Optimization of the performance characteristics in an irreversible regeneration magnetic Brayton refrigeration cycle

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1 . Article. SCIENCE CHINA Physics, Mechanics & Astronomy February 2012 Vol. 55 No. 2: doi: /s Optimization of the performance characteristics in an irreversible regeneration magnetic Brayton refrigeration cycle WANG Hao & WU GuoXing Tianhua College, Shanghai Normal University, Shanghai , China Received August 19, 2011; accepted October 24, 2011; published online January 9, 2012 A model of the irreversible regenerative Brayton refrigeration cycle working with paramagnetic materials is established, in which the regeneration problem in two constant-magnetic field processes the irreversibility in two adiabatic processes are considered synthetically. Expressions for the COP, cooling rate, power input, the minimum ratio of the two magnetic fields, etc., are derived. It is found that the influence of the irreversibility the regeneration on the main performance parameters of the magnetic Brayton refrigerator is remarkable. It is important that we have obtained several optimal criteria, which may provide some theoretical basis for the optimal design operation of the Brayton refrigerator. The results obtained in the paper can provide some new theoretical information for the optimal design performance improvement of real Brayton refrigerators. paramagnetic salt, Brayton refrigeration, irreversibility, regeneration, performance characteristics, optimum criterion PACS numbers: Sg, Mc Citation: Wang H, Wu G X. Optimization of the performance characteristics in an irreversible regeneration magnetic Brayton refrigeration cycle. Sci China-Phys Mech Astron, 2012, 55: , doi: /s The conventional vapor compression refrigerator widely used in many fields has many disadvantages such as the energy efficiency being difficult to improve further, the gases used enhance the greenhouse effect or deplete the ozone layer, so on. Magnetic refrigeration is an environment-safe refrigeration technology which has been studied for a long time [1 5] for its potential use in refrigeration cycles as an alternative to gases, especially at low temperatures. As more more attention is paid to environmental protection investigations on room-temperature magnetic refrigeration materials, magnetic refrigeration theory technology have had a great development [6 11]. Optimal analysis design of magnetic refrigeration cycles [12 17] have become one of the interesting research subjects for people working in thermodynamics statistical physics. Moreover, most of the literature has only researched the optimal performance characteristics of a magnetic Stirling [18 20] or Ericsson refrigeration cycle [21 24]. Corresponding author shnuwh@163.com The magnetic Brayton refrigeration cycle is one of the important refrigeration cycles has some distinct merits that are noteworthy in the research manufacture of roomtemperature magnetic refrigerators. Although the analysis design of some magnetic Brayton refrigeration cycles has been undertaken [25 27], research on the performance of the magnetic Brayton refrigeration cycle is relatively rare [28 30]. In fact, the magnetic Brayton refrigeration cycle is also a significant regenerative cycle. Therefore, it is necessary to consider both the regenerative other irreversible effects on a magnetic Brayton refrigeration cycle. In the article, we investigate the main characteristics of an irreversible regenerative magnetic Brayton refrigeration cycle. The paper is organized as follows. In sect. 1, the main thermodynamic properties of a paramagnetic salt are analyzed. In sects. 2 3, an irreversible regenerative magnetic Brayton refrigeration cycle is established expressions of the COP, cooling rate power input are derived. The characteristic curves between these three important performance parameters the temperatures of the working substance at the two important state points are plotted. In sect. 4, by c Science China Press Springer-Verlag Berlin Heidelberg 2012 phys.scichina.com

2 188 Wang H, et al. Sci China-Phys Mech Astron February 2012 Vol. 55 No. 2 means of numerical calculation, some important performance bounds the optimal operating regions of the magnetic refrigeration cycle are further analyzed. 1 Main thermodynamic properties of magnetic systems It may be shown by using statistical mechanics that the partition function of a simple paramagnetic salt system is given by [31] sinh 2 j+1 2 j x Z = sinh 1 2 j x N, 1 where x = gμ B jh, j is the quantum number of the angular momentum, g is known as Le s factor, μ B is the Bohr kt magneton, k is the Boltzmann constant, H is the magnetic field, T is the absolute temperature N is the number of magnetic moments. Using eq. 1 the Helmholtz free energy A = kt ln Z, we can calculate the entropy, internal energy, magnetization heat capacity at constant field as: A S = T H { [ = Nk ln sinh C H = 2 j j ] [ ] } 1 x ln sinh 2 j x xb j x, 2 U = A + TS = NkTxB j x, 3 M = U T H = Nkx 2 A = Ngμ B jb j x, 4 H T 2 j j + 1 x csc h 2 x 2 j 2 j j x csc h 2 2 j x, 5 where B j x = 2 j j + 1 coth x j 2 j j coth j x is the Brillouin function. At high temperatures or weak magnetic fields, x << 1, eq. 4 may be simplified as: M = CH T, 6 where C = Ng2 μ B j j + 1 is the Curie constant. In such a 3k case, eqs. 2, 3 5 may be simplified, respectively, as: S = A = Nk ln 2 j + 1 CH2 T H 2T 2, 7 U = A + TS = CH2 T, 8 U C H = = CH2 T H T. 9 2 Based on the above equations, the performance characteristics of an irreversible regenerative magnetic Brayton refrigerator will be analyzed evaluated. 2 An irreversible regenerative magnetic Brayton refrigeration cycle An irreversible regenerative magnetic Brayton refrigeration cycle consists of two adiabatic processes two magnetic processes at constant magnetic field. Its temperature-entropy diagram is shown in Figure s 1 are the two constant-field processes at field values H 1 H 2 H 1 > H 2, respectively. 1 2s 4 5s are the two reversible adiabatic processes; are the two irreversible adiabatic processes; the regeneration is carried out during the processes ; Q H Q L are the amounts of heat exchange between the working substances the heat reservoirs; Q R is the amount of regeneration; T H T L are the temperatures of the high- low-temperature reservoirs, the temperatures of state points 1, 2, 2s,3,4,5,5s 6 are represented by T 1, T 2, T 2s,, T 4, T 5, T 5s, respectively. The temperatures of the working substance have the following relation: T 5s < T 5 < < T L < T 1 < T 4 < T H < < T 2s < T S Q L 5 5s H 2 6 T L 1 Q R 4 T Figure 1 The temperature-entropy diagram of an irreversible regenerative magnetic Brayton refrigeration cycle. T H Q H 3 H 1 2s 2

3 Wang H, et al. Sci China-Phys Mech Astron February 2012 Vol. 55 No According to the cycle model mentioned above eq. 9, Q R, Q H Q L may be expressed, respectively, as: Q R = Q 34 = Q 61 = CH T 4 1 = CH T 1, 11 Q H = Q 23 = CH T 2 1, 12 Q L = Q 56 = CH 2 2 Based on eq. 7 we have 1 T T 2s T 1 = T 4 T 5s = H 1 H 2, 14 where = H 1 /H 2 > 1 is the ratio of the two magnetic fields. For a real magnetic refrigerator, the related processes of the cycle are generally irreversible. In order to describe the irreversibility in the two adiabatic processes, we introduce the compression expansion efficiencies η e = T 4 T 5 T 4 T 5s, 15 η c = T 2s T 1 T 2 T By using eqs , the temperatures at state points 1, 2, 4 5 of the working substance may be expressed as: T 1 = αt H = α τ T L, 17 T 2 = α + η c 1 T L, 18 η c τ T 4 = α 2 α + α 2 1τ T L, 19 T 5 = α η e + η e α + α 2 1τ T L, 20 where τ = T L /T H is the temperature ratio of the two reservoirs α = T 1 /T H is a regeneration factor. It is easily found from Figure 1 that, when α = 1, the cycle is operated in the case with a maximum regeneration, 1 i.e., Q R = CH2 2 1 ;whenα = τ, the cycle is operated in the case without regeneration. Thus, the range of α T L T H is τ α 1. Substituting eqs into eqs gives Q H = C 2 H τη c α + η c 1 T L, 21 Q L = CH 2 2 α + α 2 1τ α η e + η e T L 3 Expressions for several important parameters The COP, cooling rate power input are three important performance parameters which are often considered in the optimal design theoretical analysis of refrigerators. In order to optimize the power input cooling rate of the Brayton refrigeration cycle, we have to calculate the cycle period. Owing to heat transfer irreversibility, the temperatures of state points 3 6 are different from T H T L.Because of heat resistance, the temperature should be higher than T H, the temperature should be lower than T L,as shown in Figure 1. It is often assumed that the heat transfer between the working substance the heat reservoirs obeys the Newtonian law [18,32], such that the amount of heat transferred in the two iso-magnetic field processes, Q H Q L, may be expressed as [33,34]: Q H = k 1 LMTD 1 t 1, 23 Q L = k 2 LMTD 2 t 2, 24 the heat transfer in the regenerator is given by [35] Q R = Q 34 = Q 61 = k R LMTD R t R, 25 where k 1, k 2, k 3 k 4 are the thermal conductances between the cyclic working substance the heat reservoirs at temperatures T H T L, t 1, t 2 t R are the corresponding heatexchange times, LMTD 1 = T 2 T H T H ln T, 26 2 T H T H LMTD 2 = T L T 5 T L ln T, 27 L T 5 T L LMTD R = T 1 T 4 ln T T 1 T 4 Note that there is no heat exchange in the two adiabatic processes, thus, the time of the adiabatic processes is so small compared with that of the iso-magnetic field processes that it may be neglected. For this reason, the period of an irreversible regenerative magnetic Brayton refrigerator is approximately equal to the sum of t 1, t 2 t R, i.e., where t = t 1 + t 2 + t R, 29 { [ CH 2 TH α 1 + ηc α 1 ] } 2 2 η c ln η c T H t 1 =, αk 1 T H + η c 1

4 190 Wang H, et al. Sci China-Phys Mech Astron February 2012 Vol. 55 No. 2 t 2 = t R = CH 2 2 α τ + α 2 τ [ ] τt H 1 + αη e η e α τ+α ln 2 τ τt H, αβτk 2 T H η e + η e α τ + α CH2 2 2 τ 2 1 αt H ln α 2 τt H α 2 τt H α τ+α 2 τ [ 2 ], τ k R + αt H α τ + α 2 τ + 1 or 1 CH2 2 1 αt H ln αt H α 2 τt H α τ+α t R = 2 τ [ ]. 2 τ k R + αt H α τ + α 2 τ + 1 Using eqs one can derive the mathematical expressions for the COP, cooling rate power input for the irreversible regenerative magnetic Brayton refrigeration cycle, respectively, as: α + τα 2 τ R = Q L t P = W t = ε = Q L W = ατt H [ 1 ηe ] α+α 2 τ τ τη e η e αt H 3 η c +η c 1, 30 α + τα 2 τ ατt H η e + η e 1 A 2 η c αk 1 T H η c B α τ + α 2 τ, 31 α τ+α 2 E2 τ ατk 2 T H + η e η e + α 2 τt H 1 [ k R T3 + αt 2 τ H α τ+α 2 τ + 1] [ α + α 2 τ τ αt H τ η e η e 3 ] η c + η c 1 = A 2 η c αk 1 T H η c B α τ + α 2 τ, 32 α τ+α 2 E2 τ ατk 2 T H + η e η e + α 2 τt H 1 [ k R T3 + αt 2 τ H α τ+α 2 τ + 1] where { [ TH α 1 + ηc α 1 ] } A = ln, T H η c [ τt H 1 + α η ] e η e α τ + α B = ln 2 τ τt H αt H E = ln. α 2 τt H α τ + α 2 τ It can be seen from eqs that, besides the thermal conductances k 1, k 2, k 3 k 4, the adiabatic irreversibility factors η c η e, the temperatures, the regeneration factor α the ratio of the two magnetic fields also play important roles in the performance of the irreversible regenerative magnetic Brayton refrigeration cycle. Hence, it is necessary to discuss deeply the impacts of the parameters on the optimal performance characteristics. From eqs , we can obtain three-dimensional performance characteristic curves of the COP, cooling rate power input varying with for other given parameters, as shown in Figures 2 4, where R = R/kT H is the dimensionless cooling rate P = P/kT H is the dimensionless power input. It can be seen from Figure 2 that the cooling rate first increases then decreases as the temperature increases, while it is only a monotonically increasing function of. This indicates that there exists a maximum value for the cooling rate when attain certain values Figure 2 Color online The three-dimensional graph of R varying with the variables for the parameters T H = 250 K, τ = 0.90, α = 0.98, η c = η e = 0.98, k 1 = k 2 = k 3 = k 4 = 2.5. R

5 Wang H, et al. Sci China-Phys Mech Astron February 2012 Vol. 55 No α τ + α 2 τ + = ατk 2 T H τt H η e + η e ε Figure 3 Color online The three-dimensional graph of ε varying with the variables for the parameters T H = 250 K, τ = 0.90, α = 0.98, η c = η e = 0.98, k 1 = k 2 = k 3 = k 4 = τ where F = + αt H α τ + α 2 τ + 1. This implies that when the cooling rate of the irreversible regenerative magnetic Brayton refrigeration cycle attains its optimal value, of the working substance at state points 3 6 are not arbitrary must satisfy eq. 33. It should be pointed out that eq. 33 is a transcendental equation. Based on eqs , by means of a numerical approach, the optimal performance of the irreversible regenerative magnetic Brayton refrigerator will be discussed further in the following sections. P Figure 4 Color online The three-dimensional graph of power input P varying with the variables for the parameters T H = 250 K, τ = 0.90, α = 0.98, η c = η e = 0.98, k 1 = k 2 = k 3 = k 4 = 2.5. Moreover, Figures 3 4 show the COP is only a monotonically increasing function of power input is only a monotonically decreasing function of. This implies that when related thermodynamic parameters including satisfy eqs , an irreversible regenerative magnetic Brayton refrigeration cycle employing paramagnetic materials as the working substance will operate in the optimal operating condition. Using a refrigerator, one always wants to obtain a cooling rate as large as possible for a given power input. For this reason, substituting eq. 31 into the extreme condition R/ = 0, one can obtain t 1 + t 2 + t R T α τ + α 2 τ ατt H η e + η e 1 α τ + α 2 τ 2 1 α 2 τt H Fk R α 2 τt H α τ + α 2 τ B α τ + α 2 τ + ατk 2 T 2 6 T H η e η e t R F 4 Several important criteria 4.1 The ratio of the two magnetic fields on the cyclic performance For a refrigerator, both the COP ε cooling rate R of the cycle are always required to be larger than zero. One can easily prove that, from eqs , the feasible region for the ratio of the two magnetic fields should be η e + > [ ] η 2 e + 4α τ T H 1 ηe T H 2 [ 1 η e T H ] min. 34 Consequently, min is one important parameter of the irreversible regenerative magnetic Brayton refrigeration cycle determines the minimum ratio of the two magnetic fields. By using eqs. 30, 31 33, one can also plot R opt - ε R - optimum characteristic curves as shown in Figures 5 6. It can be seen from Figures 5 6 that the optimal cooling rate the corresponding ε R decrease as the ratio of the two magnetic fields decreases in the region of < ε, while the optimal cooling rate R opt the corresponding ε R decrease as the compression ratio of two isobaric processes increases in the region > R. So the ratio of the two magnetic fields for an irreversible regenerative magnetic Brayton refrigerator should be ε R, 35 where ε R are two important performance parameter bounds of the irreversible regenerative magnetic Brayton refrigerator. They determine the lower upper bounds on the ratio of the two magnetic fields. It also can be seen in Figures 5 6 that the cooling rate COP decrease as the regeneration factor α increases when the refrigeration cycle is operated in the optimal region. That is to say, the larger the amount of regeneration, the smaller the cooling rate COP. In addition, both the

6 192 Wang H, et al. Sci China-Phys Mech Astron February 2012 Vol. 55 No. 2 ε R ε Rmax α =1.00 ηc= η e =0.90 R opt R optmax R optm 0.04 α =1.00 η 0.02 c= η e = R R ε Rm ε R ε Rmax 0.0 ε Figure 5 Color online The COP ε R versus ratio of magnetic fields = H 1 /H 2 curves for the parameters T H = 250 K, τ = 0.90 k 1 = k 2 = k 3 = k 4. R opt R optmax R α =1.00 ηc= η e = Figure 6 Color online The optimal dimensionless cooling rate R opt versus ratio of magnetic fields = H 1 /H 2 curves for the parameters T H = 250 K, τ = 0.90 k 1 = k 2 = k 3 = k 4. cooling rate COP always increase as the adiabatic irreversibility factors η c η e increase. 4.2 The characteristics of R opt, ε R the performance bounds By using eqs. 30, 31 33, one can plot the R opt -ε R curves. Figure 7 indicates that, for an irreversible regenerative magnetic Brayton refrigerator, there exists a maximum optimal cooling rate R opt max with the corresponding COP ε Rm a maximum COP ε R max with the corresponding optimal cooling rate R optm. The optimally operating region of the Brayton refrigerator should be situated in R optm R opt R opt max, 36 ε Rm ε R ε R max, 37 Figure 7 Color online The optimal dimensionless cooling rate R opt versus the COP ε R curves for the parameters T H = 250 K, τ = 0.90 k 1 = k 2 = k 3 = k 4. where R opt max, R optm, ε R max ε Rm are four important performance bounds of an irreversible regenerative magnetic Brayton refrigeration cycle. Furthermore, for the irreversible regenerative magnetic Brayton refrigerator, the optimal cooling rate the COP depend closely on the adiabatic irreversibility factors η c η e the regeneration factor α besides the temperature ratio τ of the two reservoirs. From Figure 7 we can also find that R opt max, R optm, ε R max ε Rm all decrease as the adiabatic irreversibility factors η c η e decrease. However, R opt max, R optm, ε R max ε Rm decrease as the regeneration factor α increases. 4.3 The characteristics of P opt, P optr, P optε the performance bounds The optimal power input P opt is an important thermodynamic parameter of an irreversible regenerative magnetic Brayton refrigerator, as shown in Figures 8 9. It can be clearly seen from Figure 8 that there exists a maximum optimal cooling rate R opt max when P opt = P optr. When R opt < R opt max there exist two different optimal power inputs for a given optimal cooling rate R opt, where one is smaller the other is larger than P optr. Obviously, regardless of COP ε R, the optimal power input should not be chosen to be larger than P optr because the optimal cooling rate R opt decreases as the optimal power input P opt increases when P opt > P optr. Thus, the irreversible regenerative magnetic Brayton refrigerator should be operated in the reasonable region of P opt P optr. Similarly, by analyzing Figure 9, we can see that there exists a maximum COP ε R max when P opt = P optε. If only the COP is considered, the optimal region of the power input should be P opt P optε. / In general, based on P optε = R optm εr max, P optr = / R opt max εrm the optimal operating region mentioned in eqs , we should appropriately control the optimal power input the optimal values of the power input should be located in P optε P opt P optr. 38

7 Wang H, et al. Sci China-Phys Mech Astron February 2012 Vol. 55 No P opt α =1.00 ηc= η e =0.90 performance improvement of real Brayton refrigerators. This work was supported by the Program for Excellent Young Teachers of Shanghai, China Grant No. thc P optr R opt R optmax Figure 8 Color online The optimal dimensionless power input P opt versus optimal dimensionless cooling rate R curves for the parameters opt T H = 250 K, τ = 0.90 k 1 = k 2 = k 3 = k 4. P opt P opt ε α =1.00 ηc= η e = ε R ε Rmax Figure 9 Color online The optimal dimensionless power input P opt versus the COP ε R curves for the parameters T H = 250 K, τ = 0.90 k 1 = k 2 = k 3 = k 4. 5 Conclusions We have established a model of the irreversible regenerative Brayton refrigeration cycle working with paramagnetic materials, in which the regeneration problem in two constantmagnetic field processes the irreversibility in two adiabatic processes are considered synthetically. Expressions for the COP, cooling rate, power input, the minimum ratio of the two magnetic fields, etc., are derived. It is found that the influence of the irreversibility the regeneration on the main performance parameters of the magnetic Brayton refrigerator is remarkable. It is important that we have obtained several optimal criteria, which may provide some theoretical basis for the optimal design operation of the Brayton refrigerator. The results obtained in the present paper can provide some new theoretical information for the optimal design 1 Warburg E. Magnetische untersuchungen. I. Uber einige wirkungen der coercitivkraft. Ann Phys, 1881, 13: Yan Z, Chen J. The characteristics of polytropic magnetic refrigeration cycles. J Appl Phys, 1991, 70: Huang W, Teng C. A simple magnetic refrigerator evaluation model. J Magn Magn Mater, 2004, 282: Siddikov B, Wade B. Numerical simulation of the active magneticregenerator. Comput Math Appl, 2005, : Brück E. Developments in magnetocaloric refrigeration. J Phys D: Appl Phys, 2005, 3823: Zimm C, Jastrab A, Sternberg A, et al. Description performance of a near-room temperature magnetic refrigerator. Adv Cryog Eng, 1998, 43: Tegus O, Brück E, Buschow K, et al. Transition-metal-based magnetic refrigerants for room-temperature applications. Nature, 2002, 415: Brück E, Tegus O, De Boer F. Magnetic refrigeration towards roomtemperature applications. Phys B, 2003, 319: Yu B, Gao Q, Zhang B, et al. Review on research of room temperature magnetic refrigeration. Int J Refrig, 2003, 26: Li P, Gong M Q. A practical model for analysis of active magnetic regenerative refrigerators for room temperature applications. Int J Refrig, 2006, 29: Yu B F, Zhang Y. Research on performance of regenerative room temperature magnetic refrigeration cycle. Int J Refrig, 2006, 29: Hakuraku Y. Thermodynamic simulation of a rotating Ericsson cycle magnetic refrigerator without a regenerator. J Appl Phys, 1987, 62: Chen L, Yan Z. Main characteristics of a Brayton refrigeration-cycle of paramagnetic salt. J Appl Phys, 1994, 75: Wang J T, Chen J. Influence of several irreversible losses on the performance of a ferroelectric Stirling refrigeration-cycle. Appl Energy, 2002, 72: He J, Chen J. Regenerative characteristics of electrocaloric Stirling or Ericsson refrigeration cycles. Energy Convers Manage, 2002, 43: Kitanovski A, Egolf P W. Thermodynamics of magnetic refrigeration. Int J Refrig, 2006, 29: Brown G V. Magnetic heat pumping near room temperature. J Appl Phys, 1976, 47: Wu F, Chen L, Sun F, et al. Optimization of irreversible magnetic Stirling cryocoolers. Int J Eng Sci, 2001, 39: Lin G, Tegus O, Zhang L, et al. General performance characteristics of an irreversible ferromagnetic Stirling refrigeration cycle. Phys B, 2004, 344: Tyagi S K, Lin G, Kaushik S C, et al. Thermoeconomic optimization of an irreversible Stirling cryogenic refrigerator cycle. Int J Refrig, 2004, 27: He J, Chen J, Wu C. Optimization on the performance characteristics of a magnetic Ericsson refrigeration-cycle affected by multiirreversibilities. J Energy Resour Technol, 2003, 125:

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