Problem 23 - A Heat Transfer Textbook

Question

Water at \(7^{\circ} \mathrm{C}\) flows at \(0.38 \mathrm{~m} / \mathrm{s}\) across the top of a \(0.207 \mathrm{~m}\)-long, thin copper plate. Methanol at \(87^{\circ} \mathrm{C}\) flows across the bottom of the same plate, at the same speed but in the opposite direction. Make the obvious first guess as to the temperature at which to evaluate physical properties. Then plot the plate temperature as a function of position. (Do not bother to correct the physical properties in this problem, but note Problem 6.24.) With everything that varies along the plate, determine where the local heat flux would be least.

Step-by-Step Solution

Answer

Evaluate properties at 47°C; plot temperature distribution; minimum heat flux occurs where gradient is smallest.

Step 1: Identify and List Given Values

We have water at 7°C flowing at 0.38 m/s across a copper plate's top, and methanol at 87°C flowing at the same speed across the bottom in the opposite direction. The copper plate is 0.207 m long.

Step 2: Determine the Mean Temperature

Take the average of water and methanol temperatures to find the mean temperature for evaluating physical properties:\[T_{mean} = \frac{(7 + 87)}{2} = 47^{\circ}C.\]

Step 3: Set Up Heat Transfer Problem

Assume one-dimensional heat transfer across the plate. Use the mean temperature to evaluate properties such as thermal conductivity and heat capacities.

Step 4: Establish Boundary Conditions

The top of the plate is in contact with 7°C water and the bottom with 87°C methanol. Temperature gradient across the plate drives the heat transfer.

Step 5: Heat Transfer Analysis

Compute heat flux using Fourier's Law:\[q = -k \frac{dT}{dx},\]where \(k\) is the thermal conductivity of the copper plate. Integrate along the plate's length to find temperature distribution.

Step 6: Graph Temperature vs Position

Using your computations, construct a plot of temperature as a function of position along the plate. The x-axis will represent position, while the y-axis represents temperature.

Step 7: Find Location of Minimum Heat Flux

Identify the point along the plate where the temperature gradient is smallest, as this corresponds to the lowest local heat flux.

Key Concepts

Thermal Conductivity

Thermal conductivity is a property that describes a material's ability to conduct heat.
It is an essential factor in determining how heat flows through a material, such as the copper plate in our exercise. Copper is known for its high thermal conductivity, which makes it an excellent conductor of heat.
Factors influencing thermal conductivity include:

• Material composition: Metals typically have higher thermal conductivity than non-metals.
• Temperature: As temperature changes, the thermal conductivity of a material might also change.
• Phase: Solids, liquids, and gases conduct heat differently.

In our problem, the thermal conductivity of copper allows the heat to transfer efficiently between the water and methanol.
Understanding this concept helps us calculate the heat flux across the plate using Fourier's Law.

Temperature Distribution

Temperature distribution describes how temperature varies across a material.
In our example, it refers to the temperature change from one end of the copper plate to the other.
This distribution is crucial for understanding and predicting how heat spreads in a system. The process to determine temperature distribution involves:

• Establishing boundary conditions: Knowing the temperature on both sides of the plate.
• Assuming a steady-state condition: Assuming temperature does not change with time.
• Solving the heat equation: To find how temperature changes with position.

By analyzing temperature distribution along the plate, we can identify areas with higher or lower temperatures.
This analysis gives us insights into heat flow dynamics in the system.

Fourier's Law

Fourier's Law is a fundamental principle for understanding heat flow within a material.
It states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the material's thermal conductivity.Expressed mathematically:\[ q = -k \frac{dT}{dx}, \]where:

• \( q \) is the heat flux, or rate of heat transfer per unit area,
• \( k \) is the thermal conductivity of the material, and
• \( \frac{dT}{dx} \) is the temperature gradient across the material.

In applying Fourier's Law to our problem, we can calculate the heat flux across the copper plate.
By understanding how variables like thermal conductivity and temperature gradient affect heat flow, a deeper comprehension of the temperature distribution and heat transfer efficiency can be attained.