Problem 53 - College Physics

Question

A pot with a steel bottom 8.50 \(\mathrm{mm}\) thick rests on a hot stove. The area of the bottom of the pot is 0.150 \(\mathrm{m}^{2} .\) The water inside the pot is at \(100.0^{\circ} \mathrm{C},\) and 0.390 \(\mathrm{kg}\) are evaporated every 3.00 min. Find the temperature of the lower surface of the pot, which is in contact with the stove.

Step-by-Step Solution

Answer

The temperature of the lower surface of the pot is approximately 123.21°C.

Step 1: Identify Given Information

We have a steel pot with a bottom thickness of 8.50 mm, which we convert to meters as 0.00850 m. The bottom area of the pot is 0.150 m². Water evaporates at a rate of 0.390 kg every 3.00 minutes at a boiling point of 100.0°C.

Step 2: Calculate Heat Needed to Evaporate Water

The heat required to vaporize the water is given by the equation \( Q = mL \), where \( m = 0.390 \text{ kg} \) is the mass, and \( L = 2.26 \times 10^6 \text{ J/kg} \) is the latent heat of vaporization for water. Thus, \( Q = 0.390 \times 2.26 \times 10^6 = 881400 \text{ J} \) in 3.00 minutes.

Step 3: Find Heat Transfer Rate

To find the heat transfer rate \( \dot{Q} \), we divide the total heat by time: \( \dot{Q} = \frac{Q}{t} \), where \( t = 3 \times 60 = 180 \text{ s} \). Therefore, \( \dot{Q} = \frac{881400}{180} \approx 4897.22 \text{ W} \).

Step 4: Apply Fourier's Law of Heat Conduction

Fourier's Law is \( \dot{Q} = \frac{kA(T_{surface} - T_{water})}{d} \), where \( k \) for steel is approximately 50.2 W/m·K. Plugging in the values: \( 4897.22 = \frac{(50.2)(0.150)(T_{surface} - 100)}{0.0085} \).

Step 5: Solve for Temperature of Lower Surface

Rearrange to solve for \( T_{surface} \):\[ T_{surface} = \frac{4897.22 \times 0.0085}{50.2 \times 0.150} + 100. \] Calculating the expression gives \( T_{surface} \approx 123.21^{\circ} \text{C} \).

Key Concepts

Fourier's Law

Fourier's Law of heat conduction is a fundamental principle in thermal physics. It describes how heat energy is transferred through a material due to a temperature difference. According to Fourier's Law, the rate of heat transfer (\( \dot{Q} \)) through a material is proportional to the negative gradient of temperature and the area perpendicular to the gradient. The formula can be expressed as:
\[\dot{Q} = -kA \frac{dT}{dx}\]Here, \( k \) is the thermal conductivity of the material, \( A \) is the area through which heat is conducted, and \( \frac{dT}{dx} \) is the temperature gradient.

• In the exercise, Fourier's Law helps us calculate how much heat is transferred through the steel bottom of the pot.
• It involves a simple rearrangement to solve for the temperature at one side of the steel sheet.
• The equation allows us to understand the influence of material properties like thermal conductivity on heat transfer.

latent heat of vaporization

The latent heat of vaporization is the heat required to convert a unit mass of a liquid into vapor without a change in temperature. This concept is crucial when studying processes like boiling and evaporation. In terms of physics, it involves the breaking of molecular bonds in a liquid state to form a gas.

• The latent heat of vaporization for water is approximately \( 2.26 \times 10^6 \) J/kg.
• This value indicates the amount of energy needed to turn 1 kilogram of water into steam, maintaining a constant temperature of 100°C.
• In the exercise, this energy conversion is calculated to understand how much heat is absorbed from the stove to vaporize the water in the pot, which is measured over a period of 3 minutes.

Understanding the latent heat of vaporization helps in calculating energy changes in heating processes and designing heating systems and equipment.

thermal conductivity

Thermal conductivity (\( k \)) is a property of a material that determines how well it can conduct heat. Higher thermal conductivity means the material is a better conductor of heat. It's a crucial factor in calculations involving heat transfer through solids.

• For example, steel, which is used in the pot in this exercise, has a thermal conductivity of approximately \( 50.2 \) W/m·K.
• Materials with high thermal conductivity transfer heat more efficiently compared to insulating materials with lower values of \( k \).
• In the given problem, we use this property of steel to find how heat flows from the stove through the pot to the boiling water.

Having knowledge about thermal conductivity allows engineers and scientists to design systems that manage temperature and energy consumption effectively.

temperature difference

The concept of temperature difference is essential in understanding heat flow. Heat naturally flows from areas of higher temperature to areas of lower temperature. The magnitude of this temperature difference greatly affects the rate of heat transfer.

• In the exercise, the temperature difference between the stove (lower surface) and the boiling water (upper surface) drives the conduction process.
• Calculating the temperature difference is crucial in determining the heat transfer rate as per Fourier's Law.
• Analyzing temperature differences helps us set the necessary conditions for achieving desired thermal balance in various industrial applications.

Temperature difference is a simple yet powerful concept that allows one to predict and manipulate heat flow effectively.