Question

In a thin-walled double-pipe heat exchanger, when is the approximation \(U=h_{i}\) a reasonable one? Here \(U\) is the overall heat transfer coefficient and \(h_{i}\) is the convection heat transfer coefficient inside the tube.

Short answer

Answer: The approximation \(U=h_{i}\) is reasonable when the resistance from convection inside the tube is much larger than the sum of the convective resistance outside the tube and the conductive resistance through the tube wall. This typically occurs when the inner fluid has a very low thermal conductivity or when there is strong turbulence inside the tube.

Step-by-step solution

Identifying the Heat Transfer Mechanisms

In a double-pipe heat exchanger, heat is transferred between two fluids through the walls of the tubes. This process involves three heat transfer mechanisms: convection inside the tube, conduction through the tube wall, and convection outside the tube. The overall heat transfer coefficient (\(U\)) accounts for all three mechanisms, while the convection heat transfer coefficient inside the tube (\(h_i\)) accounts for only the convection inside the tube.

Understanding the Overall Heat Transfer Coefficient

The overall heat transfer coefficient (\(U\)) combines the effects of convection (\(h_i\) and \(h_o\)) and conduction (\(k_w\)) in a series of resistances. By accounting for the length of the exchanger tube (\(L\)) and the thickness of the tube wall (\(x_w\)), we can relate \(U\) to the three mechanisms using the following equation: \(1/U = (1/h_i) + (x_w/k_w) + (1/h_o)\) Here, \(h_o\) is the convection heat transfer coefficient outside the tube, and \(k_w\) is the thermal conductivity of the tube wall material.

Determining When U = h_i is Reasonable

The approximation \(U = h_i\) is reasonable when the resistance from both the conduction through the tube wall and the convection outside the tube are negligible compared to the resistance from the convection inside the tube. This condition can be expressed mathematically as follows: \(1/U \approx 1/h_i \Rightarrow (1/h_i) >> (x_w/k_w) + (1/h_o)\) In practice, this approximation is reasonable when the convective resistance inside the tube (\(1/h_i\)) is much larger than the sum of the convective resistance outside the tube (\(1/h_o\)) and the conductive resistance through the tube wall (\(x_w/k_w\)). This typically occurs when the inner fluid has a very low thermal conductivity or when there is strong turbulence inside the tube, resulting in a high convective resistance.

Conclusion

In a thin-walled double-pipe heat exchanger, the approximation \(U=h_{i}\) is reasonable when the resistance from convection inside the tube dominates over the other two resistances (conduction through the tube wall and convection outside the tube). This may occur when the inner fluid has a low thermal conductivity or under conditions of strong turbulence within the tube.

Key concepts

Overall heat transfer coefficient

The overall heat transfer coefficient, denoted as \(U\), is a measure that combines different modes of heat transfer happening in a heat exchanger. In cases like a double-pipe heat exchanger, heat moves from a hot fluid to a cold one through the wall of a pipe, involving convection and conduction.
Here's how it works:

• Convection inside the tube: Heat transfer through fluid motion inside the tube.
• Conduction through the tube wall: Heat transfer through the solid metal tube's thickness.
• Convection outside the tube: Heat transfer from the outer pipe wall to the surrounding fluid.

The overall heat transfer coefficient is crucial because it tells you how effectively heat is being transferred across the entire heat exchanger. It is literally a combination of all the resistances from these mechanisms. This is why it is useful to understand how each resistance contributes to the total heat transfer.

Convection heat transfer

Convection heat transfer involves the movement of fluid that carries heat from one place to another. It is characterized by the convection heat transfer coefficient, denoted as \(h_i\) for inside the tube and \(h_o\) for outside the tube.
This process is divided into two types:

• Free or natural convection: Occurs spontaneously due to the natural movement of fluid resulting from density differences caused by temperature variations.
• Forced convection: It’s induced by an external source like a pump or fan, providing higher heat transfer rates compared to free convection.

In a double-pipe heat exchanger, controlling the flow conditions like the fluid velocity can significantly affect convective heat transfer rates. Better flow management can help in achieving higher \(h_i\) or \(h_o\), thereby influencing the overall heat transfer coefficient \(U\).

Conduction resistance

Conduction resistance is the opposition to heat flow through a material. In the context of a double-pipe heat exchanger, it involves the metal wall of the tube, which separates the hot and cold fluids.
The resistance to heat conduction can be described as: \[R_{cond} = \frac{x_w}{k_w}\]where:

• \(x_w\) is the thickness of the tube wall.
• \(k_w\) is the thermal conductivity of the tube material.

It represents how the different materials and thicknesses of the walls resist heat flow. Low resistance means the material is a good conductor, such as copper; while, high resistance materials like stainless steel reduce heat conduction. Thus, understanding conduction resistance helps in determining when the assumption \(U = h_i\) can be applied, particularly when the conduction resistance is negligible.

Thermal conductivity

Thermal conductivity, denoted as \(k\), is a property of materials that indicates how well they conduct heat. In a heat exchanger, the thermal conductivity of the tube wall, \(k_w\), plays a significant role in determining heat conduction.
Consider the following about thermal conductivity:

• Materials with high thermal conductivity, like metals, transfer heat efficiently.
• Materials with low thermal conductivity, like insulation, resist heat transfer.

In a thin-walled double-pipe heat exchanger, if the wall material has low thermal conductivity, the assumption \(U = h_i\) becomes plausible as the conduction resistance becomes negligible. Therefore, choosing the right materials with suitable thermal conductivities is crucial for efficient heat exchanger design.

Turbulence in heat exchangers

Turbulence in heat exchangers refers to the chaotic and irregular fluid motion that can enhance heat transfer. In the interior of the tubes in a double-pipe heat exchanger, inducing turbulence can significantly increase the convection heat transfer coefficient \(h_i\).
Here's why turbulence is important:

• Higher turbulence: Leads to more uniform temperature distribution and disrupts the boundary layer, increasing heat transfer rate.
• Flow dynamics: High fluid velocities create turbulence, which improves convective heat transfer by mixing the fluid thoroughly.

In scenarios where turbulence is high, the convection resistance inside the tube becomes dominant. This can make the approximation \(U = h_i\) more reasonable, as the effects of conduction and outer convection are minimized. Therefore, engineering designs often aim to create turbulent flows to enhance overall heat transfer capabilities.