Nikodemus s S. answered • 03/18/23
What thickness of ice will form on the surface of the lake in time t if the temperature of the enviroment is TA is constant and the surface temperature of the lake is equal to the temperature of the enviroment. The water temperature is T0. The coefficient of thermal conductivity of ice is K, the specific heat of fusion is λ, and the density is ρ.
The thickness of ice that will form on the surface of the lake in time t can be calculated using the following formula:
d = [(2K(tλ))/ρL]^(1/2)
where:
d is the thickness of the ice
K is the coefficient of thermal conductivity of ice
t is the time
λ is the specific heat of fusion
ρ is the density of ice
L is the latent heat of fusion, which is equal to λ
In this scenario, the temperature of the environment and the surface temperature of the lake are equal, which means that the heat transfer from the lake to the environment will be constant. Therefore, the rate of ice formation will also be constant.
Assuming that the initial temperature of the lake is also equal to TA, we can calculate the amount of heat that needs to be removed from the water to freeze it into ice:
Q = ρV0(T0-TA)
where:
Q is the amount of heat required to freeze the water
V0 is the volume of water to be frozen
Once we know the amount of heat that needs to be removed, we can calculate the time required to freeze the water:
t = Q/(2Kd)
Combining the two equations, we get:
d = [(2KλρV0)/(ρL)(T0-TA)]^(1/2)
Substituting this into the equation for t, we get:
t = [(ρL)(T0-TA)V0]/[2Kλ]
Therefore, the thickness of ice that will form on the surface of the lake in time t is:
d = [(2KλρV0)/(ρL)(T0-TA)]^(1/2)
and the time required to form this thickness of ice is:
t = [(ρL)(T0-TA)V0]/[2Kλ]
Note that this calculation assumes that the water temperature remains constant at T0 during the freezing process. If the water temperature drops during the process, the calculation will need to be modified accordingly.
