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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
2Astronautics Corp of America Astronautics Corp of America
DRIVE MOTOR
MAGNET
VALVE ASSEMBLY
Rotary Magnet Magnetic Refrigerator (BB2)
TORQUEMETER
BED ASSEMBLY
3Astronautics Corp of America Astronautics Corp of America
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.
4Astronautics Corp of America Astronautics Corp of America
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
5Astronautics Corp of America Astronautics Corp of America
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
6Astronautics Corp of America Astronautics Corp of America
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
7Astronautics Corp of America Astronautics Corp of America
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
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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
290.870 K0-field Curie point 5th layer
288.292 K0-field Curie point 4th 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
9Astronautics Corp of America Astronautics Corp of America
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
10Astronautics Corp of America Astronautics Corp of America
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
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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
12Astronautics Corp of America Astronautics Corp of America
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
13Astronautics Corp of America Astronautics Corp of America
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
14Astronautics Corp of America Astronautics Corp of America
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
15Astronautics Corp of America Astronautics Corp of America
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
16Astronautics Corp of America Astronautics Corp of America
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
17Astronautics Corp of America Astronautics Corp of America
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
Theory T(Hot) = 24 C
Msmt T(Hot) = 24 C
Theory T(Hot) = 20 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)
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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)
Theory T(Hot) = 30 C
Msmt T(Hot) = 30 C
Theory T(Hot) = 24 C
Msmt T(Hot) = 24 C
Theory T(Hot) = 20 C
Msmt T(Hot) = 20 C