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A Novel Sensible Heat Pump Scheme for Industrial Heat Recovery Chao Fu* and Truls Gundersen Department of Energy and Process Engineering, Norwegian University of Science and Technology, Kolbjoern Hejes vei 1.A, Trondheim NO-7491, Norway ABSTRACT: Most industrial heat pumps are based on the reversed Rankine cycle. According to Pinch Analysis, heat pumps should be placed across pinch. A “sharp” pinch with corresponding low temperature lift is required in order to achieve good performance. In addition, as will be demonstrated in this paper, the compression and expansion are not properly placed in the reversed Rankine cycle. A novel sensible heat pump scheme for industrial applications based on the reversed Brayton cycle is developed in this paper. Both compression and expansion are correctly placed above and below pinch, respectively. A systematic methodology for design of the heat pump will be presented. Operating parameters such as inlet temperatures of compression and expansion, heat capacity flow rate of the working fluid, and the pressure ratio for compression are investigated based on thermodynamic and mathematical analyses. Three examples are used to illustrate the design methodology and energy saving potential for using the novel sensible heat pump.

1. INTRODUCTION Low temperature heat can be upgraded to useful (high temperature) heat by heat pumps. Many types of heat pumps have been developed for industrial applications, as shown in Figure 1. Heat pumps can be classified into closed and open types.1,2 The working fluid in closed heat pumps is different from the process fluids to be heated or cooled, while a process fluid is used as the working fluid in open heat pumps. Heat pumps can also be classified into the reversed Rankine and Brayton cycles3 base on the thermodynamic cycles used. In the reversed Rankine cycle, the working fluid has phase change (between vapor and liquid) and thus mainly latent heat is exchanged in the evaporator and condenser. The reversed Brayton cycle heat pump operates in vapor phase without phase change, and only sensible heat is exchanged. Since the turbomachinery losses are larger, the reversed Brayton cycle has worse performance and is less used.3 By choosing optimum operating pressure ranges for real gases, the reversed Brayton cycle could be competitive, particularly for large scale high temperature nonisothermal heat generation.3 When heat pumps are used for industrial heat recovery, an important design guideline has been derived from Pinch Analysis (PA), which has been a well-established technology for heat exchanger network (HEN) design since the 1970s.4,5 For a given set of process streams (supply and target temperatures and heat capacity flow rate), the hot and cold streams can be grouped into hot and cold composite curves in a temperature (T)−enthalpy (H) diagram. The points where minimum temperature difference (ΔTmin) for heat transfer occur are called Pinch temperatures. Heat should be exchanged separately in the region above and below the pinch in order to minimize energy consumption. Any heat transfer across pinch increases both hot and cold utility consumption. For the reversed Rankine cycle heat pump, heat is added to the evaporator at a lower temperature and removed from the condenser at a higher temperature. It is thus important to place the evaporator below pinch and the condenser above pinch (i.e., the heat pump should be placed across pinch). This rule is © 2016 American Chemical Society

referred to as appropriate placement (or correct integration) of heat pumps.6 Coefficient of performance (COP), defined as the ratio between the heat delivered and work consumed, is an important energy performance indicator for heat pumps. The COP is larger when the difference between the average temperatures for heat addition and removal is smaller. The Grand Composite Curve (GCC) provides useful insights for heat pump design.6 Modified temperatures (T′) are used in GCCs, which means that T′ = T + 0.5ΔTmin for cold streams and T′ = T − 0.5ΔTmin for hot streams. In order to achieve a large COP, the reversed Rankine cycle heat pump should only be used when there is a “sharp” pinch in the GCC,7 as shown in Figure 2, where T0′ and TPI ′ are modified ambient and pinch temperatures respectively, and QHU,0 and QCU,0 are the original heating and cooling demands without a heat pump being used. Although Pinch Analysis provides useful insights for heat pump design, it is a considerable challenge to identify the optimal operating conditions such as pressures and heat loads of the condenser and evaporator. Mathematical optimization is normally required to develop cost-effective heat pumps.8−10 The placement of compressors and expanders has been discussed recently by Gundersen and co-workers.11,12 Two heuristic rules have been proposed based on thermodynamic observations:11 (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 have been stated more specifically in such a way that both compression and expansion should start at pinch temperatures.12 These studies provide useful insights and guidelines for mathematical optimization studies on HENs, including compressors and expanders.13,14 More recently, a set of theorems for the integration of compressors and expanders have been proposed based on thermodynamic and mathematReceived: Revised: Accepted: Published: 967

July 3, 2015 December 22, 2015 January 14, 2016 January 14, 2016 DOI: 10.1021/acs.iecr.5b02417 Ind. Eng. Chem. Res. 2016, 55, 967−977

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Industrial & Engineering Chemistry Research

Figure 1. Types of heat pumps: (a) closed reversed Rankine cycle; (b) open reversed Rankine cycle; (c) closed reversed Brayton cycle; and (d) open reversed Brayton cycle.

2. SENSIBLE HEAT PUMP This section presents a sensible heat pump scheme for industrial applications. The following operating parameters should be determined for the scheme: inlet temperatures for compression and expansion, pressure ratio for compression, and the heat capacity flow rate of the working fluid. This section mainly studies the optimal inlet temperatures for compression and expansion. Other parameters will be investigated in section 3. 2.1. Process Description. In order to minimize exergy consumption, both compression and expansion should start at pinch temperatures.12,15 On the basis of the reversed Brayton cycle, a sensible heat pump scheme is developed and shown in Figure 3. The working fluid (in gas phase) is compressed after

Figure 2. GCC with a “sharp” pinch for applying reversed Rankine cycle heat pumps.

ical analyses.15 Compression/expansion at pinch temperatures has been proven to be a preferred scheme with respect to minimum exergy consumption. All these studies are performed by realizing the importance of heat and work integration (e.g., efficient recovery of compression heat16,17 and expansion work)18 and energy savings in low temperature processes such as cryogenic air separation2 and liquefied natural gas.11 When reversed Rankine cycles are used in industry, according to the guidelines for the placement of heat pumps,6 the evaporator and condenser should be placed below and above pinch, respectively. However, direct compression of the working fluid at the evaporation temperature obviously violates the rule12 that compression should start at the pinch temperature. A large portion of the compression heat has been added to the below pinch region and not been utilized. In addition, the Joule−Thomson expansion starts above pinch temperature and no work is recovered from this expansion. In order to recover the compression heat and expansion work efficiently, a sensible heat pump scheme based on the reversed Brayton cycle is developed in this paper. The optimal operating parameters for this heat pump scheme are investigated.

Figure 3. A sensible heat pump scheme for industrial applications.

being heated from the expander outlet temperature (Texp,PI) to the pinch temperature (TPI) and is then cooled from the compressor outlet temperature (Tcomp,PI) to TPI before expansion. The cooling process after compression provides heat and thus reduces heating demand for other process streams (the region above pinch), and the heating from Texp,PI to TPI is satisfied by low-temperature heat in the region below pinch. The cooling demand is thus also reduced. A qualitative comparison with the reversed Rankine cycle heat pump can be performed at this stage. The sensible heat pump has the following advantages. (1) The compressor and expander are correctly integrated. All the compression heat can be utilized above pinch, and no cooling is introduced to the above pinch region by the expansion. In the reversed Rankine cycle, the compression starts at evaporation temperature (below pinch) and thus a portion of the compression heat is not utilized. In addition, the expansion starts at condensation temperature (above pinch) and cooling is thus introduced to the region above pinch (i.e., a portion of the heat available from the working fluid is not utilized). (2) Sensible heat is exchanged with process streams. In many cases where the total heat 968

DOI: 10.1021/acs.iecr.5b02417 Ind. Eng. Chem. Res. 2016, 55, 967−977

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Industrial & Engineering Chemistry Research capacity flow rate for the Grand Composite Curve below and above pinch is small (without phase change), the GCC does not show a “sharp” pinch. When a reversed Rankine cycle with pure working fluid is used, the temperature is constant during evaporation and condensation. The large temperature difference for heat transfer between the working fluid and the process streams causes considerable irreversibilities. There are three solutions proposed in literature: (i) the working fluid is operated at supercritical state so that the temperature is not constant during heat transfer,19 (ii) a mixture of working fluids is used20 and again the temperature varies during heat transfer, and (iii) a series of heat pumps being operated at different constant temperatures are used10,21 so that the irreversibilities related to heat transfer are reduced. All the three alternatives increase process complexity. When the sensible heat pump is used, the irreversibilities related to heat transfer are inherently smaller if the working fluid and the process streams have similar heat capacity flow rates. The sensible heat pump has the following challenges. (1) The size of equipment (particularly heat exchangers) increases since sensible heat is exchanged, and (2) the inefficiency of compressors and expanders may cause large irreversibilities. However, the sensible heat pump may be more competitive for the cases where the operating temperature range is large and there is no “sharp” pinch on the GCC. It should be noted that the compression can also start at the pinch temperature when a reversed Rankine cycle is used. This can be achieved by preheating the working fluid to the pinch temperature using a recuperative heating process (sensible heat pumps with recuperative heating will be illustrated in example 2). However, a large portion of the heat is delivered in sensible form and a large desuperheater is required. The difference between the reversed Rankine cycle and the sensible heat pump is small in this case. 2.2. Two Test Examples. This section presents two examples (with and without recuperative heating, respectively) to illustrate the application of the sensible heat pump. Example 1. The stream data is shown in Table 1, where Ts and Tt are the supply and target temperatures respectively, mcp

Figure 4. GCC for Example 1.

For Case A, the heat delivered (condensation heat) is QH = Qcond = 600 kW at temperature Tcond = 403.2 K. The evaporation temperature Tevap should be as high as possible in order to have a larger COP; however, the available heat at Tevap is limited by the heat surplus below pinch. The perfect match is that the evaporation heat QC = Qevap is exactly equal to the heat surplus at T′evap (= Tevap + 10 K). Assuming reversible operation and using the Carnot equations for determining COP, the following results can be obtained in this simple case: Tevap ′ = 348.1 K and Qevap = 503 kW. The net work consumption is thus W = 600−503 = 97 kW. For case B, the mcp of the working fluid should not be larger than the mcp of the GCC above and below pinch [i.e., min (30, 20) = 20 kW/K]. In order to reduce the temperature difference for heat transfer and thus irreversibilities, the mcp of the working fluid is taken as its maximum value, (i.e., 20 kW/K). Since the heating demand (QH = 600 kW) should be completely satisfied by the heat pump, the compressor outlet temperature is determined to be Tcomp,PI = 383.2 + 600/20 = 413.2 K. The compressor inlet temperature is equal to TPI = 363.2 K for cold streams since this is the highest temperature the working fluid can reach when utilizing the surplus heat below pinch. The pressure ratio for compression (pr) is thus determined to be 1.57 according to eq 1 where (nc − 1)/nc = (κ − 1)/(κη∞,comp). The outlet temperature of expansion is determined to be 336.8 K according to Equation 2, where (ne − 1)/ne = η∞,exp(κ − 1)/κ.

Table 1. Stream Data for Example 1 stream

Ts, K

Tt, K

mcp, kW/K

ΔH, kW

H1 C1 Hot utility Cold utility

383.2 363.2 403.2 288.2

343.2 383.2 403.2 288.2

20 30 − −

800 600 − −

Tcomp,PI = TPIpr(nc − 1)/ nc

(1)

Texp,PI = TPI(1/pr )(ne − 1)/ ne

(2)

For case C, in order to reduce the irreversibilities caused by heat transfer, an open heat pump is used by directly compressing process stream H1. Similar to case B, the compressor outlet temperature is determined to be 413.2 K, and pr is determined to be 1.30 according to eq 1. Note that here TPI = 383.2 K for hot stream H1. Texp,PI can then be determined to be 355.4 K. The HEN designs are shown in Figure 5. The representations of the heat pumps (heating QH and cooling QC from the heat pumps) are included in Figure 4. Note that (1) all the curves actually intersect at the pinch temperature (TPI ′ = 373.2 K); (2) the curves for case B and case C in the region above pinch are on top of each other, and the curves for the GCC, case B and case C in the region below pinch are also on top of each other. However, the drawings show slight differences for the curves in order to distinguish them. For case A, the heat

is the heat capacity flow rate, and ΔH is the enthalpy change of a stream. The following assumptions are made for all examples in this paper: (1) polytropic efficiency for compressors and expanders, η∞,comp = η∞,exp = 1, (2) ΔTmin = 20 K, (3) T0 = 288 K, and (4) the fluid to be compressed or expanded is ideal gas with constant specific heat ratio κ = 1.4. The GCC is shown in Figure 4. The pinch temperature is 383.2/363.2 K, and the minimum hot and cold utility requirements are QHU,0 = 600 kW and QCU,0 = 800 kW, respectively. The heating demand can be satisfied by the hot utility (case O) or using a heat pump. The following three cases are studied: case A, a closed reversed Rankine cycle is used; case B, a closed sensible heat pump is used; and case C, an open sensible heat pump is used. In all the three cases, the heating demand is assumed to be completely satisfied by using the heat pumps. 969


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Figure 5. HEN designs for example 1: (a) case O; (b) case A; (c) case B; and (d) case C.

working fluid and the process streams. The exergy consumption for case C is reduced by 74.3% compared to case O where the heating demand is completely satisfied by hot utility. The COP for case C is much higher than case A. However, the investment cost in case C should also be higher, since the load (and size) of the compressor is larger and an expander is required. Detailed cost analysis is beyond the scope of this paper.

pump receives heat QC,A = 503 kW from stream H1 at a constant temperature of 338.1 K (T′ = 348.1 K), and it delivers heat QH,A = 600 kW to stream C1 at a constant temperature of 403.2 K (T′ = 393.2 K). For case B, the closed heat pump receives heat QC,B = 528 kW from stream H1 and delivers heat QH,B = 600 kW to stream C1. For case C, the open heat pump (stream H1 is compressed) delivers heat QH,C = 600 kW to stream C1 and cooling (stream H1 is expanded) QC,C = 528 kW to stream H1. In both cases B and C, the heat is transferred at varying temperatures since the fluids of heat pumps do not have phase changes. Obviously the temperature differences for heat transfer between the working fluid of sensible heat pumps (cases B and C) and process streams are smaller compared to case A. The performance comparison is presented in Table 2. For case O, the heating and cooling demands are satisfied by

E = Q (1 − T0/T )

Example 2. This example illustrates the application of sensible heat pumps with recuperative heating. The stream data is shown in Table 3, and the corresponding GCC is shown in Table 3. Stream Data for Example 2

Table 2. Performance Comparison for Example 1 cases

O

A

B

C

hot utility demand, kW cold utility demand, kW compression work, kW expansion work, kW net work consumption, kW exergy consumption, kW COP

600 800 − − − 171.1 −

0 297 97 − 97 97 6.19

0 272 1000 928 72 72 8.33

0 244 600 556 44 44 13.64

(3)

stream

Ts, K

Tt, K

mcp, kW/K

ΔH, kW

H1 C1 hot utility cold utility

403.2 403.2 473.2 288.2

373.2 453.2 473.2 288.2

15 10 − −

450 500 − −

Figure 6. There are two pinch points at 423.2/403.2 K and 403.2/383.2 K, respectively. For case O without a heat pump being used, the HEN design is shown in Figure 7a. Similar to example 1, the heating demand can be completely satisfied by using heat pumps and the following three cases are studied: case A, a closed reversed Rankine cycle is used; case B, a closed sensible heat pump is used; and case C, an open sensible heat pump is used. For case A, the heat delivered (condensation heat) is Qcond = 500 kW at Tcond = 473.2 K. Similar to the study of case A in example 1, the following results can be obtained for the evaporation process: T′evap = 368.0 K and Qevap = 378 kW. The HEN design is shown in Figure 7b. For case B, the mcp of the working fluid is taken as the smaller one of the mcp of the GCC above and below pinch [i.e.,

utilities. The exergy content (E) for a given amount of heat (Q) at temperature (T) is calculated by eq 3. The exergy consumption is reduced by 43.3% in case A where the reversed Rankine cycle heat pump is used. Further reduction is achieved in case B where the sensible heat pump is used. The reason is that the compressor and expander are appropriately placed in case B. The net work consumption for case C is even lower due to reduced irreversibilities for heat transfer between the 970


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portions: one portion (10 kW/K) is operated according to the heat pump cycle and another portion is cooled to target temperature. The design of the open heat pump cycle is straightforward (similar to the design in case B) and is thus not presented. The HEN design is shown in Figure 7d. Similar to example 1, in order to complement the HEN designs, the representations of the heat pumps are included in the GCC as shown by Figure 6 and are not explained in detail. The performance comparison is shown in Table 4. Compared to case O where the utilities are used, the exergy Table 4. Performance Comparison for Example 2 Figure 6. GCC for example 2.

min (15, 10) = 10 kW/K]. The heating demand (500 kW) is completely satisfied by the heat pump. The compressor outlet temperature is thus determined to be Tcomp,PI = 423.2 + 500/10 = 473.2 K. The compressor should start at the higher pinch (423.2/403.2 K), since heat is added by compression. Similarly, the expander should start at the lower pinch (403.2/383.2 K) since heat is removed by expansion. The compressor inlet temperature is thus equal to TPI = 403.2 K for the cold stream. This temperature can not be achieved by the surplus heat below pinch, instead a recuperative heating process is required [i.e., the cooling of the working fluid to the lower pinch (403.2 K for hot stream)] before expansion can be used for such heating process. The pressure ratio for compression is thus determined to be pr = 1.75. The outlet temperature of expansion at the lower pinch (403.2 K for hot stream) is determined to be Texp,PI = 343.6 K. The corresponding HEN design is shown in Figure 7c. Note that a recuperative heating process between the two pinch points is used in order to place the compressor and expander at the correct pinch temperatures. Also note that the working fluid in the heat pump changes its identity (as hot or cold) [i.e., it is used as a cold stream before compression (after expansion) and a hot stream after compression (before expansion)]. For case C, process stream H1 is used as the working fluid for the heat pump. Since the mcp of the heat pump is limited to 10 kW/K (according to case B), stream H1 is split into two

cases

O

A

B

C

hot utility demand, kW cold utility demand, kW compression work, kW expansion work, kW net work consumption, kW exergy consumption, kW COP

500 450 − − − 195.5 −

0 72 122 − 122 122 4.10

0 54 700 596 104 104 4.81

0 24 700 626 74 74 6.76

consumption is reduced by 37.6% (case A) and 46.8% (case B), respectively, when the closed reversed Rankine cycle and the closed sensible heat pump are used. The largest COP is again achieved in case C where an open sensible heat pump is used. The work (exergy) consumption is reduced by 39.3% compared to case A. Due to the existence of two pinch points, a recuperative heating process has been used in the sensible heat pump in order to ensure that the compressor and expander start at correct pinch temperatures. As discussed in section 2.1, the compression of the working fluid in the reversed Rankine cycle can also start at the pinch temperature using a preheating process. However, a large portion of the heat is then delivered in sensible form, and the difference between the reversed Rankine cycle and the sensible heat pump becomes smaller. A self-heat recuperative technology has been developed by Kansha et al.22 The sensible heat of an effluent stream is recuperated and used for heating the feed stream by compressing the effluent stream at its supply temperature.

Figure 7. HEN designs for example 2: (a) case O; (b) case A; (c) case B; and (d) case C. 971


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used to completely or partly satisfy the heating demand. The exergy consumption for the two cases are compared since both heat and work are involved. The exergy content of the cold utility is neglected (since the temperature is near ambient). The exergy consumption in case A is EA = QHU,0(1 − T0/THU). For case B, the working fluid of the heat pump is assumed to be ideal gas with constant specific heat ratio κ. The compression and expansion should start at the higher (TPI,c) and lower (TPI,e) pinch temperatures, respectively. The pressure ratio for compression is assumed to be pr. The compression and expansion work can be easily derived as long as the outlet temperatures for compression and expansion are determined using eqs 1 and 2). The compression heat above pinch is assumed to be completely utilized (i.e., no new pinch is created by integrating the compression heat). In addition, the cooling introduced by the expansion is assumed to be no more than the cooling demand, otherwise a new pinch is created below TPI,e and the heating demand increases. The heating demand is thus QB = QHU,0 − Wcomp,B. The exergy consumption is EB = QB(1 − T0/THU) + Wcomp,B − Wexp,B. The exergy consumption for the two cases is compared in the following way:

The feed stream (as cold) and the effluent stream (as hot) have equal heat capacity flow rates. Even though it has not been explicitly stated by the authors, both compression and expansion actually start at pinch temperatures in the self-heat recuperative scheme. Incidentally, the scheme is in line with the open sensible heat pump (case C) presented in this example except that the mcp values of the GCC above and below pinch are not necessarily the same (stream splitting is thus used in this study).

3. PROCESS DESIGN WITH RESPECT TO MINIMUM EXERGY CONSUMPTION This section investigates the conditions when a sensible heat pump can be used to save exergy, and the optimal operating parameters such as the heat capacity flow rate of the working fluid and pressure ratio for compression. 3.1. Conditions for Exergy Savings. Examples 1−2 show that considerable energy savings may be achieved when a sensible heat pump is used in heat recovery problems. This section investigates the conditions when the sensible heat pump can be used to save exergy. Figure 8 shows the GCC for a heat recovery problem. There are two pinch points (TPI,c ′ and ′ ). The following two cases are compared: case A, the TPI,e heating demand is satisfied by the hot utility at a constant temperature THU; case B, the sensible heat pump (Figure 3) is

EA − E B = mcp(TPI,e − Texp,PI,e) − mcp(Tcomp,PI,c − TPI,c) (T0/THU) = (mcp/THU){TPI,eTHU[1 − pr −(ne − 1)/ ne ] − TPI, cT0[pr(nc − 1)/ nc − 1]}

(4)

The exergy consumption for case B is less than case A when TPI,eTHU[1 − pr−(ne − 1)/ne] > TPI,cT0[pr(nc − 1)/nc − 1], which can be rewritten in the following form: mcpTHU[1 − pr −(ne − 1)/ ne ] mcpT0[pr

(nc − 1)/ nc

− 1]

=

Wexp,HU Wcomp,0

>

TPI,c TPI,e

(5)

where Wexp,HU is the work produced from expansion at THU and Wcomp,0 is the work consumed from compression at T0. Eq 5 can be used to determine whether a sensible heat pump (assuming that pr is given) can be used to reduce exergy consumption. For example, if there is only one pinch on the GCC (T′PI,c = T′PI,e), case B has less exergy consumption when the work produced from expansion at THU is larger than the work consumed from compression at T0. Also note that the larger the ratio (Wexp,HU/Wcomp,0) is, the more exergy is saved by using the sensible heat pump. 3.2. Maximum Heat Capacity Flow Rate of the Working Fluid. It has been assumed that no new pinch is created by compression or expansion in the above analysis. This assumption is actually in line with a set of theorems for the placement of compressors and expanders in HENs.15 The maximum portions that can be compressed and expanded are thus limited. A temperature is defined as a Potential Pinch Point if it may create a new pinch after the heating/cooling effect caused by compression/expansion is included. As illustrated in Figure 8, the following temperatures are Potential Pinch Points: (i) the convex kink points (see definition below) on the GCC in the region between T′ = T′comp,PI,c and T′ = Texp,PI,c ′ (such as points a, b, a′, and b′); (ii) the points T′ = Tcomp,PI,c ′ and T′ = Texp,PI,e ′ on the GCC (points c and c′), or the point on the line T′ = T′comp,PI,c with H = QHU,0 if T′comp,PI,c is higher than the highest temperature on the GCC, and the point on the line T′ = Texp,PI,e ′ with H = QCU,0 if Texp,PI,e ′ is lower than the lowest temperature on the GCC; (iii) the intersection point

Figure 8. GCC for the application of sensible heat pumps. 972


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Industrial & Engineering Chemistry Research between the constant temperature line T′ = T′exp,PI,e (or T′ = T′comp,PI,c) and a pocket (point e′) on the GCC. A convex kink 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 portion that can be compressed at TPI,c, (mcp)comp,PI,c,max, is determined by the following steps: starting ′ ), draw lines between the pinch point at the pinch point (TPI,c and Potential Pinch Points (above pinch) and extend the line with the largest slope until it intersects with the constant ′ ). The corresponding heating temperature line (T′ = Tcomp,PI,c demand at the intersection (point d in Figure 8), Qcomp,max, is equal to the maximum work resulting from the compression, and (mcp)comp,PI,c,max can thus be determined by eq 6. The maximum portion that can be expanded at TPI,e, (mcp)exp,PI,e,max, is determined in a similar way: starting at the pinch point (T′PI,e), draw lines between the pinch point and Potential Pinch Points (below pinch) and extend the line with the largest negative slope until it intersects with the constant temperature ′ ). The corresponding cooling demand Qexp,max line (T′ = Texp,PI,e at the intersection (point f ′ in Figure 8) is equal to the maximum expansion work, and (mcp)exp,PI,e,max is determined by eq 7. In order to reduce the irreversibilities caused by heat transfer, the mcp of the working fluid in the sensible heat pump should be as large as possible but limited by the values determined above (i.e., min [(mcp)comp,PI,c,max, (mcp)exp,PI,e,max]). (mcp)comp,PI,c,max = Q comp,max /(T ′comp,PI,c − T ′PI,c )

(6)

(mcp)exp,PI,e,max = Q exp,max /(T ′PI,e − T ′exp,PI,e )

(7)

dEB = (mcp/THU){TPI,cT0pr −1/ nc (1 − 1/nc) dpr − TPI,eTHUpr −2 + 1/ ne (1 − 1/ne)}

⎛ dE ⎞ d⎜ dpB ⎟ ⎝ r⎠ dpr

(9)

= (mcp/THU){TPI,cT0(1 − 1/nc)( −1/nc)pr −1/ nc − 1 − TPI,eTHU(1 − 1/ne)( −2 + 1/ne)pr −3 + 1/ ne } (10)

pr can be determined by specifying the value of eq 9 to be zero, as shown by eq 11: ncne /2ncne − nc − ne ⎡ TPI,eTHU ne − 1 nc ⎤ ⎥ × × pr = ⎢ ⎢⎣ TPI,cT0 ne nc − 1 ⎥⎦

(11)

By setting the first-order derivative in eq 9 to zero and utilizing this, the second-order derivative in eq 10 can be simplified as shown in eq 12. ⎛ dE ⎞ d⎜ dpB ⎟ ⎝ r⎠ dpr

= (mc p/THU)TPI,cT0(1 − 1/nc)pr −1/ nc − 1 (2 − 1/nc − 1/ne)

(12)

Since nc,ne > 1 for adiabatic compression and expansion processes, it is concluded that the second-order differential is positive. This means that a local minimum value is obtained for EB when mcp is fixed. As discussed in section 3.2, for any given pr, mcp should be as large as possible in order to reduce the irreversibilities for heat transfer, and the actual value can be determined using the concept of Potential Pinch Points (Figure 8). The value determined from eq 11 can be regarded as an upper bound (pr,u) for pr. This is explained in the following way: For pr,u (assumed to be given by eq 11), the corresponding maximum heat capacity flow rate, (mcp)u can be determined using the concept of Potential Pinch Points (i.e., point a as shown in Figure 9). For any pr > pr,u, the corresponding mcp should not

3.3. Optimal Pressure Ratio for Compression. The pressure ratio (pr) for compression has been assumed to be specified in the above analysis. A small pr normally results in a large COP for the heat pump since the temperature lift is small; however, the amount of useful heat delivered may be too small to justify the use of the heat pump. For heat pumps based on the reversed Rankine cycle, a considerable challenge is to determine the optimal operating parameters such as pressures and loads for the condensers and evaporators. Similarly, it is a challenge for sensible heat pumps to determine the optimal pr, and this is of course case-dependent. Rather than using comprehensive mathematical optimization models, a near optimal value can be determined either by mathematical analysis or by sensitivity analysis within given boundaries. The following section presents an analytical way to specify reasonable boundaries. The objective is to minimize exergy consumption. The exergy consumption for case B (section 3.1) is rewritten in eq 8. There are two variables: mcp and pr. With the assumption that the mcp of the working fluid in the sensible heat pump is fixed, EB has a local minimum when the first-order differential value is zero and the second-order differential value is positive, as determined by eqs 9 and 10. E B = Q B(1 − T0/THU) + Wcomp,B − Wexp,B = Q HU,0(1 − T0/THU) + (mcp/THU) {TPI,cT0[pr(nc − 1)/ nc − 1] − TPI,eTHU[1 − pr −(ne − 1)/ ne ]} (8)

Figure 9. GCC for illustrating the upper bound for the pressure ratio. 973


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373.2/353.2 K. The hot and cold utility demands are 981.6 and 900 kW, respectively. The exergy consumption related to external heating and cooling is calculated to be 419.4 kW for case (O) without using a heat pump. The HEN design is shown in Figure 11a. In case A where a sensible heat pump is being used, a pressure ratio pr = 2.92 determined by eq 11 is applied. The following parameters are used: TPI,e = 373.2 K, TPI,c = 353.2 K, and nc = ne = 1.4. The condition for using this heat pump can be examined by eq 5: Wexp,HU/Wcomp,0 and TPI,c/TPI,e are calculated to be 1.286 and 0.946, respectively, thus exergy can be saved by using the heat pump. In accordance with eqs 1 and 2, Tcomp,PI,c and Texp,PI,e are determined to be 479.7 and 274.8 K, respectively. The corresponding modified temperatures are Tcomp,PI,c ′ = 469.7 K and Texp,PI,e ′ = 284.8 K. Based on Figure 10a and using the concept of Potential Pinch Points, Qcomp,max and Qexp,max are determined to be 683.0 and 900 kW, respectively. (mcp)comp,PI,c,max and (mcp)exp,PI,e,max are then determined to be 6.41 and 11.48 kW/K using eqs 6 and 7. The mcp of the working fluid in the heat pump should take the lower value (i.e., 6.41 kW/K). The GCC is shown in Figure 10b. A new pinch is created at 410/390 K (the Potential Pinch Point in case O). The HEN including the heat pump can then be designed and is shown in Figure 11b. The heating demand has not been completely covered by the heat pump. Similar studies can be performed for pr = 2, 2.3, 2.6, 3.2 and 3.5; however, details from these cases are not presented. T′comp,PI,c calculated from all these values is above the lower Potential Pinch Point (T′ = 375 K). If the heating demand is designed to be completely satisfied by the heat pump (case B), it is found that T′comp,PI,c = 516.3 K for mcp = 6.41 kW/K using Figure 10a. The corresponding pr is determined to be 4.04. The GCC is shown in Figure 10c, which indicates that heating is not required. Any pr larger than 4.04 will reduce the mcp of the working fluid and thus increase the irreversibilities for heat transfer (since larger temperature differences will be used). These cases (pr > 4.04) should not be investigated.

be more than (mcp)u, otherwise a new pinch point will be created (point a) and this should be avoided. For any mcp < (mcp)u, according to eq 11, the local optimum (minimum exergy consumption) is obtained at pr,u. Note that pr,u is independent of mcp as long as it is fixed. It can thus be concluded that pr should not be higher than pr,u (i.e., pr,u can be used as the upper bound). Actually pr,u is an optimal value for pr except for some cases where pr < pr,u and the corresponding mcp is much larger than (mcp)u. In these cases, even less exergy consumption may be achieved. For example, as shown in Figure 9, when T′ = Tcomp,PI ′ intersects with the GCC at point b, the corresponding maximum mcp is larger than (mcp)u. The corresponding pr may result in less exergy consumption compared to the case where pr,u is used. Thus, pr,u is the optimal value for pr in many cases and can be used as an upper bound in sensitivity analysis for other cases where there are Potential Pinch Points between T′ = T′exp,PI,e and T′ = T′comp,PI,c (calculated when pr = pr,u) on the GCC that give larger mcp than (mcp)u. It should be noted that only one heat pump is assumed used in the above analysis. The exergy consumption is of course lower if more heat pumps are used; however, the capital cost and plant complexity could be considerably increased. 3.4. Illustrative Example. This example is used to illustrate the optimal design of a closed sensible heat pump in heat recovery problems with respect to minimum exergy consumption. Example 3. The stream data is shown in Table 5, and the corresponding GCC is shown in Figure 10a. The pinch point is Table 5. Stream Data for Example 3 stream

Ts, K

Tt, K

mcp, kW/K

ΔH, kW

H1 H2 C1 C2 Hot utility Cold utility

373.2 410 353.2 390 503.2 288.2

328.2 390 390 483.2 503.2 288.2

20 25 20 8 − −

900 500 736 745.6 − −

Figure 10. GCCs for example 3: (a) case O; (b) case A; (c) case B; and (d) case C. 974


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Figure 11. HEN designs for example 3: (a) case O and (b) case A.

Table 6. Performance Comparison for Example 3 pr

1.35

2

2.3

2.6

2.92

3.2

3.5

4.04

mcp, kW/K hot utility demand, kW cold utility demand, kW compression work, kW expansion work, kW net work consumption, kW exergy consumption, kW COP

20 745.8 684.0 635.8 616.4 19.5 338.1 12.11

6.41 614.0 598.1 495.8 429.8 66.1 328.4 5.57

6.41 501.5 521.8 608.3 506.6 101.7 316.0 4.72

6.41 399.1 456.4 710.7 571.5 139.2 309.7 4.19

6.41 298.8 397.5 811.0 630.9 180.1 307.7 3.79

6.41 217.3 351.9 892.5 676.4 216.1 308.9 3.54

6.41 135.4 308.4 974.4 719.8 254.6 312.5 3.32

6.41 0 241.1 1110.0 787.0 322.9 322.9 3.03

In all the above cases, mcp = 6.41 kW/K has been used since this value is determined using the same potential pinch point (T′ = 400 K). When pr is small and T′comp,PI,c < 400 K, the Potential Pinch Point changes to T′ = 375 K due to the heat pocket. The corresponding heating demand is 236 kW. Assuming that T′comp,PI,c = 375 K and the heating demand (236 kW) is satisfied by using the heat pump, the following values can be determined for the heat pump: pr = 1.35 and mcp = 20 kW/K. The corresponding GCC for this case (C) is shown in Figure 10d. The performance for the above cases is shown in Table 6. The minimum exergy consumption is achieved when the boundary value (2.92) determined by eq 11 is used. This is in line with the previous statement that pr,u is an optimal value for pr in many cases. The largest COP is achieved when pr = 1.35 (the smallest value among the cases). However, the exergy consumption is actually the largest. The reason is that a very small portion (24.0%) of the heating demand (981.6 kW) has been covered by the heat pump. Compared to case O, the exergy consumption is reduced by 26.6% when the optimal value (pr = 2.92) is used. However, according to Figure 11, more units are required: two heat exchangers, one compressor, and one expander. Without a detailed economic analysis, it is impossible to conclude whether such a heat pump should be installed and what the optimal value for pr is if it is installed. Economic analysis is beyond the scope of this paper.

overestimated, since the exergy of heat assumes reversible processes (i.e., the Carnot cycle). Rather than using exergy, a conversion factor between heat and work could have been used instead of the Carnot factor (1 − T0/THU). A similar analysis could then have been performed; however, this is not presented in the paper. Although exergy does not necessarily reflect the relative quality or price of heat and work, it is an important parameter that indicates the maximum improvement potential from a purely thermodynamic point of view. It should be noted that heat can even be more expensive than work in some plants. Of course, in such cases, the installation of heat pumps becomes quite attractive. It should be noted that the reversed Rankine cycle is more attractive when there is a “sharp” pinch on the Grand Composite Curve for an industrial heat recovery problem, since the heat capacity flow rate of the working fluid and thus the equipment sizes can be much smaller. As shown in examples 1 and 2, the sensible heat pump may be more attractive than the reversed Rankine cycle in some cases where the temperature range is relatively large and there is no “sharp” pinch on the GCC. In reversed Rankine cycles, the fluids work in two phases. In order to deliver heat to higher temperature sinks during condensation, elevated pressure is necessary. The selection of working fluids for a specific operating temperature, and thus pressure range is an important consideration. For reversed Brayton cycles, the fluids work in a single phase (gas). The operating temperature is not necessarily related to the pressure. Theoretically, elevated pressure is not a consideration. Atmospheric gases could potentially be used as working fluids. However, the cycle performance may be improved by proper selection of working fluids in practice compared to cases where ideal gas is assumed to be used as in this paper. The design methodology developed in this paper focuses on cases where only one heat pump is used. More than one heat pump can of course be applied to reduce the irreversibilities

4. DISCUSSION Section 3.1 derives the conditions when a sensible heat pump can be used to save exergy. The upper bound for pressure ratio has been derived in section 3.3. This value can actually be used as the optimal value in many cases. The use of exergy as a performance indicator and objective function in optimization has two limitations. First, the relative prices of heat and power do not always follow the second Law of Thermodynamics. Second, when trading off heat and work, the quality of heat is 975


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and work is used instead of exergy. This can account for discrepancies between the Second Law of Thermodynamics and the actual pricing of different energy types, as well as the fact that exergy of heat is based on unrealistic conversion processes between heat and power.

related to heat transfer; however, the capital cost and process complexity are much larger. The polytropic efficiencies for compressors and expanders vary with temperature; however, they are assumed to be constant for simplicity in this study. The variation of turbomachine efficiency with temperature should be considered for a detailed engineering design. There may also be problems related to operating compressors at high pinch temperatures, thus it is necessary to develop compressors that can run at high temperatures in order to implement sensible heat pump technologies. Finally, it should be noted that this novel sensible heat pump scheme is under conceptual development. Similar to the reversed Rankine cycle heat pumps, the increased capital cost for installing heat pumps may not be justified by the energy savings. Plant size and reliabilities should also be to be taken into consideration. More important, although reversed Brayton cycles are commonly used in low-temperature processes such as natural gas liquefaction, to the authors’ knowledge, no heat pumps have been installed based on the reversed Brayton cycles. Very limited experience is thus available for the installation of sensible heat pumps. However, the sensible heat pump scheme may be an attractive alternative when energy saving is an important concern, particularly for cases where hot utilities are expensive.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +47 73592799. Fax: +47 73593580. E-mail: fuchao83@ hotmail.com. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE

Roman Letters

cp = specific heat capacity at constant pressure, kJ/(kgK) E = exergy, kW H = enthalpy, kW m = mass flow rate, kg/s nc = polytropic index for compression ne = polytropic index for expansion p = pressure, bar pr = pressure ratio for compression Q = heat, kW T = temperature, K T′ = modified temperature, K ΔTmin = minimum temperature difference, K W = work, kW

5. CONCLUSIONS Industrial applications of heat pumps based on the reversed Rankine cycle are limited to cases where there are “sharp” pinch points on the Grand Composite Curve (GCC). However, many process streams do not have phase change, and the corresponding GCC does not have “sharp” pinch points. In addition, neither compression nor expansion has been correctly placed in the reversed Rankine cycle. On the basis of the reversed Brayton cycle, a novel sensible heat pump scheme has been developed in this paper. Both compression and expansion start at pinch temperatures. This ensures that the compression work has been completely converted into savings in hot utility, and expansion does not increase the heating demand. This is in line with the appropriate placement rules presented in recent literature for compressors and expanders, with the objective of minimum exergy consumption. In addition, since sensible heat is exchanged between the working fluid and process streams, the temperature differences for heat transfer and thus the irreversibilities are reduced. It is not required to have a “sharp” pinch point on the GCC for process streams. The conditions when exergy can be saved by using sensible heat pumps have been derived. The determination of the optimal pressure ratio for compression is a considerable challenge, since it is closely related to the process streams in various cases. The upper bound of the pressure ratio has been derived based on thermodynamic and mathematical analyses. This value is also the optimum with respect to minimum exergy consumption in many cases. The value can be used as the upper bound for sensitivity analysis in order to find near optimal value for other cases. Three examples have been used to illustrate the design and applications of sensible heat pumps, including closed and open types with/without recuperative heating, all resulting in considerable energy savings. The placement of compressors above pinch temperature may cause operational problems, and further development of high-temperature compressors is required. Even though exergy consumption has been used as an objective parameter, a similar procedure can be derived for cases where a conversion factor between heat

Greek Letters

η∞,comp = compressor polytropic efficiency η∞,exp = expander polytropic efficiency κ = specific heat ratio

Subscripts

c = compression (at pinch temperature) comp = compression cond = condensation CU = cold utility e = expansion (at pinch temperature) exp = expansion evap = evaporation HU = hot utility max = maximum min = minimum PI = pinch s = supply t = target u = upper bound 0 = ambient Abbreviations

■

COP = coefficient of performance GCC = grand composite curve HEN = heat exchanger network PA = pinch analysis

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