During a cold winter, the surface of a river develops a layer of ice of unknown

Question

During a cold winter, the surface of a river develops a layer of ice of unknown thickness L. Known are the water temperature Tw = 4 degreeree C, the atmospheric air temperature Ta= - 30 degreeree C, and the temperature of the underside of the ice layer T0= 0 degreeree C. The thermal conductivity of ice is k = 1.88 W/m-K. The convective heat transfer coefficients on the water and air sides of the ice layer are hw =100 W/m2-K and ha = 50 W/m2-K, respectively. Calculate the temperature of the upper surface of the ice layer, T and the ice thickness L.

Explanation / Answer

equate the heat flow at all the interfaces ,the rate of heat flow remains constant at the steady state.

for the water ice interface = >
convective heat transfer = conductive heat transfer
hw x A x T = KAT/x

considering the heat transfer per unit area

A= 1

hw x (4 - Ti) = K x (Ti - 0)/L

100 x (4 - Ti ) = 1.88 Ti/L

400L - 100Ti x L = 1.88 Ti

Ti = 400L/(100L + 1.88) ...................... 1

now,

considering the conductive heat transfer = convective heat transfer at the upper ice interface

KAT/x = ha A T

as A =1

1.88 x (Ti - o)/L = 50 x (0 - -30)

1.88 Ti/L = 1500

Ti = 797.87L .................2

but Ti = 400L/(100L + 1.88)

Ti = 400L/(100L + 1.88) = 797.87L

solving for L weget

L = .02 m

Ti = 2.06 degress

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