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Determining the optimal configuration of a heat exchanger (with a two-phase refrigerant) using exergoeconomics
Applied Energy 71 (2002) 191–203 www.elsevier.com/locate/apenergy
Determining the optimal configuration of a heat exchanger (with a two-phase refrigerant) using exergoeconomics M. Dentice d’Accadiaa, A. Ficherab,*, M. Sassoa, M. Vidirib a DETEC-Universita` degli Studi di Napoli Federico II, P. le Tecchio 80-80125, Naples, Italy Dipartimento di Ingegneria Industriale e Meccanica, Universita` di Catania, Viale A. Doria 6-95125, Catania, Italy
b
Received 12 October 2001; received in revised form 19 December 2001; accepted 5 January 2002
Abstract In this paper, the exergoeconomic theory is applied to a heat exchanger for optimisation purposes. The investigation was referred to a tube-in-tube condenser with the single-phase fluid to be heated flowing in the inner annulus and the two-phase refrigerant flowing in the external annulus. First, the irreversibility due to heat transfer across the stream-to-stream temperature-difference and to frictional pressure-drops is calculated as a function of two design variables: the inner-tube’s diameter and the saturation temperature of the refrigerant, on which the heat-exchange area directly depends. Then, a cost function is introduced, defined as the sum of two contributions: the amortisation cost of the condenser under study and the operating cost of the conventional electric-driven heat-pump in which this component will have to work. The latter contribution is directly related to the overall exergy destruction rate in the plant, whereas the amortisation cost mainly depends on the heat-exchange area. So, design optimisation of the device can be performed by minimising this cost function with respect to the selected design variables. The so-called structural approach (Coefficient of Structural Bond) is used in the optimisation, in order to relate the local irreversibility in the condenser to the overall exergy destruction rate in the heat-pump plant. A numerical example is discussed, in which, for a commercial heat-exchanger, the design improvements needed to obtain a cost-optimal configuration are investigated. The results show that significant improvements can be obtained with respect to devices based on conventional values of the design parameters. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermoeconomics; Condensation; Heat pump; Irreversibility; Design optimisation
* Corresponding author. Tel./fax: +39-095-337994. E-mail addresses: afi[email protected] (A. Fichera), [email protected] (M. Dentice d’Accadia), [email protected] (M. Sasso). 0306-2619/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(02)00007-7
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Nomenclature A heat-exchange area, m2 cF unit cost of electric energy, LIT/kWh CSB coefficients of structural bond D diameter, m F fuel, W h heat-transfer coefficient, W/ m2 K H mean operation time, h/year I irreversibility, W LIT Italian lire : m mass flow rate, kg/s p pressure, Pa P: product, W Q: heat-transfer rate, W Q0 heat-transfer rate per unit length, W/m s: thickness, m S entropy generation rate, W/K T temperature, K UA overall thermal-conductance x axial coordinate directon y design variable Z cost, MLIT
Subscripts b bulk gen generated i variable index in inner k device index o reference value r refrigerant sat saturated tot total w water wl wall Superscripts = matrix mean value Greek symbols difference amortisation factor, years1 l thermal conductivity, W/m K coefficient of external cost, LIT/W total cost, LIT/year density, kg/m3
1. Introduction Optimisation of heat exchangers has been widely studied using different approaches, such as Entropy Generation Minimisation, Exergy Analysis, Life Cycle Analysis and Thermoeconomic Analysis [1–4]. The evaluation of the optimal trade-off between fluid friction and heat transfer irreversibility and the analysis of augmentation techniques to increase the wall-fluid heat transfer, without causing a damaging increase in pumping power is significant. The aim of this paper was to perform the design optimisation of a heat-pump condenser, i.e. a heat exchanger in which a phase-change occurs, considering both thermodynamic and economic points-of-view (i.e. thermoeconomic and exergoeconomic optimisations). In order to do this, it was first necessary to evaluate the entropy generation due to heat exchange and pressure losses as a function of the design variables selected for the optimisation analysis. Then, it was necessary to evaluate the relation between exergy destruction rate and
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193
operation cost of the electric-driven vapour-compression heat-pump in which the condenser has to operate. This was done using the so-called structural approach, that allows one to take into account, in a simple way, the influence of the local process irreversibility on that of the plant as a whole [5]. Finally, by correlating the design variables of the problem to the economic cost of the heat exchanger, the optimum trade-off between operation and amortisation costs was investigated. In the following sections, the methods for the calculation of entropy production in the condenser and thermoeconomic optimisation followed in the paper are described. Then, a numerical example is developed.
2. Entropy generation The overall irreversibility of the process, i.e. the exergy destruction rate in the condenser, was calculated as the product of the environment or ‘‘dead-state’’ temperature, T0, by the entropy generation rate, according to the Gouy–Stodola theorem [5]: : Itot ¼ T0 Sgen;tot
ð1Þ
The heat exchanger under analysis was a tube-in-tube type, made-up by two coaxial tubes arranged in circular coils. The cold fluid (water) flows in the inner tube whereas the hot fluid (refrigerant) flows in the annulus. The diameters of both coaxial tubes were assumed constant. Moreover, the following assumptions were considered: (i) Newtonian non-reacting pure fluids; (ii) operation in steady-state conditions; (iii) mono-dimensional flow conditions; (iv) negligible thermal interaction with the environment; and (v) negligible streamwise conduction in the wall of the heat exchanger. : The overall entropy generation rate in the condenser, Sgen;tot was calculated as the sum of three components: two contributions due to the cold and the hot fluids flowing in the condenser and one due to heat transfer across the tube wall: : : : : Sgen;tot ¼ Sgen;r þ Sgen;w þ Sgen;wl
ð2Þ
where: : S: gen;r =entropy production rate in the control volume containing the refrigerant; S: gen;w =entropy production rate in the control volume containing the cold fluid; Sgen;wl =entropy production rate due to the heat exchanged through the inner wall of the heat exchanger. : : Each of the terms Sgen;r and Sgen;w was calculated as the sum of two contributions: (i) irreversibility due to friction and (ii) irreversibility due to heat transfer across a finite temperature-difference between the wall temperature and the bulk temperature of the fluid. So, for each side of the heat exchanger, and for an elementary section of
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length dx, located at the generic abscissa, x, along the path of the fluids, the entropy generation in the control volume occupied by the fluid was calculated as [1]: : : : dSgen;i Q02 mi dpi ¼ þ ; dx Dhi Tb;i Tb;i dx
i ¼ w; r
ð3Þ
where D is the inner or outer diameter, for water (i=w) and refrigerant (i=r), respectively. The first term on the right-hand side of Eq. (3) represents the contribution due to heat transfer, while the second term is the irreversibility due to friction. Finally the: entropy production due to the heat exchanged through the inner wall of the tube, Sgen;wl , was evaluated according to: : : dSgen;wl 1 1 ¼ Q0 ð4Þ Tw þ Twwl Tr Trwl dx where the local temperature–differences between fluid and wall were defined as: Twwl
: Q0 ¼ Din hw
:
Trwl
Q0 ¼ ðDin þ 2sÞhr
ð5Þ
Property evaluation for both fluids was based on data available in the literature [6]. Two different fluid-flow models were considered to describe the behaviour of the two-phase and the single-phase flow, respectively. Specific relations, quoted in the literature as able to ensure satisfactory fitting with experimental results, were used to address the evaluation of the heat-transfer coefficients and the pressure drops [7,8]. The overall entropy-generation rate was obtained by integrating the elementary contributions calculated from Eqs. (2), (3) and (4) over the heat exchanger.
3. Structural thermoeconomic optimisation Exergoeconomic optimisation of the condenser, that will be presented in the next section, was based on the so-called structural method, which is briefly summarised as follows [5]. The exergy destruction rate for the plant as a whole, Itot, can be written as: Itot y ¼ Ftot y P
ð6Þ
where P is the product of the process, represented by the increase of the physical exergy of the water in the condenser, Ftot is the external resource, or fuel, consumed by the system—here represented by the electric power required by the heat-pump compressor—and y is the matrix of all design variables affecting plant performance, whose component yi,k represents the i-th variable directly affecting the performance
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of device k. If maintenance and other accessory costs are neglected, the economic cost of owning and operating the plant can be expressed as: X tot y ¼ cF Ftot y H þ Zk ðy k Þ
ð7Þ
k
where cF is the cost per unit exergy of the fuel, H is the equivalent time of full-load operation assumed for the plant on a yearly base, is the amortisation factor and the terms of the sum, Zk, represent the costs of the components that make-up the plant. In order to determine the optimal condition with respect to a chosen design-variable, say yi,k, when holding fixed all the others, it is necessary to differentiate Eq. (7) with respect to yi,k and to impose that the derivative of the total cost is nil: X @Zi =@yi;k ¼ 0 @tot =@yi;k ¼ cF @Ftot =@yi;k H þ
ð8Þ
i
Assuming a constant overall product, i.e. P=cost, we have: Itot ¼ Ftot
ð9Þ
and differentiating with respect to the design variable to be optimised: @Itot =@yi;k ¼ @Ftot =@yi;k
ð10Þ
Combining Eqs. (8) and (10) it is possible to express the optimal condition as: X @tot =@yi;k ¼ CSBi;k cF @Ik =@yi;k H þ ð@Zi =@xi Þ ¼ 0
ð11Þ
i
where the coefficient of structural bond, CSBi,k, has been introduced and defined as [5]: CSBi;k ¼ @Itot =@yi;k = @Ik =@yi;k
ð12Þ
where Ik is the exergy–destruction rate in the k-th component of the plant. Eq. (8) can be also expressed as: @tot =@yi;k ¼ CSBi;k cF @Ik =@yi;k H þ @Zk =@yi;k þ i;k @Ik =@yi;k ¼ 0
ð13Þ
where the contribution due to the k-th component directly influenced by variable yi,k has been separated from those referred to the other components and the coefficient of external costs:
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i;k ¼
X @Zi =@Ik
ð14Þ
i6¼k
has been introduced. The coefficient i,k can fall to zero both when the variation of yi,k does not influence significantly the cost of components different from the k-th and when these latter components are available in various standard sizes and the variation of Zi induced by varying the design variable yi,k is so low that size variations can be neglected. In the case herein studied, the coefficient i,k was zero, leading to the following final expression for optimal conditions: @tot =@yi;k ¼ CSBi;k cF @Ik =@yi;k H þ @Zk =@yi;k ¼ 0
ð15Þ
4. An example The methodology described in the previous sections was applied to a commercial heat-exchanger to be used as the condenser in a conventional electric-driven vapourcompression heat-pump, assuming the heat-exchange area as the main design variable to be optimised. As previously stated, the heat exchanger is a tube-in-tube type, made-up by two coaxial tubes arranged in circular coils, as shown in Fig. 1. The fluid to be heated is water and flows in the inner tube, and the refrigerant R22 flows in the annulus. In order to focus on the condensation process, a de-superheating was neglected, and therefore the refrigerant is always assumed in two-phase condition. In other terms, we assumed, for simplicity, that the optimisation would refer only to the condensation process, leaving apart the de-superheating. Inner and outer tubes were copper and steel, respectively. The thickness of the copper wall that separates hot and cold fluids is s=1.5 mm, with a mean thermal conductance l=398 W/m K. Some of the main characteristics of the condenser herein studied are reported in Table 1. The following assumptions were made in the optimisation: The water inlet temperature was Tw,in=35.0 C. The water mass flow rate was fixed and equal to 1.00 kg/s. The heat flow rate to be provided was fixed and equal to 41.9 kW, so that the approximate outlet temperature for water was also fixed (Tw,out ffi 45.0 C, neglecting the pressure losses, that depend on the variables to be optimised). The refrigerant was R22. The unit cost of the fuel required for moving the compressor, cF, was used as an external parameter, and ranged from 150 to 250 LIT/kWh (1936 LIT ffi 1.0 Euro ffi US$ 1.0). The exergy destruction was evaluated from entropy generation [Eq. (1)] assuming an environmental temperature equal to 280 K.
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Fig. 1. Scheme of the condenser.
Table 1 Main characteristics of the condenser under analysis Dimension A Dimension B Dimension D Dimension E Dimension R1 (refrigerant in) Dimension R2 (refrigerant out) Dimensions W1, W2 (water in/out) Net weight Heat duty
500 mm 339 mm 435 mm 48 mm 18 mm 16 mm 28 mma 27 kg 41.9 kW
a This value refers to the reference configuration (commercial model); in the following, this dimension is assumed as a design variable.
The refrigerant saturation temperature, Tsat, at the beginning of the condensation process was one of the design variables to be optimised, since it directly affects the heat exchange area; it is assumed to vary in the range 48– 60 C (the minimum theoretical value was obviously 45 C, corresponding to the temperature of water leaving the condenser). The inner diameter of the inner tube, Din (indicated as W1=W2 in Fig. 1), that also affects the heat exchange area (since both refrigerant and water
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heat-exchange coefficients depend on it), was considered as a further design variable; it was assumed to vary in the range from 25 to 40 mm. The refrigerant mass flow rate was considered as a function of the initial saturation temperature, Tsat, and was calculated, for : any value of this temperature, as the ratio of the condenser heat duty, Q, to the enthalpy of condensation hsat corresponding to that temperature: : : m r ffi Q =ðhsat Þ
ð16Þ
The result provided by Eq. (16) is approximate, since pressure drops induce a slight variation in the difference between inlet and outlet enthalpies with : respect to hs , but the error in mr was always less than 0.5%. The calculation of the heat-exchange area, A, was based on the following procedure: first, the overall thermal conductance (UA) is approximately calculated—neglecting all pressure-losses—as the ratio of the condenser heat : duty, Q, to the mean logarithmic temperature-difference in the heat exchanger corresponding to the value assumed for Tsat and Din: then, the overall length of the condenser, L, is calculated as: Din þ 2s ln 6 1 1 Din þ L ¼ ðUAÞ 6 þ 4 2l hw Din hr ðDin þ 2sÞ 2
3 7 7 5
ð17Þ
where h w and h r are the mean heat exchange coefficients for water and refrigerant, calculated on the basis of the values of Tsat, Din and mass flow rates previously fixed; finally, the heat exchange area, from the cold side (water), is obviously expressed as A ¼ Din L
ð18Þ
The CSBi,k referred to the condenser was considered as an external parameter, and assumed equal to 1.7; this value was based on the results provided by a simulation programme for the type of plants under analysis and for refrigerant R22 [9]. The mean operation time of the plant, H, was set equal to 1000 h per year. The cost of the condenser was expressed as a function of the heat-exchange area, measured from the cold side of the heat exchanger, according to the following relation: Z ¼ Zo
A Ao
n ð19Þ
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where Z0=2.0 MLIT and A0=2.0 m2 are the reference cost and area, respectively, and the exponent n was chosen equal to 0.9. Finally, a discount rate of 5% and a time horizon of 10 years was assumed, leading to a value of the amortisation factor =0.1295 years1. For given values of the design variables Tsat and Din, the refrigerant-side and water-side heat exchange coefficients, the pressure losses and the heat-exchange area, A, needed to provide the required heat duty, can be evaluated. The overall irreversibility in the condenser, is as previously discussed. So, all terms of Eqn. (15) can be calculated as functions of any design variable, and optimal design conditions with respect to Tsat for a fixed Din—or inversely—can be directly found out by solving this equation. As an example, in Figs. 2 and 3, the solution of Eq. (15) with respect to the design variable Tsat is shown by plotting the derivative @/@Tsat for fixed values of Din, assuming cF equal to 150 and 250 LIT/kWh, respectively. The dependence of optimal conditions upon Din is not very strong, since the optimal saturation temperature always falls within a range of about 1 C when the inner diameter varies: as shown in the figures, optimal values of Tsat range between about 50.9 and 52.0 C for cF=150 LIT/kWh, and between 48.2 and 49.2 C for cF=250 LIT/kWh. Optimal conditions appear more sensitive to the value of cF: Figs. 4 and 5 show the solution of the optimisation problem for different values of cF, with Din fixed (25 and 39 mm, respectively). When cF increases, the optimal saturation temperature,
Fig. 2. Graphical results for structural optimisation with respect to Tsat for various diameters and for cF=150 LIT/kWh.
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Fig. 3. Graphical results for structural optimisation with respect to Tsat for various diameters and for cF=250 LIT/kWh.
Fig. 4. Graphical results for structural optimisation with respect to Tsat for various cF and Din=25 mm.
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for any given Din, decreases. In order to find out the overall optimal conditions, with respect to both design variables, for a fixed cF, a simple iterative procedure was followed: an initial value of Din in the range of interest was assigned; the corresponding optimal value of Tsat was calculated; the optimal value of Din corresponding to this latter value of Tsat was calculated and compared with the initial value; the procedure was iterated until satisfactory convergence was attained. In Table 2 the overall optimal conditions for three different values of cF are reported. It appears clear that an increase in the unit cost of the electric energy used by the heat-pump plant involves an increase in the optimal heat-exchange area, mainly obtainable by lowering the initial temperature of condensation. In fact, it is
Fig. 5. Graphical results for structural optimisation with respect to Tsat for various values of cF and Din=40 mm. Table 2 Overall optimal values of Tsat, Din and A for different values of cF Optimal values Design variable
Tsat ( C) Din (mm) A (m2)
cF=150 LIT/kWh
cF=200 LIT/kWh
cF=250 LIT/kWh
51 36 2.51
49 35 3.26
48 35 3.73
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more and more profitable to invest money in greater heat exchange areas in order to save energy and therefore money during the heat-pump operation. It may also be interesting to compare the final results shown in Table 2 with those obtainable by neglecting the influence of the local condenser irreversibility on the overall exergy destruction, and then on the electric consumption of the heat pump. It was calculated that, in this case, the optimal values of the saturation temperature were from 6 to 8% higher than those reported in Table 2, whereas the optimal heat-exchange areas were from 40 to 45% smaller. This result outlines the importance of taking into account structural effects when optimising single components of thermal systems.
5. Conclusions Particular attention must be given to improving energy savings of refrigerators and heat pumps due to the their increasing influence in global energy consumption. In the paper, an exergoeconomic analysis has been used to search for the optimal design of a condenser to be used in a conventional vapour-compression heat pump. In particular, by varying the inner tube diameter and the refrigerant saturation temperature, the optimal heat-exchanger area, A, was calculated, for a fixed condenser heat duty and for different values of the cost of electric energy, following the so-called structural approach. The optimal values obtained for A appear significantly greater than those usually adopted in commercial models, especially for high values of the electric energy cost. This example shows how the use of simple thermodynamic and thermoeconomic optimisation methodologies in refrigeration could contribute to determining the a correct design of new equipment, in a sector characterised by emerging technologies and by new working fluids.
Acknowledgements The final version of Margherita Vidiri’s thesis has facilitated the composition of this paper. M. Dentice d’Accadia, A. Fichera and M. Sasso wish to thank Margherita for her joyful presence.
References [1] Bejan A. Entropy generation through heat and flow. New York: Wiley, 1982. [2] Aceves-Saborio S, Ranasinghe J, Reistad GM. ASME Journal of Heat Transfer 1989;3:29–36. [3] Cornelissen RL, Hirs GG. Exergetic optimisation of a heat exchanger. Energy Convers 1997;38(15– 17):1567–76. [4] London AL, Shah RK. Costs of irreversibilities in heat exchanger design. Heat-Transfer Engineering 1983;4(2). [5] Kotas TJ. The exergy method of thermal plant analysis. FL:Krieger, 1995. [6] Hewitt GF. Heat-exchanger design handbook. New York: Begell House, 1998.
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[7] Stephan K. Heat transfer in condensation and boiling. Berlin: Springer-Verlag, 1992. [8] Gabrielii C, Vamling L. Replacement of R22 in tube and shell condensers. Int J Refrig 1997;20(3): 165–78. [9] de Rossi F, Mastrullo R, Mazzei P, Sasso M. EASY, exergetic analysis of vapour compression systems. Napoli (Italy): Liguori Editore, 1999.
Determining the optimal configuration of a heat exchanger (with a two-phase refrigerant) using exergoeconomics M. Dentice d’Accadiaa, A. Ficherab,*, M. Sassoa, M. Vidirib a DETEC-Universita` degli Studi di Napoli Federico II, P. le Tecchio 80-80125, Naples, Italy Dipartimento di Ingegneria Industriale e Meccanica, Universita` di Catania, Viale A. Doria 6-95125, Catania, Italy
b
Received 12 October 2001; received in revised form 19 December 2001; accepted 5 January 2002
Abstract In this paper, the exergoeconomic theory is applied to a heat exchanger for optimisation purposes. The investigation was referred to a tube-in-tube condenser with the single-phase fluid to be heated flowing in the inner annulus and the two-phase refrigerant flowing in the external annulus. First, the irreversibility due to heat transfer across the stream-to-stream temperature-difference and to frictional pressure-drops is calculated as a function of two design variables: the inner-tube’s diameter and the saturation temperature of the refrigerant, on which the heat-exchange area directly depends. Then, a cost function is introduced, defined as the sum of two contributions: the amortisation cost of the condenser under study and the operating cost of the conventional electric-driven heat-pump in which this component will have to work. The latter contribution is directly related to the overall exergy destruction rate in the plant, whereas the amortisation cost mainly depends on the heat-exchange area. So, design optimisation of the device can be performed by minimising this cost function with respect to the selected design variables. The so-called structural approach (Coefficient of Structural Bond) is used in the optimisation, in order to relate the local irreversibility in the condenser to the overall exergy destruction rate in the heat-pump plant. A numerical example is discussed, in which, for a commercial heat-exchanger, the design improvements needed to obtain a cost-optimal configuration are investigated. The results show that significant improvements can be obtained with respect to devices based on conventional values of the design parameters. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermoeconomics; Condensation; Heat pump; Irreversibility; Design optimisation
* Corresponding author. Tel./fax: +39-095-337994. E-mail addresses: afi[email protected] (A. Fichera), [email protected] (M. Dentice d’Accadia), [email protected] (M. Sasso). 0306-2619/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(02)00007-7
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Nomenclature A heat-exchange area, m2 cF unit cost of electric energy, LIT/kWh CSB coefficients of structural bond D diameter, m F fuel, W h heat-transfer coefficient, W/ m2 K H mean operation time, h/year I irreversibility, W LIT Italian lire : m mass flow rate, kg/s p pressure, Pa P: product, W Q: heat-transfer rate, W Q0 heat-transfer rate per unit length, W/m s: thickness, m S entropy generation rate, W/K T temperature, K UA overall thermal-conductance x axial coordinate directon y design variable Z cost, MLIT
Subscripts b bulk gen generated i variable index in inner k device index o reference value r refrigerant sat saturated tot total w water wl wall Superscripts = matrix mean value Greek symbols difference amortisation factor, years1 l thermal conductivity, W/m K coefficient of external cost, LIT/W total cost, LIT/year density, kg/m3
1. Introduction Optimisation of heat exchangers has been widely studied using different approaches, such as Entropy Generation Minimisation, Exergy Analysis, Life Cycle Analysis and Thermoeconomic Analysis [1–4]. The evaluation of the optimal trade-off between fluid friction and heat transfer irreversibility and the analysis of augmentation techniques to increase the wall-fluid heat transfer, without causing a damaging increase in pumping power is significant. The aim of this paper was to perform the design optimisation of a heat-pump condenser, i.e. a heat exchanger in which a phase-change occurs, considering both thermodynamic and economic points-of-view (i.e. thermoeconomic and exergoeconomic optimisations). In order to do this, it was first necessary to evaluate the entropy generation due to heat exchange and pressure losses as a function of the design variables selected for the optimisation analysis. Then, it was necessary to evaluate the relation between exergy destruction rate and
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193
operation cost of the electric-driven vapour-compression heat-pump in which the condenser has to operate. This was done using the so-called structural approach, that allows one to take into account, in a simple way, the influence of the local process irreversibility on that of the plant as a whole [5]. Finally, by correlating the design variables of the problem to the economic cost of the heat exchanger, the optimum trade-off between operation and amortisation costs was investigated. In the following sections, the methods for the calculation of entropy production in the condenser and thermoeconomic optimisation followed in the paper are described. Then, a numerical example is developed.
2. Entropy generation The overall irreversibility of the process, i.e. the exergy destruction rate in the condenser, was calculated as the product of the environment or ‘‘dead-state’’ temperature, T0, by the entropy generation rate, according to the Gouy–Stodola theorem [5]: : Itot ¼ T0 Sgen;tot
ð1Þ
The heat exchanger under analysis was a tube-in-tube type, made-up by two coaxial tubes arranged in circular coils. The cold fluid (water) flows in the inner tube whereas the hot fluid (refrigerant) flows in the annulus. The diameters of both coaxial tubes were assumed constant. Moreover, the following assumptions were considered: (i) Newtonian non-reacting pure fluids; (ii) operation in steady-state conditions; (iii) mono-dimensional flow conditions; (iv) negligible thermal interaction with the environment; and (v) negligible streamwise conduction in the wall of the heat exchanger. : The overall entropy generation rate in the condenser, Sgen;tot was calculated as the sum of three components: two contributions due to the cold and the hot fluids flowing in the condenser and one due to heat transfer across the tube wall: : : : : Sgen;tot ¼ Sgen;r þ Sgen;w þ Sgen;wl
ð2Þ
where: : S: gen;r =entropy production rate in the control volume containing the refrigerant; S: gen;w =entropy production rate in the control volume containing the cold fluid; Sgen;wl =entropy production rate due to the heat exchanged through the inner wall of the heat exchanger. : : Each of the terms Sgen;r and Sgen;w was calculated as the sum of two contributions: (i) irreversibility due to friction and (ii) irreversibility due to heat transfer across a finite temperature-difference between the wall temperature and the bulk temperature of the fluid. So, for each side of the heat exchanger, and for an elementary section of
194
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length dx, located at the generic abscissa, x, along the path of the fluids, the entropy generation in the control volume occupied by the fluid was calculated as [1]: : : : dSgen;i Q02 mi dpi ¼ þ ; dx Dhi Tb;i Tb;i dx
i ¼ w; r
ð3Þ
where D is the inner or outer diameter, for water (i=w) and refrigerant (i=r), respectively. The first term on the right-hand side of Eq. (3) represents the contribution due to heat transfer, while the second term is the irreversibility due to friction. Finally the: entropy production due to the heat exchanged through the inner wall of the tube, Sgen;wl , was evaluated according to: : : dSgen;wl 1 1 ¼ Q0 ð4Þ Tw þ Twwl Tr Trwl dx where the local temperature–differences between fluid and wall were defined as: Twwl
: Q0 ¼ Din hw
:
Trwl
Q0 ¼ ðDin þ 2sÞhr
ð5Þ
Property evaluation for both fluids was based on data available in the literature [6]. Two different fluid-flow models were considered to describe the behaviour of the two-phase and the single-phase flow, respectively. Specific relations, quoted in the literature as able to ensure satisfactory fitting with experimental results, were used to address the evaluation of the heat-transfer coefficients and the pressure drops [7,8]. The overall entropy-generation rate was obtained by integrating the elementary contributions calculated from Eqs. (2), (3) and (4) over the heat exchanger.
3. Structural thermoeconomic optimisation Exergoeconomic optimisation of the condenser, that will be presented in the next section, was based on the so-called structural method, which is briefly summarised as follows [5]. The exergy destruction rate for the plant as a whole, Itot, can be written as: Itot y ¼ Ftot y P
ð6Þ
where P is the product of the process, represented by the increase of the physical exergy of the water in the condenser, Ftot is the external resource, or fuel, consumed by the system—here represented by the electric power required by the heat-pump compressor—and y is the matrix of all design variables affecting plant performance, whose component yi,k represents the i-th variable directly affecting the performance
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of device k. If maintenance and other accessory costs are neglected, the economic cost of owning and operating the plant can be expressed as: X tot y ¼ cF Ftot y H þ Zk ðy k Þ
ð7Þ
k
where cF is the cost per unit exergy of the fuel, H is the equivalent time of full-load operation assumed for the plant on a yearly base, is the amortisation factor and the terms of the sum, Zk, represent the costs of the components that make-up the plant. In order to determine the optimal condition with respect to a chosen design-variable, say yi,k, when holding fixed all the others, it is necessary to differentiate Eq. (7) with respect to yi,k and to impose that the derivative of the total cost is nil: X @Zi =@yi;k ¼ 0 @tot =@yi;k ¼ cF @Ftot =@yi;k H þ
ð8Þ
i
Assuming a constant overall product, i.e. P=cost, we have: Itot ¼ Ftot
ð9Þ
and differentiating with respect to the design variable to be optimised: @Itot =@yi;k ¼ @Ftot =@yi;k
ð10Þ
Combining Eqs. (8) and (10) it is possible to express the optimal condition as: X @tot =@yi;k ¼ CSBi;k cF @Ik =@yi;k H þ ð@Zi =@xi Þ ¼ 0
ð11Þ
i
where the coefficient of structural bond, CSBi,k, has been introduced and defined as [5]: CSBi;k ¼ @Itot =@yi;k = @Ik =@yi;k
ð12Þ
where Ik is the exergy–destruction rate in the k-th component of the plant. Eq. (8) can be also expressed as: @tot =@yi;k ¼ CSBi;k cF @Ik =@yi;k H þ @Zk =@yi;k þ i;k @Ik =@yi;k ¼ 0
ð13Þ
where the contribution due to the k-th component directly influenced by variable yi,k has been separated from those referred to the other components and the coefficient of external costs:
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i;k ¼
X @Zi =@Ik
ð14Þ
i6¼k
has been introduced. The coefficient i,k can fall to zero both when the variation of yi,k does not influence significantly the cost of components different from the k-th and when these latter components are available in various standard sizes and the variation of Zi induced by varying the design variable yi,k is so low that size variations can be neglected. In the case herein studied, the coefficient i,k was zero, leading to the following final expression for optimal conditions: @tot =@yi;k ¼ CSBi;k cF @Ik =@yi;k H þ @Zk =@yi;k ¼ 0
ð15Þ
4. An example The methodology described in the previous sections was applied to a commercial heat-exchanger to be used as the condenser in a conventional electric-driven vapourcompression heat-pump, assuming the heat-exchange area as the main design variable to be optimised. As previously stated, the heat exchanger is a tube-in-tube type, made-up by two coaxial tubes arranged in circular coils, as shown in Fig. 1. The fluid to be heated is water and flows in the inner tube, and the refrigerant R22 flows in the annulus. In order to focus on the condensation process, a de-superheating was neglected, and therefore the refrigerant is always assumed in two-phase condition. In other terms, we assumed, for simplicity, that the optimisation would refer only to the condensation process, leaving apart the de-superheating. Inner and outer tubes were copper and steel, respectively. The thickness of the copper wall that separates hot and cold fluids is s=1.5 mm, with a mean thermal conductance l=398 W/m K. Some of the main characteristics of the condenser herein studied are reported in Table 1. The following assumptions were made in the optimisation: The water inlet temperature was Tw,in=35.0 C. The water mass flow rate was fixed and equal to 1.00 kg/s. The heat flow rate to be provided was fixed and equal to 41.9 kW, so that the approximate outlet temperature for water was also fixed (Tw,out ffi 45.0 C, neglecting the pressure losses, that depend on the variables to be optimised). The refrigerant was R22. The unit cost of the fuel required for moving the compressor, cF, was used as an external parameter, and ranged from 150 to 250 LIT/kWh (1936 LIT ffi 1.0 Euro ffi US$ 1.0). The exergy destruction was evaluated from entropy generation [Eq. (1)] assuming an environmental temperature equal to 280 K.
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Fig. 1. Scheme of the condenser.
Table 1 Main characteristics of the condenser under analysis Dimension A Dimension B Dimension D Dimension E Dimension R1 (refrigerant in) Dimension R2 (refrigerant out) Dimensions W1, W2 (water in/out) Net weight Heat duty
500 mm 339 mm 435 mm 48 mm 18 mm 16 mm 28 mma 27 kg 41.9 kW
a This value refers to the reference configuration (commercial model); in the following, this dimension is assumed as a design variable.
The refrigerant saturation temperature, Tsat, at the beginning of the condensation process was one of the design variables to be optimised, since it directly affects the heat exchange area; it is assumed to vary in the range 48– 60 C (the minimum theoretical value was obviously 45 C, corresponding to the temperature of water leaving the condenser). The inner diameter of the inner tube, Din (indicated as W1=W2 in Fig. 1), that also affects the heat exchange area (since both refrigerant and water
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heat-exchange coefficients depend on it), was considered as a further design variable; it was assumed to vary in the range from 25 to 40 mm. The refrigerant mass flow rate was considered as a function of the initial saturation temperature, Tsat, and was calculated, for : any value of this temperature, as the ratio of the condenser heat duty, Q, to the enthalpy of condensation hsat corresponding to that temperature: : : m r ffi Q =ðhsat Þ
ð16Þ
The result provided by Eq. (16) is approximate, since pressure drops induce a slight variation in the difference between inlet and outlet enthalpies with : respect to hs , but the error in mr was always less than 0.5%. The calculation of the heat-exchange area, A, was based on the following procedure: first, the overall thermal conductance (UA) is approximately calculated—neglecting all pressure-losses—as the ratio of the condenser heat : duty, Q, to the mean logarithmic temperature-difference in the heat exchanger corresponding to the value assumed for Tsat and Din: then, the overall length of the condenser, L, is calculated as: Din þ 2s ln 6 1 1 Din þ L ¼ ðUAÞ 6 þ 4 2l hw Din hr ðDin þ 2sÞ 2
3 7 7 5
ð17Þ
where h w and h r are the mean heat exchange coefficients for water and refrigerant, calculated on the basis of the values of Tsat, Din and mass flow rates previously fixed; finally, the heat exchange area, from the cold side (water), is obviously expressed as A ¼ Din L
ð18Þ
The CSBi,k referred to the condenser was considered as an external parameter, and assumed equal to 1.7; this value was based on the results provided by a simulation programme for the type of plants under analysis and for refrigerant R22 [9]. The mean operation time of the plant, H, was set equal to 1000 h per year. The cost of the condenser was expressed as a function of the heat-exchange area, measured from the cold side of the heat exchanger, according to the following relation: Z ¼ Zo
A Ao
n ð19Þ
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where Z0=2.0 MLIT and A0=2.0 m2 are the reference cost and area, respectively, and the exponent n was chosen equal to 0.9. Finally, a discount rate of 5% and a time horizon of 10 years was assumed, leading to a value of the amortisation factor =0.1295 years1. For given values of the design variables Tsat and Din, the refrigerant-side and water-side heat exchange coefficients, the pressure losses and the heat-exchange area, A, needed to provide the required heat duty, can be evaluated. The overall irreversibility in the condenser, is as previously discussed. So, all terms of Eqn. (15) can be calculated as functions of any design variable, and optimal design conditions with respect to Tsat for a fixed Din—or inversely—can be directly found out by solving this equation. As an example, in Figs. 2 and 3, the solution of Eq. (15) with respect to the design variable Tsat is shown by plotting the derivative @/@Tsat for fixed values of Din, assuming cF equal to 150 and 250 LIT/kWh, respectively. The dependence of optimal conditions upon Din is not very strong, since the optimal saturation temperature always falls within a range of about 1 C when the inner diameter varies: as shown in the figures, optimal values of Tsat range between about 50.9 and 52.0 C for cF=150 LIT/kWh, and between 48.2 and 49.2 C for cF=250 LIT/kWh. Optimal conditions appear more sensitive to the value of cF: Figs. 4 and 5 show the solution of the optimisation problem for different values of cF, with Din fixed (25 and 39 mm, respectively). When cF increases, the optimal saturation temperature,
Fig. 2. Graphical results for structural optimisation with respect to Tsat for various diameters and for cF=150 LIT/kWh.
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Fig. 3. Graphical results for structural optimisation with respect to Tsat for various diameters and for cF=250 LIT/kWh.
Fig. 4. Graphical results for structural optimisation with respect to Tsat for various cF and Din=25 mm.
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for any given Din, decreases. In order to find out the overall optimal conditions, with respect to both design variables, for a fixed cF, a simple iterative procedure was followed: an initial value of Din in the range of interest was assigned; the corresponding optimal value of Tsat was calculated; the optimal value of Din corresponding to this latter value of Tsat was calculated and compared with the initial value; the procedure was iterated until satisfactory convergence was attained. In Table 2 the overall optimal conditions for three different values of cF are reported. It appears clear that an increase in the unit cost of the electric energy used by the heat-pump plant involves an increase in the optimal heat-exchange area, mainly obtainable by lowering the initial temperature of condensation. In fact, it is
Fig. 5. Graphical results for structural optimisation with respect to Tsat for various values of cF and Din=40 mm. Table 2 Overall optimal values of Tsat, Din and A for different values of cF Optimal values Design variable
Tsat ( C) Din (mm) A (m2)
cF=150 LIT/kWh
cF=200 LIT/kWh
cF=250 LIT/kWh
51 36 2.51
49 35 3.26
48 35 3.73
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more and more profitable to invest money in greater heat exchange areas in order to save energy and therefore money during the heat-pump operation. It may also be interesting to compare the final results shown in Table 2 with those obtainable by neglecting the influence of the local condenser irreversibility on the overall exergy destruction, and then on the electric consumption of the heat pump. It was calculated that, in this case, the optimal values of the saturation temperature were from 6 to 8% higher than those reported in Table 2, whereas the optimal heat-exchange areas were from 40 to 45% smaller. This result outlines the importance of taking into account structural effects when optimising single components of thermal systems.
5. Conclusions Particular attention must be given to improving energy savings of refrigerators and heat pumps due to the their increasing influence in global energy consumption. In the paper, an exergoeconomic analysis has been used to search for the optimal design of a condenser to be used in a conventional vapour-compression heat pump. In particular, by varying the inner tube diameter and the refrigerant saturation temperature, the optimal heat-exchanger area, A, was calculated, for a fixed condenser heat duty and for different values of the cost of electric energy, following the so-called structural approach. The optimal values obtained for A appear significantly greater than those usually adopted in commercial models, especially for high values of the electric energy cost. This example shows how the use of simple thermodynamic and thermoeconomic optimisation methodologies in refrigeration could contribute to determining the a correct design of new equipment, in a sector characterised by emerging technologies and by new working fluids.
Acknowledgements The final version of Margherita Vidiri’s thesis has facilitated the composition of this paper. M. Dentice d’Accadia, A. Fichera and M. Sasso wish to thank Margherita for her joyful presence.
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