Cryogenics 52 (2012) 111–118

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Cryogenics

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Thermodynamics of active magnetic regenerators: Part I

Andrew Rowe ⇑Department of Mechanical Engineering, Institute for Integrated Energy Systems, University of Victoria, 3800 Finnerty Rd., Victoria, B.C., Canada V8W 3P6

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 April 2011Received in revised form 8 August 2011Accepted 15 September 2011Available online 21 September 2011

Keywords:AMR cycleThermodynamicsMagnetocaloric effectMagnetic refrigeration

0011-2275/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cryogenics. 2011.09.005

⇑ Tel.: +1 250 721 8920.E-mail address: arowe@uvic.ca

Cycle-averaged relationships for heat transfer, magnetic work, and temperature distribution are derivedfor an active magnetic regenerator cycle. A step-wise cycle is defined and a single equation describing thetemperature as a function of time and position is derived. The main assumption is that the convectiveinteraction between fluid and solid is large so that thermal equilibrium between fluid and solid existsduring a fluid flow phase (regeneration). Relations for the temperatures at each step in the cycle aredeveloped assuming small regenerative perturbations and used to derive the net cooling power and mag-netic work at any location in the AMR. The overall energy balance expression is presented with transfor-mations needed to relate the boundary conditions to effective operating temperatures. An expression isderived in terms of operating parameters and material properties when each location is regenerativelybalanced; this relation indicates needed conditions so the local energy balance will satisfy the assumedcycle. By solving the energy balance expression to determine temperature distribution one can calculatework, heat transfer, and COP.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The active magnetic regenerator (AMR) uses distributed mag-netic work to modulate the temperature of a heat transfer fluid.The intent is to create a steady-state cycle with the ability to gen-erate net heat transfer over a range of temperatures spanning theAMR. The operation of an AMR is governed by thermal interactionsbetween the magnetocaloric material and a heat transfer fluid. Thedetermination of performance usually requires knowledge of themagnetic and thermal properties of the solid, geometric andhydraulic properties of the regenerator, and the ability to deter-mine magnetization as a function of applied field and temperature;both of these variables have spatial and temporal variations. Ana-lyzing the AMR cycle can be a complex problem.

Attempts at understanding the thermodynamics of the AMR cy-cle often fall into two very different levels of resolution: low-level,or first-order methods which are somewhat heuristic and mathe-matically simple [1–6]; and, higher-order methods, usually involv-ing the solution of coupled, non-linear, partial differentialequations for the refrigerant and fluid [7–15]. Each approachbrings different value to understanding AMR cycles. First ordermethods can reveal broad criteria or limitations; however, theyare so simple they tend to provide little guidance in the actual con-struction or operation of an AMR. Higher order methods may bringmore accuracy, but, so far, require detailed thermodynamic proper-ties for magnetocaloric materials. Modeling of the higher order

ll rights reserved.

problem is time consuming, non-trivial due to the highly non-linear behavior of magnetocaloric materials, and results are oftenspecific to the magnetocaloric material being used. Consideringthat permanent magnet, near room-temperature AMR devices areunlikely to be commercially competitive using a single magnetoc-aloric material, the complexity of the development problem isamplified. The need to choose multiple materials and then createan AMR composition that works efficiently over a large tempera-ture span at low-fields currently requires a significant investmentin computational, material, and experimental resources.

An approximate analysis of the AMR cycle that captures moredetail than first-order approaches, but provides more general de-sign guidelines than higher-order techniques and is mathemati-cally tractable, is needed. The ability to do rapid calculations andsensitivity studies would engage more researchers and help focusknowledge on a broader range of technology development activi-ties such as component sizing, cost analysis, manufacturability,market assessments, magnet designs, and cycle optimization. Fur-thermore, a simplified tool for AMR performance analysis wouldassist material scientists by providing a more accurate frameworkfor material design and selection, and at the same time help engi-neers by reducing the need for magnetic material expertise andallowing more focus on device issues.

The purpose of this paper is to develop approximate expres-sions describing AMR thermodynamics and performance usingthe key parameters specific to the AMR cycle. Equations are de-rived for the following thermodynamic quantities: temperaturedistribution, net heat transfer, and work rate. The coefficient ofperformance is determined using the last two parameters. The

Nomenclature

a, b, c, d discrete points in a cycle, –A surface area, m2

c specific heat, J kg K�1

C volumetric specific heat, J m�3 K�1

f defined parameters, –h convection coefficient, W m�2 K�1

H magnetic field, A m�1

k thermal conductivity, W m�1 K�1

K thermal conductance, W K�1

L length, mm mass, kgM magnetization per unit mass, A m2 kg�1

Q heat transfer, net enthalpy flux, WR thermal mass ratio, –s entropy, J kg�1 K�1

t time coordinate, sU utilization, – (Eq. 5)x non-dimensional spatial coordinate

Greeka porosity, –b balance, –

d differential variation, –j non-dimensional conductance, –l magnetic permeability, T A�1 m�1

q density, kg m�3

r symmetry, –s period, sD low field to high field change, –U utilization, –(Eq. 6)

SubscriptB constant magnetic field (l0H), or blow, –C cold or low-field, –eff effective, –f fluid, –H hot or high-field, –M magnetic, –p constant pressure, parasitic, –s solid, –

Superscripth per unit length, –n MCE scaling exponent, –

112 A. Rowe / Cryogenics 52 (2012) 111–118

development begins with the governing partial differential equa-tions describing the time-dependent energy balance for the fluidand solid. These expressions are then simplified using parametergroupings that arise in the exact analysis. Using the assumptionof good thermal contact between the fluid and solid allows thephysics of the AMR process to be reduced to a single differentialequation. The thermodynamic expressions describing AMR perfor-mance are derived by applying this single governing equation to astep-wise AMR cycle and performing a piece-wise integration intime.

2. Governing equations

Thermal regenerator modeling includes or neglects variousterms so as to balance accuracy with simplicity. Models of AMRsoften assume plug flow prevails and one spatial dimension is suf-ficient to model the energy interactions (assumes internal temper-ature gradients are insignificant.) The effects of viscous dissipationon the thermal energy balance are often considered negligible;thus, the simplified relations describing the energy change of theheat transfer fluid (f) and the solid (s) as a function of space andtime can be written [13]:

m0f cp@Tf

@~tþ _mcp

@Tf

@~x¼ @

@~xAkf ;eff

@Tf

@~x

� �þ hA0ðTs � Tf Þ; ð1Þ

m0scB@Ts

@~tþm0sTs

@M@T

� �H

@l0H@~t¼ @

@~xAks;eff

@Ts

@~x

� �þ hA0ðTf � TsÞ: ð2Þ

m0f is the mass of fluid per unit length of the regenerator entrainedin the pores and m0s is the mass of magnetocaloric material per unitlength, M is the magnetization per unit mass, c is mass specific heat,A0 is the surface area of the matrix per unit length, keff is effectiveconductivity, and A is the cross-sectional area of the regenerator.

With a prescribed regenerator geometry, material, and applied-field and flow waveforms one may determine the transient and spa-tial nature of the temperatures throughout the AMR. However, thenon-linear nature of the material properties and the use of coupledpartial differential equations make it difficult to discern the impacts

of basic design and operating parameters on AMR performance. Asimpler representation of the physics would help with our under-standing of the cycle; however, a simplified representation of anAMR should take into account the fact that the AMR is physicallydescribed by the coupled interaction of the fluid and solid.

2.1. Idealized cycle equation

A simplified equation describing AMR behavior can be obtainedin the limit of very high convection coefficient when it may be as-sumed that local thermal equilibrium exists, Tf ffi Ts = T(x, t). In thiscase, a single temperature represents both fluid and solid and thetwo energy equations can be added to produce a single partial dif-ferential equation:

m0f cp þm0scB

� � @T@~tþ _mcp

@T@~x� @

@~xkeff A

@T@~x

� �

¼ �m0sT@M@T

� �H

@l0H@~t

: ð3Þ

In Eq. (3) the solid and fluid thermal conductivities have been re-placed by a single effective value, keff. In practice, empirical relationsincluding an effective static conductivity and a component due tofluid dispersion are used to represent the diffusive flux of energy[16].

Normalizing all terms in Eq. (3) by the thermal mass of refriger-ant per unit length and non-dimensionalizing time by the blowperiod, ~t ¼ sBt, and space by regenerator length, ~x ¼ Lx, results inthe following equation describing an AMR:

@T@tþ U

@T@x� 1

m0sLcB

sB

R@

@xkeff A

L@T@x

� �¼ 1

R@Tad

@tð4Þ

where

U � UR

ð5Þ

U �_mcpsB

m0sLcBð6Þ

Ha

a

c d

b

HH

HLa'

c'

A. Rowe / Cryogenics 52 (2012) 111–118 113

R � 1þm0f cp

m0scBð7Þ

@Tad

@t� � T

cB

@M@T

� �H

@l0H@t

: ð8Þ

Eq. (4) makes use of the fact that the grouping on the right is thedifferential isentropic temperature change of the refrigerant dueto a differential change in field, i.e., the magnetocaloric effect, asrepresented by Tad (Eq. (8)). The total magnetocaloric effect, DT, is,

DT ¼ �Z HH

HL

TcB

@M@T

� �Hl0 dH: ð9Þ

Eqs. (5) and (6) are two forms of utilization – a key parameter forregenerator operation [17]. They differ by the thermal mass onenormalizes the fluid flow with. They are equivalent when the ther-mal mass of fluid in the pores or a regenerator is insignificant nextto the thermal mass of the solid. The parameter R defined by Eq. (7)results from the ratio of total thermal mass to the solid thermalmass. Assuming the diffusion term is negligible in comparison tothe other terms in Eq. (4), the energy balance can be written,

@T@t¼ �U

@T@xþ 1

R@Tad

@tð10Þ

and is an idealized, time-dependent energy equation for an AMR.The impact of entrained fluid thermal mass reduces the effectivemagnetocaloric effect and is an intrinsic irreversibility. This equa-tion is assumed to be valid during a blow when the convection coef-ficient is large.

Because U and R are both functions of temperature and mag-netic field, a constant reference specific heat can be useful. Assum-ing a reference specific heat, cS, the following parameters arise,

cRX �cBXðT;HÞ

cSð11Þ

URX �_mcpsB

m0sLcS¼ cRXUX ð12Þ

where X becomes C for the low-field, or cold blow, condition and Hfor the high-field (hot blow) condition. Eq. (11) defines the referencesolid specific heat ratio and Eq. (12) gives a reference utilization interms of either the hot or cold blow. Other useful parameters are band r, where

b � ð_mcpsBÞCð _mcpsBÞH

; ð13Þ

r � cC

cH¼ cBðT;H ¼ 0Þ

cBðT;H ¼ HHÞ: ð14Þ

Due to the field and temperature dependence of the solid, andthe possibility of having different amounts of fluid blown throughthe regenerator at high field and low field, Eqs. (5), (6), (7), (11) and(12) will also use C and H as subscripts to differentiate betweencold and hot blow conditions.

In the following section Eq. (10) is used to determine the peri-odic steady-state conditions describing the temperature at anylocation in an AMR.

t

0

mH

m

Cm−

Cτ

τ ΒΗCτ Β

tCτ

Fig. 1. Applied field and mass flow for a discrete four-step AMR cycle.

3. AMR cycle

The AMR cycle is assumed to be a discrete four-step processconsisting of: (1) adiabatically increasing the magnetic field, (2)displacing heat transfer fluid through the AMR from the cold sideto the hot side, (3) adiabatically decreasing the field, and, finally,(4) displacing fluid in the opposite direction. These steps can over-lap; however, for the following analysis, the discrete four-step pro-cess will be assumed.

Fig. 1 shows the assumed field and flow waveforms over a com-plete cycle. The total period is sC and the duration of the fluid flowwhile at low and high field are respectively sBC and sBH(the sub-script B represents the ‘‘blow’’.) It should be noted that the massflow rates and blow periods are not necessarily equal; the pre-ferred relationship between these parameters will be discussed la-ter. The assumed orientation of the regenerator is with the coldend at the origin and the warm end at x = L, on the x-axis. Thus,the mass flow is in the positive direction during the high field,and in the negative direction during the low field. The peak appliedfield is HH and the low field strength is HL. For what follows, thelow field strength is assumed zero.

The representation of the idealized AMR cycle (Eq. (10)) at somelocation in the bed is shown on a T–s diagram for the refrigerant inFig. 2. The sequence of letters corresponds to the steps shown inFig. 1.

Eq. (10) tells us that if R = 1, the adiabatic increase in field fromb–c will result in a temperature change equal to the magnetocalo-ric effect of the material (because U = 0 during magnetization.) If Ris greater than one, the temperature change will be lower due tothe parasitic thermal mass of the fluid.

A general design goal is to operate an AMR at high frequency sothat the cooling power per unit volume of AMR is large. Assumingthis is true, the field change will be rapid. In addition, becausethere is no fluid flow during b–c or d–a, it is assumed that the ther-mal coupling during those steps is small and the resulting temper-ature change of the solid is equal to the magnetocaloric effect.During the short delay after the field change and before a blow be-gins, the solid and fluid are assumed to come to thermal equilib-rium. In Fig. 2 the dashed lines from b–c and d–a suggest whatthe fluid temperature may be during the magnetization anddemagnetization steps. For example, the fluid and solid tempera-tures are equal during the cold blow and are at b when the blowfinishes. The magnetic field is then quickly increased, and becausethe thermal coupling between solid and fluid is weak, the solidtemperature increases to c0 while the fluid remains at b. The solidand fluid then come to thermal equilibrium (while the field is con-stant) resulting in the temperature of the solid and fluid beingequal at point c. Referring to Fig. 2, the impact of fluid thermalmass in the AMR is to increase the area of the T–s diagram resultingin more magnetic work while simultaneously reducing the effec-tive temperature lift (b–c versus b–c0.)

Based on the above assumptions, relationships between thetemperatures b–c0–c and d–a0–a can be derived by a simple energybalance. For b–c0–c,

m0scHðTc0 � TcÞ ¼ m0f cpðTc � TbÞ; ð15Þ

and, with some manipulation,

s

HHT

HL

a

c'c

a'

b

d

CTδ

HTδ

( )aT TΔ

C

d TT T

dTδΔΔ +

Fig. 2. Idealized AMR cycle at an arbitrary location in the regenerator.

114 A. Rowe / Cryogenics 52 (2012) 111–118

Tc ¼Tc0 þ ðRH � 1ÞTb

RH: ð16Þ

The subscript, H, on c and R indicates the specific heat and thermalmass ratio at high field. Note that Eq. (16) is consistent with Eq. (10)in that,

Tc � Tb ¼Tc0 þ ðRH � 1ÞTb

RH� Tb ¼

DTb

RH: ð17Þ

A similar energy balance for d–a0–a gives,

m0scC Ta � Ta0ð Þ ¼ m0f cpðTd � TaÞ ð18Þ

and,

Ta ¼Ta0 þ ðRC � 1ÞTd

RC: ð19Þ

Again, the subscript, C, indicates the specific heat or thermal massratio at low field.

3.1. Step-wise temperatures

Using Eqs. (10), (16) and (19), relationships between all of thetemperatures in Fig. 2 are derived. It is assumed that the magnet-ocaloric effect is continuous in temperature and can be determinedby a first order Taylor series expansion about point a as shown inFig. 2. The temperatures in Fig. 2 can be determined in referenceto preceding points in the cycle as follows:

ðbÞ Tb ¼ Ta þ dTC ð20Þðc0Þ Tc0 ¼ Tb þ DTb ð21Þ

ðcÞ Tc ¼Tc0 þ ðRH � 1ÞTb

RHð22Þ

ðdÞ Td ¼ Tc þ dTH ð23Þða0Þ Ta0 ¼ Td � DTa0 ð24Þ

ðaÞ Ta ¼Ta0 þ ðRC � 1ÞTd

RCð25Þ

Using Eq. (10) and noting that during the high and low fieldblows the magnetic field change is zero (so the term representingthe magnetocaloric effect is zero,) the temperature changes due toregeneration can be determined by integrating over the blowperiod,

dT ffi �U@T@x: ð26Þ

In writing Eq. (26) it is assumed that the perturbation is small en-ough so that the local utilization and temperature gradient can beconsidered constant. The larger the perturbation due to regenera-tion, the less accurate this approximation becomes. Noting the signconventions for the hot and cold blows, the temperature perturba-tions for cold and hot blows are respectively:

dTC ffi UC@T@x

ð27Þ

dTH ffi �UH@T@x

����H

: ð28Þ

The temperature gradient at high field can be written in termsof the gradient at low field and the magnetocaloric effect (seeAppendix A),

@T@x

����H

¼ 1þ 1RH

dDTdT

� �@T@x

����C

ð29Þ

making Eq. (28),

dTH ffi �UH 1þ 1RH

dDTdT

� �@T@x: ð30Þ

The preceding relations provide expressions to determine thelocal temperature at all points in the cycle in reference to the tem-perature at point a, Ta.

4. Heat transfer

The idealized AMR cycle developed in Section 3 results in a nettransfer of energy from the cold side to the warm side of the regen-erator. An expression is now derived to determine the impact ofoperating parameters and material properties on the coolingpower.

4.1. Local energy transfer

Over a complete cycle, the energy transfer along the AMR by thefluid at some location in the AMR is equal to the cycle-averaged en-thalpy flux,

Qa ¼1sC

I_mhd~t: ð31Þ

The enthalpy of the fluid can be determined in reference to Ta

by,

hðTÞ ¼ ha þ cpðT � TaÞ ð32Þ

where ha is the fluid enthalpy at Ta.Eq. (31) is written as a piece-wise integration,

Qa ¼1sC

Z b

a� _mCcp

ha

cpþ T � Ta

� �d~t þ

Z d

c

_mHcpha

cpþ T � Ta

� �d~t

" #

ð33Þ

and, rearranging using non-dimensional time,

Qa ¼ð _mcpsBÞH

sCð1� bÞ ha

cp� Ta

� �þZ d

cT dt � b

Z b

aT dt

" #: ð34Þ

The last two integrals can be solved when Eq. (10) is integratedover time for each isofield process,Z b

aT dt ¼

Z 1

0UC

dTdx

t þ Ta

� �dtZ d

cT dt ¼

Z 1

0�UH 1þ 1

RH

dDTdT

� �dTdx

t þ Tc

� �dt

ð35Þ

and, solving,Z b

aT dt ¼ UC

2dTdxþ TaZ d

cT dt ¼ �UH

21þ 1

RH

dDTdT

� �dTdxþ Tc

ð36Þ

A. Rowe / Cryogenics 52 (2012) 111–118 115

Using the expressions derived earlier to write Tc in terms of Ta

gives,

Tc ¼ Ta þDTRHþ 1þ 1

RH

dDTdT

� �UC

dTdx; ð37Þ

and combining with Eq. (36),Z d

cT dt ¼ Ta þ

DTRHþ UC

UH� 1

2

� �1þ 1

RH

dDTdT

� �UH

dTdx: ð38Þ

The last part of Eq. (34) can now be written,Z d

cT dt � b

Z b

aTdt ¼ ð1� bÞTa þ

DTRHþ f1UH

dTdx

ð39Þ

where f1 is defined as,

f1 �UC

UH� 1

2

� �1þ 1

RH

dDTdT

� �� b

2UC

UH: ð40Þ

So, Eq. (31) can be written as,

Q a ¼M0

SLcS

sCURH ð1� bÞha

cpþ DT

RHþ f1UH

dTdx

� �: ð41Þ

4.2. Net heat transfer

The net enthalpy flux is reduced by parasitic heat leaks. A con-duction type loss is assumed to occur as follows (with non-dimen-sional space coordinate),

Q k ¼ �KdTdx

ð42Þ

where K is an overall thermal conductance, i.e.

K ¼ kAL: ð43Þ

The net local cooling power is then,

Q ¼ Q a þ Q k; ð44Þ

or,

Q ¼ m0sLcS

sCURH ð1� bÞha

cpþ DT

RHþ f1UH

dTdx

� �� K

dTdx: ð45Þ

Eq. (45) can be written using a non-dimensional conductance,

j � KsC

m0SLcSð46Þ

so that the cycle-averaged net heat transfer at a section of the AMRis,

Q ¼ m0sLcS

sCURH ð1� bÞha

cpþ DT

RHþ f1UH �

jURH

� �dTdx

� �: ð47Þ

L0

T

xδ

( )p B Hmc τ

( )p B Cmc τ0aT

1aT

Q dQQ x

dxδ+

( )pW Q xδ′ ′+

( )aT x

x

Fig. 3. A schematic representation of steady-state operation in an AMR.

5. Magnetic work

At a location in the regenerator the local rate of magnetic workper unit length of the AMR is

W 0M ¼ m0sB

dMdt

ð48Þ

where B is the applied field strength, l0H [18]. The sign conventionbeing that work performed on the magnetic material is positive.

The time-averaged magnetic work per unit length is,

W 0M ¼ m0s

1sC

IB

dMdt

dt ¼ m0sBdMdt: ð49Þ

Expanding the magnetization change in terms of temperature andfield as follows:

dM ¼ @M@T

� �B

dT þ @M@B

� �T

dB; ð50Þ

produces the following relation for the time-averaged local work,

W 0M ¼ m0s

@M@T

� �BB

dTdtþ @M

@B

� �TB

dBdt

" #: ð51Þ

Eq. (51) is written using the average partial derivatives. Thisapproximation becomes less valid as the utilization increases oras the difference between low and high field becomes large. Thesecond time-averaged term in the square brackets is equal to zeroresulting in the following relationship describing the local work,

W 0M ¼ m0s

@M@T

� �B

BdTdt: ð52Þ

As shown in Appendix B,

BdTdt¼�BH

sCðRH�1Þ 1� n

nþ11r

dDTdT

� �DTRHþðf2þðRH�1Þf3ÞUH

dTdx

� �:

ð53Þ

The explicit dependence on magnetization and field is removedfrom Eq. (52) by using Eq. (C.4). The expression for local work inthe AMR becomes,

W 0M¼

m0scS

sC

�DTT

cRH ðRH�1Þ 1� nnþ1

1r

dDTdT

� �DTRHþ f2þðRH�1Þf3ð ÞUH

dTdx

� �:

ð54Þ

The total magnetic work rate can be found by integrating Eq. (54)over the length of the AMR;

WM ¼Z L

0W 0

Md~x ¼ LZ 1

0W 0

M dx: ð55Þ

6. Energy balance

As shown schematically in Fig. 3, at any location in the AMR anenergy balance must exist between heat transfer and magneticwork in a periodic steady-state condition. Work is also needed todisplace the fluid through the regenerator; although the work isdissipated in the regenerator it is assumed that this thermal loadis small relative to the regenerative energy exchange betweenthe fluid and solid.

Besides magnetic work input, the possibility of an external heatload is modeled using,

Q 0p ¼ aðTamb � TaÞ ð56Þ

116 A. Rowe / Cryogenics 52 (2012) 111–118

where Tamb is the ambient temperature and a is the effective ther-mal conductance per unit length.

Using Fig. 3, the local energy balance is,

W 0M þ Q 0p ¼

dQd~x

: ð57Þ

Eq. (57) determines the temperature distribution throughout theAMR for specified boundary conditions. Once temperature distribu-tion is determined, the cooling power and magnetic work can beexplicitly calculated. Other loss mechanisms, such as pressure dropand imperfect convective heat transfer, may be considered sepa-rately which and then combined to estimate the overall COP.

6.1. Boundary conditions

The calculated temperature profile is the distribution of tem-peratures for point a in the local cycles i.e. Ta(x). The boundary con-ditions are the temperatures on each side of the AMR, Ta0 and Ta1,

Ta0 � Taðx ¼ 0Þ ð58ÞTa1 � Taðx ¼ 1Þ: ð59Þ

As these temperatures are for a specific point in the cycle, theyshould not be considered the reservoir temperatures, TC and TH, thatheat is being pumped across.

Various possibilities exist for choosing the effective cold andwarm temperatures. Experimental devices often measure the fluidtemperature just outside of the AMR, and the average tempera-tures are used to define the operating conditions. An average fluidtemperature is defined as,

Tf �14ðTa þ Tb þ Tc þ TdÞ: ð60Þ

Using Eqs. (20)–(23), the average fluid temperature becomes,

Tf ¼12

2Ta þDTRHþ 1þ 1

RH

dDTdT

� �UC

dTdx

� �ð61Þ

where the following has been used to simplify the expression,

ðdTC ; dTHÞ � ð2Ta;2TcÞ: ð62Þ

The relationship between Eqs. (58) and (59) and the averagefluid temperatures at each end are determined by evaluating Eq.(61) at the appropriate end of the regenerator. Thus, the averagetemperature of the fluid on the cold end is,

Tf jx¼0 ¼ Ta0 þ12

DTRHþ 1þ 1

RH

dDTdT

� �UC

dTdx

� �x¼0

ð63Þ

and, the average fluid temperature on the warm end is,

Tf jx¼1 ¼ Ta1 þ12

DTRHþ 1þ 1

RH

dDTdT

� �UC

dTdx

� �x¼1: ð64Þ

Eqs. (63) and (64) map the boundary conditions on Ta to measuredconditions.

6.2. Local cycle completion

Constraints on operating parameters arise due to the assumedcycle and the need for the AMR to interact with thermal reservoirsat the cold and warm ends. One can imagine that Eqs. (27) and (30)must be related if the cycle depicted in Fig. 2 is to be realized. If theregenerator blows were arbitrary then the cycle may not completeaccording to the assumed pseudo-Brayton process. A constraintrelating the two blow processes can be determined using Eqs.(20)–(25), (27) and (30).

Assuming the cycle starts at a, and progresses in a counter-clockwise fashion, the temperature at point d can be written as,

Td ¼ Ta þDTRHþ 1þ 1

RH

dDTdT

� �dTC þ dTH: ð65Þ

Another path to point d from a is a–a0–d, given by,

Td ¼ Ta0 þ DT � dDTdT

Ta � Ta0ð Þ: ð66Þ

Using Eq. (25), Eq. (66) can be written in terms of a only,

Td ¼ Ta þ RC þ ðRC � 1ÞdDTdT

� ��1

DT: ð67Þ

If the cycle is to complete as assumed, Eq. (65) must equal Eq. (67);thus,

1þ 1RH

dDTdT

� �dTC þ dTH ¼ RC þ ðRC � 1ÞdDT

dT

� ��1

� 1RH

!DT:

ð68Þ

Using Eqs. (27) and (30) for the regenerative temperature perturba-tions and simplifying gives,

UC � UH ¼ RC þ ðRC � 1ÞdDTdT

� ��1

� 1RH

" #1þ 1

RH

dDTdT

� ��1

� DTdT=dx

: ð69Þ

Expanding the terms on the left side gives,

ð _mcpsBÞCm0sLcC

1RC�ð

_mcpsBÞHm0sLcH

1RH¼ RCþðRC�1ÞdDT

dT

� ��1

� 1RH

" #1þ 1

RH

dDTdT

� ��1 DTdT=dx

:

ð70Þ

Eq. (70) provides a necessary condition for the step-wise cycle to besatisfied at every location in the AMR. An example of the implica-tions of this constraint is most easily shown for the limiting caseof RC = RH = 1 which is approached when the fluid in the pores ofthe regenerator has a much smaller thermal mass than the solid.In this case, the term in square brackets on the right side of Eq.(70) is equal to zero; therefore, the following must be satisfied,

ð _mcpsBÞCð _mcpsBÞH

¼ cC

cH: ð71Þ

This result was previously found for an ideal AMR with insignificantfluid thermal mass in the pores [6]. The ratio on the left side of Eq.(71) is b (balance) and the ratio on the right side is r (symmetry). So,to create the assumed cycle, the following must be true:

b ¼ r ðwhen RC ¼ RH ¼ 1Þ: ð72Þ

In general, Eq. (70) will not be satisfied at all locations in anAMR because of the temperature and field-dependent specific heatof the refrigerant.

7. Discussion

The following equations describing the thermodynamics of anAMR cycle have been derived in terms of the non-dimensionalspace co-ordinate, x:

QðxÞ ¼ m0sLcS

sCURH ð1� bÞha

cpþ DT

RHþ f1UH �

jURH

� �dTdx

� �; ð73Þ

W 0MðxÞ¼

m0scS

sC

DTT

cRH ðRH�1Þ 1� nnþ1

1r

dDTdT

� �DTRHþðf2þðRH�1Þf3ÞUH

dTdx

� �:

ð74Þ

Eq. (73) describes the net transfer of heat at any location in theAMR, and Eq. (74) is the local rate of magnetic work. The parame-ters f1–f3 are:

A. Rowe / Cryogenics 52 (2012) 111–118 117

f1 �UC

UH� 1

2

� �1þ 1

RH

dDTdT

� �� b

2UC

UH; ð75Þ

f2 � 1þ 1RH� n

nþ 1UC

UH

� �dDTdT

; ð76Þ

f3 �UC

UH

1RH� n

nþ 11r

UC

UH� 1

� �1þ 1

RH

dDTdT

� �� �dDTdT

: ð77Þ

The total magnetic work input is,

WM ¼Z L

0W 0

M d~x ¼ LZ 1

0W 0

M dx; ð78Þ

and, the temperature distribution is determined as that which sat-isfies the periodic steady-state energy balance,

W 0M þ Q 0p ¼

1L

dQdx

: ð79Þ

The key assumptions in the derivation are: (1) thermal couplingbetween the solid and fluid is sufficiently large to assume a singletemperature represents both substances during a blow; (2) therefrigerant properties are continuous; (3) the power dissipatedby fluid displacement is small relative to the thermal regenerationbetween solid and fluid; and, (4) the temperature perturbation dueto regeneration is small so that the local temperature gradient andutilization are constant over a blow. The last assumption is increas-ingly less valid as utilization and temperature gradient increase. Inaddition, the validity of perfect thermal coupling during a blowperiod between the fluid and solid (assumption 1) decreases nearthe ends of the regenerator. Thus, the theory developed here isapproximate and should not be expected to replace a completesolution of the coupled energy equations using actual propertydata. However, the approximate nature of the thermodynamicdescription lends itself to easy computation and interpretationwhile also providing a generic approach to understanding AMR de-sign, operation and the effects of different magnetocaloricmaterials.

8. Conclusions

Relationships describing the thermodynamic quantities of anidealized AMR cycle are derived by assuming the convective inter-action is large so that thermal equilibrium between fluid and solidexists during regeneration. In Section 3, a step-wise cycle is definedand a single equation describing the temperature as a function oftime and position is derived. Relations for the temperatures at eachstep in the cycle are also developed. In Section 7 a relationship giv-ing the net cooling power at any location in the AMR is derived,and in Section 8 a relationship for local work rate is developed.In Section 6.1, the overall energy balance expression is presentedwith transformations needed to relate the boundary conditionsto effective operating temperatures. In Section 6.2 a balanceexpression is derived in terms of operating parameters and mate-rial properties; this relation indicates conditions when the localenergy balance would satisfy the assumed cycle. The main thermo-dynamic expressions for the AMR cycle are summarized in Section7. By solving the energy balance expression to determine temper-ature distribution (Eq. (79)) one can calculate work, heat transfer,and COP. This can done using real refrigerant properties with tem-perature and magnetic field-dependence included, or, approximatesolutions using idealized conditions are possible. Some examplesof the latter are presented in part II of this work.

Acknowledgments

The support of the Natural Sciences and Engineering ResearchCouncil of Canada and the NSERC Hydrogen Canada (H2Can) Stra-tegic Network is greatly appreciated.

Appendix A

The temperature distribution throughout the AMR at point a inthe cycle is T(x). Assuming that the temperature gradients duringthe hot and cold blows are constant over the duration of the blows,the gradient at point c in the cycle can be written in terms of pointa. At point c in the cycle, the temperature gradient at high field canbe written,

dTdx

����H

ffi Tc2 � Tc1

dxðA:1Þ

where 1 and 2 are two locations in the AMR dx apart. Eq. (A.1) iswritten in terms of the low field temperatures at b using,

Tc ¼Tc0 þ ðRH � 1ÞTb

RH¼ Tb þ DTb þ ðRH � 1ÞTb

RH: ðA:2Þ

Thus, Eq. (A.1) becomes,

dTdxHffi 1

dxDTb2 þ RH2Tb

RH2� DTb1 þ RH1Tb1

RH1

� �: ðA:3Þ

RH is assumed to be a weak function of temperature so it is assumedthat RH2 = RH1 = RH.

The temperatures at point 2 are related to point 1 by,

Tb2 ¼ Tb1 þdTdx

dx; ðA:4Þ

and,

DTb2 ¼ DTb1 þdDTdT

� �dTdx

dx: ðA:5Þ

Using the above in Eq. (A.3) and simplifying gives,

dTdx

����H

¼ 1þ 1RH

dDTdT

� �dTdx: ðA:6Þ

Appendix B

To create an explicit relation for the local work the followingrelationship must be evaluated,

BdTdt¼ 1

sC

IB

dTdt

� �dt: ðB:1Þ

Using Fig. 1 and the relations derived in Section 3.1, Eq. (B.1) can beevaluated by piece-wise integration,

BdTdt¼ 1

sC

Z b

a|{z}ðiÞ

þZ c0

b|{z}ðiiÞ

þZ d

c0|{z}ðiiiÞ

þZ a0

d|{z}ðivÞ

þZ a

a0|{z}ðvÞ

26664

37775: ðB:2Þ

Each of the integrals in the summation corresponds to the labeledsteps in Fig. 1 and Fig. 2. For a–b and a0–a the field strength is as-sumed constant at BL = 0, therefore terms (i) and (v) are equal tozero.

Assuming the magnetocaloric effect is a function of the appliedfield according to,

Tad ¼ DTB

BH

� �n

; ðB:3Þ

118 A. Rowe / Cryogenics 52 (2012) 111–118

for process b–c0 the refrigerant temperature is determined by Eq.(B.3) and term (ii) can be written,Z c0

bBdTad ¼

Z BH

0

BBH

nDTbB

BH

� �n�1

dB ¼ nnþ 1

DTbBH: ðB:4Þ

At b the total adiabatic temperature change is found by expansionabout a, thus, Eq. (B.4) becomes,

ðiiÞ ¼ nnþ 1

BH DT þ dDTdT

dTC

� �: ðB:5Þ

Likewise, for d–a0,

ðivÞ ¼ � nnþ 1

BH DT � dDTdT

dTa0a

� �: ðB:6Þ

From c0 to d the field is constant at BH and term (iii) is,

ðiiiÞ ¼ BH dTc0c þ dTHð Þ: ðB:7Þ

Using Eqs. (B.5)–(B.7),

BdTdt¼ BH

sCdTH þ

nnþ 1

dDTdT

dTC

� �þ dTc0c þ

nnþ 1

dDTdT

dTa0a

� �� �:

ðB:8Þ

Eq. (B.8) is now written using the relationships derived in Section3.1. The first group in parentheses is,

�f2UHdTdx: ðB:9Þ

where the parameter, f2, is,

f2 � 1þ 1RH� n

nþ 1UC

UH

� �dDTdT

: ðB:10Þ

The second group is,

� RH � 1RH

� �DT þ dDT

dTdTC

� �

þ nnþ 1

RC � 1RH

� �RHðdTC þ dTHÞ þ DT þ dDT

dTdTC

� �dDTdT

; ðB:11Þ

or, after substituting and simplifying,

ðRH � 1Þ �f3UHdTdx� 1� n

nþ 11r

dDTdT

� �DTRH

� �ðB:12Þ

where the parameter f3 is,

f3 �UC

UH

1RH� n

nþ 11r

UC

UH� 1

� �1þ 1

RH

dDTdT

� �� �dDTdT

: ðB:13Þ

Eq. (B.8) can now be written using Eqs. (B.9) and (B.13),

BdTdt¼ �BH

sCðRH � 1Þ 1� n

nþ 11r

dDTdT

� �DTRH

�

þðf2 þ ðRH � 1Þf3ÞUHdTdx

�: ðB:14Þ

Appendix C

The isothermal magnetic entropy change is related to magneti-zation via,

DsMðTÞ ¼Z HH

0

@MðT;HÞ@T

� �Hl0dH ðC:1Þ

or, in terms of the average isofield variation in magnetization withtemperature from an applied field of H = 0 to H = HH,

DsMðTÞ ¼@M@T

� �B

BH ðC:2Þ

where the applied field, l0H, is written using B. It is useful to be ableto transform between magnetic entropy change and adiabatic tem-perature change. The following equivalence is assumed (a goodapproximation for a second-order material near room-temperaturelike gadolinium and field changes of �2 T),

DsMðTÞ ffi �cHðTÞDTðTÞ

T: ðC:3Þ

where cH is the mass specific heat at high field and T. Eqs. (C.2) and(C.3) give,

cHDTT¼ � @M

@T

� �B

BH: ðC:4Þ

References

[1] Wood ME, Potter WH. General analysis of magnetic refrigeration and itsoptimization using a new concept: maximization of refrigerant capacity.Cryogenics 1985;25:667–83.

[2] Cross CR, Barclay JA, Degregoria AJ, Jaeger SR, Johnson JW. Optimaltemperature–entropy curves for magnetic refrigeration. Adv Cryogenic Eng1988;33:767.

[3] Chen J, Yan Z. Regenerative characteristics of magnetic or gas Stirlingrefrigeration cycle. Cryogenics 1993;33:863–7.

[4] Reid CE, Barclay JA, Hall JA, Sarangi S. Selection of materials for an activemagnetic regenerative refrigerator. J Alloys Compnds 1994:366–71.

[5] Hall JL, Reid CE, Spearing IG, Barclay JA. Thermodynamic considerations for thedesign of active magnetic regenerative refrigerators. Adv Cryogenic Eng1996;41:1653–63.

[6] Rowe AM, Barclay JA. Ideal magnetocaloric effect for active magneticregenerators. J Appl Phys 2003;93:1672–6.

[7] Matsumoto K, Hashimoto T. Thermodynamic analysis of magnetically activeregenerator from 30 to 70 K with a Brayton-like cycle. Cryogenics 1990;30:840–5.

[8] Carpetis C. An assessment of the efficiency and refrigeration power of magneticrefrigerators with ferromagnetic refrigerants. Adv Cryogenic Eng 1994;39:1407–15.

[9] DeGregoria AJ. Modeling the active magnetic regenerator. Adv Cryogenic Eng1992;37 B:867–73.

[10] Hu JC, Xiao JH. New method for analysis of active magnetic regenerator inmagnetic refrigeration at room temperature. Cryogenics 1995;35:101–4.

[11] Smaı̈li A, Chahine R. Thermodynamic investigations of optimum activemagnetic regenerators. Cryogenics 1998;38:247–52.

[12] Engelbrecht KL, Nellis GF, Klein SA, Boeder AM. Modeling active magneticregenerative refrigeration systems. In: Proceedings of 1st internationalconference on magnetic refrigeration at room temperature; 2006. p. 265–74.

[13] Dikeos J, Rowe A, Tura A. Numerical analysis of an active magnetic regenerator(AMR) refrigeration cycle. Adv Cryogenic Eng 2006;51A and B(823):993–1000.

[14] Nielsen KK, Bahl CRH, Smith A, Pryds N, Hattel J. A comprehensive parameterstudy of an active magnetic regenerator using a 2D numerical model. Int JRefrig 2010;33:753–64.

[15] Tagliafico G, Scarpa F, Canepa F. A dynamic 1-D model for a reciprocatingactive magnetic regenerator; influence of the main working parameters. Int JRefrig 2010;33:286–93.

[16] Wang J, Carson JK, North MF, Cleland DJ. A new structural model of effectivethermal conductivity for heterogeneous materials with co-continuous phases.Int J Heat Mass Transfer 2008;51:2389–97.

[17] Nellis GF, Klein SA. Regenerative heat exchangers with significant entrainedfluid heat capacity. Int J Heat Mass Transfer 2006;49:329–40.

[18] Callen HB. Thermodynamics and an introduction to thermostatistics. 2nded. New York: John Wiley and Sons; 1985.