Energy Transport

Energy Transport Energy Transport Focus on Æheat transfer Heat Transfer Mechanisms: Heat Transfer Mechanisms: • Conduction • Conduction • Radiation • Radiation • Convection (mass movement of fluids) • Convection (mass movement of fluids)

Conduction Conduction Conduction heat transfer occurs only when there is physical contact between bodies (systems) at different temperatures by molecular motion. Heat transfer through solid bodies is by conduction alone, whereas the heat may transfer from a solid surface to a fluid partly by conduction and partly by convection.

Fourier’s Law of Thermal Conduction Fourier’s Law of Thermal Conduction ∂T qy =−k ∂y temperature the heat flux in thermal gradient in the the y direction conductivity y-direction 2 (K/m)

The proportionality ratio, k, is called thermal conductivity

Thermal Conductivity (k) Thermal Conductivity (k) The thermal conductivity of a substance is defined as the heat flow per unit area per unit time when the temperature decreases by one degree in unit distance. The thermal conductivity is a material property which reflects the relative ease or difficulty of the transfer of energy through the material. It depends on the bonding and structure of the material.

Thermal Diffusivity (α) Thermal Diffusivity (α) ƒThe thermal diffivity is a fundamental quantity. It is analogous to momentum and mass diffusivities. α=k/ρCv where ρand Cv are the density and specific heat of the material, respectively. 2 -1 ƒIn mks system, the unit of thermal diffusivity is m2.s-1, In mks system, the unit of thermal diffusivity is m .s , 2 -1 2 -1 while in the cgs system it is usually cm .s . while in the cgs system it is usually cm .s .

Thermal Radiation Thermal Radiation Thermal radiation is the energy radiated from hot surfaces as electromagnetic waves. It does not require medium for its propagation. Heat transfer by radiation occur between solid surfaces, although radiation from gases is also possible. Solids radiate over a wide range of on certain wavelengths only. The energy flux emitted by an ideal radiator is proportional to the fourth power of its absolute temperature. 4 e =σT b where eb is the emissive power and σ Boltzmann constant.

When thermal radiation strikes a body, it can be absorbed by the body, reflected from the body, or transmitted through the body. The fraction of the incident radiation which is absorbed by the body is called absorptivity ( ) Other fractions of incident radiation which are reflected and transmitted are called reflectivity (symbol ) and transmissivity, The sum of these fractions should be unity i.e.

Convection Convection Convection is the heat transfer within a fluid, involving gross motion of the fluid itself. Fluid motion may be caused by differences in density as in free convection. Density differences are a direct result of temperature differences between the fluid and the solid wall surface. In forced convection, the fluid motion is produced by mechanical means, such as a domestic fan-heater in which a fan blows air across an electric element.

Heat Transfer Coefficient (h) Heat Transfer Coefficient (h) When a moving fluid at one temperature is in contact with a solid at a different temperature, heat exchanges between the solid and the fluid by conduction at a rate given by Fourier’s law. Under such cases the distribution of temperature within the fluid and the heat flux at the wall can be determined by using heat transfer coefficient, h, h=q /(T -T ) =-[k(dT/dy) ]/(T -T ) 0 s f 0 s f where Ts, the surface temperature Tf, bulk fluid temperature q , heat flux at the wall 0 (dT/dy)0, the temperature gradient in the fluid normal to the wall at the fluid-solid interface. k, the conductivity of the fluid

Thermal Conductivity of Gases-I Thermal Conductivity of Gases-I Conduction of energy in a gas phase is primarily by transfer of translational energy from molecule to molecule as the faster moving (higher energy) molecules collide with the slower ones. C Vλ v k = 3 where C , the heat capacity per unit volume, V, the average v speed, λ, the mean free path. κ 3T) 1/ 2 1 k = 2  B3  d  π m  where m, mass of fluid molecules, KB, the Boltzmann constant, T, absolute temperature, d, the center to center distance of two molecules. ƒThe thermal conductivity of gases is independent of pressure an depends only on temperature. This conclusion is valid up to about ten atmospheres (1.0133 x 105 Pa)

Thermal Conductivity of Gases-II Thermal Conductivity of Gases-II Eucken developed the following equation for the thermal conductivity of polyatomic gases at normal pressures,  1.25R η k = C +   p M  where M, molecular weight, C , the heat capacity at constant p pressure.

This figure is valid up to about ten atmospheres.

Thermal Conductivity of Gas Mixtures Thermal Conductivity of Gas Mixtures The thermal conductivity of gas mixtures can be estimated within a few percent by the following equation ∑X k M 1 / 3 i i i k = i mix 1 / 3 ∑X Mi i i where X is the mole fraction of component i having molecular i weight, M , and intrinsic thermal conductivity k . i i

Thermal Conductivity of Solids-I Thermal Conductivity of Solids-I Solids transmit thermal energy by two modes: ƒelastic vibrations of the lattice moving through the crystal in the form of waves ƒfree electrons moving through the lattice also carry energy similar to the case in gases (this is observed in metals)

Thermal Conductivity of Solids-II Thermal Conductivity of Solids-II Each lattice vibration (there is always a spectrum of vibrations) may be described as a traveling wave carrying energy and obeyin the laws of quantum mechanics. By analogy with light theory, the waves in a crystal exhibit the characteristics of particles and are called phonons. Two types of phonon-phonon interaction are observed in solids: ƒNormal or N-type ƒUmklapp (U-process) collision  1.25R η k = C +   p M 

Thermal Conductivity of Solids-III Thermal Conductivity of Solids-III Since the number of phonons increases with temperature and the wavelength of phonons λph is proportional to 1/T. At room temperature and above, molar heat capacity Ĉv for most materials is roughly constant Æthe thermal conductivity of a solid which conducts energy only by phonons, decreases with increasing temperature. 2 λ = 20T d/ γ T ph m where Tm, melting point, T = absolute temperature, d, crystal- lattice dimension, and γ, Gruneisen’s constant (~2 for most solids at ordinary temperatures). THIS IS GENERALLY OBSERVED IN ELECTRICALLY INSULATING SUBSTANCES SUCH AS OXIDES (BUT NOT IN THE FORM OF POROUS, BULK MATERIALS).


Thermal Conductivity of Solids-IV Thermal Conductivity of Solids-IV Phonons are also scattered by ƒdifferences in isotopic masses ƒchemical impurities ƒdislocations ƒsecond phases

Thermal Conductivity of Solids-V Thermal Conductivity of Solids-V In electrical conductors, in addition to phonons, conduction electrons contribute to thermal conductivity. The electronic contribution to the thermal conductivity, kel, 2 2 π n K Tλ k e B el = el 3m V e f 3 where n , the number of free electrons per cm , λ the mean free e el path of electron, Vf, electron velocity at the Fermi surface and me, electron mass. Wiedmann-Franz law and the Lorentz number, L, are utilized to determine what is the dominant mechanism for thermal conduction k L = el = 2 .45 x 10 −8 W ohm K – 2 σ T e

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