Heat Flow Through A Compound Media or Wall

Bodies in series

Let us consider a compound media like a slab or a wall made up of two different materials A and B having thickness x₁ and x₂ as shown in (Fig. 1)

Fig. (1) Bodies in series

Let the temperature of two end faces be θ₁ and

θ 2

Let the temperature of the common surface be

θ

Let K₁ and K₂ be the thermal conductivities of A and B respectively.

If we consider θ₁

>θ 2

, so that heat flow will be from A to B and when the steady state is reached

eqn (1)

[latex] \theta=\frac{\mathrm{K}_{1} \mathrm{~A}\left(\theta_{1}-\theta\right)}{x_{1}} [/latex]

eqn (2)

and for B medium, [latex] \text { and for B medium, } \theta=\frac{\mathrm{K}_{2} \mathrm{~A}\left(\theta-\theta_{2}\right)}{x_{2}} [/latex]

Now at steady state,

we have, [latex] \frac{\mathrm{K}_{1} \mathrm{~A}\left(\theta_{1}-\theta\right)}{x_{1}}=\frac{\mathrm{K}_{2} \mathrm{~A}\left(\theta-\theta_{2}\right)}{x_{2}} [/latex]

(i.e.)

[latex] \frac{\mathrm{K}_{1} \theta_{1}}{x_{1}}-\frac{\mathrm{K}_{1} \theta}{x_{1}}=\frac{\mathrm{K}_{2} \theta}{x_{2}}-\frac{\mathrm{K}_{2} \theta_{2}}{x_{2}} [/latex]

(i.e.) [latex] \left(\frac{\mathrm{K}_{1}}{x_{1}}+\frac{\mathrm{K}_{2}}{x_{2}}\right) \theta=\left(\frac{\mathrm{K}_{1} \theta_{1}}{x_{1}}+\frac{\mathrm{K}_{2} \theta_{2}}{x_{2}}\right) [/latex]

(i.e.) [latex] \theta=\frac{\frac{\mathrm{K}_{1} \theta_{1}}{x_{1}}+\frac{\mathrm{K}_{2} \theta_{2}}{x_{2}}}{\frac{\mathrm{K}_{1}}{x_{1}}+\frac{\mathrm{K}_{2}}{x_{2}}} [/latex]

eqn (3)

(i.e.) [latex] \theta=\frac{\sum \frac{\mathrm{K}_{1} \theta_{1}}{x_{1}}}{\sum \frac{\mathrm{K}_{1}}{x_{1}}} [/latex]

Putting the value of $θ$ in eqn (1) or (2) we can get the quantity of heat flowing through the compound media.

Read More Topics

De-Broglie’s concept of matter

Laws for explaining the energy distribution

Characteristics of De-Broglie waves