Thermal Behavior of Materials

ME 2105Dr. R. Lindeke

Some Definitions

• Heat Capacity: the amount of heat (energy) required to raise a fundamental quantity of a material 1 K˚

• The quantity is usually set at 1 gm-atom (elements) or 1 gm-mole (compounds)• Given by the foumula: C=q/(mT) in units of J/gm-

atom* K˚ or J/gm-mole* K˚

• Specific Heat: a measure of the amount of heat energy to raise a specific mass of a material 1 K˚

Heat Capacity

• Heat capacity is reported in 1 of two ways: • Cv – the heat capacity when a constant volume of

material is considered• Cp – the heat capacity when a constant pressure is

maintained while higher than Cv these values are nearly equal for most materials• Cp is most common in engineering applications (heat

stored or needed at 1 atm of pressure)• At temperature above the Debye Temperature Cv 3R

Cp

Definitions• Thermal Expansion is the “growth” of materials due to

increasing vibration leading to larger inter-atomic distances and increasing vacancy counts for materials as temperature increases

Thermal Expansion

• Linear thermal expansion is given by this model:

• As an Example:• A gold ring (diameter = 12.5 mm) is worn by a person,

they are asked to wash the dishes at their apartment – water temperature is 50˚C – how big is the ring while it is submerged?

dL

LdT

Thermal Expansion is “Temperature Dependent”

Solving:

0

0 0

37.5 27 37.537.5

37.5 27

37.5

50

6

60

data from table 7.2

* 12.5*3.14159 39.2699

14.1

16.5 14.1

14.15

similarly for 50 : 14.21

14.21 14.1514.18*102

14.18*10 *39.2

avg

avg

L L T

L d

mmmm C

L L T

7*12.5 0.007

39.27 .00712.502f

mm

D mm

Definition:

• Thermal Conductivity: the transfer of heat energy through a material (analogous to diffusion of mass)

• Modeled by:• Note, k is a function of

temperature (like was)

at steady state conditions:

dQdtk

dTA dx

Qtk

TA x

• Modeling Fourier’s Law of Thermal Conduction (heat flow thru a bounded area)

Thermal Conductivity

• Involves two primary (atomic level) mechanisms:• Atomic vibrations – in ceramics and polymers this dominates• Conduction by free electrons – in metals this dominates

• Focusing on Metals: • thermal conductivity decreases as temperature increases

since atomic vibration disrupt the primary free electron conduction mechanism

• Adding alloying “impurities” also disrupts free electron conduction so alloys are less conductive than pure metals

Thermal Conductivity

• Focusing on Ceramics and Polymers: • Atomic/lattice vibrations are “wave-like” in nature and

impeded by structural disorder• Thermal conductivity will, thus, drop with increasing

temperature• In some ceramics, which are “transparent” to IR radiation,

TC will eventually rise at elevated temperatures since radiant heat transfer will begin to dominate “mechanical” conduction

• Porosity level has a dramatic effect of TC (pores are filled with low TC gases which limits overall TC for a structure (think fiberglass insulation and ‘stryo-foam’ cups)

Continuing Table 7.4:

Figure 7.5 Thermal conductivity of several ceramics over a range of temperatures.

(From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)

Definition:

• Thermal Shock: it is simply defined as the fracture of a material (usually a brittle ceramic) as the result of a (sudden) temperature change and is dependent of the interplay of the two material behaviors: thermal expansion and thermal conductivity

• Thermal Shock can be explained in one of two ways:• Failure stress can be built up by constrained thermal

expansion• Rapid temperature changes lead to internal temperature

gradients and internal residual stresses – finite thermal conductivity reasoning – see figure 7.7

By Constrained Thermal Expansion:

Figure 7.6 Thermal shock resulting from constraint of uniform thermal expansion. This process is equivalent to: a. free expansion followed by; b. mechanical compression back to the original length.

Let’s Consider an Example:

• A 400 mm long ‘rod’ of Stabilized ZrO2 ( = 4.7x10-6 mm/mm˚C) is subject to a thermal cycle in a ‘ceramic’ engine – it’s the crank shaft! – from RT (25˚C) to 800˚C. Determine the induced stress and determine if it is likely to fail?

• E for Stabilized ZrO2 is 150 GPa – table 6.4

• MOR for Stabilized ZrO2 is 83 MPa -- table 6.4

exp

0

6

where:

4.7*10 * 800 25 0.00364

150 *0.00364 0.546 546

TI TI

TTI

TI

TI

E

ll

mmT mmmmMPa GPa MPamm

Since the Inducted Compressive Stress exceeds the MOR (from Table 6.4) one might expect the ‘rod’ to fail or rupture – unless it is allowed to expand into a designed in ‘pocket’ built into the engine block to accept the shaft’s expansion

By Thermal Conductivity (induced)Temperature Gradients:

Figure 7.7 Thermal shock resulting from temperature gradients created by a finite thermal conductivity. Rapid cooling produces surface tensile stresses and Griffith Crack Generation.

Figure 7.8 Thermal quenches that produce failure by thermal shock are illustrated. The temperature drop necessary to produce fracture (T0 − T) is plotted against a heat-transfer parameter (rmh). More important than the values of rmh are the regions corresponding to given types of quench (e.g., water quench corresponds to an rmh around 0.2 to 0.3).

(From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)