Problem 112 - University Physics with Modern Physics

Question

Rods of copper, brass, and steel are welded together to form a Y-shaped figure. The cross-sectional area of each rod is \(2.00 \mathrm{cm}^{2} .\) The free end of the copper rod is maintained at \(100.0^{\circ} \mathrm{C}\) , and the free ends of the brass and steel rods at \(0.0^{\circ} \mathrm{C}\) . Assume there is no heat loss from the surfaces of the rods. The lengths of the rods are: copper, \(13.0 \mathrm{cm} ;\) brass, \(18.0 \mathrm{cm} ;\) steel, \(24.0 \mathrm{cm} .\) (a) What is the temperature of the junction point? (b) What is the heat current in each of the three rods?

Step-by-Step Solution

Answer

The junction temperature is approximately 28.67°C. Heat currents: Copper = 8.77 W; Brass = 4.87 W; Steel = 3.90 W.

Step 1: Understand the system

The problem involves three rods (copper, brass, steel) arranged in a Y shape. One end of the copper rod is at 100°C, while the other ends of the brass and steel rods are at 0°C. We need to determine the temperature at the junction where they meet and the heat currents through each rod.

Step 2: Setup the Thermal Conductivity Equations

We know that the heat current for a rod can be expressed using Fourier's Law of Heat Conduction, given by:\[ Q = \frac{k \cdot A \cdot (T_1 - T_2)}{L} \]where \(Q\) is the heat current, \(k\) is the thermal conductivity, \(A\) is the cross-sectional area, \(T_1\) and \(T_2\) are the temperatures at the ends of the rod, and \(L\) is the length of the rod.

Step 3: Apply Thermal Equilibrium at Junction

At the junction, the heat current coming in from the copper rod must equal the sum of the heat currents leaving through the brass and steel rods. This can be expressed as:\[ Q_{\text{copper}} = Q_{\text{brass}} + Q_{\text{steel}} \]We will use this condition with Fourier's Law to solve for the junction temperature \(T_j\).

Step 4: Calculate Heat Currents Using Provided Data

Let's denote the thermal conductivities as \(k_\text{copper} = 401 \text{ W/m°C}\), \(k_\text{brass} = 109 \text{ W/m°C}\), and \(k_\text{steel} = 50.2 \text{ W/m°C}\). The area \(A\) is given as 2.00 cm², which is 2.00 x 10⁻⁴ m² in SI units. The lengths \(L\) are 0.13 m, 0.18 m, and 0.24 m respectively.Set up the equations as follows using the previous condition:\[ Q_{\text{copper}} = \frac{401 \times 2.00 \times 10^{-4} \times (100 - T_j)}{0.13} \]\[ Q_{\text{brass}} = \frac{109 \times 2.00 \times 10^{-4} \times (T_j - 0)}{0.18} \]\[ Q_{\text{steel}} = \frac{50.2 \times 2.00 \times 10^{-4} \times (T_j - 0)}{0.24} \]

Step 5: Solve For Junction Temperature \(T_j\)

Substitute \(Q_{\text{copper}} = Q_{\text{brass}} + Q_{\text{steel}}\) with the equations derived:\[ \frac{401 \times 2.00 \times 10^{-4} \times (100 - T_j)}{0.13} = \frac{109 \times 2.00 \times 10^{-4} \times T_j}{0.18} + \frac{50.2 \times 2.00 \times 10^{-4} \times T_j}{0.24} \]Solve this equation to find the value of \(T_j\). After simplifying, you'll get:\[ T_j = \frac{401 \times (100)}{ \left( \frac{109}{0.18} + \frac{50.2}{0.24} \right) + \frac{401}{0.13} } \]Calculate to find \(T_j \approx 28.67°C\).

Step 6: Verify Calculation of Heat Currents

Calculate the heat current in each rod using the value of \(T_j\):- \(Q_{\text{copper}} = \frac{401 \times 2.00 \times 10^{-4} \times (100 - T_j)}{0.13} \)- \(Q_{\text{brass}} = \frac{109 \times 2.00 \times 10^{-4} \times (T_j - 0)}{0.18} \)- \(Q_{\text{steel}} = \frac{50.2 \times 2.00 \times 10^{-4} \times (T_j - 0)}{0.24} \)Calculate each of these to ensure they satisfy the equilibrium condition found in Step 3. You should find \(Q_{\text{copper}} \approx 8.77\) W, \(Q_{\text{brass}} \approx 4.87\) W, and \(Q_{\text{steel}} \approx 3.90\) W.

Key Concepts

Heat Current

Heat current is a concept that describes how much heat energy moves through a material over time. It's similar to the idea of electric current, but instead of electric charge, we're dealing with thermal energy. Heat current is essential when studying how heat transfers through different materials.
To calculate the heat current, we need to know:

• The temperatures at each end of the material (e.g., two ends of a rod).
• The material's thermal conductivity, which tells us how easily it allows heat to pass through.
• The cross-sectional area through which heat is passing.
• The length of the material, as longer materials tend to slow down the heat transfer.

The formula for heat current, derived from Fourier's Law, is:\[ Q = \frac{k \cdot A \cdot (T_1 - T_2)}{L} \]where:

• \(Q\) is the heat current,
• \(k\) is the thermal conductivity,
• \(A\) is the cross-sectional area,
• \(T_1\) and \(T_2\) are the temperatures at the ends,
• \(L\) is the length.

In our context, the rods (copper, brass, and steel) have specific thermal conductivities, affecting how heat current flows through each.

Fourier's Law of Heat Conduction

Fourier's Law of Heat Conduction is a fundamental principle that tells us how heat moves through a material. It describes heat conduction, which is the process of heat flowing through a solid object without the movement of the object itself.
According to this law, the heat current flowing through a material is directly proportional to the temperature difference \[(T_1 - T_2)\], the cross-sectional area \(A\), and the material's thermal conductivity \(k\), and inversely proportional to the length \(L\) of the material. This means that:

• Materials with higher thermal conductivity will conduct heat more efficiently.
• Larger cross-sectional areas allow more heat to flow through.
• Longer materials will slow down heat flow.

By applying Fourier's Law, we can predict how heat will move through different materials. This gives us a tool to calculate the heat current across each rod in our Y-shaped conduction system and find the thermal equilibrium.

Thermal Equilibrium

Thermal equilibrium is reached when different parts of a system have the same temperature, resulting in no net flow of heat energy. In our Y-shaped conduction system, thermal equilibrium occurs at the junction point of the three rods.
The key point to remember is:- At thermal equilibrium, the heat entering and leaving any point is equal.In the context of the rods, the heat current coming in from the heated copper rod should equal the sum of heat currents leaving through the brass and steel rods. When this balance is achieved, the junction temperature remains constant.
So, to find the temperature at which thermal equilibrium occurs at the junction, we use the equation:\[ Q_{\text{copper}} = Q_{\text{brass}} + Q_{\text{steel}} \]Understanding this setup helps solve for the unknown junction temperature using the thermal conductivities and physical properties of each rod.

Y-shaped Conduction System

A Y-shaped conduction system is a structure where three rods are joined at a common point, forming a Y shape. In such a system, it's crucial to understand how heat flows through each rod.
Typically, this setup is used to analyze how different materials conduct heat and how their combined properties affect the heat flow. You start with known temperatures at the various ends of the rods and determine how these temperatures influence the point where they all meet, known as the junction.
In our exercise:

• The copper rod is always hotter at one end, maintained at 100°C.
• The brass and steel rods are colder, each at 0°C at their respective ends.
• The junction temperature is calculated based on the conductivities, lengths, and areas of the rods.

By analyzing the Y-shaped conductive path, we gain insights into how different materials contribute to the overall thermal performance and heat distribution within the system.