphtha hydrotreater (DHT), cracked naphtha hydrotreater (CNHT). The feed of these units can be considered as hydrogen demands while the outlet streams can be taken as sources. There are also two individual hydrogen sources: the catalytic reforming unit (CRU) and the steam reforming unit (SRU). Besides, the hydrogen utility with the concentration of 95 mol % is also available in this system. The detailed process data are shown in Table 1. We will treat

HCU NHT CNHT DHT SRU CRU

hydrogen concentration (mol %)

2495.0 180.2 720.7 554.4 flow (mol/s)

80.61 78.85 75.14 77.57 hydrogen concentration (mol/%)

1801.9 138.6 457.4 346.5 623.8 415.8 to be determined

75.00 75.00 70.00 73.00 93.00 80.00 95.00

Let us take purifier model type I into consideration first (Table 2). For purifier model type I, the purified product concentration is 98 mol %, while the hydrogen recovery ratio is 0.95. Step 1: Hydrogen source and sink composite curves are constructed, and the initial pinch point is found, as shown in Figure 7. Point P is the pinch point without purifier. It is on the last segment of the hydrogen source composite curve, DG. The 8543



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position where points E and A coincide. Thus the purified product line DE turns into AI, as shown in Figure 8. Then the

Table 2. The Iteration Process of Example 1 with Purifier Model Type I interval

coordinate of point P(P′)

distance

1 2 3 4

(94.175, 65.9225) (94.8547, 66.3983) (94.7993, 66.3595) (94.8040, 66.3628)

0.8297 0.0676 0.0057

Figure 8. The optimal solution of example 1 with purifier model type I.

hydrogen sink composite curve will be moved together with AI by sliding point P downward on the hydrogen source line DG until the hydrogen utility line A′H intersects the line AI at its end point I. It is clear that the coordinate of point P will change while the composite curve shifts. Since the pinch point P is still on the last segment of the hydrogen source composite curve, the feed concentration Cin to the purifier does not change. Hence the tail gas concentration Cr will also remain unchanged, which means the tail gas line EF can be slid to make the points F and P coincide. The steps described above are repeated to obtain a series of coordinate of point P. It can be shown that after the last movement of the hydrogen sink composite curve, the distance of the coordinate between the last two iterations of point P is within the tolerance of 0.01. Finally, the optimal solution to the network with purification is obtained, as shown in Figure 8. The projection of HI on the x axis is the minimum hydrogen utility consumption 196.77 mol/ s, which agrees with that obtained by Foo and Abdul Manan32 and Liao et al.29 The pinch point P is on both the tail gas line and the hydrogen source line, and Cin is at its maximum value. According to the optimal conditions, this is the overall optimal solution. The projection of DF on the x axis is the minimum flow rate which must be sent to the purifier 94.8 mol/s, and the projection of DE on the x coordinate is the purified product 64.33 mol/s. It is the same result which is obtained with an automated targeting technique by Ng et al.33 and a novel conceptual method recently proposed by Liu et al.26 In this way, the proposed graphical targeting method obtained the minimum hydrogen utility and minimum flow rate to be purified simultaneously. Next, we shift to purifier model type II, the solution and diffusion model of the membrane. The parameters of the membrane are given as follows. The permeabilities of hydrogen and methane are specified as 0.556 (m3/(m2 s Mpa)) and 4.41 × 10−3 (m3/(m2 s Mpa)), respectively. The pressure of the feed stream and purified product of the purifier is 5 MPa and 3 MPa, respectively; the area of the membrane AMEM is 200 m2. We also suppose that the coordinates of point E and F are (xE, yE)

Figure 7. The hydrogen composite curves and initial pinch point P of example 1.

projection of AB on the x axis is the minimum utility consumption without a purifier, which reads 268.82 mol/s. Step 2: The initial triangle of purifier model I is fixed. For the sake of convenience in calculation, point D is designated as the origin. Substituting the coordinate of point D, the value of R and Creg into eq 9 and 10 yields yE = 0.98x E (13) R=

yE yF

= 0.95 (14)

Point P is not only on the last segment of source composite curve DG, it is also on the purifier tail gas line which means point P coincides with point P′ (the point at which the purifier tail gas line intersects the sink composite curve) in this case. Since point F is on the segment of DG, substituting the coordinates of D and G into eq 8, we obtain yF = 0.7x F (15) Substituting the coordinate of point P (102.5231, 71.7662) into eq 7 and solving it together with eqs 13−15, we obtain the coordinates of E and F as (69.5693, 68.1779) and (102.5231, 71.7662), respectively. From the result, we can see that point P and point F coincide. This result can also be concluded from the fact that point P′ is also on the source composite curve. Therefore, we can get the coordinate of F without solving these equations in this scenario, and the coordinate of point E can be obtained by solving eqs 5 and 6 only. Step 3: The minimum hydrogen utility consumption is targeted by shifting the sink composite curve. The purified product concentration is 98 mol %, which is higher than that of the utility. It is scenario b, Creg>Cutility. DE will be shifted to the 8544



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Table 4. Process Data of Example 2a

and (xF, yF). Since point D is the origin, we can rearrange eq 10 and 11 as ⎛ y y ⎞ yE = Lg H AMEM ⎜Pin F − Preg E ⎟ 2 xE ⎠ ⎝ xF

flow (mol/s)

sink

(16)

⎡ ⎛ ⎛ y ⎞⎤ y ⎞ x E − yE = Lg CH AMEM ⎢Pin⎜1 − F ⎟ − Preg ⎜1 − E ⎟⎥ 4 ⎢⎣ ⎝ x E ⎠⎥⎦ xF ⎠ ⎝ (17)

a

It is clear that points F and P will coincide. Substituting the coordinate of point F into eqs 16 and 17, we can get the coordinate of point E. Then we can construct the initial triangle with the coordinate of point E. After the iterating process as shown in Table 3, an optimal solution is obtained, as shown in Figure 9. The projection of

hydrogen concentration (mol %)

1 2 source

400 600 flow (mol/s)

92.8 87.5667 hydrogen concentration (mol %)

1 2 utility

350 500 to be determined

91 85 99

Creg = 95%, R = 0.9.

on the x axis is the minimum hydrogen utility consumption 182.86 mol/s.

Table 3. The Iteration Process of Example 1 with Purifier Model Type II interval

coordinate of point P(P′)

distance

1 2 3

(102.5232, 71.7663) (94.8959, 66.4271) (94.8959, 66.4271)

9.3104 0

HK on the x axis is the hydrogen utility 197.66 mol/s; the projection of DF on the x axis is the feed flow rate to the purifier, which is 94.90 mol/s.

Figure 10. The hydrogen composite curves and initial pinch point P of example 2.

Step 2: Fixing the initial triangle of the purifier. Since the pinch point is also on the last segment of the hydrogen source composite curve, it can be deduced that point F will also coincide with point P (25.0507, 21.2929). For the sake of convenience in the calculations, point D is designated as the origin. Substituting the coordinate of point D, the value of R and Creg into eqs 9 and 10 yields: yE = 0.95x E (18) R= Figure 9. The optimal solution of example 1 with purification model type II.

yE yF

= 0.9 (19)

Substituting the coordinate of point F (25.0507, 21.2929) into eqs 18 and 19, the coordinates of E (20.1722, 19.1636) are obtained. Step 3: The minimum hydrogen utility consumption is targeted by shifting the sink composite curve. Since the hydrogen concentration of the purified product is lower than that of the hydrogen utility, it is scenario c. The sink composite curve will be shifted as has been described above. The optimal solution of the hydrogen network with purification is obtained as shown in Figure 11. The projection of HK on the x axis is the minimum hydrogen utility consumption 158.31 mol/s; the projection of DP on the x axis is the minimum flow rate to the purifier, 42.68 mol/s. The result agrees with that obtained by Agrawal and Shenoy22 based on the graphical method of limiting composite.

Solution-diffusion model of the membrane separation process is first integrated into a graphical targeting approach in this example. The targeting method will be more suitable for real industry problems with these mechanism based model. Example 2Two Pinch Points Coincide and on the Last Segment of Hydrogen Source Composite Curve. This example is taken from Agrawal and Shenoy.22 The detailed process data are shown in Table 4, and the purifier model type I is employed in this example. Step 1: The composite curves are constructed, and the pinch point P is located, as shown in Figure 10. The projection of AH 8545



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Figure 11. The optimal solution of example 2 with purification.

Figure 12. The composite curves without a purifier for example 3.

Example 3. Water Network with Two Separate Pinch Points. The targeting approach proposed in the present work is not limited to hydrogen networks only. It can also be used in water networks with a regenerator. There are plenty of works on the targeting of water networks with a regenerator.22,32,34−39 The regenerators in these works are modeled by one inlet and one outlet. However, water regenerators such as steam stripping and reverse osmosis are one inlet and two outlet processes. Our method is suitable for this kind of regeneration problem. In water networks, most of the water utility is fresh water with no contaminants; therefore they all belong to scenario c. The following three examples will show the application in water networks. This example is taken from Polley and Polley.40 The detailed process data are shown in Table 5.

R=

a

flow (t/h)

yF = 0.25x F

50 100 80 70 flow (t/h)

20 50 100 200 contaminant concentration (ppm)

1 2 3 4 utility

50 100 70 60 to be determined

50 100 150 250 0

x E − yE /1000 x P − yP /1000

(21)

(22)

Suppose point P′ (50, 10) is the pinch point on the tail gas line. Substituting the coordinate of point P′ into eq 7, we can get yF − 10 x F − 50

=

10 − yE 50 − x E

(23)

Solving eqs 20−23 simultaneously, we can get the coordinates of E (48.0356, 0.9607) and F (50.5745, 12.6436). Step 3: The optimal solution of the network is calculated after the shifting process of scenario c. As we can see in Figure 13, the projection of AK on the x axis is the minimum fresh water consumption of 22.18 t/h; the projection of DF on the x axis is the minimum water flow rate to the purifier, 58.1 t/h.

contaminant concentration (ppm)

1 2 3 4 source

Fin(1 − C in)

=

Suppose that point F is on the last segment of the water source composite curve, DG. Substituting the coordinates of D and G into eq 8 yields

Table 5. Process Data of Example 3a sink

Freg(1 − Creg)

Creg = 20 ppm, R = 0.95.

Step 1: The composite curves and initial pinch point are obtained first, as shown in Figure 12. The projection of AI on the x axis is the minimum fresh water consumption 70 t/h. Step 2: The initial triangle of the regenerator is fixed as the method above proposed. Substituting the coordinate of point D and the value of Creg into eq 9, we can obtain yE = 0.02x E (20) Substituting the value of R into eq 10 yields

Figure 13. The optimal solution with purification reuse for example 3. 8546



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Example 4Pinch Point Changes during the Shift Process. This is also a water network example. The detailed process data are shown in Table 6. Table 6. Process Data of Example 4a sink

a

flow (t/h)

contaminant concentration (ppm)

1 2 3 4 source

75 81.7 103.02 50 flow (t/h)

203 109 47 20 contaminant concentration (ppm)

1 2 3 4 utility

50 100 70 60 to be determined

50 100 150 250 0

Creg = 20 ppm, R = 0.95. Figure 15. Two pinch points of the composite curves.

Step 1: The composite curves are constructed, and the initial pinch point P is located. As shown in Figure 14, it should be noted that point R is not on the water source line MN. The projection of AH on the x axis is the minimum fresh water consumption 70 t/h.

Figure 16. The optimal solution with purification reuse for example 4.

consumption 31.47 t/h; the projection of DF on the x axis is the minimum flow rate to the regenerator 50.57 t/h.

7. CONCLUSION Purification is critical in reducing the utility consumption of hydrogen networks. Purifiers with different separating principles can be employed to recover hydrogen in refineries. This paper proposes an improved graphical approach to target the minimum utility consumption and the minimum feed flow rate to purifiers simultaneously by sliding the sink composite curve along the intersected source segment. The equation based generation of the purifier triangle makes this method compatible with both performance- and mechanism-based purifier models. The solution-diffusion model of the membrane separation process is first integrated into the graphical targeting method, and an example shows the effectiveness of this approach. The intersected sliding procedure ensures the minimum purifier feed target. Although the targeting procedure includes iterative calculations of the purifier triangle and the pinch point, the procedure converged efficiently as illustrated by the examples. The proposed method was also extended to freshwater minimization of water network synthesis problems.

Figure 14. The composite curves without a purifier for example 4.

Step 2: The initial triangle of the regenerator is fixed by getting the coordinates E and F. Then the next procedure is to shift the sink composite curve to get the optimal position. Since point R is near the source composite curve, the sink composite curve can be shifted by sliding point P downward along GM until point R intersects the source composite curve, as shown in Figure 15. Step 3: The sink composite curve can then be shifted by sliding point R downward on the water source line MN. The final optimal solution is shown in Figure 16. It is easy to see from Figure 16 that the pinch points are R and P′, while Cin is at its maximum value Cin,max. Since P′ is on the tail gas line, this result agrees with 1 of the optimal condition; thus the utility consumption is minimized. Because there is another pinch point R, the obtained purifier feed flow rate is also minimized. The projection of AK on the x axis is the minimum fresh water 8547



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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 The financial support provided by the National Natural Science Foundation of China (21106129), the Specialized Research Fund for the Doctoral Program of Higher Education (20110101120019), the Fundamental Research Funds for the Central Universities (2011QNA4032) and the National Basic Research Program of China (2012CB720500) are gratefully acknowledged.

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NOTATION F = flow rate Freg = flow rate of purified product Fin = flow rate of purifier inlet Fr = flow rate of purifier tail gas C = concentration Creg = concentration of purified product Cin = concentration of purifier inlet Cr = concentration of purifier tail gas Cutility = concentration of utility R = hydrogen (water) recovery ratio Lgn = permeability of component n Amem = membrane area Dn = diffusivity z = membrane thickness Preg = pressure of purified product Pin = pressure of purifier inlet Sn = solubility Cin,mm = concentration at which maximum mixing of hydrogen source streams occurs Fin,Cin,max = the maximum flow rate under given Cin Cin,mm = minimum value of Cin Cin,max = maximum value of Cin

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Pinch Sliding Approach for Targeting Hydrogen and Water Networks

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