FERMILAB-SLIDES-19-042-TD Thermal contact resistance R.C. Dhuley CSA Short Course: Property and Cooler Considerations for Cryogenic Systems Sunday, July 21 2019; CEC-ICMC 2019 at Hartford CT This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.
Technical importance in cryogenics Any cryogen-free system or a system seeking to be cryogen-free will encounter thermal contact resistance • Sub-Kelvin experiments coupled to ADRs, dilution refrigerators, etc. • Bath cooled systems seeking cryogen-independence via conductive coupling to cryocoolers Undesired consequences of large thermal contact resistance: • Long cooldown times • Poor thermal equilibrium between experiment and cooler even when heats leaks are small • Large sample-cooler temperature jump during operation (reduction in the range of operating temperatures) • Each of the above issue will worsen with decreasing temperature! Complexities: • No unified or simple models: too many governing parameters • Difficult experimental characterization 2 6/19/2019 Dhuley | Thermal contact resistance
Outline and course objectives Objectives: To understand the complexities of the problem, familiarize with existing theory to obtain rough estimates, learn how to characterize low temperature thermal contacts. Outline: ▪ Origins and mechanisms ▪ Theoretical models for metallic contacts • ‘macroscopic’ constriction resistance • ‘microscopic’ boundary resistance ▪ Characteristics of contact resistance at low temperatures ▪ Measurement techniques ▪ Contact resistance R&D at Fermilab • SuperCDMS SNOLAB sub-Kelvin cryostat • Conduction cooling of an SRF niobium cavity ▪ Examples of data from the literature 3 6/19/2019 Dhuley | Thermal contact resistance
Origins Reduction in heat transfer area - surface “waviness”, microscopic asperities (roughness) Oxide surface layer (metals) Surface films, adsorbed gases Differential thermal contraction (cryogenic case) Ref: Van Sciver, Nellis, Pfotenhauer The actual physical boundary (carrier reflection, scattering) 4 6/19/2019 Dhuley | Thermal contact resistance
Contact heat transfer mechanisms Ref: Madhusudhana • Conduction through actual solid-solid contact spots (spot or constriction resistance) - important for cryogenic applications • Conduction through interstitial medium, example air (gap resistance) - neglected if fluid is absent (eg. vacuum in cryogenic systems ) • Radiation - small unless T or Δ T are is large ( not significant at low T ) 5 6/19/2019 Dhuley | Thermal contact resistance
Spot resistance, analyses Heat flow analysis (thermal model) - constriction resistance due to “thinning” of heat flow lines - boundary reflection of heat carriers (electrons, phonons) - determines the basic premise of contact resistance Surface texture analysis (geometrical model) - surface roughness, slope of as valleys and peaks - determines number and size of contacting asperities Asperity deformation analysis (mechanical model) - Surface microhardness, elastic modulus, applied pressure/force - determines the area of ‘real’ or physical contact (the surface area available for heat transfer) 6 6/19/2019 Dhuley | Thermal contact resistance
Thermal analysis: macroscopic vs. microscopic Differentiated based on spot “Knudsen” number mean free path l , = Kn constrictionsize a , (equivalent of continuum and molecular flow regimes of gases) Ref: Prasher and Phelan Major influencers l : temperature and purity of metals (especially cryogenic conditions) a : surface finish/roughness, machining processes 7 6/19/2019 Dhuley | Thermal contact resistance
Thermal analysis: macroscopic vs. microscopic Constriction resistance Boundary resistance Ref: Madhusudhana Ref: Madhusudhana l a l a • diffusion limited thermal • ballistic/boundary scattering transport effects • macroscopic component • microscopic component dominates dominates l ~ a : both effects important 8 6/19/2019 Dhuley | Thermal contact resistance
Thermal analysis of a spot: macroscopic ▪ Macroscopic spot resistance ( ): “bulk” thermal conductivity l a holds valid at the spot (diffusion regime) Analytical solution is obtained Semi-infinite solid Ref: Madhusudhana cylinder, with a by solving the steady state heat round constriction heat flow lines diffusion in cylindrical coordinates radius >> mfp insulated ▪ Result (See textbook by C. V. Madhusudhana for analytical solution steps): 1 0.25 Spot at uniform = = • Unit is K/W R temperature macro spot , 4 ak ak • Spot condition changes the solution by 8% (often 8 0.27 Spot with uniform = = R negligible in practice) macro spot , heat flux 2 3 a k ak 9 6/19/2019 Dhuley | Thermal contact resistance
Thermal analysis of spots in parallel, joints ▪ Bounded spot ▪ Bounded joint ( / ) a b ( / ) a b = = R R C 4 ak C 2 ak s • Ψ (a/b) is constriction 2 k k = alleviation factor (<1) 1 2 k + s • Usable form is given k k 1 2 later equivalent thermal Ref: Madhusudhana conductivity Ref: Madhusudhana ▪ Contact with multiple spots = − − 1 1 Parallel sum: R R C Ci i For n contacts of average size a m and neglecting variation in Ψ : ( / ) a b = R C Ref: Madhusudhana 2 na k m s (idealized representation of contact plane) 10 6/19/2019 Dhuley | Thermal contact resistance
Surface topography (geometry) analysis ▪ The contacting surfaces are Ref: Dhuley characterized in terms of their • Roughness (height distribution of peaks and valleys) • Asperity slope (‘steepness’ of peaks and valleys) ▪ These are essentially random, but are often assumed to have Gaussian distribution • σ = standard deviation of heights • m = standard deviation of slopes ▪ Relation to typically measured surface roughness R is rms surface roughness = q R R where q a 2 R is average surface roughness a 11 6/19/2019 Dhuley | Thermal contact resistance
Surface topography (geometry) analysis ▪ Determination of surface geometry parameters • Roughness parameter (z(x) is local height/depth) L sample 1 = measured using a profilometer R z x dx ( ) a L (eq. laser scanning microscope) sample 0 • Average asperity slope L computed from profilometer sample 1 ( ) dz x = m dx measurements L dx sample 0 • Empirical correlations (find m from known σ ) Ref: Bahrami et al. Ref: Bahrami et al. 12 6/19/2019 Dhuley | Thermal contact resistance
Surface topography (geometry) analysis ▪ Equivalent roughness and surface slope are calculated as: = + 2 2 s 1 2 = + 2 2 m m m s 1 2 Ref: Bahrami et al. ▪ Average spot size (a m ) and number of spots per unit area (n) can be now be obtained as: 2 ( ) A 2 A A for circular = − = 2 1 s real real real a 4 exp erfc n a m contacts m m A A A s apparant apparent apparent Note: A real /A apparent is still unknown and is obtained via deformation analysis 13 6/19/2019 Dhuley | Thermal contact resistance
Asperity deformation analysis ▪ Asperities deform ‘heavily’ because the tiny contact area they represent supports all the applied load ▪ Deformation, whether elastic or plastic, can be determined by evaluating a plasticity index (several have been proposed) E ' = • m Greenwood index: G s H micro − 1 − − 2 2 1 1 where is effective elastic modulus in terms of the = − 1 2 E ' 2 E E 1 2 individual elastic modulus and Poisson’s ratio; and H is microhardness of the softer material. • Plastic contacts: - freshly prepared rough surfaces 1 G • 0.7 Elastic contacts: - polished surfaces; subsequent contact of G plastically deformed surfaces 14 6/19/2019 Dhuley | Thermal contact resistance
Asperity deformation analysis ▪ For a plastically deformed contact, the ratio of real contact area to apparent contact area is given by: P P A − − applied 4 2 = holds for and a constant value 10 10 applied real H A H of H micro micro apparent micro P A = applied real for larger loads + A H P apparent micro applied • H micro is Vickers microhardness; can be approximated as 3*yield strength if microhardness is not readily available. • Microhardness is indentation depth dependent and therefore a function of the surface roughness (asperity heights) Ref: Bahrami et al. 15 6/19/2019 Dhuley | Thermal contact resistance
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