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.
Rate of formation of ice = (0.0040 cal/s·cm·C° x 333 kJ/kg x 0.92 g/cm3) / (-14°C) = 0.037 cm/hour
To calculate the rate of formation of ice on the slab, we need to use the formula:
Rate of formation of ice = (Latent heat of fusion)/(Thermal conductivity * density * temperature difference)
First, let's convert the temperature difference from Celsius to Kelvin:
-14°C + 273.15 = 259.15 K
Next, let's convert the thermal conductivity from cal/s·cm·C° to J/s·m·K:
0.0040 cal/s·cm·C° = 0.0040 * 4.1868 J/s·m·K = 0.016744 J/s·m·C°
Now, let's convert the density from g/cm3 to kg/m3:
0.92 g/cm3 = 0.92 * 1000 kg/m3 = 920 kg/m3
Now, let's calculate the rate of formation of ice:
Rate of formation of ice = (333 * 1000)/(0.016744 * 920 * 259.15)
= 3.33 * 10^8 / (0.016744 * 920 * 259.15)
≈ 148.13 cm/hour
Therefore, the rate of formation of ice on the ice slab is approximately 148.13 centimeters per hour.
To calculate the rate of formation of ice, we need to determine the heat flow through the ice slab.
The heat flow can be calculated using the formula:
Q = k * A * (ΔT / Δx)
Where:
Q is the heat flow (in calories per second)
k is the thermal conductivity of ice (0.0040 cal/s·cm·C°)
A is the surface area of the ice slab (in cm^2)
ΔT is the temperature difference between the air and ice (in degrees Celsius)
Δx is the thickness of the ice slab (in cm)
First, we need to convert the thickness of the ice slab to meters (0.058 cm = 0.058 m) and calculate the surface area of the ice slab.
The surface area (A) can be calculated as:
A = L * W
Since we don't have the dimensions of the ice slab, let's assume it has a square shape, and both the length (L) and width (W) are equal.
Using simple geometry, we can say L = W = √A
Now, let's determine the temperature difference (ΔT) between the air and ice. Since the air temperature is given as -14°C, and assuming the ice slab's temperature to be 0°C, ΔT = -14°C - 0°C = -14°C.
Now, we can substitute all the values into the formula:
Q = (0.0040 cal/s·cm·C°) * A * (-14°C / 0.058 m)
Next, we need to convert the heat flow from calories per second to joules per second. 1 calorie is equal to 4.184 joules.
So, Q = (0.0040 cal/s·cm·C°) * A * (-14°C / 0.058 m) * 4.184 J/cal
Finally, let's convert the units of the rate of ice formation from m/s to cm/hour:
Rate of ice formation = Δx / Δt
Since we have 1 meter (100 cm) of ice formation in Δt seconds, we can convert it to cm/hour by multiplying by 3600 (60 minutes * 60 seconds).
Rate of ice formation (cm/hour) = (0.058 m / Δt) * 3600 cm/hour
Now, to determine the value of Δt, we need to know how long it took to form the 5.8 cm slab of ice.
Based on the information given, we can't calculate the exact rate of ice formation without knowing the time it took to form the slab.