Physics · Ch 7 — Thermal Properties of Matter

Coefficient of Thermal Conductivity

7.9.1.2

Coefficient of Thermal Conductivity

To quantify conduction precisely, consider a cube of side $x$ with two opposite faces, each of area $A$ , maintained at temperatures $T_1$ and $T_2$ ( $T_1 > T_2$ ) (Fig. 7.12(b)). Experiments in the steady state show the heat $Q$ that flows from the hot face to the cold face is: (i) directly proportional to the cross-sectional area, $Q \propto A$ ; (ii) directly proportional to the temperature difference, $Q \propto (T_1-T_2)$ ; (iii) directly proportional to the time $t$ for which heat flows, $Q \propto t$ ; and (iv) inversely proportional to the perpendicular distance $x$ between the two faces, $Q \propto 1/x$ . Combining all four:

Q = \frac{kA(T_1-T_2)t}{x} \quad \text{--- (7.34)}

where $k$ , the coefficient of thermal conductivity, is a material-dependent constant of proportionality. Setting $A=1\text{ m}^2$ , $(T_1-T_2)=1\,°\text{C (or K)}$ , $t=1\text{ s}$ and $x=1\text{ m}$ gives $Q=k$ numerically: $k$ is the quantity of heat that flows in one second between the opposite faces of a 1 m cube whose faces are kept 1 °C (or 1 K) apart. Rearranged,

k = \frac{Qx}{A(T_1-T_2)t} \quad \text{--- (7.35)}

The SI unit of $k$ is $\text{J s}^{-1}\text{m}^{-1}\,°\text{C}^{-1}$ (equivalently $\text{W m}^{-1}\text{K}^{-1}$ ), with dimensional formula $[L^1M^1T^{-3}K^{-1}]$ .

Writing the heat FLOW RATE $Q/t$ as $P_{cond}$ (in watts):

$P_{cond} = \frac{Q}{t} = \frac{kA(T_1-T_2)}{x} \quad \text{--- (7.36)}$ …

Figure 7.12aFig. 7.12(a): Section of a metal bar in the steady state

What this figure shows. A horizontal metal bar/rod is shown with one end (left) in contact with a heat source and the other end (right) exposed to a colder region. Along the length of the bar, several cross-sectional slices are marked at intervals, each labelled with its own steady, but progressively lower, temperature from the hot end towards the cold end (e.g. temperatures decreasing left to right), illustrating that in the steady state every cross-section has settled at its own constant (in time) temperature, but that temperature is different from one cross-section to the next, decreasing uniformly with distance from the ho …

Figure 7.12bFig. 7.12(b): Section of a cube in the steady state

What this figure shows. A cube of side x is drawn with two OPPOSITE faces (front and back, each of cross-sectional area A) explicitly labelled: the front face is maintained at a higher temperature T1 and the back face at a lower temperature T2 (T1 > T2), with an arrow drawn through the cube from the hot face to the cold face indicating the direction of steady-state heat flow (heat quantity Q flowing through in time t across the perpendicular distance x bet …

Table 7.7Table 7.7: Coefficient of thermal conductivity (k) of some materials

Substance | Coefficient of thermal conductivity (J s^-1 m^-1 K^-1)

Silver | 406

Copper | 385

Aluminium | 205

Steel | 50.2

Insulating brick | 0.15

Glass | 0.8

Brick and concrete | 0.8

Water | 0.8 …

Misc Ex.17Rate of energy loss per unit area through a glass window

Worked out. A 5 mm thick glass window (k(glass) = 1 W/m K) has outside temperature -20 °C and inside temperature 25 °C (a 45 K difference); using Pcond/A = k(T1-T2)/x, the example computes the rate of energy loss per square metre as 1x45/(5x10^-3) = 9x10^3 W/m^2. …

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