Question

Consider a 1000 -W iron whose base plate is made of \(0.5-\mathrm{cm}-\) thick aluminum alloy 2024-T6 \(\left(\rho=2770 \mathrm{kg} / \mathrm{m}^{3}\right.\) and \(\left.c_{p}=875 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right) .\) 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 \(22^{\circ} \mathrm{C}\). Assuming 90 percent of the heat generated in the resistance wires is transferred to the plate, determine the minimum time needed for the plate temperature to reach \(200^{\circ} \mathrm{C}\).

Short answer

The density of the aluminium alloy is 2770 kg/m³, and the base plate has a thickness of 0.5 cm and a surface area of 0.03 m². Answer: It takes approximately 81 seconds for the base plate to heat up from 22 degrees Celsius to 200 degrees Celsius.

Step-by-step solution

Calculate the Mass of the Base Plate

Given that the base plate is made of aluminium alloy 2024-T6 with density (ρ) of 2770 kg/m³, thickness of 0.5 cm, and surface area of 0.03 m², we can calculate the mass of the base plate. First, convert the thickness from cm to m, giving 0.005 m. Then apply the formula for the volume of the plate which is Area x Thickness, so it's 0.03 m² x 0.005 m = 0.00015 m³. Once we have the volume, we can calculate the mass by multiplying the volume by the density: 0.00015 m³ x 2770 kg/m³ = 0.4155 kg.

Compute the Heat Transferred to the Plate

Now calculate the heat transferred to the plate, 90% of the heat generated by the iron. This total heat production is given as the power output (in Watts), which is equivalent to Joules per second (J/s). Given that the iron has a power output of 1000W, the heat transferred to the plate each second is (1000 J/s) x 0.9 = 900 J/s.

Calculate the Required Temperature Increase

The initial temperature of the plate is 22 degrees Celsius and it needs to be heated up to 200 degrees Celsius. This gives a temperature change (ΔT) of 200 - 22 = 178 degrees Celsius.

Use the Heat Capacity Formula to Find the Time Required

The heat capacity formula Q=mcΔT allows us to calculate the time required. We want to find out how long it takes for the required heat to be transferred, given that each second the plate receives 900 Joules. Rearranging the formula, Q/t=mcΔT gives t = (m*c*ΔT) / (Q/t). Substituting the known values, we have t = (0.4155 kg * 875 J/kg°C * 178°C) / 900 J/s = 81.082 seconds. So, the plate will reach 200 degrees Celsius in approximately 81 seconds, assuming the efficiency of heat transfer remains constant.

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in physics that involves the movement of energy from one object, area, or medium to another as heat. It plays a crucial role in thermal processes and is essential for understanding how objects come to a state of thermal equilibrium. There are three main modes of heat transfer: conduction, convection, and radiation.

In the context of our exercise, the heat generated by the iron's resistance wires is primarily transferred to the base plate via conduction, where the kinetic energy is passed between particles through direct contact. The problem assumes that 90 percent of this generated heat successfully reaches the plate, which implies efficient conduction. Convection and radiation may also play a part in heat transfer but to a lesser extent in this situation. It's important to note that the efficiency of heat transfer often determines how fast an object, like the iron in our exercise, will reach a desired temperature.

Specific Heat Capacity

The specific heat capacity, often denoted as 'c_p', is an intrinsic property of materials that measures the amount of heat required to raise the temperature of one kilogram of the substance by one degree Celsius. Different substances have different specific heat capacities, meaning they require varying amounts of energy to change their temperatures.

In our exercise, aluminum alloy 2024-T6 has a specific heat capacity of 875 J/kg°C. This value allows us to calculate the total energy needed to achieve the temperature rise from an ambient 22°C to 200°C. Using the formula \( Q = mc\Delta T \), where \( m \) is mass and \( \Delta T \) is the change in temperature, we assess the energy transfer and thermal behavior of the iron's base plate. If the specific heat capacity were lower, the plate would need less energy to reach the same temperature, and vice versa.

Thermal Analysis

Thermal analysis encompasses a range of techniques and calculations used to determine how materials and systems respond to changes in temperature. In a practical sense, thermal analysis helps us predict and quantify heat transfer, energy usage, and temperature changes within a system. It's a crucial part of design and functionality in many engineering disciplines.

In the exercise, performing thermal analysis allowed us to calculate the minimum time needed for the iron's base plate to reach a specified temperature. By considering factors like the specific heat capacity of the material, the mass of the iron's base plate, and the efficiency of heat transferred from the resistance wires, we could solve for the time needed for the plate to heat up. Thermal analysis isn't just academic; it's applied in manufacturing, quality control, and even safety, ensuring that devices like irons perform as intended without causing harm or inefficiency.