A tank of water has been outdoors in cold weather, and a slab of ice 5.8 cm thick has formed on its surface (Fig. 18-46). The air above the ice is at -14°C. Calculate the rate of formation of ice (in centimeters per hour) on the ice slab. Take the thermal conductivity of ice to be 0.0040 cal/s·cm·C°, its density to be 0.92 g/cm3, and its latent heat of fusion to be 333 kJ/kg. Assume no energy transfer through the tank walls or bottom.
To calculate the rate of formation of ice on the slab, we need to determine the heat transfer through the slab of ice. This can be done using the principles of heat conduction.
The rate of heat transfer (Q) through a material can be calculated using the equation:
Q = k * A * ΔT / d
where:
- Q is the rate of heat transfer (in calories per second or watts),
- k is the thermal conductivity of the material (in cal/s·cm·°C),
- A is the cross-sectional area of the material perpendicular to the direction of heat transfer (in cm^2),
- ΔT is the temperature difference across the material (in °C),
- d is the thickness of the material (in cm).
In our case, the slab of ice is growing from the bottom up, so the temperature difference ΔT will be between the air temperature (-14°C) and the freezing point of water (0°C). Therefore, ΔT = 0 - (-14) = 14°C.
The cross-sectional area A can be calculated by multiplying the length and width of the ice slab (assumed to be the same as the tank dimensions). However, since this information is not provided, we cannot calculate A directly.
The last piece of information needed is the density of ice (ρ), which is given as 0.92 g/cm^3.
Now, let's calculate the rate of formation of ice:
1. Convert the thickness of the ice slab from centimeters to meters:
Thickness (m) = 5.8 cm / 100 = 0.058 m
2. Convert the density of ice from grams per cubic centimeter to kilograms per cubic meter:
Density (kg/m^3) = 0.92 g/cm^3 * 1000 kg/g = 920 kg/m^3
3. Calculate the mass of the ice slab:
Mass (kg) = Density (kg/m^3) * Volume (m^3)
= 920 kg/m^3 * A * 0.058 m
4. Calculate the latent heat of fusion (L) required to freeze the mass of ice:
L = Mass (kg) * Latent Heat of Fusion (J/kg)
= 920 kg * 333,000 J/kg
5. Calculate the rate of formation of ice:
Rate of Formation of Ice (m/s) = Q / L
= (k * A * ΔT / d) / (Mass (kg) * Latent Heat of Fusion (J/kg))
To convert the rate of formation of ice from meters per second to centimeters per hour, multiply the result by 100 and then by 3600:
Rate of Formation of Ice (cm/h) = Rate of Formation of Ice (m/s) * 100 * 3600
Unfortunately, without the dimensions of the ice slab (length and width), we cannot calculate the cross-sectional area A and, consequently, the rate of formation of ice.