Question

Two flow passages with different cross-sectional shapes, one circular another square, are each centered in a square solid bar of the same dimension and thermal conductivity. Both configurations have the same length, \(T_{1}\), and \(T_{2}\). Determine which configuration has the higher rate of heat transfer through the square solid bar for \((a) a=1.2 b\) and \((b) a=2 b\).

Short answer

Answer: In both cases, (a) \(a=1.2b\) and (b) \(a=2b\), the square flow passage configuration has a higher rate of heat transfer than the circular flow passage configuration.

Step-by-step solution

Calculate the cross-sectional areas

Calculate the cross-sectional area of the circular flow passage and square flow passage by using the formulas: \(A_{circle} = \pi (\frac{b}{2})^2\) and \(A_{square} = a^2\) respectively.

Comparing the areas for Case (a) \(a=1.2b\)

Substitute the values of \(a=1.2b\) into the equations for the cross-sectional areas and compare them. $$ A_{circle} = \pi (\frac{b}{2})^2 = \frac{\pi b^2}{4} $$ $$ A_{square} = (1.2b)^2 = 1.44b^2 $$ Now, compare \(A_{circle}\) and \(A_{square}\) to determine which is larger: $$ \frac{A_{square}}{A_{circle}} = \frac{1.44b^2}{\frac{\pi b^2}{4}} = \frac{1.44}{\frac{\pi}{4}} \approx 1.83 $$ Since \(A_{square} \approx 1.83 A_{circle}\), the square flow passage has a larger cross-sectional area, and hence a higher rate of heat transfer in case (a).

Comparing the areas for Case (b) \(a=2b\)

Substitute the values of \(a=2b\) into the equations for the cross-sectional areas and compare them. $$ A_{circle} = \pi (\frac{b}{2})^2 = \frac{\pi b^2}{4} $$ $$ A_{square} = (2b)^2 = 4b^2 $$ Now, compare \(A_{circle}\) and \(A_{square}\) to determine which is larger: $$ \frac{A_{square}}{A_{circle}} = \frac{4b^2}{\frac{\pi b^2}{4}} = \frac{4}{\frac{\pi}{4}} \approx 5.09 $$ Since \(A_{square} \approx 5.09 A_{circle}\), the square flow passage has a larger cross-sectional area, and hence a higher rate of heat transfer in case (b). To conclude, the square flow passage configuration has a higher rate of heat transfer in both cases, \((a) a=1.2b\) and \((b) a=2b\).

Key concepts

Cross-sectional Area Comparison

Understanding the difference in cross-sectional areas is crucial when analyzing heat transfer in different configurations. Cross-sectional area is simply the surface area of a shape when viewed from a specific direction. In terms of heat transfer, larger areas allow more space for heat flow, which typically leads to a higher rate of heat transfer.
To compare areas:

• Circular Flow Passage: The area is calculated using the formula \(A_{circle} = \pi \left(\frac{b}{2}\right)^2\). This formula derives from the area of a circle, \(\pi r^2\), where the radius \(r\) is \(\frac{b}{2}\).
• Square Flow Passage: The area uses the formula \(A_{square} = a^2\). Here, the side length \(a\) is adjusted according to the problem's description.

For both cases presented, substituting the values of \(a\) related to \(b\), indicates that the square passage always results in a larger cross-sectional area compared to the circular passage.

Thermal Conductivity

Thermal conductivity is a measure of how well a material can conduct heat. Different materials have different thermal conductivities, which means that the efficiency of heat transfer can greatly depend on the material of the flow passage.
The quantity of heat that can be conducted through a material is influenced by:

• The material's inherent thermal conductivity value, often denoted by \(k\).
• The temperature gradient across the material, as heat flow is driven by temperature differences.
• The cross-sectional area through which the heat flows.

In this exercise, since both configurations have the same thermal conductivity value, we focus on how the shape and area impact the heat transfer rate. Nonetheless, in real applications, choosing materials with higher thermal conductivity can significantly enhance heat transfer.

Flow Passage Shape

The shape of the flow passage plays a vital role in heat transfer processes. Different shapes can significantly affect the rate at which heat transfers through the material.
Here are some considerations:

• Circular Shapes: Commonly used due to their structural efficiency, but may not always provide the maximum area for heat transfer compared to other shapes.
• Square Shapes: As shown in the examples, they tend to offer more surface area for a given dimension, which enhances the rate of heat transfer.

Thus, selecting the optimal flow passage shape needs consideration of both the intended thermal application and the practical design constraints. The choice between square and circular depends not only on the maximal potential area but also the ease of manufacturing and maintenance.

Heat Transfer Rate

The heat transfer rate is an essential metric in many engineering and scientific applications. It refers to the amount of heat energy transferred per unit time through a material. This rate is influenced by multiple factors, including cross-sectional area, thermal conductivity, and shape.
Calculating and comparing the heat transfer rate in different configurations helps determine the optimal design for specific thermal applications. Larger cross-sectional areas will often support higher rates of heat transfer, assuming the material and conditions remain constant.
In the comparison of circular and square passages:

• The square passage has a significantly larger area, resulting in a proportional increase in the heat transfer rate.
• As the size of \(a\) increases relative to \(b\), the disparity in cross-sectional areas increases, further amplifying the difference in heat transfer rates.

Ultimately, optimizing heat transfer rates requires balancing the shape and material properties with functional and structural requirements.