Question

An ice skating rink is located in a building where the air is at \(T_{\text {air }}=20^{\circ} \mathrm{C}\) and the walls are at \(T_{w}=25^{\circ} \mathrm{C}\). The convection heat transfer coefficient between the ice and the surrounding air is \(h=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The emissivity of ice is \(\varepsilon=0.95\). The latent heat of fusion of ice is \(h_{i f}=333.7 \mathrm{~kJ} / \mathrm{kg}\) and its density is \(920 \mathrm{~kg} / \mathrm{m}^{3}\). (a) Calculate the refrigeration load of the system necessary to maintain the ice at \(T_{s}=0^{\circ} \mathrm{C}\) for an ice rink of \(12 \mathrm{~m}\) by \(40 \mathrm{~m}\). (b) How long would it take to melt \(\delta=3 \mathrm{~mm}\) of ice from the surface of the rink if no cooling is supplied and the surface is considered insulated on the back side?

Short answer

Answer: The refrigeration load necessary to maintain the ice at 0°C is approximately 100.235 kW, and it would take approximately 1.22 hours to melt a 3mm layer of ice from the surface of the rink if no cooling is supplied.

Step-by-step solution

Determine the area of the ice rink

Calculate the area of the ice rink with the given dimensions: 12m by 40m. A = length × width A = 12m × 40m A = 480 m^2

Calculate convection heat transfer rate

Calculate the convection heat transfer rate between the ice rink and the surrounding air using the given convection heat transfer coefficient (h) and the temperature difference between the ice and the air (T_air - T_s). q_conv = h × A × (T_air - T_s) q_conv = 10 W/m^2K × 480 m^2 × (20°C - 0°C) q_conv = 96,000 W

Calculate radiation heat transfer rate

Calculate the radiation heat transfer rate between the ice rink and the surrounding walls using the given emissivity (ε), the Stefan-Boltzmann constant (σ = 5.67 × 10^-8 W/m^2K^4), and the temperature difference between the walls and the ice (T_w - T_s). q_rad = ε × σ × A × (T_w^4 - T_s^4) q_rad = 0.95 × 5.67 × 10^-8 W/m^2K^4 × 480 m^2 × ((25+273)^4 - (0+273)^4) q_rad ≈ 4235 W

Calculate total heat transfer rate and refrigeration load

Calculate the total heat transfer rate between the ice rink and the surrounding environment by summing the convection and radiation heat transfer rates (q_total = q_conv + q_rad). The refrigeration load is equal to the total heat transfer rate. q_total = q_conv + q_rad q_total = 96,000 W + 4235 W q_total ≈ 100,235 W (or 100.235 kW)

Calculate the volume of ice to be melted

Calculate the volume of ice to be melted (3mm layer) using the given area of the ice rink and thickness of the ice layer (δ = 3mm or 0.003m). V = A × δ V = 480 m^2 × 0.003m V = 1.44 m^3

Calculate the mass of ice to be melted

Calculate the mass of ice to be melted using the given density of ice (ρ = 920 kg/m^3) and the volume calculated in Step 5. m = ρ × V m = 920 kg/m^3 × 1.44 m^3 m = 1324.8 kg

Calculate the energy needed to melt the ice

Calculate the energy required to melt the ice using the given latent heat of fusion of ice (h_if = 333.7 kJ/kg) and the mass of ice calculated in Step 6. E = m × h_if E = 1324.8 kg × 333.7 kJ/kg E = 441940.56 kJ

Calculate the time required to melt the ice

Calculate the time required to melt the ice by dividing the total energy needed to melt the ice by the refrigeration load (assuming no cooling is supplied, so the heat transfer rate remains constant). Convert the time to hours. t = E / q_total t = 441940.56 kJ / 100,235 J/s t ≈ 4405.2 s t ≈ 1.22 hours The refrigeration load necessary to maintain the ice at 0°C is approximately 100.235 kW, and it would take approximately 1.22 hours to melt a 3mm layer of ice from the surface of the rink if no cooling is supplied.

Key concepts

Convection Heat Transfer

Convection heat transfer is the process of heat transfer between a solid surface and a fluid that is in motion over it. In the case of an ice rink, the solid surface is the ice and the fluid is the air above it. The efficiency of this transfer depends greatly on the convection heat transfer coefficient, often denoted as \(h\).

To better understand, think of \(h\) as a measure of how easily heat is transported from the ice to the air. The heat transfer rate, \(q_{\text{conv}}\), can be calculated using the formula:

• \(q_{\text{conv}} = h \times A \times (T_{\text{air}} - T_{\text{s}})\)
• A = area of the ice rink
• \(T_{\text{air}} = \) air temperature
• \(T_{\text{s}} = \) ice temperature

This equation highlights the importance of both the temperature difference and the surface area of contact. Simply put, a larger temperature difference or a larger area results in more heat being transferred. In this exercise, the calculated convection heat transfer was 96,000 W, contributing to the overall refrigeration load.

Radiation Heat Transfer

Radiation heat transfer is the energy transfer that occurs through electromagnetic waves. It does not necessitate a medium, meaning it can operate through a vacuum, but in this situation, it occurs between the surfaces of the ice rink and the walls surrounding it.

The efficiency of radiation heat transfer is influenced by the emissivity of the materials involved. The emissivity \(\varepsilon\) measures how effectively a surface emits energy as radiation. For ice, it was given as 0.95, indicating it emits most of its energy as radiation.

The radiation heat transfer rate, \(q_{\text{rad}},\) is determined by:

• \(q_{\text{rad}} = \varepsilon \times \sigma \times A \times (T_{\text{w}}^4 - T_{\text{s}}^4)\)
• \(\sigma = \) Stefan-Boltzmann constant = \(5.67 \times 10^{-8} \text{W/m}^2\text{K}^4\)
• \(T_{\text{w}} = \) wall temperature
• \(T_{\text{s}} = \) ice temperature

This formula shows that the radiation heat transfer is strongly dependent on the temperatures of the surfaces in Kelvin, specifically the fourth power of these temperatures. In this exercise, the radiation heat transfer rate calculated was approximately 4235 W.

Latent Heat of Fusion

Latent heat of fusion is essential when it comes to phase changes, like ice turning into water. It represents the amount of energy required to change a kilogram of a substance from solid to liquid at the same temperature. For ice, this is represented as \(h_{\text{if}}\) and given as 333.7 kJ/kg.

When the ice on the rink begins to melt, heat energy is absorbed without a change in temperature until the ice becomes liquid. To determine the total energy needed to melt some ice, we use the formula:

• \(E = m \times h_{\text{if}}\)
• \(m = \) mass of the ice

This exercise calculated the energy required to melt a 3mm layer of ice, equating to about 441,940.56 kJ.

Ice Melting Time

The time it takes for ice to melt is determined by the energy available to facilitate that change in state. Ice melting time plays a crucial role in refrigeration load calculations, as it informs us of how long the system can support without additional cooling before the ice begins to disappear.

The melting time can be calculated by dividing the total energy needed to melt the ice by the heat transfer rate. The formula used in this exercise was:

• \(t = \frac{E}{q_{\text{total}}}\)
• Where \(E\) is the energy required to melt the ice
• \(q_{\text{total}} = \) total heat transfer rate

This total rate includes both convection and radiation components, showing how the environment's temperature and the system's refrigeration load affect melting. For this exercise, it turned out to be around 1.22 hours.