Sub-ambient heat exchanger network design including compressors

Chemical Engineering Science 137 (2015) 631–645

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Sub-ambient heat exchanger network design including compressors Chao Fu n, Truls Gundersen Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

H I G H L I G H T S







The integration of heat and work is studied. Compressors are integrated with sub-ambient heat exchanger networks. Pinch Compression is proven to be preferred. The exergy consumption is minimized.

art ic l e i nf o

a b s t r a c t

Article history: Received 15 April 2015 Received in revised form 3 July 2015 Accepted 10 July 2015 Available online 23 July 2015

Heat can be converted into work and vice versa, and this fact provides opportunities for the integration between them. This paper presents a systematic methodology for the integration of compressors into heat exchanger networks below ambient temperature. The objective is to minimize exergy consumption. Four theorems are proposed and used as the basis for the design methodology. Pinch Compression, i.e., compression starts at pinch temperature, is proven to be preferred under certain conditions. It is concluded that minimum exergy consumption can be achieved when compression starts at pinch temperatures, ambient temperature or the cold utility temperature. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Heat exchanger network Sub-ambient Pinch analysis Exergy Pinch compression

1. Introduction The synthesis of heat exchanger networks (HENs) is a central part of heat recovery problems. One of the well-established methodologies for HEN synthesis is Pinch Analysis (PA) (Linnhoff and Hindmarsh, 1983). The minimum utility consumption is established as a target using PA in an early stage of process design. For a given set of hot and cold streams (heat capacity flowrates, supply and target temperatures are fixed), the pinch temperatures are established by the minimum temperature difference ðΔT min Þ for heat transfer. Both hot and cold utilities increase if any heat is transferred across the pinch. The Grand Composite Curve (GCC) is developed for targeting and selection of utilities. PA has achieved a notable success for above ambient process design. In sub-ambient processes, both heat and work (for producing refrigeration energy) are involved. However, traditional PA only sets targets for heat loads and was thus less used below ambient (Linnhoff and Dhole, 1992). Using the Carnot factor instead of temperature as ordinate, the GCC was transformed into the Exergy Grand Composite Curve n

Corresponding author. Tel.: þ 47 73592799; fax: þ 47 73593580. E-mail addresses: [email protected], [email protected] (C. Fu).

http://dx.doi.org/10.1016/j.ces.2015.07.022 0009-2509/& 2015 Elsevier Ltd. All rights reserved.

(EGCC) (Dhole and Linnhoff, 1994; Linnhoff and Dhole, 1992), which was used for shaftwork targeting in sub-ambient processes. An exergetic efficiency factor was used to convert the exergy content of heat into shaftwork. This study focuses on the integration of both heat and work. The following two observations in sub-ambient processes provide basic insights for the study: (i) compression consumes less work at lower temperatures but refrigeration is required to achieve the low temperature; and (ii) expansion produces more work at higher temperatures but less refrigeration is produced at lower temperatures. There is thus a trade-off between cold utility consumption and work consumption/production when pressure changing equipment such as compressors/expanders are included below ambient. One application example for the integration of heat and work is the utilization of compression heat for preheating the boiler feedwater in steam cycles (Fu and Gundersen, 2013a). Another example is the self-heat recuperation technology (Kansha et al., 2010) where heat is upgraded by compression, i.e., the temperature of heat is lifted. A challenging question related to the integration of heat and work is: at what temperatures should compression and expansion start? This question is related to the concept of Appropriate Placement (also referred to as Correct Integration) and the plus/minus

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principle (Linnhoff and Parker, 1984; Linnhoff and Vredeveld, 1984) in HEN design. Aspelund et al. (2007) formulated the following two heuristic rules: (i) compression adds heat to the system and should preferably be done above pinch, and (ii) expansion provides cooling to the system and should preferably be done below pinch. The rules were stated more specifically by Gundersen et al. (2009): both compression and expansion should start at the pinch temperature. Following the rules, a recuperative vapor recompression cryogenic air distillation scheme was developed by Fu and Gundersen (2013b). The Extended Pinch Analysis and Design (ExPAnD) methodology developed by Aspelund et al. (2007) presents 10 heuristic rules for the integration of compressors and expanders into HENs. On the basis of the ExPAnD methodology, Wechsung et al. (2011) presented an MINLP optimization formulation for the synthesis of sub-ambient HENs including compression and expansion. The work was further extended by Onishi et al. (2014) using a superstructure with the objective of minimizing total annualized cost. Rather than using mathematical optimization methodologies presented by Wechsung et al. (2011) and Onishi et al. (2014), this paper, with a focus on the integration of compressors into HENs below ambient temperature, presents a straightforward graphical methodology for HEN design including compressors. Since both heat and work are involved, the objective is to minimize exergy consumption. A set of theorems are proposed and proven to assist the design.

2. Theorems The design methodology is developed based on four theorems that are introduced in this section. The following assumptions are made for the proof of Theorems 1–4: (1) supply and target states (temperature and pressure) for process streams and utilities for heating and cooling are given; (2) only one stream is compressed and only one cold utility (one temperature level) is used; (3) the compressor polytropic efficiency is constant, (4) the gas to be compressed is ideal gas with a constant specific heat ratio κ  cp =cv , and (5) the exergy content of hot utility is negligible (assumed to be near ambient conditions). 2.1. Theorem 1 Theorem 1. For sub-ambient processes, a HEN design with Pinch Compression (compression starts at the pinch temperature) consumes the smallest amount of exergy if the following conditions are satisfied: (1) the outlet temperature of compression at cold utility temperature is lower than ambient temperature, and (2) Pinch Compression does

T (oC)

not produce more heat than required, i.e., the original pinch point is not removed. 2.1.1. Proof of Theorem 1 Proof of Theorem 1. A cold stream is assumed to be compressed from ps to pt . In the case of a hot stream being compressed, a similar proof can be established. The composite curves (CCs) for the streams without including pressure manipulation (Case O) are shown in Fig. 1(a). The heating and cooling demands are Q HU;0 and Q CU;0 respectively. The pinch temperature is T PI for the cold streams. For compression above pinch, less work is consumed when the operating temperature is lower. Pinch Compression thus consumes the smallest amount of exergy if the original pinch point is not removed, i.e., the cooling demand does not increase. Comparison is then performed between Pinch Compression and compression below pinch. The following cases are compared: Case A – compression starts a temperature below pinch T A ðT CU r T A o T PI Þ; Case B – Pinch Compression is used, T B ¼ T PI . For Case A, the work is W A ¼ mcp ðT comp; A  T A Þ ¼ mcp T A ½ðpt = ps Þðnc  1Þ=nc  1, where ðnc  1Þ=nc ¼ ðκ  1Þ=ðκη1;comp Þ, η1;comp is compressor polytropic efficiency, mcp is the heat capacity flowrate of the stream being compressed, and T comp;A is the outlet temperature of compression at T A . Since compression of the cold stream increases its temperature and enthalpy, the total cooling demand increases, Q CU;A ¼ Q CU;0 þ xW A where x is the fraction compressed below pinch, 0 o x r 1. Although the compression process (pressure ratio) is not really split, the fraction compressed below pinch is referring to the temperature increase from T A to T PI and the fraction above pinch is referring to the temperature increase from T PI to T comp;A . When T comp;A rT PI , i.e., the entire compression of the stream is performed below pinch, x ¼ 1. The cooling provided by the cold stream is reduced since its enthalpy is increased by W A , and the cooling demand is thus increased by the same amount ðW A Þ. When T comp;A 4T PI , the compression of the stream operates across the pinch. This case is shown in Fig. 1(b). The dashed curves represent the original CC without compression (Case O) and the solid curves represent the CC for Case A. The compression above pinch (from T PI to T comp;A ) does not affect the cooling demand (cold utility) demand, but it reduces the heating demand (hot utility) due to the introduction of compression heat from T PI to T comp;A . The compression below pinch (from T A to T PI ), however, increases the cooling demand by mcp ðT PI  T A Þ, which is written as xW A where 0 o x ¼ mcp ðT PI  T A Þ=mcp ðT comp;A  T A Þ o 1. The corresponding cold CC in the temperature range from T A to T comp;A has smaller heat capacity flowrate compared to Case O, as shown in Fig. 1(b). The exergy content ðEQ CU Þ of a given cooling duty ðQ CU Þ at T CU (subambient) is derived by EQ CU ¼ Q CU ðT 0 =T CU  1Þ. The exergy content of a given amount of work is equivalent to the work. The total

T (oC)

T0

QHU,0

QHU,0

T0

T (oC)

QHU,0

T0

QHU,A

QHU,B Tcomp,B

Tcomp,A ΔTmin

ΔTmin

TPI

ΔTmin

TPI TA

QCU,0

QCU,0

TCU H (kW)

TB

QCU,B=QCU,0

QCU,A TCU

TPI

TCU H (kW)

Fig. 1. CCs for Theorem 1: (a) Case O, (b) Case A, (c) Case B.

H (kW)

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

exergy consumption for Case A is thus determined to be EA ¼ EQ CU;A þ W A ¼ ðQ CU;0 þ xW A ÞðT 0 =T CU 1Þ þW A . Note that the exergy content of hot utility is neglected. For Case B, W B ¼ mcp ðT comp;B T B Þ ¼ mcp T PI ½ðpt =ps Þðnc  1Þ=nc  1, Q CU;B ¼ Q CU;0 and EB ¼ EQ CU;B þ W B ¼ Q CU;0 ðT 0 =T CU 1Þ þW B . Note that the entire compression is performed above pinch (from T B to T comp;B , where T B ¼ T PI ). As shown in Fig. 1(c), the cooling demand below pinch will then not be affected by the compression as long as the original pinch point is not removed. Above pinch there is heat deficit, and the compression will result in reduced heating demand. The corresponding cold CC in the temperature range from T B to T comp;B has smaller heat capacity flowrate compared to Case O. When x ¼ 1, i.e., T comp;A rT PI , the difference of exergy consumption is found to be EB  EA ¼ W B  W A T 0 =T CU ¼ mcp ½ðpt =ps Þ ðnc  1Þ=nc 1ðT B  T A T 0 =T CU Þ. Since only one cold utility at T CU is assumed to be used, any process streams below T CU must be in heat balance (i.e., part of a heat “pocket”). Compression below T CU causes cooling deficit and the need for another cold utility, and should thus not be used, i.e., T A ZT CU . The following inequality is satisfied: T B oT 0 r T A T 0 =T CU . In addition, pt 4 ps , ðnc  1Þ=nc 4 0

and thus ðpt =ps Þðnc  1Þ=nc  1 4 0. It is then concluded that EB o EA , i.e., the exergy consumption of Case B is less than for Case A. When 0 o x o 1, i.e., T comp;A 4 T PI , xW A ¼ mcp ðT PI  T A Þ, and the exergy consumption is compared in the following way: EB  EA ¼ W B  W A  xW A ðT 0 =T CU  1Þ ¼ mcp ðT comp;PI T comp;A Þ  mcp ðT PI  T A ÞT 0 =T CU ¼ mcp ðT PI  T A Þ½ðpt =ps Þðnc  1Þ=nc  T 0 =T CU  ¼ mcp ðT PI  T A ÞðT comp;CU T 0 Þ=T CU Since T PI  T A 4 0 and T comp;CU oT 0 according to Condition (1), it is again concluded that Case B consumes less exergy. In conclusion, Theorem 1 has been proven. 2.1.2. Maximum fraction subject to Pinch compression According to Condition (2) in Theorem 1, the heat resulting from Pinch Compression should not be more than the heating demand ðQ HU;0 Þ, i.e., the original pinch point is not removed, otherwise the cooling demand increases. Fig. 2 shows the Grand Composite Curve (GCC) without pressure manipulation. Modified temperatures ðT 0 Þ are used, which means that T 0 ¼ T þ 0:5ΔT min for cold streams and T 0 ¼ T  0:5ΔT min for hot streams. The outlet temperature of Pinch Compression is T 0comp;PI . A temperature is defined as a Potential Pinch Point if it may create new pinch points after a portion of the compression heat is included. The following

T’ (K) T’0

QHU,0 Qcomp,max

d f e

c

T’comp,PI

b a

T’PI

QCU,0

H (kW) Fig. 2. GCC without pressure manipulation.

633

temperatures are Potential Pinch Points: (i) the convex kink points (see definition below) on the GCC in the range between T ¼ T 0PI and T 0 ¼ T 0comp;PI (such as points a and b); (ii) the point T 0 ¼ T 0comp;PI on the GCC (point c) or the point on the line T 0 ¼ T 0comp;PI with H ¼ Q HU;0 if T 0comp;PI is higher than the highest temperature on the GCC (not shown in Fig. 2); (iii) the intersection point between the constant temperature line T 0 ¼ T 0comp;PI and a pocket (point e) in the GCC. A convex kind point on the GCC is defined as a point where either the slope decreases without sign change or the slope increases with sign change when referring to the positive y-axis direction (i.e., modified temperature). The maximum fraction of the stream using Pinch Compression, ðmcp Þcomp;PI;max , is determined by the following steps: starting at the pinch point ðT 0PI Þ, draw lines between the pinch point and Potential Pinch Points and extend the line with the largest slope until it intersects with the constant temperature line ðT 0 ¼ T 0comp;PI Þ. The corresponding heating demand at the intersection point (f in Fig. 2), Q comp;max , is equal to the work resulting from Pinch Compression, and ðmcp Þcomp;PI;max can thus be determined to be Q comp;max =ðT 0comp;PI  T 0PI Þ. Stream splitting is thus required if the heat capacity flowrate of the stream being compressed mcp 4 ðmcp Þcomp;PI;max , and the fraction of the stream using Pinch Compression is ðmcp Þcomp;PI;max . The identity (hot or cold) of the streams to be compressed and the location of supply and target temperatures are not distinguished in the GCC application for simplicity. The corresponding errors are investigated in Appendix. The cold utility demand actually increases by ymcp ΔT min , where 0 r y r1, when the stream identity and corresponding supply and target temperatures are considered. This amount is negligible since ΔT min is normally quite small in sub-ambient processes and thus ymcp ΔT min is small compared to the total cold utility demand. In the case that the additional cold utility demand ðymcp ΔT min Þ needs to be included, a comprehensive procedure can of course be developed to include all the cases presented in Appendix. 2.1.3. Example 1 Example 1. The stream data is shown in Table 1, indicating that a cold stream (C1) is compressed from 1 bar to 3 bar. The following assumptions are made for all examples in this paper: (1) η1;comp ¼ 1, (2) ΔT min ¼ 4 K, (3) T 0 ¼ 288 K, and (4) the fluid to be compressed is ideal gas with constant specific heat ratio κ ¼ 1:4. Since T comp;CU is calculated to be T CU ðpt =ps Þðnc  1Þ=nc ¼ 164:2 K o T 0 , according to Theorem 1, Pinch Compression should be used to reduce exergy consumption. The GCC without pressure manipulation is shown in Fig. 3(a). The pinch temperature is 202/198 K. The following two cases are compared: Case A – compression at the lowest temperature, T CU (120 K); Case B – compression at T PI for the cold streams (198 K). The new stream data for C1 is shown in Table 2. In Case A, the new supply temperature (164.2 K) is the outlet temperature of compression at T CU . In Case B, Pinch Compression is used for stream C1, resulting in two new streams: one (C1_1) is heated from T s to T PI (198 K) before compression and another (C1_2) is heated from T comp;PI (271.0 K) to T t after compression. The corresponding GCCs are shown in Fig. 3(b) and (c). The original pinch point is not removed, however, the utilities change when pressure manipulation (compression) is included. Table 3 shows the performance results. Using Case O as the basis, the cold utility demand ðQ CU Þ increases by an amount equal to the compression work in Case A, however, Q CU for Case B does not change, instead Q HU increases by an amount equal to the compression work. Even though more work is consumed in Case B compared to Case A, the exergy consumption is reduced by 20.6%. The reason is that heat has been introduced by compression to the

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C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

region below pinch and thus the cold utility demand increases in Case A. 2.2. Theorem 2

Theorem 2. For sub-ambient processes, once the heating demand has been satisfied by Pinch Compression, the remaining portion (stream splitting is used) is compressed at either T CU or T 0 if the outlet temperature of compression at T CU is lower than T PI . 2.2.1. Proof of Theorem 2 Proof of Theorem 2. A cold stream is assumed to be compressed and a similar proof can be established if a hot stream is compressed. Since the heating demand can be completely satisfied by Pinch Compression, the outlet temperature of Pinch Compression, T comp;PI , should be higher than the lowest possible hot utility temperature, T HU;min , as shown in Fig. 4. When Pinch Compression is used, stream splitting is required. According to Theorem 1, afirst portion (α) should be compressed at T PI so that the heating demand is satisfied by the compression heat. Further compression of the remaining portion below T 0 increases the cooling demand by an amount equal to the compression work, and the work consumption decreases when the temperature is lower. Compression at T CU thus consumes the smallest amount of work and cooling demand, compared to other compression schemes below T 0 . The compression of the remaining portion can also be done at T 0 so that the cooling demand does not increase since there is a cooling Table 1 Stream data for Example 1. Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

H1 C1 C2 Hot utility Cold utility

288 120 198 288 120

124 284 284 288 120

3 2 4 – –

492 328 344 – –

– 1 – – –

– 3 – – –

300 280 260 240 220 200 180 160 140 120 100 0

50

100

150

200

250

300

¼ ðmcp Þγ ðT comp;CU T CU ÞT 0 =T CU  ðmcp Þγ ðT comp;0  T 0 Þ ¼ ðmcp Þγ T CU ½ðpt =ps Þðnc  1Þ=nc  1T 0 =T CU  ðmcp Þγ T 0 ½ðpt =ps Þðnc  1Þ=nc 1 ¼0 The two schemes thus have the same exergy consumption, which means that once the heating demand has been satisfied by Pinch Compression, the remaining portion (stream splitting is used) should be compressed at either T 0 or T CU . When Pinch Compression can satisfy the heating demand of the process, any compression scheme at higher temperatures will increase work consumption and even cold utility demand if the heating demand has been satisfied by the compression heat, resulting in increased exergy consumption. Thus, the comparison between compression below pinch and Pinch Compression (with stream splitting) is performed in the following way. Two cases are compared: Case B – Pinch Compression (portion α) is used to satisfy the heating demand and the remaining portion (γ) is compressed at T CU ; Case C – compression at some intermediate temperature T C ðT CU r T C o T PI Þ and the heating demand is satisfied by compression heat above pinch. Case C actually includes two sub-cases: the first case (C.i) is that the heating demand is satisfied and T comp;C Z T HU;min (the stream is not split); the second case (C.ii) is that the compression heat (above pinch) is more than required

300 280 260 240 220 200 180 160 140 120 100 0

50

100

150 H (kW)

H (kW)

T ' (K)

EB  EA

T ' (K)

T ' (K)

Stream

deficit above pinch. The two cases are distinguished by the remaining compression (portion γ) being done at T 0 (Case A) or T CU (Case B). For Case A, W A ¼ ðmcp Þα ðT comp;PI  T PI Þ þðmcp Þγ ðT comp;0 T 0 Þ, Q CU;A ¼ Q CU;0 , and EA ¼ Q CU;0 ðT 0 =T CU 1Þ þ W A . Similarly, W B ¼ ðmcp Þα ðT comp;PI T PI Þ þ ðmcp Þγ ðT comp;CU  T CU Þ, Q CU;B ¼ Q CU;0 þ ðmcp Þγ ðT comp;CU  T CU Þ, and EB ¼ Q CU;B ðT 0 =T CU  1Þ þ W B . Notice that for Case A, hot utility at T 0 is used for cooling after Ambient Compression, but this does not add to the exergy consumption since the exergy content of hot utility is at ambient conditions is zero. The exergy consumption for the two cases can thus be compared

300 280 260 240 220 200 180 160 140 120 100 0

20

40

60 H (kW)

80

100

120

Fig. 3. GCCs for Example 1: (a) Case O without pressure manipulation, (b) Case A, (c) Case B.

200

250

300

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

Table 2 Stream data for C1 in Example 1. Cases Case A C1 Case B C1_1 C1_2

635

Table 4 Stream data for Example 2.

Ts, K

Tt , K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

164.2

284

2

217.4

3

3

120 271.0

198 284

2 2

156 26

1 3

1 3

Table 3 Performance comparison for Example 1. Cases

O

A

B

Hot utility demand, kW Cold utility demand, kW Pinch temperature, K Compression work, kW Exergy consumption, kW

258 78 200 – –

258 166.4 200 88.4 321.4

112 78 200 146 255.2

Stream

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

H1 C1 C2 Hot utility Cold utility

288 120 198 288 120

124 284 284 288 120

3 2 3 – –

492 328 258 – –

1 – – – –

4 – – – –

¼ ðQ CU;B  Q CU;C ÞðT 0 =T CU  1Þ þ ðW B  W C Þ
 ðmcp Þγ ¼ mcp ðT 0 =T CU Þ ðT comp;CU  T CU Þ  ðT PI  T C Þ mcp ( ) ðnc  1Þ=nc   pr;C1 1 ðnc  1Þ=nc ðnc  1Þ=nc  T PI ðn  1Þ=n  pr;C2 ¼ mcp ðT 0 =T CU Þ T CU pr c c pr;C1 ðn  1Þ=nc

¼

c mcp ðT 0 =T CU Þðpr;C1

ðnc  1Þ=nc pr;C1

 1Þ ðn  1Þ=nc ½T CU pr c  T PI 

ðn  1Þ=n

o

T’ ( C) T’0 QHU,0

T’comp,PI

T’HU, min

T’PI

T’comp,C

T’B T’C

2.2.2. Example 2

QCU,0

Example 2. The stream data is shown in Table 4, indicating that a hot stream (H1) is compressed. Since T comp;CU ¼ T CU ðpt =

T’CU H (kW) Fig. 4. GCCs without pressure manipulation for Theorem 2.

and the stream is split: the first portion is compressed at T C so that the heating demand is satisfied, and the remaining portion is compressed at T CU . The first comparison is performed between Cases B and C.i. Assuming that the compression ratios below and above pinch are pr;C1 and pr;C2 respectively in Case C.i, obviously pr;C1 pr;C2 ¼ pr ¼ pt =ps . Since the heating demand is satisfied by the compression heat above pinch in both cases, i.e., Q HU;0 ¼ mcp ðT comp;C  T PI Þ ¼ ðmcp Þα ðT comp;PI  T PI Þ, the following mcp ratios can be derived as: ðn  1Þ=n

ðn  1Þ=n

c c c c 1 pr;C2 1 ðmcp Þα T comp;C  T PI T PI ½pr;C2 ¼ ¼ ¼ ðn  1Þ=n ðn  1Þ=n c c c c mcp T comp;PI  T PI T PI ½pr 1 pr 1

ðn  1Þ=n

ðn  1Þ=nc

c c  pr;C2 ðmcp Þγ mcp  ðmcp Þα pr c ¼ ¼ ðn  1Þ=n c c mcp mcp pr 1

For Case B, W B ¼ Q HU;0 þ ðmcp Þγ ðT comp;CU  T CU Þ, Q CU;B ¼ Q CU;0 þ ðmcp Þγ ðT comp;CU  T CU Þ, and EB ¼ Q CU;B ðT 0 =T CU  1Þ þ W B . Similarly for Case C, W C ¼ Q HU;0 þ mcp ðT PI  T C Þ, Q CU;C ¼ Q CU;0 þ mcp ðT PI  T C Þ, and EC ¼ Q CU;C ðT 0 =T CU 1Þ þ W C . The exergy consumption can thus be compared in the following way: EB  EC

c c  1 4 0 and T CU pr ðnc  1Þ=nc ¼ T comp;CU o T PI (accoSince pr;C1 rding to the condition in Theorem 2), it is concluded that EB o EC , i. e., Case B consumes less exergy. For Case C.ii, one portion is compressed at T CU . This portion can be removed from the comparison by subtracting an equal portion with compression at T CU in Case B. The previous proof can then be applied and the same conclusion is achieved. There may also be cases where the stream is split into many portions that are compressed at different temperatures below pinch, however, the compression of each portion consumes more exergy compared to the case where Pinch Compression combined with compression at T CU (if necessary) is used for the same portion. The total effect is that Case B has the minimum exergy consumption. Theorem 2 has thus been proven.

ps Þðnc  1Þ=nc ¼ 178:3 K o T 0 , according to Theorem 1, Pinch Compression should be used to reduce exergy consumption. The GCC without pressure manipulation is shown in Fig. 5(a). The pinch temperature is 202/198 K. The following three cases are compared: Case A – Pinch Compression is used for the entire stream (H1); Case B – Pinch Compression is used to satisfy the heating demand and the remaining portion is compressed at T CU ; Case C – Pinch Compression is similarly used as in Case B and the remaining portion is compressed at T 0 . Due to compression at different temperatures, the stream data for C1 changes as shown in Table 5. The design in Case A is straightforward. The GCC is shown in Fig. 5(b). The heating demand has been completely satisfied, however, the original pinch is removed, indicating that a too large portion is compressed at T PI . For Case B, T comp;PI ¼ T PI ðpt =ps Þðnc  1Þ=nc ¼ 300:2 K, the work resulting from Pinch Compression should thus be limited to Q comp;max ¼ 172 kW using the concept of Potential Pinch Points. The maximum portion (α) using Pinch Compression is calculated to be ðmcp Þcomp;PI;max ¼ Q comp;max = ðT 0comp;PI  T 0PI Þ ¼ 172=ð298:2 200Þ ¼ 1:75 kW=K. The remaining portion (γ) is then compressed at T CU þ ΔT min . As can be seen from Fig. 5(c), the original pinch is remained and the heating demand has been satisfied by the compression heat. For Case C, the portion γ is compressed at T 0 ð ¼ T s Þ and then cooled from T comp;0 to T 0 þ ΔT min by hot utility (actually acting as cold utility), and is further cooled to T t by cold utility. The GCC is

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

300 280 260 240 220 200 180 160 140 120 100

T ' (K)

T ' (K)

636

0

50

100

150

300 280 260 240 220 200 180 160 140 120 100 0

200

50

100

300 280 260 240 220 200 180 160 140 120 100

0

150

200

250

60

80

100

H (kW)

T ' (K)

T ' (K)

H (kW)

50

100

H (kW)

150

200

300 280 260 240 220 200 180 160 140 120 100 0

20

40

H (kW)

Fig. 5. GCCs for Example 2: (a) Case O, (b) Case A, (c) Case B, (d) Case C.

Table 5 Stream data for H1 in Example 2. Cases Case A H1_1 H1_2 Case B H1_α1 H1_α2 H1_γ1 H1_γ2 Case C H1_α1 H1_α2 H1_γ1 H1_γ2

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

288 300.2

202 124

3 3

258 528.6

1 4

1 4

288 300.2 288 184.3

202 124 124 124

1.75 1.75 1.25 1.25

150.5 308.4 205 75.4

1 4 1 4

1 4 1 4

288 300.2 428.0 292

202 124 292 124

1.75 1.75 1.25 1.25

150.5 308.4 136 210

1 4 4 4

1 4 4 4

by the effect of ΔT min : in Case B, the portion γ is compressed at T CU þ ΔT min (cooled by the cold utility with a temperature difference of ΔT min for heat transfer) instead of T CU , while in Case C, the portion γ is cooled from T 0 þ ΔT min (cooled by the ambient utility after compression with a temperature difference of ΔT min for heat transfer) instead of T 0 by the cold utility. 2.3. Theorem 3

Theorem 3. For sub-ambient processes, the heat resulting from compression at T CU should be used to reduce the portion with Pinch Compression, if the following conditions are satisfied: (1) Pinch Compression produces more heat than required, and (2) the outlet temperature of compression at T CU is between T PI and T 0 . 2.3.1. Proof of Theorem 3

Table 6 Performance comparison for Example 2. Cases

O

A

B

C

Hot utility demand, kW Cold utility demand, kW Pinch temperature, K Compression work, kW Exergy consumption, kW

172 78 200 – –

0 200.6 – 294.6 575.4

0 153.4 200 247.2 462.0

0 83 – 346.9 463.0

shown in Fig. 5(d). The original pinch point has been removed since the portion γ is cooled to T 0 þ ΔT min instead of T 0 by the hot utility due to a minimum temperature difference of ΔT min for heat transfer. As a result, the cooling demand increases by an amount of ðmcp Þγ ΔT min ¼ 1:25 U 4 ¼ 5 kW, which is exactly equal to the difference in cooling demand for Cases O and C. The performance results are shown in Table 6. Compared to Case A where the entire stream is compressed at T PI following the heuristic rules suggested by Gundersen et al. (2009), the exergy consumption is reduced by 19.7% in Case B where stream splitting is used. As indicated by Theorem 2, Cases B and C have almost the same exergy consumption. The slight difference (1 kW) is caused

Proof of Theorem 3. According to Theorem 2, after the heating demand has been satisfied by Pinch Compression, the compression of the remaining portion should be done at either T CU or T 0 if T comp;CU r T PI . When T comp;CU 4 T PI , heat is introduced to the region above pinch by compressing the remaining portion at T CU . This heat can be used to reduce the portion with Pinch Compression. The reduced portion can then be compressed at T CU , resulting in less compression work and cooling demand. The GCCs are shown in Fig. 6. The solid lines represent the GCC without pressure manipulation. Due to compression of the remaining portion (γ) at T 0CU , the amount of heat introduced to the region above pinch is Q γ . The dashed lines (in green) represent the new GCC above T 0PI after pressure manipulation of the portion γ has been included. The portion of heat covered by Pinch Compression (the portion α) is reduced from Q HU;0 to Q α . A portion (referred to as β) should be compressed at T 0comp;CU if this is a new pinch temperature. The following two cases exist and are discussed: Case (1). If mcp ðT 0comp;CU  T 0PI Þ o Q T 0comp;CU where Q T 0comp;CU is the heating demand at T 0comp;CU , no new pinch is created at T 0comp;CU . The portion β is thus not required since compression above pinch

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

T’ (oC) T’0


 ðmcp Þγ ðmcp Þα þ mcp ðT comp;A  T PI Þ  ðT comp;PI T PI Þ þ ðT PI T comp;CU Þ mcp mcp 8 2 3 " #9 < = ðmcp Þγ 1 1 T PI 1  ðn  1Þ=n ðT 0 =T CU Þ ¼ mcp T PI 41  ðn  1Þ=n 5  c c c c ; : mc p pr;A1 pr;B1  ðmcp Þα ðnc  1Þ=nc ðn  1Þ=nc  1  T PI ½pr c 1 þ mcp T PI ½pr;A2 mcp  ðmcp Þγ ðnc  1Þ=nc T PI ½1  pr;B2  þ mcp

QHU,0

T’comp,PI Qα

Qβ

Qγ

T’HU, min

T’comp,CU QT’comp,CU

T’PI QCU,0

T’CU H (kW) Fig. 6. GCCs for Theorem 3.

should be avoided according to Theorem 1. The portions are determined by the following equations:

α and γ

ðmcp Þα ðT 0comp;PI  T 0PI Þ þ ðmcp Þγ ðT 0comp;CU  T 0PI Þ ¼ Q HU;0 ðmcp Þα þ ðmcp Þγ ¼ mcp Similar to the Proof of Theorem 2, it is necessary to compare the compression scheme proposed above (Case B) with compression below pinch. For Case A, assume that compression starts at T A ðT CU r T A o T PI Þ and the heating demand is exactly satisfied by the compression. The pressure ratios below and above pinch are pr;A1 and pr;A2 respectively, obviously pr;A1 pr;A2 ¼ pr ¼ pt =ps , T PI ¼ ðn  1Þ=nc

c T A pr;A1

ðnc  1Þ=nc

and T comp;A ¼ T A pr

. The work consumption is

W A ¼ mcp ðT comp;A  T A Þ. The cooling demand increases due to compression below pinch, i.e., Q CU;A ¼ Q CU;0 þ mcp ðT PI  T A Þ. For Case B, assuming that the pressure ratios below and above pinch are pr;B1 and pr;B2

respectively for the portion

B2 ¼ pr ¼ pt =ps ,

ðnc  1Þ=nc T PI ¼ T CU pr;B1

γ, obviously pr;B1 pr;

and T comp;CU ¼

ðn  1Þ=nc T CU pr c .

The

work consumption is W B ¼ ðmcp Þα ðT com;PI  T PI Þ þ ðmcp Þγ ðT comp; CU  T CU Þ, and the cooling demand is Q CU;B ¼ Q CU;0 þ ðmcp Þγ ðT PI  T CU Þ. Note that the cooling duty is increased by an amount that is equal to the compression work below pinch. Compression above pinch reduces the heating demand. In both Cases A and B, the heating demand is satisfied by the compression heat, thus Q HU;0 ¼ mcp ðT comp;A  T PI Þ ¼ ðmcp Þα ðT comp;PI  T PI Þ þ ðmcp Þγ ðT comp;CU T PI Þ, i.e., ½ðmcp Þα þ ðmcp Þγ T PI ½pr; ðnc  1Þ=nc

A2ðnc  1Þ=nc  1 ¼ ðmcp Þα T PI ½pr

ðn  1Þ=nc

c  1 þ ðmcp Þγ T PI ½pr;B2

 1.

The following two heat capacity flowrate ratios can then be obtained: ðn  1Þ=n

637

ðn  1Þ=n

c c c c  pr;B2 ðmcp Þα pr;A2 ¼ ðn  1Þ=n ðnc  1Þ=nc c c mcp pr p

r;B2

ðn  1Þ=n

c pr ðnc  1Þ=nc  pr;A2

c ðmcp Þγ ¼ ðn  1Þ=n c c ðn  1Þ=n mcp c p pr c r;B2

By introducing the two heat capacity flowrate ratios, it can be derived that EA  EB ¼ 0, i.e., the two cases (A and B) have the same exergy consumption. The compression below pinch ðT CU r T A o T PI Þ includes other cases where the compression heat is more than required and stream splitting is required: the first portion is compressed at T A and the remaining portion is compressed at T CU . The heating demand is satisfied by the compression of the two portions. The portion with compression at T CU can actually be removed from the comparison by subtracting an equal portion with compression at T CU in Case B. The same conclusion is then obtained. The compression below pinch may also include cases where the stream is split into many portions that are compressed at different temperatures below pinch. The compression of each portion consumes an equal amount of exergy as a corresponding case where Pinch Compression combined with compression at T CU (if necessary) is used for the same portion. The general result is that the two cases (A and B) have the same exergy consumption. Case (2). If mcp ðT 0comp;CU  T 0PI Þ ZQ T 0comp;CU , a new pinch is created at T 0comp;CU due to a large portion of the stream (γ) being compressed at T 0CU . Compression at this new pinch (β) is then necessary. The three portions are determined by the following equations: ðmcp Þα ðT 0comp;CU  T 0PI Þ þ ðmcp Þγ ðT 0comp;CU  T 0PI Þ ¼ Q T '

comp;CU

ðmcp Þα ðT 0comp;PI  T 0PI Þ þ ðmcp Þβ ðT 0comp;T comp;CU  T 0comp;CU Þ þ ðmcp Þγ ðT 0comp;CU  T 0PI Þ ¼ Q HU;0 ðmcp Þα þðmcp Þβ þðmcp Þγ ¼ mcp A procedure similar to the proof for Case (1) can be established to prove that the combination of Pinch Compression and compression at T 0CU has the minimum exergy consumption when the heating demand is Q HU;0  Q β (accounting for the heating effect from compression of portion β). Since a new pinch is created at T 0comp;CU and according to Theorem 1, the heating demand Q β should be satisfied by compression at the new pinch temperature. The proposed scheme is thus preferred with respect to minimum exergy consumption. If Q γ is large enough, compression at T 0PI is not necessary and no feasible solution can be found for the above equations. Since a new pinch is created at T 0comp;CU , then according to Theorem 2, a combination of compression at the new pinch and compression at T CU consumes the minimum amount of exergy. The three portions are determined by the following equations: ðmcp Þα ¼ 0 ðmcp Þβ ðT 0comp;T comp;CU  T 0comp;CU Þ þ Q T '

comp;CU

¼ Q HU;0

The exergy consumption of the two cases is compared in the following way:

ðmcp Þα þðmcp Þβ þðmcp Þγ ¼ mcp

EA  EB

It should be noted that similar to Case (1), there are situations where compression below pinch ðT CU o T A o T PI Þ combined with compression at T CU (if necessary) may achieve the same minimum exergy consumption, and the capital cost could be lower since the number of stream splits may be reduced. The determination of

¼ ðQ CU;A Q CU;B ÞðT 0 =T CU 1Þ þ ðW A  W B Þ
 ðmcp Þγ ¼ mcp ðT PI  T A Þ  ðT PI  T CU Þ ðT 0 =T CU Þ mcp

638

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

Table 7 Stream data for Example 3. Stream

Ts, K

Tt , K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

H1 C1 C2 Hot utility Cold utility

288 150 198 288 150

154 284 284 288 150

5 3 4 – –

670 402 344 – –

– 1 – – –

– 5 – – –

compression inlet temperature is not graphically straightforward for complex cases where new pinches are created. The objective of this paper is to develop a straightforward design methodology with the objective of minimum exergy consumption. The consideration of cost as well as retrofit of HENs is beyond the scope and will be studied in future work. The cases with compression at some intermediate temperature ðT CU o T o T PI Þ below pinch are thus not included in the design procedure, since they do not reduce the exergy consumption that is achieved following the design procedure. 2.3.2. Example 3 Example 3. The stream data is shown in Table 7, indicating that a cold stream (C1) is compressed. Since T comp;CU ¼ T CU ðpt =ps Þðnc  1Þ=nc ¼ 237:6 K o T 0 , the exergy consumption can be reduced by using Pinch Compression. The GCC without pressure manipulation (Case O) is shown in Fig. 7(a). The pinch temperature is 202/198 K. Note that T comp;CU is above pinch. The following three cases are compared: Case A – Pinch Compression is used to satisfy the heating demand and the remaining portion is compressed at T CU ; Case B – the heating demand is satisfied by the heat from Pinch Compression and the compression at T CU (following Theorem 3); Case C – the heating demand is satisfied by compression at T C (T CU o T C o T PI ) combined with compression at T CU (if necessary). For Case A, T comp;PI is calculated to be 313.6 K and T 0comp;PI ¼ 315:6 K. Q comp;max is then determined to be 172 kW using the concept of Potential Pinch Points. The portion (α) using Pinch Compression is thus calculated to be ðmcp Þcomp;PI;max ¼ 1:49 kW=K. The remaining portion (γ) is compressed at T CU . For Case B, T comp;CU ¼ 237:6 K4 T PI , according to Theorem 3, the heat resulting from compression at T CU should thus be used to reduce the portion with Pinch Compression. Since mcp ðT 0comp;CU  T 0PI Þ ¼ 3 U ð239:6  200Þ ¼ 118:8 kW 4 Q T 0comp;CU ¼

79:2 kW, a new

pinch is created at T 0comp;CU . The outlet temperature of compression at the new pinch temperature is calculated to be T comp;T comp;CU ¼ 237:6 U 50:4=1:4 ¼ 376:3 K. The following values can be determined by solving the equations presented in Case (2) in Theorem 3: ðmcp Þα ¼ 0 kW=K, ðmcp Þβ ¼ 0:67 kW=K and ðmcp Þγ ¼ 2:33 kW=K. For Case C, according to Fig. 7(a), T 0HU;min ¼ 286 K. Assuming that T comp;C ¼ T HU;min ¼ 288 K, T C is then determined to be 181.8 K. The compression work above pinch,W PI ¼ mcp ðT comp;C  T PI Þ ¼ 3 U ð288 198Þ kW ¼ 270 kW, is larger than the heating demand

(172 kW). Stream splitting is thus required: one portion (θ) is

compressed at T C ¼ 181:8 K and another portion (γ) is compressed at T CU . Since the heating demand above the new pinch (T 0comp;CU ) is

172 kW 79.2 kW ¼92.8 kW, the compression of the portion θ should satisfy this heating demand, i.e., ðmcp Þ θ ðT comp;C  T comp;CU Þ ¼ ðmcp Þθ ð288  237:6Þ kW ¼ 92:8 kW. The following values can then be determined: ðmcp Þθ ¼ 1:84 kW=K and ðmcp Þγ ¼ 1:16 kW=K. The new stream data for C1 is shown in Table 8 and the GCCs are shown in Fig. 7. The original pinch (200 K) has been removed in

all the three cases since heat is introduced to the region above pinch when a portion (γ) is compressed at T CU . A new pinch is created at T 0comp;CU ¼ 239:6 K and the heating demand is satisfied by the compression heat in both Cases B and C. The performance results are shown in Table 9. Compared to Case A, since the portion using Pinch Compression has been reduced by utilizing the heat from compression at T CU , lower exergy consumption is achieved in Case B. The consumptions of heating, cooling, compression work and exergy are exactly the same for Cases B and C, as indicated in the proof of Theorem 3. The number of stream splits (¼2) is also the same. Since the determination of the inlet temperature for compression below pinch is not as straightforward as for Pinch Compression when the case is complex (see Section 3), Pinch Compression is suggested for a preliminary design with the objective of minimum exergy consumption. 2.4. Theorem 4

Theorem 4. For sub-ambient processes, a HEN design with compression at T CU consumes the smallest amount of exergy if the outlet temperature of compression at T CU is higher than ambient temperature. 2.4.1. Proof of Theorem 4 Proof of Theorem 4. Theorems 1–3 deal with cases where the outlet temperature of compression at T CU is lower than T 0 , while Theorem 4 is concerned with the opposite case, i.e., T comp;CU 4 T 0 . It has been explained that compression above pinch consumes more exergy than Pinch Compression in Theorem 1 since more work is consumed. It is then only necessary to investigate the compression schemes below pinch. The following two cases are compared: Case A (compression at T A , T CU r T A o T PI ) and Case B (compression at T B , T CU r T B oT PI ). For Case A, Q CU;A ¼ Q CU;0 þ xA W A ¼ Q CU;0 þ mcp ðT PI T A Þ where xA is the fraction compressed below pinch, 0 o xA o 1. The definition of x is the same as defined in Theorem 1. The cooling demand is increased due to the compression below pinch. Similarly for Case B, Q CU;B ¼ Q CU;0 þ xB W B ¼ Q CU;0 þ mcp ðT PI T B Þ. The exergy consumption is compared in the following way: EA  EB ¼ ðQ CU;A  Q CU;B ÞðT 0 =T CU  1Þ þ ðW A  W B Þ ¼ mcp ðT B  T A ÞT 0 =T CU  mcp ðT comp;B T comp;A Þ ¼ mcp ðT B  T A ÞðT 0  T CU pr ðnc  1Þ=nc Þ=T CU According to the condition in Theorem 4, T CU pr ðnc  1Þ=nc ¼ T comp;CU 4 T 0 , thus it is concluded that EA o EB when T A o T B , i.e., the exergy consumption is lower when the inlet temperature of compression is lower. Compression at T CU should thus be used. In practice, the pressure ratio can be split (multiple-stage compression) so that T comp;CU o T 0 and Pinch Compression can be applied. 2.4.2. Example 4 Example 4. The stream data is shown in Table 10. Cold stream C1 is compressed and T comp;CU ¼ T CU ðpt =ps Þðnc  1Þ=nc ¼ 308:9 K 4 T 0 , thus according to Theorem 4, the compression should start at T CU . The pinch temperature for Case O without pressure manipulation is 212/208 K. The following two cases are compared: Case A – compression at T PI for the cold streams (208 K); Case B – compression at T CU (160 K). Due to compression at different temperatures, the stream data for C1 changes as shown in Table 11. The performance comparison is shown in Table 12. Compared to Case A where Pinch Compression is used, although more cooling

350

350

300

300

250

250

T ' (K)

T ' (K)

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

200 150

639

200 150 100

100 0

50

100

150

0

200

50

100

400

350

350

300

300

T ' (K)

T ' (K)

150

200

250

150

200

250

H (kW)

H (kW)

250 200

250 200 150

150 100

100

0

50

100

150

200

0

250

50

100

H (kW)

H (kW) Fig. 7. GCCs for Example 3: (a) Case O, (b) Case A, (c) Case B, (d) Case C.

Table 8 Stream data for H1 in Example 3. Cases Case A C1_α1 C1_α2 C1_γ Case B C1_β1 C1_β2 C1_γ Case C H1_θ1 H1_θ2 H1_γ

Table 10 Stream data for Example 4.

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

Stream

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

150 314.0 237.6

198 284 284

1.49 1.49 1.51

71.5 44.1 70.1

1 5 5

1 5 5

H1 C1 C2 Hot utility Cold utility

288 160 208 288 160

164 284 284 288 160

3 2 4 – –

372 248 304 – –

– 1 – – –

– 10 – – –

150 376.3 237.6

237.6 284 284

0.67 0.67 2.33

58.7 61.8 108.1

1 5 5

1 5 5

150 288 237.6

181.8 284 284

1.84 1.84 1.16

58.5 7.36 53.8

1 5 5

1 5 5

Table 11 Stream data for C1 in Example 4. Cases Case A C1_1 C1_2 Case B C1

Table 9 Performance comparison for Example 3. Cases

O

A

B

C

Hot utility demand, kW Cold utility demand, kW Pinch temperature, K Compression work, kW Exergy consumption, kW

172 96 200 – –

0 228.5 – 304.9 515.1

0 221.0 239.6 297.0 500.2

0 221.0 239.6 297.0 500.2

demand is required in Case B, the exergy consumption is less. Compression at T CU should thus be used.

3. Design procedure On the basis of the four theorems presented in Section 2, a procedure for integrating compressors into sub-ambient HENs is proposed in this section.

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

160 401.6

208 284

2 2

96 235.2

1 10

1 10

308.9

284

2

326.6

10

10

3.1. Detailed design procedure The detailed design procedure is shown in Fig. 8. The first step is to calculate T comp;CU and compare it with T 0 . According to Theorems 1 and 4, compression starts at T CU if T comp;CU 4 T 0 and at T PI if T comp;CU r T 0 . The maximum portion using Pinch Compression, ðmcp Þcomp;PI;max , is limited by the heating demand. Using the concept of Potential Pinch Point, ðmcp Þcomp;PI;max can be determined. According to Theorem 1, Pinch Compression is used for the entire stream if its heat capacity is smaller than ðmcp Þcomp;PI;max . Otherwise, stream splitting is used and Pinch Compression is used for the portion ðmcp Þcomp;PI;max . A new GCC is then produced, where the pressure manipulation of the portion with Pinch Compression, i.e., the heating or cooling of the portion from T s to T PI before compression and from T comp;PI to T t after compression are included.

640

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

The pressure manipulation of the remaining portion should not be included. The portion available for compression at new pinch temperatures can then be determined. The procedure is repeated until the entire stream has been compressed or the heating demand has been completely satisfied (i.e., the pinch problem has become a threshold problem, ðmcp Þcomp;PI;max ¼ 0). In the latter case, according to Theorem 2, the remaining portion of the stream is compressed at T 0 or T CU if T comp;CU r T PI . Otherwise, as indicated by Theorem 3, the portion for compression at the original T PI should be reduced and an iterative procedure is required: A new GCC is produced by including pressure manipulation only for the portion of the stream with compression at T CU (the remaining portion of the stream is included in the GCC without pressure manipulation), and the procedure for implementing Pinch Compression is repeated. The procedure stops if the entire stream has been compressed, otherwise the portion for compression at T CU increases until T comp;CU is lower than the new pinch temperature (which means that the pinches below T comp;CU have been removed). When the heating demand has been completely satisfied by

compression at the new pinch point(s) above T comp;CU , the remaining portion of the stream (γ) should be compressed at T 0 or T CU . According to the design procedure, it can be observed that in order to achieve a design with minimum exergy consumption, compression of streams should be implemented at pinch temperatures, T 0 or T CU . In some cases, compression at an intermediate temperature between T PI and T CU may achieve the same minimum value for exergy consumption, however, the design procedure could be complicated since the determination of the intermediate temperatures is not straightforward. 3.2. Example 5

Example 5. The design procedure is illustrated with this example. The stream data is shown in Table 13. A cold stream (C1) is compressed and T comp;CU ¼ T CU ðpt =ps Þðnc  1Þ=nc ¼ 164:2 K o T 0 , Pinch Compression should thus be used to reduce exergy consumption. Table 13 Stream data for Example 5.

Table 12 Performance comparison for Example 4. Cases

O

A

B

Hot utility demand, kW Cold utility demand, kW Pinch temperature, K Compression work, kW Exergy consumption, kW

228 48 210 – –

0 48 210 387.2 425.6

26.2 144 210 297.8 413

Stream

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

H1 H2 C1 C2 Hot utility Cold utility

288 217 120 144 288 120

175 124 284 252 288 120

7 7 4 8 – –

791 651 656 864 – –

– – 1 – – –

– – 3 – – –

Calculate Tcomp,CU

Yes

Theorem 4

T0 < Tcomp ,CU ?

No Draw the GCC without pressure manipulation, determine ( mc p )comp , PI ,max

Compression at TCU

( mc p ) ≤

No

Yes

( mc p ) comp ,PI ,max ? The stream is split. The portion for Pinch Compression is ( mc p ) comp , PI ,max

Theorem 1 Pinch Compression is implemented

The remaining portion is compressed at TCU and added to the total portion for compression at TCU

Draw new GCC without pressure manipulation for the remaining portion, find new ( mc p ) comp , PI ,max

Theorem 3 Yes No

( mc p )comp ,PI ,max

= 0?

Yes

TPI < Tcomp ,CU ?

Theorem 2 No Compression at T0 or TCU

New GCCs are produced and pressure is manipulated only for the portion with compression at TCU, find

(mc p )comp , PI ,max The mcp is reduced by the portion compressed at TCU

Fig. 8. Design procedure for integrating compressors into sub-ambient HENs.

300 280 260 240 220 200 180 160 140 120 100

Qcomp,max =51 kW

0

T ' (K)

T ' (K)

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

T 'comp,PI =199.1 K

50

100

150

200

300 280 260 240 220 200 180 160 140 120 100 0

250

50

100

300 280 260 240 220 200 180 160 140 120 100 50

100

150

200

250

300 280 Qcomp,max,new=99 kW 260 240 220 200 180 160 140 120 100 0 50

T ' (K)

T ' (K)

300 280 260 240 220 200 180 160 140 120 100

50

100

150

200

250

T 'comp,PI,new =293.5 K

100

150

200

250

300 280 260 240 220 200 180 160 140 120 100

Qcomp,max =15 kW

T 'comp,PI =199.1 K

0

50

100

150

200

250

H (kW)

H (kW)

T ' (K)

200

H (kW)

H (kW)

0

150 H (kW)

T ' (K)

T ' (K)

H (kW)

0

641

300 280 260 240 220 200 180 160 140 120 100 0

50

100 H (kW)

150

200

Fig. 9. GCCs for Example 5: (a) Case O, (b) Case A, (c) Case B, (d) Case C.i, (e) Case C.ii, (f) Case D.i, (g) Case D.

The GCC without pressure manipulation (Case O) is shown in Fig. 9 (a). The pinch temperature is 144/148 K, which is lower than T comp;CU . The following cases are compared: Case A – the entire stream is compressed at T CU ; Case B – Pinch Compression is used for the entire stream; Case C – stream splitting is used: compression at the original and new pinches are used and the heating demand is satisfied by the compression heat, while the remaining portion is compressed at T CU (the corresponding compression heat is not utilized); Case D – the proposed design procedure is used, i.e., the difference compared to Case C is that the heat resulting from compression at T CU is utilized to reduce the portion with Pinch Compression.

For Cases A and B, the design procedure is straightforward and not presented in detail. The new stream data for H1 is shown in Table 14 and the GCCs are shown in Fig. 9(b), and (c). The original pinch (144 K) has been removed and a new pinch (215 K) is created in both cases. The reason is that the amount of heat supplied to the region above pinch exceeds the limit, Q comp;max ¼ 51 kW according to Fig. 9(a). For Case C, T comp;PI is calculated to be 197.1 K, and ðmcp Þcomp;PI;max ¼ Q comp;max =ðT 0comp;PI T 0PI Þ ¼ 0:96 kW=K. A subcase (C.i) is studied, where stream C1 is split into two portions: a portion (α ¼0.96 kW/K) is compressed at T PI and the remaining portion (O¼3.04 kW/K) is not compressed. The new GCC is shown

642

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

Table 14 Stream data for H1 in Example 5. Cases Case A C1 Case B C1_1 C1_2 Case C.i C1_α1 C1_α2 C1_O Case C.ii C1_α1 C1_α2 C1_β1 C1_β2 C1_γ Case D.i C1_γ C1_O Case D.ii C1_γ C1_O Case D C1_β1 C1_β2 C1_γ

compressing the portion

Ts, K

Tt, K

mcp , kW/K

ΔH, kW

ps , bar

pt , bar

164.2

284

4

479.2

1

3

120 197.1

144 284

4 4

96 347.6

1 3

1 3

120 197.1 120

144 284 284

0.96 0.96 3.04

23.0 83.4 498.6

1 3 1

1 3 1

120 197.1 120 291.5 164.2

144 284 213 284 284

0.96 0.96 1.26 1.26 1.78

23.0 83.4 117.2 9.5 213.2

1 3 1 3 3

1 3 1 3 3

164.2 120

284 284

1.78 2.22

213.2 364.1

3 1

3 1

164.2 120

284 284

2.46 1.54

213.2 364.1

3 1

3 1

120 291.5 164.2

213 284 284

1.26 1.26 2.74

117.2 9.5 328.3

1 3 3

1 3 3

Table 15 Performance comparison for Example 5. Cases

O

A

B

C ( ¼C.ii)

D

Hot utility demand, kW Cold utility demand, kW Pinch temperature, K Compression work, kW Exergy consumption, kW

150 72 146 – –

99 197.8 215 176.8 453.7

99 233.4 215 212.4 539.2

0 150.7 215 228.7 439.7

0 142.1 215 220.0 418.9

in Fig. 9(d). A new pinch ðT 0PI;new Þ is created at 215 K and T comp;PI;new ¼ 291:5 K. Based on the concept of Potential Pinch Point, ðmcp Þcomp;PI;max;new is determined to be 1.26 kW/K. A second subcase (C.ii) can then be studied, where stream C1 is split into three portions: the portion portion

α ¼ 0.96 kW/K is compressed at T PI , the

β ¼1.26 kW/K is compressed at T PI;new , and the remaining

portion γ ¼ 1.78 kW/K is compressed at T CU . The corresponding GCC is shown in Fig. 9(e). The original pinch has been removed due to the heat introduced to the region above pinch by compressing the portion γ at T CU . For Case D, based on the results of Case C, a first subcase (D.i) is studied, where stream C1 is split into two portions: a portion (γ ¼1.78 kW/K) is compressed at T CU and the remaining portion (O¼ 2.22 kW/K) is not compressed. The GCC is shown in Fig. 9(f). The following values for Pinch Compression (portion α) can be determined: Q comp;max ¼ 15 kW and ðmcp Þcomp;PI;max ¼ 0:28 kW=K. According to the study in Case C, a portion β ¼ 1.26 kW/K is required to be compressed at the new pinch T PI;new ¼ 213 K to satisfy the heating demand. Since

α þ β ¼1.54 kW/K, the portion to

be compressed at T CU is γ ¼2.46 kW/K. A second subcase (D.ii) is then studied in a similar way as for Case D.i: the portion γ ¼2.46 kW/K is compressed at T CU and the remaining portion O¼1.54 kW/K is not compressed. The procedure is repeated until that the original pinch has been removed andT comp;CU o T PI;new . The

remaining portion γ is then compressed at T CU after the heating demand above T PI;new has been completely satisfied by

β at T PI;new . The following values can

finally be determined for Case D: β ¼1.26 kW/K and γ ¼2.74 kW/K. The GCC is shown in Fig. 9(g). The performance comparison is shown in Table 15. Compared to Case A where compression is done at T CU , Pinch Compression for the entire stream increases the exergy consumption by 18.8%. However, the exergy consumption is reduced when Pinch Compression is used in combination with compression at T CU , and is further reduced when the design procedure shown in Fig. 8 is used. The HEN designs including pressure manipulation for Cases A (compression at T CU ) and D (the design procedure proposed) are shown in Fig. 10. Case D requires 1 more compressor, however, the exergy consumption is reduced by 7.7%. Multi-stream heat exchangers can of course be applied to substitute the individual two-fluid heat exchangers in both cases for sub-ambient processes. 4. Conclusions The integration of heat and work is investigated in this study. A systematic graphical design methodology is developed for integrating compressors into heat exchanger networks below ambient temperature. The objective is to minimize exergy consumption. The design methodology is based on four theorems that are proposed, proven and illustrated by simple examples. The theorems can be simplified by the following three statements: (1) Pinch Compression is used if the outlet temperature of compression at cold utility temperature is lower than ambient temperature (Theorem 1), otherwise compression should be done at cold utility temperature (Theorem 4); (2) after the heating demand has been completely satisfied by Pinch Compression, the compression of the remaining portion should be done at either cold utility temperature or ambient temperature if the outlet temperature of compression at cold utility temperature is lower than the pinch temperature (Theorem 2); otherwise (3) the heating resulting from compression at cold utility temperature should be utilized to reduce the portion with Pinch Compression (Theorem 3). It can be concluded that in order to achieve minimum exergy consumption, compression should be done at pinch temperatures, ambient temperature or cold utility temperature. Compression at other temperatures does not reduce the exergy consumption. The Grand Composite Curve has been used directly without paying attention to the identity of the stream to be compressed (hot or cold) and without considering the location of supply and target temperatures. This makes the design procedure considerably less complex, but unfortunately it introduces a small underestimation of utility consumption in some cases related to the need for driving forces. This small error can be quantified to ymcp ΔT min , where 0 r y r 1.

Nomenclature |Roman letters cp cv E H m nc p

specific heat capacity at constant pressure, kJ/(kg K) specific heat capacity at constant volume, kJ/(kg K) exergy, kW enthalpy, kW mass flowrate, kg/s polytropic index for compression pressure, bar

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

643

Fig. 10. HENs for Example 5: (a) Case A, (b) Case D.

Q T T0 ΔT min W x y

heat, kW temperature, K modified temperature, K minimum temperature difference, K work, kW fraction fraction

Greek letters

α β γ θ η1;comp κ

portion of stream portion of stream portion of stream portion of stream compressor polytropic efficiency specific heat ratio

Subscripts comp CU HU

compression cold utility hot utility

max

maximum

min PI s t 0

minimum pinch supply target ambient; original

Abbreviations CC EGCC ExPAnD GCC HEN MINLP

composite curve exergy grand composite curve extended pinch analysis and design grand composite curve heat exchanger network mixed integer nonlinear programming

Appendix. Errors in cold utility estimation When Pinch Compression is implemented, the maximum portion with Pinch Compression is constrained using the concept of Potential Pinch Points. The cold utility demand increases if a larger portion is compressed at T PI . The identity of streams (hot/cold) to be compressed and the location of stream temperatures (supply/

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target) were not considered when using the Grand Composite Curve (GCC) in the design procedure. This Appendix investigates the corresponding errors. With 4 different temperatures, T s , T t , T PI and T comp;PI , the total number of sequences of these temperatures is ð4!Þ ¼ 24. Since T PI o T comp;PI , the number of cases is reduced to 12. In addition, the relative position of T s and T t depends on the stream type. Thus there are 6 unique sequences for hot streams and the same number for cold streams. All these cases are shown in Table A1, from which the following 4 cases are studied in detail illustrating hot and cold streams with supply temperature above or below pinch: C.1 (a cold

Table A1 Errors in cooling demand estimation. Cases

Sub-cases

Underestimation

C.1: T s o T t o T PI o T comp;PI

(i): T comp;PI Z T PI þ ΔT min (ii): T comp;PI o T PI þ ΔT min (i): T comp;PI Z T PI þ ΔT min (ii): T comp;PI o T PI þ ΔT min – (i): T s Z T PI þ ΔT min (ii): T s o T PI þ ΔT min (i): T s Z T PI þ ΔT min (ii): T s o T PI þ ΔT min (i): T s Z T PI þ ΔT min (ii): T s o T PI þ ΔT min (i): T s r T PI  ΔT min (ii): T s 4T PI  ΔT min – – – – T t Z T comp;PI þ ΔT min T t o T comp;PI þ ΔT min

mcp ΔT min ymcp ΔT min , mcp ΔT min ymcp ΔT min , 0 mcp ΔT min ymcp ΔT min , mcp ΔT min ymcp ΔT min , mcp ΔT min ymcp ΔT min , mcp ΔT min ymcp ΔT min , 0 0 0 0 mcp ΔT min ymcp ΔT min ,

C.2: T s o T PI r T t o T comp;PI C.3: T s o T PI o T comp;PI r T t C.4: T PI r T s o T t o T comp;PI C.5: T PI r T s o T comp;PI r T t C.6: T PI o T comp;PI r T s o T t H.1: T t o T s o T PI o T comp;PI H.2: H.3: H.4: H.5: H.6:

T t o T PI r T s o T comp;PI T t o T PI o T comp;PI r T s T PI r T t o T s o T comp;PI T PI r T t o T comp;PI o T s T PI o T comp;PI r T t o T s

0oyo1 0oyo1

0ryo1 0ryo1 0oyo1 0oyo1

0ryo1

stream is compressed and T s o T PI ), C.4 (a cold stream is compressed and T s Z T PI ), H.1 (a hot stream is compressed and T s o T PI ) and H.2 (a hot stream is compressed and T s Z T PI ). Case C.1: The stream is compressed after being heated from T s to T PI , and then cooled from T comp;PI to T t . The heating from T s to T t before Pinch Compression is included when using the GCC (without pressure manipulation) to determine Q CU;0 . If T comp;PI Z T PI þ ΔT min (Sub-case C.1.i), as shown in Fig. A1(a), the cooling from T comp;PI to T PI þ ΔT min after compression is satisfied by the heat deficit above pinch that limits the maximum portion using Pinch Compression, ðmcp Þcomp;PI;max . Note that this heat deficit cannot be used to cool cold streams to T PI . The cooling of the stream from T PI þ ΔT min to T t þ ΔT min is satisfied by the heating of the stream from T t to T PI before compression (recuperative heating). Finally, the cooling of the stream from T t þ ΔT min to T t should be satisfied by cold utility, i.e., the cooling demand is underestimated by mcp ΔT min when the GCC is used to determine ðmcp Þcomp;PI;max . If T comp;PI o T PI þ ΔT min (Sub-case C.1.ii), the cooling from T comp;PI to T t þ ΔT min is satisfied by recuperative heating when T t þ ΔT min r T comp;PI , and the cooling from T t þ ΔT min to T t increases the cooling demand by mcp ΔT min , however, the heating of the stream from T comp;PI  ΔT min to T PI before compression can be used to cool other streams and thus reduce the cooling demand. As a result, the cold utility is underestimated by mcp ΔT min  mcp ½T PI  ðT comp;PI  ΔT min Þ ¼ mcp ðT comp;PI  T PI Þ, which can be written as ymcp ΔT min where 0 o yo 1. When T t þ ΔT min 4T comp;PI , recuperative heating cannot be used, and the cooling of the stream from T comp;PI to T t increases the cooling demand by ymcp ΔT min where 0 oy o 1. Case C.4: The stream is compressed after being cooled from T s to T PI , and then cooled from T comp;PI to T t after compression. By constraining ðmcp Þcomp;PI;max , the heat deficit above pinch can cool a stream with mcp from T comp;PI þ ΔT min to T PI þ ΔT min . If T s Z T PI þ ΔT min (Sub-case C.4.i), as shown in Fig. A1(b), this heat deficit (above pinch) covers three parts: (1) the cooling of the stream from

Fig. A1. Illustration of heat balance in various Cases: (a) C.1.i, (b) C.4.i, (c) H.1.i, (d) H.2.

C. Fu, T. Gundersen / Chemical Engineering Science 137 (2015) 631–645

T s to T PI þ ΔT min before compression, (2) the cooling resulting from heating the stream from T s to T t when using the GCC (without pressure manipulation) to determine Q CU;0 , and (3) the cooling from T comp;PI to T t after compression. However, the cooling of the stream from T PI þ ΔT min to T PI after compression must be done by cold utility, thus the cooling demand increases by mcp ΔT min . If T s o T PI þ ΔT min (Sub-case C.4.ii), the cooling of the stream from T s to T PI increases the cooling demand by ymcp ΔT min where 0 r y o1. Case H.1: The stream is compressed after being heated from T s to T PI , and then cooled from T comp;PI to T t after compression. By constraining ðmcp Þcomp;PI;max , the heat deficit above pinch can cool the hot stream from T comp;PI to T PI after compression. If T s r T PI  ΔT min (Sub-case H.1.i), as shown in Fig. A1(c), the cooling from T PI to T s þ ΔT min is satisfied by heating of the stream from T s to T PI  ΔT min before compression (recuperative heating). The cooling from T s to T t is part of the original GCC used to determine Q CU;0 . The cooling from T s þ ΔT min to T s increases the cooling demand by mcp ΔT min . If T s 4 T PI  ΔT min (Sub-case H.1.ii), the cooling from T PI to T s is satisfied by the cold utility, and the cooling demand is underestimated by ymcp ΔT min where 0 o y o 1. Case H.2: The stream is compressed after being cooled from T s to T PI , and then cooled from T comp;PI to T t after compression. By constraining ðmcp Þcomp;PI;max , the heat deficit above pinch can cool the hot stream from T comp;PI to T PI after compression. The cooling from T s to T PI before compression and from T PI to T t after compression is exactly equal to the cooling demand for the cooling from T s to T t when using the GCC to determine Q CU;0 . The cold utility is thus not underestimated when Pinch Compression is implemented. Similarly, all the other cases can be studied and the results are shown in Table A1. In summary, the cooling demand is

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underestimated by ymcp ΔT min where 0 r yr 1, when Pinch Compression is implemented while the identity of the stream to be compressed and the locations of T s and T t in the GCC are not considered.

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