Optimal operating conditions of a transcritical endoreversible cycle using a low enthalpy heat source

Applied Thermal Engineering 107 (2016) 379–385

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Research Paper

Optimal operating conditions of a transcritical endoreversible cycle using a low enthalpy heat source Malika Rachedi a,⇑, Michel Feidt b, Madjid Amirat a, Mustapha Merzouk c a

LMFTA, Faculté de physique, USTHB, Université des Sciences et de la Technologie Houari Boumediene, BP 32 El Alia, 16111, Bab Ezzouar, Algiers, Algeria LEMTA ENSEM, 2 Avenue de la Forêt de Haye TSA 60604, 54518 Van Doeuvre-Lès-Nancy Cedex, France c Departementt de Mécanique, Faculté des Sciences, Université Saad Dahlab de Blida, Route de la Soumaa, BP 270, Blida 9000, Algeria b

h i g h l i g h t s
Thermodynamics analysis of a finite size heat engine driven by a finite heat source.
Mathematical modelling of a transcritical endoreversible organic Rankine cycle.
Parametric study of the optimum operating conditions of transcritical cycle.
Choice of appropriate parameters could lead to very promising efficiencies.

a r t i c l e

i n f o

Article history: Received 27 January 2016 Accepted 17 June 2016 Available online 18 June 2016 Keywords: Thermodynamics Transcritical cycle Optimisation Geothermal energy

a b s t r a c t In the context of thermodynamic analysis of finite dimensions systems, we studied the optimum operating conditions of an endoreversible thermal machine. In this study, we considered a transcritical cycle, considering external irreversibilities. The hot reservoir is a low enthalpy geothermal heat source; therefore, it is assumed to be finite, whereas the cold reservoir is assumed to be infinite. The power optimisation is investigated by searching the optimum effectiveness of the heat-exchanger at the hot side of the engine. The sum of the total effectiveness and the second law of thermodynamics are used as constraints for optimisation. The optimal temperatures of the working fluid and optimum performances are evaluated based on the most significant parameters of the system: (1) the ratio of heat capacity rate of the working fluid to the heat capacity rate of the coolant and (2) the ratio of the sink temperature to the temperature of the hot source. The parametric study of the cycle and its approximation by a trilateral cycle enabled us to determine the optimum value of the effectiveness of the heat exchangers and the optimal operating temperatures of the cycle considered. The efficiencies obtained are in the range of 15–25% and was found to exceed the efficiency expected by the Curzon and Ahlborn prevision; meanwhile, the Carnot efficiency remains at a high limit. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The environmental problems related to global warming and price instability of fossil energy sources has attracted considerable scientific interest to questions of rationalization and optimisation of the energy use. In fact, in Algeria, an annual average increase has occurred totalling 5.6% in the production of electricity. This production increased from 7 TW h to 43 TW h between 1985 and 2010. Annual average consumption followed the same growth rate Fig. 1 [1]. The efficiencies of conventional plants, operating with steam turbines and gas turbines, are approximately 45% and 35%, ⇑ Corresponding author. E-mail address: [email protected] (M. Rachedi). http://dx.doi.org/10.1016/j.applthermaleng.2016.06.115 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

respectively. As a result, approximately (55–65%) of the thermal energy supplied to these plants is lost and dispersed into the surroundings, creating a form of pollution (heat) in addition to those of the emission of greenhouse gases from the combustion of fossil fuels (coal, natural gas, oil, or fuel oil), whose reserves are limited. In a recent study [2], the authors reported that approximately 70% of the initial energy used by Canadian industries is lost and dissipated into the atmosphere, contributing to the increase in the greenhouse effect. This study also proposes that recovery of this energy enables a reduction of the greenhouse effect equivalent to 2.5 times the rate targeted by the Kyoto Protocol. Hence, to improve the performance of energy systems and reduce their negative impact on the environment, the researchers were interested in the recovery and use of low temperature heat.

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Nomenclature _ p heat capacity rate (kJ/s) C ¼ mC Cp specific heat capacity (kJ/kg K) _ m flow rate (kg/s) Q heat flux (W) r ¼ TTSiL T temperature (K) W rate of work (W) Greek letters a ðor aÞ ¼ CCwL

g

efficiency

ha ¼ TTSia

Methods that can be used to convert waste heat (low temperatures 80–150 °C) to generate electricity are attracting increasing interest. These methods can also be associated with renewable energy sources, such as geothermal or solar energy. The Rankine cycle has attracted new interest over the last two decades, due to use of organic fluids; the Rankine cycle appears to be a highly promising approach for the conversion of medium and low temperatures heat sources into electricity. However, in lowtemperature Rankine cycles, the boiling temperature is much lower; thus, the use of water/steam as the working fluid is not indicated because of its low efficiency. Under these conditions, improvement of the cycle efficiency and the reduction of the size of the turbine as well as other equipment are considerable, due to the use of organic fluids. In this study, we focus on hot springs whose temperatures are between 80 and 120 °C, related to some Algerian geothermal sites. The most abundant worldwide geothermal resources are low and medium temperatures in the liquid phase. There are 200 thermal springs in northern Algeria. The temperature of some of these reservoirs is estimated to be between 100 and 120 °C [3]. Although some studies on space heating have been accomplished [4], the possible use of these sources to produce electricity required the use of organic Rankine cycles. In these installations, the heat of geothermal fluid is transferred to a secondary fluid; the geothermal fluid itself does not come into contact with the rotating parts of the installation. According to DiPippo [5], this type of installation could be very advantageous for thermal sources less than 150 °C. Note

Fig. 1. Algerian electricity consumption during the last 30 years [1].

hb ¼ TTSib

e

heat exchangers effectiveness

Subscripts a hot exchanger side b condenser side C Carnot CA Curzon-Ahlborn H hot reservoir L low reservoir si condition at source inlet condition at source outlet so TLC trilateral cycle w working fluid

that these facilities have zero emissions to the environment. The abundant case studies or realisations in the literature demonstrate that the organic Rankine cycles have low thermal efficiency (5– 10%) and the exergy efficiency is in the range (24–45%) [6]. In recent years, several studies have been conducted to highlight the factors involved in improving efficiency of these cycles. In examining the literature, we determined that the challenge lies in the following [2,7,8,9,10,11,12,13]: – The choice of the working fluid. – The special design of the cycle. – The process should have a high efficiency and the best possible exploitation of the source, taking into account the temperature of the geothermal fluid and the site climate data. – Another optimisation criterion is the minimisation of the surfaces of the heat exchangers (evaporator, condenser, and regenerator), as they constitute a large fraction of the cost of the installation. – In addition to these criteria, these fluids must meet conditions of safety and environmental protection. A detailed study was conducted by Saleh et al. [11] of 31 pure fluids for use in the organic Rankine cycles. The performances of the fluids were compared for similar operating conditions between 30 and 100 °C. These conditions can be treated as cases of low enthalpy geothermal applications. In this study the different cycles investigated are imposed by the shape of the saturation curve in the T-s diagram, which exhibits great importance. A negative slope of the saturated vapour curve involves wet steam at the turbine outlet (wet fluid), leading to superheating at turbine inlet, which reduces the performance of the cycle. In the case of a positive slope of the line (dry fluid), a regenerator can be used to improve the cycle efficiency. In the same study, a pinch analysis also revealed that, in the case of using a geothermal source where it is recommended to extract a maximum amount of heat from a minimum flow of hot fluid, for a 120 °C temperature source, maximum recovery was by a supercritical fluid. The shape of the working fluid saturation curve and their critical temperatures are therefore crucial parameters in selecting a working fluid and the configuration of a power cycle using recovered heat. With the above information in mind, Liu et al. [13] introduced the parameter n (ds/dT) to classify working fluids, where n represents the inverse of the slope of the saturation curve in the T-s diagram. Fluids are classified as follows: n > 0, wet fluid; n < 0, dry fluid; and n = 0, isentropic fluid. Based on the same parameter, Chen et al. [8] conducted a detailed investigation on 35 fluids to study the influence of their properties on the

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performances of organic Rankine cycles and supercritical ones. The parameter n, the value of the latent heat of vaporisation, the density, the heat capacity and the critical temperature play crucial roles in the choice of the working fluid. The authors of the study of Chen et al. [8] established a classification of fluids according to their distribution on a chart they called the T - n chart. From this study, the fluids R-32, R-125 and R-143a are wet fluids with critical temperatures above 320 K (without any problem regarding condensation) and have reasonable critical pressures. These fluids could be highly promising for supercritical or trans-critical Rankine cycles dedicated to the conversion of low enthalpy geothermal sources into electricity. 2. Transcritical cycle Most of the studies cited in the literature agree that the supercritical or transcritical Rankine cycle is a better alternative than the organic Rankine cycle because it does not have good coupling with the heat source. When energy is extracted from a recovered heat source (or a geothermal source), the enthalpy of this fluid source decreases, and the profile of its temperature decreases along the high-temperature heat exchanger. It is well-established that in a cycle using a finite source, better performance and more power are obtained when the profile of the working fluid temperature increases following the shape of the source temperature [7], while in the Rankine cycle, the evaporation occurs at a constant temperature, leading to further exergy destruction. Thus, determining the choice of a working fluid depends greatly on the adaptability of the operating temperatures to the source temperature profile. A comparative study between a transcritical cycle using carbon dioxide and an organic Rankine cycle operating with R 123 was performed by Chen et al. [7]; this study demonstrated that, for identical operating conditions, the transcritical cycle provides a slightly higher power than the classical organic Rankine cycle. In another study [14], carbon dioxide has been considered in the case of a cycle using a solar energy source with a temperature near 165 °C; a prototype was built by the authors of that study, and they found that the cycle showed promising results. Other studies considering lower-temperature heat sources (100 °C) Cayer et al. [15,16] focused on parametric analysis and optimisation of this type of cycle, with the fluids used being CO2, ethane and R125. These studies demonstrated that an analysis based only on the first law is not sufficient to determine the best cycle parameters and choose the most appropriate fluid. The thermal efficiency, the net power obtained, the exergy efficiency as well as UA (the product of the overall heat transfer coefficient by the area) are all parameters that guide in the choice of an appropriate fluid. Nevertheless ethane and CO2 have critical temperatures below 320 °C and could cause problems for condensation in certain regions, unlike R-125. Transcritical cycles, where the transition to the vapour phase is achieved at variable temperatures and the condensation process occurs at constant temperature, have been theoretically approximated by a triangular form (TLC: trilateral cycle). DiPippo [17] recommended comparison of the thermal efficiencies of these plants to efficiencies calculated from a triangular configuration (gTLC) because the Carnot efficiency, gC, sets an excessively high limit.

gTLC ¼

TH  TL TH þ TL

ð1Þ

TL TH

ð2Þ

gC ¼ 1 

381

3. Modelling and optimisation of a triangular cycle Producing electricity from heat requires a heat engine operating between two heat reservoirs: a heat source and a heat sink. The use of geothermal energy involves the conversion of this energy (geothermal heat) in the most useful form, i.e., work or electricity. The most efficient heat engine is the Carnot reversible (ideal) machine, which is the one with the highest efficiency. As actual cycles are less than ideal, the question of the definition of efficiency or performance arises; therefore, one must determine the share of the heat source that can be converted to useful work and the share that is lost. Heat transfer, pressure drop and the non-adiabacity of heat engines are all parameters that make energy conversion systems non-ideal systems. Bejan [19] highlighted in many of his works the importance of a joint analysis between thermodynamics, heat transfer and fluid flow for modelling and optimisation of energy systems to locate the entropy generation sources and to minimise them. The degree of non-ideality of the system is therefore closely related to its physical characteristics. The work of Chambadal and Novikov at the late 1950s and more recent works attributed to Curzon and Ahlborn have shown that the thermal efficiency of an endoreversible Carnot engine at maximum power is less than the efficiency of the ideal Carnot engine, with both engines operating between the same reservoirs based on a linear law of heat transfer between the hot reservoir and the thermal machine. The current efficiency expression attributed to these authors is given by:

gCA

sffiffiffiffiffiffi TL ¼1 TH

ð3Þ

The endoreversible cycle has been extensively treated in the literature [20–24]; although this cycle is more realistic than the Carnot cycle, the authors cited above, according to each study, have taken into account more criteria to refine the optimisation of power generation cycles. In addition to external irreversibility, internal irreversibilities and non-adiabacity of the system are taken into account. The authors also emphasised the importance of the effect of heat transfer laws; the most commonly used heat transfer laws are the radiative and convective laws. 3.1. Cycle description The cycle considered is a classical Rankine cycle. The corresponding system is comprised of an evaporator, a turbine, a condenser and a pump. The working fluid is raised to a pressure above its critical pressure when it passes through the pump, and then its temperature increased to a temperature above its critical temperature during the heat recovery process from the geothermal fluid in the evaporator. Subsequently, the working fluid is subjected to expansion in the turbine and condenses at a constant temperature in the condenser in the last step (Fig. 2). The cycle of Fig. 3 is an approximation of the cycle described above, wherein the expansion in the turbine is assumed to be isentropic. We also assume that the energy consumed in the pump is negligible compared to that recovered during the expansion in the turbine, the exchangers are counter flow heat exchangers, and the system is adiabatic (without heat losses). 3.2. Analysis

Schuster et al. [18] found that the TLC operating with organic fluids exhibits an efficiency of 8% higher than the same fluid operating in organic Rankine cycle systems.

3.2.1. Evaluation of the heat fluxes

  _ p geo ðT Si  T So Þ Q H ¼ mC

ð4Þ

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2Q H Q  L¼0 Ta þ Tb Tb

ð11Þ

Considering that: a ¼ CCwL ; ha ¼ TTSia ; hb ¼ TTSib ; r ¼ TTSiL and eH þ eL ¼ eT 6 2 Taking into account Eqs. (6)–(8), (10) and (11) are then written as follows:

_ ¼ eH ð1  hb Þ  eT  eH ðhb  r Þ W

a

2a

eH 1  hb hb  r  ¼0 hb ðeT  eH Þ ð2  eH Þhb þ eH

ð12Þ ð13Þ

_ ¼ W . where W C w T Si After many algebraic calculations and substituting the solutions hb of Eq. (13) in Eq. (12), the non-dimensional power output is given by:

   1=2 _ keH þ2rr eH Þ2 þ4r eH ð2þ2akeH ÞÞ  W ¼ eH 1  ð2akeH þ2rreH Þðð2a2ð2þ2 akeH Þ    1=2 ð2akeH þ2rr eH Þðð2akeH þ2rr eH Þ2 þ4r eH ð2þ2akeH ÞÞ eT  eH r a 2ð2þ2akeH Þ

Fig. 2. Transcritical cycle.

ð14Þ where k ¼ eTeHeH The optimum effectiveness eH of the hot heat-exchanger at maximum power is obtained by taking the derivative of Eq. (14) with respect to eH and setting it to zero; a, r and eT are parameters.   _ Obtaining the root of @@eWH ¼ 0 , allows for determination of eH(opt) and the optimal values of the operating temperatures of the cycle, namely, (ha, hb). Note that the solutions with physical meaning must also meet the following conditions:

rhhb hha h1;

eH 6 1 and eL 6 1

4. Results and discussion _ temperatures ha, The variations of the dimensionless power W, hb of the cycle, and the performance are studied for different values of the parameters a, r and eT. Initially, attention was given to the sensitivity analysis of the total effectiveness eT of the heat exchang_ according to eH, we ers, with a and r being fixed. Calculating W

Fig. 3. Triangular cycle.

determine the value of eT required for different values of the ratio

  _ p w ðT a  T b Þ Q H ¼ mC

ð5Þ

  _ p w eH ðT Si  TbÞ Q H ¼ mC

ð6Þ

  _ p L eL ðT b  T L Þ Q L ¼ mC

ð7Þ

Equality between Eqs. (5) and (6) leads to

T a ¼ T b þ eH ðT Si  T b Þ

ð8Þ

_ p ¼ C, where C is the heat In further calculation, we set mC capacity rate. 3.2.2. The cycle power production The first law of thermodynamics allows us to write:

W ¼ QH  QL W ¼ C w eH ðT si  TbÞ  C L eL ðT b  T L Þ

ð9Þ ð10Þ

If the cycle is assumed to be endoreversible, then the second law implies the entropy balance as:

_ giving the optimum eh, for a = 0.001 and r = 0.74 for different Fig. 4. Evolution of W eT (eT = 1 is the one which give a possible solution that is eh < 1).

M. Rachedi et al. / Applied Thermal Engineering 107 (2016) 379–385

_ for a = 0.01 eT = 1.05 is the acceptable value. Fig. 5. Evolution of W,

_ for a = 0.1 eT = 1.30 is the adequate solution. Fig. 6. Evolution of W,

Fig. 7 shows that large values of a require a larger eT; however, increasing eT does not result in a higher maximum power. It can be seen by considering a range of a varying between 0.001 and 0.200 that the value of the total effectiveness eT of the heat-exchangers increases by 40%, whereas the maximum power decreases by approximately 50%. Fig. 8 shows that, for fixed values of eT and r, increasing a from 0.001 to 0.010 leads to an approximately 10% decrease of the maximum power. In addition, the quotient TL/TSi also has great importance in the generated power level; the decrease of r indicates the availability of a heat source of higher temperature or the existence of lowest temperature for the cold reservoir. Fig. 9 shows the power levels generated for different values of r, with eT being fixed and a varying between 0.001 and 0.008. All the above-mentioned results are in agreement with those observed for the curves of ha and hb. The difference between the temperature ha at the inlet of the turbine and the condensing temperature hb is proportional to the power provided by the cycle, as shown in Figs. 10 and 11. According to the results observed earlier, we find that the difference between the values of ha and hb is diminishing as a increases (lowest power) and the difference increases as r decreases (higher power). Finally, we are also interested in evaluating trilateral cycle efficiency (gTLC) according to different values of r: 0.60, 0.74 and 0.80. It is found, as expected, that the maximum efficiency is obtained for the lowest values of a, with the efficiency increasing from 10.5% to 24%, as shown in Fig. 12. The comparison of the three efficiencies of Carnot, C&A and TLC has highlighted an interesting result. As predicted, the Carnot efficiency is the highest, whereas, gCA is less than gTLC, even if it is estimated at maximum power. The ideal triangular cycle efficiency is more important because, in that cycle, the profile of the working fluid temperature, by following the source temperature, allows for less destruction of exergy, as shown in Fig. 13. Both the Carnot efficiency and the Curzon and Ahlborn efficiency (gC and gCA) are constant because they are evaluated with respect to TH and TL. To extract more information of our results, the non-dimensional power was plotted versus (gTLC) for r equal to 0.6 and (a) ranging from 0.001 to 0.2. These curves are plotted together in Fig. 14 to visualise the optimal values of powers and the corresponding efficiency, for each exploited value of (a). The data in the literature estimated that organic Rankine cycle’s efficiency facilities are approximately 5%; our calculations showed that the efficiencies

_ evolution for different values of a and the eT required for constant value Fig. 7. W of r 0.74.

a. These calculations led, for each value of a considered and r being fixed, to obtain eT and the optimum value of eH. Figs. 4–6 show that for small values of a, eT is smaller for the optimal configuration, which leads to installations having heat-exchangers of small sizes.

383

_ for different value of a, eT = 1 and r = 0.74. Fig. 8. Comparison of W

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Fig. 9. Effect of the value of r on the non-dimensional power.

Fig. 12. Effect of different values of r on the evolution of gTLC with respect of a.

Fig. 10. Optimum non-dimensional temperatures ha and hb with respect to a for constant value of r.

Fig. 13. Comparison of gTLC to gC and gCA.

_ and the corresponding optimum gTLC. Fig. 14. Visualization of the maximum W Fig. 11. Comparison of optimum non-dimensional temperatures ha and hb with respect to a for two values of r (0.60 and 0.74).

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of a transcritical cycle may vary between 15% and 25%. Fig. 14 confirms that maximum power and maximum efficiency are highly similar, as well. 5. Conclusions A thermodynamics analysis of a finite size heat engine driven by geothermal finite heat source was performed, considering external irreversibilities. Due to the nature of the source envisaged, a transcritical endoreversible cycle was considered. A mathematical model was established to determine the optimum operating conditions based on the most significant parameters of the cycle. For all values of the ratio of heat capacity rate, we achieved an optimum value of eH and the corresponding maximum power. This approach enabled us to assess the optimal operating temperature and the performance of the trilateral cycle. The change in the ratio of the temperature of the cold reservoir to the temperature of the heat source demonstrated the importance of local conditions. This finding indicated that the climate conditions and the source temperature level play a crucial role in the performance of power cycles. Moreover, the modelling of the transcritical cycle and its approach by a triangular endoreversible cycle suggests that in fact, the efficiency can be between 15% and 25% according to the values of the parameters of the cycle. This type of cycle would be very advantageous for low enthalpy geothermal sources, to meet the energy needs of remote communities. Validation and extension of the proposed model will be performed in due course. References [1] R. Boudries, Analysis of solar hydrogen production in Algeria: case of an electrolyzer-CPV system, Int. J. Hydrogen Energy 38 (2013) 11507–11518. [2] N. Galanis, E. Cayer, P. Roy, E.S. Denis, M. Desilets, Electricity generation from low temperature sources, J. Appl. Fluid Mech. 2 (2) (2009) 55–67. [3] F.Z. Kedaid, Database on the geothermal resources of Algeria, Geothermics 36 (3) (2007) 265–275. [4] M. Rachedi, Design of a district heating system for the Hammam Righa spa, Algeria (No. 3), 1989. . [5] R. DiPippo, Small geothermal power plants: design, performance and economics, GHC Bull. 20 (2) (1999) 1–8. .

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