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Parametric study of exergetic efficiency of a small-scale cogeneration plant incorporating a heat pump
Applied Thermal Engineering 23 (2003) 459–472 www.elsevier.com/locate/apthermeng
Parametric study of exergetic efficiency of a small-scale cogeneration plant incorporating a heat pump W. Malinowska a, L. Malinowski
b,*
a
b
Institute of Agricultural Engineering, Agricultural Academy of Szczecin, ul. Papie_za Pawła VI, 1/3, 71-459 Szczecin, Poland Faculty of Maritime Technology, Technical University of Szczecin, Al. Piast ow 41, 71-065 Szczecin, Poland Received 15 May 2002; accepted 1 November 2002
Abstract A comparative parametric analysis is carried out of a small-scale combined heat and power plant incorporating a heat pump and the conventional system in which heat is produced in a hot water boiler and electrical energy is drawn from the power grid. Relative exergetic efficiency is defined as the quotient of exergetic (rational) efficiencies of the cogeneration plant and the related conventional system. Dependence of this efficiency on the power-to-heat ratio for chosen values of parameters characterizing the compared systems is calculated and shown pictorially. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Cogeneration; Exergetic analysis; Relative exergetic efficiency; Parametric study
1. Introduction Combined heat and power generation, or cogeneration, has two main advantages over a standard energy system. Firstly, primary energy is used more efficiently in this case, secondly, much less exergy is lost during the primary energy transformation into power and heat. In the case of the standard energy system, the production of power is accompanied by a considerable waste of heat, whereas, the production of heat does not take advantage of the large difference between the combustion gas and heated medium temperatures. Transmission losses related to the standard system are not miningless as well. On the other hand, a drawback of cogeneration systems is the dependence of heat production on electrical power required by the energy consuming facility. To *
Corresponding author. Tel.: +48-91-449-4827; fax: +48-91-449-4737. E-mail address: [email protected] (L. Malinowski).
1359-4311/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 2 ) 0 0 2 1 6 - 8
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W. Malinowska, L. Malinowski / Applied Thermal Engineering 23 (2003) 459–472
Nomenclature cpf cpw CB CHP E_ e E_ i E_ iL E_ w1 E_ w2 m_ f Q_ C Q_ F Q_ H Q_ HP Q_ iH Q_ iL Q_ iW T0 Tc Te TfB Tf THP Tw1 Tw2 Twm W_ W_ HP a b d DTB DTf DTHP
specific heat at constant pressure of engine exhaust gas, J/(kg K) specific heat at constant pressure of water, J/(kg K) coefficient defined by Eq. (28) coefficient defined by Eq. (27) desired exergy output, W exergy input for conventional system, W exergy input for cogeneration plant, W exergy flow rate of water returning from dwelling, W exergy flow rate of water delivered to dwelling, W flow rate of engine exhaust gas, kg/s rate of cooling heat of set engine/generator, W rate of exhaust heat, W overall rate of heat transferred to hot water system, W rate of heat transferred to hot water in heat pump condenser, W rate of combustion heat for heating purposes in conventional system, W rate of combustion heat in cogeneration plant, W rate of combustion heat for power purposes in conventional system, W reference (atmospheric environment) temperature, K condensation temperature of heat pump agent, K evaporation temperature of heat pump agent, K temperature of hot combustion gas in boiler, K temperature of exhaust gas at engine outlet, K minimum temperature of medium transferring heat to evaporator of heat pump (in particular THP ¼ T0 ), K temperature of water returning from dwelling, K temperature of water at outlet of water heater or boiler, K temperature of water at outlet of heat pump condenser, K electric power delivered to dwelling, W electric driving power of heat pump, W degree of utilization of exhaust heat defined by Eq. (20) coefficient of performance of heat pump relative exergetic efficiency difference between combustion gas outlet temperature and water inlet temperature in boiler, K difference between exhaust gas outlet temperature and water inlet temperature in water heater, K difference between condensing refrigerant temperature and water outlet temperature in heat pump condenser, Tc Twm , as well as difference between minimum temperature of heat reservoir of heat pump and temperature of evaporating refrigerant, THP Te , K
W. Malinowska, L. Malinowski / Applied Thermal Engineering 23 (2003) 459–472
uL uQ uW U UHP gB gE gEL x w
461
chemical exergy to NCV for fuel in cogeneration plant chemical exergy to NCV for fuel consumed by boiler in conventional system chemical exergy to NCV for fuel consumed by power plant in conventional system power-to-heat ratio, W_ =Q_ H ratio of Q_ HP to Q_ H efficiency of boiler overall efficiency of production of electricity in conventional system, taking into account transmission losses overall efficiency of production of electricity in cogeneration plant ratio of Q_ C to Q_ F exergetic efficiency
make a cogeneration system more flexible, an idea of incorporating a vapour compression heat pump into the plant was put forward [1–4]. In small-scale cogeneration plants, intended for domestic applications, an electrical motor driven heat pump seems a better choice than an absorption heat driven heat pump. The electric motor driven heat pump increases electrical demand and allows the generator engine to run efficiently at a very low electrical demand of the dwelling [1]. Moreover, when electrical demand is very low, there is not enough waste heat to feed the absorption pump. A cogeneration plant equipped with a heat pump characterizes the power-toheat ratio ranging from zero (all power drives the heat pump) to approximately 0.5–1 (only the heat of exhaust gas and engine cooling heat are utilized, no power is delivered to the heat pump). In the present paper, we compare a small-scale cogeneration system, equipped with a vapour compression heat pump, with the conventional system in which heat is produced in a hot water boiler, and electrical energy is supplied by the power grid. We introduce a relative efficiency of the cogeneration system defined as the quotient of exergetic (rational) efficiencies of the cogeneration plant and the related conventional system. The cogeneration plant considered in this paper is shown schematically in Fig. 1. Our analysis is performed using the following assumptions: – desired exergy outputs for the compared systems are equal, – electrical power delivered to the dwelling and the rate of heat transferred to the hot water system are equal for both systems, respectively, – engine cooling heat is not directly transferred to the hot water but it can be transported to the heat pump evaporator (see Fig. 1). The first assumption is difficult to accomplish in practice [3]. We adopted it with the aim of comparison of equivalent cases in our theoretical analysis. 2. Analysis We define the relative exergetic efficiency of the local cogeneration plant as w d¼ L w
ð1Þ
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Fig. 1. Schematic diagram of a small-scale combined heat and power plant analysed in the paper.
where wL and w are the exergetic or rational efficiencies [5] of the local cogeneration plant and the corresponding conventional system, respectively wL ¼ w¼
E_ e E_ iL
E_ e E_ i
ð2Þ
ð3Þ
where the desired exergy output, E_ e , is assumed to be equal for both considered cases E_ e ¼ W_ þ E_ w2 E_ w1
ð4Þ
Substituting Eqs. (2) and (3) into Eq. (1) we have d¼
E_ i E_ iL
ð5Þ
The rates of exergy delivered to the local cogeneration plant and the related conventional system are respectively E_ iL ¼ uL Q_ iL
ð6Þ
E_ i ¼ uQ Q_ iH þ uW Q_ iW
ð7Þ
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where Q_ iL ¼ ðW_ þ W_ HP Þ=gEL
ð8Þ
Q_ iH ¼ Q_ H =gB
ð9Þ
Q_ H ¼ m_ w cpw ðTw2 Tw1 Þ
ð10Þ
Q_ iW ¼ W_ =gE
ð11Þ
Substituting Eqs. (6) and (7) into Eq. (5) we can see that the relative exergetic efficiency, d, has important practical sense. For the same fuel used in the compared systems, e.g. natural gas, there is uL ¼ uQ ¼ uW , and d is equal to the ratio of fuel consumptions by the standard and local systems respectively. When different fuels are used, d approximately equals the ratio of combustion heats required by the two systems, as the differences between the values of u for various fuels usually do not exceed 5% [6]. Eqs. (6)–(11) are substituted into Eq. (5) with the result d¼
uW W_ =ðgE Q_ H Þ þ uQ =gB uL ðW_ þ W_ HP Þ=ðgEL Q_ H Þ
ð12Þ
The coefficient of performance of a heat pump is defined as b ¼ Q_ HP =W_ HP
ð13Þ
The heat pump increases the temperature of hot water (heating agent) from Tw1 to Twm Q_ HP ¼ m_ w cpw ðTwm Tw1 Þ
ð14Þ
We define the power-to-heat ratio U ¼ W_ =Q_ H
ð15Þ
and the ratio Q_ HP to Q_ H UHP ¼
Q_ HP m_ w cpw ðTwm Tw1 Þ Twm Tw1 ¼ ¼ m_ w cpw ðTw2 Tw1 Þ Tw2 Tw1 Q_ H
ð16Þ
Substituting Eqs. (13), (15) and (16) into Eq. (12) we get d¼
uW U=gE þ uQ =gB uL ðU þ UHP =bÞ=gEL
ð17Þ
The relative exergetic efficiency, d, is a function of U and depends on a number of dimensionless parameters characterizing the system. To calculate d from Eq. (17) we have to know the temperature Twm as both UHP and b depend on it. To determine Twm , we use the energy balance equation for the set engine/generator Q_ iL ¼ W_ þ W_ HP þ Q_ F þ Q_ C
ð18Þ
The flux of heat delivered to the hot water, Q_ H , is the same in both systems, i.e. cogeneration and conventional
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Q_ H ¼ Q_ HP þ aQ_ F
ð19Þ
where the coefficient a < 1 determines the degree of utilization of engine exhaust heat a¼
m_ f cpf ðTf Twm DTf Þ Tf Twm DTf ¼ m_ f cpf ðTf T0 Þ Tf T0
ð20Þ
If the exhaust gas were cooled to the atmospheric temperature, T0 , the coefficient a would be equal to unity. The exhaust gas raises the temperature of the water from Twm to Tw2 aQ_ F ¼ m_ w cpw ðTw2 Twm Þ
ð21Þ
Taking advantage of Eqs. (8), (10), (13), (14), (15), (16), (20) and (21) we transform Eq. (18) into the following form: U¼
1þx Tf T0 Tw2 Twm 1 Twm Tw1 1=gEL 1 Tf Twm DTf Tw2 Tw1 bðTwm Þ Tw2 Tw1
ð22Þ
where x ¼ Q_ C =Q_ F
ð23Þ
is a known coefficient characterizing the engine. Eq. (22) enables us to calculate Twm for given U, x, gEL , Tf , DTf , Tw1 , Tw2 and known dependence of b on Twm .
Fig. 2. Simple vapour-compression heat pump cycle.
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To determine the function bðTwm Þ we calculated the coefficient of performance, bs , for the simple vapour-compression heat pump cycle (see Fig. 2) for a number of values of the evaporation, Te , and condensation, Tc , temperatures. We used the data for the solkane 134a refrigerant, reported in [7]. Based on 72 values of bs we calculated, by the method of the least squares, the coefficients: a ¼ 0:900254, b ¼ 284:6681, and c ¼ 0:991686 in the following approximate formula: bsa ¼ a þ bðTc Te Þc
ð24Þ
where (see Fig. 2) Tc ¼ Twm þ DTHP
ð25Þ
Te ¼ THP DTHP
ð26Þ
The maximum difference between values of the coefficient of performance calculated from the approximate formula (24) and those based on the cycle is of the order of 5% for the range of
Fig. 3. Coefficient of performance of simple vapour-compression heat pump cycle as a function of difference between condensation and evaporation temperatures. (+++) exact values based on the cycle; (––) approximate values calculated from Eq. (24).
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Te ¼ 273–313 K and Tc ¼ 278–353 K (see Fig. 3). The coefficient of correlation and the variance explained are equal to 0.999 and 0.998, respectively. Furthermore, we assumed that ð27Þ
b ¼ CHP bsa
where the coefficient CHP takes account of all differences between a real heat pump and the simple cycle. We also consider the effect of Tw1 on the boiler efficiency, gB , in the conventional system g B ¼ CB
TfB Tw1 DTB TfB T0 DTB
ð28Þ
where the coefficient CB equals the boiler efficiency in the case of Tw1 ¼ T0 (see Fig. 4). Eq. (22) can be rewritten in the following form: U¼
ð1 þ xÞUF UHP 1=gEL 1 b
ð29Þ
where UF ¼
Q_ F 1 Tw2 Twm ¼ Q_ H a Tw2 Tw1
ð30Þ
In our analysis we assume that the minimum value of Q_ H equals aQ_ F (available exhaust heat is utilized in 100 times a percent). Therefore, for the case considered in this work, the maximum value of U equals Umax ¼
1þx að1=gEL 1Þ
Fig. 4. Temperature profiles in hot water boiler in conventional system.
ð31Þ
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as in this case UHP ¼ 0 (Twm ¼ Tw1 ) and UF ¼ 1=a. For example, for x ¼ 0:1, a ¼ 0:893 and gEL ¼ 0:35, Umax equals 0.66.
3. Results of calculations We used expression (17) to investigate the effect of the power-to-heat ratio, U, on the relative exergetic efficiency, d, for chosen values of parameters characterizing the compared systems, such as the temperature of water returning from the dwelling, Tw1 , the temperature of water at the outlet of the water heater or boiler, Tw2 , the minimum temperature of medium transferring heat to the heat pump evaporator, THP , the overall efficiency of production of electricity in the cogeneration plant, gEL , the overall efficiency of production of electricity in the conventional system, gE , and the coefficient x, defined by Eq. (23). The overall efficiency of production of electricity in the conventional system, gE , takes into account the transmission losses which can reach 9–10% of gross generation of electricity, thus gE can be lower than the overall efficiency of a power plant even by 9–10%. Calculations were performed for the following base values of the parameters:
Fig. 5. Relative exergetic efficiency versus power-to-heat ratio for various temperatures of water returning from dwelling.
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Tw1 ¼ 310 K, Tw2 ¼ 338 K, THP ¼ 290 K, T0 ¼ 297 K, Tf ¼ 700 K, TfB ¼ 1400 K, DTHP ¼ 6, DTf ¼ 30, DTB ¼ 30, gEL ¼ 0:35, gE ¼ 0:35, x ¼ 0:1, CHP ¼ 0:6, CB ¼ 0:85, uL ¼ 1:04, uQ ¼ 1:04, uW ¼ 1:08. The values of the above-cited independent parameters are chosen based on the literature, mainly on the quoted references. We endeavoured to use typical, realistic values. The dependent variables, e.g. Twm , b, gB , etc., are calculated from our theoretical model. The results are shown in Figs. 5–10 in the form of families of curves d ¼ dðUÞ. The curves belonging to a family are calculated for the base value of a chosen parameter and additionally for three or four more values of the same parameter. So, in all these figures there is one common curve present that is calculated for the base values of parameters. It is seen in Figs. 5–10 that for the data used, the relative exergetic efficiency, d, is greater then unity, it approximately ranges from 1.35 to 2.3, which means that from the exergetic point of view the local cogeneration plant is always superior to the conventional system. From the practical point of view (see paragraph 2), this means that the heat consumption (or fuel cost if the same fuel is used in both systems) is lower by about 26–57% in the cogeneration system. To understand the course of curves d ¼ dðUÞ in the respective figures, we have to remember that as the power-to-heat ratio, U, increases, the contribution of the heat
Fig. 6. Relative exergetic efficiency versus power-to-heat ratio for various temperatures of water at outlet of water heater or boiler.
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Fig. 7. Relative exergetic efficiency versus power-to-heat ratio for various temperatures of medium transferring heat to heat pump evaporator.
pump to the heating of water in the cogeneration plant decreases. For Umax (see Eq. (31)), the water is heated solely by the engine exhaust heat. Reduction of participation of the heat pump in the heating of water results in the decrease of water temperature at the outlet of the heat pump condenser, Twm . The lower Twm , the larger the coefficient of performance of the heat pump is (see Eqs. (24)–(26) and Fig. 3). In Fig. 5 the temperature of water returning from the dwelling, Tw1 , is a changeable parameter. It is seen that d decreases as Tw1 rises, which is caused by the decrease of the coefficient of performance of the heat pump, b. The transfer of a given amount of heat through a higher temperature difference requires more work. As the power-to-heat ratio, U, increases, the influence of Tw1 on d reduces because the participation of the heat pump in the heating of water reduces to naught. In Fig. 6 the effect of U on d for various values of the temperature at the outlet of the water heater or boiler, Tw2 , is shown. In this case, the dependence of U on d is more complex than in the case of Tw1 . For high values of Tw2 , the curves d ¼ dðUÞ have maximums. It is seen in Fig. 7 that the relative exergetic efficiency, d, rises with increasing temperature of heat source for the heat pump, THP , which results from the increase of b with increasing THP . Rise in the overall efficiency
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Fig. 8. Relative exergetic efficiency versus power-to-heat ratio for various overall efficiencies of production of electricity in cogeneration plant.
of production of electricity in the cogeneration plant, gEL , and the drop in the overall efficiency of production of electricity in the conventional system, gE , bring about an increase in d (see Figs. 8 and 9). As U ! 0, the curves for various values of gE converge because in such a case no electrical power is delivered to the dwelling. This is not the case for the gEL curves as here electrical energy is consumed by the heat pump. When gE is small, an increase in U can cause a rise in d because the positive effect of small gE on the value of d rises with increasing U. The curves gE ¼ 0:45 and gE ¼ 0:50 in Fig. 9 relate to the most efficient gas-fired combined cycle plants. When electrical demand is high, U ¼ 0:65, d drops to the value of 1.35 at gE ¼ 0:50 and gEL ¼ 0:35. Although gE gEL , the fuel consumption in the cogeneration plant can be still by about 26% lower in comparison with the combined plant that results from the utilization of the engine exhaust gas heat in the cogeneration case. Shown in Fig. 10 is the effect of the coefficient x ¼ Q_ C =Q_ F on d. As seen in this figure, d increases with x decreasing because for smaller values of x more heat from the engine exhaust gas is transported to the water. The cooling heat of the set engine/generator is not directly included to the model; it can be taken into account through the value of temperature THP .
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Fig. 9. Relative exergetic efficiency versus power-to-heat ratio for various overall efficiencies of production of electricity in conventional system (gE takes into account transmission losses).
4. Conclusions A small-scale combined heat and power plant incorporating a heat pump has been compared with the conventional system in which heat is produced in a hot water boiler, and electrical energy is drawn from the power grid. From the exergetic point of view the local cogeneration plant is always superior to the conventional system. The rational exergetic efficiency of the local plant can be over twice higher than the efficiency of the corresponding conventional system. The differences between the rational efficiencies of both systems depend on parameters characterizing the systems, such as the power-to-heat ratio, the temperature of water returning from the dwelling, the temperature of water at the outlet of the water heater or boiler, the minimum temperature of medium transferring heat to the heat pump evaporator, the overall efficiency of production of electricity in the cogeneration plant, and the overall efficiency of production of electricity in the conventional system. The relative exergetic efficiency defined in the paper additionally expresses the relative heat (or fuel) consumption by the compared systems. We have shown that the primary energy demand can be reduced by about 26–57% in the case of the cogeneration plant incorporating a vapour compression heat pump. The reduction of carbon dioxide emission can also be significant
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Fig. 10. Relative exergetic efficiency versus power-to-heat ratio for various coefficients x ¼ Q_ C =Q_ F .
in this case. A drawback of the cogeneration system is the considerable investment costs that should be balanced by the reduction of operating costs.
References [1] P.C. Few, M.A. Smith, J.W. Twiddel, Modelling of a combined heat and power (CHP) plant incorporating a heat pump for domestic use, Energy 22 (7) (1997) 651–659. [2] M.A. Smith, P.C. Few, Modelling of a domestic-scale co-generation plant thermal capacitance considerations, Applied Energy 68 (1) (2001) 69–82. [3] M.A. Smith, P.C. Few, Second law analysis of an experimental domestic scale cogeneration plant incorporating a heat pump, Applied Thermal Engineering 21 (1) (2001) 93–110. [4] K. Lucas, Efficient energy systems on the basis of cogeneration and heat pump technology, International Journal of Thermal Sciences 40 (4) (2001) 338–343. [5] T.J. Kotas, The Exergy Method of Thermal Plant Analysis, Krieger Publishing Company, Malabar, Florida, 1995. [6] J. Szargut, D.R. Morris, F.R. Steward, Exergy Analysis of Thermal, Chemical, and Metallurgical Processes, Hemisphere Publishing Corporation, New York, 1988. [7] W.Z. Black, J.G. Hartley, Thermodynamics, HarperCollins College Publishers, New York, 1996.
Parametric study of exergetic efficiency of a small-scale cogeneration plant incorporating a heat pump W. Malinowska a, L. Malinowski
b,*
a
b
Institute of Agricultural Engineering, Agricultural Academy of Szczecin, ul. Papie_za Pawła VI, 1/3, 71-459 Szczecin, Poland Faculty of Maritime Technology, Technical University of Szczecin, Al. Piast ow 41, 71-065 Szczecin, Poland Received 15 May 2002; accepted 1 November 2002
Abstract A comparative parametric analysis is carried out of a small-scale combined heat and power plant incorporating a heat pump and the conventional system in which heat is produced in a hot water boiler and electrical energy is drawn from the power grid. Relative exergetic efficiency is defined as the quotient of exergetic (rational) efficiencies of the cogeneration plant and the related conventional system. Dependence of this efficiency on the power-to-heat ratio for chosen values of parameters characterizing the compared systems is calculated and shown pictorially. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Cogeneration; Exergetic analysis; Relative exergetic efficiency; Parametric study
1. Introduction Combined heat and power generation, or cogeneration, has two main advantages over a standard energy system. Firstly, primary energy is used more efficiently in this case, secondly, much less exergy is lost during the primary energy transformation into power and heat. In the case of the standard energy system, the production of power is accompanied by a considerable waste of heat, whereas, the production of heat does not take advantage of the large difference between the combustion gas and heated medium temperatures. Transmission losses related to the standard system are not miningless as well. On the other hand, a drawback of cogeneration systems is the dependence of heat production on electrical power required by the energy consuming facility. To *
Corresponding author. Tel.: +48-91-449-4827; fax: +48-91-449-4737. E-mail address: [email protected] (L. Malinowski).
1359-4311/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 2 ) 0 0 2 1 6 - 8
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W. Malinowska, L. Malinowski / Applied Thermal Engineering 23 (2003) 459–472
Nomenclature cpf cpw CB CHP E_ e E_ i E_ iL E_ w1 E_ w2 m_ f Q_ C Q_ F Q_ H Q_ HP Q_ iH Q_ iL Q_ iW T0 Tc Te TfB Tf THP Tw1 Tw2 Twm W_ W_ HP a b d DTB DTf DTHP
specific heat at constant pressure of engine exhaust gas, J/(kg K) specific heat at constant pressure of water, J/(kg K) coefficient defined by Eq. (28) coefficient defined by Eq. (27) desired exergy output, W exergy input for conventional system, W exergy input for cogeneration plant, W exergy flow rate of water returning from dwelling, W exergy flow rate of water delivered to dwelling, W flow rate of engine exhaust gas, kg/s rate of cooling heat of set engine/generator, W rate of exhaust heat, W overall rate of heat transferred to hot water system, W rate of heat transferred to hot water in heat pump condenser, W rate of combustion heat for heating purposes in conventional system, W rate of combustion heat in cogeneration plant, W rate of combustion heat for power purposes in conventional system, W reference (atmospheric environment) temperature, K condensation temperature of heat pump agent, K evaporation temperature of heat pump agent, K temperature of hot combustion gas in boiler, K temperature of exhaust gas at engine outlet, K minimum temperature of medium transferring heat to evaporator of heat pump (in particular THP ¼ T0 ), K temperature of water returning from dwelling, K temperature of water at outlet of water heater or boiler, K temperature of water at outlet of heat pump condenser, K electric power delivered to dwelling, W electric driving power of heat pump, W degree of utilization of exhaust heat defined by Eq. (20) coefficient of performance of heat pump relative exergetic efficiency difference between combustion gas outlet temperature and water inlet temperature in boiler, K difference between exhaust gas outlet temperature and water inlet temperature in water heater, K difference between condensing refrigerant temperature and water outlet temperature in heat pump condenser, Tc Twm , as well as difference between minimum temperature of heat reservoir of heat pump and temperature of evaporating refrigerant, THP Te , K
W. Malinowska, L. Malinowski / Applied Thermal Engineering 23 (2003) 459–472
uL uQ uW U UHP gB gE gEL x w
461
chemical exergy to NCV for fuel in cogeneration plant chemical exergy to NCV for fuel consumed by boiler in conventional system chemical exergy to NCV for fuel consumed by power plant in conventional system power-to-heat ratio, W_ =Q_ H ratio of Q_ HP to Q_ H efficiency of boiler overall efficiency of production of electricity in conventional system, taking into account transmission losses overall efficiency of production of electricity in cogeneration plant ratio of Q_ C to Q_ F exergetic efficiency
make a cogeneration system more flexible, an idea of incorporating a vapour compression heat pump into the plant was put forward [1–4]. In small-scale cogeneration plants, intended for domestic applications, an electrical motor driven heat pump seems a better choice than an absorption heat driven heat pump. The electric motor driven heat pump increases electrical demand and allows the generator engine to run efficiently at a very low electrical demand of the dwelling [1]. Moreover, when electrical demand is very low, there is not enough waste heat to feed the absorption pump. A cogeneration plant equipped with a heat pump characterizes the power-toheat ratio ranging from zero (all power drives the heat pump) to approximately 0.5–1 (only the heat of exhaust gas and engine cooling heat are utilized, no power is delivered to the heat pump). In the present paper, we compare a small-scale cogeneration system, equipped with a vapour compression heat pump, with the conventional system in which heat is produced in a hot water boiler, and electrical energy is supplied by the power grid. We introduce a relative efficiency of the cogeneration system defined as the quotient of exergetic (rational) efficiencies of the cogeneration plant and the related conventional system. The cogeneration plant considered in this paper is shown schematically in Fig. 1. Our analysis is performed using the following assumptions: – desired exergy outputs for the compared systems are equal, – electrical power delivered to the dwelling and the rate of heat transferred to the hot water system are equal for both systems, respectively, – engine cooling heat is not directly transferred to the hot water but it can be transported to the heat pump evaporator (see Fig. 1). The first assumption is difficult to accomplish in practice [3]. We adopted it with the aim of comparison of equivalent cases in our theoretical analysis. 2. Analysis We define the relative exergetic efficiency of the local cogeneration plant as w d¼ L w
ð1Þ
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Fig. 1. Schematic diagram of a small-scale combined heat and power plant analysed in the paper.
where wL and w are the exergetic or rational efficiencies [5] of the local cogeneration plant and the corresponding conventional system, respectively wL ¼ w¼
E_ e E_ iL
E_ e E_ i
ð2Þ
ð3Þ
where the desired exergy output, E_ e , is assumed to be equal for both considered cases E_ e ¼ W_ þ E_ w2 E_ w1
ð4Þ
Substituting Eqs. (2) and (3) into Eq. (1) we have d¼
E_ i E_ iL
ð5Þ
The rates of exergy delivered to the local cogeneration plant and the related conventional system are respectively E_ iL ¼ uL Q_ iL
ð6Þ
E_ i ¼ uQ Q_ iH þ uW Q_ iW
ð7Þ
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463
where Q_ iL ¼ ðW_ þ W_ HP Þ=gEL
ð8Þ
Q_ iH ¼ Q_ H =gB
ð9Þ
Q_ H ¼ m_ w cpw ðTw2 Tw1 Þ
ð10Þ
Q_ iW ¼ W_ =gE
ð11Þ
Substituting Eqs. (6) and (7) into Eq. (5) we can see that the relative exergetic efficiency, d, has important practical sense. For the same fuel used in the compared systems, e.g. natural gas, there is uL ¼ uQ ¼ uW , and d is equal to the ratio of fuel consumptions by the standard and local systems respectively. When different fuels are used, d approximately equals the ratio of combustion heats required by the two systems, as the differences between the values of u for various fuels usually do not exceed 5% [6]. Eqs. (6)–(11) are substituted into Eq. (5) with the result d¼
uW W_ =ðgE Q_ H Þ þ uQ =gB uL ðW_ þ W_ HP Þ=ðgEL Q_ H Þ
ð12Þ
The coefficient of performance of a heat pump is defined as b ¼ Q_ HP =W_ HP
ð13Þ
The heat pump increases the temperature of hot water (heating agent) from Tw1 to Twm Q_ HP ¼ m_ w cpw ðTwm Tw1 Þ
ð14Þ
We define the power-to-heat ratio U ¼ W_ =Q_ H
ð15Þ
and the ratio Q_ HP to Q_ H UHP ¼
Q_ HP m_ w cpw ðTwm Tw1 Þ Twm Tw1 ¼ ¼ m_ w cpw ðTw2 Tw1 Þ Tw2 Tw1 Q_ H
ð16Þ
Substituting Eqs. (13), (15) and (16) into Eq. (12) we get d¼
uW U=gE þ uQ =gB uL ðU þ UHP =bÞ=gEL
ð17Þ
The relative exergetic efficiency, d, is a function of U and depends on a number of dimensionless parameters characterizing the system. To calculate d from Eq. (17) we have to know the temperature Twm as both UHP and b depend on it. To determine Twm , we use the energy balance equation for the set engine/generator Q_ iL ¼ W_ þ W_ HP þ Q_ F þ Q_ C
ð18Þ
The flux of heat delivered to the hot water, Q_ H , is the same in both systems, i.e. cogeneration and conventional
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Q_ H ¼ Q_ HP þ aQ_ F
ð19Þ
where the coefficient a < 1 determines the degree of utilization of engine exhaust heat a¼
m_ f cpf ðTf Twm DTf Þ Tf Twm DTf ¼ m_ f cpf ðTf T0 Þ Tf T0
ð20Þ
If the exhaust gas were cooled to the atmospheric temperature, T0 , the coefficient a would be equal to unity. The exhaust gas raises the temperature of the water from Twm to Tw2 aQ_ F ¼ m_ w cpw ðTw2 Twm Þ
ð21Þ
Taking advantage of Eqs. (8), (10), (13), (14), (15), (16), (20) and (21) we transform Eq. (18) into the following form: U¼
1þx Tf T0 Tw2 Twm 1 Twm Tw1 1=gEL 1 Tf Twm DTf Tw2 Tw1 bðTwm Þ Tw2 Tw1
ð22Þ
where x ¼ Q_ C =Q_ F
ð23Þ
is a known coefficient characterizing the engine. Eq. (22) enables us to calculate Twm for given U, x, gEL , Tf , DTf , Tw1 , Tw2 and known dependence of b on Twm .
Fig. 2. Simple vapour-compression heat pump cycle.
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To determine the function bðTwm Þ we calculated the coefficient of performance, bs , for the simple vapour-compression heat pump cycle (see Fig. 2) for a number of values of the evaporation, Te , and condensation, Tc , temperatures. We used the data for the solkane 134a refrigerant, reported in [7]. Based on 72 values of bs we calculated, by the method of the least squares, the coefficients: a ¼ 0:900254, b ¼ 284:6681, and c ¼ 0:991686 in the following approximate formula: bsa ¼ a þ bðTc Te Þc
ð24Þ
where (see Fig. 2) Tc ¼ Twm þ DTHP
ð25Þ
Te ¼ THP DTHP
ð26Þ
The maximum difference between values of the coefficient of performance calculated from the approximate formula (24) and those based on the cycle is of the order of 5% for the range of
Fig. 3. Coefficient of performance of simple vapour-compression heat pump cycle as a function of difference between condensation and evaporation temperatures. (+++) exact values based on the cycle; (––) approximate values calculated from Eq. (24).
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Te ¼ 273–313 K and Tc ¼ 278–353 K (see Fig. 3). The coefficient of correlation and the variance explained are equal to 0.999 and 0.998, respectively. Furthermore, we assumed that ð27Þ
b ¼ CHP bsa
where the coefficient CHP takes account of all differences between a real heat pump and the simple cycle. We also consider the effect of Tw1 on the boiler efficiency, gB , in the conventional system g B ¼ CB
TfB Tw1 DTB TfB T0 DTB
ð28Þ
where the coefficient CB equals the boiler efficiency in the case of Tw1 ¼ T0 (see Fig. 4). Eq. (22) can be rewritten in the following form: U¼
ð1 þ xÞUF UHP 1=gEL 1 b
ð29Þ
where UF ¼
Q_ F 1 Tw2 Twm ¼ Q_ H a Tw2 Tw1
ð30Þ
In our analysis we assume that the minimum value of Q_ H equals aQ_ F (available exhaust heat is utilized in 100 times a percent). Therefore, for the case considered in this work, the maximum value of U equals Umax ¼
1þx að1=gEL 1Þ
Fig. 4. Temperature profiles in hot water boiler in conventional system.
ð31Þ
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as in this case UHP ¼ 0 (Twm ¼ Tw1 ) and UF ¼ 1=a. For example, for x ¼ 0:1, a ¼ 0:893 and gEL ¼ 0:35, Umax equals 0.66.
3. Results of calculations We used expression (17) to investigate the effect of the power-to-heat ratio, U, on the relative exergetic efficiency, d, for chosen values of parameters characterizing the compared systems, such as the temperature of water returning from the dwelling, Tw1 , the temperature of water at the outlet of the water heater or boiler, Tw2 , the minimum temperature of medium transferring heat to the heat pump evaporator, THP , the overall efficiency of production of electricity in the cogeneration plant, gEL , the overall efficiency of production of electricity in the conventional system, gE , and the coefficient x, defined by Eq. (23). The overall efficiency of production of electricity in the conventional system, gE , takes into account the transmission losses which can reach 9–10% of gross generation of electricity, thus gE can be lower than the overall efficiency of a power plant even by 9–10%. Calculations were performed for the following base values of the parameters:
Fig. 5. Relative exergetic efficiency versus power-to-heat ratio for various temperatures of water returning from dwelling.
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Tw1 ¼ 310 K, Tw2 ¼ 338 K, THP ¼ 290 K, T0 ¼ 297 K, Tf ¼ 700 K, TfB ¼ 1400 K, DTHP ¼ 6, DTf ¼ 30, DTB ¼ 30, gEL ¼ 0:35, gE ¼ 0:35, x ¼ 0:1, CHP ¼ 0:6, CB ¼ 0:85, uL ¼ 1:04, uQ ¼ 1:04, uW ¼ 1:08. The values of the above-cited independent parameters are chosen based on the literature, mainly on the quoted references. We endeavoured to use typical, realistic values. The dependent variables, e.g. Twm , b, gB , etc., are calculated from our theoretical model. The results are shown in Figs. 5–10 in the form of families of curves d ¼ dðUÞ. The curves belonging to a family are calculated for the base value of a chosen parameter and additionally for three or four more values of the same parameter. So, in all these figures there is one common curve present that is calculated for the base values of parameters. It is seen in Figs. 5–10 that for the data used, the relative exergetic efficiency, d, is greater then unity, it approximately ranges from 1.35 to 2.3, which means that from the exergetic point of view the local cogeneration plant is always superior to the conventional system. From the practical point of view (see paragraph 2), this means that the heat consumption (or fuel cost if the same fuel is used in both systems) is lower by about 26–57% in the cogeneration system. To understand the course of curves d ¼ dðUÞ in the respective figures, we have to remember that as the power-to-heat ratio, U, increases, the contribution of the heat
Fig. 6. Relative exergetic efficiency versus power-to-heat ratio for various temperatures of water at outlet of water heater or boiler.
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Fig. 7. Relative exergetic efficiency versus power-to-heat ratio for various temperatures of medium transferring heat to heat pump evaporator.
pump to the heating of water in the cogeneration plant decreases. For Umax (see Eq. (31)), the water is heated solely by the engine exhaust heat. Reduction of participation of the heat pump in the heating of water results in the decrease of water temperature at the outlet of the heat pump condenser, Twm . The lower Twm , the larger the coefficient of performance of the heat pump is (see Eqs. (24)–(26) and Fig. 3). In Fig. 5 the temperature of water returning from the dwelling, Tw1 , is a changeable parameter. It is seen that d decreases as Tw1 rises, which is caused by the decrease of the coefficient of performance of the heat pump, b. The transfer of a given amount of heat through a higher temperature difference requires more work. As the power-to-heat ratio, U, increases, the influence of Tw1 on d reduces because the participation of the heat pump in the heating of water reduces to naught. In Fig. 6 the effect of U on d for various values of the temperature at the outlet of the water heater or boiler, Tw2 , is shown. In this case, the dependence of U on d is more complex than in the case of Tw1 . For high values of Tw2 , the curves d ¼ dðUÞ have maximums. It is seen in Fig. 7 that the relative exergetic efficiency, d, rises with increasing temperature of heat source for the heat pump, THP , which results from the increase of b with increasing THP . Rise in the overall efficiency
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Fig. 8. Relative exergetic efficiency versus power-to-heat ratio for various overall efficiencies of production of electricity in cogeneration plant.
of production of electricity in the cogeneration plant, gEL , and the drop in the overall efficiency of production of electricity in the conventional system, gE , bring about an increase in d (see Figs. 8 and 9). As U ! 0, the curves for various values of gE converge because in such a case no electrical power is delivered to the dwelling. This is not the case for the gEL curves as here electrical energy is consumed by the heat pump. When gE is small, an increase in U can cause a rise in d because the positive effect of small gE on the value of d rises with increasing U. The curves gE ¼ 0:45 and gE ¼ 0:50 in Fig. 9 relate to the most efficient gas-fired combined cycle plants. When electrical demand is high, U ¼ 0:65, d drops to the value of 1.35 at gE ¼ 0:50 and gEL ¼ 0:35. Although gE gEL , the fuel consumption in the cogeneration plant can be still by about 26% lower in comparison with the combined plant that results from the utilization of the engine exhaust gas heat in the cogeneration case. Shown in Fig. 10 is the effect of the coefficient x ¼ Q_ C =Q_ F on d. As seen in this figure, d increases with x decreasing because for smaller values of x more heat from the engine exhaust gas is transported to the water. The cooling heat of the set engine/generator is not directly included to the model; it can be taken into account through the value of temperature THP .
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Fig. 9. Relative exergetic efficiency versus power-to-heat ratio for various overall efficiencies of production of electricity in conventional system (gE takes into account transmission losses).
4. Conclusions A small-scale combined heat and power plant incorporating a heat pump has been compared with the conventional system in which heat is produced in a hot water boiler, and electrical energy is drawn from the power grid. From the exergetic point of view the local cogeneration plant is always superior to the conventional system. The rational exergetic efficiency of the local plant can be over twice higher than the efficiency of the corresponding conventional system. The differences between the rational efficiencies of both systems depend on parameters characterizing the systems, such as the power-to-heat ratio, the temperature of water returning from the dwelling, the temperature of water at the outlet of the water heater or boiler, the minimum temperature of medium transferring heat to the heat pump evaporator, the overall efficiency of production of electricity in the cogeneration plant, and the overall efficiency of production of electricity in the conventional system. The relative exergetic efficiency defined in the paper additionally expresses the relative heat (or fuel) consumption by the compared systems. We have shown that the primary energy demand can be reduced by about 26–57% in the case of the cogeneration plant incorporating a vapour compression heat pump. The reduction of carbon dioxide emission can also be significant
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Fig. 10. Relative exergetic efficiency versus power-to-heat ratio for various coefficients x ¼ Q_ C =Q_ F .
in this case. A drawback of the cogeneration system is the considerable investment costs that should be balanced by the reduction of operating costs.
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