During a winter day, the atmospheric temperature falls to -10^ ^ ^ C , and ice begins to form on the surface of a lake. Determine the rate of increase of the ice thickness when 10 cm of ice has already formed. Assume

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Quantrex Academy · Accepted Answer

Correct answer: D. 5.06 × 10^ -7 m s^ -1

The rate of heat conduction through the ice layer is given by:

\dfrac{dQ}{dt} = \dfrac{KA(T_w - T_a)}{x}

The rate of heat released due to the formation of ice of thickness

dx

is:

\dfrac{dQ}{dt} = \rho A L \dfrac{dx}{dt}

Equating the two rates of heat transfer:

\rho A L \dfrac{dx}{dt} = \dfrac{KA(T_w - T_a)}{x}

\dfrac{dx}{dt} = \dfrac{K(T_w - T_a)}{\rho L x}

Substituting the given values,

K = 1.7 \text{ W m}^{-1} ^{}^{{}^{\circ}} \textC^{-1}

,

T_w = 0^{}^{{}^{\circ}} \text{C}

,

T_a = -10^{}^{{}^{\circ}} \text{C}

,

\rho = 1000 \text{ kg m}^{-3}

,

L = 3.36 \times 10^5 \text{ J kg}^{-1}

, and

x = 0.1 \text{ m}

:

\dfrac{dx}{dt} = \dfrac{1.7 \times (0 - (-10))}{1000 \times 3.36 \times 10^5 \times 0.1}

\dfrac{dx}{dt} = \dfrac{17}{3.36 \times 10^7}

\dfrac{dx}{dt} \approx 5.06 \times 10^{-7} \text{ m s}^{-1}

During a winter day, the atmospheric temperature falls to -10^ ^ ^...

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