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Dimensionless numerical model for simulation of active magnetic regenerator refrigerator
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Dimensionless numerical model for simulation of active magnetic regenerator refrigerator A. Sarlah*, A. Poredos Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, SI-1000 Ljubljana, Slovenia
article info
abstract
Article history:
In order to obtain a better reliability, consistency and accuracy of results obtained with
Received 4 December 2007
a numerical simulation of an AMRR (active magnetic regenerator refrigerator), a dimen-
Received in revised form
sionless numerical model was developed, which can equally be used for determination of
20 January 2010
regenerator’s heat transfer coefficient and simulation of passive heat regenerators or
Accepted 5 April 2010
AMRR operation. Regenerator’s heat transfer coefficient af, is a crucial input parameter in
Available online 9 April 2010
the simulation of AMRR operation and has a primal effect on the outcome of a solution. This paper deals with a derived dimensionless model and discusses errors involved when
Keywords:
using different models for heat transfer coefficient and AMRR operation simulation. ª 2010 Elsevier Ltd and IIR. All rights reserved.
Magnetic refrigerator Modelling Simulation Heat transfer Regenerator
Mode`le nume´rique adimensionnel pour la simulation d’un re´frige´rateur magne´tique a` re´ge´ne´rateur actif Mots cle´s : Re´frige´rateur magne´tique ; Mode´lisation ; Simulation ; Transfert de chaleur ; Re´ge´ne´rateur
1.
Introduction
It has been 10 years since the discovery of the giant magnetocaloric effect in the Gd5(Si2Ge2) compound (Gschneidner et al., 2005), which started an era of increased research in magnetic refrigeration. A decent number of prototypes (Gschneidner et al., 2005; Zimm et al., 2005; Okamura et al., 2005; Rowe et al., 2005) have been built and tested during this time, all of them running at near-room temperature more or less successfully. In order to theoretically verify
experimental results of the operation of prototypes, various numerical models have also emerged (Engelbrecht et al., 2005; Petersen et al., 2007; Sarlah and Poredos, 2005). Putting them all together, there seems to be an increasing need to develop a model which could accurately simulate operation of an AMR refrigerator and that could allow a better comparison of various refrigerators using different magnetic materials, operational setups, magnetic field strengths, etc. Such model could significantly contribute to the pre-design analysis, thus cutting the time and money needed to find an
* Corresponding author. Tel.: þ386 1 4771 446; fax: þ386 1 2518 567. E-mail address: [email protected] (A. Sarlah). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.04.003
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Nomenclature packing area density (m2 m3) heat transfer area (m2) cross-section area (m2) specific heat (J kg1 K1) hydraulic diameter (m) dispersion coefficient (-) modified Eckert number (-) friction factor (-) magnetic field strength (A m1) Colburn factor (-) length (m) mass flow (kg s1) M magnetization (A m1) NTU number of transferred units (-) p pressure (Pa) PeL (PeLf) (modified) Peclet number (-) Ph Ph dimensionless number (-) Pr modified Prandtl number (-) aspect ratio (L/dh) (-) Ra Re Reynolds number (-) Sth Sth dimensionless number (-) t time (s) T temperature (K) u internal energy (J) fluid interstitial velocity (m s1) vf
ap Aht Ar c dh Dd Ec f H j L _ m
optimal AMR design, which can later be manufactured and experimentally tested. One crucial parameter that directly determines efficiency of the AMR is heat transfer coefficient, which needs to be experimentally tested and numerically evaluated in order to determine its thermal e j(Re) e and hydraulic e f(Re) e properties. Doing so, it is reasonable to use the same tool e same numerical model, which includes all influential thermo-dynamical and hydro dynamical properties e for both, determination of heat transfer coefficient and simulation of AMRR operation. Any discrepancies between them can result in initial errors which are then further escalated in the numerical simulation. Heggs and Burns (1988) and Shah and Zhou (1997) have made a comparison of various techniques for prediction of heat transfer coefficients when performing a single-blow experiment with compact heat exchangers. They have concluded that there are significant differences between them, i.e. results obtained by shape factor, maximum slope, differential fluid enthalpy method or direct curve matching method (using least squares matching technique) can differ for as much as 20% (Heggs and Burns, 1988), which is significantly more than the uncertainty of particular method (between 4 and 10%). Furthermore, a lot of data on heat transfer coefficients of various heat exchangers, which is available in publications, is for certain regenerator geometries (packed bed, honeycomb,.), which can differ from the ones used in magnetic refrigerators e since introduction of AMR refrigerators there has been a constant need to discover new, advanced regenerator geometries with higher efficiencies. Available data was also obtained using certain
vs V x
specific volume (m3 kg1) volume (m3) coordinate (m)
Greek a e F G l m0 r q s J x
heat transfer coefficient (W m2 K1) porosity (-) utilization factor (-) dimensionless mag. field (-) conductivity (W m1 K1) permeability of vacuum (Wb A1 m1) density (kg m3) dimensionless temperature (-) dimensionless time (-) inlet forcing function (-) dimensionless length (-)
Subscripts ad-H/ad-C adiabatic heating/cooling C/H cold/hot HHEX/CHEX hot/cold heat exch. eff effective exp experimental f fluid H magnetic field init initial r regenerator s solid
experimental methods (single blow, heating-only,.) and numerical models (Heggs and Burns, 1988; Kays and London, 1964), which in many cases did not include all physical phenomena that are important and significant for description of AMRR operation; i.e. such as longitudinal heat conduction in solid and fluid dispersion. Objective of this work is to present a dimensionless model which can, with certain assumptions, be equally used for both purposes: determination of regenerator’s heat transfer coefficient and simulation of AMR refrigerator.
2.
Determination of heat transfer coefficient
Heat transfer coefficient of compact heat exchangers can only be determined by using transient test techniques together with transient methods, developed for certain range of core NTU numbers. Most common test technique is singleblow technique, which is described in details in several publications together with proposed features of the experimental setup (Heggs and Burns, 1988; Kaviany, 1995; Shah , 2003; Mullisen and Loehrke, 1986) and thus and Sekulic will not be further discussed here. However, single-blow techniques must be accompanied with a transient method, which evaluates experimental data in order to obtain proper heat transfer coefficients. There exist several methods (Heggs and Burns, 1988; Shah and Zhou, 1997; Mullisen and Loehrke, 1986), each with its advantages and disadvantages, while mostly recommended is one proposed by Mullisen and Loehrke (1986). They proposed a method,
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 0 6 1 e1 0 6 7
Temperature
Tfin − exp Tfout − exp Tfout − num
Time Fig. 1 e Single-blow experiment: matching of experimental and numerical data.
drop. For a purpose of determination of heat transfer coefficient we can replace vp/vx in Eq. (1) with Dpexp/L [Eq. (14)], which can be measured during the experiment. Pressure drop across regenerator is in most models neglected, since its significance is small e however, it should be accounted when dealing with high velocity gradients and large viscosity, i.e.: high pressure drop exchangers or high viscous fluids. In our case it is included in derivation in order to be consistent with a later discussed model of an active magnetic regenerator in which pressure drop is important due to its impact on efficiency. In derivation of Eqs. (1) and (2) we have assumed constant fluid mass flow rate during the experiment and constant properties of the solid. Both equations can be written in dimensionless form with introduction of dimensionless length [Eq. (3)], time [Eq. (4)] and temperature [Eq. (5)], defined as: x¼
which allows arbitrary fluid inlet temperature variation (inlet forcing function) and determines appropriate heat transfer coefficient by performing iterative process of matching experimentally and numerically obtained fluid outlet temperature curves until they match within a certain agreement (Fig. 1), i.e. minimum of least squares between theoretical and experimental data. Their method allows use of various numerical models, which can be individually adjusted to an observed physical problem. Basis of a current model for determination of the heat transfer coefficient of regenerators is a well-known equation for a change of internal energy of a material, expressed with enthalpies (Byrd et al., 2002). With below mentioned assumptions, both equations e for fluid and solid e can, according to Fig. 2, be written as: vTf vTf _ þ mðtÞ$c þ af $ap $Ar Tf Ts rf Tf $cf Tf $Ar $3$ f Tf $ vt vx _ mðtÞ vp $ ¼ rf Tf vx
vTs v Ts þ af $ap $Ar $ Ts Tf leff $Ar $ 2 ¼ 0 rs $cs $Ar $ð1 3Þ vt vx
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x L
(3)
_ f $cf Tf vf m x x ¼ t $ t s ¼ F Tf $ rs $cs $ð1 eÞ$Vr L vf vf qi ¼
Ti ðx; tÞ Tmin Tmax Tmin
i ¼ f; s
(4)
(5)
Doing so we obtain: vqf þ NTU qf $ qf qs ¼ Ph vx
(6)
2 v2 qs vqs 1 v qs $ þ NTU qf $ qs qf 2$F qf $ 2 vs vx$vs PeLf qf vx 2 v2 qs þ F qf $ 2 ¼ 0 vs where following dimensionless numbers are used:
(1)
2
(2)
In Eq. (1) we have added a term which sums up irreversible rate of internal energy increase by viscous dissipation, and have left out heat conduction of a fluid (fluid dispersion) e the latter is included in the equation for a solid [Eq. (2)] as leff [Eqs. (11) and (12)], while viscous dissipation term was replaced with a term representing a result of acting viscous forces: pressure
NTU e number of transferred units: af $ap $Vr NTU qf ¼ _ f $cf qf m F e utilization factor: rf qf $cf qf $3 F qf ¼ rs $cs $ð1 3Þ
(8)
(9)
PeLf e modified Peclet (PeL) number: 1 1 1 leff 1 ¼ $ ¼ $ PeLf qf ReL qf $Preff qf 3 vf $L$rf qf $cf qf 3
(10)
where leff is effective heat conductivity defined for plate [Eq. (11)] or packed bed [Eq. (12)] heat exchangers:
Fig. 2 e Differential segment of the regenerator with all the energy terms.
leff ¼ ls $ð1 3Þ þ lf $3
(11)
leff ¼ ls þ lf $Dd ðPeÞ
(12)
Ph e dimensionless number representing a ratio of pressure drop across regenerator and a volumetric enthalpy change between initial and end time of the experiment:
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Ph qf ¼
Dpexp ¼ 2$f Ref $Ec qf $Ra rf qf $cf qf $ðTmax Tmin Þ
dp Dpexp z L dx
(13)
(14)
In Eq. (7) second and third term in the bracket can be neglected in case gas is a heat transfer fluid since their contribution is, due to a low utilization factor, very small. For purpose of a numerical calculation we need to determine initial and boundary conditions [Eqs. (15) and (16)] and inlet forcing function [Eq. (17)] e T(t) curve for fluid inlet temperature, which is obtained from the experiment: qf ðx; s ¼ 0Þ ¼ qs ðx; s ¼ 0Þ ¼ qinit
(15)
vqs vqs vqs vqs F$ F$ ¼ ¼0 vx vs x¼0 vx vs x¼1
(16)
qf ðx ¼ 0; sÞ ¼ JðsÞjx¼0 ¼
TðtÞjx¼0 Tmin Tmax Tmin
(17)
With a proper iterative numerical procedure (implicit finite difference method with sufficiently small time and length steps) we can determine heat transfer coefficient af for each experimental run. In order to obtain a final result, average of for runs (two heating and two cooling single-blow runs) is computed and presented as Nu (Re, Pr) or j (Re) correlations. Such procedure for obtaining heat transfer coefficient is in close agreement with the nature of AMR operation, i.e. alternating hot/cold blows of heat transfer fluid, which makes it favourable for analysing active magnetic regenerators. Providing proper experimental setup, both correlations can be obtained with an accuracy of up to 2% (Shah and Zhou, 1997). For a purpose of this investigation an experimental setup was prepared according to the features proposed by Mullisen and Loehrke (1986). Using method described above, we have obtained j(Re) correlations (Fig. 3) for gas flow through two most common heat exchanger geometries: randomly packed bed with spheres and (isosceles) triangular flow passages. −1
10
j [−]
Kays & London [10]
Correlation for spheres: j = 2.452*Re−0.9429
Correlation for isosceles: j = 0.2267*Re−0.2944
Numerical model for AMR
Magnetic refrigeration works by exploiting a physical phenomenon called magneto-caloric effect, which can be mathematically described either by a change in temperature or entropy. For purpose of determining AMR operation, these magnetic properties of the magneto-caloric material need to be included into the abovementioned model. Three possibilities are available here: a) The simplest and most straightforward way of determining adiabatic temperature change during magnetization/demagnetization is to correlate it from appropriate DT (T ) diagrams (Tishin and Spichkin, 2003; Pecharsky and Gschneidner, 1997; Bru¨ck et al., 2003), where a pulsed e on/off magnetic field e is assumed (Fig. 4). From such diagrams heating and cooling curves can be determined according to Kitanovski and Egolf (2006), which allow us to calculate DTad-H and DTad-C e adiabatic temperature change during heating/cooling periods. If material with a Curry temperature TCu is magnetized, its temperature increases by DTad-H to a higher temperature TH. On the other hand, if material with temperature TCu is demagnetized, its temperature decreases by DTad-C to a lower temperature TC (Fig. 5). It should be noted that both DTad-H and DTad-C have different values due to a shift between the curves! Mentioned diagrams are widely attainable in the literature for almost all magnetic materials and can be easily incorporated in the numerical model. Drawback of this possibility is that it does not appropriately describe the phenomenon of magneto-caloric effect from thermo-dynamical point of view and can thus only be used as a first approximation of AMR simulation when comparing different second order magnetic materials e first order materials include a significant portion of latent heat, which in this simple model is not included and can as such present a substantial error!
dT T vM dH ¼ m0 $vs dt cH vT H dt
−3
10
3.
b) Shir et al. (2004), proposed following equation for magnetization dependant temperature change of an isentropic AMR system:
−2
10
10
Details on both geometries are listed in Table 1. As it can be seen from the figure, our proposed correlation data slightly differs (up to 9%) from the reference correlations given by Kays and London (1964). Due to a larger uncertainty pertained to a mass flow meter, error of proposed correlations was 4.6%.
2
10
3
Re [−]
Fig. 3 e Comparison of experimental data and correlation proposed by Kays and London (1964) for regenerator of isosceles triangles and packed bed of spheres.
(18)
Eq. (18) takes into account time variable magnetic field (and magnetization) and as such better corresponds with a physical phenomenon e it allows us to simulate an arbitrary magnetic field distribution (dH/dt). However, it requires knowledge of partial derivative of magnetization versus temperature, which can be obtained in relevant publications (Tishin and Spichkin, 2003).
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Table 1 e Details of tested compact heat exchanger geometries. Property Size of regenerator [mm] 3 [e] ap [m2 m3] dh [mm] Thickness (diameter) [mm]
Isosceles (copper) Spheres (steel) 20 16 60 0.518 4421 0.391 0.20
20 16 60 0.398 2126 0.749 2.00
c) Engelbrecht et al. (2005), proposed that magnetic work transfer and internal energy of the material are combined into total entropy change of the material due to the applied magnetic field, which can then be further divided into magnetic and temperature components (Engelbrecht et al., 2005; Tishin and Spichkin, 2003):
Ts
vss vus vðvs MÞ vss vm H vTs ðTs ; m0 HÞ 0 þ cH ðTs ; m0 HÞ ¼ m0 H ¼ Ts vt vt vm0 H vt vt vt (19)
The latter one seems the best description of the magnetocaloric effect and is as such thermodynamically most consistent. For this reason we will use it for further implementation of the model. However, this possibility has also its own drawbacks e in order to obtain properties of the magnetic material in Eq. (19) proper data needs to be available, i.e. s(T, H ): entropyetemperature dependency at various magnetic field strengths. Since numerical procedure for obtaining appropriate properties depends on the quality of available data, significant errors can be produced with this method if not handled properly. In order to adapt the model of a passive regenerator [Eqs. (6)e(17)] for use in AMRR together with magnetic properties, we need to introduce two new dimensionless parameters: Dimensionless temperature now becomes (according to Fig. 4): q¼
T TC TH TC
(20)
Dimensionless magnetic field: H Hmin Hmax Hmin
(21)
TH 1
T ad-H
2
Ts=TCu 4 3
Tad-C
TC Time t
Fig. 5 e Determination if maximum temperature span of passive magnetic refrigerator.
Introducing dimensionless parameters Eqs. (20) and (21) into Eqs. (6) and (7), assuming that magnetic field density is constant along the length of the regenerator and that fluid mass flow rate is kept constant during the entire period of hot/cold blow, we obtain following equations for fluid and solid phase: vqf þ NTU qf $ qf qs ¼ Ph qf vx
(22)
2 v2 qs vqs 1 v qs $ þ NTU qf $ qs qf 2$F qf ; qs ; G $ 2 vs vx$vs PeLf qf vx 2 v2 qs vG ð23Þ þ F qf ; qs ; G $ 2 þ Sthðqs ; GÞ$ ¼ 0 vs vs where Sth is a dimensionless number consisting of a heat capacity at constant field [Eq. (25)] and partial derivative of entropy with respect to dimensionless magnetic field: SthðTs ; m0 HÞ ¼
Ts Hmax Hmin vss $ $ðTs ; m0 HÞ $ cH ðTs ; m0 HÞ TH TC vm0 H T
(24)
Specific heat at constant field and partial derivative of entropy with respect to magnetic field can be obtained from experimental curves of s(T ) dependency for different magnetic field strengths H: vs cH ¼ T vT H
(25)
Eq. (24) can be further simplified by insertion of dimensionless numbers defined in Eqs. (20) and (21) into partial differentials in Eqs. (24) and (25), in which case we obtain a final form of Sth dimensionless number:
Adiabatic temperature change ΔTad [K]
G¼
Temperature T
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 0 6 1 e1 0 6 7
Sthðqs ; GÞ ¼
TC TCu
TH Temperature T [K]
Fig. 4 e Heating and cooling curves of adiabatic temperature change.
vqs vss ðqs ; GÞ $ vss H vG qs
(26)
As is evident from Eq. (24) and (26), in order to use proposed procedure we need data (experimental curves) in the form of s (T,H ), which must be numerically evaluated (interpolated and then differentiated). Such procedure does take off some of the accuracy if available data is not reliable and consistent, but allows us to closely follow thermo-dynamical phenomena. Initial [Eq. (15)] and boundary [Eq. (17)] conditions now change to:
1066 qf ðx; s ¼ 0Þ ¼ qs ðx; s ¼ 0Þ ¼ qinit
(27)
qf ðx ¼ 0; sÞjvf >0 ¼ qCHEX
(28)
qf ðx ¼ 1; sÞjvf <0 ¼ qHHEX
(29)
Dimensionless temperature [−]
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 0 6 1 e1 0 6 7
where qCHEX and qHHEX, are hot/cold reservoir temperatures. And boundary conditions on the solid side [Eq. (16)] become: vqs vqs ¼ vqs F qf ; qs ; G vqs ¼ 0 F qf ; qs ; G vx vs x¼0 vx vs x¼1
(30)
Utilization factor in Eqs. (23) and (30) now depends on fluid and solid temperatures and dimensionless magnetic field due to their impact on specific heat of solid. Initial and boundary conditions in Eqs. (27)e(29) can be adjusted to a specific problem that we are trying to simulate. Eqs. (22) and (23) can now be numerically implemented, using finite difference scheme, in order to obtain final result of the AMRR simulation. Above presented model requires that regenerator’s geometry satisfy condition: Bi < 0.1, which enables us to use lumped capacitance model at all. Mentioned condition was met in case of both, a packed bed regenerator (sphere size: 2 mm) and isosceles triangles (material thickness: 0.2 mm), which were used during our study.
4.
Results
Described dimensionless numerical model can be equally used for simulating transient and steady state operation of AMR refrigerator. It enables us to make a quick, quantitative comparison of different magnetic refrigerators, using various magnetic materials, heat transfer fluids and operating conditions (magnetic field strength, frequency, mass flow,.). From the definition of dimensionless temperature e Eq. (20) and Figs. 4 and 5 e we can see that an ideal passive magnetic refrigerator operating at the Currie temperature of the magnetic material can achieve a maximum theoretical temperature span that corresponds to a dimensionless value of 1. This temperature span is achieved in four steps as it can be seen in Fig. 5: 1. magnetization, 2. cooling, 3. demagnetization and 4. heating of the magnetic material under the influence of room temperature. Active magnetic regenerator on the other hand, can due to its regenerative operation, operate at a larger temperature span. Increase in temperature span, due to AMR operation, can thus easily be read from the final result (Fig. 6) e in presented simulation (Gd in shape of spherical particles of size 0.5 mm was used as a magnetic material and helium as a heat exchange fluid) it sums up to about 25% increase as compared to a passive magnetic regenerator. Sensitivity analysis performed with the model input parameters shows that the highest impact on the final solution of the simulation (temperature span) have heat transfer coefficient and magnetic properties of the material. 10% higher heat transfer coefficient (which is a maximum difference between proposed and reference correlations) yields about 4.4% higher temperature span and thus cooling power of the magnetic refrigerator, while magnetic properties
1.2 1 0.8 0.6 0.4 0.2 0 −0.2
0
20
40
60
80
100
120
140
160
180
Dimensionless time [−] Fig. 6 e Transient response of the AMR refrigerator in dimensionless temperatureetime dependency.
amount to about 5.2%. Other input parameters, such as longitudinal heat conduction and fluid dispersion show a significantly lower impact on temperature span (both around 1%), while fluid properties amount to about 1.5% change in temperature span. While uncertainty of magnetic properties depends solely on the data obtained from references and on the procedures used to evaluate it, we can on the other hand significantly reduce the impact of error in heat transfer coefficient, if we combine a single-blow experiment with a simulation of AMR refrigerator using a consistent model for both. Proposed numerical model can specially be of great value when developing new regenerator geometries of which few data is available in the publications.
5.
Conclusions
A consistent dimensionless model for determination of both, regenerator’s heat transfer coefficient and operation of AMR refrigerator is introduced. It comprises of all parameters that have effect on determination of the coefficient or operation of the magnetic refrigerator. In order to verify the method, j(Re) correlations were determined with the experiment and compared with some reference correlations e proposed correlation slightly differs from reference ones (5e10%). Numerical model for determination of regenerator’s heat transfer coefficient is later further developed in order to be used as a tool for simulation of AMR refrigerator. Such model can yield several advantages from the aspect of consistency between a numerical model used for heat transfer coefficient determination and a model for AMR simulation. Sensitivity analysis showed that a 10% higher heat transfer coefficient yields a 4.4% higher temperature span of the AMRR e this uncertainty can be avoided if using a consistent model for both. At the same time, dimensionless model improves the comparability of various magnetic refrigerators (different magnetic materials, magnetic field strengths, regenerator geometries,.), which are otherwise hard to compare due to several parameters which crucially influence the operation of AMRR.
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references
Bru¨ck, E., Tegus, O., Li, X.W., de Boer, F.R., Buschow, K.H.J., 2003. Magnetic refrigeration e towards room-temperature applications. Phys. B Condens. Matter. 327, 431e437. Byrd, R.B., Stewart, W., Lightfoot, E.N., 2002. Transport Phenomena, second ed. John Wiley & Sons, Inc., USA, 336 pp. Engelbrecht, K.L., Nellis, G.F., Klein, S.A., Boeder, A.M., 2005. Modeling active magnetic regenerative refrigeration systems. In: 1st International Conference on Magnetic Refrigeration at Room Temperature e Conference Proceedings, Montreux, Switzerland, pp. 265e273. Gschneidner Jr., K.A., Pecharsky, V.K., Tsokol, A.O., 2005. Recent developments in magnetocaloric materials. Rep. Progr. Phys. 68, 1479e1539. Heggs, P.J., Burns, D., 1988. Single-blow experimental prediction of heat transfer coefficients. Exp. Therm. Fluid Sci. 1 (Issue 3), 243e251. Kaviany, M., 1995. Principles of Heat Transfer in Porous Media, second ed. Springer. Kays, W.M., London, A.L., 1964. Compact Heat Exchangers, second ed. McGraw-Hill Book Company, USA. Kitanovski, A., Egolf, P., 2006. Thermodynamics of magnetic refrigeration. Int. J. Refrigeration 29 (Issue 1), 3e21. Mullisen, R.S., Loehrke, R.I., 1986. A transient heat exchanger evaluation test for arbitrary fluid inlet temperature variation and longitudinal core conduction. ASME J. Heat Tran. 108, 370e376. Okamura, T., Yamada, K., Hirano, N., Nagaya, S., 2005. Performance of a room-temperature rotary magnetic refrigerator. In: 1st International Conference on Magnetic Refrigeration at Room Temperature e Conference Proceedings, Montreux, Switzerland, pp. 319e324.
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Pecharsky, V.K., Gschneidner Jr., K.A., 1997. Effect of alloying on the giant magnetocaloric effect of Gd5(Si2Ge2). J. Magn. Magn. Mater. 167 (No. 3), 179e184. Petersen, T.F., Pryds, N., Smith, A., Bahl, C.R.H., 2007. A numerical analysis of a reciprocating active magnetic regenerator with a parallel-plate regenerator geometry. In: 2nd IIF-IIR International Conference on Magnetic Refrigeration at Room Temperature e Conference Proceedings, Portoroz, Slovenia, pp. 271e279. Rowe, A., Dikeos, J., Tura, A., 2005. Experimental studies of a near room-temperature magnetic refrigeration. In: 1st International Conference on Magnetic Refrigeration at Room Temperature e Conference Proceedings, Montreux, Switzerland, pp. 325e334. Sarlah, A., Poredos, A., 2005. Regenerator for magnetic cooling in shape of a honeycomb. In: 1st International Conference on Magnetic Refrigeration at Room Temperature e Conference Proceedings, Montreux, Switzerland, pp. 283e290. , D.P., 2003. Fundamentals of Heat Exchanger Shah, R.K., Sekulic Design. John Wiley & Sons, Inc., USA. Shah R.K., Zhou S.Q., 1997, Experimental techniques for obtaining design data of compact heat exchanger surfaces. In: Proceedings of the International Conference on Compact Heat Exchangers for the Process Industries, Snowbird, Utah, pp. 365e379. Shir, F., Mavriplis, C., Bennett, L.H., Torre, E.D., 2004. Analysis of room temperature magnetic regenerative refrigeration. Int. J. Refrigeration 28 (No. 4), 616e627. Tishin, A.M., Spichkin, Y.I., 2003. The Magnetocaloric Effect and Its Applications. Institute of Physics Publishing, Bristol and Philadelphia, 475 pp. Zimm, C, Boeder, A, Chell, J, Sternberg, A, Fujita, A, Fujieda, S, Fukamichi, K. Design and performance of a permanent magnet rotary refrigerator. In: 1st International Conference on Magnetic Refrigeration at Room Temperature e Conference Proceedings, Montreux, Switzerland, 2005, pp. 367e373.
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journal homepage: www.elsevier.com/locate/ijrefrig
Dimensionless numerical model for simulation of active magnetic regenerator refrigerator A. Sarlah*, A. Poredos Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, SI-1000 Ljubljana, Slovenia
article info
abstract
Article history:
In order to obtain a better reliability, consistency and accuracy of results obtained with
Received 4 December 2007
a numerical simulation of an AMRR (active magnetic regenerator refrigerator), a dimen-
Received in revised form
sionless numerical model was developed, which can equally be used for determination of
20 January 2010
regenerator’s heat transfer coefficient and simulation of passive heat regenerators or
Accepted 5 April 2010
AMRR operation. Regenerator’s heat transfer coefficient af, is a crucial input parameter in
Available online 9 April 2010
the simulation of AMRR operation and has a primal effect on the outcome of a solution. This paper deals with a derived dimensionless model and discusses errors involved when
Keywords:
using different models for heat transfer coefficient and AMRR operation simulation. ª 2010 Elsevier Ltd and IIR. All rights reserved.
Magnetic refrigerator Modelling Simulation Heat transfer Regenerator
Mode`le nume´rique adimensionnel pour la simulation d’un re´frige´rateur magne´tique a` re´ge´ne´rateur actif Mots cle´s : Re´frige´rateur magne´tique ; Mode´lisation ; Simulation ; Transfert de chaleur ; Re´ge´ne´rateur
1.
Introduction
It has been 10 years since the discovery of the giant magnetocaloric effect in the Gd5(Si2Ge2) compound (Gschneidner et al., 2005), which started an era of increased research in magnetic refrigeration. A decent number of prototypes (Gschneidner et al., 2005; Zimm et al., 2005; Okamura et al., 2005; Rowe et al., 2005) have been built and tested during this time, all of them running at near-room temperature more or less successfully. In order to theoretically verify
experimental results of the operation of prototypes, various numerical models have also emerged (Engelbrecht et al., 2005; Petersen et al., 2007; Sarlah and Poredos, 2005). Putting them all together, there seems to be an increasing need to develop a model which could accurately simulate operation of an AMR refrigerator and that could allow a better comparison of various refrigerators using different magnetic materials, operational setups, magnetic field strengths, etc. Such model could significantly contribute to the pre-design analysis, thus cutting the time and money needed to find an
* Corresponding author. Tel.: þ386 1 4771 446; fax: þ386 1 2518 567. E-mail address: [email protected] (A. Sarlah). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.04.003
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Nomenclature packing area density (m2 m3) heat transfer area (m2) cross-section area (m2) specific heat (J kg1 K1) hydraulic diameter (m) dispersion coefficient (-) modified Eckert number (-) friction factor (-) magnetic field strength (A m1) Colburn factor (-) length (m) mass flow (kg s1) M magnetization (A m1) NTU number of transferred units (-) p pressure (Pa) PeL (PeLf) (modified) Peclet number (-) Ph Ph dimensionless number (-) Pr modified Prandtl number (-) aspect ratio (L/dh) (-) Ra Re Reynolds number (-) Sth Sth dimensionless number (-) t time (s) T temperature (K) u internal energy (J) fluid interstitial velocity (m s1) vf
ap Aht Ar c dh Dd Ec f H j L _ m
optimal AMR design, which can later be manufactured and experimentally tested. One crucial parameter that directly determines efficiency of the AMR is heat transfer coefficient, which needs to be experimentally tested and numerically evaluated in order to determine its thermal e j(Re) e and hydraulic e f(Re) e properties. Doing so, it is reasonable to use the same tool e same numerical model, which includes all influential thermo-dynamical and hydro dynamical properties e for both, determination of heat transfer coefficient and simulation of AMRR operation. Any discrepancies between them can result in initial errors which are then further escalated in the numerical simulation. Heggs and Burns (1988) and Shah and Zhou (1997) have made a comparison of various techniques for prediction of heat transfer coefficients when performing a single-blow experiment with compact heat exchangers. They have concluded that there are significant differences between them, i.e. results obtained by shape factor, maximum slope, differential fluid enthalpy method or direct curve matching method (using least squares matching technique) can differ for as much as 20% (Heggs and Burns, 1988), which is significantly more than the uncertainty of particular method (between 4 and 10%). Furthermore, a lot of data on heat transfer coefficients of various heat exchangers, which is available in publications, is for certain regenerator geometries (packed bed, honeycomb,.), which can differ from the ones used in magnetic refrigerators e since introduction of AMR refrigerators there has been a constant need to discover new, advanced regenerator geometries with higher efficiencies. Available data was also obtained using certain
vs V x
specific volume (m3 kg1) volume (m3) coordinate (m)
Greek a e F G l m0 r q s J x
heat transfer coefficient (W m2 K1) porosity (-) utilization factor (-) dimensionless mag. field (-) conductivity (W m1 K1) permeability of vacuum (Wb A1 m1) density (kg m3) dimensionless temperature (-) dimensionless time (-) inlet forcing function (-) dimensionless length (-)
Subscripts ad-H/ad-C adiabatic heating/cooling C/H cold/hot HHEX/CHEX hot/cold heat exch. eff effective exp experimental f fluid H magnetic field init initial r regenerator s solid
experimental methods (single blow, heating-only,.) and numerical models (Heggs and Burns, 1988; Kays and London, 1964), which in many cases did not include all physical phenomena that are important and significant for description of AMRR operation; i.e. such as longitudinal heat conduction in solid and fluid dispersion. Objective of this work is to present a dimensionless model which can, with certain assumptions, be equally used for both purposes: determination of regenerator’s heat transfer coefficient and simulation of AMR refrigerator.
2.
Determination of heat transfer coefficient
Heat transfer coefficient of compact heat exchangers can only be determined by using transient test techniques together with transient methods, developed for certain range of core NTU numbers. Most common test technique is singleblow technique, which is described in details in several publications together with proposed features of the experimental setup (Heggs and Burns, 1988; Kaviany, 1995; Shah , 2003; Mullisen and Loehrke, 1986) and thus and Sekulic will not be further discussed here. However, single-blow techniques must be accompanied with a transient method, which evaluates experimental data in order to obtain proper heat transfer coefficients. There exist several methods (Heggs and Burns, 1988; Shah and Zhou, 1997; Mullisen and Loehrke, 1986), each with its advantages and disadvantages, while mostly recommended is one proposed by Mullisen and Loehrke (1986). They proposed a method,
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 0 6 1 e1 0 6 7
Temperature
Tfin − exp Tfout − exp Tfout − num
Time Fig. 1 e Single-blow experiment: matching of experimental and numerical data.
drop. For a purpose of determination of heat transfer coefficient we can replace vp/vx in Eq. (1) with Dpexp/L [Eq. (14)], which can be measured during the experiment. Pressure drop across regenerator is in most models neglected, since its significance is small e however, it should be accounted when dealing with high velocity gradients and large viscosity, i.e.: high pressure drop exchangers or high viscous fluids. In our case it is included in derivation in order to be consistent with a later discussed model of an active magnetic regenerator in which pressure drop is important due to its impact on efficiency. In derivation of Eqs. (1) and (2) we have assumed constant fluid mass flow rate during the experiment and constant properties of the solid. Both equations can be written in dimensionless form with introduction of dimensionless length [Eq. (3)], time [Eq. (4)] and temperature [Eq. (5)], defined as: x¼
which allows arbitrary fluid inlet temperature variation (inlet forcing function) and determines appropriate heat transfer coefficient by performing iterative process of matching experimentally and numerically obtained fluid outlet temperature curves until they match within a certain agreement (Fig. 1), i.e. minimum of least squares between theoretical and experimental data. Their method allows use of various numerical models, which can be individually adjusted to an observed physical problem. Basis of a current model for determination of the heat transfer coefficient of regenerators is a well-known equation for a change of internal energy of a material, expressed with enthalpies (Byrd et al., 2002). With below mentioned assumptions, both equations e for fluid and solid e can, according to Fig. 2, be written as: vTf vTf _ þ mðtÞ$c þ af $ap $Ar Tf Ts rf Tf $cf Tf $Ar $3$ f Tf $ vt vx _ mðtÞ vp $ ¼ rf Tf vx
vTs v Ts þ af $ap $Ar $ Ts Tf leff $Ar $ 2 ¼ 0 rs $cs $Ar $ð1 3Þ vt vx
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x L
(3)
_ f $cf Tf vf m x x ¼ t $ t s ¼ F Tf $ rs $cs $ð1 eÞ$Vr L vf vf qi ¼
Ti ðx; tÞ Tmin Tmax Tmin
i ¼ f; s
(4)
(5)
Doing so we obtain: vqf þ NTU qf $ qf qs ¼ Ph vx
(6)
2 v2 qs vqs 1 v qs $ þ NTU qf $ qs qf 2$F qf $ 2 vs vx$vs PeLf qf vx 2 v2 qs þ F qf $ 2 ¼ 0 vs where following dimensionless numbers are used:
(1)
2
(2)
In Eq. (1) we have added a term which sums up irreversible rate of internal energy increase by viscous dissipation, and have left out heat conduction of a fluid (fluid dispersion) e the latter is included in the equation for a solid [Eq. (2)] as leff [Eqs. (11) and (12)], while viscous dissipation term was replaced with a term representing a result of acting viscous forces: pressure
NTU e number of transferred units: af $ap $Vr NTU qf ¼ _ f $cf qf m F e utilization factor: rf qf $cf qf $3 F qf ¼ rs $cs $ð1 3Þ
(8)
(9)
PeLf e modified Peclet (PeL) number: 1 1 1 leff 1 ¼ $ ¼ $ PeLf qf ReL qf $Preff qf 3 vf $L$rf qf $cf qf 3
(10)
where leff is effective heat conductivity defined for plate [Eq. (11)] or packed bed [Eq. (12)] heat exchangers:
Fig. 2 e Differential segment of the regenerator with all the energy terms.
leff ¼ ls $ð1 3Þ þ lf $3
(11)
leff ¼ ls þ lf $Dd ðPeÞ
(12)
Ph e dimensionless number representing a ratio of pressure drop across regenerator and a volumetric enthalpy change between initial and end time of the experiment:
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Ph qf ¼
Dpexp ¼ 2$f Ref $Ec qf $Ra rf qf $cf qf $ðTmax Tmin Þ
dp Dpexp z L dx
(13)
(14)
In Eq. (7) second and third term in the bracket can be neglected in case gas is a heat transfer fluid since their contribution is, due to a low utilization factor, very small. For purpose of a numerical calculation we need to determine initial and boundary conditions [Eqs. (15) and (16)] and inlet forcing function [Eq. (17)] e T(t) curve for fluid inlet temperature, which is obtained from the experiment: qf ðx; s ¼ 0Þ ¼ qs ðx; s ¼ 0Þ ¼ qinit
(15)
vqs vqs vqs vqs F$ F$ ¼ ¼0 vx vs x¼0 vx vs x¼1
(16)
qf ðx ¼ 0; sÞ ¼ JðsÞjx¼0 ¼
TðtÞjx¼0 Tmin Tmax Tmin
(17)
With a proper iterative numerical procedure (implicit finite difference method with sufficiently small time and length steps) we can determine heat transfer coefficient af for each experimental run. In order to obtain a final result, average of for runs (two heating and two cooling single-blow runs) is computed and presented as Nu (Re, Pr) or j (Re) correlations. Such procedure for obtaining heat transfer coefficient is in close agreement with the nature of AMR operation, i.e. alternating hot/cold blows of heat transfer fluid, which makes it favourable for analysing active magnetic regenerators. Providing proper experimental setup, both correlations can be obtained with an accuracy of up to 2% (Shah and Zhou, 1997). For a purpose of this investigation an experimental setup was prepared according to the features proposed by Mullisen and Loehrke (1986). Using method described above, we have obtained j(Re) correlations (Fig. 3) for gas flow through two most common heat exchanger geometries: randomly packed bed with spheres and (isosceles) triangular flow passages. −1
10
j [−]
Kays & London [10]
Correlation for spheres: j = 2.452*Re−0.9429
Correlation for isosceles: j = 0.2267*Re−0.2944
Numerical model for AMR
Magnetic refrigeration works by exploiting a physical phenomenon called magneto-caloric effect, which can be mathematically described either by a change in temperature or entropy. For purpose of determining AMR operation, these magnetic properties of the magneto-caloric material need to be included into the abovementioned model. Three possibilities are available here: a) The simplest and most straightforward way of determining adiabatic temperature change during magnetization/demagnetization is to correlate it from appropriate DT (T ) diagrams (Tishin and Spichkin, 2003; Pecharsky and Gschneidner, 1997; Bru¨ck et al., 2003), where a pulsed e on/off magnetic field e is assumed (Fig. 4). From such diagrams heating and cooling curves can be determined according to Kitanovski and Egolf (2006), which allow us to calculate DTad-H and DTad-C e adiabatic temperature change during heating/cooling periods. If material with a Curry temperature TCu is magnetized, its temperature increases by DTad-H to a higher temperature TH. On the other hand, if material with temperature TCu is demagnetized, its temperature decreases by DTad-C to a lower temperature TC (Fig. 5). It should be noted that both DTad-H and DTad-C have different values due to a shift between the curves! Mentioned diagrams are widely attainable in the literature for almost all magnetic materials and can be easily incorporated in the numerical model. Drawback of this possibility is that it does not appropriately describe the phenomenon of magneto-caloric effect from thermo-dynamical point of view and can thus only be used as a first approximation of AMR simulation when comparing different second order magnetic materials e first order materials include a significant portion of latent heat, which in this simple model is not included and can as such present a substantial error!
dT T vM dH ¼ m0 $vs dt cH vT H dt
−3
10
3.
b) Shir et al. (2004), proposed following equation for magnetization dependant temperature change of an isentropic AMR system:
−2
10
10
Details on both geometries are listed in Table 1. As it can be seen from the figure, our proposed correlation data slightly differs (up to 9%) from the reference correlations given by Kays and London (1964). Due to a larger uncertainty pertained to a mass flow meter, error of proposed correlations was 4.6%.
2
10
3
Re [−]
Fig. 3 e Comparison of experimental data and correlation proposed by Kays and London (1964) for regenerator of isosceles triangles and packed bed of spheres.
(18)
Eq. (18) takes into account time variable magnetic field (and magnetization) and as such better corresponds with a physical phenomenon e it allows us to simulate an arbitrary magnetic field distribution (dH/dt). However, it requires knowledge of partial derivative of magnetization versus temperature, which can be obtained in relevant publications (Tishin and Spichkin, 2003).
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Table 1 e Details of tested compact heat exchanger geometries. Property Size of regenerator [mm] 3 [e] ap [m2 m3] dh [mm] Thickness (diameter) [mm]
Isosceles (copper) Spheres (steel) 20 16 60 0.518 4421 0.391 0.20
20 16 60 0.398 2126 0.749 2.00
c) Engelbrecht et al. (2005), proposed that magnetic work transfer and internal energy of the material are combined into total entropy change of the material due to the applied magnetic field, which can then be further divided into magnetic and temperature components (Engelbrecht et al., 2005; Tishin and Spichkin, 2003):
Ts
vss vus vðvs MÞ vss vm H vTs ðTs ; m0 HÞ 0 þ cH ðTs ; m0 HÞ ¼ m0 H ¼ Ts vt vt vm0 H vt vt vt (19)
The latter one seems the best description of the magnetocaloric effect and is as such thermodynamically most consistent. For this reason we will use it for further implementation of the model. However, this possibility has also its own drawbacks e in order to obtain properties of the magnetic material in Eq. (19) proper data needs to be available, i.e. s(T, H ): entropyetemperature dependency at various magnetic field strengths. Since numerical procedure for obtaining appropriate properties depends on the quality of available data, significant errors can be produced with this method if not handled properly. In order to adapt the model of a passive regenerator [Eqs. (6)e(17)] for use in AMRR together with magnetic properties, we need to introduce two new dimensionless parameters: Dimensionless temperature now becomes (according to Fig. 4): q¼
T TC TH TC
(20)
Dimensionless magnetic field: H Hmin Hmax Hmin
(21)
TH 1
T ad-H
2
Ts=TCu 4 3
Tad-C
TC Time t
Fig. 5 e Determination if maximum temperature span of passive magnetic refrigerator.
Introducing dimensionless parameters Eqs. (20) and (21) into Eqs. (6) and (7), assuming that magnetic field density is constant along the length of the regenerator and that fluid mass flow rate is kept constant during the entire period of hot/cold blow, we obtain following equations for fluid and solid phase: vqf þ NTU qf $ qf qs ¼ Ph qf vx
(22)
2 v2 qs vqs 1 v qs $ þ NTU qf $ qs qf 2$F qf ; qs ; G $ 2 vs vx$vs PeLf qf vx 2 v2 qs vG ð23Þ þ F qf ; qs ; G $ 2 þ Sthðqs ; GÞ$ ¼ 0 vs vs where Sth is a dimensionless number consisting of a heat capacity at constant field [Eq. (25)] and partial derivative of entropy with respect to dimensionless magnetic field: SthðTs ; m0 HÞ ¼
Ts Hmax Hmin vss $ $ðTs ; m0 HÞ $ cH ðTs ; m0 HÞ TH TC vm0 H T
(24)
Specific heat at constant field and partial derivative of entropy with respect to magnetic field can be obtained from experimental curves of s(T ) dependency for different magnetic field strengths H: vs cH ¼ T vT H
(25)
Eq. (24) can be further simplified by insertion of dimensionless numbers defined in Eqs. (20) and (21) into partial differentials in Eqs. (24) and (25), in which case we obtain a final form of Sth dimensionless number:
Adiabatic temperature change ΔTad [K]
G¼
Temperature T
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 0 6 1 e1 0 6 7
Sthðqs ; GÞ ¼
TC TCu
TH Temperature T [K]
Fig. 4 e Heating and cooling curves of adiabatic temperature change.
vqs vss ðqs ; GÞ $ vss H vG qs
(26)
As is evident from Eq. (24) and (26), in order to use proposed procedure we need data (experimental curves) in the form of s (T,H ), which must be numerically evaluated (interpolated and then differentiated). Such procedure does take off some of the accuracy if available data is not reliable and consistent, but allows us to closely follow thermo-dynamical phenomena. Initial [Eq. (15)] and boundary [Eq. (17)] conditions now change to:
1066 qf ðx; s ¼ 0Þ ¼ qs ðx; s ¼ 0Þ ¼ qinit
(27)
qf ðx ¼ 0; sÞjvf >0 ¼ qCHEX
(28)
qf ðx ¼ 1; sÞjvf <0 ¼ qHHEX
(29)
Dimensionless temperature [−]
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 0 6 1 e1 0 6 7
where qCHEX and qHHEX, are hot/cold reservoir temperatures. And boundary conditions on the solid side [Eq. (16)] become: vqs vqs ¼ vqs F qf ; qs ; G vqs ¼ 0 F qf ; qs ; G vx vs x¼0 vx vs x¼1
(30)
Utilization factor in Eqs. (23) and (30) now depends on fluid and solid temperatures and dimensionless magnetic field due to their impact on specific heat of solid. Initial and boundary conditions in Eqs. (27)e(29) can be adjusted to a specific problem that we are trying to simulate. Eqs. (22) and (23) can now be numerically implemented, using finite difference scheme, in order to obtain final result of the AMRR simulation. Above presented model requires that regenerator’s geometry satisfy condition: Bi < 0.1, which enables us to use lumped capacitance model at all. Mentioned condition was met in case of both, a packed bed regenerator (sphere size: 2 mm) and isosceles triangles (material thickness: 0.2 mm), which were used during our study.
4.
Results
Described dimensionless numerical model can be equally used for simulating transient and steady state operation of AMR refrigerator. It enables us to make a quick, quantitative comparison of different magnetic refrigerators, using various magnetic materials, heat transfer fluids and operating conditions (magnetic field strength, frequency, mass flow,.). From the definition of dimensionless temperature e Eq. (20) and Figs. 4 and 5 e we can see that an ideal passive magnetic refrigerator operating at the Currie temperature of the magnetic material can achieve a maximum theoretical temperature span that corresponds to a dimensionless value of 1. This temperature span is achieved in four steps as it can be seen in Fig. 5: 1. magnetization, 2. cooling, 3. demagnetization and 4. heating of the magnetic material under the influence of room temperature. Active magnetic regenerator on the other hand, can due to its regenerative operation, operate at a larger temperature span. Increase in temperature span, due to AMR operation, can thus easily be read from the final result (Fig. 6) e in presented simulation (Gd in shape of spherical particles of size 0.5 mm was used as a magnetic material and helium as a heat exchange fluid) it sums up to about 25% increase as compared to a passive magnetic regenerator. Sensitivity analysis performed with the model input parameters shows that the highest impact on the final solution of the simulation (temperature span) have heat transfer coefficient and magnetic properties of the material. 10% higher heat transfer coefficient (which is a maximum difference between proposed and reference correlations) yields about 4.4% higher temperature span and thus cooling power of the magnetic refrigerator, while magnetic properties
1.2 1 0.8 0.6 0.4 0.2 0 −0.2
0
20
40
60
80
100
120
140
160
180
Dimensionless time [−] Fig. 6 e Transient response of the AMR refrigerator in dimensionless temperatureetime dependency.
amount to about 5.2%. Other input parameters, such as longitudinal heat conduction and fluid dispersion show a significantly lower impact on temperature span (both around 1%), while fluid properties amount to about 1.5% change in temperature span. While uncertainty of magnetic properties depends solely on the data obtained from references and on the procedures used to evaluate it, we can on the other hand significantly reduce the impact of error in heat transfer coefficient, if we combine a single-blow experiment with a simulation of AMR refrigerator using a consistent model for both. Proposed numerical model can specially be of great value when developing new regenerator geometries of which few data is available in the publications.
5.
Conclusions
A consistent dimensionless model for determination of both, regenerator’s heat transfer coefficient and operation of AMR refrigerator is introduced. It comprises of all parameters that have effect on determination of the coefficient or operation of the magnetic refrigerator. In order to verify the method, j(Re) correlations were determined with the experiment and compared with some reference correlations e proposed correlation slightly differs from reference ones (5e10%). Numerical model for determination of regenerator’s heat transfer coefficient is later further developed in order to be used as a tool for simulation of AMR refrigerator. Such model can yield several advantages from the aspect of consistency between a numerical model used for heat transfer coefficient determination and a model for AMR simulation. Sensitivity analysis showed that a 10% higher heat transfer coefficient yields a 4.4% higher temperature span of the AMRR e this uncertainty can be avoided if using a consistent model for both. At the same time, dimensionless model improves the comparability of various magnetic refrigerators (different magnetic materials, magnetic field strengths, regenerator geometries,.), which are otherwise hard to compare due to several parameters which crucially influence the operation of AMRR.
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