Conduction - Coefficient of Thermal Conductivity

Fig. 7.12(a): Section of a metal bar in steady state. Fig. 7.12(b): Section of a cube in steady state.

Consider a cube of side x and each face of cross-sectional area A. The opposite faces are maintained at temperatures T₁ and T₂ (T₁ > T₂), as shown in Fig. 7.12(b).

Experiments show that under steady-state conditions, the quantity of heat Q that flows from the hot face to the cold face is:

1. Directly proportional to the cross-sectional area A of the face:
   Q ∝ A
2. Directly proportional to the temperature difference between the two faces:
   Q ∝ (T₁ − T₂)
3. Directly proportional to the time t (in seconds) for which heat flows:
   Q ∝ t
4. Inversely proportional to the perpendicular distance x between the hot and cold faces:
   Q ∝ \[\frac {1}{x}\]

The coefficient of thermal conductivity of a material is defined as the quantity of heat that flows in one second between the opposite faces of a cube of side 1 m, the faces being kept at a temperature difference of 1°C (or 1 K).

Combining the above four factors:

\[Q\propto\frac{A(T_1-T_2)t}{x}\]

\[{Q=\frac{kA\left(T_1-T_2\right)t}{x}}\]   ---(1)

where k is a constant of proportionality called the coefficient of thermal conductivity. Its value depends upon the nature of the material.

From Eq. (1):

\[{k=\frac{Qx}{A\left(T_1-T_2\right)t}}\]

SI unit of k:

\[k=\frac{J\cdot m}{m^2\cdot^\circ C\cdot s}=Js^{-1}m^{-1o} C^{-1}\]

SI unit of k = Js^-1m^-1°C^-1 or Js^-1m^-1K^-1

Dimensions of k:

[k] = [L¹ M¹ T⁻³ K⁻¹]

Form Eq. (1)

\[\frac{Q}{t}=\frac{kA(T_1-T_2)}{x}\] ---(2)

The quantity \[\frac {Q}{t}\], denoted by P_cond, is the time rate of heat flow (i.e., heat flow per second) from the hotter face to the colder face at right angles to the faces.

• SI unit of P_cond = watt (W)
• Therefore, the SI unit of k can also be written as:

SI unit of k = W m⁻¹ ∘C⁻¹ or W m⁻¹ K⁻¹

Using calculus, Eq. (2) may be written as:

\[{\frac{dQ}{dt}=-kA\frac{dT}{dx}}\]

where \[\frac {dT}{dx}\] is the temperature gradient.

• The negative sign indicates that heat flow is in the direction of decreasing temperature.

• If A = 1 m² and \[\frac {dT}{dx}\] $=1$, then \[\frac {dQ}{dt}\] $=k$ (numerically).

• Under steady state, heat flow Q depends on A, (T₁ − T₂), t, and 1/x.
• The coefficient k is a material property — it does not depend on the shape or size of the body.
• SI unit of k: W m⁻¹ K⁻¹ (or J s⁻¹ m⁻¹ K⁻¹)
• Dimensions of k: [M¹L¹T⁻³K⁻¹]
• The differential form \[\frac {dQ}{dt}\] = -kA\[\frac {dT}{dx}\] applies when temperature varies continuously; the negative sign enforces the hot-to-cold direction of heat flow.