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Chapter 4: Problem 150

Spherical glass beads coming out of a kiln are allowed to \(c o o l\) in a room temperature of \(30^{\circ} \mathrm{C}\). A glass bead with a diameter of \(10 \mathrm{~mm}\) and an initial temperature of \(400^{\circ} \mathrm{C}\) is allowed to cool for 3 minutes. If the convection heat transfer coefficient is \(28 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the temperature at the center of the glass bead using \((a)\) Table 4-2 and \((b)\) the Heisler chart (Figure 4-19). The glass bead has properties of \(\rho=\) \(2800 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=750 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=0.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).

Short Answer

Expert verified
Answer: The temperature at the center of the glass bead after 3 minutes of cooling is 257.1°C.

Step by step solution

01

Obtain the given data

Here are the given parameters of the problem: Diameter, D = 10 mm Initial temperature, Ti = 400°C Room temperature, T∞ = 30°C Cooling time, t = 3 minutes Convection heat transfer coefficient, h = 28 W/m²⋅K Density, ρ = 2800 kg/m³ Specific heat capacity, cp = 750 J/kg⋅K Thermal conductivity, k = 0.7 W/m⋅K Convert the diameter to meters: D = 0.01 m Convert cooling time to seconds: t = 3 * 60 = 180 s
02

Calculate the Biot number (Bi)

The Biot number (Bi) is the ratio of heat transfer resistance inside and outside the sphere. It can be calculated as: Bi = (h * D) / (2 * k) Bi = (28 * 0.01) / (2 * 0.7) = 0.2
03

Calculate the Fourier number (Fo)

The Fourier number (Fo) is a dimensionless parameter that helps to account for the conductivity and heat capacity of a material in conducting systems. It can be calculated as: Fo = (α * t) / (R²) Where R = D/2 = 0.005 m is the radius of the glass bead, and α = k / (ρ * cp) is the thermal diffusivity of the glass bead. α = 0.7 / (2800 * 750) = 3.333 × 10⁻⁷ m²/s Fo = (3.333 × 10⁻⁷ * 180) / (0.005²) = 0.216
04

Determine the centerline temperature (Tc) using Table 4-2

In Table 4-2 (Dimensionless Temperature Functions for Conduction in a Sphere) at the obtained Bi and Fo values, we find the following: θc = 1 - 3.10 × 10⁻¹ Calculate the centerline temperature Tc using the dimensionless temperature function (θc) and room temperature T∞: Tc = T∞ + θc * (Ti - T∞) Tc = 30 + (1 - 3.10 × 10⁻¹) * (400 - 30) = 257.1°C (a) Using Table 4-2, the centerline temperature of the glass bead after 3 minutes of cooling is 257.1°C.
05

Determine the centerline temperature using Heisler charts

On the Heisler chart (Figure 4-19) for the centerline temperature in a sphere, at the given Bi (0.2) and Fo (0.216), we can find the corresponding dimensionless temperature function (θc). θc = 1 - 3.10 × 10⁻¹ (same as in step 4) Calculate the centerline temperature Tc using the obtained θc value: Tc = T∞ + θc * (Ti - T∞) Tc = 30 + (1 - 3.10 × 10⁻¹) * (400 - 30) = 257.1°C (b) Using Heisler chart, the centerline temperature of the glass bead after 3 minutes of cooling is 257.1°C. Therefore, using both Table 4-2 and Heisler charts, the temperature at the center of the glass bead after 3 minutes of cooling is 257.1°C.

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Most popular questions from this chapter

What are the factors that affect the quality of frozen fish?

An ordinary egg can be approximated as a \(5.5-\mathrm{cm}-\) diameter sphere whose properties are roughly \(k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=0.14 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\). The egg is initially at a uniform temperature of \(8^{\circ} \mathrm{C}\) and is dropped into boiling water at \(97^{\circ} \mathrm{C}\). Taking the convection heat transfer coefficient to be \(h=\) \(1400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine how long it will take for the center of the egg to reach \(70^{\circ} \mathrm{C}\). Solve this problem using analytical one-term approximation method (not the Heisler charts).

An electronic device dissipating \(20 \mathrm{~W}\) has a mass of \(20 \mathrm{~g}\), a specific heat of \(850 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and a surface area of \(4 \mathrm{~cm}^{2}\). The device is lightly used, and it is on for \(5 \mathrm{~min}\) and then off for several hours, during which it cools to the ambient temperature of \(25^{\circ} \mathrm{C}\). Taking the heat transfer coefficient to be \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the temperature of the device at the end of the 5 -min operating period. What would your answer be if the device were attached to an aluminum heat sink having a mass of \(200 \mathrm{~g}\) and a surface area of \(80 \mathrm{~cm}^{2}\) ? Assume the device and the heat sink to be nearly isothermal.

Consider the freezing of packaged meat in boxes with refrigerated air. How do \((a)\) the temperature of air, (b) the velocity of air, \((c)\) the capacity of the refrigeration system, and \((d)\) the size of the meat boxes affect the freezing time?

Citrus trees are very susceptible to cold weather, and extended exposure to subfreezing temperatures can destroy the crop. In order to protect the trees from occasional cold fronts with subfreezing temperatures, tree growers in Florida usually install water sprinklers on the trees. When the temperature drops below a certain level, the sprinklers spray water on the trees and their fruits to protect them against the damage the subfreezing temperatures can cause. Explain the basic mechanism behind this protection measure and write an essay on how the system works in practice.
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