PHY862 – Fall 2018 Cryogenic Systems Pete Knudsen This material is based upon work supported by the U.S. Department of Energy Office of Science under Cooperative Agreement DE-SC0000661, the State of Michigan and Michigan State University. Michigan State University designs and establishes FRIB as a DOE Office of Science National User Facility in support of the mission of the Office of Nuclear Physics. Introduction .Processes that provide cooling to practical superconducting devices presently require cryogenic process conditions, namely temperatures generally below -150 ºC (~123 K) .Typically this involves working fluids that have a normal boiling point less than this temperature .Process cycles used to produce the required cooling are quite different from the more conventional vapor compression cycle Week 1 | Slide 2 Introduction (cont.) .Key aspect to appreciate is the energy intensiveness •e.g., a typical large 4.5 K helium refrigerator requires three orders of magnitude more energy input (for the same cooling) than a normal home cooling system; a 2 K helium refrigerator requires three and often closer to four times what is required for 4.5 K refrigeration .In numbers: a typical large 4.5 K helium refrigerator requires ~250 W of input power per 1 W of cooling .A typical 2 K (~30 mbar) helium refrigerator requires ~1000 W/W, although it should be possible to achieve ~750 W/W for a well matched and well-designed system Week 1 | Slide 3 So Why Superconductors & Why Helium? .Superconductor cables, implemented as electro-magnets, are commonly used in particle accelerators and colliders, MRI’s, plasma confinement (fusion research), and more .Practical type II superconductors (i.e., those made from alloys) used as electro-magnets, such as Nb-Ti and Nb3Sn, have transition temperatures of 9.6 K and 18.1 K, respectively .Transition temperatures correspond to zero current density at zero magnetic field, but some margin is needed for actual operation •Note: The temperature, magnetic field and current (density) with which the superconductor transitions from normal to superconducting is a three-dimensional surface Week 1 | Slide 4 Review of Thermodynamics .Thermodynamic Laws – Describe what you cannot do in regards to work, heat, and temperature: •Zeroth Law: you cannot break any of the rules…even if you want to •First Law: you cannot win – the best you can even dream about is breaking even •Second Law: so much for your dream…because actually you cannot break even either...unless you somehow manage to reach absolute zero •Third Law: the final dream crusher…you cannot reach absolute zero (temperature), unless you’re immortal (Adapted from Wei Cai, Stanford Univ.) Week | Slide 5 Review of Thermodynamics (cont.) .Extensive properties – scale linearly with amount of substance(s) within the system; e.g., by ‘amount’ it is meant the mass (), number of moles (), or number of molecules •So, for example, the volume (), internal energy (), and entropy () are extensive properties .Intensive properties – are independent of amount of substance(s) within the system •e.g., pressure (), temperature (), and electro-chemical potential () .If extensive properties like volume, internal energy, and entropy are scaled to the amount of substance(s) within the system (i.e., divide by the mass, moles, or molecules), then it becomes an intensive property •e.g., ⁄ , ̅ ⁄ , ⁄ , ⁄ , ⁄ , ̅ ⁄ Week | Slide 6 Zeroth Law .Zeroth Law – introduces temperature as an (intensive, macroscopic) state quantity, and defines thermodynamic equilibrium and is the basis of temperature measurement •Thermal equilibrium – closed system who’s macroscopic state quantities are not changing with time Week | Slide 7 First Law .Establishes the total energy as a state variable; also, called the “conservation of energy” .To make sense of the First Law, the concept of (process) ‘path’ independence and conversely, (process) ‘path’ dependence must be established. .Consider an equilibrium state ‘1’ and another (different) equilibrium state ‘2’ for a particular system; and (for simplicity) two different and arbitrary (process) ‘paths’ that we will call ‘a’ and ‘b’ .Since the system energy is an (extensive) property, the change in energy is simply, ∆ •i.e., it does not depend on the ‘path’, ‘a’ or ‘b’ Week | Slide 8 First Law (cont.) .For energy to be conserved, it must be true (according to the First Law) that the change in energy is equal to the work input and the heat input into the system; e.g., ∆ .Where, is the path dependent, or inexact differential, of the work into the system, and, is the path dependent, or inexact differential, of the heat into the system .That is, for path ‘a’ will not necessarily be equal for path ‘b’. Likewise for, .In differential form, , noting that the energy is an exact, or path independent, differential Week | Slide 9 First Law (cont.) .Work and heat are NOT state properties! .Note: some texts consider work positive if it leaves the system. The sign convention of work and heat is arbitrary, but its implementation must be consistent Week | Slide 10 Second Law .Establishes entropy as a state variable describing the ‘quality’ or ‘availability’ of energy .If paths ‘a’ and ‘b’ are “reversible”, then we know experimentally that, ⁄ •How would we empirically know this (an example)?...will come back to that (Clausius-Clapeyron equation) .Or, in integral form (Clausius equality), ∆ .Where, is the (inexact) differential heat input from a “reversible” path Week | Slide 11 Second Law (cont.) .As much as we can look at this as a definition of entropy as a state variable, we can also look at this as a definition of a reversible (process) path .For a path ‘c’, between equilibrium states ‘1’ and ‘2’, which is not reversible, ∆ .Which, referring back to the differential form, tells us that, .It should not escape one’s attention that in the First and Second Laws, inexact (path dependent) differentials yield exact (path independent) differentials; and visa-versa! Week | Slide 12 The Traditional Carnot Cycle .The Carnot cycle can be a heat engine, transferring heat from a high temperature reservoir to a lower temperature reservoir with a net work output .It can also be a refrigerator, operating in reverse and requiring a net work input .The Carnot cycle does not convert heat energy! QH QH High Temperature Reservoir Log(T0) Log(T0) T=constant Wnet= Wc -Wx Wx Wc Expander Compressor s = constant s = constant Log(T) T=constant Log(T) Low Temperature QL Reservoir QL s s s Week | Slide 13 The Carnot Cycle (cont.) .So, from our discussion above, we know that, ∆ ⁄ ⁄ .And, from the First Law, we know that since we start and end at the same state point for a cycle, ∆ 0 , .Where, , is the net work input; i.e., total input work () minus total output work () , ∆ .For an (isothermal) refrigerator, the coefficient of performance is defined as, ≡ 1 , Week | Slide 14 The Carnot Cycle (cont.) .More commonly in cryogenics, we refer to the inverse of the coefficient of performance, as it is more representative of the energy intensiveness of such processes; i.e., ratio of net input work to cooling provided to the load 1 .Note that to arrive at this result, we did not have to assume anything about the process between the reversible isothermal heat transfer steps, except that the entropy difference was constant at a given temperature Week | Slide 15 The ‘Generalized’ Carnot Cycle .On the right is a generalized Carnot cycle, comprised of reversible isothermal heat rejection and reversible isothermal heat adsorption as does the cycle on the left •However, the only requirement for the ‘vertical’ paths connecting these is that at any given temperature, the entropy difference is the same as for the heat rejection and adsorption •So, for example, if the fluid were an ideal gas, these paths could be both at constant pressure or both at constant specific volume QH QH Log(T ) Log(T ) s 0 s 0 Log(T) s Log(T) s Q L QL s s Week | Slide 16 Third Law – Nernst’s Theorem .As a system approaches absolute zero temperature, all processes cease and the entropy of the system approaches a minimum value (which can be defined as zero) .It is not possible to reach absolute zero using any (physical) process, even if it were ideal .Example: since constant pressure lines converge at zero temperature, it would take an infinite number of isentropic expansions with perfect counter-flow heat exchange to reach T=0 (Van Sciver, Helium Cryogenics, p. 15) Week | Slide 17 Third Law – Nernst’s Theorem (cont.) .Coldest temperature ever recorded at ground level on Earth: 184 K (Soviet Vostok Station in Antarctica) .Temperature of the moon (dark side), ~120 K .Deep outer space background temperature: 2.7 K .He-3 / He-4 dilution refrigerator: ~1x10-3 K .Adiabatic demagnetization using nuclear paramagnets: 1.5x10-6 K (lattice) .MIT (2015): 4.5x10-10 K (Bose-Einstein condensate) Week | Slide 18 Fundamental Relations .The following are fundamental relations; i.e., thermodynamic potentials •Internal energy , , •Helmholtz free energy , , •Enthalpy , , •Gibbs free energy , , .Note: •Ideal gas law, , is NOT a fundamental relation (its like a mechanical equation of state – EOS) • is NOT a fundamental relation (its like a thermal equation of state – EOS) 19 Derived Thermodynamic Properties .Commonly encountered thermodynamic properties and their definitions: •Note: ⁄ , ⁄ , ⁄ , 1⁄ ; where, is the mass .Specific heat at constant pressure, ≡