Preliminary Synthesis of Fuel Gas Networks to Conserve Energy and

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Preliminary Synthesis of Fuel Gas Networks to Conserve Energy and Preserve the Environment M. M. Faruque Hasan, I. A. Karimi,* and Cory Matthew Avison Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576 ABSTRACT: Many chemical plants produce a variety of hydrocarbon gases with fuel value. A fuel gas network (FGN) integrates and uses these fuel gases appropriately to make best use of them. FGNs are critical components of many chemical plants including liquefied natural gas (LNG) plants and refineries. However, a systematic approach to design and operate realistic FGNs is not currently available in the literature. We address the optimal synthesis of an FGN with many practical features such as auxiliary equipment (valves, pipelines, compressors, heaters/coolers, etc.), nonisobaric and nonisothermal operation, nonisothermal mixing, nonlinear fuel-quality specifications, fuel/utility costs, disposal and treatment costs, and emission standards. We develop a nonlinear program (NLP) based on a novel superstructure that embeds plausible alternatives for heating/cooling, moving, mixing, and splitting. We successfully apply our model to three real-life case studies from the LNG and refinery industries to demonstrate that an FGN can save 4050% of the total energy costs of a plant and reduce the fuel-from-feed or fuel-from-product consumptions by similar amounts. This work represents an important contribution toward conserving energy, preserving the environment, and improving plant economics using advanced techniques of process systems optimization.

1. INTRODUCTION The world is facing two critical and inseparable issues regarding energy and climate change. In 2008, the world primary energy consumption1 was about 11295 million tonnes of oil equivalent, with the Asia-Pacific region accounting for more than one-third of the total share. The global energy demand is predicted2 to rise by as much as 57% from 2004 to 2030. The fossil fuels (oil, coal, and natural gas) that supply over 85% of the current primary worldwide demand will continue to be the major source for the near future. However, coal, oil, and natural gas emit 208, 164, and 117 lb of CO2 per one million British thermal units (Btu) of energy, respectively. Because the cleanest energy is the energy that is never used, immediate measures to reduce energy usage through conservation are extremely critical in arresting the dreadful impact on the environment due to greenhouse gas (GHG) emissions. One contributing factor behind the increase in energy demand is the rapid growth of chemical industries in China, India, and the Middle East, among other regions. Even in the United States, the chemical industry consumes nearly 20% of the total industrial energy consumption.3 Energy contributes as much as 40% to the total operating costs of a chemical plant.4 It is a major component of the daily operating costs in many plants such as refineries, petrochemicals, gas processing, bulk chemicals, cement, iron and steel, and aluminum. Such energy is used in the form of steam, heat, or electricity to run the movers and processing units. Most plants buy electricity or fuel externally to satisfy their energy demands. However, some consume a portion of their raw materials, products, or byproducts as fuel. For instance, refineries (Figure 1) use products such as vaporized liquefied petroleum gas (LPG) and fuel oil as fuel supplements. This is called FFP or fuel from product. On the other hand, base-load liquefied natural gas (LNG) plants (Figure 2) use the feed (natural gas) as a fuel to run their gas turbine drivers and generators. This is called FFF or r 2011 American Chemical Society

Figure 1. Fuel gas system in a typical refinery.

fuel from feed. Obviously, FFF and FFP directly reduce plant capacity and revenue, and measures to reduce such consumptions are certainly desirable. To this end, an attractive option is to use the, typically many, impurity, waste, surplus, byproduct, purge, and/or side streams with some heating value as fuel, instead of sending them to flare. In general, many plants have units that can consume such fuels. Table 1 lists several plants including refineries and LNG plants where such fuel sources and potential consumers exist. Although the list is by no means exhaustive, it shows that many plants have one or more sources of fuel that can be utilized to partially or even completely satisfy the fuel demands of various plant Received: February 8, 2011 Accepted: April 26, 2011 Revised: April 20, 2011 Published: April 26, 2011 7414

dx.doi.org/10.1021/ie200280m | Ind. Eng. Chem. Res. 2011, 50, 7414–7427

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Table 1. Fuel Sources and Consumers in Various Process Plants fuel source

sink Refinery

crude distillation unit

boilers

reformer fluid catalytic cracker

turbines hydrotreating

demethanizer/expander

dehydrogenation

tail or refinery gases LNG Plant

Figure 2. Fuel gas system in a typical LNG plant.

high-pressure flash gas

boilers

boil-off gas

SRU and AGR units

fuel from feed

gas turbine generators

end flash gas

gas turbine drivers off-site facilities

equipment. For instance, the separation processes in a refinery generate substantial quantities of flash gas, off-gas, purged gas, and so on that contain combustible hydrocarbon gases. The gases from the stabilizer drums in the naphtha stabilizer, top of the atmospheric distillation columns, and vacuum ejectors of the vacuum distillation units are rich in C1C4 components and potential sources of fuel. An LNG plant produces various tail gases such as high-pressure flash gas (HPFG), boil-off gas (BOG), end flash gas (EFG), and so on. Tail gases are leftover gases that are neither product streams nor suitable for recycling back to the liquefaction process. They contain mainly methane, ethane, and nitrogen. Some can contain as much as 75% methane. Unless utilized somewhere fruitfully, the tail gases become production losses. BOG produced due to heat leaking into LNG tanks can contain as much as 95% methane. Purge and off-gases from ammonia and methanol production containing mainly hydrogen, methane, carbon monoxide, and so on can be of much value as fuels.5,6 Syngas is a residue/product of gasification and a mixture of carbon monoxide, carbon dioxide, hydrogen, nitrogen, and water vapor. It is normally used as an intermediate in ammonia, hydrogen, and methanol production and many petrochemical processes. However, if present in excess, syngas can be a potential energy source, as it has about 50% of the energy density of natural gas. All of these gases are attractive sources of useful energy that can reduce FFF and/or FFP consumption. Instead of just being vented or flared, they can be used to power boilers, turbines, fired heaters, furnaces, reformers, and so on. Heavy-duty turbines such as those from General Electric can consume a variety of fuel gases ranging from natural gas, LNG, and LPG to air- and oxygen-blown syngases, tail gases, and process gases comprising varying mixtures of methane, hydrogen, carbon monoxide, carbon dioxide, and so on.7 Good-quality fuel gases can be economically transported even up to 500 km.8 Often, the various fuel gas streams in a plant can be used in an ad hoc or one-to-one manner to satisfy the fuel needs of selected equipment. One could also burn the purge or tail gas and recover the energy from the flue gas9 through heat integration in a heat exchanger network. Whereas the former approach can be suboptimal, the latter requires capital expenditures in the form of additional furnaces and exchangers. The best approach would be to design a systematic fuel integration network that collects fuels and fuel gases from all sources, mixes them in optimal proportions, and supplies them to the various consumers (boilers,

Ammonia Plant boilers purge gas recovery

turbines

off-gas

steam reformer secondary reformer hydrodesulfurization Methanol Plant boilers

purged gas fuel

turbines reformer distillation Hydrogen Plant

hydrogen

reformer

gas from PSA unit

boilers

fuel from product

turbines Steel Plant

blast furnace gases

boilers

coke oven gases COREX gases

turbines

turbines, fired heaters, furnaces, etc.). Such a network is known as a fuel gas network (FGN). An FGN burns fuel only in sinks (e.g., combustion chamber of a boiler, turbine, or fired heater) and needs no additional furnaces. An optimal FGN can supply the fuel to generate heat, power, and steam to support onsite and offsite plant operations in the best possible manner. Its design and operation depend on the flows, compositions, and qualities of the available fuel gases and the requirements of the various consumers. With the increasing demand for chemical and petrochemical products, decreasing natural resources, and stringent restrictions on GHG emissions, Integrating Energy, Environment, and Economics (IEEE in Figure 3) is the key to sustainability. The use of a well-designed and well-operated FGN is an excellent example of IEEE. It can yield “greener and leaner” operation through reduced emissions arising from reduced flaring and substantial savings arising from greater fuel efficiency. In addition to reduced fuel usage, waste, and pollution, the benefits10 of an FGN can also include higher profit margins, increased plant energy security, augmented product quality, and even stable operations. These 7415

dx.doi.org/10.1021/ie200280m |Ind. Eng. Chem. Res. 2011, 50, 7414–7427


Figure 3. IEEE (integrating energy, environment, and economics) for sustainability.

benefits complement the popular corporate initiatives such as zero-flare policies, sustainable development, waste heat recovery, flue gas utilization, purge reduction, mass and energy integration, and various measures for monetizing all molecules. The fact that the best steps toward a sustainable process are neither extreme nor obvious makes decisions complex and challenging and points to the beneficial role of advanced optimization techniques. Meyer and Floudas11 considered a wastewater treatment network with flows between treatment plants that are essentially mixers followed by treatment units with fixed removal ratios. Pham et al.12 proposed a discretization approach using implicit enumeration or discretization of qualities to avoid nonconvex bilinear terms.11,1315 However, they used linear qualities only and did not consider treatment units. Whereas the qualities are critical for safe and satisfactory sink operation, emission risks are important from an environmental perspective. Misener et al.16 included several environmental standards from the U.S. Environmental Protection Agency (EPA) in large-scale pooling problems such as reformulated gasoline blending. Considerable work also has been published on the blending and scheduling of refinery materials such as crude oil17 and gasoline.18 Limited work exists on the integration of “fuel streams” (versus refinery fuels) per se to reduce energy costs as an issue. De Carli et al.10 addressed the intelligent management and control of a fuel network. Wicaksono et al.19 proposed a mixed-integer nonlinear programming (MINLP) model for integrating various fuel sources in an LNG plant. They then extended this approach20 to integrate JBOG (jetty BOG) as an additional source. Hasan et al.21 addressed the optimal synthesis of FGNs and presented two models based on two superstructures, one with one-stage mixing and the other with two-stage mixing. However, all of these works focused primarily on mass and property integration to meet customer needs and demands. They assumed isothermal and isobaric systems and isothermal mixing. These are limiting assumptions, as pressure and temperature variations are significant and often necessary in most plants. For instance, EFG and BOG are at nearly 160 °C in an LNG plant, whereas other potential fuel streams would be at high temperatures. Moreover, because streams usually vary in pressures, an FGN must equalize pressures for mixing various streams. This, in turn, would cause temperature changes in gaseous streams. Thus, accounting for changes in temperatures and pressures is critical in a gas network.22 The existing literature also neglects utility costs associated with such changes, benefits from utilizing potential sources, and costs associated with disposing useless residual gases from a gas network. Finally, even those works that addressed FGNs neither rigorously

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accommodated various fuel-quality requirements of the consumers nor provided a comprehensive model to optimize FGN synthesis. In this article, we address the optimal design of an FGN. We show this to be a fuel gas mixing and distribution problem with features that have not been considered in the literature so far. These features include pressure and temperature changes (nonisobaric and nonisothermal operation), nonisothermal mixing, nonlinear fuel-quality specifications, utility costs, benefits/ costs for utilization or nonutilization, treatment costs, and flue gas emission standards. We develop a nonlinear program (NLP) based on a superstructure that includes essential equipment such as compressors, valves, and utility heaters/coolers in addition to sources and sinks. Thus, it embeds plausible alternatives for heating/cooling, moving, mixing, and splitting. We also address the issue of retrofitting new sources of fuel (e.g., integrating JBOG in an LNG plant) into an existing FGN. Finally, using several examples, we highlight the utility of optimal FGN synthesis in reducing the energy costs of fuel-using and energyintensive chemical plants. In this work, we consider an FGN with a dedicated pool (header) for each fuel-consumer. We begin with a detailed description of the problem and then define various quality requirements relevant to gaseous fuels. Next, we develop an NLP model that synthesizes an FGN to extract the most savings by integrating various fuel sources. Finally, we present three industrial case studies: The first considers an LNG plant application, the second extends this application to integrate JBOG, and the third addresses a refinery example.

2. PROBLEM DESCRIPTION A typical FGN has I source streams (i = 1, 2, ..., I), J pools (j = 1, 2, ..., J), and K fuel sinks (k = 1, 2, ..., K) with a header for each sink. In addition, it can have several auxiliary units such as utility heaters, utility coolers, valves, compressors, and pipelines. S specifications (s = 1, 2, ..., S) measure the quality of a fuel gas. A source stream is any gas stream with a potential for mass integration and a nonzero heating (or fuel) value. The stream can have a fixed or varying flow, composition, specification, pressure, and temperature. It can be internal or external to the plant. Multiple source streams with similar qualities can be merged into a larger source stream. HPFG, BOG, EFG, and natural gas feed are some examples of source streams in an LNG plant. We divide the source streams into three types: waste/purge, feed/product/ byproduct, and standard fuel. The first is an internal stream that has some heating value but no commercial value as a byproduct. Unless utilized, the plant would flare or discharge it as waste. Let i = 1, 2, ..., I1 denote such streams. Gas streams such as BOG, producer gas, and blast furnace gas (BFG) are examples. These streams might not be sufficient to satisfy fully the energy needs of a plant. Thus, the plant might utilize a feed or product/byproduct stream in addition. We call these approaches fuel from feed (FFF) and fuel from product (FFP), respectively. An FFF stream comes at a price, as the plant must purchase and purify the feed appropriately before using it. However, its cost will be lower than those of a standard fuel from an external supplier such as natural gas. An FFP stream also involves costs in terms of lost revenue. It reduces plant output. Let i = I1 þ 1, I1 þ 2, ..., I2 denote the FFF/FFP streams. If the waste/purge and FFF/FFP streams are not sufficient to meet the energy needs of a plant, then the plant must buy commercial fuel from an external 7416


Industrial & Engineering Chemistry Research supplier. Let i = I2 þ 1, I2 þ 2, ..., I represent these standard fuel streams. We measure the energy demands of various plant units in terms of the lower heating value (LHV). A pool is a mixing unit followed by a treatment unit that receives and mixes the fuel streams from one or more sources, treats the resulting mixture, and then feeds it to one or more of the K headers. Although the fuel streams entering any pool can be at different temperatures, they must be at the same pressure. An FGN might or might not have pools. In this work, we assume K dedicated pools, one for each sink, and merge them with the headers. In other words, we have one-stage mixing at the headers with no intermediate pools. A sink is any entity that can usefully consume a fuel gas to produce heat/work. Turbines, boilers, furnaces, fired heaters, and so on are typical sinks. We define two types of sinks, fixed and flexible. Fuel gas consumers such as driver gas turbines and fired heaters are normally dedicated to perform tasks with fixed duties or energy demands. In other words, they have no freedom or flexibility to consume more than their designated duties. We call these consumers fixed sinks (k = 1, 2, ..., K1). In contrast, fuel users such as electricity generating gas turbines and steam generating boilers have some flexibility in consuming fuel over and above their normal duties. They can use the extra energy to generate and sell electricity for additional revenue. We call these consumers flexible sinks (k = K1 þ 1, K1 þ 2, ..., K). The header before each sink is essentially a mixer that mixes the streams from various sources and/or treats them to make a feed with acceptable quality, flow, pressure, temperature, and composition. The streams entering any header can be at different temperatures, but they must be at the same pressure. With this summary, we state our FGN problem as follows: Given (1) I sources with known types, pressures, temperatures, flows, compositions, and qualities; (2) K fuel sinks (K1 fixed and K K1 flexible) with known allowable ranges for temperatures, pressures, flows, qualities, and energy demands (LHV); (3) S fuel-quality specifications and their mixing rules (if applicable); (4) operating characteristics of the valves, compressors, and utility heaters/coolers; (5) economic (cost, price, value, etc.) data for utilizing, treating, and disposing various streams; and (6) costs of heating and cooling various streams and running the movers determine (1) all network flows, temperatures, and pressures; (2) existence, duties, and temperatures of various compressors and heaters/coolers; and (3) flow and specifications of the fuel to each sink assuming that (1) operation is in the steady state with no chemical reactions or phase changes; (2) sources (sinks) with identical properties or attributes are lumped into a single source (sink); (3) LHVs of fuel components do not change with temperature; (4) all streams are ideal gases; (5) no expander turbines are used, only valves; (6) all compressions are single-stage and adiabatic; (7) all expansions are JouleThompson expansions;

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(8) all streams are below their inversion temperatures for JouleThompson expansions, and no stream is sufficiently pure hydrogen to have a negative Joule Thompson coefficient; (9) unlimited utilities are available at any desired temperature; and (10) zero pressure drops are found in heaters, coolers, headers, and transfer lines. The aim of the problem is to minimize the total annualized cost of the fuel gas network. This includes the costs of various fuels, products, byproducts, utility, and treatment/disposal, and so on, as well as annualized CAPEX (capital expenditure) and OPEX (operating expenditure) costs of auxiliary equipment such as utility heaters/coolers, valves, compressors, pipelines, and movement. Synthesis of an optimal FGN poses several challenges. First, the source streams can differ in supply flows, compositions, densities, pressures, and temperatures. Furthermore, their characteristics might even vary dynamically with plant operation. For instance, the amounts of jetty BOG (JBOG) coming from the storage and loading facilities of an LNG plant vary with time. Excess JBOG can be produced during the loading of an LNG ship, but little otherwise. In a refinery, the amounts of fuel gases can vary with the production plan, operating strategy, and crude oil characteristics. The same holds true for fuel sinks. They can differ in their energy demands (LHVs) and fuel qualities such as heating value (HV), Wobbe index (WI), temperature, and pressure. Even for a static scenario with a known FGN design in which the characteristics of the sources and sinks are stationary, it is not easy to determine the best configuration and policy that an FGN should employ to match, mix, and distribute the fuels. Many alternatives might exist, and not all might be technically feasible or economical. Second, mere availability of a fuel gas does not guarantee its usability. Use of a low-quality fuel gas in a turbine or boiler could violate emission standards and/or cause problems, even to the extreme of plant shutdowns. Low-temperature fuel streams are occasionally heat integrated with process streams before they enter the FGN or flares. Fuel gases from flash drums are saturated vapors. The presence of such streams in a mixer might result in unwanted condensation. Such condensation is even more serious if it occurs inside the feed system of a turbine/boiler. Thus, one must ensure that temperatures remain above the dew-point temperatures (DPTs) in an FGN. Furthermore, mixing must occur at the same pressure. This means that operating constraints on DPTs and pressures are necessary not only for the sinks, but also for the pools or headers. Finally, the nonconvexity arising from the nonlinear fuelquality specifications makes the optimization task even more challenging. Our interactions with LNG plant operators suggest that the common practice is to propose and evaluate a few promising scenarios and select the best one on a case-by-case basis, which generally fails to garner the full benefits.

3. FUEL GAS SPECIFICATIONS The flow, quality, and composition of a fuel feed have significant impacts on the operation and life of the equipment (e.g., gas turbine, boiler, furnace, etc.) that uses it. Every such sink has specific acceptable ranges for flow, quality, temperature, pressure, and composition. The composition can vary from predominantly methane, hydrogen, carbon monoxide, and other 7417



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hydrocarbons to high amounts of diluents such as nitrogen and CO2. To measure fuel quality, industry uses various specifications or allowable limits for fuel properties and contaminants. Fuel properties include high and low heating values (HHV and LHV, respectively), specific gravity (SG), Wobbe index (WI), methane number (MN), superheat, dew point temperature (DPT), hydrocarbon dew point (HDP), moisture dew point (MDP), flammability ratio (FR), constituent limits, contaminant levels, pressure, temperature, etc. Some of these properties are interrelated. Possible contaminants include particulates, liquids, sulfur, and trace metals.7 3.1. Lower Heating Value (LHV). LHV is a key specification that depends on gas composition and measures energy content per unit mass or volume of fuel. For gaseous fuels, LHV per unit volume is more important, because most drivers have a constant volume intake.23 We measure the energy demands of various units in terms of LHV. 3.2. Wobbe Index (WI). Whereas LHV gives a direct measure of energy content, WI is a good indicator of fuel interchangeability.24 It is defined as sffiffiffiffiffiffi LHV TR WI ¼ pffiffiffiffiffiffi ð1Þ SG Tg where Tg is the fuel temperature and TR is a reference temperature (usually 288 K). Because LHV is a measure of energy content and the flow of a fuel gas into a sink is inversely proportional to (SG)1/2, WI is a direct measure of the energy flow into a sink. WI captures the effects of composition, temperature, and LHV. Two fuels with different temperatures, compositions, and LHVs can have the same WI value (i.e., the same energy flow), and two fuels with the same value of WI exhibit similar burning characteristics. 3.3. Methane Number (MN). Whereas WI indicates fuel interchangeability, methane number (MN) measures the composition variability and knock resistance of fuel gases. It is used for gas turbines. The knock resistance is measured by comparing the compression ratio at which the fuel knocks to that of a reference fuel gas that knocks at the same compression ratio. The reference fuel gas is a mixture of methane and hydrogen. The knock resistance of pure methane is defined as MN = 100. Hydrogen, on the other hand, has a low knock resistance, and its MN is 0. MN for a fuel gas whose knock resistance is equivalent to that of a gas with 80% methane and 20% hydrogen is defined as 80. One correlation for MN is25 MN ¼ 1:624½406:14 þ 508:04RHCR 173:55ðRHCRÞ2 þ 20:17ðRHCRÞ3 119:1

ð2Þ

where RHCR is the reactive hydrogen to carbon ratio (H/C) and is given as RHCR ¼

4xCH4 þ 6xC2 H6 þ 8xC3 H8 þ 10xC4 H10 þ 12xC5 H12 þ 14xC5þ xCH4 þ 2xC2 H6 þ 3xC3 H8 þ 4xC4 H10 þ 5xC5 H12 þ 6xC5þ ð3Þ

Here, x represents the alkane composition in the gas mixture. This correlation is valid for H/C > 2.5 only. The minimum MN requirement for gas turbines is usually 80. From eq 2, one can calculate that an RHCR value of 3.758 gives MN = 80. Thus, we

can replace the constraint of minimum MN = 80 by minimum RHCR = 3.758 for gas turbines. 3.4. Temperatures. Fuel gases must be superheated before entry into a sink to ensure that the supply is completely free of liquids. However, high temperatures inside a turbine can cause a sudden trip. Therefore, all sinks have limits (TLg e Tg e TU g ) on fuel temperature. Apart from these limits, the limits on LHV and WI can also constrain Tg as follows: 2 3 2 3 !2 !2 L U LHV 1 LHV 1 5 e Tg e min4T U , 5 max4TgL , g SGU SGL WIU WIL ð4Þ The fuel to fired heaters, boilers, and combustion turbines must not contain any liquid droplet for several reasons. Liquid can severely damage the equipment, pose control problems, and block injectors.24 Thus, it is imperative to keep the fuel temperature above the DPT in any FGN. The pressure drop in a turbine feed system can eventually lead to a temperature below the DPT because of JouleThompson expansion. Thus, it is also critical that the fuel gas be sufficiently superheated. Standard ASME B133.726 recommends a superheating value of 28 K (i.e., Tg g DPT þ 28 K) to prevent any liquid dropouts in a fuel system. In addition, the presence of water and condensable hydrocarbons in a fuel gas can lead to the formation of hydrates, hydrogen sulfide, or acidic carbon dioxide.24 Although 28 K is the superheating requirement in general, specific superheating values for moisture and hydrocarbons are7 Pg 5 Tsm ¼ 5:15 312 ð5Þ 9 100

Tsh

" # 2 Pg Pg 5 ¼ 2:33 2:8 305 9 100 100

ð6Þ

where Tsm (K) is the superheating requirement for moisture, Tsh (K) is that for hydrocarbons, and Pg (psia) is the gas pressure. 3.5. Pressure. The feed pressures permissible at sinks vary with unit and combustor types. Adequate fuel pressure is critical to ensure flow controllability and maintain required pressure ratios across the combustion nozzles. This is especially important for turbines, because turbines are sensitive to pressure variations. Because turbine temperature can increase with pressure, a sudden pressure change can result in a potential trip. 3.6. Flammability Ratio (FR). FR is defined as the ratio of richto-lean flammability limits of a fuel gas to that of natural gas. Excessive hydrogen and/or carbon monoxide can increase the rich-to-lean flammability limit ratio.7 Therefore, FR > 1 for a gas with high hydrogen or carbon monoxide content, and FR < 1 for a gas with high nitrogen or carbon dioxide content. FR can be defined on the basis of weight or volume. A large FR value can cause problems. It can damage a combustion system, or the resulting high flame temperature can cause explosion due to the autoignition of the fuel at turbine exhaust. Thus, a fuel with a high hydrogen content might require addition of inert components to reduce these risks. However, any inert additive or nitrogen in a fuel gas reduces the flue gas temperature. Hydrogen also has an interesting behavior during expansions. Below a certain inversion temperature, most gases cool during adiabatic expansion, except for hydrogen. 7418



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3.7. Contaminants. Depending on process plants and fuel types, “raw” fuel gases can contain contaminants such as tar, coke, salt water, sand, clay, rust, iron sulfide, scrubber oil or liquid, compressor lube oil, naphthalene, gas hydrates, and trace metals,7 as well as environmental hazards such as sulfur, nitrous oxides (NOx), volatile organic compounds (VOCs), toxic compounds (TOX), and benzene. An FGN must ensure proper conditioning and/or mixing and distribution such that contaminant levels in fuels remain acceptable.

4. PROBLEM FORMULATION Our FGN has several novel features. First, our FGN includes auxiliary equipment such as compressors, valves, heaters, and coolers, along with their capital and utility costs. This allows us to include hitherto neglected cryogenic fuel sources into an FGN. Second, our FGN allows nonisothermal mixing, temperature changes in heaters and coolers, compressions, and expansions, and pressure changes effected by valves and compressors to ensure isobaric mixing. Third, it imposes several quality requirements to ensure practically safe and feasible operation and prevent material loss and unwanted fuel discharge from the network. Fourth, it includes some highly nonlinear quality specifications such as WI and DPT. Finally, the network economics/objective includes source stream utilization, source stream disposal, treatment, CAPEX and OPEX costs for auxiliary equipment, and fuel substitution. This demonstrates the full use of IEEE for sustainable process development. Although these factors might have been considered piecemeal in standard pooling or blending problems, we incorporate them all together in this work. We now present a nonlinear programming (NLP) formulation for the above problem. Unless stated otherwise, all indices such as i, j, k, and s assume the full ranges of their valid values in all constraints. In general, we use F for flow, T and θ for temperature, and P for pressure. We first propose a superstructure (Figure 4) for our FGN. It employs one-stage mixing in the header/pool before each sink. Consider a source stream (i) in Figure 4. It enters a splitter (Si) that generates K substreams, one for each header k. Each such substream originating from Si passes through at most two utility exchangers (heater and/or cooler) and at most one mover (valve or compressor) and terminates in header k. A substream cannot use both a compressor and a valve at the same time, and we fix their exact locations after obtaining the best FGN, and not in advance. Header k mixes I substreams at a pressure Pk and treats them to make a feed acceptable to sink k. The auxiliary units (heaters, coolers, valves, transfer lines, and compressors) described in the above superstructure might or might not exist in the optimal FGN. 4.1. Mass and Energy Flows. Let Fi denote the known available flow of stream i. The FGN need not use all of the available flow. Hence, we define Φi (eFi) as the flow of stream i entering Si. The remaining flow (Fi Φi) does not enter the FGN and might have an alternate use (e.g., feed or byproduct), appropriate disposal (e.g., internal waste stream), or zero cost (e.g., external fuel). Let Fik denote the flow from source i to sink k. Clearly K X k¼1

Fik ¼ Φi

ð7Þ

Figure 4. Proposed intermediate pools.

superstructure

for

an

FGN

with

no

The flow into each sink must respect some given flow limits (FLk , FU k) FkL e

I X i¼1

Fik e FkU

ð8a,bÞ

One must (can) meet (exceed) the energy demands of the fixed (flexible) sinks exactly I X

Fik LHV i ¼ Dk ,

k ¼ 1, 2,

... , K1

ð9aÞ

i¼1 I X

Fik LHV i g Dk ,

k ¼ K1 þ 1, K1 þ 2, ... , K

ð9bÞ

i¼1

where Dk is the slated energy demand of sink k. 4.2. Temperatures and Pressures. Let Ti (Pi) be the known temperature (pressure) of source stream i, Tk (Pk) be the temperature (pressure) of header k, and TLk e Tk e TU k and PLk e Pk e PU k be the acceptable ranges of temperature and pressure, respectively, for header/sink k. We define SSik as the substream of source stream i originating from Si and entering header k. Let Tik (TLi e Tik e TU i ) denote the temperature of SSik as it enters header k, where TLi and TU i are the lowest and highest allowable temperatures, respectively, for source i. An SSik, if it exists, undergoes a temperature change of (TikTi) and a pressure change of (PkPi) from splitter Si to header k. The various auxiliary units (compressor, valve, heater, or cooler) that might exist along SSik effect these changes. Now, whether an auxiliary unit exists or not, we define a temperature change for each possible unit along SSik as Tik ¼ Ti þ ΔTikB ΔTikV þ ΔTikH ΔTikC

ð10Þ

where ΔTBik g 0 is the temperature change across the compressor (blower), ΔTVik g 0 is that across the valve, ΔTH ik g 0 is that across the heater, and ΔTCik g 0 is that across the cooler. Because both compressor and valve cannot exist simultaneously, one or both of ΔTBik or ΔTVik must be zero. 7419



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The temperature change in SSik due to JouleThompson expansion through a valve is given by μi(Pi Pk), where μ is the JouleThomson coefficient (JTC), defined as μ = (δT/δP)H. If μ is positive (negative), then the gas cools (heats) on expansion. The temperature at which μ changes sign from positive to negative is called the inversion temperature. Usually, the inversion temperatures are high for most gases (except hydrogen). Thus, hydrogen usually heats during adiabatic expansion. Using this, we can compute ΔTVik as ΔTikV g μi ðPi Pk Þ

where ψks is a known treatment factor for specification s in header k. Equations 15a,b are applicable for qualities such as LHV, 1/SG, heat capacity, and contaminant concentration. However, WI is a nonlinear quality index that needs special attention. We define LHV as s = 1, and 1/SG as s = 2. Then, using eq 1, the allowable limits on WI, and linear mixing rules for s = 1 and s = 2, we obtain ðWILk Þ2

ð11Þ

Because the change in temperature across a valve depends only on the change in pressure, its existence, but not its location, affects the heater/cooler duties. In other words, the energy requirements for SSik are independent of the location of a valve, and we can locate the valve after obtaining the FGN solution. The case of a compressor is more complex. The power required by a compressor depends strongly on its inlet temperature.27 Although our superstructure does not explicitly define a location for the compressor, we can identify the optimal location relative to the heater/cooler (if any) based on the compressor inlet temperature. Normally, compressors have the highest CAPEX and OPEX in a process. As a general rule, compression is more expensive than heating or cooling. Therefore, it is in our interest to minimize the inlet temperature of a compressor. Because the minimum possible temperature along SSik is (Ti ΔTCik), we argue that the compressor must be located at this temperature along the substream. In other words, we can take (Ti ΔTCik) as the inlet temperature of the compressor with no loss of optimality. Using this value, we can compute ΔTBik as " # ni C ðT ΔT Þ P i k ik 1 ð12Þ ΔTikB g ηi Pi

TkL

I X

Fik Cpi e

i¼1

I X i¼1

Fik Cpi Tik e TkU

I X

Fik Cpi

ð14a,bÞ

i¼1

4.3. Quality Specifications. As discussed earlier, S specifications measure the quality of a fuel gas in the FGN. Furthermore, each sink (fired heater, turbine, boiler, etc.) has acceptable limits on these specifications. Let qis denote the value of specification s for source stream i, and let qLks e qks e qU ks be the acceptable range of specification s for sink k. For specifications that are linearly additive based on mixture composition or have appropriate linear indices, one can compute a mixture specification in a linear manner. For instance, we use 1/SG as a linear quality index for specific gravity. Thus, the following expression ensures acceptable linear quality for each sink k

qLks

I X i¼1

Fik e ψks

I X i¼1

Fik qis e qUks

I X i¼1

Fik

ð15a,bÞ

i¼1

Fik Cpi

i¼1 I X

Fik Cpi Tik ð16a,bÞ

i¼1

To ensure RHCR g 3.758 for gas turbine sinks, we use eq 3 to obtain 0:242 ψk, CH4

I X i¼1

Fik qi, CH4 g 1:516 ψk, C2 H6

þ 3:274 ψk, C3 H8 þ 6:79 ψk, C5 H12

I X i¼1 I X i¼1

I X i¼1

Fik qi, C2 H6

Fik qi, C3 H8 þ 5:032 ψk, C4 H10

Fik qi, C5 H12 þ 8:548 ψk, C5þ

I X i¼1

I X i¼1

Fik qi, C4 H10

Fik qi, C5þ

ð17a,bÞ Finally, the minimum (moisture and hydrocarbon) superheating requirements (eqs 5 and 6) give the following lower bounds on Tk X I I X 5 Pk MDPk þ 5:15 Fik Cpi e Fik Cpi Tik 312 100 9 i¼1 i¼1 ð18aÞ (

" #) 2 5 Pk Pk HDPk þ 2:33 2:8 305 9 100 100 I X

ð13Þ

Finally, the energy balance across header k and TLk e Tk e TU k gives

I X

Fik Cpi Tik e qk1 2 qk2 TR e ðWIUk Þ2

where ηi is the adiabatic compression efficiency for source i and ni is the adiabatic compression coefficient, which is given by R/Cpi, where Cpi is the heat capacity. The lowest possible temperature of SSik at any intermediate point must remain above TLi ; therefore Ti ΔTikC ΔTikV g TiL

I X

i¼1

Fik Cpi e

I X

Fik Cpi Tik

ð18bÞ

i¼1

where MDPk and HDPk are the given moisture and hydrocarbon dew points, respectively, at sink k. 4.4. Objective. The total annualized cost of an FGN includes the following components: (1) utility costs for the auxiliary units (heater, cooler, valve, compressor), (2) annualized CAPEX costs for auxiliary units, (3) piping costs, (4) purchase cost of using a source stream, (5) disposal cost of unused source stream, and (6) treatment cost in each header. We assume that both the annualized capital cost (CAPEX) and OPEX for the auxiliary units (heaters, coolers, compressors, and valves) are linear functions of their energy duties. Although the duty computations for heaters and coolers are straightforward, the same is not true for valves and compressors. We first consider a compressor and then a valve. The work required for adiabatic compression is given by " # ni F θ C P ik ik pi k 1 WikB ¼ ηi Pi where θik is the inlet temperature. Therefore, using eq 12, we get WBik = FikCpiΔTBik. 7420



ARTICLE

Although a valve does not consume any utility, we penalize the pressure loss in a valve by imposing a penalty. We compute this penalty in terms of the work required to reverse the expansion through the valve. To this end, we define ΔTEik g 0 as " # ðTi ΔTikC ΔTikV Þ Pi ni E 1 ð19Þ ΔTik g ηi Pk Then, the energy required to regain the lost pressure through the valve in SSik is FikCpiΔTEik. The FGN might not use a source stream completely. Thus, we define two costs for each source stream: Ri is the cost of a unit flow of stream i, and βi is the cost for not using it. βi is the cost of dilution, transport, disposal, incineration, and so on, if the FGN does not use a source stream. Ri = 0 and βi g 0 for i = 1, 2, ..., I1 (waste/purge source streams). Ri > 0 and βi = 0 for an FFF/FFP stream or a standard fuel. For sinks, we define δk as the cost [$/(MMscf/h), where MMscf stands for million standard cubic feet, a common measure for volumes of gas] of treating a unit flow in header k, and νk (k = K1 þ 1, K1 þ 2, ..., K) as the revenue ($/kWh) from the surplus energy consumption in sink k. With this definition, we write our objective function as ! I K I X X X min ½Ri Φi þ βi ðFi Φi Þ vk LHV i Fik Dk i¼1

þ þ þ

K X I X

k ¼ K1 þ 1

δk Fik þ

k¼1 i¼1 I X K X

πik Fik þ

BBik Fik Cpi ΔTikB þ BCik Fik Cpi ΔTikC

i¼1 k¼1 I X K X i¼1 k¼1

Hik ¼ Cpi Ti Fik þ ΔHikB ΔHikV þ ΔHikH ΔHikC

ð21Þ

ΔHikV g μi Cpi ðPi Pk ÞFik HikB

ðCpi Ti Fik ΔHikC Þ g ηi

ð22Þ

" # Pk ni 1 Pi

ð23Þ

Cpi Ti Fik ΔHikC ΔHikV g Cpi TiL Fik

ð24aÞ

ΔHikB þ ΔHikV e Cpi TiU Fik

ð24bÞ

Cpi TiL Fik e Hik e Cpi TiU Fik TkL

i¼1

I X K X

i¼1 k¼1

i¼1 k¼1 I X K X i¼1 k¼1

I X K X

it might be better to define these products as independent variables. As one can see below, this reduces the nonlinear complexity of our model considerably. We use the heat content, H = FCpT, of a stream as a variable to replace the product terms mentioned in the preceding paragraph. We multiply eqs 1013 by FikCpi and define FikCpiTik as Hik, FikCpiΔTBik as ΔHBik, FikCpiΔTVik as ΔHVik, FikCpiΔTEik as ΔHEik, H C C FikCpiΔTH ik as ΔHik , and FikCpiΔTik as ΔHik. This simplifies eqs 1014a,b, 16a,b, 18a, 18b, and 20 as follows

BEik Fik Cpi ΔTikE ðWILk Þ2

H BH ik Fik Cpi ΔTik

I X

Cpi Fik e

i¼1

I X i¼1

I X

I X

Hik e TkU I X

Hik e qk1 2 qk2 TR

i¼1

i¼1

ð24c,dÞ ð25a,bÞ

Cpi Fik

i¼1

Cpi Fik e ðWIUk Þ2

I X

Hik

i¼1

ð26a,bÞ

ð20Þ

C where πik, BBik, BEik, BH ik , and Bik are appropriate constants. The first sum in eq 20 is the total cost of source streams including purchase and disposal costs. The second sum is the revenue from the surplus energy production by the flexible sinks. The third is the cost of treatment11 in the headers. The fourth is the annualized CAPEX þ OPEX of pipelines. The fifth is the annualized CAPEX of valves plus the penalty for expansion through valves. The remaining terms represent the annualized CAPEX þ OPEX of compressors, heaters, and coolers, respectively. This completes our NLP formulation (M1), which includes eqs 720. Note that optimization will make eqs 11, 12, and 19 active and, thus, equalities at the optimal solution. M1 is a complex nonconvex nonlinear model, where eq 12 involves products of fractional powers of pressures and temperatures, eqs 14a,b have bilinear terms, eqs 16a,b have bilinear and quadrilinear28 terms, eqs 18a and 18b have bilinear and product terms of pressures and flows, and eq 20 is a nonconvex objective function with bilinear terms. Therefore, we now present a clever reformulation to reduce its nonlinear complexity.

5. REFORMULATION A careful examination of our model suggests that most of the bilinear (and thus nonconvex) terms arise as products of flow (Fik) and temperature (Tik). They are present in eqs 14a,b, 16a,b, and 20. Considering the frequency of these terms in the model,

X I I X 5 Pk MDPk þ 5:15 Cpi Fik e Hik 312 9 100 i¼1 i¼1 ð27aÞ

(

" #) 2 I I X X 5 Pk Pk 305 HDPk þ 2:33 2:8 Cpi Fik e Hik 9 100 100 i¼1 i¼1

ð27bÞ ΔHikE

min

I X i¼1

þ

ðCpi Ti Fik ΔHikC ΔHikV Þ g ηi

½Ri Φi þ βi ðFi Φi Þ

K X I X k¼1 i¼1

þ

I X K X i¼1 k¼1

K X

vk

k ¼ K1 þ 1

δk Fik þ

I X K X

πik Fik þ

i¼1 k¼1 I X K X

BBik ΔHikB þ

i¼1 k¼1

" # Pk ni 1 Pi I X

ð28Þ !

LHV i Fik Dk

i¼1

I X K X i¼1 k¼1

H BH ik ΔHik þ

BEik ΔHikE

I X K X i¼1 k¼1

BCik ΔHikC ð29Þ

Note that eq 24b is an additional equation, and eq 24c,d are bounds on Hik. Equations 24a, 24b, and 24c,d ensure that the heat content variables are zero when flows are zero. 7421



ARTICLE

Table 2. Source Data for CS1 and CS2 specification/parameter

EFG

HPFG

TBOG

FFF

JBOG

flow (MMscf/h)

2.8

0.16

0.55

80

>80

>80

>80

>65

MDP (K)

277

277

277

277

277

HDP (K) LHV (Btu/MMscf)

277 30010000

277 30010000

277 30010000

277 30010000

277 30010000

1/SG (28.96/Mol.Wt.)

1.02.4

1.02.4

1.02.4

1.02.4

1.02.4

FR

1.010

1.010

1.010

1.010

1.010

methane (%)

>85.0

>85.0

>85.0

>85.0

>65.0

ethane (%)

Preliminary Synthesis of Fuel Gas Networks to Conserve Energy and

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