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Chapter 1: Problem 145

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

Expert verified
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

01

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
02

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
03

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
04

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)
05

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
06

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
07

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
08

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.

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Most popular questions from this chapter

A hollow spherical iron container with outer diameter \(20 \mathrm{~cm}\) and thickness \(0.2 \mathrm{~cm}\) is filled with iced water at \(0^{\circ} \mathrm{C}\). If the outer surface temperature is \(5^{\circ} \mathrm{C}\), determine the approximate rate of heat loss from the sphere, in \(\mathrm{kW}\), and the rate at which ice melts in the container. The heat of fusion of water is \(333.7 \mathrm{~kJ} / \mathrm{kg}\).

While driving down a highway early in the evening, the air flow over an automobile establishes an overall heat transfer coefficient of \(18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The passenger cabin of this automobile exposes \(9 \mathrm{~m}^{2}\) of surface to the moving ambient air. On a day when the ambient temperature is \(33^{\circ} \mathrm{C}\), how much cooling must the air conditioning system supply to maintain a temperature of \(20^{\circ} \mathrm{C}\) in the passenger cabin? (a) \(670 \mathrm{~W}\) (b) \(1284 \mathrm{~W}\) (c) \(2106 \mathrm{~W}\) (d) \(2565 \mathrm{~W}\) (e) \(3210 \mathrm{~W}\)

Four power transistors, each dissipating \(12 \mathrm{~W}\), are mounted on a thin vertical aluminum plate \(22 \mathrm{~cm} \times 22 \mathrm{~cm}\) in size. The heat generated by the transistors is to be dissipated by both surfaces of the plate to the surrounding air at \(25^{\circ} \mathrm{C}\), which is blown over the plate by a fan. The entire plate can be assumed to be nearly isothermal, and the exposed surface area of the transistor can be taken to be equal to its base area. If the average convection heat transfer coefficient is \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the temperature of the aluminum plate. Disregard any radiation effects.

Determine a positive real root of this equation using \(E E S\) : $$ 3.5 x^{3}-10 x^{0.5}-3 x=-4 $$

An electronic package in the shape of a sphere with an outer diameter of \(100 \mathrm{~mm}\) is placed in a large laboratory room. The surface emissivity of the package can assume three different values \((0.2,0.25\), and \(0.3)\). The walls of the room are maintained at a constant temperature of \(77 \mathrm{~K}\). The electronics in this package can only operate in the surface temperature range of \(40^{\circ} \mathrm{C} \leq T_{s} \leq 85^{\circ} \mathrm{C}\). Determine the range of power dissipation \((\dot{W})\) for the electronic package over this temperature range for the three surface emissivity values \((\varepsilon)\). Plot the results in terms of \(\dot{W}(\mathrm{~W})\) vs. \(T_{s}\left({ }^{\circ} \mathrm{C}\right)\) for the three different values of emissivity over a surface temperature range of 40 to \(85^{\circ} \mathrm{C}\) with temperature increments of \(5^{\circ} \mathrm{C}\) (total of 10 data points for each \(\varepsilon\) value). Provide a computer generated graph for the display of your results and tabulate the data used for the graph. Comment on the results obtained.
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