Q. 17

A semicircular rod is joined at its end to a straight rod of the same material and the same cross-sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Fid the ratio of the heat transferred through a cross-section of the semicircular rod to the heat transferred through a cross-section of the straight rod in a given time.

Answer :


Given:

Temperature at junction 1: T1
Temperature at junction 2: T2
Length of the rod 1: x= d=2r
Where d is the diameter and r is the radius.
Length of the rod 2: x’ = circumference = πr
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.
As both the rods have same material, their thermal conductivity is the same.
(1)
Rate of heat flowing through rod 1

(2)
Rate of heat flowing through rod 2

Ratio of the rate of heat transferred from semi circular rod to straight rod is



Hence, the ratio of the heat transferred through a cross-section of the semicircular rod to the heat transferred through a cross-section of the straight rod in a given time is 2:π


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