Question

In a production facility, 1.6 -in-thick \(2-\mathrm{ft} \times 2\) -ft square brass plates \(\left(\rho=532.5 \mathrm{lbm} / \mathrm{ft}^{3} \text { and } c_{p}=0.091 \mathrm{Btu} / \mathrm{lbm} \cdot^{\circ} \mathrm{F}\right)\) that are initially at a uniform temperature of \(75^{\circ} \mathrm{F}\) are heated by passing them through an oven at \(1500^{\circ} \mathrm{F}\) at a rate of 300 per minute. If the plates remain in the oven until their average temperature rises to \(900^{\circ} \mathrm{F}\), determine the rate of heat transfer to the plates in the furnace.

Short answer

Answer: The rate of heat transfer to the brass plates in the furnace is 6,424,140 Btu/min.

Step-by-step solution

Calculate mass of a brass plate

First, we need to calculate the mass of a single brass plate. Since the plate has a volume, we can use the density formula: Mass = Density × Volume The volume of a square brass plate can be calculated using the formula: Volume = width × height × thickness For our brass plate: width = 2 ft, height = 2 ft, and thickness = 1.6 in (converted to ft: 1.6 in / 12 = 0.1333 ft). Volume = (2 ft) × (2 ft) × (0.1333 ft) = 0.5326 ft^3 Now we can calculate the mass using the density of the brass: Density = 532.5 lbm/ft^3 Mass = Density × Volume = (532.5 lbm/ft^3) × (0.5326 ft^3) = 283.37 lbm

Calculate heat transfer to a single plate

Next, we will calculate the heat transfer to a single plate using the equation: Q = mc_pΔT Where Q is the heat transfer, m is the mass of the plate, c_p is the specific heat at constant pressure, and ΔT is the change in temperature. In our problem: - mass (m) = 283.37 lbm - specific heat (c_p) = 0.091 Btu/lbm·°F - initial temperature (T_initial) = 75 °F - final temperature (T_final) = 900 °F First, we calculate the change in temperature (ΔT): ΔT = T_final - T_initial = 900 °F - 75 °F = 825 °F Now we can calculate the heat transfer (Q) to a single plate: Q = (283.37 lbm) × (0.091 Btu/lbm·°F) × (825 °F) = 21413.8 Btu

Calculate the rate of heat transfer

Finally, we need to calculate the rate of heat transfer to the plates in the furnace. Since there are 300 plates passing through the oven per minute, we can find the total rate of heat transfer by multiplying the heat transfer for one plate (Q) by the rate of 300 plates/minute: Rate of heat transfer = Q × Rate of plates = (21413.8 Btu) × (300 plates/min) = 6424140 Btu/min So, the rate of heat transfer to the brass plates in the furnace is 6,424,140 Btu/min.

Key concepts

Specific Heat

The concept of specific heat is critical when calculating the amount of thermal energy, or heat, required to change the temperature of a material. Specific heat, symbolized as c_p, is the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius or one degree Fahrenheit. It is expressed in units such as Btu/(lbm·°F) or J/(kg·°C).

Every material has a unique specific heat value, which reflects how sensitively it reacts to the addition or removal of heat. In the case of our brass plates with a specific heat of 0.091 Btu/(lbm·°F), it suggests that brass does not require much thermal energy to change its temperature compared to materials with higher specific heat values.

To improve understanding of this concept, one might consider how different materials respond to heat. For example, water, with a high specific heat, takes longer to heat up and cool down, while metals like brass, with lower specific heats, will do so more quickly. This is why a metal bench feels much colder on a cool day compared to a wooden one, even though both are technically at the same temperature.

Thermal Energy Transfer

Thermal energy transfer refers to the movement of heat from one object or substance to another. This transfer can occur through different processes: conduction, convection, and radiation. Conduction happens when heat moves through a solid material, like our brass plates in the oven. Convection occurs when heat is transferred through fluids (liquids or gases) that are moving, and radiation is heat transfer through electromagnetic waves.

Understanding the rate of thermal energy transfer is essential in many scientific and engineering applications, such as the operation of the furnace in our problem. Mathematically, the equation Q = mc_pΔT encapsulates how the thermal energy transfer (Q) can be calculated when the mass (m), specific heat (c_p), and temperature change (ΔT) are known.

Real-world applications

Calculating thermal energy transfer is not only integral in industrial processes like heating brass plates but is also crucial in areas like climate control within buildings, efficiency of thermal machines, and even in culinary arts where understanding heat transfer can mean the difference between a well-cooked meal and an under or overcooked one.

Mass and Temperature Change

The concepts of mass and temperature change are deeply intertwined in thermal calculations. Mass refers to the amount of matter in an object and is typically measured in kilograms (kg) or pounds mass (lbm). When heat is added to or removed from an object, the temperature change is a direct result of that heat transfer.

In the context of our brass plate problem, the mass of each plate influences how much heat is needed to achieve the desired temperature change. In our example, a 283.37 lbm mass will require a significant amount of heat to increase in temperature substantially. Conversely, if the mass were lower, less thermal energy would be needed for the same temperature change.

To facilitate better comprehension, one could envisage temperature change like a journey where the mass is the vehicle's size. A larger vehicle (greater mass) needs more fuel (thermal energy) to complete a journey (temperature change), while a smaller vehicle would require less. Delving into this analogy helps illustrate the proportional relationship between the amount of material and the energy necessary to alter its temperature.