Thermoeconomic analysis of sensible heat, thermal energy storage systems

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Applied Thermal Engineering Vol. 18, No. 8, pp. 693±704, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-4311(97)00045-8 1359-4311/98 $19.00 + 0.00

THERMOECONOMIC ANALYSIS OF SENSIBLE HEAT, THERMAL ENERGY STORAGE SYSTEMS Roman DomanÂski and Giuma Fellah* Warsaw University of Technology, Institute of Heat Engineering, Nowowiejska 25, 00-665 Warsaw, Poland (Received 25 April 1997) AbstractÐThe paper considers the advantages of employing a thermoeconomic analysis for describing the complete charging±discharging cycle of sensible heat, thermal energy storage systems. The main task is to ®nd the performance of the storage systems at the minimum total cost of owning, maintaining, and operating such systems. The e€ect of di€erent monetary values on optimum number of heat transfer units, charging time, and second-law eciency are presented in graphical form. Comparison with results that are based on exergy analysis alone shows the advantages of adopting such an approach for designing and operating thermal energy storage units. # 1998 Elsevier Science Ltd. All rights reserved. KeywordsÐThermoeconomic analysis, exergy, storage systems, sensible heat.

NOMENCLATURE A B1, B2, B3, B4 c cP cv G  G k m_ c…d† M N Ntu P Pr _ Q Qactual Qideal R S S_ t T U yc(d) Z1, Z2

surface area (m2) constants in Equations (17), (18) and (20) heat capacity (J kgÿ1 Kÿ1) constant pressure speci®c heat (J kgÿ1 Kÿ1) constant volume speci®c heat (J kgÿ1 Kÿ1) ÿ2 ÿ1 mass ¯ow rate per unit cross-sectional p area (kg m s ) dimensionless mass velocity (G… RT0 =P0 †) speci®c heat ratio (cp/cv) mass ¯ow rate of gas during charging (discharging) (kg sÿ1) mass of the storage material (kg) entropy generation number number of heat transfer units (UA=mc _ p) pressure (N mÿ2) Prandtl number rate of heat transfer (W) actual storage capacity of a storage unit (J) ideal storage capacity of a storage unit (J) gas constant (J/(kg K)) entropy (J Kÿ1) entropy rate (W Kÿ1) time (s) temperature (K) overall heat transfer coecient (W mÿ2 Kÿ1) heat exchanger parameters, charging (discharging) constants in Equation (22)

Greek symbols b C y G_ s_ l z_ 0 a_

operational parameter exergy (J) dimensionless time total cost rate (dollars sÿ1) cost rate of owning and maintaining the storage system (dollars sÿ1) cost of irreversibility (dollars kJÿ1) cost rate of maintenance and others (dollars sÿ1) _ T ), (kW mÿ2) (s=l

*Author to whom correspondence should be addressed. 693

R. DomanÂski and G. Fellah

694 g e_ t Z1st

(lP =lT ) (z_ 0 =lT ), (kW) dimensionless temperature (…T ÿ T0 †=T0 ) ®rst law eciency

Subscripts 0 c CV d ec ed F I gen P Q T

ambient condition charging control volume discharging exit during charging exit during discharging ®nal initial generation due to pressure drop due to cooling the heat transfer ¯uid to ambient condition due to temperature di€erence

INTRODUCTION

Bejan [1] suggested that the actual purpose of a thermal energy storage system is not to store energy, but to store exergy, which is the quantity of real value. He presented a treatment of a sensible heat thermal energy storage unit during the process of charging. Modeling of an entire charging±discharging cycle of a sensible heat storage system was reported by [2, 3]. It was found that a typical optimum system destroyed about 70 to 90% of the supplied exergy. Contributions of other authors [4±6] to the subject show the advantages of utilizing the exergy concept for designing and operating thermal energy storage units. An analysis of sensible heat, thermal energy storage systems based on thermoeconomic consideration was reported by references [7, 8]. The analysis was performed for a process of charging only, and showed the in¯uence of some parameters on the optimum number of heat transfer units and charging time. The present study extends their analysis to model the entire charging±discharging cycle. The performance of the storage system at minimum total cost is sought. MODELING THE STORAGE SYSTEM

The system consists of a storage medium of mass M and heat capacity c, at initial temperature of TI and to be heated to a ®nal temperature of TF. It operates in a thermodynamic cycle with a single cycle being composed of a charging process followed by a discharging one, see Fig. 1. During the charging process, both valves ``A'' and ``B'' are opened while ``C'' and ``D'' are closed. A mass m_ c of air at Tc and Pc passes through the well insulated storage unit, and is to be cooled to a ®nal state of Tec and Pec. The air is then cooled to the environmental condition at T0 and P0. During the discharging process valves ``A'' and ``B'' are closed, while ``C'' and ``D'' are opened. A cold mass m_ d of air at Td and Pd passes through the storage unit, and is to be heated to a ®nal state of Ted and Ped. The recovered air will be used for other purposes that are out of the range of the present analysis. The storage medium is at uniform temperature throughout the cycle, with no phase change, and with constant properties. Air behaves as an ideal gas with constant cP. The environment is at T0=298 K and P0=0.1 MPa. FIRST-LAW ANALYSIS

The storage material is initially at a temperature TI, that is greater than the discharging temperature Td such that TI ˆ Td …1 ‡ b†: b is a factor greater than 0.0.

…1†

Thermoeconomic analysis of storage systems

695

In dimensionless form Equation (1) is written as tI ˆ …1 ‡ td †…1 ‡ b† ÿ 1,

…2†

where tˆ

T ÿ T0 : T0

The process of charging Energy balance. The process of charging is depicted in Fig. 2. For the control volume 1, we ®nd yc ˆ

Tc ÿ Tec …t† ˆ 1 ÿ eÿNtuc , Tc ÿ T…t†

…3†

T…y† ÿ TI ˆ 1 ÿ eÿyc y , Tc ÿ TI

…4†

Tec …y† ÿ TI ˆ 1 ÿ yc eÿyc y : Tc ÿ TI

…5†

A dimensionless time y is de®ned as m_ c cP t : Mc

…6†

  1 Tc ÿ TI ln : yc Tc ÿ T…y†

…7†

yˆ y can be written as yˆ

Fig. 1. The storage system.

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696

Fig. 2. The process of charging.

First-law eciency. It can be written as Z1st ˆ

actual amount of energy stored during charging , ideal amount of energy stored during charging Qactual ˆ Mc…Tc ÿ TI †…1 ÿ eÿyc yc †:

…8† …9†

The ideal amount of stored energy is estimated by assuming the ®nal temperature of the storage material (TF) reaches theoretically the charging temperature (Tc), that is Qideal ˆ Mc…Tc ÿ TI †,

…10†

Z1st ˆ 1 ÿ eÿyc yc :

…11†

then, we get

Control volume 2 is a hypothetical control volume, and simulates cooling the working ¯uid to ambient condition, where _ 0 ˆ m_ c cP …Tec ÿ T0 †: Q

…12†

The process of discharging Energy balance. The process of discharging is shown in Fig. 3. It can be shown that yd ˆ

Td ÿ Ted …t† ˆ 1 ÿ eÿNtud , Td ÿ T…t†

…13†

T…y† ÿ TF ˆ 1 ÿ eÿ…m_ d =m_ c †yd y , Td ÿ TF

…14†

Ted …y† ÿ TF ˆ 1 ÿ yd eÿ…m_ d =m_ c †yd y : Td ÿ TF

…15†

  1 TF ÿ Td ln : …m_ d =m_ c †yd T…y† ÿ Td

…16†

The discharging time is given by yˆ

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697

Fig. 3. The process of discharging.

SECOND-LAW ANALYSIS

The process of charging For the control volume 1, the rate of entropy generation is …y TF k ÿ 1 Pc ln ‡ …ln…1 ÿ B1eB2y †dy, ‡ yc …Sc,gen †CV1 ˆ ln k TI P0 0

…17†

'' indicates a dimensionless quantity. k ˆ …cP =cv †, and `` Here we have assumed Pec=P0. Equation (17) can be divided into:(i) entropy generation due to thermal e€ects   …y TF …18† ‡ …ln…1 ÿ B1eB2y †dy: …Sgen,c,T †CV1 ˆ ln TI 0 (ii) entropy generation due to frictional e€ects …Sgen,c,P †CV1 ˆ yc

  kÿ1 Pc ln , k P0

ÿ
B1 ˆ yc 1 ÿ TTFI and B2 = ÿ yc. For control volume 2, the rate of entropy generation can be written as   …y …Sgen,Q †CV2 ˆ yc tc ‡ …tc ÿ tI †…eÿyc y ÿ 1† ÿ …B3 ÿ B4eB2y †dy , 0

…19†

…20†

B3 = 1 + tc and B4 = yc(tcÿtI).The total entropy generation during charging is then …Sgen,c † ˆ …Sgen,c,T †CV1 ‡ …Sgen,c,P †CV1 ‡ …Sgen,Q †CV2 :

The process of discharging It can be shown the entropy generation due to thermal e€ects is   … 1 ‡ tI m_ d yd ln…1 ‡ Z1eZ2y †dy, ‡ …Sgen,d,T † ˆ ln m_ c 0 1 ‡ tF and due to friction is

Z1 ˆ yd then

ÿ tF ÿtd
1‡td

  m_ d k ÿ 1 Pd ln yd …Sgen,d,P † ˆ , m_ c P0 k

…21†

…22†

…23†

_d and Z2 ˆ ÿ m m_ c yd :The total entropy generation during the discharging process is

…Sd,gen † ˆ …Sgen,d,T † ‡ …Sgen,d,P †:

…24†

R. DomanÂski and G. Fellah

698

_ Fig. 4. Dependence of Ntu, N, and Z2nd on a.

Hence, total entropy generation for the complete charging±discharging cycle is …Sgen † ˆ …Sc,gen † ‡ …Sd,gen †:

…25†

The analysis is limited to turbulent ¯ow in a smooth circular pipe, then the pressure drop may be given by the following equations [2].   Pc  2 Ntuc 0:5 , ˆ 0:5 ‡ 0:25 ‡ Pr2=3 …1 ‡ tc †G P0

…26†

   0:5 _ d 1:8 Pd 2=3 m 2  ˆ 0:5 ‡ 0:25 ‡ Pr …1 ‡ td †G Ntud : P0 m_ c

…27†

A dimensionless supplied exergy during a charging process is developed as    kÿ1 Pc  ln , Cc ˆ yc tc ÿ ln…1 ‡ tc † ‡ k P0

…28†

and for the discharging process as

    d ˆ m_ d yd td ÿ ln…1 ‡ td † ‡ k ÿ 1 ln Pd , C k m_ c P0

…29†

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_ Fig. 5. Dependence of yc, yyc, and Z1st on a.

C ˆ Cc ‡ Cd ,

…30†

 ˆ C :The entropy generation number is de®ned as where C McT0 N ˆ1ÿ

T0 Sgen : C

…31†

The second law eciency is then Z2nd ˆ 1 ÿ N:

…32†

THERMOECONOMIC ANALYSIS

The objective of the present work is to reduce the total cost G_ (dollars sÿ1) of owning, maintaining, and operating the storage system. Here, the de®nition of G_ as given by [7, 8] is adopted: _ ‡ lT T0 S_ T ‡ lP T0 S_ P ‡ z_ 0 : G_ ˆ sA

…33†

700

R. DomanÂski and G. Fellah

Fig. 6. In¯uence of the charging temperature on the second-law eciency.

_ s(dollars mÿ2 sÿ1) is the cost attributed to owning and maintaining the storage system. z_ 0 (dolÿ1 lars s ) is the maintenance cost and any other extraneous costs that apply to the storage system as a whole. Equation (33) is put in more convenient form, such that:

or

_ ‡ T0 …S_ T ‡ gS_ P † ‡ e_ Š, G_ ˆ lT ‰aA

…34†

  _ P _ _ _G ˆ lT a_ Ntu mc ‡ T0 …ST ‡ gSP † ‡ e_ , U

…35†

where a_ ˆ ls_T …kW mÿ2 †, g ˆ llTP , and e_ ˆ lzT0 …kW†:At minimum G_ dG_ ˆ 0: d Ntu _ cP, U, and since z_ 0 is not a function in the size of the storage unit, we ®nd For given m,   _ P d Ntu mc _ _ ‡ T0 …ST ‡ gSP † ˆ 0: ˆ a_ U d Ntu

…36†

…37†

_ In¯uence of g on G_ is important only The values of S_ T and S_ P are calculated at minimum G. _ when the contribution of SP to total exergy loss is very large, a case which is not likely to exist in practice. From the foregoing presentation, we conclude that the parameter a_ is the only one _ which contributes to the performance of the storage system at minimum values of G. _ Equation (37) indicates real values of lT, lP, and z0 are not required to perform the analysis, as a lack of data might exist.

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701

Fig. 7. In¯uence of the charging temperature on Ntu.

RESULTS AND DISCUSSION

For the next discussion we assume the working ¯uid is air with ideal gas behavior, also we assume there is no change of phase of the storage media. The cost of the storage medium is included in the ®xed cost of the storage unit. For convenience we let the mass ¯ow ratio (m_ c =m_ d ) = 1.0. In¯uence of a_ on the performance of the storage unit at the minimum total price  ˆ 0:05 For the next discussion values of practical interest are selected, such that: G ÿ2 ÿ1 (G = 17.1 kg m s ); b = 0.1 (TI=1.1 T0); Td=T0=298 K; charging temperature = 0.2 (Tc=1.2 T0); cP=1.0 kJ kgÿ1 Kÿ1; Pr = 0.71. A storage medium with M equal to 1000 kg, and c = 4.2 kJ kgÿ1 Kÿ1 is selected. The bene®t of such selection will enable us to compare the results with that given by Krane [2] and DomanÂski and Fellah [3] An analysis based on exergy concept alone results in an optimum Ntu of 5.6, optimum Z2nd of 0.2663 and optimum yc of 0.86, with corresponding Z1st of 0.5768. Those values are in excellent agreement with that given by [2, 3], and are shown as horizontal lines, in Fig. 4(a) and (c), Fig. 5(a) and (c) respectively.Values of a_ greater than 1.0 indicate the contribution of the capital cost to the total cost is more than that due to exergy loss by thermal e€ects. Thus, smaller _ Fig. 4(a). Consequently, substantial values of Ntu would be selected with an increase in a, increases in NQ and in yc are obtained, Fig. 4(b) and Fig. 5(a). The changes in NT and NP are _ small, resulting in a decrease in the second law with an increase in a. A slight reduction in yyc with a_ (Fig. 5(b)) causes a modest increase in Z1st, this is predicted by Equation (11), and shown in Fig. 5(c). That means, as we elongate the charging time, the ®nal storage temperature approaches the charging one. More energy is stored at the expense of the exergy loss.

R. DomanÂski and G. Fellah

702

Fig. 8. In¯uence of ¯ow rate on Ntu.

For values of a_ less than 1.0, a storage unit with higher Ntu should be used to reduce the destruction of exergy. Better heat transfer characteristics are obtained, resulting in a substantial increase in Z2nd. In¯uence of charging temperature The foregoing values have been adopted for the next discussion, except the charging temperature which varies between 0.4 (1.4 T0) and 2 (3 T0). For a given Ntu an increase in the charging temperature would increase irreversibility due to the temperature di€erence between the working ¯uid and the storage medium. To reduce this e€ect a larger Ntu should be employed. However, this may be restricted by the cost of the storage unit itself. Figures 6 and 7 show the e€ect of the charging temperature at the minimum total price on the Ntu and the second-law eciency _ For instance when Tc=0.8 we ®nd: for distinct values of a. _ for a=0.5, Ntu = 2.4 and Z2nd=21.34%, _ for a=1, Ntu = 1.7 and Z2nd=17.51%, _ for a=2, Ntu = 1.0 and Z2nd=10.64%. _ and we may reach the condition where Both Ntu and Z2nd are degrading with an increase in a, owning a storage unit is unreasonable. In¯uence of the mass ¯ow rate A reasonable charging temperature for solar energy applications is selected for the next discussion, that is tc=0.4 (Tc=417.2 K). The mass ¯ow rate varies between G = 0.01 and 0.1. All other parameters are selected as before. In¯uence of ¯ow rate on Ntu and Z2nd at minimum total cost is shown in Figs 8 and 9. A _ value of a=0.5 produces Ntu 0 1.0 and Z2nd010%; larger values of a_ would suggest even smaller Ntu and Z2nd.

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703

Fig. 9. In¯uence of the ¯ow rate on the second-law eciency.

For G = 0.05, we ®nd _ for a=0.1, Ntu = 2.5 and Z2nd=19.47%, _ for a=0.25, Ntu = 1.7 and Z2nd=15.70%, _ for a=0.5, Ntu = 1.0 and Z2nd=9.47%. _ and then there must be a limit Again both Ntu and Z2nd are degrading with an increase in a, where storing energy becomes impractical from a thermoeconomic viewpoint. CONCLUSION

The study focuses on the advantages of utilizing thermoeconomic aspects in designing and operating thermal energy storage systems. The trade-o€ between cost of the irreversibility rate and that of the storage system, formed the basis for the technique of thermoeconomic optimization. Large values of a_ implies owning a thermal storage unit with small surface area. When the ¯ow condition is pre-speci®ed, small value of Ntu and low second-law eciency may be obtained. Optimum values of Ntu and Z2nd that are obtained from both exergy and thermoeconomic analyses approach each other only when the contribution of the capital cost to total cost is negligible. The results are quite consistent with reality. However, the analysis provides an important tool for the engineer to select the right storage unit for a given application. REFERENCES 1. A. Bejan, Two thermodynamic optima in the design of sensible heat units for energy storage. ASME J. Heat Transfer 100, 708±712 (1978). 2. R. J. Krane, A second law analysis of the optimum design and operation of thermal energy storage systems. Int. J. Heat Transfer 30, 43±57 (1987). 3. R. DomanÂski and G. Fellah, Exergy as a tool for designing and operating thermal storage units. Biuletyn Instytutu Technik Cieplnej Politechniki Warszawskiej 81, 24±45 (1995).

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4. H. BjurstroÈm and B. Carlson, An exergy analysis of sensible and latent heat storage. Heat Recovery Systems 5, 233± 250 (1985). 5. G. A. Adebiyi and L. D. Russell, A second law analysis of phase-change thermal energy storage systems. ASME, HDT 80, 9±20 (1987). 6. S. A. Saborio, H. Nakamura and G. M. Reistad, Optimum eciencies and phase change temperatures in latent heat storage systems. ASME J. Energy Resources Technol. 116, 79±86 (1994). 7. M. A. Badar, S. M. Zubair and A. A. Al-Farayedhi, Second-law-based thermoeconomic optimization of a sensible thermal energy storage system. Energy 18, 641±649 (1993). 8. M. A. Badar and S. M. Zubair, On thermoeconomics of a sensible heat, thermal energy storage system. ASME J. Solar Energy Engineering 117, 255±259 (1995).