Heat conduction in solids - Thermal Conductivity

HEAT CONDUCTION IN SOLIDS ‒ THERMAL CONDUCTIVITY

The conducting property vary from material to material, so it is essential to know about the conducting property of a material.

Let us consider a solid material slab having an area of cross section 'A', thickness 'x' as shown in Fig 3.1. Let the temperature at the hot end be θ₁ and that of cold end be θ₂. Now the amount of heat(Q) flowing from hot end to the cold end is directly proportional to

(i) The area of cross section 'A' (i.e) Q α A

(ii) The temperature difference (i.e) Q α (θ₁‒ θ₂)

(iii) The time of conduction (i.e) Q α t and

(iv) The amount of heat flowing from hot end to cold end is inversely proportional to the thickness of the slab

(i.e) Q α 1/x

 ∴ Combining all the factors we can write that the amount of heat conducted Q α A(θ₁ ‒θ₂)t / x

 (or)

Q = [ KA(θ₁ ‒θ₂)t ] / x

Where K is the proportionality constant, known as coefficient of thermal conductivity, which depends on the materials

From eqn.(1) we can write,

if A = 1 cm²;

θ₁ ‒θ₂=1° C;

t=1 sec;

 x = 1 cm

then Q=K

Definition: The coefficient of thermal conductivity is defined as the amount of heat conducted per second, normally across unit area of cross section, maintained at unit temperature gradient.

Temperature gradient:

The quantity (θ₁ ‒θ₂)/x of fall of temperature with respect to the distance or thickness which is known as temperature gradient.

For smaller values (θ₁ ‒θ₂) /x is written as ‒dθ/dx

∴ eqn.(1) can be written as

Q = ‒KA (dθ/dx) t

The negative sign indicates the decrease in temperature as the distance increases.

Thermal Diffusivity (h):

Before the steady state is reached, the rate of heat flow is determined by a quantity called thermal diffusively (h) or thermometric conductivity. It is defined as the ratio of thermal conductivity to the thermal capacity per unit volume [The product of specific heat capacity (S) and the density (ρ)] of the material.

(i.e) h = Thermal Conductivity / Thermal Capacity

(i.e)

 h =  K / ρS

Note: Under steady state condition, thermal conductivity alone measure the rate of heat flow through the materials.

Specific heat capacity:

It is defined as the amount of heat required to raise the temperature of unit mass of the substance through one kelvin.

 S = Q/mθ Jkg^‒1 K^‒1