Find the rate of heat flow through a cross-section of the rod shown in figure (θ₂> θ₁). Thermal conductivity of the material of the rod is K.

Question

Find the rate of heat flow through a cross-section of the rod shown in figure (θ 2 > θ 1 ). Thermal conductivity of the material of the rod is K.

Philoid · Accepted Answer

Let’s redraw the diagram

From the above diagram we can say that ΔABE is similar to ΔACD.

By the property of similar triangles-

=

⇒ x = r₁ + (r₂ – r₁ )

Lets assume-

a =

⇒ r = ax + r₁ (1)

Thermal resistance is given by –

dR =

Now area

dR =

=

Solving above integral

R =

⇒ R =

–

Rate of heat flow =

q = Kπr₁r₂