An Upgraded Graphical Method for the Synthesis of Direct Work

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX-XXX

pubs.acs.org/IECR

Upgraded Graphical Method for the Synthesis of Direct Work Exchanger Networks Yu Zhuang,† Linlin Liu,† Lei Zhang,† and Jian Du*,†,‡ †

Institute of Chemical Process Systems Engineering, School of Chemical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China ‡ State Key Laboratory of Fine Chemicals, Chemical Engineering Department, Dalian University of Technology, Dalian 116024, Liaoning, China ABSTRACT: Numerous studies on heat integration have made significant progress in reducing external thermal energy in chemical plants. However, the notion of work exchange between high- and low-pressure streams to reduce the costly mechanical energy has not been fully explored. In this paper, an upgraded graphical method is proposed for the synthesis of direct work exchanger networks (WEN) using direct work exchangers under any operating conditions, such as isothermal, isentropic, and polytropic. In this method, the improved composite curves of work sources and work sinks are plotted in the pressure index (μ) versus work (W) diagram. On the basis of thermodynamic and numerical analysis, two improved linear-approximation auxiliary lines for every work sink are presented to assist identifying the feasible match. Several matching rules are proposed to accurately identify the feasible matches between work sinks and work sources. From the proposed method, the maximum amount of mechanical energy recovery, the minimum amount of the external utility, as well as the corresponding network design can be determined. Two examples are presented, and the results prove the correctness and generality of the developed method for synthesizing direct WEN.

1. INTRODUCTION

tion. Additionally, the advances in energy efficiency can also facilitate the reduction of carbon emissions. On the basis of a straightforward extension of the well-known HENs, work integration was first introduced by Huang and Fan in 1996, in which the mechanical energy recovery was achieved by work exchanger networks (WEN) synthesis.9 The worktransfer unit in this WEN is referred to as a direct work exchanger.10 From the viewpoint of thermodynamic analysis on how to achieve a feasible match between SR and SK with different pressures, volumetric flow rates and the amount of work, a pressure−work (P−W) diagram was proposed to characterize the feasibility of work exchange. Moreover, some matching conditions and rules were presented for synthesizing a thermodynamically feasible WEN. Due to intensively increasing energy consumption for pressure manipulation, work integration has attracted sufficient attention over the past few years. Using indirect work exchangers, such as single-shaft-turbine-compressor (SSTC), Shin et al.11 formulated a mixed integer linear programming (MILP) model to target the minimum total average energy consumption in an LNG production. Regarding LNG system as well, Hasan et al.12 focused on the compressor operations with the objective of lower total energy cost; Del Nogal et al.13−15 established a mixed integer nonlinear programming (MINLP) model to obtain the optimal power system for utility networks.

Energy usage is a major global concern due to the increasing carbon emissions and the rapid depletion of fossil fuels reserves. The manufacturing sectors account for about 52% energy loss and up to 46% of the total emissions.1 Hence, it is significantly fundamental to reduce energy consumption in industrial processes by applying innovative strategies for energy conservation and efficiency improvement. During the past few decades, heat integration has been responsible for thermal energy recovery basically through heat exchanger networks synthesis (HENs) combined with heat pump and heat engine.2−5 As another form of energy frequently available in chemical plants, mechanical energy is utilized at a lower energy efficiency, 30% of which is lost in production.1 However, how to conserve the more expensive mechanical energy effectively has not been thoroughly explored. On the basis of thermodynamic viewpoint, to recover mechanical energy, work needs to be transferred from high-pressure streams (work sources, SR) to low-pressure streams (work sinks, SK) when a pressure difference exists, which is similar to the heat integration that depends on the temperature difference among process streams. In chemical production processes, particularly for synthetic methanol, ammonia synthesis and offshore LNG production,6−8 varieties of pressurized streams consume work for compression while depressurized streams produce work by expansion. Obviously, if the pressures of SR are high enough, their available mechanical energy can be used to compress the SK, thus considerably reducing the external utility consump© XXXX American Chemical Society

Received: Revised: Accepted: Published: A

August 10, 2017 November 14, 2017 November 15, 2017 November 15, 2017 DOI: 10.1021/acs.iecr.7b03319 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research Afterward, Razib et al.7 introduced a superstructure-based MINLP formulation to achieve mechanical energy recovery. Recent developments in work integration have heightened the significance for interaction between work and heat. Onishi et al.16,17 characterized the heat integration for optimal pressure recovery by MINLP formulations. Later they proposed a multiobjective optimization model for synthesis of work and heat exchanger networks (WHENs) to trade-off economic and environmental performance.18 Huang and Karimi19 proposed MINLP optimization models for WHENs with the minimum total annual cost. Fu and Gundersen20−22 introduced a graphical design procedure to integrate turbines and compressors into HEN to reveal the interaction between work and heat. However, their studies did not consider the direct work exchangers which have higher operational efficiency than indirect work exchangers.23 To recover more mechanical energy, several attempts have been made to investigate direct WEN synthesis. Deng et al.24 proposed a basically thermodynamic analysis method for the gas−gas work exchanger based on pressure-enthalpy diagram. Specific to the ammonia synthesis process, Chen and Feng25 settled a WEN problem using a P−W diagram. Liu et al.26 proposed a graphical integration method for only synthesizing the direct WEN under isothermal condition. Currently, Aida and Huang27 developed a thermodynamic modeling and analysis method to accurately predict the maximum mechanical energy recovery prior to WEN synthesis. However, these methods are incapable of synthesizing direct WEN under adiabatic condition based on network design. As mentioned above, the open researches have clearly indicated that the WEN, composed of direct or indirect work exchangers, expanders, and compressors, can be used to efficiently recover mechanical energy. However, there are still no existing methods on synthesizing a direct WEN applied to both isothermal and adiabatic condition. In this paper, an upgraded graphical method based on a new μ−W diagram is developed for synthesis of direct WEN focusing on the identification of feasible matches between SR and SK, which contributes to the maximum amount of mechanical energy recovery and the minimum amount of external utility. In the next section, a brief introduction of pressure index is provided, which facilitates constructing improved composite curves of SR and SK plotted in the μ−W diagram and further deriving the improved linear-approximation auxiliary lines of work sinks in the succeeding section. On the basis of this, the feasible matches can be identified, and the corresponding matching rules are proposed in the subsequent section. Apparently, the proposed graphical method can be employed to achieve the synthesis of direct WEN operated under isothermal and adiabatic condition. To demonstrate methodological effectiveness, two example studies are conducted and the solutions are compared to that obtained by other methods.

SK should be accurately identified. As stated, the WEN is composed of direct work exchangers, stand-alone compressors, and stand-alone expanders. The stand-alone compressors and expanders are regarded as utility compressors and expanders, which consumes external pressurized utility and external depressurized utility, respectively. For the purpose of simplifying this synthesis process, we make the following assumptions:28 (1) all process streams are considered as ideal gas. (2) The process operational efficiency is 100%, which means that the mechanical energy is entirely transferred from SR to SK in the work-transfer units without energy losses. (3) Each work-transfer unit is reciprocating without clearance volume. (4) Heat exchange between process streams is not considered since temperature constraints are not taken into account. On the basis of these assumptions, it is aware that the direct WEN synthesis problem can be dealt with using an upgraded graphical method. To construct the improved composite curves of work sources and work sinks, which will be plotted in the μ− W diagram in the next section, the pressure index29 should be introduced prior to the description of the μ−W diagram. Note that the compression work required for SK and the expansion work produced by SR are governed by their initial and final states, respectively, as well as their volumetric flow rates and pressures. For isothermal compression, the work can be expressed as eq 1; similarly, the work through isothermal expansion is expressed as eq 2. ⎛ ⎛ P ⎞⎞ Wcomp iso = PinVin⎜⎜ln⎜ out ⎟⎟⎟ ⎝ ⎝ Pin ⎠⎠

(1)

⎛ ⎛ P ⎞⎞ Wexp iso = PinVin⎜⎜ln⎜ in ⎟⎟⎟ ⎝ ⎝ Pout ⎠⎠

(2)

On the basis of the characteristic of an isothermal process, eq 3 can be derived: (3)

PV = constant

Therefore, the work expressed in eq 1 and eq 2 can also be written as ⎛⎛ ⎛ P ⎞⎞ ⎛ ⎛ P ⎞⎞ ⎞ Wcomp iso = V0 × ⎜⎜⎜⎜P0 ln⎜ out ⎟⎟⎟ − ⎜⎜P0 ln⎜ in ⎟⎟⎟⎟⎟ ⎝ P0 ⎠⎠ ⎝ ⎝ P0 ⎠⎠⎠ ⎝⎝

(4)

⎛⎛ ⎛ P ⎞⎞ ⎛ ⎛ P ⎞⎞⎞ Wexp iso = V0 × ⎜⎜⎜⎜P0 ln⎜ in ⎟⎟⎟ − ⎜⎜P0 ln⎜ out ⎟⎟⎟⎟⎟ ⎝ P0 ⎠⎠ ⎝ ⎝ P0 ⎠⎠⎠ ⎝⎝

(5)

where V0 denotes the standard volumetric flow rate of the compressed streams and P0 is the standard pressure, which equals 100 kPa. Pin and Pout represent the inlet and outlet

( )

pressures, respectively. Besides, the quantitiesP0 ln

2. PROBLEM DEFINITION In this section, the problem for synthesizing direct WEN can be stated as follows. Consider a set of work sources (SRi, i = 1,2, ..., NSR) and a set of work sinks (SKj, j = 1,2, ..., NSK), together with their supply and target pressures (i.e., PSRis, PSRit, PSKjs, and PSKjt), volumetric flow rates (i.e., VSRi and VSKj), and the minimum pressure difference (i.e., ΔPmin). This paper aims at synthesizing a WEN with the maximum mechanical energy recovery, where the desirably feasible matches between SR and

( ) can be defined as μ

andP0 ln

Pin P0

out

Pout P0

and μin, respectively,

which are defined as the pressure index for isothermal compression or expansion. With regard to the adiabatic/polytropic compression or expansion, the process is governed by the following expression: PV γ = constant

(6)

Where γ denotes the adiabatic/polytropic exponent, assumed to be constant which is equal to 1.41 in this article. B

DOI: 10.1021/acs.iecr.7b03319 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

3. IMPROVED COMPOSITE CURVES AND LINEAR-APPROXIMATION AUXILIARY LINES Inspired by the graphical method for WEN synthesis developed by Liu et al.,26 which is only applied to isothermal process, the improved composite curves and linear-approximation auxiliary lines plotted in a novel μ−W diagram will be proposed for synthesizing direct WEN, which can be applied to isothermal, adiabatic, and polytropic operations. 3.1. Improved Composite Curves of Work Sources and Work Sinks. On the basis of the analysis in section 2, it can be derived that W has linear relationship with μ under isothermal, adiabatic, and polytropic conditions, where differences among these three conditions are the different relationships between pressure index and pressure. In other words, when a process stream is depressurized or pressurized from its supply pressure to target pressure, the work can be expressed by eq 11 and eq 12. Therefore, for the work-transfer units operated under any condition, the first step is to convert the pressure to pressure index using eq 13, thus the amount of work shows a linear correlation with the pressure index. In accordance with the two linear relationships, the compression and expansion process can be represented by a directed line segment in the μ−W diagram, as shown in Figure 1. From this figure, we can see that the horizontal range

Note that polytropic compression or expansion is an actual irreversible process with energy losses, which can be generally used to describe the thermodynamic characteristics of gas compression or expansion. For the adiabatic compression or expansion, it is a special process of the polytropic compression or expansion, where energy losses are ignored. Since the adiabatic/polytropic processes have a certain similarity, the compression work and the expansion work are given in eq 7 and eq 8, respectively. Wcomp

Wexp

adi/poly

γ − 1/ γ ⎛⎛ ⎞ Pout ⎞ γ ⎜ = − 1⎟⎟ PinVin⎜⎜ ⎟ γ−1 ⎝⎝ Pin ⎠ ⎠

(7)

γ − 1/ γ ⎛⎛ ⎞ Pin ⎞ γ ⎜ PinVin⎜⎜ = − 1⎟⎟ ⎟ γ−1 ⎝⎝ Pout ⎠ ⎠

adi/poly

(8)

As stated, these two equations can also be written as γ − 1/ γ ⎞ ⎛⎛⎛ γ ⎞ ⎛ Pout ⎞ ⎟ Wcompadi/poly = V0 × ⎜⎜⎜⎜ P ⎜ ⎟ ⎟ ⎜ ⎝ γ − 1 ⎟⎠ 0⎝ P0 ⎠ ⎠ ⎝ ⎝ γ − 1/ γ ⎞⎞ ⎛⎛ γ ⎞ ⎛ Pin ⎞ ⎜ ⎟⎟ − ⎜⎜ ⎟P0⎜ ⎟ ⎟⎟ γ − 1 P ⎝ ⎠ ⎝ ⎠ 0 ⎝ ⎠⎠

(9)

γ − 1/ γ ⎞ ⎛⎛⎛ γ ⎞ ⎛ Pin ⎞ ⎟ Wexpadi/poly = V0 × ⎜⎜⎜⎜ P0⎜ ⎟ ⎟ ⎟ ⎜ ⎝ γ − 1 ⎠ ⎝ P0 ⎠ ⎠ ⎝⎝ γ − 1/ γ ⎞⎞ ⎛⎛ γ ⎞ ⎛ Pout ⎞ ⎟⎟ − ⎜⎜⎜ P ⎜ ⎟ ⎟0 ⎟⎟ γ − 1 P ⎝ ⎠ ⎝ 0 ⎠ ⎝ ⎠⎠

Similarly, the quantities γ − 1/ γ

( )P ( ) γ γ−1

0

Pin P0

(10) γ − 1/ γ

( )P ( ) γ γ−1

0

Pout P0

and

Figure 1. Directed line segment representation for work sources.

can also be denoted as μout and μin,

between the two end points of any line segments denotes the amount of work provided by SR (or demanded by SK), and the vertical range represents the pressure index difference. Similar to that in the temperature versus enthalpy diagram for HEN synthesis, shifting the directed line segment horizontally will not affect the physical significance of the stream; however, it cannot be shifted vertically. To synthesize a WEN integrated with the whole SR and SK, the improved composite curves should be plotted combining SR and SK in the μ−W diagram. By taking the three work sources in Figure 1 as an example, the work source composite curve construction procedure is specified as the following steps. Step 1: the entire pressure index region is divided into different pressure index intervals (i.e., 1, 2, ···, K, wherein K is the total number of pressure index intervals) in the descending order according to the two end points of each work source, as shown by the dashed lines in Figure 1. Step 2: beginning with the first pressure index interval, the work source composite curve is constructed at each pressure index interval. If only one work source exists, just shift it to the corresponding position horizontally, such as AB, CD, and EF shown in Figure 2. Step 3: if there are two work sources, shift the left work source to the position where its right end point and the left end

respectively, which represents the pressure index for adiabatic/polytropic compression and expansion. Therefore, the compression and expansion work can be expressed as Wcomp = V0(μout − μin )+

(11)

Wexp = V0(μin − μout )+

(12)

Wherein ⎧ ⎛ ⎞ ⎪ P0 ln⎜ P ⎟ isothermal condition ⎪ ⎝ P0 ⎠ ⎪ μ=⎨ γ − 1/ γ ⎪⎛ γ ⎞ ⎛ P ⎞ adiabatic/polytropic ⎟P ⎜ ⎟ ⎪⎜ ⎪ ⎝ γ − 1 ⎠ 0⎝ P0 ⎠ ⎩ condition

(13)

Comparing eq 1−eq 10, it can be seen that, although W has an extremely complicated relationship with P under adiabatic/ polytropic condition, the relationship between W and μ is linear. This is a fundamentally theoretical basis to achieve successful implementation of the graphical method for the synthesis of direct WEN. C


Article

Industrial & Engineering Chemistry Research Δμmin,1 = μSR in − μSK out =

⎛ P out + ΔPmin ⎞γ − 1/ γ ⎛ P out ⎞γ − 1/ γ γ γ P0⎜ SK P0⎜ SK ⎟ − ⎟ γ−1 ⎝ γ − 1 ⎝ P0 ⎠ P0 ⎠

(20)

From these two equations it can be seen that the minimum pressure index difference has a highly nonlinear relationship with the outlet pressure of SK, even though the minimum pressure difference is assumed to be constant. In order to identify the feasible match in the μ−W diagram, the linear approximation of the two equations above should be conducted.

Figure 2. Composite curve of work sources.

In accordance with eq 19,

point of the right work source are on the same vertical line, such as BC and DE shown in Figure 2. Step 4: if more work sources exist, construct the composite curve following the same procedure as step 3. By connecting these directed line segments at each pressure index interval in the descending order, the improved work source composite curve can be derived, as shown in Figure 2. Using the same construction procedure, the improved work sink composite curve can also be obtained by combining all the line segments of SK. 3.2. Improved Linear-Approximation Auxiliary Lines of Work Sinks. According to Cheng et al.,10 the work exchange operation through direct work exchangers occurs with four consecutive steps. On the basis of it, the inlet pressure of SR should be higher than the outlet pressure of SK, and the inlet pressure of SK should be higher than the outlet pressure of SR, in order to achieve a continuous operation. To ensure a relatively higher work transfer rate, the minimum pressure difference should be required for work exchange. Hence, the necessary condition for feasible work exchange can be expressed by the following equations: PSR in − PSK out ≥ ΔPmin

(14)

PSK in − PSR out ≥ ΔPmin

(15)

d2Δμmin,1 (dPSK out)2

under isothermal

condition can be derived by eq 21: d2Δμmin,1 (dPSK out)2

= P0

(ΔPmin)2 + 2PSK outΔPmin (PSK out)2 (PSK out + ΔPmin)2

>0 (21)

2

Similarly,

d Δμmin (dPSK out)2

of adiabatic process can also be expressed by

eq 22. −1/ γ (PSK out)γ + 1/ γ − (PSK out + ΔPmin)γ + 1/ γ 1⎛ 1 ⎞ = − ⎜ ⎟ out 2 γ ⎝ P0 ⎠ (dPSK ) [(PSK out)(PSK out + ΔPmin)]γ + 1/ γ

d2Δμmin,1

>0

(22)

Since the second derivative under isothermal or adiabatic condition is always positive, the function curve between the minimum pressure index difference and the outlet pressure of SK is concave, as shown in Figure 3 and Figure 4, where PSK,out denotes PSKout in Figure 3 and Figure 4.

where ΔPmin is taken as 70 kPa in this article, from a recommended value range between 30 and 75 kPa.9 In accordance with eq 13−eq 15, another necessary condition for feasible work exchange based on the pressure index can be derived as follows. μSR in − μSK out ≥ Δμmin,1

(16)

μSK in − μSR out ≥ Δμmin,2

(17)

Figure 3. Relationship between Δμmin,1 and PSKout under isothermal condition.

From these two figures, it can be seen that the relationship between Δμmin,1 and PSKout has the same variation trend under isothermal or adiabatic condition, which shows that the minimum pressure index difference decreases with the increase

Comparing eqs 14 and 15 with eqs 16 and 17, it can be derived that both of Δμmin,1 and Δμmin,2 have a complex relationship with ΔPmin. For a feasible match between SR and SK, the lowerbound constraint can be represented by eq 18. PSR in − PSK out = ΔPmin

(18)

Then, from the combination of eq 13 and eq 16, the minimum pressure index difference, Δμmin,1 can be derived, as shown by eq 19 and eq 20, which are applied to isothermal condition and adiabatic condition, respectively. Δμmin,1 = μSR in − μSK out ⎛ P out + ΔPmin ⎞ ⎛ P out ⎞ = P0 ln⎜ SK ⎟ − P0 ln⎜ SK ⎟ P0 ⎝ ⎠ ⎝ P0 ⎠

Figure 4. Relationship between Δμmin,1 and PSKout under adiabatic condition.

(19) D


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Figure 5. Linear-approximation auxiliary lines of a work sink.

the upgraded graphical method. Inspired by Aida and Huang,27 the directed line segments of SR and SK should cross each other if a feasible match exists. Moreover, in the μ−W diagram, the necessary and sufficient condition for a feasible match is that the directed line segment of SR should not only cross the directed line segment of SK but also intersect with the two linear-approximation auxiliary lines of SK simultaneously. This is because the pressure index differences at two end points of SK are equal or greater than their respective minimum pressure index difference in this situation. Hence, SR can provide mechanical energy to SK through a direct work exchanger. Using the necessary and sufficient condition, the feasible match can be correctly identified. For a specific SR-SK match as shown in Figure 6, GH denotes the SR, while AB denotes SK,

of the outlet pressure of SK. As stated, since the function curve is concave, thus the complex function curve in Figure 3 and Figure 4 can be linearly approximated by the straight line connecting any two points, as shown by the dashed line in Figure 5a. The relationship between Δμmin,2 and PSKout has the identical trend with that between Δμmin,1 and PSKout using the same analytical method. This means that if the pressure index difference is satisfied with eq 16 and eq 17 at two end points, the pressure index difference at other points in the straight line segment will also meet the requirement. Therefore, for a directed line segment of SK in the μ−W diagram, only the minimum pressure index differences at the two end points of SK are calculated respectively according to eq 19 and eq 20, and then the two linear-approximation auxiliary lines of SK can be determined by simply connecting their respective minimum pressure index difference at the end points of directed line segment of each work sink, as shown by the dashed lines in Figure 5b. In summary, the linear-approximation auxiliary lines of SK can facilitate identifying the feasible matches between SR and SK, on account that the complex nonlinear relationship between the minimum pressure index difference and the outlet pressure is simplified to be linear according to the above analysis. However, due to this simplification where only the minimum pressure index difference at the end points is considered, the larger pressure index difference between the two end points is needed, which results in less mechanical energy recovery.

Figure 6. Identification of feasible match between work source and work sink.

4. IDENTIFICATION OF FEASIBLE MATCHES AND MATCHING RULES On the basis of the above introduction of improved composite curves and linear-approximation auxiliary lines of SK, a feasible match can be identified using the graphical method in the μ−W diagram. The main goal of this method is to identify all the feasible matches, to achieve the maximum mechanical energy recovery. For this purpose, the necessary and sufficient conditions for a feasible match are first analyzed in order to establish the identification procedure and strategies for the feasible matches in the succeeding section. In the meantime, the general and rigorous matching rules will be proposed to assist identifying the feasible matches. The specific synthesis process is described as follows. 4.1. Necessary and Sufficient Conditions for a Feasible Match. As mentioned above, the basic necessary condition for feasible work exchange has been expressed by eq 14 and eq 15. Then the derived eq 16 and eq 17 are regarded as another necessary condition for identifying feasible matches by

the two linear-approximation auxiliary lines of which is represented by CD and EF. Apparently, in this situation, GH and AB can be matched, which indicates that GH can supply mechanical energy to AB in the overlapping region and the horizontal projection of this overlapping region is exactly equal to the amount of work (i.e., MN). It is also obvious to discover that G1H1 and G2H2 are limiting cases where GH is horizontally shifted to the left and right end, respectively. Furthermore, it can also be derived that the slope of GH should be larger than that of AB, which signifies that the standard volumetric flow rate of the matched SR should be less than that of the matched SK since the slope is equal to the reciprocal of the standard volumetric flow rate. Therefore, another necessary and sufficient condition for a feasible match can be obtained as expressed by eq 23. V0,SR < V0,SK E

(23) DOI: 10.1021/acs.iecr.7b03319 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research From Figure 6, it can be seen that the pressure index difference among the two auxiliary lines at two end points is 2Δμmin,1 and 2Δμmin,2 respectively. In accordance with Figure 3 and Figure 4, 2Δμmin,1 should be less than 2Δμmin,2. Combined with the introduced necessary and sufficient conditions, it can be concluded that the inlet and outlet pressure index difference of a matched SR should be greater than 2Δμmin,1. For instance, as illustrated in Figure 7, IJ (Δμ < 2Δμmin,1) and MN (Δμ =

more than that demanded by SK, the SR can be split into several parallel subsources, as shown in Figure 8. GH is split into two parallel branches, GH1 and G1H, in which the former one can be matched with AB. Similarly, if the mechanical energy demanded by SK is more than that provided by SR, the SK should be split into several series subsinks, as shown in Figure 9. AB is split into two series branches, AK and KB, where the former one can be matched with GH.

Figure 7. Infeasible match between work source and work sink.

Figure 9. Splitting work sinks into series branches with identical volumetric flow rate.

2Δμmin,1) can never be matched with AB because they cannot cross the auxiliary lines simultaneously. In other words, no mechanical energy is provided. 4.2. Identification Procedure and Strategies for Feasible Matches. As stated, the main difference between HEN and WEN is that the inlet pressure index of SK should be greater than the outlet pressure index of SR for a feasible match. Thus, the pressure index intervals should be determined by the composite curves of SK prior to identification of feasible matches, which is different from the division of temperature intervals based on both the temperatures of the cold and hot streams. Afterward, the matching feasibility will be analyzed at each interval according to the above analysis. Note that the ideal case is that the mechanical energy provided by the matched SR is equal to that demanded by the matched SK. For instance, as shown in Figure 8, GH1 is exactly matched with AB, where the mechanical energy of GH1 is entirely recovered by AB. However, generally, the supplied energy by SR is not equal to the demanded energy by SK. In this case, splitting work sources or work sinks into subsources or subsinks with different volumetric flow rates can help achieve feasible matches. If the mechanical energy provided by SR is

Interestingly, it should be noted that the composite curve of SR can be as a whole to match the SK at one specific pressure index interval when it is satisfied with the introduced necessary and sufficient conditions for a feasible match. This will be specified in Example 2. 4.3. Brief Summary for the Matching Rules. On the basis of the above analysis, the matching rules for feasible matches between SR and SK can be specifically summarized as follows. (1) A feasible match should be satisfied with three necessary and sufficient conditions: the directed line segment of SR should cross the directed line segment of SK and the two linear-approximation auxiliary lines of SK simultaneously; the standard volumetric flow rate of the matched SR should be less than that of the matched SK; the inlet and outlet pressure index difference of a matched SR should be greater than 2Δμmin,1. (2) If the mechanical energy provided by SR is more than that demanded by SK, SR should be split into several parallel subsources; otherwise, SK should be split into several series subsinks. (3) The composite curve of SR can be as a whole to match SK at a specific pressure index interval when it is satisfied with the necessary and sufficient conditions for a feasible match. In summary, the procedure for direct WEN synthesis using the proposed upgraded graphical method is illustrated in Figure 10.

5. CASE STUDIES In this section, two examples from open literature are studied to demonstrate the significance and effectiveness of the proposed method. As described above, the upgraded graphical method is used to identify all feasible matches between SR and SK for achieving the maximum mechanical energy recovery based on the network design. 5.1. Example 1. Example 1 taken from Liu et al.26 involves two work sources and four work sinks. Their detailed stream data is listed in Table 1. As stated, the minimum admissible pressure difference for a feasible work exchange is 70 kPa. It is

Figure 8. Splitting work sources into parallel branches with different volumetric flow rates. F


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Figure 11. SR and SK composite curves for example 1.

Figure 10. Flow diagram of the developed upgraded graphical method. Figure 12. SR-SK match in the pressure index interval EF [368.89, 329.21].

assumed that the work-transfer units in this example are operated under isothermal conditions. By following the procedure shown in Figure 10, the first step is to convert the pressure values to the pressure index values. In accordance with eq 13, the pressure index values can be obtained, as shown in column 3 and 5 of Table 1. In the following step, based on the aforementioned construction procedure of composite curves, the SR and SK composite curves can be plotted, as shown in Figure 11. In addition, the auxiliary lines of SK can also be plotted in Figure 11, according to the detailed description in section 3.2. On the basis of this figure, the feasible matches between SR and SK will be identified. Next, to identify the feasible matches clearly, the SK composite curve is divided into several pressure index intervals taking each inflection point as the dividing point. Therefore, from Figure 11, it can be seen that there are seven pressure index intervals: EF [368.89, 329.21], FG [329.21, 290.69], GH [290.69, 288.48], HI [288.48, 74.19], IJ [74.19, 33.65], JK [33.65, 0.00], and KL [0.00, −35.67]. Note that since a small pressure index difference exists in GH, thus the segment cannot be seen clearly in Figure 11. Afterward, the feasible matches can be identified as follows: (1) SK in the pressure index interval EF [368.89, 329.21]. The SR segment AB cannot be matched with the SK in EF because its inlet and outlet pressure index (449.65 and 443.08) are both higher than the upper bound of EF. Nevertheless, the segment BC [443.08, 109.86] can be matched with the SK in EF because it is satisfied with the proposed necessary and sufficient conditions for a feasible match. Since the energy provided by SR are more than that demanded by SK, BC can be split into two parallel subsources,

Figure 13. SR-SK match in the pressure index interval GI [290.69, 74.19].

Figure 14. SR-SK match in the pressure index interval IJ [74.19, 33.65].

Table 1. Process Stream Data for Example 1 stream no.

Ps (kPa)

μs (kPa)

Pt (kPa)

μt (kPa)

V0 (N m3 s−1)

W (kW)

SR1 SR2 SK1 SK2 SK3 SK4

8400 8970 100 70 2690 140

443.08 449.65 0.00 −35.67 329.21 33.65

100 300 210 1790 4000 1830

0.00 109.86 74.19 288.48 368.89 290.69

13.58 2.72 88.58 18.43 60.03 4.65

6018.10 925.40 6572.40 5975.00 2381.60 1194.80

G


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Figure 15. Flowsheet of integrated work exchanger network for example 1: (a) the solution of the proposed method and (b) the solution obtained by Liu et al.26 (The legend refers to Aida and Huang.27)

Table 2. Comparison of Solutions by Different Methods for Example 1 this work Liu et al.26 changing rate (%)

WR (kW)

Wcomp (kW)

Wexp (kW)

no. of WE

no. of comp

no. of exp

6925 5889 +17.6

9190 10226 −10.1

18 1044 −

7 7 −

5 5 −

1 3 −

B1C and BC1, and the former one can be matched with SK in EF, as shown in Figure 12. The recovered mechanical energy is 2381.6 kW. (2) Sink in the pressure index interval FG [329.21, 290.69]. From Figure 11, it can be seen that FG is a vertical segment, which means that there is no sink in this interval.

Hence, it is unnecessary to identify the feasible match in FG. (3) As mentioned above, GH is a relatively smaller pressure index interval with less required mechanical energy, thus it is not necessary to identify the feasible match. However, GH and HI can be merged into a new interval GI [290.69, 74.19] in H


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Industrial & Engineering Chemistry Research Table 3. Process Stream Data for Example 2 stream no.

Ps (kPa)

μs (kPa)

Pt (kPa)

μt (kPa)

V0 (N m3 s−1)

W (kW)

SR1 SR2 SR3 SK1 SK2

850 960 800 100 100

640.75 663.84 629.56 343.90 343.90

100 160 300 510 850

343.90 394.27 473.34 552.31 640.75

3.52 4.86 2.67 6.65 6.80

1044.49 1308.80 417.80 1386.60 2019.49

order to recover more mechanical energy by a feasible match, which also meets the requirement of the necessary and sufficient conditions for feasible matches. Similar to the above analysis, BC can also be matched in this interval. As the mechanical energy required by SK is more than that provided by SR, thus GI should be split into two series subsinks, GN and NI, and the former one can match BC, as shown in Figure 13. The mechanical energy recoverable is 3051.8 kW. (4) Sink in the pressure index interval IJ [74.19, 33.65]. As stated earlier, the supplied energy is less than the required energy, so IJ is also divided into two series subsinks, IQ and QJ, and the former one can be matched with the source segment CD that is satisfied with the matching feasibility, as shown in Figure 14. The recovered mechanical energy is 1492.7 kW. For the final two intervals, JK [33.65, 0.00] and KL [0.00, −35.67], apparently it is unsatisfied with the pressure index difference constraints. Therefore, no feasible matches will exist in these two intervals. On the basis of the above analysis, it can be derived that all the feasible matches have been identified. This means that the WEN has been successfully synthesized using our developed method, as shown in Figure 15a. Moreover, in the derived network, the maximum amount of mechanical energy recovery (6,925 kW) using the presented method is achieved, which is 17.6% more than that obtained by Liu et al.26 This also indicates that the minimum amount of external pressurized and depressurized utility consumption can remarkably decrease. Besides, the number of expanders used in our solutions is smaller than that used in their solutions. The detailed comparison is summarized in Table 2. From Figure 15a, it can be seen that SR1 has the most available mechanical energy of the two work sources, where five direct work exchangers extract the whole energy (6018.10 kW) from SR1 to the four work sinks while one utility turbine extracts 514.8 kW from SR1 that is not recovered in Figure 15b. This is because the source segment BC can both be matched with the sink segments EF and GN, and its mechanical energy is exactly recovered by the two sink segments. Similarly, only one utility turbine extracts 17.89 kW from SR2 in Figure 15a, while SR2 delivers 529.59 kW to two utility turbines. Moreover, the proposed method can be applied to any operating conditions, while only WEN of the isothermal condition can be obtained by Liu et al.26 5.2. Example 2. Example 2 is selected from Aida and Huang.27 Table 3 lists the detailed process stream data. The minimum admissible pressure difference for a feasible work exchange is still taken as 70 kPa. It is assumed that the worktransfer units in the example are operated under adiabatic condition. By following the same procedure shown in Figure 10, using eq 13 converts the pressure values to the pressure index values, as illustrated in columns 3 and 5 of Table 3. Subsequently, similar to the aforementioned analysis in example 1, the SR and SK composite curves and the two auxiliary lines of SK can be plotted, as shown in Figure 16. On

Figure 16. SR and SK composite curves for example 2.

Figure 17. SR-SK match in the pressure index interval GH [640.75, 552.31].

Figure 18. SR-SK match in the pressure index interval HI [552.31, 343.90].

Figure 19. SR-SK match in the pressure index interval HI [552.31, 392.47]. I


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Figure 20. Flowsheet of integrated work exchanger network for example 2: (a) the solution of the proposed method and (b) the solution obtained by Aida and Huang.27

Table 4. Comparison of Solutions by Different Methods for Example 2 this work Aida and Huang27 changing rate (%)

WR (kW)

Wcomp (kW)

Wexp (kW)

no. of WE

no. of comp

no. of exp

1997 1778 +12.3

1409 1628 −13.5

774 993 −22.1

4 4 −

3 3 −

2 3 −

the basis of this figure, the feasible matches between work sources and work sinks will be identified. In the succeeding step, to explicitly show the identification of feasible matches, the SK composite curve in Figure 16 is divided into two pressure index intervals: GH [640.75, 552.31] and HI [552.31, 343.90]. Afterward, the feasible matches between SR and SK can be identified as follows: (1) SK in the pressure index interval GH [640.75, 552.31]. Since the SR segment AB and BC both have a relatively smaller pressure index difference, it is insufficient to match any SK. However, if

AB, BC, and CD are regarded as a new interval as a whole, where it is satisfied with the necessary and sufficient conditions, then the new interval can be matched with SK in GH, as shown in Figure 17. Additionally, because the energy provided by SR are more than that demanded by SK, the new interval can be split into two parallel subsources, A1D and AD1, and the latter one can be matched with SK in GH, as shown in Figure 17. The recovered mechanical energy is 601.7 kW. (2) Sink in the pressure index interval HI [552.31, 343.90]. It should be noted that the supplied energy (1846.3 kW) is much less than the J


Article


Although the derived WEN using our introduced method is better in energy performance than that in the open literature, the optimal solution cannot be guaranteed herein. Due to no consideration of the capital expenditure of work-transfer units, there may be some other network configurations which are also able to recover mechanical energy with the objective of the lowest total annual cost. However, the development on work integration simultaneously considering operational and capital expenditure requires a more sophisticated method that is beyond the scope of this article. In addition, for a WEN operated under the adiabatic or polytropic condition, it should be noted that stream temperature will be changed after compression or expansion, and interestingly, the amount of work transfer should be affected by the change of temperature. In this situation, it is of upmost significance to investigate the interaction between work integration and heat integration. Due to vital importance for major energy efficiency improvement, it is conceivable that some remarkable progress on the studies about simultaneous synthesis of work and heat exchanger networks will be made in our future work.

required energy (2804.4 kW). Therefore, to recover more mechanical energy, this interval is classified into two sections: [552.31, 343.90] and [552.31, 392.47], where the feasible matches are identified, respectively. For the former one, it can be matched with HR1, where the first SK segment HI is split into two series subsinks, HR1 and R1I, as shown in Figure 18. For the latter one, it can be matched with HR2, in which the second SK segment HI is also split two series subsinks, HR2 and R2I, as shown in Figure 19. Comparing these two figures, it can be seen that the matched SR undergoes great changes. The reason is that one work source will not match any work sink once its mechanical have been entirely recovered, which can be illustrated in Figure 20. On the basis of all the feasible matches obtained above, the WEN synthesis has been indeed achieved using our developed method, as shown in Figure 20. Furthermore, as shown in Table 4, the total amount of mechanical energy recovery obtained by the presented method is 1997 kW, 12.3% more than that derived by Aida and Huang.27 In further comparison with their solutions, the minimum amount of external pressurized and depressurized utility consumption in our solutions also achieve significantly decreasing percents, 13.5% and 22.1%, respectively. In addition, the number of expanders used in our solutions is only one more than that used in their solutions. From Figure 20a, it can be seen that the three work sources both deliver mechanical energy to two work sinks through direct work exchangers while one utility turbine extracts 417.8 kW from SR3 that is not recovered in Figure 20b. This is because the SK1 is split into two substreams with different flow rates, one of which matches SR1 and the other matches SR3, where network design is not considered in the open literature. Moreover, the proposed method can achieve synthesizing a better WEN with the maximum mechanical energy recovery, while only the amount of mechanical energy without consideration of optimal process flowsheet development can be obtained by Aida and Huang.27

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 411 84986301. Fax: +86-411-84986201. E-mail: [email protected]. ORCID

Lei Zhang: 0000-0002-7519-2858 Jian Du: 0000-0001-7667-4835 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Natural Science Foundation of China (no. 21776035 and no. 21576036) for providing the research fund for this project.

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6. CONCLUSIONS Mechanical energy recovery has a significant impact on energy efficiency improvement in chemical processes. Since the operating mode of work exchangers is dramatically different from that of traditional heat exchangers, Pinch Analysis, which is widely applicable for HENs, cannot be applied to work integration. Therefore, to facilitate direct WEN synthesis, an upgraded thermodynamic graphical methodology is proposed for synthesizing direct WEN operated under isothermal and adiabatic condition in this article. In this method, the improved linear-approximation auxiliary lines and three matching rules are proposed to assist identifying the feasible matches, aiming at the optimal work exchanger network design, together with maximum mechanical energy recovery and minimum external utility consumption. The methodology is rigorous and general for work integration through network design using work exchange operations under any conditions, such as isothermal, adiabatic, and polytropic conditions. Two examples are investigated to justify the efficacy of the proposed method. For the first example, it gives a WEN with 17.6% increased mechanical energy recovery than the literature method that is only applied to isothermal operation; for the second example, 12.3% more work is recovered compared with the literature model, which focused on the mechanical energy recovery prior to network design.

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K

ABBREVIATIONS HENs = heat exchanger networks synthesis LNG = liquefied natural gas MILP = mixed-integer linear programming MINLP = mixed-integer nonlinear programming SSTC = single-shaft-turbine-compressor WEN = work exchanger networks WHENs = work and heat exchanger networks NOMENCLATURE comp = compressors exp = expanders K = the total number of pressure index intervals NSK = the number of work sources NSR = the number of work sinks P = pressure, kPa P0 = standard pressure, kPa ΔPmin = the minimum driving force demanded for work exchange, kPa V = volumetric flow rate, m3 s−1 V0 = standard volumetric flow rate, m3 s−1 W = reversible shaft work, kW Wcomp = the minimum amount of pressurized utility, kW Wexp = the minimum amount of depressurized utility, kW WR = the maximum amount of mechanical energy recovery, kW DOI: 10.1021/acs.iecr.7b03319 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


(17) Onishi, V. C.; Ravagnani, M. A. S. S.; Caballero, J. A. Retrofit of heat exchanger networks with pressure recovery of process streams at sub-ambient conditions. Energy Convers. Manage. 2015, 94, 377−393. (18) Onishi, V. C.; Ravagnani, M. A. S. S.; Jiménez, L.; Caballero, J. A. Multi-objective synthesis of work and heat exchange networks: Optimal balance between economic and environmental performance. Energy Convers. Manage. 2017, 140, 192−202. (19) Huang, K.; Karimi, I. A. Work-heat exchanger network synthesis (WHENS). Energy 2016, 113, 1006−1017. (20) Fu, C.; Gundersen, T. Integrating compressors into heat exchanger networks above ambient temperature. AIChE J. 2015, 61, 3770−3785. (21) Fu, C.; Gundersen, T. Correct integration of compressors and expanders in above ambient heat exchanger networks. Energy 2016, 116, 1282−1293. (22) Fu, C.; Gundersen, T. Heat and work integration: Fundamental insights and applications to carbon dioxide capture processes. Energy Convers. Manage. 2016, 121, 36−48. (23) Chen, Z.; Wang, J. Heat, mass and work exchange networks. Front. Chem. Sci. Eng. 2012, 6, 484−502. (24) Deng, J.; Shi, J.; Zhang, Z.; Feng, X. Thermodynamic analysis on work transfer process of two gas streams. Ind. Eng. Chem. Res. 2010, 49, 12496−12502. (25) Chen, H.; Feng, X. Graphical difference for targeting work exchange networks. World Academy of Science, Enginerring and Technology 2012, 6, 977−981. (26) Liu, G.; Zhou, H.; Shen, R.; Feng, X. A graphical method for integrating work exchange network. Appl. Energy 2014, 114, 588−599. (27) Aida-Rankouhi, A.; Huang, Y. Prediction of maximum recoverable mechanical energy via work integration: A thermodynamic modeling and analysis difference. AIChE J. 2017, 63, 4814. (28) Zhuang, Y.; Liu, L.; Liu, Q.; Du, J. Step-wise synthesis of work exchange networks involving heat integration based on the transshipment model. Chin. J. Chem. Eng. 2017, 25, 1052−1060. (29) Bandyopadhyay, S.; Chaturvedi, N. D.; Desai, A. Targeting Compression Work for Hydrogen Allocation Networks. Ind. Eng. Chem. Res. 2014, 53, 18539−18548.

WE = work exchangers γ = the adiabatic/polytropic exponent μ = pressure index, kPa Δμmin = the minimum pressure index difference, kPa Subscripts/Superscripts

adi comp exp i in iso j out poly s SK SR t

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adiabatic condition compression expansion index for work source inlet of pressure stream isothermal condition index for work sink outlet of pressure stream polytropic condition supply state of a work source or work sink work sink work source target state of a work source or work sink

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An Upgraded Graphical Method for the Synthesis of Direct Work

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