Normal Cryogenic Fluid Behavior (emphasis on helium, but not superfluid, which will be covered later) Tom Peterson, SLAC January 2017 Outline Cooling modes for superconducting devices Forced flow cooling Two-phase flow and pool boiling Fluid dynamics Saturated bath thermodynamics

January, 2017 USPAS Cryogenic Fluids Tom Peterson 2 Large-scale cooling of superconducting devices Physicists and engineers designing a large-scale liquid helium system typically must design the cooled components (magnets or RF cavities, their containers, and the interfaces to them) Cooling mode, heat transfer, pressure drops, cooldown, warm-up and non-steady or upset system operations all must be considered as part of the component design

The cooled devices must be viewed as part of the cryogenic system January, 2017 USPAS Cryogenic Fluids Tom Peterson 3 Cooling modes in large-scale cryogenic systems recently in operation Pool boiling helium I used in superconducting RF for HERA (DESY), LEP (CERN), KEKB (KEK, Japan), CESR (Cornell) Forced flow of subcooled or supercritical helium I for cooling superconducting magnets (Tevatron, HERA, SSC, RHIC, ITER) Stagnant, pressurized helium II (the Tore Supra tokamak in France demonstrated the technology, LHC magnets)

Saturated helium II (CEBAF, TTF at DESY, SNS at Oak Ridge, and others, foreseen for FRIB, Project X, ILC, and more) This list also illustrates the extent to which superconductivity and cryogenics have become standard technology for accelerators January, 2017 USPAS Cryogenic Fluids Tom Peterson 4 Helium phase diagram (S. W. VanSciver, Helium Cryogenics, p. 54) Critical point 5.2 K, 2.245 atm

Lambda transition at 1 atm 2.172 K SRF -- HERA, LEP, KEKB, CESR Magnets -- HERA, Tevatron Magnets -- SSC Magnets -- Tore Supra, LHC SRF -- CEBAF, TTF, SNS, ILC January, 2017 USPAS Cryogenic Fluids Tom Peterson 5 Cooling modes -- magnets vs RF

Accelerator magnets are often cooled with subcooled liquid Typically working near the limit of the superconductor with large stored energy Ensure complete liquid coverage and penetration Superconducting RF cavities are generally cooled with a saturated bath Large surface heat transfer in pool boiling for local hot spots Very stable pressures, avoid impact of pressure variation on cavity tune January, 2017 USPAS Cryogenic Fluids Tom Peterson

6 Cooling modes--surface heat flux Boiling helium I (normal helium) 1 W/cm2 in nucleate boiling with 0.5 K temperature rise to the object surface so equivalent to 2 W/cm2K Forced convection helium I Convection coefficients on the order of 0.1 W/cm2K Saturated helium II (superfluid helium, SF) 1 W/cm2 heat transport to the surface without bubbles Pressurized helium II Kapitza conductance about 0.6 W/cm2K January, 2017 USPAS

Cryogenic Fluids Tom Peterson 7 Pressurized versus pool boiling Pressurized helium (normal or superfluid) gives maximum penetration of helium mass in magnet coils, which may be a factor in stability if not also heat transfer. But heat flow results in a temperature rise. Pool boiling gives pressure stability (important for superconducting RF), provides maximum local heat transfer, and provides nearly isothermal cooling. January, 2017 USPAS

Cryogenic Fluids Tom Peterson 8 Heat transport through channels-pressurized normal helium in SSC SSC dipole nominal operating temperature was to be 4.35 K, tightly constrained for magnet quench performance Allowable temperature rise of only 0.050 K allowed heat absorption of about 4 J/gK x 0.050 K = 0.20 J/g and forced high flow rate (100 g/s) as well as use of recoolers Forced flow of supercritical helium periodically recooled by heat exchange with a saturated bath January, 2017 USPAS

Cryogenic Fluids Tom Peterson 9 Recooler flow scheme January, 2017 USPAS Cryogenic Fluids Tom Peterson 10 Heat transport through channels-pressurized normal helium This plot of helium enthalpy versus T

H(J/g) illustrates the large amount of heat absorbed (20+ J/g) if one can tolerate 6.5 K or even more Nominally 5 K thermal intercept flow may take advantage of this heat capacity January, 2017 USPAS 40 30 20 10

Cryogenic Fluids Tom Peterson 5 6 7 T (K) 11 Convective heat transfer Convective heat transfer heat transfer from a solid surface into a moving fluid A complex sequence of heat transfer from the surface to a boundary layer and into the bulk fluid, a combination of conduction and mass transport

We analyze convection with the equation January, 2017 USPAS Cryogenic Fluids Tom Peterson 12 Evaluating hc This equation defines the convection coefficient, hc Empirical and semi-empirical methods are used to find approximate hc Formulations for liquids and gases work reasonably well for normal helium

Several dimensionless parameters are commonly used January, 2017 USPAS Cryogenic Fluids Tom Peterson 13 Reynolds number, Re Re provides a ratio of fluid inertia to viscosity Re < 2000 in a tube is generally laminar flow January, 2017 USPAS

Cryogenic Fluids Tom Peterson 14 Nusselt number, Nu Correlations of Nu with other parameters have proven useful in evaluation of convection coefficients January, 2017 USPAS Cryogenic Fluids Tom Peterson 15

Prandtl number, Pr Prandtl number is a ratio of fluid properties Relates velocity profile (kinematic viscosity is a sort of momentum diffusivity) to the temperature profile (thermal diffusivity) from the surface into the fluid January, 2017 USPAS Cryogenic Fluids Tom Peterson 16 Correlations For gases (0.6 < Pr < 0.8, for example Pr for helium gas = 0.66) in a long pipe with fully

developed velocity profile January, 2017 USPAS Cryogenic Fluids Tom Peterson 17 Convection summary Far too large a topic to cover here Many correlations, depending on whether free convection, laminar, or turbulent, fluid properties, etc. Entrance effects, surface and boundary layer effects Nevertheless, the classical correlations

generally work well for normal helium January, 2017 USPAS Cryogenic Fluids Tom Peterson 18 More about 2-phase helium flow Baker plot published in 1954 based on data for air and water and applied to oil and gas in pipes In 1960 - 1961, work at Los Alamos suggested the diagram could be applied to 2-phase hydrogen Papers published in 1985 and 1987 at the CEC described experimental results showing that the Baker plot does not apply to 2-phase helium flow For practical pressure drops and flow velocities with

normal helium, one may assume that 2-phase helium flow is separated CEA Grenoble studies of 1.9 K 2-phase flow for CERN found that a vapor flow of about 5 m/sec begins to entrain liquid droplets January, 2017 USPAS Cryogenic Fluids Tom Peterson 19 Plot from Simultaneous Flow of Oil and Gas, by Ovid Baker (1954) -- Do not use for helium! January, 2017 USPAS

Cryogenic Fluids Tom Peterson 20 Do not use Baker Plot for helium January, 2017 USPAS Cryogenic Fluids Tom Peterson J. C. Theilacker and C. H. Rode, An Investigation into Flow Regimes for Two-phase Helium Flow, Advances

In Cryogenic Engineering, Vol. 33, pp. 391-398, 1988. 21 Pool boiling and 2-phase flow Considerations for pool boiling systems Control of liquid levels, long time constants, inventory management Forced convection for warm-up and cool-down Two-phase flow Liquid and vapor phases separate with any acceptably low pressure drop Baker Plot does not apply! January, 2017 USPAS

Cryogenic Fluids Tom Peterson 22 Boiling Heat Transfer for Oxygen, Nitrogen, Hydrogen, and Helium, by E.G. Brentari, et al, NBS Technical Note 317, Boulder, CO, 1965. January, 2017 USPAS Cryogenic Fluids Tom Peterson 23 Helium boiling curves Note the transition from nucleate to film

boiling at about 1 K delta-T Working delta-T for nucleate boiling such as in a helium subcooler (pressurized helium cooled by boiling helium) is ~0.1 K. January, 2017 USPAS Cryogenic Fluids Tom Peterson 24 Boiling Heat Transfer for Oxygen, Nitrogen, Hydrogen, and Helium, by E.G. Brentari, et al, NBS Technical Note 317, Boulder, CO, 1965. More boiling curves

January, 2017 USPAS Cryogenic Fluids Tom Peterson 25 Some simple analytic formulas for fluid flow In designing cryogenic piping, we generally have low pressure drop and may assume steady-state conditions for normal operational conditions Emergency venting may be very dynamic, non-steady, but we often do conservative analyses assuming worstcase as if a steady-state condition Cryogenic liquids and vapors (except for Helium II) behave like normal liquids and gases

Standard engineering pressure drop and heat transport equations may be used January, 2017 USPAS Cryogenic Fluids Tom Peterson 26 January, 2017 USPAS Cryogenic Fluids Tom Peterson 27

January, 2017 USPAS Cryogenic Fluids Tom Peterson 28 The point of this little derivation is to show that for sections of pipe with large enough pressure drop that density and velocity changes are significant, iterating pressure drop calculations to come up with a linear average density through the section of constant cross section gives a good estimate of pressure drop. January, 2017 USPAS Cryogenic Fluids

Tom Peterson 29 Pressure drop analysis, working formula for round pipes This is a form of the D'Arcy-Weisbach formula. With pressure drop expressed as head loss, this is sometimes called simply the Darcy formula. (Note that delta-P changed signs here, to a positive number.) January, 2017 USPAS Cryogenic Fluids Tom Peterson 30

Crane Technical Paper #410 Flow of Fluids through Valves, Fittings, and Pipes January, 2017 USPAS Cryogenic Fluids Tom Peterson 31 For example from previous list Where P is pressure drop in psi, V is the specific volume (in3/lbm), K is the total resistance coefficient = fL/d so is dimensionless, W is the mass flow rate (lbm/hr), and d is the pipe inner diameter (in). Compare to

from slide 34 -- no unit conversions, and a different definition of friction factor. Note! Some sources define f based on hydraulic radius and some on diameter, a factor 4 difference for pipes! January, 2017 Cryogenic Fluids USPAS Tom Peterson 32 Example pressure drop analyses See Excel file C5_39_relief_calcs-TJP.xls Illustrates relief venting calculation with stepwise reassessment of Re, friction factor, and fittings losses for constant mass flow

See Excel file PressureDropLongPipeDec2008.xls Pipe divided into sections for reassessment of properties January, 2017 USPAS Cryogenic Fluids Tom Peterson 33 Heat capacity discussion The following plots illustrate the fact that the heat capacity of metals becomes vanishingly small at liquid helium temperatures Cool-down to ~80 K is dominated by removal

of heat from the solid materials Cool-down below ~20 K is dominated by removal of heat from the helium January, 2017 USPAS Cryogenic Fluids Tom Peterson 34 Heat capacity Heat capacity per unit mass Heat capacity (J / gK) 10.00000

1.00000 0.10000 0.01000 Stainless Steel Helium 0.00100 0.00010 0.0 50.0 100.0 150.0

200.0 250.0 300.0 350.0 Temperature (K) January, 2017 USPAS Cryogenic Fluids Tom Peterson 35

Heat capacity per unit volume Heat capacity (J / cc-K) 10.0000 1.0000 0.1000 Stainless Steel Helium 0.0100 0.0010 0.0001

1.0 10.0 100.0 1000.0 Temperature (K) January, 2017 USPAS Cryogenic Fluids Tom Peterson 36

Provisions for cool-down and warm-up Cool-down Return vapor may block liquid supply flow in the same channel; a simple fill from the top or one end might not work. A cool-down vent and/or a bottom-fill port may be required. Warm-up Flow will stratify. Local electric heat, a bottom vent port, or other feature to force heat down to the lower parts of a cold mass may be required. The small capillary tubes connected to a manifold and providing helium to the bottoms of helium vessels in TESLAstyle cryomodules were included primarily with warm-up in mind January, 2017 USPAS Cryogenic Fluids

Tom Peterson 37 Subcooling in a liquid bath Saturated liquid (in equilibrium with vapor at its surface) has a higher pressure below the surface by virtue of the weight of the liquid. This pressure provides the possibility for a slightly elevated temperature below the surface without boiling and/or some subcooling below the vapor pressure at that higher pressure. The pressure under a head of liquid is

January, 2017 USPAS Cryogenic Fluids Tom Peterson 38 Pressure head of helium column Thus, for helium at 4.5 K, Now, it would be good to understand the relationship of this elevated pressure below the surface to a new saturation temperature at that pressure. This new temperature, higher than the saturation temperature at the surface, tells us how much delta-T is available for heat transfer without the onset of

January, 2017 Cryogenic Fluids 39 boiling. USPAS Tom Peterson Clapeyron Equation January, 2017 USPAS Cryogenic Fluids Tom Peterson 40 Basis for Clapeyron Equation

The Clapeyron Equation comes from two substances in equilibrium over a phase transition satisfying G=0 G=0 Where G = Gibbs free energy is defined as G = H TS January, 2017 USPAS Cryogenic Fluids Tom Peterson 41 Delta-T available under a head of liquid helium at 4.5 K

Note that 10.8 mK/meter, although a small number, implies a significant saturation temperature increase at, for example, 10 meters depth, for example down to an accelerator tunnel or experimental hall. January, 2017 USPAS Cryogenic Fluids Tom Peterson 42 Delta-T available under a head of liquid helium at 2.0 K Note that 0.14 mbar/cm or 14 mbar/meter is a significant delta-P relative to the total pressure of 30 mbar at 2.0 K. January, 2017

USPAS Cryogenic Fluids Tom Peterson 43 Saturated bath of liquid Argon Pure liquid argon is a common component of high energy physics detectors Liquid argon calorimetry Time Projection Chamber (TPC) In both cases, especially in the latter, purity and lack of bubbles are important for minimal noise and good signal TPC depends on electron drift to a charged plate, and electron lifetime is critical

January, 2017 USPAS Cryogenic Fluids Tom Peterson 44 Looking at maximum heat flux via free convection (no boiling) for D0 liquid argon calorimeter back in 1988. Analytical process: Calculate available T T as a function of depth. Given T T, calculate

free convective heat flux for various geometries and Orientations. January, 2017 USPAS Cryogenic Fluids Tom Peterson 45 Liquid Argon TPC Heat input from supports and other heat sources to the liquid argon bath should not produce bubbles Heat transport by free convection without nucleation of bubbles

Free convection driven by liquid density differences, due to temperature differences Temperature differences limited by saturation temperature at depth January, 2017 USPAS Cryogenic Fluids Tom Peterson 46 Total pressure vs vapor pressure Consider a glass of water open to normal air and at the same temperature as the air 100% humidity in air water vapor in air in equilibrium with liquid in the glass Now warm the water slightly saturation pressure of

liquid higher than vapor pressure of water in air Why does it not boil? Answer: bubbles must overcome total pressure, not just vapor pressure of the water in air Typically our cryogenic systems involve pure helium and nitrogen, so vapor pressure = total P January, 2017 USPAS Cryogenic Fluids Tom Peterson 47 Inhibit boiling with total pressure One could accomplish the same sort of thing for a liquid argon bath

Pressurize the system with helium gas Total pressure much higher than vapor pressure of LAr (Problem for detectors: argon purity with some helium dissolving in LAr) January, 2017 USPAS Cryogenic Fluids Tom Peterson 48 References S. W. VanSciver, Helium Cryogenics, Plenum Press, 1986. R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot, Transport Phenomena, John Wiley &Sons, 1960. Ovid Baker, "Design of Pipelines for the Simultaneous Flow of Oil

and Gas," Oil and Gas Journal (July 26, 1954) p. 185-195 J. C. Theilacker and C. H. Rode, An Investigation into Flow Regimes for Two-phase Helium Flow, Advances in Cryogenic Engineering, Vol. 33, pp. 391-398, 1988. E.G. Brentari, et al, Boiling Heat Transfer for Oxygen, Nitrogen, Hydrogen, and Helium, NBS Technical Note 317, Boulder, CO, 1965. Crane Technical Paper #410 Flow of Fluids through Valves, Fittings, and Pipes January, 2017 USPAS Cryogenic Fluids Tom Peterson 49