A uniform copper rod of 50 cm length is insulated on the sides and has its ends exposed to ice and steam respectively. If there is a layer of water 1 mm thick at each end, the temperature gradient in ° C m - 1 in the bar is (assume that the thermal conductivity of copper is 400 W m - 1 K - 1 and water is 0 .4 W m - 1 K - 1 )

Question

A uniform copper rod of 50 cm length is insulated on the sides and has its ends exposed to ice and steam respectively. If there is a layer of water 1 mm thick at each end, the temperature gradient in ° C m - 1 in the bar is (assume that the thermal conductivity of copper is 400 W m - 1 K - 1 and water is 0 .4 W m - 1 K - 1 ), 60, 40, 50, 65

MARKS App · Accepted Answer

Given, K c = 400 W M - 1 K - 1 , K w = 0.4 W M - 1 K - 1 θ 1 = 100 ° C , θ 2 = 0 ° C In steady-state, the flow of heat will be the same throughout the whole system K w A 100 - θ 10 - 3 = K c A θ - θ ′ 50 100 Or 100 - θ = K c K w × 10 - 3 × θ - θ ′ 0.5 = 400 0.4 × 10 - 3 × θ - θ ′ 0.5 100 - θ = 2 θ - θ ′ ( 100 − θ ) K w A 10 − 3 = ( θ ′ − 0 ) K w A 10 − 3 100 − θ = θ ′ Or 100 - θ = 4 θ - 200 ⇒ 5 θ = 300 ⇒ θ = 60 ° C ∴ θ ′ = 100 - θ = 100 - 60 = 40 ° C Hence, the temperature gradient along the bar = θ - θ ′ 0.5 = 60 - 40 0.5 = 40 ° C m - 1

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