1Cryogenic EngineeringMagnetic Work and the Magnetocaloric
EffectMech 445 Cryogenic Engineering - Lecture 19 1Magnetic Cycles
If we have a way of altering entropy, we have a way of creating a
cooling cycle. For gases, entropy is a function of temperature and
pressure( ) , s s T p =1423 a significant entropy change can be
induced by a pressure variation. If the total entropy is constant
(ds = 0)( ) , s s T psT Ap Ts sds dT dpT p| | c c | |= + | |c c \ .
\ .pTc sds dT dpT p| | c= + |c\ .s T sdT dp| | c= |c\ .2ps T sT dp|
| cA = } |c\ .Mech 445 Cryogenic Engineering - Lecture 19 2 When we
magnetize a substance, we alter entropy (magnetic entropy change)
How can we use this to create a magnetic cycle?sp Tc p |c\ .( ) ,
es s T B =1 p p Tc p |c\ .2Magnetism Review Maxwells eqn for
magnetic flux: Flux lines have no beginning or end there are no
magnetic monopoles (unlike charge.)0 B V = The functional form of
the permeability as a function of H defines different magnetic
materials. i.e. The concept of magnetization, M, arises from the
impact of specific materials on flux density Can be explained by
applying Biot-Savart to atoms Electrons are charges in motion and,
therefore, generate magnetic moments just as a coil of wire
carrying a current will generate a magnetic moment Moment is
determined when placed in external field measure Lorentz force0 B
V( ) H =Mech 445 Cryogenic Engineering - Lecture 19 3 Moment is
determined when placed in external field measure Lorentz
forceMagnetic Phenomena The magnetic moment arising from the
current loop is, An electron orbiting an atom will generate a
moment due to orbital angular dipole moment Area of loop I = g g
gmomentum In addition, a moment is generated by the spin moment of
the electron The magnetic moment due to electron spin is defined as
a Bohr magneton, The net magnetic moment, J, is found by summing
the spin, S, and orbital, L, moments,27 29.27 10 A m2ee hm| | |= =
|\ .J S L = + Mech 445 Cryogenic Engineering - Lecture 19 4 The
behaviour of the magnetic dipoles of atoms or ions determines the
macroscopic behaviour of the material Three general types
Diamagnetic, paramagnetic, ferromagnetic (many variations,
ferrimagnetic, antiferromagnetic.)3Magnetic Phenomena The general
constitutive relation (relating flux density to field) for any
material can be written in many ways:( )0B H M = + Mis due to
internal currents, H is developed by internal and external
currents, M is the magnetization and is the total magnetic moment
of a sample (sum of all atomic moments) divided by the volume of
the sample. M is a function of the applied field and can be
described by the susceptibility, ( )0B H M +imMV( ) ( ) M H H H _
=Mech 445 Cryogenic Engineering - Lecture 19 5 Susceptibility is
related to permeability by,So, Or, in terms of relative
permeability, 0(1 ) _ = +( )01 B H _ = +( )01r _ = +0 rB H
=Magnetic Materials The three general magnetic material behaviours
can be described by their susceptibilities or permeabilities:
Diamagnetic material the flux density in a di i i l i l h ld i
iMferromagneticdiamagnetic material is less than would exist in the
same region of space if the material were not there. Susceptibility
is small and negative. Or, equivalently, examples copper, bismuth,
silver superconductors are perfect diamagnets, Paramagnetic
material the flux density is higher than with free space, but still
have small susceptibilities,1r HparamagneticdiamagneticMech 445
Cryogenic Engineering - Lecture 19 6 Examples aluminum, platinum,
many metals Ferromagnetic flux density is greatly enhanced, very
high permeabilities, Examples iron, cobalt, nickel, rare earth
metals and alloys. Strong coupling between microscopic moments
causes non-linear response to applied field.1r >>4Magnetic
Materials Some ferromagnetic materials will retain their net
magnetization when the field is removed Bpermanentmagnet These are
called permanent magnets and the remaining flux density, is called
the remnance, Br When a large enough field in a direction opposite
to the magnetization vector is applied, the bulk magnetization will
return to zero. The field strength required for this is called the
coercive field New, rare-earth permanent magnets have l d i fi
ldHferromagneticparamagneticdiamagnetic HcBr1r >1r >Mech 445
Cryogenic Engineering - Lecture 19 7very large remnance and
coercive fields. The strength of a permanent magnet is usually
specified in terms of the energy stored. The area in the second
quadrant is an indication of thisgcMagnetic Materials Besides being
a function of H, the magnetization can also be (is) a function of
temperature: Why? The atomic/molecular dipoles want to align with
an external field, but thermal vibrations act to prevent full
alignment (saturation.) As temperature decreases, greater alignment
is possible. For some materials, when a critical temperature is
reached, the material will change from paramagnetic to
ferromagnetic - called the Curie temperature.( , ) M M T H =Mech
445 Cryogenic Engineering - Lecture 19 8p This phenomenon leads to
the magnetocaloric effect and allows one to create a magnetic
cycle.5Magnetocaloric Effect Reversible temperature change when
adiabatic change of magnetic field discovered by Warburg ~1881. Not
to be confused with eddy current heating due to Faradays Law = i
ibl irreversible. Total entropy is a function of temperature and
field, s(T,H).1 2T2 BaHTfMech 445 Cryogenic Engineering - Lecture
19 9( , , )LMCE T T B B A A1 2s1BaLTiMagnetic Work First, we need
to understand the concept of magnetic work Imagine a magnetic body
inside a superconducting solenoid. If current is constant, then the
emf across the battery is zero When the current is not zero a field
is created by the coil and the material inside becomes magnetized.
Assume that the magnetization is a single valued function of
current Ie. there is no hystersis ( ) , I = M M rMech 445 Cryogenic
Engineering - Lecture 19 10 If the body were not inside the
solenoid, the current would produce a magnetic flux density which
is a linear function of current This is the external magnetic
field, Be ( )e I = B b rPosition in systemDepends on shape of
coil6Work due to field change Want to equate work done by power
supply to magnetization of the system. If the current is increased,
the external field increases and the magnetic moment changes in
response Th b tt d k t d thi Th t f k i i b The battery does work
to do this. The rate of work is give by The voltage (back emf)
arises from two sources. One is the change in magnetic flux, Be for
an empty solenoid the magnetic work is equal to the change in
energy of the magnetic fieldcurrent x voltagedWIVdt = =212 edW d B
dV| |= } |\ .Mech 445 Cryogenic Engineering - Lecture 19 11 And the
integral is over the entire volume of the solenoid field. The
second contribution to the work is due to the magnetization of the
system inside the solenoid.02\ .Magnetic Work Consider a an
elementary dipole at some position r. A current loop with current i
and an area a. The magnetic moment of the dipole is then If the
current in the solenoid is I, the field produced by the solenoid at
i = m a, p ythe dipole is, The field creates a flux linkage through
the small current loop given by The grouping ba is the mutual
inductance (by definition) and using Faradays law(Mutual inductance
relates the voltage induced in one coil to the current change in a
second coil)( ) voltage didt= ( b r a( ) B A I u = = b r a( )e I =
B b r21 12v diLdt = ` )Mech 445 Cryogenic Engineering - Lecture 19
12) And, We can rewrite the above as, And the work done by the
battery is,( ) voltage ddt= mb rvoltage e dI dt= B mmagedW dvIdt
dt= = mB mag edW d = B m7Magnetic Work Where does this leave us
with an expression for magnetic work? The previous result applies
for any single valued magnetic body not just an elementary dipole.
The magnetic moment of the system can be determined by integrating
the magnetization over the entire volume of the system So, Or The
total work done by the power supply is thus( )T dV = }m M rmag Te
edW d ddVdt dt dt= = }m MB B( )mag edW d dV = } B M Magnetization
per unit massMech 445 Cryogenic Engineering - Lecture 19 13 The
total work done by the power supply is thus, And, the work
performed by the magnetic material is,( )20work on magneticwork to
develop material external field12mag e edW d B dV d dV| |= + } } |\
. B Mm edw dm = BWork per unit massInternal Energy For a simple
magnetic substance (only work mode is magnetic) we can write the
the fundamental thermodynamic relation asdu q w o o = m edu Tds dm
= + B Therefore using Maxwells relations,m su udu ds dms mc c | | |
|= + | |c c \ . \ .m e edh u m Tds m d = = B Bm edu Tds dm = + B( )
, u u s m =u u| | | |c c c c | | | | ee BsT mB s| | c c | |= | |c c
\ .\ .m mdu q w o oMagnetic energy ( ) , eh h s = B Magnetic
enthalpym edu Tds dm + BMech 445 Cryogenic Engineering - Lecture 19
14m ss mu um s s m| | | |c c c c | | | |= | | | |c c c c \ . \ .\ .
\ .es mB Tm sc c | | | | = | |c c \ . \ .( ) , eg g T = Bm edg h Ts
sdT m d = = Bee BTs mB T| | c c | |= | |c c \ .\ .Magnetic gibbs
energy8Magnetic Entropy For a material with entropy as a function
of temperature and field,( , )es T B ee B Ts sds dT dBT B| | c c |
|= + | |c c \ . \ . Using the definition of heat capacity And
Maxwells relations Therefore, for an isentropic field changee e B
T\ . \ .Bee Tc sds dT dBT B| | c= + |c\ .BeBc mds dT dBT Tc | |= +
|c \ .0 Bc mdT dBc | | | T mdT dBc | | |Mech 445 Cryogenic
Engineering - Lecture 19 15 Integrating from initial field strength
to final gives the magnetocaloric effect, MCE0 BeBdT dBT T| |= + |c
\ . eB BdT dBc T= |c \ .fiBeB B BT mMCE dBc Tc | | } |c\
.Magnetocaloric materials Where would MCE be highest? Paramagnets
have a e B Bd T T mdB c TA c | | |c\ . Paramagnets have a
constitutive relation of the following form- Ferromagnetic
materials near the Curie temperature show a large variation in M as
a f ti f t t d, C "Curie constant"eCBmT= =0MGdMech 445 Cryogenic
Engineering - Lecture 19 16function of temperature and applied
field strength. Much more complicated expression to determine
magnetization9Magnetocaloric Effect Rare earth elements and alloys
demonstrate high magnetocaloric effects. g0-5 TMech 445 Cryogenic
Engineering - Lecture 19 17V. Pecharsky and K. Gschneidner, Adv.
Cry. Eng. 43 (1998) p. 1729 V.K. Pecharsky, K.A. Gschneidner Jr. /
Journal of Magnetism and Magnetic Materials 200 (1999)
44}56First-order materials Previous materials have second-order
phase change (2ndderivative of Helmholtz energy is discontinuous)
heat capacity is continuous, but has a peak Some new alloys undergo
a first-order magnetic ordering (phase change) (1stderivative of
Helmholtz energy is discontinuous) heat capacity becomes infinite
derivative of Helmholtz energy is discontinuous) heat capacity
becomes infinite They all have hysteresis not good.Mech 445
Cryogenic Engineering - Lecture 19 1810Heat Capacity Magnetic
materials have an additional energy storage mode (mode of
ordering). ( g) The total entropy is a function of lattice
(vibration + expansion), electronic, and magnetic. For materials
that magnetically( ) ( ) ( ) ( ) , ,tot e latt elec mag es T B s T
s T s T B = + +Mech 445 Cryogenic Engineering - Lecture 19 19 For
materials that magnetically order at low temperatures, the magnetic
heat capacity can be much larger than all other
components.Materials Disadvantages: MCE small (~2 K/T) Localized
near magnetic phase transition H t it f ti f T d B Heat capacity
function of T and BGd3456MCE (K)GdTbGd0.74Tb0.24 Gd0.85Er0.15
(Dan'kov)0-2 TMech 445 Cryogenic Engineering - Lecture 19 20200 220
240 260 280 300 320012Temperature
(K)M(UQTR)(G&P)(Tishin)11Single shot cooling MCE has been used
for a long time to achieve very low achieve very low temperatures.
Precool paramagnetic material in a high field, thermally isolate
(thermal switch) then remove the fieldMech 445 Cryogenic
Engineering - Lecture 19 21e ove e e d ~10-5K possibleBatch
Magnetic Cycles Can imagine different cycles analogous to
traditional gas cycles. T BHCarnotT BHEricsson Like gas cycles,
recuperation can increase temperature span. Advantages: Solid
refrigerant (compact) Reversible materials (efficiency) Inherent
work recovery T BHBraytonsBLsBLT BHR-BraytonMech 445 Cryogenic
Engineering - Lecture 19 22y(efficiency x 2) Benign materials
(sustainable)sBLsBL