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A lake surface is exposed to an atmosphe...

A lake surface is exposed to an atmosphere where the temperature is `lt 0^(@)C`. If the thickness of the ice layer formed on the surface grows form `2cm` to `4cm` in `1` hour. The atmospheric temperature, `T_(a)` will be- (Thermal conductivity of ice `K = 4 xx 10^(-3) cal//cm//s//.^(@)C`, density of ice `= 0.9 gm//ice.` Latent heat of fustion of ice `= 80 cal//m`. Neglect the change of density during the state change. Assume that the water below the ice has `0^(@)` temperature every where)

A

`-20 .^(@)C`

B

`0.^(@)C`

C

`-30.^(@)C`

D

`15.^(@)C`

Text Solution

Verified by Experts

The correct Answer is:
C
The rate of heat flow through the layer of the ice,

`i_(H) = (0-(-theta))/((y//KA)) = (theta KA)/(y) = (dQ)/(dt) "…."(i)`
Also, rate of heat gain during fusion of ice,
`(dQ)/(dt) = L(dm)/(dt) = I.rho.A(dy)/(dt) "……"(ii)`
From equation (i) and (ii), we get
`(KAtheta)/(y) = rhoAL = (dy)/(dt)`
`overset(4)underset(2)(int) ydy = underset(0)overset(3600)(int) ((Ktheta)/(rhoL))dt`
`[(y^(2))/(2)]_(2)^(4) = (Ktheta)/(rhoL)[t]_(0)^(3600)`
`rArr 1/2 xx p[16xx 4] = (4 xx 10^(-3) xx theta xx (3600 - 0))/(0.9 xx 80)`
`rArr theta = 1/2 xx (12 xx 0.9 xx 80)/(4 xx 3600 xx 10^(-3)) = 30 .^(@)C`
`:.` The atmospheric temperature `= - theta = - 30 .^(@)C`.
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