Question

A 30 -cm-long cylindrical aluminum block $\left(\rho=2702 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=0.896 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=236 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\(, and \)\left.\alpha=9.75 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right), 15 \mathrm{~cm}$ in diameter, is initially at a uniform temperature of \(20^{\circ} \mathrm{C}\). The block is to be heated in a furnace at \(1200^{\circ} \mathrm{C}\) until its center temperature rises to \(300^{\circ} \mathrm{C}\). If the heat transfer coefficient on all surfaces of the block is $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine how long the block should be kept in the furnace. Also, determine the amount of heat transfer from the aluminum block if it is allowed to cool in the room until its temperature drops to $20^{\circ} \mathrm{C}$ throughout. Solve this problem using the analytical one-term approximation method.

Short answer

Question: Calculate the time required for the center temperature of an aluminum block to rise to 300°C when it is heated in a furnace at 1200°C. Also, determine the amount of heat transfer when the block is allowed to cool down again. The properties of the aluminum block and its surroundings are given. Use the analytical one-term approximation method. Answer: The time required for the center temperature of the aluminum block to rise to 300°C is _____ seconds, and the amount of heat transferred during the cooling process is _____ Joules.

Step-by-step solution

List the given information

We have the following given information about the block and its surroundings: - Aluminum block properties: - Length (L) = 30 cm - Diameter (D) = 15 cm - Density (ρ) = 2702 kg/m³ - Specific heat (c_p) = 0.896 kJ/kg·K - Thermal conductivity (k) = 236 W/m·K - Alpha (α) = 9.75 x 10⁻⁵ m²/s - Initial temperature (T_initial) = 20°C - Furnace temperature (T_furnace) = 1200°C - Required center temperature (T_center) = 300°C - Heat transfer coefficient (h) = 80 W/m²·K Since our answers need to be in seconds and Joules, we'll also convert some given units. c_p = 896 J/kg·K

Calculate the Bi number and find the approximation for the given scenario

Using the analytical one-term approximation method, calculate the Biot number to decide the suitable estimation. Bi = (h * L) / k Now, if Bi < 0.1, we can use the one-term approximation.

Calculate the time required for the center temperature to rise to 300°C

Using analytical one-term approximation, the time required for the center of the block to reach the target temperature can be calculated using the following equation: t = (L² * θ) / (π² * α) where θ = [(T_center - T_initial) / (T_furnace - T_initial)]

Calculate the heat transferred during the cooling process

When the aluminum block is cooled to the room temperature, we need to find the amount of heat transferred. We'll use the following equation: Q = m * c_p * (T_center - T_initial) First, find the mass of the block using its volume and density: m = ρ * V Then, calculate the heat transferred (Q) using the mass and the difference in temperature.

Summarize the results

Summarize the found time required for the center of the block to reach the target temperature, and the amount of heat transferred during the cooling process.