Question

Stainless steel ball bearings \(\left(\rho=8085 \mathrm{kg} / \mathrm{m}^{3}\right.\) and \(\left.c_{p}=0.480 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) having a diameter of \(1.2 \mathrm{cm}\) are to be quenched in water at a rate of 800 per minute. The balls leave the oven at a uniform temperature of \(900^{\circ} \mathrm{C}\) and are exposed to air at \(25^{\circ} \mathrm{C}\) for a while before they are dropped into the water. If the temperature of the balls drops to \(850^{\circ} \mathrm{C}\) prior to quenching, determine the rate of heat transfer from the balls to the air.

Short answer

Answer: The rate of heat transfer from the ball bearings to the air is 269.2 W.

Step-by-step solution

Calculate the volume and mass of a ball bearing

First, let's calculate the volume of a single ball bearing using the formula for the volume of a sphere: \(V = \frac{4}{3}\pi r^{3}\). The diameter of a ball bearing is given as \(1.2 \mathrm{cm}\), so its radius is \(r = 0.6 \mathrm{cm}\) or \(0.006\mathrm{m}\). Now, we can calculate the volume: \(V = \frac{4}{3}\pi (0.006)^{3} = 9.057\times10^{-7} \mathrm{m}^{3}\) Next, we can calculate the mass of the ball bearing using its density, ρ: \(mass = \rho \times V = 8085 \mathrm{kg/m^3} \times 9.057\times10^{-7}\mathrm{m}^{3} = 0.00732\mathrm{kg}\)

Calculate the heat transfer for one ball bearing

Now that we know the mass of a single ball bearing, we can calculate the heat transfer using the formula \(Q=mc_p\Delta T\): Since \(Q>0\), we have \(Q = (0.00732\mathrm{kg})\times(480\mathrm{J/kg\,K})\times (850-900)^\circ\mathrm{C} = -20.19\,\mathrm{J}\) The negative sign indicates heat is transferred from the ball (hotter) to the air (cooler).

Calculate the rate of heat transfer for all ball bearings

Finally, we need to determine the total rate of heat transfer for all 800 ball bearings per minute. We can do this by multiplying the heat transfer for one ball bearing by the rate of cooling (800 per minute): \(Q_{total}= 800 \times (-20.19\,\mathrm{J}) = -16152\,\mathrm{J/min}\) Since we are asked for the rate of heat transfer, we can divide by 60 to get the answer in Joules per second: \(Q_{total} = \frac{-16152\,\mathrm{J/min}}{60\,\mathrm{min/s}} = -269.2\,\mathrm{W}\) So the rate of heat transfer from the ball bearings to the air is \(269.2\,\mathrm{W}\) (neglecting the negative sign, which indicates the direction of heat transfer).

Key concepts

Thermodynamics

Thermodynamics is an essential branch of physics that deals with the relationship between heat and other forms of energy. Specifically, it focuses on how heat transfer can be converted to and from other forms of energy and how it affects the properties of substances.

In the context of our exercise, thermodynamics governs how the heat is transferred from the hot stainless steel ball bearings to the cooler air. This heat transfer results in a temperature change of the ball bearings, which can be calculated using principles derived from the laws of thermodynamics. The rate at which heat is transferred is central to understanding how rapidly the ball bearings lose their heat when exposed to air, which is directly related to the principles of heat transfer, a core area in thermodynamics.

Heat Transfer Calculation

One of the key computations in thermodynamics is the heat transfer calculation. This involves determining the amount of heat energy transferred from one substance to another as a result of a temperature difference.

To calculate heat transfer, the formula we use is: \( Q = mc_p\triangle T \), where \( Q \) is the heat energy transferred, \( m \) is the mass of the object, \( c_p \) is the specific heat capacity, and \( \triangle T \) is the change in temperature.

Applying This to Our Exercise

In our problem, we needed to calculate the rate of heat transfer from the ball bearings to the air. By understanding the heat transfer calculation, we can accurately determine how much heat energy has been lost by the ball bearings before they are quenched in water.

Quenching Process

The quenching process is a crucial technique in materials engineering where a hot object is rapidly cooled by immersing it in a quenching medium such as water, oil, or air. This rapid cooling alters the microstructure of the material, often increasing the hardness and reducing the brittleness.

For the stainless steel ball bearings in our exercise, air quenching is used initially where the high temperature balls are exposed to air at room temperature before they are dropped into water, which is another stage of quenching. Quenching affects the thermal properties and structural integrity of the ball bearings, a factor that is particularly important in industrial processes where such properties can determine the functionality and durability of engineered products.

Specific Heat Capacity

Specific heat capacity is a property that describes how much heat energy is required to raise the temperature of a substance by a single degree Celsius (or Kelvin). It is a crucial factor in performing heat transfer calculations because it helps predict the amount of heat absorbed or released during temperature changes.

The specific heat capacity of a substance is usually denoted by \(c_p\) in the context of the heat transfer formula given earlier. It allows us to compute the precise heat energy change as part of the quenching process or any other scenario involving temperature shifts. The stainless steel ball bearings in the exercise have a specific heat capacity of \(0.480 \mathrm{kJ} / \mathrm{kg} \times^{\circ} \mathrm{C}\), which we use to calculate the amount of energy needed to change their temperature.