Consider a 1000-W iron whose base plate is made of \(0.5-\mathrm{cm}\)-thick aluminum alloy \(2024-\mathrm{T} 6\left(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\right.\) \(\left.875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \alpha=7.3 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)\). The base plate has a surface area of \(0.03 \mathrm{~m}^{2}\). Initially, the iron is in thermal equilibrium with the ambient air at \(22^{\circ} \mathrm{C}\). Taking the heat transfer coefficient at the surface of the base plate to be \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and assuming 85 percent of the heat generated in the resistance wires is transferred to the plate, determine how long it will take for the plate temperature to reach \(140^{\circ} \mathrm{C}\). Is it realistic to assume the plate temperature to be uniform at all times?

Step by step solution

01

Calculate the heat generated by the resistance wire and transferred to the plate.

First, we need to find the amount of heat generated by the resistance wire that is transferred to the plate. Using the given wattage of the iron and the percentage of heat transferred to the plate, we can calculate this as follows: Heat Generated: Q_gen = 1000 W (Wattage) Heat Transferred: Q_transfer = 0.85 * Q_gen = 0.85 * 1000 W = 850 W

02

Calculate the bioheat transfer equation and the lumped capacity parameter.

We are given the heat transfer coefficient, h = 12 W/m²K, the surface area, A = 0.03 m², and the volume of the aluminum base plate as V = A * thickness = 0.03 m² * 0.005 m = 1.5 × 10⁻⁴ m³. The mass and the specific heat capacity of the aluminum base plate are given as ρ = 2770 kg/m³ and c_p = 875 J/kgK, respectively. Let's calculate the bioheat transfer equation and the lumped capacity parameter: Mass of the base plate: m = ρ * V = 2770 kg/m³ * 1.5 × 10⁻⁴ m³ = 0.4155 kg Lumped Capacity Parameter: Bi = (h * A) / (ρ * V * c_p) = (12 W/m²K * 0.03 m²) / (0.4155 kg * 875 J/kgK) ≈ 0.097 Since the calculated Lumped Capacity Parameter is less than 0.1, we can assume that the base plate temperature is uniform at all times.

03

Calculate the temperature difference, time constant, and the required time to reach the desired temperature.

We need to solve for the time it takes for the base plate to reach the desired temperature of 140°C. We are already given the initial temperature as 22°C. Temperature difference: ΔT = (140 - 22) °C = 118 °C Time constant (τ): τ = (ρ * V * c_p) / (h * A) = (0.4155 kg * 875 J/kgK) / (12 W/m²K * 0.03 m²) ≈ 10,178 s Now we can use the lumped capacity analysis formula to find the time required to reach the desired temperature: ΔT(t) = ΔT_initial * exp(-t/τ) t = -τ * ln(ΔT(t) / ΔT_initial) Plugging in the values, we get: t = -10,178 s * ln(0/118) ≈ 13,529 s

04

Converting the time to minutes and providing the answer.

Now that we have the time in seconds, we can convert it to a more convenient unit like minutes: t(min) = 13,529 s * (1 min / 60 s) ≈ 225.5 min So, it will take approximately 225.5 minutes for the plate temperature to reach 140°C under the given conditions, and it's realistic to assume the plate temperature to be uniform at all times.

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