Question

Spherical glass beads coming out of a kiln are allowed to cool 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 \(3 \mathrm{~min}\). 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 the analytical one-term approximation method. 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

Answer: The temperature at the center of the spherical glass bead after cooling for 3 minutes is 318.42°C.

Step-by-step solution

Convert dimensions and time to SI units

First, we need to convert the dimensions and time given in SI units. Diameter (d) = 10 mm = 0.01 m Cooling Time (t) = 3 min = 180 s

Calculate the Biot number (Bi)

The Biot number represents the relative importance of internal temperature gradients to external convection. It's given by the formula: Bi = hLc/k where h is the convection heat transfer coefficient, Lc is the characteristic length, and k is the thermal conductivity. For a sphere, the characteristic length (Lc) is equal to the radius (r) divided by 3. Lc = (d/2)/3 = 0.005/3 m Bi = (28 W/m²·K)(0.005/3 m)/(0.7 W/m·K) = 0.0667

Calculate the Fourier number (Fo)

The Fourier number represents the ratio of heat conduction rate to the heat storage rate. It's given by the formula: Fo = αt/Lc^2 where α is the thermal diffusivity and t is the cooling time. The thermal diffusivity (α) can be calculated as: α = k/(ρ·cp) Now, we can calculate the α using the given values: α = (0.7 W/m·K)/(2800 kg/m³ * 750 J/kg·K) = 3.3333×10⁻⁷ m²/s Next, we compute the Fourier number (Fo): Fo = (3.3333×10⁻⁷ m²/s)(180 s)/(0.005/3 m)² = 0.71998

Obtain the one-term approximation solution

For the analytical one-term approximation method to solve for the temperature at the center of the sphere (Tc), we need the equation: Tc = T∞ + (Ti - T∞) × Φ where T∞ is the room temperature, Ti is the initial temperature, and Φ is the dimensionless center temperature, which is given by: Φ = (3/2) * e^(-Bi×ξ₁²×Fo)×ξ₁*sin(ξ₁)/(ξ₁+cos(ξ₁)) The first eigenvalue, ξ₁, for a sphere can be found in heat transfer tables or through numerical methods and it is approximately 3.52. Now, we can calculate the dimensionless center temperature: Φ = (3/2) * e^(-0.0667×3.52²×0.71998)×3.52*sin(3.52)/(3.52+cos(3.52)) = 0.75284 Finally, we can find the temperature at the center of the glass bead (Tc): Tc = 30°C + (400°C - 30°C) × 0.75284 = 318.42°C

The result

The temperature at the center of the spherical glass bead after cooling for 3 minutes using the analytical one-term approximation method is 318.42°C.