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**Thermal Behavior of Materials**

ME 2105 Dr. R. Lindeke

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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 formula: 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˚

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**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

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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

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**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?

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**Thermal Expansion is “Temperature Dependent”**

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Solving:

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**Note, k is a function of temperature (like was)**

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)

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

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**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 vibrations disrupt the primary free electron conduction mechanism Adding alloying “impurities” also disrupts free electron conduction so alloys are less conductive than pure metals

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**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 on TC (pores are filled with low TC gases which limits overall TC for a structure (think fiberglass insulation and ‘strya-foam’ cups)

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And Continuing:

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**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.)

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Definition: Thermal Shock: it is simply defined as the fracture of a material (often a brittle ceramic) as the result of a (sudden) temperature change and is dependent on 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 based in finite thermal conductivity reasoning

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**By Constrained Thermal Expansion:**

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.

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**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 MOR for Stabilized ZrO2 is 83 MPa

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Since the Inducted Compressive Stress exceeds the MOR 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

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**By Thermal Conductivity (induced)Temperature Gradients:**

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

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**(From W. D. Kingery, H. K. Bowen, and D. R**

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

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**Thermal Shock Resistance**

• Occurs due to: uneven heating/cooling. • Ex: Assume top thin layer is rapidly cooled from T1 to T2: s rapid quench resists contraction tries to contract during cooling T2 T1 Tension develops at surface Temperature difference that can be produced by cooling: Critical temperature difference for fracture (set s = sf) set equal • Result: • Large thermal shock resistance when is large.

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**Thermal Shock Resistance Parameter**

Where: f is fracture strength of a material l is coeff. Of linear thermal expansion k is thermal conductivity of material E is modulus of elasticity

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