Problem 64 - University Physics with Modern Physics

Question

You are asked to design a cylindrical steel rod 50.0 cm long, with a circular cross section, that will conduct 190.0 J/s from a furnace at 400.0\(^\circ\)C to a container of boiling water under 1 atmosphere. What must the rod’s diameter be?

Step-by-Step Solution

Answer

The rod's diameter must be approximately 2.13 cm.

Step 1: Understand the Problem

We need to find the diameter of a cylindrical steel rod that conducts heat at a rate of 190.0 J/s from a furnace at 400.0\(^\circ\)C to boiling water at 100.0\(^\circ\)C. The rod is 50.0 cm long, and we will use the formula for heat conduction.

Step 2: Use the Heat Conduction Formula

The heat conduction formula is \( Q/t = \frac{kA(T_1 - T_2)}{L} \), where \( Q/t \) is the heat transfer rate (190.0 J/s), \( k \) is the thermal conductivity of steel, \( A \) is the cross-sectional area, \( T_1 \) and \( T_2 \) are the temperatures of the furnace and water, respectively, and \( L \) is the length of the rod.

Step 3: Insert Known Values

Insert the known values into the equation: \( 190.0 = \frac{kA(400.0 - 100.0)}{0.5} \). Here, \( k \) for steel is approximately 50.2 W/m·K. Convert 50.0 cm to meters as 0.5 m.

Step 4: Solve for Cross-Sectional Area

Rearrange the formula to solve for the cross-sectional area \( A \): \( A = \frac{(190.0 \times 0.5)}{(50.2 \times 300)} \). Calculate \( A \).

Step 5: Calculate the Diameter

The cross-sectional area \( A \) of a circle is given by \( A = \pi (d/2)^2 \). Solve for the diameter \( d \) by rearranging the formula: \( d = 2 \times \sqrt{\frac{A}{\pi}} \). Substitute the value of \( A \) from the previous step to find \( d \).

Key Concepts

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. It indicates how easily heat can pass through a material.
The higher the thermal conductivity, the better the material is at conducting heat.
In the context of a cylindrical steel rod, it is represented as the constant \( k \) in the heat conduction formula.Understanding this concept is crucial for applications where efficient heat transfer is required.

• Steel has a moderate thermal conductivity, meaning it is reasonably good at transferring heat.
• It is measured in watts per meter per kelvin (\( W/m \cdot K \)). In this exercise, steel's thermal conductivity is given as 50.2 \( W/m \cdot K \).

This value helps determine how much heat will be conducted over a given length of time.

Cylindrical Steel Rod

A cylindrical steel rod is a long, circular object often used for structural and mechanical purposes.
In heat conduction problems, its shape and material are significant factors.
The length of the rod influences how much heat can be transferred from one end to the other. In real-life applications, these rods are commonly used in heating systems and engineering projects.

• The rod in this exercise is described as being 50.0 cm long, which should be converted to meters as 0.5 m for calculation purposes.
• The rod's material, steel, affects both its mechanical properties and its thermal performance.

A solid understanding of these aspects is essential for effectively employing these rods in practical situations.

Cross-Sectional Area

The cross-sectional area of a rod is the area of its circular face that is perpendicular to its length.
It plays a crucial role in determining how much heat it can conduct.
The larger the cross-sectional area, the more heat it can carry from one end to the other.The formula for calculating the area is crucial:

• For the circular cross section of a cylindrical rod, the area \( A \) is calculated as \( \pi \left( \frac{d}{2} \right)^2 \), where \( d \) is the diameter of the rod.

In heat conduction calculations, this area determines how much thermal energy is transferred over the rod's length and impacts the overall efficiency of the heat transfer process.

Temperature Difference

Temperature difference is defined as the difference in temperature between two points.
For heat conduction, it greatly influences the rate of thermal energy transfer.
The greater the difference in temperature, the higher the rate of heat transfer.This parameter is represented as \( T_1 - T_2 \) in the formula, where \( T_1 \) is the temperature of the hot side (furnace) and \( T_2 \) is the cooler side (boiling water):

• In the given exercise, the temperature difference is 400.0°C - 100.0°C = 300.0°C.
• This difference drives the heat transfer process down the steel rod.

Understanding the impact of temperature difference is essential in designing systems that efficiently manage heat flow.

Circular Cross Section

A circular cross section refers to the round shape of the rod's face when sliced perpendicular to its length.
This shape is common in cylindrical objects and is significant in calculations of area.This simple geometric property simplifies many engineering equations:

• A circle's properties, such as its constant radius across all angles, facilitate straightforward calculations.
• This shape helps in evenly distributing stress and avoiding sharp or weak points, a reason why it's favored in construction and engineering.

In the context of the given exercise, the circular cross section allows for the application of the formula \( A = \pi (d/2)^2 \). It is used to compute the area needed for heat conduction.