From 60f81709c2fcf2099b6ebb56ea2783e50033bec3 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Tue, 21 Dec 2021 20:25:57 +0000 Subject: [PATCH] add notes on thermal expansion, thermal conductivity --- ...ot-2021-12-21T19:51:58,667328620+00:00.png | Bin 0 -> 130551 bytes mechanical/mmme1029_materials.md | 140 ++++++++++++++++++ 2 files changed, 140 insertions(+) create mode 100644 mechanical/images/vimscrot-2021-12-21T19:51:58,667328620+00:00.png diff --git a/mechanical/images/vimscrot-2021-12-21T19:51:58,667328620+00:00.png b/mechanical/images/vimscrot-2021-12-21T19:51:58,667328620+00:00.png new file mode 100644 index 0000000000000000000000000000000000000000..60404edc2620f89325c04708ea2f10f22445e90c GIT binary patch literal 130551 zcmV)%K#jkNP)f8PB9{`$QkKiA zb-8Tq+Ew;?Yis}5l4^IoskYW$+u9XHi8e)%A}LPda0VEpK^o*n>IOQY13Kr|FWr03 zIs39L!mR9BD~d8=i}Mp2fkl~6 z-VFltsr%9(gbmlPD953QgyVrN%8B~aISc#Ai)O_~?S%gn=x5ldpEAVfFbqWr`b07o 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Kevlar/Epoxy | 1200 | 20 (All units in MPa) + +# Thermal Properties of Materials + +## Specific Heat Capacity + +How much heat energy is required to raise the temperature of a body by one unit: + +$$ C_p = \frac{\Delta E}{m\Delta T} $$ + +where $c$ is specific heat capacity. + +It is measured at a constant pressure, usually $1.013\times 10^5$ Pa. + +## Molar Heat Capacity + +$$C_pm = \frac{\Delta E}{n\Delta T}$$ + +

+ + +#### What is a mole? + + + +> The mole (symbol: mol) is the base unit of amount of substance in the International System of +> Units (SI). +> It is defined as exactly $6.02214076\times 10^{23}$ elementary entities ("particles") + +~ [Wikipedia: Mole (unit)](https://en.wikipedia.org/wiki/Mole_(unit)) + +
+ +
+ + +#### How Much Does a mol of Something weigh? + + + +A mol of an element weighs its relative atomic mass ($A_r$) but in grams. +For example, Carbon-12 has an $A_r$ of 12 (as it's made of 6 neutrons, 6 protons, and 6 electrons +which have negligible mass) so a mol of Carbon-12 has a mass of 12 g. + +
+ +## Thermal Expansion + +
+ + +### Origin of Thermal Expansion + + + +All atomic bonds vibrate, on the magnitude of gigahertz. +The bonds vibrate about a mean positoin and the vibration is a simple harmonic motion. + +From the graph below you can see that as energy (in the form of heat) is supplied to the bonds, +the amplitude of the vibrations get larger and larger. +You can also see the mean position of the bond gets further and further away, meaning the volume +of the material also is increasing. +The mean position of the bond is what dictates the volume, as this means the inter-atomic +separation increases. + +![Morse Potential Graph](./images/vimscrot-2021-12-21T19:51:58,667328620+00:00.png) + +Morse potential is the energy well between 2 bonded atoms. +The graph is asymmetric due to the repulsion experienced by atoms as they apporach. + +
+ +### Linear Coefficient of Thermal Expansion + +$$\alpha_L = \frac{\Delta L}{L_0 \Delta T}$$ + +where $L$ is the sample length. + +
+ + +#### Example 1 + +A 1 m long bar of aluminium metal cools in the solid state from 660 \textdegree{}C to +25 \textdegree{}C. +Calculate the length of the bar after it cools down, given $\alpha_L = 25\times10^{-6}$ K$^{-1}$. + + + +\begin{align*} +l_0 &= 1 \\ +\Delta T &= T_f - T_0 = 25 - 660 = -635 \\ +\\ +\alpha_L &= \frac{l_f - l_0}{l_0 \Delta T} \\ +\alpha_L l_0 \Delta T &= l_f - l_0 \\ +l_f &= \alpha_L l_0 \Delta T + l_0 = 0.984 +\end{align*} + +
+ +### Linear Thermal Expansion and Isotropism + +Since isotropic solids have the same properties in all directions, you can say that for an +isotropic solid: + +$$\alpha_V = 3\alpha_L = \frac{\Delta V}{V_0 \Delta T}$$ + +### Reasons to Care About Thermal Expansion + +- A coating on a material may fail if the thermal expansion coefficients do not match +- A brittle material may thermally shock and fracture due to thermal expansion mismatch between + the ouside and inside, especially if the material is not very thermally conductive + +## Thermal Conductivity + +Thermal conductivity is the rate at which heat power is transferred through a material. + +$$\frac{Q}{A} = k \frac{\Delta T}{\Delta x}$$ + +where $Q$ is heat power, $A$ is area of the surface, $\frac{\Delta T}{\Delta x}$ is the +temperature gradient, and $k$ is the thermal conductivity constant. + +
+ + +### Origin of Thermal Conductivity + + + +Heat is transferred through materials by electrons (and partially by atomic vibrations) + +Metals have high thermal conductivity as their delocalised 'sea' electrons are about to move about +easily. +This makes them excellent conductors of heat and electricity. + +Ceramics, glasses, and polymers do not have delocalised electrons and are therefore poor conductors +of heat and electricity (they are insulators). + +Polymer foams are even better insulators because they have holes which lowers their density. + +