zimm delft 08

1Astronautics Corp of America Astronautics Corp of America

Modeling Magnetic Refrigeration

Dr. Steven Jacobs (primary author) [email protected]. Carl Zimm (presenter) [email protected]

Astronautics Corporation of AmericaMadison and Milwaukee, WI USA

Delft Days on Magnetocalorics, Oct 30-31, 2008

machines, materials, modeling


DRIVE MOTOR

MAGNET

VALVE ASSEMBLY

Rotary Magnet Magnetic Refrigerator (BB2)

TORQUEMETER

BED ASSEMBLY


Modeling the Magnetic Refrigeration System1

xCold

ReservoirHot Reservoir

Porous bed of MCM

Fluid flow

x = 0 x = L

( )t

B

B

STTTha

x

Tk

xt

TC bbbf

bb

bbb ∂

∂∂∂−−−+

∂∂

∂∂=

∂∂− ρερε )1()1(

( ) Fx

TC

ATTha

x

Tk

xt

TC f

ff

fbf

ff

ff +∂

∂Φ−−+

∂∂

∂∂=

∂∂ ρ

ερ

Rate of change of internal energy

Energy transport via conduction

Energy exchange between fluid and

solid

Magnetic work done on MCM

Energy transport via flow

Energy generation from viscous dissipation

1Engelbrecht 2008 Ph.D. Thesis, UW ME Dept.


Solving The MR Thermal Profile Equations

• ACA developed new and extremely efficient method for solving the thermal profile equations

• Easily handles layered beds with discontinuous properties

• Exploits smoothness of temperature profiles within layer by using high-order, rapidly converging techniques

• On the order of 100 times faster than other published methods– In some cases (low flow rate, small span) ~ 500 x faster

• Evaluation of refrigerator performance fast enough that:– Equation solver can be coupled to numerical optimization software

to perform automated optimal design of refrigeration system– Large scale parameter studies can be performed


Software ValidationSystem with 5 layers of LaFeSiH (36% porosity), 120 RPM, 4 lit/min, 1.4 T field

Convergence

153.661254170.072670402

153.661255170.072670322

153.661279170.072665202

153.660588170.072145102

153.646353170.05815362

153.624981170.03647142

Cooling Load (W)

Heat Exhaust (W)

Number of Terms

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

0.01 0.10

Relative Mesh Size

Rel

ativ

e E

rro

r in

En

erg

y C

on

serv

atio

n

Energy Conservation

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.01 0.10 1.00

Relative Mesh Size

RM

S E

rro

r in

Bed

Eq

uat

ion C-to-H

Demag

H-to-C

Mag

Mesh Size^4

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.01 0.10 1.00

Relative Mesh Size

Rel

ativ

e E

rro

rin

Flu

id P

DE C-to-H

Demag

H-to-C

Mag

Mesh Size^4

Satisfaction of Equations

Theoretical rate of convergence

Theoretical rate of convergence


Modeling Magnetocaloric Properties

• Magnetocaloric properties– Cb(B,Tb), S(B,Tb)

• High-order methods employed in solution of thermal profile equations requires smooth dependence on B, Tb

– Precludes interpolation

• Should satisfy thermodynamic constraints

– S = T integral of C/T– S decreasing function of B

• ACA developed new analytic representation for C such that S can be expressed in closed form and satisfies thermodynamic constraints

• Free parameters in representation chosen to fit to data and enforce thermodynamic constraints

• Excellent fit to Cb data for a variety of MCM (first-order, second-order)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370

T (K)

C (

J/K

g/K

)

Data

Fit

B = 0

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370

T (K)

C (

J/K

g/K

)

Data

Fit

B = 2 Tesla


Optimized System Design – Engineering Prototype (BB3)

• 5 layers of LaFeSiH with 37% porosity, particle diameter fixed at 205 microns

• Bed dimensions fixed by size of existing magnet

• Peak magnet field = 1.4 T• Goal: maximize cooling

power with pressure drop ≤15 PSI

298.15 (upper bound)Hot reservoir temp (K)

2.00000 (upper bound)Frequency (Hz)

3.07974Flow rate (L/min)

294.4460-field Curie point 5th layer (K)

291.2710-field Curie point 4th layer (K)

287.9850-field Curie point 3rd layer (K)

284.6700-field Curie point 2nd layer (K)

281.6530-field Curie point 1st layer (K)

ValueSystem Parameter

≥ 10

≤ 15

≥ 5

≥ 750

Target

14Span (K)

15.053Pressure drop (PSI)

8.671COP (W/W)

1476.4Total cooling load (W)

ValuePerformance Parameter


Optimized System Design – Supplemental Electronics Cooler (SEC)

• Layered bed of LaFeSiH• Bed dimensions can be

chosen by optimization• Peak magnet field = 1.5 T• Hot reservoir temperature

fixed at 44 C• Goal: minimize cold outlet

temperature with pressure drop ≤ 20 PSI, COP ≥ 10, and cooling power ≥ 3 kW

293.258 K0-field Curie point 6th layer



285.671 K0-field Curie point 3rd layer

283.066 K0-field Curie point 2nd layer

280.803 K0-field Curie point 1st layer

6Number of layers

6.5812 sq. cmBed cross-sectional area

40.03 mmBed length

304.911 KCold reservoir temperature

4.791 lit/minFlow rate

upper bound250 micronsParticle diameter

upper bound2.0 HzMagnet rotation frequency

upper bound0.37Porosity

NotesValueSystem parameter

≤ 33 C29.5 CCold outlet temperature

≤ 20 PSI20.15 PSIPressure drop

≥10 W/W10.37 W/WCOP

≥ 3 kW2.999 kWTotal cooling power

Design TargetValuePerformance Parameter


The Advantages of a Layered BedTHot = 25 C 120 RPM 3.1 lit/min Bed Volume = 15.6 cm3 1.4 T Field

0

2

4

6

8

10

12

14

16

18

20

22

0 200 400 600 800 1000 1200 1400 1600 1800

Cooling Load (W)

Te

mpe

ratu

re S

pan

(C

)

5-layer LaFeSiH bed

Gd bed

1-layer LaFeSiH bed

5-layer bed optimized for performance at span = 14 C

*Note how rapidly the 5-layer bed performance deteriorates as the span goes above its design span


Sensitivity of Optimized Designs to Random Variation in Curie Temperatures

• Lack of control during fabrication will cause variation in Curie temperatures

– Strong sensitivity to Curie temperatures → low fabrication yield

• Curie temperatures may change over time while in service

– Strong sensitivity to Curie temperatures → short service life

• Use Monte-Carlo analysis to evaluate sensitivity in performance to random Curie point variation of up to ± 0.5 K

• Cooling load dropped < 8% over 400 Monte-Carlo trials

• No trial exceeded the cooling load of the design – establishes validity of optimization process

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

2411

2442

2473

2504

2535

2566

2597

2628

2659

2690

2721

2752

2783

2814

2845

2876

2907

2938

2969

3000

Cooling Load (W)

Fra

ctio

n o

f M

C T

rial

s

At design Curie points

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

1302

1314

1326

1338

1350

1362

1374

1386

1398

1410

1422

1434

1446

1458

1470

1482

1494

1506

1518

1530

Cooling Load (W)

Fra

ctio

n o

f M

C T

rial

s At design Curie points

Engineering prototype

SEC


Sensitivity of Optimized designs to Flow Imbalance

Pump

Bed 1

Bed 2

Φ

C H

C H

Cold-to-hot flowΦ

Hot-to-cold flow

Φ/2 + ∆ Φ/2 −∆

Φ/2 + ∆Φ/2 –∆

Flow Imbalance in a 2-Bed System

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Relative Flow Imbalance

Rel

ativ

e C

oo

ling

Lo

ad

SEC

BB3

Systems are sensitive to flow imbalance

• 5% imbalance reduces cooling load by 27% for SEC, 37% for BB3

• Must design flow control system to minimize any possible flow imbalance


The Effect of Eddy Currents

• Bed and metal shell used to seal bed and attach it to plenum are subjected to strong, time-varying magnetic field which generates eddy currents

• Eddy currents cause Joule heating as they flow in resistive shell and bed

• Does this effect need to be included in thermal profile equations?

• Can metal with higher conductivity be used for bed shell, other bed parts?

Bed of MCM

Metal shell fits over bed


Eddy Currents in the Bed Shell

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.2 0.4 0.6 0.8 1.0

Time (s)

Oh

mic

Po

wer

(W)

BB3 ShellPeak Field = 1.4 T

Power reaches peak as fringe field sweeps over shell

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10

Magnet Rotation Frequency (Hz)

Ave

rag

e P

ow

er o

ver

Cyc

le (

W) Stainless Steel

Titanium alloy

Instantaneous power developed in stainless steel bed shell at 2 Hz

Average power developed in bed shell over one cycle

At desired operating frequency (2 Hz) Joule

heating is negligible with either metal


Eddy Currents in the Bed

• Difficult to model eddy current generation in bed exactly because electrical conductivity of porous bed is complicated function of position

• Get an upper bound on the eddy current generation by treating bed as solid block of material with conductivity (1 – ε) × σ where ε = bed porosity, σ = conductivity of spherical particles composing the bed

• Assume σ = 1.4 × 10-6 mhos/m (same as stainless steel)

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10

Magnet Rotation Frequency (Hz)

Ave

rag

e P

ow

er o

ver

Cyc

le (

W) SEC

BB3

At desired operating frequency, Joule heating in bed is negligible


Simulating the Effect of Dead Volume

• Dead volume is the volume at each end of the bed where substantial mixing can occur between inlet and outlet fluids, which are generally at different temperatures

• Model dead volume by adding a layer at each end of the bed composed of packed, inert particles

– Particles are thermally conductive but have no magnetocaloric effect

– Dead volume layers have same porosity as active portion of bed

• Particles serve to “mix” the inlet and outlet fluids, simulating the enhanced thermal interaction between them due to their complex flow pattern

• The “Dead Volume Ratio” is the ratio of the volume of one dead volume layer to the fluid volume of the active bed layers = l / L

Cold Reservoir

Hot Reservoir

Po

rou

s bed

of

MC

M

Fluid flow

x = 0

x = L

x = L+l

x = -l Dead volume

Dead volume


The Effect of Dead Volume on Optimized System Performance

0

400

800

1200

1600

2000

2400

2800

3200

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Dead Volume Ratio

Co

olin

g L

oad

(W)

SEC

BB3

The presence of dead volume can significantly degrade machine performance

Dead volume has a substantial detrimental effect at small temperature spans where the cooling load is largest and therefore the temperature difference between inlet and outlet fluids is largest

As the span increases and cooling load decreases, the inlet and outlet temperature difference decreases so the effect of dead volume decreases


Theoretical and Experimental Performance of BB2 with Gd Beds60 RPM 0.75 lit/min 1.4 T Peak Field

0

2

4

6

8

10

12

14

16

18

20

22

24

26

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Cooling Load (W)

Tem

per

atu

re S

pan

(C

)

Theory T(Hot) = 30 C

Msmt T(Hot) = 30 C


Msmt T(Hot) = 24 C


Msmt T(Hot) = 20 C

Excellent agreement at zero span precludes existence of

dead volume

Over-performance of machine at zero span probably due to

convective cooling to ambient at this relatively high temperature

Under-performance of machine at large span is consistent

with 4% flow imbalance (see next

slide)


Theoretical and Experimental Performance of BB2 with Gd Beds4% Flow Imbalance

0

2

4

6

8

10

12

14

16

18

20

22

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Cooling Load (W)

Tem

per

atu

re S

pan

(C)


Msmt T(Hot) = 30 C


Msmt T(Hot) = 24 C


Msmt T(Hot) = 20 C

zimm delft 08

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