Article pubs.acs.org/IECR
Targeting Compression Work for Hydrogen Allocation Networks Santanu Bandyopadhyay,* Nitin Dutt Chaturvedi, and Amit Desai Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India ABSTRACT: Hydrogen management is important in refineries in achieving cost competitiveness to produce clean and lowsulfur fuels. Other than minimizing the fresh hydrogen requirement, it is extremely important to reduce the compression work, required in any hydrogen allocation network (HAN). In this paper, a rigorous targeting methodology to minimize compression work in a HAN is proposed. The proposed methodology is algebraic and addresses flow, concentration, and pressure constraints in HAN. On the basis of the rigorous mathematical arguments, it has been proved that problem of minimizing compression work requirement in a HAN with multiple pressure levels can be broken into various subproblems having two pressure levels only and the compression work requirement for two pressure levels can be minimized via minimizing cross-flows between these pressure levels. The applicability of the proposed algorithm is demonstrated through illustrative examples. header. Umana et al.18 integrated empirical model for hydrogen requirement in hydrotreaters with the refinery hydrogen network to minimize overall hydrogen requirement. These approaches, however, do not include compression work for the hydrogen network. Pressure is one of the major parameters in gas networks as direct mixing at different pressures is not allowed. Hydrogen can be supplied only from higher pressure to lower pressure without any compression work. Any stream flowing from a low pressure source to a higher pressure sink requires compression work. Compression work increases the operating cost of a HAN significantly.19 In recent years, various mathematical programming based methods have been proposed to optimize HAN with pressure constraints. Hallale and Liu19 proposed a mixed-integer nonlinear programming (MINLP) formulation to optimize HAN to include flow rate, hydrogen purity, pressure, purifiers, and other economic aspects. Liu and Zhang20 developed a systematic method to select purification processes and integrate them into HAN. Fonseca et al.21 discussed hydrogen resource saving potential for a refinery case study. Khajehpour et al.22 proposed a reduced superstructure of Hallale and Liu.19 Ahmad et al.23 presented a multiperiod approach for designing flexible HAN. Kumar et al.24 developed both nonlinear programming (NLP) and MINLP formulations considering the compressor inlet and outlet pressures as variable. Jiao et al.25 proposed multiobjective formulation to include minimization of the operating cost along with minimization of the investment cost. Ding et al.26 proposed a methodology by constructing the average pressure profiles of hydrogen sources and sinks which assists in determining the need of compression work. Wu et al.27 proposed a mathematical model in order to minimize total exergy consumption to include hydrogen resource as well as compressor work. Jiao et al.28 proposed a MINLP formulation to optimize flexible HAN. Shariati et al.29 proposed a new targeting methodology to determine the minimum fresh
1. INTRODUCTION Hydrogen is an important utility in refineries and is required for the production of clean fuels such as low-sulfur gasoline and diesel. Between 2010 and 2030, the global requirement of hydrogen for the refining industry is expected to be 14 trillion standard cubic feet.1 Onsite hydrogen production capacity has expanded by 64% globally between 1982 and 2012.1 The shortage of hydrogen resources motivates the integrated management of hydrogen allocations networks (HAN). In recent years, various techniques based on physical insight as well as mathematical optimization have been reported in the literature for hydrogen management in chemical process plants. Alves and Towler2 addressed the problem of hydrogen pinch analysis through hydrogen surplus diagram to identify the minimum external hydrogen resource requirement in a HAN. El-Halwagi et al.3 and Prakash and Shenoy4 independently proposed a material recovery pinch diagram that can be employed to target the minimum external hydrogen requirement in a HAN. Bandyopadhyay5 proposed an algebraic methodology, and Foo et al.6 extended the concept of cascade analysis for targeting the minimum resources in any resource allocation network (RAN). Agrawal and Shenoy7 developed a unified conceptual approach for HAN incorporating hydrogen purification. Zhao et al.8 proposed a graphical method for the integration of hydrogen distribution systems and later extended it to include the effect of multiple impurities.9 Liao et al.10,11 developed a rigorous targeting procedure to optimize HAN with and without hydrogen purification system. Liu et al.12 proposed a methodology for identifying the minimum hydrogen resource and the optimal flow rate to the purifier. Liu et al.13,14 analyzed the effect of variation in purification feed flow rates on the hydrogen resource requirement. Lou et al.15 proposed a graphical method to target the minimum fresh hydrogen requirement and the minimum feed flow rate to purifiers simultaneously. Zhang et al.16 proposed a graphical approach for minimizing fresh hydrogen requirement in HAN with purification reuse and introduced the concepts of maximum hydrogen surplus and mass transfer triangle. Deng et al.17 proposed a mathematical programming model for the synthesis of hydrogen network with intermediate hydrogen © 2014 American Chemical Society
Received: Revised: Accepted: Published: 18539
August 30, 2014 November 2, 2014 November 13, 2014 November 13, 2014 dx.doi.org/10.1021/ie503429q | Ind. Eng. Chem. Res. 2014, 53, 18539−18548
Industrial & Engineering Chemistry Research
Article
hydrogen requirement in petrochemical complexes and a superstructure based optimization approach to reduce overall fresh hydrogen requirement. Zhou et al.30 proposed a MINLP model including economic and environmental aspect of HAN. Jhaveri et al.31 compared five different mathematical models to optimize HAN. It may be noted that no algebraic methodology has been proposed to minimize overall compression work in a HAN. In this paper, an algebraic methodology is proposed to calculate the overall minimum compression work requirement for a HAN. The proposed algorithm is mathematically rigorous and therefore, it guarantees the optimum solution.
Due to concentration constraint at every demand, the hydrogen mass-load requirement for any internal demand may be mathematically expressed as follows: Ns
∑ fij qsi + f frj qfr ≤ Fdjqdj Nd
Ffr =
+ fiw = Fsi
i=1
+ f frj = Fdj
⎛ ⎛ P ⎞⎞ Wisothermal = PinFin⎜⎜ln⎜ out ⎟⎟⎟ ⎝ ⎝ Pin ⎠⎠
(5)
Applying the characteristic of an isothermal process: PF = constant
(6)
The work expressed in eq 5 can also be expressed as Wisothermal = F0*((P0 ln(Pout /P0)) − (P0 ln(Pin /P0)))
(7)
Where, F0 is the standard volumetric flow rate being compressed and P0 is the pressure under standard conditions. The compressor inlet and outlet pressure are Pin and Pout, respectively. The quantities P0 ln(Pout/P0) and P0 ln(Pin/P0) can be denoted as μin and μout, respectively, and may be called the pressure index for isothermal compression. For polytropic compression, the process is governed by the following equation: PF n = constant
(8)
Where, n is the polytropic index. Work required for polytropic compression work is given as Wpoly
∀i ∈ (1, 2, ... Ns)
∀ j ∈ (1, 2, ... Nd)
(4)
The Minimum fresh hydrogen requirement (eq 4), subject to constraints (eqs 1−3), can be calculated using any one of the established methodologies for minimization of resource requirement in a resource allocation network (RAN), e.g., material recovery pinch diagram,3,4 source composite curve,5 cascade analysis,6 etc. It may be noted that after the fresh hydrogen requirement has been minimized, the fresh hydrogen source and waste flow can be treated as entities of the sources and demands sets without any loss of generality. Let the fresh hydrogen and waste flow be denoted by i = 0 and j = 0 in the source and demand sets, respectively. With these new modified sets of sources and demands with the inclusion of fresh resource and waste flows, second part of optimization problem is formulated. The compression power required for a flow stream is governed by the initial and final states (volumetric flow and pressure) as well as the process followed for compression. For isothermal compression, the work done can be expressed as
n − 1/ n ⎞ ⎛⎛ Pout ⎞ ⎛ n ⎞ ⎟ ⎜ ⎟P F =⎜ − 1 ⎟ ⎜ ⎟ ⎝ n − 1 ⎠ in in⎜⎝ Pin ⎠ ⎠ ⎝
(9)
(1)
Using eq 8, work for polytropic compression can be expressed as
(2)
n − 1/ n ⎞ ⎛ ⎛ n ⎞ 1/ n n − 1/ n ⎜⎛ Pout ⎞ ⎜ ⎟ = P0 Pin F0⎜⎜ − 1⎟⎟ ⎟ ⎝ n − 1⎠ ⎠ ⎝⎝ Pin ⎠
Ns
∑ fij
∑ f frj j=1
Nd j=1
(3)
The total fresh hydrogen requirement is
2. PROBLEM DEFINITION In this section, the problem for targeting the minimum compression work in a HAN is defined mathematically. Consider a process that consists of a set of demands and a set of sources that are described as follows: • The set of Ns sources is defined where each source i {1, 2, ..., Ns} has a fixed flow Fsi with a given percentage concentration csi of pure hydrogen in the flow stream and is available at a given pressure psi. The quality of a source is denoted by the percentage contaminant level qsi (= 100 − csi). • The set of Nd demands is defined where each demand j {1, 2, ..., Nd} has a specific flow requirement Fdj at a minimum acceptable percentage concentration cdj of pure hydrogen in the flow stream and the flow is received at a minimum pressure of pdj. The quality of a demand is denoted by the percentage contaminant level qdj (= 100 − cdj). • A fresh (external) hydrogen resource is also available at a specified percentage concentration cf r (equivalently, quality qfr = 100 − cf r) and pressure pfr that can be used to supplement the use of sources to meet the demands. The excess flows from the sources that are not supplied to any demand are termed as waste. It may be noted that the waste is typically used in the fuel system of the plant. • Flow from a source at higher pressure can be passed to any demand with lower pressure requirement without any work. However, compressors are required to pass flows from lower pressure sources to higher pressure demands. The objective is to develop a procedure to calculate the minimum total compression work in a feasible HAN while satisfying minimum external hydrogen resource requirement. Let f ij denotes the flow transferred from source i {1, 2, ..., Ns} to demand j {1, 2, ..., Nd}. Let f frj and f iw represent the flow transferred from external resource to demand j and flow transferred from source i to waste, respectively. The flow balance for each internal source demand may be written as follows:
∑ fij
∀ j ∈ (1, 2, ... Nd)
i=1
Wpoly 18540
(10)
dx.doi.org/10.1021/ie503429q | Ind. Eng. Chem. Res. 2014, 53, 18539−18548
Industrial & Engineering Chemistry Research
Article
⎛⎛⎛ n ⎞ n − 1/ n ⎞ ⎟P (P / P ) ⎟ Wpoly = F0⎜⎜⎜ 0 out 0 ⎝ ⎠ ⎠ ⎝ ⎝ n−1 ⎞ ⎛⎛ n ⎞ n − 1/ n ⎞ ⎟ P (P / P ) ⎟⎟ − ⎜⎜ 0 in 0 ⎝⎝ n − 1 ⎠ ⎠⎠
the flows available from the sources might not match the flows required by the demands in each individual network, and hence, it might either require additional flows from other networks or supply excess waste flows to other networks or a combination of both. Without loss of generality, let the network with the higher pressure level be referred to as HAN-p2 and the network with lower pressure level be referred to as HAN-p1 (Figure 1). As
(11)
The quantities [n/(n − 1)]P0(Pout/P0) and [n/(n − 1)] P0(Pin/P0)[(n−1)/n] can be denoted as μin and μout, respectively, and represent the pressure index for polytropic compression. The compressors in the hydrogen allocation network have been assumed to follow an isothermal or a polytropic process, and hence, the compression work for a compressor in the network can be expressed as [(n−1)/n]
Wij = F(μj − μi )+
(12)
Where ⎧ P0 ln(Pi /P0) ⎪ ⎪ ⎪ μi = ⎨ ⎛ n ⎞ (n − 1)/ n ⎟ P (P / P ) ⎪⎜ 0 i 0 ⎝ ⎠ − n 1 ⎪ ⎪ ⎩
isothermal compression Figure 1. Representation of problem with two pressure levels (p1 < p2).
polytropic compression
(13)
fresh hydrogen and waste flows are included the in the sources and demands sets, respectively, the following equation can be written for overall flow balance
The positive sign in eq 12 indicates that only positive work by the compressor is to be taken in to account. It can be observed from eq 13, pi < pj implies μi < μj. Consider three pressure levels denoted by pi, pj, and pk such that pi < pj < pk. Wik = F(μk − μi )
(14)
Wik = F(μk − μj ) + F(μj − μi )
(15)
Fsp1 + Fsp2 = Fdp1 + Fdp2
The interplant cross-flow from HAN-p1 to HAN-p2 is defined as the flow from the sources in HAN-p1 to the demands in HAN-p2 and is denoted by Fp1p2. Correspondingly, the interplant cross-flow from HAN-p2 to HAN-p1 is denoted by Fp2p1. Together, Fp1p2 and Fp2p1 represent the total cross-flows between HAN-p1 and HAN-p2. From overall flow balance for each HAN, following equation can be written:
Equation 15 confirms that the compressor work for successive pressure levels obeys the following additive property: Wik = Wij + Wjk
Δ = Fp1p2 − Fp2p1 = Fsp1 − Fdp1 = Fdp2 − Fsp2
(16)
(17)
The total compression work required in a feasible network is Ns
W=
Nd
∑ ∑ fij λij i=0 j=0
(20)
In eq 20, Δ is a constant for given sets of HANs and hence, Fp1p2 and Fp2p1 are not independent. This proves the following result: Lemma 1: Minimization of overall interplant cross-f lows between two HANs is equivalent to the minimization of crossf low f rom either of the two HANs to the other. Based on the above lemma, the following theorem can be established: Theorem 1: Minimization of overall compression work in a HAN having two pressure levels is equivalent to the minimization of overall cross-f low between the individual HANs of two pressure levels. Proof: Without loss of generality, let sources {i = 1, 2, ..., k} be available in HAN-p1 and sources {i = k + 1, ..., Ns} be available in HAN-p2. Likewise let the demands {j = 1, 2, ..., m} and demands {j = m + 1, ..., Nd} be present in HAN-p1 and HAN-p2, respectively. The compression work can be expressed as
This implies that the compressor work is a linear function of the pressure index. The quantity (μj − μi)+ may be defined as the compression work index and can be denoted by λij.
λij = (μj − μi )+
(19)
(18)
The objective is to minimize the total work (W) subject to the constraints given by eqs 1−3 and the minimum external hydrogen requirement. It can be observed that all the constraints and the objective function are linear and hence this is a linear programming problem.
3. MATHEMATICAL RESULTS AND ANALYSIS According to pressure levels, two general network configurations are possible: case (1) there are only two pressure levels and case (2) there more than two pressure levels. These cases are discussed separately in the following subsections. 3.1. Case 1: Two Pressure Levels. Sources and demands at a single pressure level can be viewed as an independent network with a common pressure level. It may be noted that
k
W=
m
k
i=0 j=0 Ns
+
i=0 j=m+1
Ns
fij λij +
m
∑ ∑ fij λij i=k+1 j=0
Nd
∑ ∑ i=k+1 j=m+1
18541
Nd
∑ ∑ fij λij + ∑ ∑ fij λij
(21)
dx.doi.org/10.1021/ie503429q | Ind. Eng. Chem. Res. 2014, 53, 18539−18548
Industrial & Engineering Chemistry Research
Article
Figure 2. Compression work requirement for flows in HAN with two pressure levels.
W = Fp1p1λp1p1 + Fp1p2λp1p2 + Fp2p1λp2p1 + Fp2p2λp2p2
pH‑1. Let us consider all sources and demands up to pH‑2 to be a single HAN (HANpH‑2). Now sources and demands at pressure level pH‑1 along with the additional sources and demands, due to interplant flow transfer between pH to pH‑1, are considered as other HAN (HANm-pH‑1). As suggested by Theorem 1, the minimum interplant flows between these two HANs and the minimum compression work between these two pressure levels (WPH‑2/H‑1) can be determined along with additional sources and demands at pH‑2. This process can be continued until the last two pressure levels are reached. Total compression work required can be expressed as
(22)
Work requirements for all of the possible flows are shown in Figure 2. No compressor work is required to transfer flows either at same pressure (i.e., from p1 to p2 or from p2 to p2) or from higher pressure to lower pressure (i.e., from p2 to p1). Therefore, the first, third, and fourth terms on the RHS of eq 22 are zero for all feasible network configurations (Figure 2). W = Fp1p2λp1p2
(23)
Since λp1p2 is a constant, minimization of W is equivalent to minimization of Fp1p2, the cross-flow from HAN-p1 to HAN-p2. As per lemma 1, minimizing Fp1p2 is equivalent to minimizing overall cross-flow between HAN-p1 and HAN-p2. This proves the theorem. It may be noted that for targeting the minimum interplant cross-flow, methodology proposed by Sahu and Bandyopadhyay32 can be directly adopted. In this paper, a slightly modified approach is adopted. A limiting composite curve (LCC) for one plant and a reflected limiting composite curve of the other plant are plotted on the same quality−quality load diagram. These two LCCs touch each other at the resource quality, at the pinch quality, as well as at the highest quality. These two LCCs never cross each other.32 Any piecewise linear curve, bounded by these two LCCs, represents a feasible interplant cross-flow. As proved by Sahu and Bandyopadhyay,32 the interplant cross-flow curve with the least distance represents minimum interplant cross-flow. 3.2. Case 2: Multiple Pressure Levels. Let p1 be the lowest pressure and pH be the highest. Let p2, p3, ..., pH‑1 be the intermediate pressure levels (p1 < p2
