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Lecture 6

Thermal Properties of Materials

o Heat Capacity o Thermal Conductivity o Thermal Expansion o Thermal Stresses

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Objectives

Explain the temperature dependence of heat capacity & thermal conductivity of metals and amorphous solids

Explain modes of heat transfer in metals and amorphous solids

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4.1 Heat Capacity (c)

C is a measure of ability of material to absorb thermal energy

C is measured as specific heat capacity, c (= Quantity of heat needed to cause unit temp rise in 1 kg of material)

C is measured under conditions of constant volume (Cv) or pressure (CP).

dT

dQC

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When a material is heated, the absorbed Thermal energy (dQ) increases the KE of the molecules thus raising the internal energy (U) of material.

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.U K E NRT N = Number of molecules

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)1..(....................3RdT

dU

dT

dQC

VV

V

Dulong Petit Law

For a constant volume process, the system does no work on the sorrounding i.e.,

By applying 1ST law of Thermodynamics, (dQ = dU + dW), Specific heat capacity at const volume (Cv) becomes (NB dW = 0)

( 0)dQ dU dW

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Temp Dependence of Heat Capacity

Eqn (1) Cv of solids is const. It is observed that

(i) Cv of solids obey Dulong petit law only at high T > D, where D = Debye temp.

(ii) As T 0, Cv T3 and fails to obey Dulong Petit Law

T

Cv Dulong Petit Law

Exp data for most solids

3R

D

D > room temp

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The Low temp behavior of Cv is explained using quantum theory – Einstein 1916

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Fig 1. Temp Dependence of Heat Capacity of materials

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Cv of Metals

Heat capacity of metals is due to contributions from phonons and electrons

Phonons are elastic waves that are emitted when atoms vibrate about their equilibrium positions. They travel at speed of sound

A phonon is a quantum of lattice vibrational wave with energy given by

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E

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Phonon contribution to Cv is ~ AT3 at T →

0K.

Electron contribution to Cv T and becomes significant (for metals only) as T → 0K.

Thus

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ho 3Metals P nons Elect

V V VC C C T T

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Cv of Ceramics

Compared to metals, ceramics have high specific heats and high MP due to strong bonds (mix of ionic & covalent) they can absorb a lot of heat and withstand high temps

Ceramics are used in high temp applications e.g., spark plugs

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6.2 Thermal Conductivity

It is a measure of the material’s ability to transmit heat from regions of high to low temperature.

Defined by

dx

heat

T2 T1

A

' .....(2)dT

Q K Fourier s Lawdx

o Q = Heat flux (Quantity of thermal energy flowing through a unit area per unit time)

o dT/dx = temperature gradient, k = Thermal conductivity

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Note the similarity of Eqn (2) to the Fick’s first law for atomic diffusion where the diffusion flux is concentration gradient:

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dx

dCDJ x

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Thermal Conductivity of Metals

K of metals is by a combination of phonons and electrons i.e.,

KMetals = Ke + KP

KP & Ke are phonon and electronic thermal conductivities.

Phonon conductivity is due to atomic vibrations while Electron conductivity is due to Free (conduction band) electrons

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Ke >> KP since electron mean free path >> phonon mean free path.

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Factors affecting K of metals

KMetals is affected by

(i) Impurities (Alloying) –This introduces scattering thereby reducing electron

mean free path Ke & KP reduces with increase in impurities

(ii) Temp:- Increase in temp increases scattering

Cu-Zn alloy

Wt% Zn

K

of

Cu

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T

metals

K

Metal melt &

Become

amorphous

800oC

Generally, KMetals 1/T

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Note:

Since free electrons are responsible for both electrical () and thermal conduction (K) in metals, the two conductivities are related to each other by the Wiedemann-Franz law:

where L = const

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T

KL

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Conductivity of non-metals

In crystalline solids (e.g NaCl) & amorphous solids (ceramics), thermal conductivity is due to phonons & occurs at 300K

At much higher T, phonon conductivity reduces due to increase in phonon scattering resulting from increased frequency of vibrations Kcryst 1/T.

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KCryst 1/T and KAmorp T

KMetals > KCrystalline > KCeramics.

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0 200 400(2K) 800 T(K)

Amorphous solids

Crystalline Solids & Dielectrics

K

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Ceramics lack the orderly lattice arrangements S.T Phonon transmission is interrupted by microstructural defects e.g., porosity, grain size & grain boundaries etc leading to low thermal conductivity compared to crystalline solids

KCeramics reduces with increase in structural defects

At very high temp, KCeramics increases due to radiant heat transfer (through pores)

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SEM of clay ceramics showing effect of Porosity on KCeramics (Photos Adapted from Nyongesa et al, 2002)

Grain boundaries reduces KCeramics

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Thermal Applications of ceramics

Ceramics are thermally insulative Used as insulators in high temp applications e.g.,

(ii) space shuttle

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(ii) Insulation in wall – using thin brick walls

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Research References

Ogacho A. A., Aduda B.O. and Nyongesa F.W., (2006). Thermal Shock Behaviour of a kaolinite refractory prepared using a natural organic binder., Journal of Material Science., 41, 8276 - 8283.

Ogacho A. A., Aduda B.O. and Nyongesa F.W., (2003). Thermal Conductivity of a Kaolinite Refractory; Effects of an Orgarnic Binder., Journal of Material Science., 38[11], 2293 - 2297.

(iii) Ceramic lining in “jikos” & furnaces – to

prevent heat loss

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4.3 Thermal Expansion

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The energy addition to a material in the form of heat increases the thermal vibration of the atoms in their lattice sites.

The thermal expansion is a direct result of a greater separation distance between the centers of adjacent atoms as the thermal vibration of individual atoms increases with increasing temperature.

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α for ceramics and glasses < α for metals due to the shape of the energy well.

The ceramics and glasses generally have deeper wells (higher bonding energies) due to their ionic and covalent bond natures; therefore less atomic separation with the increasing temperature.

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expansion thermal oft coefficienlinear ]./[ CmmmmLdT

dL o

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The elastic modulus is directly related to the derivative of the bonding energy curve near the bottom of the well. The deeper the well, the larger derivative and greater the elastic modulus.

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Linear thermal expansion coefficients of some ceramic oxides. [from W. D. Kingery]

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Front Row Left to Right: Junping Shao, Derek Stampone, Dan Clark, Francis Mutuku, Darshana Weerawarne, Joshua Hewlett, Jeremiah Dederick, Patrick OBrien Next Row standing Left to Right: Linda Wangoh, Imran Khan, Matt Wahila, Jonathan Li, Connor Harrison, Nick Quackenbush, Matt Gochan, Gavin Osterhoudt, Felix Sauoma, Ning Kang Last Row Standing, Left to Right: Calford Otieno, Greg Parks, Austin Faucett, Steve Button, Justin Leshen, Shawn Sallis, Faramarz Hadian, William Thompson.

Graduate Students- fall 2012- 2013

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6.4 Thermal Shock

Thermal shock: Fracture of materials due to differential contraction or expansion caused by sudden cooling/heating.

Thermal shock inducing surface cracks leading to failure

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1fR T

E

Capacity of material to withstand failure from Thermal Shock is called thermal Shock resistance (TSR)

TSR is defined by Hassleman parameters R, R’ and R’’’

= Measures Resistance of material to initiation of fracture by thermal shock = Min temp drop (T) necessary to produce fracture

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1f KR

E

= Measures of possibility

for crack propagation

2 1f

ER

= Measures ability of material to resist crack propagation

f = flexural strength, K = thermal conductivity

= coeff of thermal expansion, E = Elastic Modulus

= Poisson’s ratio, T = temp difference between material and quenchant temp

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Fig 1. TRS parameters vs “mrenda” binder concentration in kaolinite ceramics used in ceramic “jikos”

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Ogacho A., Aduda B., Nyongesa F.W., (2006)., J. Mater Sci., 41, 8276

Aduda B.O., Nyongesa F.W. and Njogu M. S., (2008)., J. Mater Sci., 43, 4107

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Activity

What can you infer from Fig 1?

What conclusions and recommendations can you draw from Fig 1?

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Fig 2. R′′′ vs Thermal Quench Cycles (N) for ceramic refractories made from various clay particle sizes

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• F. W. Nyongesa, N. Rahbar, S. K. Obwoya, J. Zimba, B. O. Aduda, and W.O. Soboyejo., (2011), ISRN Mech Eng, DOI 10.5402/2011/816853.

•N. Rahbar, S. K. Obwoya, J. Zimba, B. O. Aduda, F. W. Nyongesa, I. Yakub and W.O. Soboyejo., (2010)., Experimental Mechanics DOI 10.1007/s

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Activity

What can you infer from Fig 2?

What conclusions and recommendations can you draw from Fig 2?

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Examples of thermal shock failure

Teeth cracks when exposed to sudden temp gradient

Ceramic cooking stove cooled by pouring water will fail

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RESEACH REFERENCES

Ogacho A. A., Aduda B.O. and Nyongesa F.W., (2006). Thermal Shock Behaviour of a kaolinite refractory prepared using a natural organic binder., Journal of Material Science., 41, 8276 - 8283.

F. W. Nyongesa, N. Rahbar, S. K. Obwoya, J. Zimba, B. O. Aduda, and W.O. Soboyejo., (2011). An investigation of Thermal Shock in Porous Clay Ceramics, ISRN Mechanical Engineering, DOI 10.5402/2011/816853 – March 2011.

N. Rahbar, S. K. Obwoya, J. Zimba, B. O. Aduda, F. W. Nyongesa, I. Yakub and W.O. Soboyejo., (2010). Thermal Shock Resistance of a Kyanite-Based (Aluminosilicate) Ceramic, Experimental Mechanics DOI 10.1007/s – April 2010

Failure due to thermal shock can be prevented by increasing R’ through Increasing f and K

Decreasing thermal gradient by changing temperature slowly

Decreasing E

Increase Toughness trough toughening mechanisms

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Lecture -Evaluation

Explain why Cv is virtually independent of temp at temp far removed from 0K

Explain factors that influence thermal conductivity of metals & ceramics

Explain the temp dependence of thermal conductivity of Metals

Explain Thermal Shock and how to prevent failure from Thermal Shock