Improved Synthesis of Hydrogen Networks for Refineries - Industrial

Article pubs.acs.org/IECR

Improved Synthesis of Hydrogen Networks for Refineries Anoop Jagannath,† A. Elkamel,*,‡ and I. A. Karimi*,† †

Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117585 Chemical Engineering Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

‡

S Supporting Information *

ABSTRACT: Hydrogen supplies constitute a significant cost for refineries. Thus, managing hydrogen flows and consumption in an integrated and cost-effective manner is critical. This work presents a systematic framework for modeling key units in a refinery hydrogen network. It proposes an improved superstructure and a simpler mixed-integer nonlinear programming model for synthesizing such a network with minimum total annualized cost. In contrast to the existing literature, it allows dedicated compressors, realistic cost correlations, temperature effects, stream-dependent properties, fuel gas specifications, heating, cooling, and valve expansions. Furthermore, it avoids the many bilinear and posynomial terms present in the existing models; thus it is easier to solve. Our tests with several literature examples confirm that our model gives better and more realistic solutions than the previous models, and it is also suitable for retrofit synthesis.

1. INTRODUCTION Petroleum refineries are major consumers of hydrogen. They treat/react their intermediates and products with hydrogen to obtain cleaner and more efficient fuels and meet the everstringent environmental regulations. The worldwide refining industry will need 14 trillion standard cubic feet (SCF) of onpurpose hydrogen between 2010 and 2030.1 The Asia Pacific and the Middle East will represent 40% of this demand. Several units in a refinery (e.g., continuous catalytic reformer, semiregenerated catalytic re-former, and gas processing units) produce hydrogen-containing streams. However, a refinery’s hydrogen demand generally exceeds what these units may supply; hence, many refineries may produce hydrogen onsite using steam methane re-forming (SMR), steam naphtha re-forming (SNR), or partial oxidation (POX) of natural gas. Such onsite hydrogen capacity has expanded by 59% globally between 1982 and 2007 at an average rate of about 1.2% per annum and reached 64% in 2012.2 This trend is expected to continue, as the increasing restrictions on the aromatic content of gasoline reduce the hydrogen output of catalytic re-former units (CRUs). In addition to producing more hydrogen, a refinery also has the option of recovering hydrogen from lean byproducts. The cost of hydrogen has a direct impact on the thin margins of a refinery. Faced with the options of using hydrogen from multiple sources with varying specifications and recovering it from lean streams, a state-of-the-art refinery needs an effective and optimal strategy for managing the blending, purification, distribution, consumption, and disposal of its hydrogencontaining streams. A systematically designed hydrogen network aims to facilitate this, and its synthesis has been receiving some attention in the past decade. The literature has successfully leveraged the concepts and approaches for heat and mass integration, namely, the pinch analysis3,4 and mathematical programming,5,6 to address the optimal synthesis and design of hydrogen networks. The former uses a largely sequential approach and provides powerful insights into thermodynamic targets such as minimum hydrogen consumption. While it still serves as an important tool for the design and © XXXX American Chemical Society

debottlenecking of different aspects, its ability to address economic targets and operational constraints is somewhat limited. The mathematical programming approach is more versatile and suited for simultaneous synthesis using economic targets. A superstructurebased optimization approach, while computationally more expensive and intuitively more obscure, accommodates richer synthesis options, complex operating constraints, and diverse performance metrics and, in general, yields better quality solutions. Therefore, we prefer the mathematical programming approach for this work. The mathematical programming approach involves the development and optimization of a superstructure. Hallale and Liu5 introduced an efficient mathematical method for refinery hydrogen networks and pointed out the drawbacks of pinch technology. Their model also allowed retrofitting of purifiers and new compressors into an existing network to improve hydrogen recovery. Zhang et al.7 developed a simultaneous optimization strategy for the overall refinery by integrating the hydrogen network and utilities with refinery processing and also investigated the strong interactions among them. Liu and Zhang6 developed a systematic methodology for selecting appropriate purifiers to increase the purity of the hydrogen fed to the network and minimized total annualized cost (TAC). They used linear relaxation of bilinear terms to obtain a relaxed solution to their original mixed-integer nonlinear program (MINLP) model. Fonseca et al.8 addressed the problem of actual hydrogen distribution at the Porto Refinery of the GALP ENERGIA network by using an adapted linear programming (LP) method that combined the traditional conceptual approach with mathematical optimization. Khajehpour et al.9 solved the MINLP model for the refinery hydrogen network using a reduced superstructure approach. Their approach reduced Special Issue: Jaime Cerda Festschrift Received: February 4, 2014 Revised: April 30, 2014 Accepted: July 10, 2014

A

dx.doi.org/10.1021/ie5005042 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematic configurations of units in a hydrogen network: (a) hydrogen source; (b) processing unit; (c) purification unit; (d) fuel gas sink.

that of Li et al.19 Recently, Jiao et al.20 presented a hydrogen network design model using stochastic programming to account for uncertainties in some parameters. In this work, we address the following four drawbacks of the existing work on refinery hydrogen networks. (1) Most previous work ignores the need or presence of stream conditioning equipment for heating, cooling, and expansion. Only compressors are present in their superstructures. (2) Most work uses shared rather than dedicated compressors. As we see later, this increases compression costs. (3) Most mathematical optimization models are nonlinear and too large and complex for the commercial solvers. (4) The fuel gas sinks (FGSs) in most work simply receive unutilized gas streams without any feed specifications. Recent works21,22 on fuel gas networks suggest that gas turbines, boilers, flares, fired heaters, and incinerators can be useful gas sinks, and they do have specific requirements for feed temperature, pressure, and quality. We begin by stating the grassroots synthesis problem for hydrogen networks. We then classify and model the key units in such networks and develop a new superstructure for the hydrogen network with dedicated compressors and conditioning equipment. We then derive an MINLP model for the new superstructure and show its application on several literature examples. We also show the benefits of our new superstructure and model by comparing our model with a representative existing model of Elkamel et al.12

the nonconvexity, size, and computational times of the original superstructure models by using engineering insights. They applied genetic algorithm (GA) to solve their model and used the data from a refinery in Iran to show significant savings. Liao et al.10 integrated purifiers in their retrofit study of a refinery in China and minimized total annualized cost. They compared some retrofit scenarios in their state-space superstructure model. Kumar et al.11 worked on the optimal distribution of hydrogen in a refinery network by using LP, nonlinear program (NLP), mixed-integer linear program (MILP) and MINLP models and evaluated the best among them for minimum utility and total annualized cost. Elkamel et al.12 developed a refinery hydrogen network model allowing retrofit with new compressor and purification units. They integrated that model with an overall refinery planning model to find the TACs for various planning scenarios. Ahmad et al.13 extended the model of Liu and Zhang6 and developed a multiperiod MINLP model to account for the changing operating conditions and their effects on the hydrogen network. They showed that the TAC of such a multiperiod network was lower than that of a single-period network. Salary et al.14 used process integration principles to design a hydrogen network for a refinery and proposed a systematic design hierarchy and some heuristic rules. Their procedure reduced hydrogen consumption and total network cost. Jeong and Han15 determined hydrogen consumption and recovery via pinch analysis and network optimization, while allowing the recycling of byproduct hydrogen between sources and sinks within a petrochemical complex. Jia and Zhang16 recognized the presence of components other than hydrogen and methane in their NLP model for a hydrogen network. Jiao et al.17 considered multiobjective optimization of hydrogen networks. They18 also developed a multiperiod MINLP model for hydrogen scheduling in a refinery. They showed that effective scheduling can ensure stable operation, reduce operating costs, and provide important strategies for efficient hydrogen management in a refinery. They used an MILP and NLP iterative solution methodology to avoid the composition discrepancy arising in a hydrogen scheduling model similar to

2. PROBLEM STATEMENT A refinery hydrogen network consists of I sources (i = 1, 2, ..., I), M processing units (m = 1, 2, ..., M), N purification units (n = 1, 2, ..., N), and J fuel gas sinks (j = 1, 2, ..., J). In addition, it may have conditioning units such as valves, compressors, heaters, and coolers that bring process streams to their desired conditions of pressure and temperature. A source is any entity or unit that can supply hydrogen to the network. A refinery may import hydrogen from an external B



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Lastly, a conditioning unit is a unit that changes the temperature or pressure of a stream along a transfer line. We allow four conditioning units. Valves/compressors change pressure, and heaters/coolers change temperature. Since the temperatures and pressures of various units can vary significantly in a refinery, the conditioning units are essential to enable flows within the network and also meet the feed temperature requirements of various units. The transfer lines among network units and associated conditioning units can mean substantial capital expenditure (CAPEX) and operating expenditure (OPEX). Thus, the need for and presence of conditioning units and transfer lines cannot be ignored in a realistic economic optimization. The previous works on the hydrogen networks has accounted only for the cost of compressors, but heaters, coolers, valves, and transfer lines have not been included. While valves may be relatively inexpensive, they can impact gas temperature significantly via the Joule−Thomson expansion. With the above understanding, we now state the hydrogen network synthesis problem as follows.

supplier, or it may have SMR, SNR, POX, or CRU units that produce it onsite. All these units and suppliers constitute the I sources (i = 1, 2, ..., I). Each source i has some known flow, purity, pressure, temperature, and price for its hydrogen stream, and it may supply to any of the processing, purification, or fuel gas sink units. The hydrogen source is said to be comprised of two units, namely, the hydrogen source and the splitter, as shown in Figure 1a. A processing unit (m = 1, 2, ..., M) is any unit that needs hydrogen as a reactant. Hydrotreaters, hydrocrackers, isomerizers, and olefin saturators are examples of such units in a refinery. We assume each processing unit to comprise four units, as shown in Figure 1b. These are mixer (M), reactor, separator, and splitter (S) in that sequence. The mixer combines the various hydrogen inputs to the processing unit and makes one feed stream for the reactor. These inputs may include streams from any processing units (including itself as a recycle stream), hydrogen streams from any purification units described later, and hydrogen streams from any sources described earlier. The reactor converts hydrogen into various products and byproducts. The separator recovers all of the unreacted or unutilized hydrogen from the reactor output as one hydrogen stream. The separator may produce other streams, but we ignore those not having hydrogen. The splitter recycles the hydrogen stream back to the processing units, fuel gas sinks, and/or purification units. A purification unit (n = 1, 2, ..., N) purifies or upgrades a lowpurity hydrogen stream to a high-purity stream. The most common purification units in a refinery use techniques such as pressure swing adsorption, membrane separation, or cryogenic separation. We allow the refinery to have at most N purification units. Some may already exist, and others may be installed during network synthesis. We model each purification unit to comprise four units (Figure 1c). These are mixer, purifier, hydrogen splitter, and residue splitter. The mixer combines the low-purity hydrogen streams from within the network and feeds them in a single stream to the purifier. The purifier separates this feed into a hydrogen-rich stream called raffinate or hydrogen stream and a hydrogen-lean stream called extract or residue stream. The hydrogen splitter distributes the hydrogen stream to the processing and purification units (except its own unit). The residue splitter distributes the residue stream to the fuel gas sinks only. We define a fuel gas sink (j = 1, 2, ..., J) as any unit in the refinery that can consume or dispose a gas with some fuel content. It serves as the destination for any unutilized stream in the network. We model each sink as a mixer followed by a consumer. Typical consumers are turbines, boilers, furnaces, incinerators, and flares. Boilers, turbines, and furnaces may produce heat, steam, and power, whereas flares and incinerators simply burn the combustibles. Thus, the former may generate some net returns for the network, but the latter involve cost. The mixer receives inputs from the various hydrogen sources, processing units, and purification units (residue streams only) and combines them into one stream. Figure 1d shows the schematic for fuel gas sinks. A transfer line (for example, SSpq) is the pipeline to move a gas stream from an origin unit (for instance, p = 1, 2, ..., P) to a destination unit (for instance, q = 1, 2, ..., Q) in the network. Any unit that can supply (receive) a gas stream to (from) another unit or itself in the hydrogen network is an origin (destination) unit. Hydrogen sources, processing units, and purification units are potential origin units. Fuel gas sinks, processing units, and processing units are potential destination units.

Given: (1) I hydrogen sources (i = 1, 2, ..., I) with known flows, temperatures, pressures, and purities (2) M processing units (m = 1, 2, ..., M) with known reactor feed flows, reactor feed pressures, minimum reactor feed temperatures, minimum hydrogen contents of reactor feeds, per pass conversions of hydrogen in the reactors, hydrogen stream temperatures, hydrogen stream pressures, and hydrogen stream purities (3) at most, N purification units (n = 1, 2, ..., N) with known pressures, temperatures, flow ranges, and purity ranges for the separator feeds; pressures, temperatures, flow ranges, hydrogen recoveries, and hydrogen purities in the hydrogen streams; and temperatures and pressures for the residue streams (4) J fuel gas sinks (j = 1, 2, ..., J) with known temperatures, pressures, flow ranges, purity ranges, and quality (density, contaminants, and heating value) ranges for the consumer feeds (5) CAPEX and OPEX data on the conditioning units (6) OPEX for each purification unit and economic returns from using hydrogen in each fuel gas sink Determine: (1) amount of hydrogen required by the refinery (2) structure of the hydrogen network with flows, purities, temperatures, and pressures at all points and units (3) existence, types, and duties of the equipment, conditioning units, and transfer lines Aim to minimize the TAC of the hydrogen network by including three components in TAC: (1) annualized CAPEX, including the capital costs of all conditioning units, purification units, and transfer lines (2) OPEX, including the cost of hydrogen supplies and the operating costs of the purification units, fuel gas sinks, and conditioning units (3) costs/savings for using gas streams in the fuel gas sinks Assume: (1) All non-hydrogen components in each stream are lumped into one single component. Methane, C



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Figure 2. Superstructure of the refinery hydrogen network.

(2) (3) (4) (5) (6)

Thus, the sequence of the condition units in each transfer line is cooler, valve, compressor, and then heater. With the preceding discussion and superstructure as our basis, we now develop a model that describes the synthesis of a refinery hydrogen network. First, we address the synthesis decisions related to the existence of purification units, transfer lines, and conditioning units. We define the following binary variable to model the existence of a purification unit n = 1, 2, ..., N.

carbon dioxide, ethane, nitrogen, and other lighter hydrocarbons are examples of non-hydrogen components. If and when necessary, we assume this single component to be methane for computing stream properties. No uncertainties exist. All compression processes are single stage and adiabatic with no restrictions on their inlet and outlet temperatures. All expansions are Joule−Thompson expansions via valves; no expander turbines exist. There are zero pressure drops across heaters, coolers, and transfer lines. Each possible network stream has a dedicated transfer line and conditioning units.

⎧1 if purification unit n should be installed in the network un = ⎨ ⎩ 0 otherwise

If a unit already exists in the plant, then we set un = 1. Furthermore, we can limit the number of purification limits as follows, which would allow one to select from possible options. N

3. MODEL FORMULATION

∑ un ≤ N U

Figure 2 shows our proposed superstructure for the refinery hydrogen network. A source or processing unit may feed any of the processing, purification, or fuel sink units via M + N + J split streams. The hydrogen stream from a purification unit may feed processing and purification units (except itself) via M + N − 1 split streams. The residue stream from a purification unit may feed the fuel gas sinks via J split streams. Thus, a processing unit or fuel gas sink may receive M + N + I, and a purification unit may receive M + I + N − 1 split streams. All the preceding potential connections are transfer lines. A transfer line may have one or more of cooler, compressor, valve, and heater, whose relative positions along the transfer line are important (see Jagannath et al.22). Clearly, both valve and compressor cannot exist on one line. Thus, their sequence is immaterial, and we place the compressor after the valve. To reduce the energy usage of the compressor, we place the cooler before the valve to cool the stream. Finally, we place the heater after the compressor to adjust the final stream temperature.

n=1

(1)

A potential transfer line (for example, SSpq) may or may not exist in the final network; therefore, we define another binary variable and limit the number of transfer lines to V. ⎧1 if transfer line SSpq should have flow vpq = ⎨ ⎩ 0 otherwise ⎪

⎪

P

Q

∑ ∑ vpq ≤ V p=1 q=1

(2)

All transfers to a destination unit q must attain the pressure (for instance, PINq) of that unit. Thus, if the outlet pressure (for example, POUTp) of an origin unit p exceeds (is below) PINq, then SSpq should have a valve (compressor). Since POUTp and PINq are known, we can know from vpq if a valve or compressor would exist on a transfer line. However, the same is not true for heaters and coolers. The feeds to a unit may have different D



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while the same across the outlet splitter after processing unit m gives us

temperatures, so we need to define the following binary variables for the existence of heaters and coolers. ⎧1 if SSpq should have a heater hpq = ⎨ ⎩ 0 otherwise

I

⎪

Gm =

⎪

j=1

⎧1 if SSpq should have a cooler cpq = ⎨ ⎩ 0 otherwise

hpq + cpq ≤ vpq

vijFijL ≤ Fij ≤ min(FiU , FjU)vij

(5)

L vimFim ≤ Fim ≤ min(FiU , Fm)vim

(6)

vimFinL ≤ Fin ≤ min(FiU , FnU)vin

(7)

FLim,

I

Fm =

i=1

n=1 I

Fmxm =

⎛⎡ F x (1 − χ ) ⎤ ⎞ m ⎥ L ⎜⎢ m m Gmm , Fm⎟vmm ′ ′vmm ′ ≤ Gmm ′ ≤ min⎜ ⎥⎦ ⎟⎠ ym ⎝⎢⎣

(15)

GLmn,

I

GLmm′

M

N

M

N

J

∑ Fin + ∑ Gmn + ∑ R n′ n = ∑ R nm + ∑ R nn′ + ∑ Enj i=1

m=1

n ′= 1

m=1

n ′= 1

n ′≠ n

j=1

n ′≠ n

(16) I

rn(∑ Finzi + i=1

M

N

M

N

∑ Gmnym + ∑ R n′ n ypn) = ypn( ∑ R nm + ∑ R nn′) m=1

n ′= 1

m=1

n ′= 1

n ′≠ n

Gm ′ m

N

M

(1 − rn)ypn( ∑ R nm +

M

∑ Fimzi +

∑ R nm ypn +

∑

i=1

n=1

m ′= 1

n ′≠ n

(17) (8)

m ′= 1

Gm ′ mym

m=1

N

(10) E

J

∑ R nn′) = rn yr n(∑ Enj) n ′= 1 n ′≠ n

(9)

The hydrogen balance across the processing unit gives us χm = (Fmxm − Gmym )/Fmxm

(14)

Fn =

M

∑ Fim + ∑ R nm + ∑

(13)

where and are the minimum flows required for the transfer lines to exist. For each purification unit (n = 1, 2, ..., N), we define the following: Rn′n, flow from purification unit n′; Enj, flow of the residue stream to fuel gas sink j; rn, known recovery of hydrogen in the hydrogen stream; yrn, mole fraction of hydrogen in the residue stream. The purification unit may receive feeds from processing units, hydrogen sources, and other purification units. It is unlikely to receive purified hydrogen from another purification unit, because that would mean back-mixing with its lowerpurity hydrogen feed. However, if a purification unit requires a feed with some minimum purity or involves a second stage upgrade, then such a feed would make sense. Each purification unit has two product streams, namely, a hydrogen-rich stream and a hydrogen-lean stream. The hydrogen-rich stream (product) is sent to the processing units and other purification units in the network, whereas the hydrogen-lean stream (residue) is sent to fuel gas sinks. The overall mass and hydrogen balances for the purification unit give us

FLin

N

⎞ Fj⎟⎟vmj ⎠

⎛⎡ F x (1 − χ ) ⎤ ⎞ m m m ⎥ L Gmn vmn ≤ Gmn ≤ min⎜⎜⎢ , Fn⎟⎟vmn ⎢ ⎥ y ⎣ ⎦ ⎝ ⎠ m

GLmj,

where and are the minimum flows required for the corresponding transfer lines to exist. The upper limits on the flows are set by the minimum of flows into or out of various destination and origin units. For a processing unit (m = 1, 2, ..., M), we define the following: Gm, flow out of processing unit; Gmm′, flow to processing unit m′; Rnm, hydrogen stream flow from purification unit n; Gmj, flow to fuel gas sink j; Gmn, flow to purification unit n; χm, known per pass conversion of hydrogen in the reactor; xm, known minimum required mole fraction of hydrogen in the reactor feed; ym, known mole fraction of hydrogen in the exit hydrogen stream; ypn, known mole fraction of hydrogen in the hydrogen stream from purification unit n. The overall mass and hydrogen balances for the feed mixer at processing unit m are FLij ,

Gmm ′)

m ′= 1

⎛⎡ F x (1 − χ ) ⎤ m m m ⎥ L Gmj vmj ≤ Gmj ≤ min⎜⎜⎢ , ⎢ ⎥⎦ y ⎝⎣ m

N

(4)

n=1

(12)

∑ Fij + ∑ Fim + ∑ Fin n=1

M

∑ Gmn + ∑

The bounds on the flows in various transfer lines are as follows:

(3)

m=1

(11)

N

j=1

3.1. Flow Balances. We assume all flows to be molar flows (kg mol s−1) and purities to be hydrogen mole fractions. We define the following for a hydrogen source (i = 1, 2, ..., I): Fi, actual flow (FLi ≤ Fi ≤ FUi ; for a source with fixed supply, we set FLi = Fi = FUi , and for a source (e.g., SMR or external supplier) with flexible supply, we assign appropriate bounds based on hydrogen availability); zi, known mole fraction of hydrogen; Fij, flow from source i to fuel gas sink j; Fim, flow from source i to processing unit m; Fin, flow from source i to purification unit n; Fj, total flow entering sink j (FLj ≤ Fj ≤ FUj ); Fm, known required feed flow into processing unit m; Fn, total flow entering purification unit n (FLn ≤ Fn ≤ FUn ). Then, the mass balance and flow limits for source i are

j=1

Gmm ′

m ′= 1

Fmxm(1 − χm ) = ym (∑ Gmj +

Since we have assumed unrestricted inlet and outlet temperatures for the compressors, we can safely assume that both heaters and coolers may not exist in a transfer line.

M

n=1

I

⎪

J

M

Eliminating Gm from eqs 10 and 11, we get

⎪

Fi =

N

∑ Gmj + ∑ Gmn + ∑

j=1

(18)

FnLun ≤ Fn ≤ FnUun

(19)

L U U vnn ′R nn ′ ≤ R nn ′ ≤ min(Fn , Fn ′≠ n)vnn ′

(20)



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L vnmR nm ≤ R nm ≤ min(FnU , Fm)vnm

(21)

vnjEnjL ≤ Enj ≤ min(FnU , FjU)vnj

(22)

The temperature and pressure of a transfer stream will change along SSpq due to the conditioning units, if any. Hasan et al.21 and Jagannath et al.22 modeled the temperature (Tpq, TLpq ≤ Tpq ≤ TUpq) along SSpq in terms of H ≡ FCT. Here, F is the gas flow, T is the temperature, and C is the constant specific heat. Using their approach, we write the following for all SSpq.

Since the purity of the hydrogen-lean stream is unknown, bilinear terms are present in eq 18. We can eliminate these bilinear terms by assuming a two-component system of hydrogen and methane mentioned earlier. We define E1nj as hydrogen flow and E2nj as methane flow. Therefore, Enj = E1nj + E2nj. Replacing Enj in eqs 16, 18, and 22, we get I

Fn =

M

HINpq ⎧C TOUT F − ΔH C − ΔHV + ΔH H if POUT > PIN p pq pq pq pq p q ⎪ p ⎪ C B H = ⎨Cp TOUTpFpq − ΔH pq + ΔHpq + ΔHpq if POUTp < PINq ⎪ ⎪C TOUT F − ΔH C + ΔH H if POUTp = PINq p pq pq pq ⎩ p

N

∑ Fin + ∑ Gmn + ∑ R n′ n i=1

m=1

n ′= 1

(30)

n ′≠ n

M

=

N

J

∑ R nm + ∑ R nn′ + ∑ E1nj + ∑ E2nj m=1

n ′= 1

j=1

j=1

n ′≠ n

M

N

(1 − rn)ypn( ∑ R nm + m=1

(23)

(31)

J

U HINpq ≤ CpTpq Fpq

∑ R nn′) = rn ∑ E1nj n ′= 1

j=1

n ′≠ n

vnjEnjL

⎧C TOUT F − ΔH C − ΔHV if POUT > PIN p pq pq pq p q ⎪ p L CpTpq Fpq ≤ ⎨ C ⎪Cp TOUTpFpq − ΔH pq if POUTp ≤ PINq ⎩

J

≤ (E1nj + E2nj) ≤

min(FnU ,

FjU)vnj

(24)

⎧ if POUTp > PINq ⎪ μ CpFpq(POUTp − PINq) p V ΔHpq =⎨ ⎪ ⎩0 otherwise

(25)

For each fuel sink (j = 1, 2, ..., J), we define the following: Fj, actual feed flow (FLj ≤ Fj ≤ FUj ); xj, mole fraction of hydrogen in the feed (xLj ≤ xj ≤ xUj ). Then, the mass balance equations for fuel sink j are I

Fj =

M

i=1

m=1

I

M

i=1

B ΔHpq

CL L CpΔT pq Fpqcpq

N

m=1

n=1

I

M

N

ΔHBpq

i=1

m=1 I

FjxjL ≤

n=1 M

n=1

i=1

m=1

(28)

N

∑ Fijzi + ∑ Gmjym + ∑ E1nj ≤ FjxjU n=1

otherwise

≤

C ΔH pq

≤ Cp(TOUTp −

L U Tpq )Fpq cpq

(35) (36)

ΔHHpq

where is the change in H during compression, is the change during heating, ΔHCpq is the change during cooling, and ΔHVpq is the change during expansion. Furthermore, η is the efficiency of adiabatic compression, ncp and μp are the adiabatic index and average Joule−Thompson coefficient of the stream along SSpq, FUpq (FLpq) is the maximum (minimum) possible flow HL along SSpq, and ΔTCL pq (ΔTpq ) is the minimum acceptable temperature drop (rise) along SSpq. Equation 30 gives the temperature along the transfer line SSpq written in terms of the variable H (explained earlier) based on the pressures of the origin and destination units. Equations 30−32 ensure that temperature along transfer line Tpq is within its limits of TLpq ≤ Tpq ≤ TUpq. Equations 33 and 34 model the changes due to gas expansion and compression along SSpq. They are to be substituted appropriately into eq 30. Equation 35 and 36 set the minimum acceptable heat duties for the existence of coolers and heaters, respectively. Equations 30−33 are valid only for a positive Joule−Thompson coefficient. In gas streams involving hydrogen, this coefficient can be negative under certain conditions. Therefore, we need the following in place of eqs 30, 31, and 33, when μp < 0.

N

∑ Fij + ∑ Gmj + ∑ E1nj + ∑ E2nj

⎤ − 1⎥ if POUTp < PINq ⎥⎦

HL L H U L U CpΔTpq Fpqhpq ≤ ΔHpq ≤ Cp(Tpq − Tpq )Fpq hpq

(27)

As discussed by Hasan et al.21 and Jagannath et al.,22 a fuel gas sink may demand some feed quality specifications such as lower heating value, specific gravity, gas composition, and density, etc. For the sake of simplicity, we consider xLj ≤ xj ≤ xUj as the only purity specifications. Equation 27 has bilinear terms Fjxj and Enjyrn. To eliminate them, we replace Enj by (E1nj + E2nj) and Enjyrn in eq 27 by E1nj. In general, the purity of the hydrogen gas entering a fuel gas sink must satisfy some bounds, so we rewrite eqs 26 and 27 as follows with all linear terms. Fj =

⎧ c ⎡⎛ (CpTpFpq − ΔH pq ) PINq ⎞ ⎪ ⎢⎜ ⎟ ⎪ ⎜ ⎢⎣⎝ POUTp ⎟⎠ =⎨ η ⎪ ⎪ ⎩0

ncp

(34)

(26)

n=1

∑ Fijzi + ∑ Gmjym + ∑ Enj yr n

Fjxj =

(33)

N

∑ Fij + ∑ Gmj + ∑ Enj

(32)

(29)

3.2. Pressures and Temperatures. As stated before, stream pressures and temperatures across a hydrogen network do vary and must be modeled adequately. While we need not address the changes that occur inside various units, we must effect those that occur along the transfer lines by installing appropriate conditioning units. Because all transfer lines in our network have the same superstructure of conditioning units, we can write down these constraints for a generic transfer line SSpq that moves gas from an origin unit p (p = 1, 2, ..., P) to a destination unit (q = 1, 2, ..., Q). TOUTp (TINq) and POUTp (PINq) are defined as the known exit (inlet) temperature and pressure of unit p (q).

HINpq ⎧C TOUT F − ΔH C + ΔHV + ΔH H if POUT > PIN p pq pq pq pq p q ⎪ p ⎪ C B H = ⎨Cp TOUTpFpq − ΔH pq + ΔHpq + ΔHpq if POUTp < PINq ⎪ ⎪C TOUT F − ΔH C + ΔH H if POUTp = PINq p pq pq pq ⎩ p

(30a) F


Industrial & Engineering Chemistry Research L CpTpq Fpq

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⎧C TOUT F − ΔH C if POUT > PIN p pq pq p q ⎪ p ≤⎨ C ⎪Cp TOUTpFpq − ΔH pq if POUTp ≤ PINq ⎩

(2) Processing units have no costs that are meaningful for the hydrogen network, because they already exist and operate at fixed rates. (3) For computing the CAPEX and OPEX of various network units, we define a characteristic dimension for each unit. This dimension is feed rates for purification units and transfer lines, ΔHVpq/(|μp|Cp) (essentially the product of flow and pressure drop) for valves, ΔHBpq for compressors, ΔHCpq for coolers, and ΔHHpq for heaters. (4) The CAPEX of all units have fixed and size-dependent components. The latter obey power law correlations with the characteristic dimensions. If a purification unit already exists, then its cost coefficients are zero. The same holds for all fuel gas sinks, as they also exist. (5) The OPEX for purification units, transfer lines, compressors, valves, heaters, and coolers vary linearly with their characteristic dimensions. (6) The OPEX of fuel gas sinks has two parts. The first is the cost of running the sink, which is a linear function of the total flow into the sink. This is zero for sinks that produce energy, such as gas turbine, boiler, and heater, etc., but positive for disposal sinks such as flare or incinerator. The second is the economic value or surplus revenue generated by the sink. This is negative for sinks, such as boiler, turbine, and heater, and a linear function of hydrogen flow (versus total flow) into the sink. (7) The TAC of the hydrogen network is given by TAC = AF × CAPEX + OP × OPEX, where OP represents the operating hours of the refinery in a year and AF is the annualization factor. Then, we write TAC as

(31a)

⎧ if POUTp > PINq ⎪ − μ CpFpq(POUTp − PINq) p V ΔHpq =⎨ ⎪ ⎩0 otherwise (33a)

When streams coming from different units mix to form the feed for unit q, eqs 30, 33, and 34 ensure that the mixing occurs at PINq. However, the feed must also have the required temperature of TINq. To ensure this, we use the following constraint. P

P

TINq ∑ CpFpq = p=1

∑ HINpq (37)

p=1

As discussed earlier, processing units may not require a certain feed temperature, so we do not write eq 37 for them. However, it must be written for each purification unit and fuel gas sink. 3.3. Objective Function: Total Annualized Cost. We use minimum TAC as the objective for our network synthesis model. We include the following in TAC: (1) cost of hydrogen supplies; (2) total CAPEX and OPEX for purification units, transfer lines, and conditioning units; and (3) OPEX for or revenue from fuel gas sinks. Furthermore, we assume the following. (1) The price of hydrogen from source i is pi, which is normally positive for on-purpose hydrogen producers and external hydrogen suppliers and zero for in-house hydrogen producers such as the catalytic re-forming unit. The network may or may not consume all of the gas available from a source. ⎡ ⎢ ⎢ N TAC = AF⎢ ∑ anun + ⎢ n=1 ⎢ ⎣

N

I

M

N

∑ bn(∑ Fin + ∑ Gmn + ∑ R n′ n)d n=1

i=1

m=1

n

n ′= 1 n ′≠ n

⎡ ⎢⎡ ⎧ aV if POUT < PIN ⎫⎤ P Q ⎢⎢ T p q ⎪⎥ ⎪ pq T H C T ⎬⎥vpq + apq (Fpq)dpq + ∑ ∑ ⎢⎢apq + ⎨ hpq + apq cpq + bpq B ⎪ apq if POUTp > PINq ⎪ p = 1 q = 1 ⎢⎢ ⎩ ⎭⎥⎦ ⎣ ⎢ ⎣ V ⎤⎤ ⎧ ⎫ dpq V ⎥⎥ ⎪ V ⎡ ΔHpq ⎤ ⎪ ⎥ ⎥ if POUTp < PINq ⎪ ⎪bpq⎢ H C ⎥ H H dpq C C dpq ⎬ + bpq (ΔHpq ) + bpq (ΔH pq ) ⎥⎥ + ⎨ ⎢⎣ (|μp |Cp) ⎥⎦ ⎪ ⎪ ⎥⎥ ⎪ (bB ΔH B )dpqB ⎪ ⎥⎥ if POUTp > PINq ⎭ ⎩ pq pq ⎦⎦

⎡ I ⎢ + OP⎢ ∑ pF + i i ⎢⎣ i = 1

N

I

M

N

J

J

n=1

i=1

m=1

n ′= 1

M

N

j=1

j=1

i=1

m=1

n=1

n ′≠ n

V ⎛ ⎞⎤ ΔHpq B B H H C C ⎟⎥ T V ⎜ + opqΔHpq + opqΔHpq + opqΔH pq⎟⎥ + ∑ ∑ ⎜opqFpq + opq (|μp |Cp) p=1 q=1 ⎝ ⎠⎥⎦ P

I

∑ on(∑ Fin + ∑ Gmn + ∑ R n′ n) + ∑ γjFj − ∑ βj(∑ Fijzi + ∑ Gmjym + ∑ E1nj )

Q

G

(38)



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In the preceding equation, the first CAPEX term is for purification units and the second is for all transfer lines and their conditioning units. The first OPEX term is the cost of hydrogen from various sources and the operating cost of purification units, the second is the operating cost of fuel gas sinks, and the third is the OPEX of all transfer lines and their conditioning units. This completes our MINLP formulation (M) for hydrogen network synthesis. It consists of eqs 1−15, 17, 19−21, 22−25, 28, 29, and 30−37 with eq 38 as the objective to be minimized. The most significant aspect of the model is that all its constraints are linear with only the objective function being nonlinear. We now consider some examples from the literature to illustrate the application of our model and highlight its unique features.

rise to posynomial terms in the compressor energy constraints. Besides, M1 allows multiple streams to share a compressor using an input mixer and an outlet splitter. These also result in bilinear mixing constraints. In contrast, M does not allow such mixing or splitting and hence saves on bilinear constraints. While M1 does not consider any conditioning equipment such as valves, heaters, and coolers, so ignores the temperature effects within the network, M accounts for temperature variations in the network. Lastly, M allows data, properties, and parameters that can be tailored for various units, streams, and conditions. In contrast, M1 assumes some uniform values that are stream-independent and condition-independent. The preceding differences necessitate changes in M and M1 to ensure a fair comparison. First, we ignore the temperature effects in M by eliminating heaters and coolers. We drop eqs 35 and 36 and allow only the compression and expansion terms only in eqs 30 and 31. Second, we ignore the costs of the fuel gas sink and valves and assume only one FGS, namely, a flare without any purity restrictions and flaring penalty. Third, we assume constant stream-independent properties for all of the streams in the network. We define this modified M as reduced M or RM. Since M1 in the literature has been primarily used for retrofit synthesis, we need to modify it for grassroot synthesis. First, we allocate one dedicated compressor to each processing unit to enable the recycle of unused hydrogen. Since the operating conditions of the processing units are known a priori, the inlet/ outlet pressures of these recycle compressors become known, and their capacities depend on the flows through the processing units. For transfers other than these recycles, M1 allows a shared set of makeup compressors with unlimited capacities and unknown inlet and outlet pressures. Finally, we add the capital costs of all these compressors (both makeup and recycle) in the network. We call this modified form of M1 as M2. M2 is inherently more complex and difficult than M1. The Supporting Information accompanying this work includes details of both M1 and M2. We consider four examples (examples 1−4) to illustrate the application of our model for hydrogen network design and synthesis. Examples 1−3 are for grassroot synthesis, whereas example 4 is for retrofit synthesis. First, we compare RM and M2 using examples 1−3 to show the better results from our model. For this comparison, we omit all costs except those for hydrogen, compressors, and purification units. For these, we use the cost coefficients from Hallale and Liu5 as listed in Supporting Information Table S2. Supporting Information Table S2 also lists the physical properties used for the comparison. The limitations of M2 necessitate that these physical properties be the same for all streams. Then, we also solve examples 1−3 with M. All of the tables are given in the Supporting Information. Supporting Information Table S1 lists the cost coefficients for M. We compare the solutions from RM and M. The differences in model features and cost coefficients cause them to be different. Furthermore, stream-specific properties in M as opposed to uniform or constant stream properties in RM also cause significant variations in their networks. For all examples, Supporting Information Table S3 gives the single-pass hydrogen conversions or recoveries and quality specifications on sources and feeds to various units; Supporting Information Table S4 gives the specifications on product streams from various units and sources. Supporting Information Table S5

4. EXAMPLES We assume the following for all the examples. (1) The allowable temperature range for streams is TLpq = 250 K and TUpq = 1000 K. (2) The annualization factor for the capital cost is 0.1. (3) The cost coefficients in Supporting Information Table S1 apply to all examples. (4) Plant on-stream time is 365 days (8760 h). (5) All purification units employ pressure swing adsorption (PSA). (6) Gas turbines (GT) and flares (FL) are the only fuel gas sinks. The flares have no purity, temperature, or flow restrictions. We used GAMS/DICOPT as the MINLP solver with multiple NLP solvers and GAMS/CPLEX v12.5.0.1 as the MILP solver in GAMS23 24.1.0. The NLP solvers were GAMS/ IPOPT, GAMS/SNOPT, GAMS/MINOS, and GAMS/ CONOPT. We chose the best from their solutions. As observed by Ahmetovic and Grossmann,24 the concave univariate terms in the objective may pose convergence issues with SNOPT, IPOPT, and MINOS, as their derivatives become unbounded at zero. Thus, we added a small perturbation (ε = 0.0001) to the variable in each concave univariate term. The resulting solution was used as an initial point, and the model was resolved without the small perturbation term using GAMS/ DICOPT with CONOPT as the NLP solver. For all of the examples, 0.0001 unit was used as the minimum flow and relative optimality tolerance (Optcr = 0 in GAMS) as zero. Apart from applying our model M to literature examples, we also compare it with the model M1 of Elkamel et al.12 to demonstrate its unique features. M1 is representative of several literature models5,10,13 on hydrogen network synthesis. While these models differ in some aspects, they all employ a set of shared compressors to enable flows from low-pressure origin units to high-pressure destination units. M1 differs from M in four respects: (1) shared vs dedicated compressors; (2) absence vs presence of conditioning equipment such as valves, heaters, and coolers; (3) absence vs presence of temperature effects; and (4) uniform, stream-independent, and constant vs streamspecific parameters and physical properties that can vary with process conditions. M uses a dedicated compressor for each transfer line. This allows us to assign its inlet and outlet pressures a priori. This is significant because the constraints for computing energy needs of the compressors become linear. In contrast, M1 forces streams to select compressors from a shared set. So, the inlet and outlet pressures of all compressors are unknown and give H



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Figure 3. Best network from RM for example 1. All flows in kg mol s−1.

Figure 4. Best network from M for example 1. All flows in kg mol s−1.

cooler before the compressor in the supply to A. The heater helps satisfy the feed temperature limits of FGS, and the cooler reduces compressor duty. Such stream conditioning features have been absent in the literature on hydrogen network design to the best of our knowledge. We also see that the annual CAPEX, OPEX, and TAC of RM and M differ only slightly. This is primarily due to the similar stream properties (Supporting Information Table S2 for RM and Supporting Information Table S4 for M) and also the use of linear compressor cost from Supporting Information Table S2. 4.2. Example 2. This is from Elkamel et al.12 A refinery has two sources: Hydrogen plant (HP) and catalytic re-forming unit (CR). It has five processing units: hydrocracker (HC), gas oil hydrotreater (GOHT), diesel hydrotreater (DHT), residue hydrotreater (RHT), and naphtha hydrotreater (NHT). No purification unit exits, but a disposal sink in the form of a flare exists. FL demands no quality or flow specifications and has no operating costs. All flows have been converted from MMscfd to kg mol s−1. Figure 5 (Figure 6) shows the best network from RM (M2). The fresh hydrogen supply and the gas disposal into FL are the same for both networks. The hydrogen supply matches the minimum thermodynamic requirement of 0.890 kg mol s−1. As expected, the network from RM has more compressors (nine vs five) than that from M2. Figure 6 has three recycle (R1, R3, and R4 for HC, DHT, and RHT, respectively) and two makeup compressors (MA and MB); whereas Figure 5 has four recycle (R1, R2, R3, and R4 for HC, GOHT, DHT, and RHT, respectively) and five makeup compressors (MA, MB, MC, MD, and ME). However, the network from RM still costs less. The TAC of $50,275,000/year is 6.3% lower than the $53,644,000/year for M2, the OPEX of $48,782,000/year is 3.9% lower, and the annual CAPEX of $1,493,000/year is 48.2% lower than the $2,884,000/year for M2. Clearly, RM gives a better network than M2. The lower cost from RM is largely due to the smaller compressors. Supporting Information Table S6 gives the duty, pressures, flows, CAPEX, and OPEX of the compressors in RM and M2. As we can see, the annual CAPEX and OPEX depend

summarizes the comparison of model statistics and costs for RM, M2, and M for examples 1−3. 4.1. Example 1. This is a small literature example5 with one hydrogen source (HP), two processing units (A and B), and one flare (FGS) with no inlet purity specifications. We have converted the flows from million standard cubic feet per day (MMscfd) to kg mol s−1. RM and M2 both give the same network (Figure 3) with identical flows, pressures, CAPEX, OPEX, and TAC for this small example. The annual CAPEX, OPEX, and TAC are $5,162,000/year, $140,617,000/year, and $145,779,000/year, respectively, for both RM and M2. The network requires 2.537 kg mol s−1 of hydrogen supply, the thermodynamic minimum for this example. While their solutions are identical, key differences exist between RM and M2 from an optimization perspective. First, RM is much smaller than M2. RM needs fewer continuous variables (49 vs 106), fewer constraints (71 vs 190), and fewer binary variables (9 vs 50) than M2. Second, RM is nicer and easier to solve than M2. For the case of a linear objective, RM is an MILP, while M2 is a nonconvex MINLP with bilinear and posynomial terms. Thus, RM guarantees global optimality, but M2 cannot. For instance, M2 gave local solutions of $146,583,000/year and $168,782,000/year as well. BARON could not guarantee a solution even after 1000 CPUs. The bilinear terms in M2 arise from stream mixing at the makeup compressor inlets, and the posynomial terms arise due to the unknown pressures at the compressor inlets and outlets. These make M2 solution more difficult. Such nonlinear terms do not exist in RM, because streams do not mix, and the conditions at the origin and destination units prefix the pressures at the compressors and valves. These advantages of RM over M2 remain valid even for examples 2 and 3. We now solve example 1 with M. For compressors, we use the linear costs from Supporting Information Table S2. M has 98 continuous variables, 125 constraints, and 27 binary variables. Figure 4 shows the best network with TAC = $145,693,000/year, annual CAPEX = $5,001,000/year, and OPEX of $140,692,000/year. The network is identical to that in Figure 3 from RM except for a heater for the feed to FGS and a I



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strongly on the flows, duties, and pressures. In Figure 6, MA compresses the feed to 2000 psia, even though GOHT and DHT need only 500 and 600 psia, respectively. This results in large duty, annual CAPEX, and OPEX for MA as seen in Supporting Information Table S6. However, in Figure 5, no feed stream is compressed beyond the required pressure. Thus, in spite of having more compressors, the annual CAPEX for the compressors from RM is $1,493,000/year (48.2% lower) compared to $2,884,000/year from M2. Similarly, the OPEX of the compressors for RM is $1,912,000/year or 50.8% lower than that for M2. We now solve example 2 with M. M has 461 continuous variables, 571 constraints, and 98 binary variables. Its structure and flows are similar to those from RM (Figure 5) except for an extra heater and a cooler. The heater is placed on stream from RHT to FL, and the cooler is placed on the stream from HP to HC. The best TAC is $48,965,000/year with an annual CAPEX of $157,000/year and OPEX of $48,808,000/year. The hydrogen supply remains at the thermodynamic minimum of 0.89 kg mol s−1. The HC with its relatively higher purity requirement uses the majority of the supply of hydrogen from HP, and NHT needs no supply from HP. We see that the OPEX from M is marginally more than RM. This is mainly due to the need for conditioning. However, the CAPEX is smaller in M than RM. This is mainly because M uses the nonlinear correlations for compressor costs from Supporting Information Table S1, while RM uses the linear correlations from Supporting Information Table S2. To see the benefits of integration, consider the base scenario of no integration, in which the refinery uses only pure hydrogen

from HP to satisfy all its needs. For this base case, the refinery would consume 163,987 kg mol day−1 of hydrogen. Integration via M reduces this consumption by 54%. This translates into lower natural gas usage and carbon dioxide emissions from HP. Thus, a well-designed hydrogen network has benefits from the viewpoints of energy, economics, and the environment. 4.3. Example 3. We modify example 2 by adding two sinks in the forms of a GT and a FL and allowing a potential purification unit (PSA). For comparing RM and M2, we merge both sinks into a single fuel gas sink. Figures 7 and 8 show the best networks from RM and M2, respectively. Both include one PSA unit. The TAC of $47,102,000/year for RM is 3.7% lower than the $48,919,000/year for M2, the OPEX of $45,522,000/ year for RM is 2.3% lower than $46,588,000/year for M2, and the annual CAPEX of $1,579,000/year for RM is 31.5% lower than the $2,308,000/year for M2. Figure 8 has four recycle (R1, R2, R3, and R5 for HC, GOHT, DHT, and NHT, respectively) and three makeup compressors (MA, MB, and MC). In contrast, Figure 7 has four recycle (R1, R2, R3, and R4 for HC, GOHT, DHT, and RHT, respectively) and seven makeup compressors (MA, MB, MC, MD, ME, MF, and MG). As in example 2, some streams in the network from M2 are compressed more than their required pressures. For instance, MA compresses the feeds to GOHT and DHT to 2000 psia, while they need only 500 and 600 psia, respectively. MC compresses the PSA feed to 500 psia, while only 300 psia is needed. Hence, RM has lower annual compressor CAPEX and OPEX than M2 by 32.3% and 35.5% respectively. While hydrogen supply to the network and gas discharge to the FGS are the same for RM and M2, the load on the purification unit J



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Figure 6. Best network from M2 for example 2. All flows in kg mol s−1.

4.4. Example 4. Here, we apply M for retrofit synthesis. Figure 10 shows a slightly modified existing hydrogen network from Alves and Towler.3 We modify their network slightly to resemble a structure suitable for M. For all transfers that occur in their structure, we dedicate transfer lines along with the required compressors or valves. This automatically eliminates the mixing at compressor inlets. These modifications do not alter the hydrogen flow or purity to (from) any unit. The refinery has three hydrogen sources. Steam re-forming unit (SRU) and catalytic re-forming unit (CRU) are internal. Hydrogen import (HI) is external. No costs were associated with the hydrogen stream from existing SRUs and CRUs. The refinery has four processing units: hydrocracking unit (HCU), NHT, cracked naphtha hydrotreater (CNHT), and DHT. It has a fuel gas sink with no positive or negative costs. We ignore temperature specifications, so no heaters and coolers are needed. The aim is to retrofit this network with a purification unit and minimize OPEX. All flows are in kg mol s−1. Other parameters and cost coefficients are from Supporting Information Table S2. We first fix the capacities of all the existing transfer lines, compressors, and valves and set their capital costs to zero. The OPEX for the existing network is $17,625,000/year. We now use M to minimize OPEX. Figure 11 shows the retrofitted network. The dotted lines represent the new connections. Supporting Information Table S7 compares the costs of the two networks. M reduces OPEX by $4,070,000/year (23.1%). The retrofit demands an investment of $2,718,000 with a payback period of 0.7 years.

is 3.8% lower for RM. All of these result in a 3.7% lower TAC for RM than M2. M for example 3 has 528 continuous variables, 740 constraints, and 190 binary variables. Figure 9 gives the best network with a TAC of $45,573,000/year, an annual CAPEX of $2,33,000/year, and OPEX of $45,339,000/year. The network from M is different from that from RM, but hydrogen consumption is the same at 0.828 kg mol s−1. The network from M has four recycle (R1, R2, R3, and R4 for HC, GOHT, DHT, and RHT, respectively) and nine makeup compressors (MA, MB, MC, MD, ME, MF, MG, MH, and MF) along with two heaters and one cooler. Out of the 0.039 kg mol s−1 of gas going to the fuel gas sink, 0.007 kg mol s−1 goes to GT, whereas the rest goes to FL. The supply to GT is limited due to purity restrictions. As mentioned previously, it may not be fair to compare the costs from RM and M due to significant differences in data, stream properties, and cost correlations. Since RM needs only 11 compressors, and M needs 13 compressors, we look to see if M indeed gives the solution with the least TAC. To this end, we fixed the network from RM into M. On minimizing TAC, we get a TAC of $45,583,000/year, annual CAPEX of $240,000/year, and OPEX of $45,343,000/ year. This is marginally higher than our best solution; thus we confirm that M gives a better solution. From examples 1−3, we conclude that M is simpler, more efficient, and better than the previous models. Supporting Information Table S5 also suggests that hydrogen and compressor costs are the most dominant among OPEX and CAPEX, respectively. K



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Figure 8. Best network from M2 for example 3. All flows in kg mol s−1. L



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Figure 9. Best network from M for example 3. All flows in kg mol s−1.

Figure 10. Existing network for example 4. All flows in kg mol s−1. M



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Figure 11. Optimal retrofitted network for example 4. All flows in kg mol s−1.

5. CONCLUSION Hydrogen is a valuable commodity in the refining industry. We presented a new mathematical model for synthesizing a hydrogen network with minimum TAC. The model is built on a new superstructure that accommodates several practical features to yield more realistic and economical designs. It is the first model in the literature to accommodate temperature specifications and consider temperature variations via conditioning equipment such as cooler, valve, and heater. It allows fuel gas sinks with feed specifications and revenue generation. Finally, it is a smaller and computationally more robust model with linear constraints and a nonlinear objective function. The literature models are larger and more nonlinear. On several literature examples, our model yielded substantial capital and operational cost savings (in some cases up to 50−60%) as compared to literature models. Our model is also generic, in that it could be applied to any chemical facility that produces and consumes hydrogen. The model with dedicated compressors, however, does have one disadvantage, that it may need more compressors. While one could use engineering judgment to modify the network with dedicated compressors to reduce the number of compressors by “merging” some streams with nearly identical conditions, this will definitely affect the network cost. However, finding such mergeable streams may not be always possible.

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costs of existing and retrofitted networks and text describing mathematical models for M1 and M2. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*(A.E.) E-mail: [email protected]. Tel.: +1-519-888-4567 ext. 37157. Fax: +1-519-746-4979. *(I.A.K.) E-mail: [email protected]. Tel.: +65 6516-6359. Fax: +65 6779-1936. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Department of Foreign Affairs, Trade and Development, Canada for their financial support through the Canadian Commonwealth Scholarship Program. We acknowledge AspenTech for granting us the academic license for the use of AspenHysys in estimating the properties of gases. We also acknowledge IBM for giving us the license to use their solver CPLEX which was useful for this research. Finally we acknowledge National University of Singapore and University of Waterloo for providing us with the necessary facilities for carrying out this research.

ASSOCIATED CONTENT

S Supporting Information *

■

NOTATION

i m n j

source processing unit purification unit fuel gas sink

Indices

Tables S1−S7 listing CAPEX and OPEX parameters, data and correlations for comparing RM and M2 in examples 1−3, hydrogen conversions and recoveries and quality specifications, model statistics and costs, operating variables and costs, and N



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on

p origin unit q destination unit

operating expenditure of purification unit n operating expenditure of compressor present between origin p and destination q oBpq oCpq operating expenditure of cooler present between origin p and destination q oHpq operating expenditure of heater present between origin p and destination q oTpq operating expenditure of a transfer line present between origin p and destination q oVpq operating expenditure of valve present between origin p and destination q PINq inlet pressure of destination unit q POUTp exit pressure of origin unit p pi cost associated with source i RLnm lower bound on flow for connection between purification unit n and processing unit m to exist RLnn lower bound on flow for connection between purification unit n and other purification unit n′ to exist rn recovery of purification unit n TLpq, TUpq lower and upper bounds on temperature of transfer line connecting origin p and destination q TINq inlet temperature of destination unit q TOUTp exit temperature of origin unit p V scalar to represent the maximum number of connections between origin p and destination q xLj ,xUj lower and upper bounds on purity of gas stream in fuel gas sink j xm inlet purity requirement at processing unit m ym exit purity of processing unit m ypn known purity of product or hydrogen stream from purification unit n zi purity of gas stream out of source i βj cost coefficient for revenue generated from surplus output by fuel gas sink j γj cost coefficient associated with disposal of gas streams in fuel gas sink j η scalar to show the efficiency of compressor μp Joule−Thompson expansion coefficient from origin p χm known per pass conversion of hydrogen in the reactor

Parameters

an aBpq aCpq aHpq aTpq aVpq AF bn bBpq bCpq bHpq bTpq bVpq Cp dn dBpq dCpq dHpq dTpq dVpq ELnj FLi , FUi FLij FLim FLin FLj , FUj FLn , FUn FLpq, FUpq GLmj GLmm′ GLmn NU ncp OP

fixed capital expenditure of purification unit n fixed capital expenditure of compressor present between origin p and destination q fixed capital expenditure of cooler present between origin p and destination q fixed capital expenditure of heater present between origin p and destination q fixed capital expenditure of a transfer line present between origin p and destination q fixed capital expenditure of valve present between origin p and destination q annualization factor variable capital expenditure of purification unit n variable capital expenditure of compressor present between origin p and destination q variable capital expenditure of cooler present between origin p and destination q variable capital expenditure of heater present between origin p and destination q variable capital expenditure of a transfer line present between origin p and destination q variable capital expenditure of valve present between origin p and destination q constant specific heat capacity of stream from origin p exponent factor for variable capital expenditure of purification unit n exponent factor for variable capital expenditure of compressor present between origin p and destination q exponent factor for variable capital expenditure of cooler present between origin p and destination q exponent factor for variable capital expenditure of heater present between origin p and destination q exponent factor for variable capital expenditure of a transfer line present between origin p and destination q exponent factor for variable capital expenditure of valve present between origin p and destination q lower bound on flow for connection between purification unit n and fuel gas sink j to exist lower and upper bounds out of source i lower bound on flow for connection between source i and fuel gas sink j to exist lower bound on flow for connection between source i and processing unit m to exist lower bound on flow for connection between source i and purification unit n to exist lower and upper bounds on flow into fuel gas sink j lower and upper bounds on flow into purification unit n lower and upper bounds on flow for connection between origin p and destination q to exist lower bound on flow for connection between processing unit m and fuel gas sink j to exist lower bound on flow for connection between processing unit m and other processing unit m′ to exist lower bound on flow for connection between processing unit m and purification unit n to exist scalar to represent the maximum number of purification units n adiabatic index of gas stream from origin p operating period of a refinery

Nonnegative Variables

Enj E1nj E2nj Fi Fij Fim Fin Fj Fm Fn Fpq Gmj Gmm′ Gmn HINpq Rnm O

flow of residue stream from purification unit n to fuel gas sink j flow of hydrogen component from purification unit n to fuel gas sink j flow of non-hydrogen component from purification unit n to fuel gas sink j flow out of source i flow from source i to fuel gas sink j flow from source i to processing unit m flow from source i to purification unit n flow into fuel gas sink j flow into processing unit m flow into purification unit n flow from origin p to destination q flow from processing unit m to fuel gas sink j flow from processing unit m to other processing unit m′ (including itself) flow from processing unit m to purification unit n variable to represent the product of flow (Fpq), specific heat (Cp), and temperature (Tpq) of a gas stream in transfer line SSpq connecting origin p and destination q flow from purification unit n to processing unit m dx.doi.org/10.1021/ie5005042 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research Rnn′ TAC Tpq yrn ΔHBpq

ΔHHpq

ΔHCpq

ΔHVpq

Article

flow from purification unit n to other processing unit n′ total annualized cost temperature of transfer line connecting origin to destination q purity of residue stream from purification unit n variable to represent the product of flow (Fpq), specific heat (Cp), and temperature change (ΔTpq) due to presence of compressor along the transfer line SSpq connecting origin p and destination q variable to represent the product of flow (Fpq), specific heat (Cp), and temperature change (ΔTpq) due to presence of heater along the transfer line SS pq connecting origin p and destination q variable to represent the product of flow (Fpq), specific heat (Cp), and temperature change (ΔTpq) due to presence of cooler along the transfer line SS pq connecting origin p and destination q variable to represent the product of flow (Fpq), specific heat (Cp), and temperature change (ΔTpq) due to presence of valve along the transfer line SSpq connecting origin p and destination q

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Binary Variables

cpq hpq un vij vim vin vmj vmm′ vmn vnm vnn′ vpq

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binary variable to show the existence of cooler in transfer line connecting origin p and destination q binary variable to show the existence of heater in transfer line connecting origin p and destination q binary variable to show the existence of purification unit n binary variable to show the existence of connection between source i and fuel gas sink j binary variable to show the existence of connection between source i and processing unit m binary variable to show the existence of connection between source i and purification unit n binary variable to show the existence of connection between processing unit m and fuel gas sink j binary variable to show the existence of connection between processing unit m and processing unit m′ binary variable to show the existence of connection between processing unit m and purification unit n binary variable to show the existence of connection between purification unit n and processing unit m binary variable to show the existence of connection between purification unit n and purification unit n′ binary variable to show the existence of connection between origin p and destination q

REFERENCES

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Improved Synthesis of Hydrogen Networks for Refineries - Industrial

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