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Understanding Heat Transfer Phenomena: Theory and Applications

Explore the fundamentals of heat transfer, including conduction, radiation, and convection, through the perspective of continuum hypothesis and local thermodynamic equilibrium. Discover how heat energy is transported within different mediums and learn about crystal vibrations and thermal radiation characteristics.

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Understanding Heat Transfer Phenomena: Theory and Applications

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  1. INTRODUCTION • What is Heat Transfer ? • Continuum Hypothesis • Local Thermodynamic Equilibrium • Conduction • Radiation • Convection • Energy Conservation

  2. Crystal : a three-dimensional periodic array of atoms WHAT IS HEAT ? In a solid body Oscillation of atoms about their various positions of equilibrium (lattice vibration): The body possesses heat. Conductors: free electrons ↔ Dielectics

  3. us-1 us us+1 us+2 us-1 us+1 us+2 us us+3 s-1 s s+1 s+2 s+3 Vibration of crystals with an atom Longitudinal polarization vs. Transverse polarization The energy of the oscillatory motions: the heat-energy of the body More vigorous oscillations: the increase in temperature of the body

  4. average kinetic energy vibrational state electronic state 2 dissociation energy for state 2 Energy kB = 1.3807 × 10-23 J/K rotational state electronic state 1 atT = 300 K, airM = 28.97 kg/kmol = 468.0 m/s dissociation energy for state 1 Internuclear separation distance (diatomic molecule) In a gas The storage of thermal energy: molecular translation, vibration and rotation change in the electronic state intermolecular bond energy

  5. HEAT TRANSFER Heat transfer is the study of thermal energy transport within a medium or among neighboring media by • Molecular interaction: conduction • Fluid motion: convection • Electromagnetic wave: radiation resulting from a spatial variation in temperature. Energy carriers: molecule, atom, electron, ion, phonon (lattice vibration), photon (electro-magnetic wave)

  6. local value of density CONTINUUM HYPOTHESIS Ex) density microscopic uncertainty macroscopic uncertainty (3×107 molecules at sea level, 15°C, 1atm)

  7. Mean free path of air at STP (20°C, 1atm) lm = 66 nm, • microscopic uncertainty due to molecular random motion • macroscopic uncertainty due to the variation associated with spatial distribution of density In continuum, velocity and temperature vary smoothly. → differentiable bulk motion vs molecular random motion

  8. LOCAL THERMODYNAMIC EQUILIBRIUM hot wall at Th adibatic wall L adibatic wall gas cold wall at Tc a) lm << L : normal pressure b) lm ~ L : rarefied pressure c) lm >> L

  9. CONDUCTION Gases and Liquids • Due to interactions of • atomic or molecular • activities • Net transfer of energy by random • molecular motion • Molecular random motion→ diffusion • Transfer by collision of random • molecular motion

  10. Solids • Due to lattice waves induced by • atomic motion • In non-conductors (dielectrics): • exclusively by lattice waves • In conductors: • translational motion of free electrons • as well

  11. Fourier’s Law heat flux [J/(m2s) = W/m2] k: thermal conductivity [W/m·K] As Dx → 0,

  12. Notation Q: amount of heat transfer [J] q : heat transfer rate [W], : heat transfer rate per unit area [W/m2] : heat transfer rate per unit length [W/m]

  13. Heat Flux vector quantity

  14. Ex) T = constant line or surface: isothermal lines or surfaces (isotherms)

  15. T(x, y, z) = constant • temperature : driving potential of • heat flow • heat flux : normal to isotherms along the surface of T(x, y, z) = constant

  16. Steady-State One Dimensional Conduction T = T(x) only steady-state

  17. or As Dx → 0, When k = const.,

  18. 0.4 0.7 ultra violet infrared visible thermal radiation RADIATION Thermal Radiation 10-2m 10-1m 1m 10 m 102m 103m

  19. Ex) ice lens ice lens black carbon paper Characteristics of Thermal Radiation • Independence of existence and temperature of medium

  20. diffusion • solid: lattice vibration (phonon) free electron 2. Acting at a distance Ex) sky radiation • electromagnetic wave or photon • photon mean free path • ballistic transport • volume or integral phenomena conduction • fluid: molecular random motion diffusion or differential phenomena as long as continuum holds

  21. surface emission Blackbody spectral emissive power 3. Spectral and Directional Dependence • quanta • history of path

  22. Two Points of View • Electromagnetic wave • Maxwell’s electromagnetic theory • Useful for interaction between radiation and matter 2. Photons • Planck’s quantum theory • Useful for the prediction of spectral properties of absorbing, emitting medium

  23. Radiating Medium • Transparent medium • ex: air • Participating medium • emitting, absorbing and scattering • ex: CO2, H2O • Opaque material

  24. Stefan by experiment (1879): Boltzmann by theory (1884): Stefan-Boltzmann’s law a perfect absorber • Blackbody: • Blackbody emissive power

  25. C0: speed of light in vacuum: 2.9979×108 m/s h: Planck constant: 6.6260755×10-34 J•s k: Boltzmann constant: 1.380658×10-23 J/K Planck’s law (The Theory of Heat Radiation, Max Planck, 1901) spectral distribution of hemispherical emissive power of a blackbody in vacuum

  26. El,b (W/m2.mm) For a real surface, Wavelength, l (mm) e : emissivity Blackbody spectral emissive power

  27. Surface Radiation G Kirchihoff’s law Ray-tracing method vs Net-radiation method G: irradiation [W/m2] J J: radiosity [W/m2] r : reflectivity e diffuse-gray surface at T a : absorptivity

  28. aG esT4 e diffuse-gray surface at T

  29. [W] when Tsur q Ts, e, a, A Ex) a body in an enclosure T2, e2, A2 q1 T1, e1, A1 e = a Surrounding can be regarded as a blackbody.

  30. Why is the irradiation on the small object the same as ? A1 A2

  31. A2 A1 F: view factor Reciprocity:

  32. CONVECTION energy transfer due to bulk or macro- scopic motion of fluid bulk motion: large number of molecules moving collectively • convection: random molecular motion • + bulk motion • advection: bulk motion only

  33. solid wall • hydrodynamic (or velocity) • boundary layer • thermal (or temperature) • boundary layer at y = 0, velocity is zero: heat transfer only by molecular random motion

  34. solid wall When radiation is negligible, h : convection heat transfer coefficient [W/m2.K] Newton’s Law of Cooling

  35. Convection Heat Transfer Coefficient not a property: depends on geometry and fluid dynamics • forced convection • free (natural) convection • external flow • Internal flow • laminar flow • turbulent flow

  36. control volume ENERGY CONSERVATION First law of thermodynamics • control volume (open system) • material volume (closed system) In a time interval Dt: steady-state:

  37. sur. Ex) Surface Energy Balance

  38. Find: • Surface emissive power E and irradiation G • Pipe heat loss per unit length, Example 1.2 air • Assumptions: • Steady-state conditions • Radiation exchange between the pipe and the room is between a small surface in a much larger enclosure. • Surface emissivity = absorptivity

  39. air 1. Surface emissive power and irradiation

  40. air 2. Heat loss from the pipe

  41. Q: Why not ? Example 1.2 air Conduction does not take place ?

  42. T r air

  43. Anode: (exothermic) Cathode: The convection heat coefficient, h Example 1.4 Hydrogen-air Proton Exchange Membrane (PEM) fuel cell Three-layer membrane electrode assembly (MEA) • Role of electrolytic membrane • transfer hydrogen ions • serve as a barrier to electron • transfer Membrane needs a moist state to conduct ions. Liquid water in cathode: block oxygen from reaching cathode reaction site → need to control Tc

  44. Find: The required cooling air velocity, V, needed to maintain steady state operation at Tc = 56.4ºC. • Assumptions: • Steady-state conditions • Negligible temperature variations within the fuel cell • Large surroundings • Insulated edge of fuel cell • Negligible energy flux by the gas or liquid flows

  45. Energy balance on the fuel cell

  46. Assumptions: 1) Inner surface of wall is at through the process. 2) Constant properties 3) Steady-states, 1-D conduction through each wall 4) Conduction area of one wall = Example 1.5 section A-A A A cubical cavity ice of mass M at the fusion temperature Find: Expression for time needed to melt the ice,tm

  47. : latent heat of fusion section A-A

  48. Find: 1) Cure temperature T for 2) Effect of air flow on the cure temperature for Value of h for which the cure temperature is 50°C. • Assumptions: • Steady-state conditions • Negligible heat loss from back surface of plate • Plate is very thin and a small object in large surroundings, • coating absorptivity w.r.t. irradiation from the surroundings Example 1.7 air Coating to be cured

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