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Heat Transfer

Heat Transfer. G.Vandoni CERN, AT Division. A detour in basic thermodynamics. W refrigeration work Q heat to extract at T and reject at T a. A refrigerator extracts heat at a temperature T below ambient and rejects it at a Tambient. Second law of thermodynamics :.

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Heat Transfer

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  1. Heat Transfer G.Vandoni CERN, AT Division Academic Training 2005

  2. A detour in basic thermodynamics W refrigeration work Q heat to extract atT and reject at Ta A refrigerator extracts heat at a temperature T below ambient and rejects it at a Tambient. Second law of thermodynamics : Minimize thermal loads: boundary temperatures fixed, heat transfer rate minimization seeked Maximize heat extraction: heat transfer rate fixed, minimize temperature difference Academic Training 2005

  3. The 3 modes of heat transfer • Conduction: heat transported in solids or fluids at rest FOURIER’s law: • Convection: heat transport produced by flow of fluid Convection exchange: • Radiation: heat carried by electromagnetic radiation Stefan-Boltzmann’s law: Academic Training 2005

  4. Electrical analogy T1 T2 Q V1 V2 motor I flow thermal impedance • Valid in the three cases for a small DT (linearization of Stefan-Boltzmann’s law) • series/parallel impedances • Basis for modelling and numerization above 1D Academic Training 2005

  5. Cryogenic heat transfer modes PeakNucleateBoilingFlux increase of Re for decreasing T increase of Gr for decreasing T, h~T-1/2 T3 k~T0.7 Academic Training 2005

  6. Time-independent conduction Th Tt L A Tc Heat flux reduction by intermediate temperature thermalization: Temperature profile T(x) of st.steel bar with thermalization 2/3 of length at 80K 1D, constant A Academic Training 2005

  7. Intermediate heat interception Th Tt Stainless steel pure Copper T T 300 K L A 77 K 4 K Tc x x Purely conductive T(x) profile over the whole length Thermalization (=fixing the temperature) at Tt Larger Q evacuated at Tt, but smaller at Tc => optimization possible with exergy function Academic Training 2005

  8. Thermal conductivity integrals Tc=4 K • Reduction of heat flow to the cold boundary temperature by thermal interception at intermediate temperature Academic Training 2005

  9. Time-dependent conduction Energy conservation dv difference between heat entering and leaving dv internal heat source density rate of temperature increase (thermal inertia) characterizes the propagation of a thermal transient… …through a characteristic time depending on the object’s dimension ro Diffusivity D=k/rC Academic Training 2005

  10. Diffusivity and time regimes T late regime Late regime: exponential decay x x x ro early regime b = hS/(rC V) time constant of the system bt Academic Training 2005

  11. Internal versus external resistance T x surface thermal resistance internal thermal resistance x Biot number: ro Exponential Under some circumstances, the decay is exponential starting from t=0 Lumped capacitance model applies starting t=0 Academic Training 2005

  12. Conductivity of solids -> form for pure and alloyed metals -> st.steel -> increase with T Academic Training 2005

  13. Conductivity of solids T behaviour well known • Heat carriers: phonons (k~T3) and electrons (k~T) • Good electrical conductors = good thermal conductors(but not the best ones !) • Hinder heat transmission at low T ? DEFECTS difference between pure and alloyed effect of modification of the defect content: magnetic impurities, annealing, cold work • Hinder heat transmission at high T ? Phonon-phonon Phonon-electron no difference between pure and alloyed metals Academic Training 2005

  14. Metal’s conductivity Wiedemann-Franz: free-electron metal RRR parametrization (next slide) Superconductor’s conductivity Electronic above Tc, phononic below Tc: Pb: knormal/ksupra=45/T2 In : knormal/ksupra=1/T2 => Thermally switch between conducting and isolating by applying a magnetic field>critical field… useful at low T but wrong over whole T range For Cu, Fe, Al, W Academic Training 2005

  15. RRR parametrization of k(T) Valid over a broad range of RRR, ~10% exactness A similar parametrization also available for r(RRR,T) Academic Training 2005

  16. Diffusivity of common materials Cv(T) decreases faster than k(T): small equilibration times at low T D=k/rCv Diffusivity larger for conductors than insulators Academic Training 2005

  17. Specific heat of structural materials Cv heat capacity per kg mole approximately described by the Debye function Nb: Tc/qD=0.04 qD Debye temperature, a material’s property Academic Training 2005

  18. Conductivity of gases: 2 regimes L mean free path vs wall distance L viscous: molecular: [Pa.s],[Pa],[cm] k~T0.7 q proportional to p q independent from L q independent from p q=kSDT/L k predicted by kinetic theory of gases Academic Training 2005

  19. Viscous regime Thermal conductivity k [Wcm-1 K-1] @ 1 atm Academic Training 2005

  20. Molecular regime: Kennard’s law g=Cp/Cv R ideal gas constant a accomodation coefficient 0<a<1, degree of thermal equilibrium between molecules and wall, a~0.7-1 for heavy gases. H2 He N2 for simple geometries, (parallel plates, coaxial cylinders, spheres) 300K->77K Academic Training 2005

  21. Contact resistance Temperature discontinuity at the interface: - phonon scattering (Kapitza) - spot-like contact points • Features: • Proportional to FORCE, not to pressure (constant spot area, number of contact points increases with force) • For metals, saturates above 30N @ 300K • Hysteresis upon loading cycles (plastic deformations) • Can be reduced by fillers, grease, In, coatings • For el. conductors, Rh~Rel • Rh-1=Kh increases with T then saturates • Approximately proportional to microhardness/k Academic Training 2005

  22. Contact resistances Academic Training 2005

  23. Thermal switches Switch from normal (thermally conducting) to superconducting (thermally insulating) with applied magnetic field heat sink device SCOPE: Good thermal contact for cooldown BUT Thermal insulation once cold REALIZATION: • Exchanger gas: long time for evacuation • Gas heat exchanger: short time for evacuation • Superconducting switch (Pb or In) • Polycristalline graphite: k~T3 up to 100K Academic Training 2005

  24. RADIATION BLACK-BODY: The whole incident radiation is absorbed: a=1 r+a+t=1 r+a=1 Energy conservation Opaque medium incidentP reflected rP absorbed aP transmitted tP Any surface T>0K absorbs (a) and emits (e) energy as electromagnetic radiation: depending on direction and wavelength Academic Training 2005

  25. Black-body radiation Planck’s law for energy flux emitted by a cavity [W/cm3] Wien’s law Integral over l: Stefan-Boltzmann’s law for black body s=5.67 10-8 W m-2 Academic Training 2005

  26. Heat exchange between two black surfaces A1, T1, A2, T2 Geometrical FORM FACTOR F12 F12= (radiation leaving A1 intercepted by A2) / (radiation leaving A1 in all directions) = integral of solid angle under which A1 sees A2 F12 tabulated for several useful geometries Academic Training 2005

  27. From a blackbody to a real body Definition of (total hemispherical) emissivity e < 1: black-body real-body Monochromatic directional emissivity diffuse-body (e independent of q) grey-body (e independent of l) APPROXIMATIONS Academic Training 2005

  28. Kirchoff’s law From energy conservation in a cavity: For black-body and diffuse grey body: a(T)= e (T) Practical use: a can be estimated from e provided the incident radiation and the surface have the same temperature In reality, a(l,q,T)≠e (l,q,T) Academic Training 2005

  29. Electrical analogy for real (diffuse/grey) surfaces q12 sT14 sT24 q1 q2 resistance between two blackbodies internal resistance of the surface to black-body emission motor flux total thermal impedance Blackbody form factors can be used for real diffuse-grey surfaces Academic Training 2005

  30. Heat transfer between 2 real surfaces Parallel plates Spheres and long cylinders self-contained, not concentrical/coaxial (A1<A2) effective emissivity (emissivities + view factor) A2>>A1 equivalent to A2 black:black-body radiation fills the cavity between the two surfaces and is collected by A1 proportionnally to e1 Academic Training 2005

  31. Emissivity and materials Drude law for ideal metal Real emissivities depend on direction and wavelength • Polished metals: small e • Insulators: large e • e = e(T): for real metals, e~T at low T • Coatings: since e related to surface, not bulk, resistance, => lower limit on thickness of reflectors (r1 above ~40nm) Academic Training 2005

  32. Emissivity and materials –2- Academic Training 2005

  33. Radiative heat transfer in cryogenics negligible effect of Tcold reduction of heat flux by one cooled screen Blackbody radiation from 290 K to 80 K: 399 W/m2 Blackbody radiation from 290 K to 4.2 K : 401 W/m2 Blackbody radiation from 290 K to 4.2 K: 401 W/m2 Blackbody radiation from 80 K to 4.2 K : 2.3 W/m2 Academic Training 2005

  34. Floating radiation screens Tw Tc e e T n Floating = not actively cooled, they operate at a temperature determined by heat balance Academic Training 2005

  35. Multi-layer Insulation reflector spacer blanket 1. Heat transfer parallel to the layers ~1000 times greater than normal to the layers Stacking of “reflectors” separated by insulating “spacers” Reflector: low emittance radiation shield polyester film, 300-400 A pure Al coating, usually double face Spacer: insulating, lightweight material paper, silk, polyester net thermal coupling between blanket edges and construction elements may dominate heat rate. 2. Heat transfer very sensitive to layer density single local compression affects the T profile over the entire blanket, substantially degradating heat loss (factors 2-3 more !) Academic Training 2005

  36. MLI: effective conductivity W/m2 Optimal density: 10-20 cm-1 layers/cm Low boundary heat transfer rate determined byaT, not by temperature: radiation 1 single aluminized foil is sufficient in high vacuum in bad vacuum, MLI provides sufficient insulation 77 K-> 4K High boundaryheat transfer rate determined by radiationtemperature:important reduction with layer’s number bad vacuum: radiation dominates anyway 300 K-> 77K Effective conductivity k=aT+ bT3 Heat transfer rate q=k/e DT, e = thickness Academic Training 2005

  37. MLI: number of layers 30 layers, 300K-> 77K, 0.5 W/m2 10 layers, 77K-> 4K, 20 mW/m2 N = 15 cm-1 Tc= 4.2K, e = 0.03 Academic Training 2005

  38. MLI and residual pressure Kennard’s law MLI constitutes a supplementary protection against vacuum rupture, only at low boundary temperature: at high boundary temperature, radiation dominates anyway interstitial gas: nitrogen 300 K -> 77 K 77 K -> 4.2 K 300 K -> 77 K 77 K -> 4.2 K 300 K -> 77 K 77 K -> 4.2 K 300 K -> 77 K 77 K -> 4.2 K Academic Training 2005

  39. Passive cooling by radiators Cooling in space applications towards the cosmic background radiation at 2.7K Figure: the NGST (next generation space telescope) solar screen • Requires large surface-to-volume ratio + large emissivity • Radiation cooling to a cold screen -> cool down without contact • Black silicon paints compatible with high vacuum from the space industry (cooling of CERN antiproton collector’s mobile electrodes) Academic Training 2005

  40. Free and forced CONVECTION Q transferred heat, A surface area Tf Ts h: heat transfer coefficient, function of fluid properties, flow velocity and channel geometry Scope: determine h Analysis: dimensionless groups, EMPIRICAL correlations Free (natural) convection : the fluid movement is due to expansion upon heating, reduction of density and buoyancy(kettle, fireplace) Forced convection: the fluid is set into movement by external action (pressure difference, mechanical action, elevation difference) Academic Training 2005

  41. Convection exchange coefficient Boiling, water Boiling organic liquids Condensation, water vapors Condensation, organic vapors Liquid metals, forced convection Water, forced convection Organic liquids, forced convection Gases 200atm, forced convection Gases 1atm, forced convection Gases, natural convection 1 10 102 103 104 105 106 h (W/m2K) Convective heat transfer in cryogenic fluids not different from any other, except He II Boiling HeI, N2 peak nucleate flux (PNBF): 104 W/m2K Academic Training 2005

  42. Dimensionless groups flow character fluid characteristics defines convection exchange like Re for free convection Academic Training 2005

  43. Reynolds number and flow character Viscous forces are stabilizing: laminar flow Inertial forces are de-stabilizing: turbulent flow r density, V fluid average velocity, d hydraulic diameter, m dynamic viscosity Inertia forces compared to viscous forces Laminar: low heat transfer coefficient Turbulent: high heat transfer coefficient In free convection, Gr plays the role of Re: buoyancy versus viscosity Academic Training 2005

  44. Free (natural) convection Nu = a (Gr .Pr) n= a . Ra n Empirical form Nu = function(Gr,Pr) General relation d to be used to calculate: diameter (horizontal cylinder), height (vertical plates/cylinders), smallest exchange dimension (horizontal plates), distance between walls (enclosures) Academic Training 2005

  45. Free convection in gases and air important increase at low temperature Air close to ambient conditions vertical plates Watt m-2 K-1 horizontal plates Watt m-2 K-1 common gases: h~p½, h~T-½. cold helium gas (80K, 1 bar): Nu~3.65 (laminar) Academic Training 2005

  46. 1phase forced convection Colburn formula Sieder & Tate formula Nu = f (Re,Pr) = aF Rem Prn Empirical relation Academic Training 2005

  47. Steps to solve a convection problem 1. Calulate Re to determine flow character: laminar/turbulent hydraulic calculation of pressure drop f=Fanning, function of Re 2. Evaluate Pr (fluid characteristics) 3. Choose the appropriate formula for Nu -> h 4. In doubt about the importance of free convection: calculate Gr Academic Training 2005

  48. Boiling heat transfer in He I Increase of heat transfer up to a Peak Nucleate Boiling Flux: He I: 1 W cm-2 @ 1K superheat N2: 10 W cm-2 @ 10K H2O: 100 W cm-2 @ 30K Hysteresis: cooling path not the same as warming path Positive consequence for safety: limit to the highest flux released by a warm object (quenching magnet, human skin) Academic Training 2005

  49. Two-phase convection heat transfer = bubble formation and motion near the walls + direct sweeping of the heated surface by the fluid Instabilitiesof density-wave type: pressure waves increase locally the heat transfer rate, the fluid expands => decrease in conductivity and heat transfer rate How to avoid them: • Maintain low vapor quality • Not too large differences in elevation • No downstream flow restrictions -> destabilizing • Introduce upstream flow restrictions -> stabilizing Academic Training 2005

  50. Refrigeration properties of cryogens highly effective for self-sustained vapor cooling! Working domain close to critical point: properties of liquid and vapor phase are similar low vaporization heat Low viscosity hence excellent leaktightness required for He *at normal boiling point Academic Training 2005

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