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Question

One end of a copper rod of uniform cross-section and length 1.5 meters is in contact with melting ice and the other end with boiling water. At what point along its length should a temperature of 200oC be maintained, so that in steady state, the mass of ice melting is equal to that of steam produced in the same interval of time? Assume that the whole system is insulated from the surroundings.

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Solution

Melting ice at 0°
Boiling water at 100°C
Let the distance be x from point A
Where the temperature is 200°C
We know that,
dθdt=KAΔTl (1)
Where,
dθdt= Rate of flow of heat.
K=Thermal conductivity
A=Area
ΔT=Temperature difference
l= Length of material
We know,
θ=mL [heat=θ,m=mass,L=latent heat]
Differentiating on both sides
dθdt=dmdtL (2)
From 1 and 2
dmdt=KAΔTLl (3)
At steady state
Rate at which ice is melting= Rate at which steam is produced
(dmdt)ice=(dmdt)steam
Using 3
KA(2000)Licex=KA(200100)(1.5x)Lsteam
2x×80=(1(1.5x)540)
14x=(154(1.5x))
54×1.554x=4x
54×1.5=58x
x=54×1.558
=1.4 meters




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