Heat Transfer Lecture I

8/10/2019 Heat Transfer Lecture I

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Energy TransportEnergy Transport

Focus on heat transfer

Heat Transfer Mechanisms:Heat Transfer Mechanisms:•• ConductionConduction

•• RadiationRadiation•• Convection (mass movement of fluids)Convection (mass movement of fluids)



ConductionConduction

Conduction heat transfer occurs only when there is

physical contact between bodies (systems) at differenttemperatures by molecular motion.

Heat transfer through solid bodies is by conductionalone, whereas the heat may transfer from a solidsurface to a fluid partly by conduction and partly by

convection.



FourierFourier’’s Law of Thermal Conductions Law of Thermal Conduction

y

T

k q y ∂

∂

−=

the heat flux inthe y directionW/m2

thermalconductivityW/m.K

temperaturegradient in the

y-direction(K/m)



The proportionality ratio, k , is called thermal conductivity of the materi



Thermal Conductivity (k)Thermal Conductivity (k)

The thermal conductivity of a substance is defined as the heatflow 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 throughthe material. It depends on the bonding and structure of thematerial.



Thermal Diffusivity (Thermal Diffusivity ( ))

The thermal diffivity is a fundamental quantity. It is

analogous to momentum and mass diffusivities.

=k/ρ

Cv

where ρ and Cv are the density and specific heat of thematerial, respectively.

InIn mksmks system, the unit of thermal diffusivity is msystem, the unit of thermal diffusivity is m22.s.s--11,,while in thewhile in the cgscgs system it is usually cmsystem it is usually cm22.s.s--11..



Thermal RadiationThermal Radiation

Thermal radiation is the energy radiated from hotsurfaces as electromagnetic waves. It does not requiremedium for its propagation. Heat transfer by radiationoccur between solid surfaces, although radiation fromgases is also possible. Solids radiate over a wide range of

wavelengths, while some gases emit and absorb radiatioon certain wavelengths only.The energy flux emitted by an ideal radiator is

proportional to the fourth power of its absolutetemperature.

eb=σ

T4

where eb is the emissive power and σ is the StefaBoltzmann 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 iscalled 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.



ConvectionConvection

Convection is the heat transfer within a fluid, involving grossmotion of the fluid itself.

Fluid motion may be caused by differences in density as in freeconvection. Density differences are a direct result of temperaturedifferences between the fluid and the solid wall surface.

In forced convection, the fluid motion is produced by mechanicalmeans, such as a domestic fan-heater in which a fan blows airacross an electric element.



Heat Transfer Coefficient (h)Heat Transfer Coefficient (h)

When a moving fluid at one temperature is in contact with asolid at a different temperature, heat exchanges between thesolid and the fluid by conduction at a rate given by Fourier’s

law. Under such cases the distribution of temperature withinthe fluid and the heat flux at the wall can be determined byusing heat transfer coefficient, h,

h=q0/(Ts-Tf ) =-[k(dT/dy)0]/(Ts-Tf )whereT

s,

the surface temperatureTf , bulk fluid temperatureq0, heat flux at the wall(dT/dy)0, the temperature gradient in the fluid normal to thewall at the fluid-solid interface.k, the conductivity of the fluid



Thermal Conductivity of GasesThermal Conductivity of Gases--II

where m, mass of fluid molecules, K B, the Boltzmann constant, T,absolute temperature, d , the center to center distance of twomolecules.The thermal conductivity of gases is independent of pressure andepends only on temperature. This conclusion is valid up to aboutten atmospheres (1.0133 x 105 Pa)

Conduction of energy in a gas phase is primarily by transfer oftranslational energy from molecule to molecule as the fastermoving (higher energy) molecules collide with the slower ones.

2/1

3

3

2

)1

=m

T

d k B

π

κ

3λ V C k v

=where C v, the heat capacity per unit volume, V , the average

speed, λ, the mean free path.



Thermal Conductivity of GasesThermal Conductivity of Gases--IIII

where M , molecular weight, Cp, the heat capacity at constantpressure.

Eucken developed the following equation for the thermalconductivity of polyatomic gases at normal pressures,

+=

M

25.1

RC k pη



This figure is valid up to about ten atmospheres.



Thermal Conductivity of Gas MixturesThermal Conductivity of Gas Mixtures

where Xi is the mole fraction of component i having molecular

weight, M i , and intrinsic thermal conductivity k i.

The thermal conductivity of gas mixtures can be estimated withina few percent by the following equation

∑

∑=

i ii

i

iii

mix M X

M k X

k 3/1

3/1



Thermal Conductivity of SolidsThermal Conductivity of Solids--II

Solids transmit thermal energy by two modes:

elastic vibrations of the lattice moving through the crystal in

the form of waves

free electrons moving through the lattice also carry energysimilar to the case in gases (this is observed in metals)



Thermal Conductivity of SolidsThermal Conductivity of Solids--IIII

Two types of phonon-phonon interaction are observed in solids: Normal or N-type

Umklapp (U-process) collision

Each lattice vibration (there is always a spectrum of vibrations)may be described as a traveling wave carrying energy and obeyinthe laws of quantum mechanics. By analogy with light theory, thewaves in a crystal exhibit the characteristics of particles and arecalled phonons.

+=

M

25.1

RC k pη



Thermal Conductivity of SolidsThermal Conductivity of Solids--IIIIII

Since the number of phonons increases with temperature and thewavelength of phonons λph is proportional to 1/T. At roomtemperature and above, molar heat capacity Ĉv for most

materials is roughly constant the thermal conductivity of asolid which conducts energy only by phonons, decreases withincreasing temperature.

λph = 20Tmd/ 2T

where Tm

, melting point, T = absolute temperature, d, crystal-lattice dimension, and , Gruneisen’s constant (~2 for most solidsat ordinary temperatures).

THIS IS GENERALLY OBSERVED IN ELECTRICALLY

INSULATING SUBSTANCES SUCH AS OXIDES (BUT NOTIN THE FORM OF POROUS, BULK MATERIALS).



OXIDES

POROUSOXIDES



Thermal Conductivity of SolidsThermal Conductivity of Solids--IVIV

Phonons are also scattered by differences in isotopicmasses chemical impurities dislocations second phases



Thermal Conductivity of SolidsThermal Conductivity of Solids--VV

In electrical conductors, in addition to phonons, conductionelectrons contribute to thermal conductivity. The electroniccontribution to the thermal conductivity, k el,

f e

el Be

el V m

T K nk 3

22 λ π =

where ne, the number of free electrons per cm3

, λel the mean freepath of electron, Vf , electron velocity at the Fermi surface andme, electron mass.

Wiedmann-Franz law and the Lorentz number, L, are utilized todetermine what is the dominant mechanism for thermalconduction

2-8 K ohmW10x45.2 −==T

k Le

el

σ

Heat Transfer Lecture I

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