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