A hole of radius r1 is made centrally in a uniform circular disc of thickness d and radius r2. The inner surface (a cylinder of length d and radius r1) is maintained at ta temperature θ1 and the outer surface (a cylinder of length d and radius r2) is maintained at a temperature θ2(θ1 > θ2). The thermal conductivity of the material of the disc is K. Calculate the heat flowing per unit time through the disc.
Given:
Radius of the inner cylinder: r1
Length of the cylinder= thickness of the disc: d
Radius of the disc: r2
Temperature of inner cylinder: θ1
Temperature of outer surface: θ2
The thermal conductivity of the material of the disc : K
Formula used:
Rate of amount of heat flowing or heat current is given as:
Here, Δθ is the amount of heat transferred, ΔT is the temperature difference, K is the thermal conductivity of the material, A is the area of cross section of the material and x is the thickness or length of the material.
Consider an imaginary cylinder of radius r and thickness dr between r1 and r2.
We will integrate considering this imaginary cylinder to get total heat transferred.
In differential form heat flow is
Here q is the rate of heat flowing.
Negative sign indicates the decrease in rate of heat flow with increase in the thickness of the imaginary tube.
We know that area of the cylinder is:
A = 2πrd
Where r is the radius of the cylinder and d is the length of the cylinder.
Substituting we get,
Integrating both the sides we get the total rate of heat flow through the disc. Taking radius from r1 to r2 and temperature from θ1 to θ2.
Hence, the heat flowing per unit time through the disc is q.