A scheme for concurrently heating separate water and air...

A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance gas-side heat transfer, annular fins of rectangular profile are attached to the outer tube surface. Attachment is facilitated with a dielectric adhesive that electrically isolates the fins from the current-carrying tube wall. (a) Assuming uniform volumetric heat generation within the tube wall, obtain expressions for the heat rate per unit tube length $(\mathrm{W} / \mathrm{m})$ at the inner $\left(r_{i}\right)$ and outer $\left(r_{o}\right)$ surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, $T_{s, i}$ and $T_{s, e}$, and other pertinent parameters. (b) Obtain expressions that could be used to determine $T_{s, i}$ and $T_{s, o}$ in terms of parameters associated with the water- and air-side conditions. (c) Consider conditions for which the water and air are at $T_{\infty, i}=T_{\infty, o}=300 \mathrm{~K}$, with corresponding convection coefficients of $h_{i}=2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $h_{o}=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Heat is uniformly dissipated in a stainless steel tube $\left(k_{w}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$, having inner and outer radii of $r_{i}=25 \mathrm{~mm}$ and $r_{o}=30$ $\mathrm{mm}$, and aluminum fins $\left(t=\delta=2 \mathrm{~mm}, r_{t}=55\right.$ $\mathrm{mm}$ ) are attached to the outer surface, with $R_{t, c}^{\prime \prime}=$ $10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. Determine the heat rates and temperatures at the inner and outer surfaces as a function of the rate of volumetric heating $\dot{q}$. The upper limit to $\dot{q}$ will be determined by the constraints that $T_{s, i}$ not exceed the boiling point of water $\left(100^{\circ} \mathrm{C}\right)$ and $T_{s, o}$ not exceed the decomposition temperature of the adhesive $\left(250^{\circ} \mathrm{C}\right)$.

A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance gas-side heat transfer, annular fins of rectangular profile are attached to the outer tube surface. Attachment is facilitated with a dielectric adhesive that electrically isolates the fins from the current-carrying tube wall.
(a) Assuming uniform volumetric heat generation within the tube wall, obtain expressions for the heat rate per unit tube length $(\mathrm{W} / \mathrm{m})$ at the inner $\left(r_{i}\right)$ and outer $\left(r_{o}\right)$ surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, $T_{s, i}$ and $T_{s, e}$, and other pertinent parameters.
(b) Obtain expressions that could be used to determine $T_{s, i}$ and $T_{s, o}$ in terms of parameters associated with the water- and air-side conditions.
(c) Consider conditions for which the water and air are at $T_{\infty, i}=T_{\infty, o}=300 \mathrm{~K}$, with corresponding convection coefficients of $h_{i}=2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $h_{o}=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Heat is uniformly dissipated in a stainless steel tube $\left(k_{w}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$, having inner and outer radii of $r_{i}=25 \mathrm{~mm}$ and $r_{o}=30$ $\mathrm{mm}$, and aluminum fins $\left(t=\delta=2 \mathrm{~mm}, r_{t}=55\right.$ $\mathrm{mm}$ ) are attached to the outer surface, with $R_{t, c}^{\prime \prime}=$ $10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. Determine the heat rates and temperatures at the inner and outer surfaces as a function of the rate of volumetric heating $\dot{q}$. The upper limit to $\dot{q}$ will be determined by the constraints that $T_{s, i}$ not exceed the boiling point of water $\left(100^{\circ} \mathrm{C}\right)$ and $T_{s, o}$ not exceed the decomposition temperature of the adhesive $\left(250^{\circ} \mathrm{C}\right)$.

Key Concepts

Design Constraints and Material Limitations

In engineering heat transfer systems, the operating conditions must respect material limits and process constraints—in this case, the boiling point of water and the decomposition temperature of the adhesive. These constraints define the upper limits on the volumetric heating rate by setting maximum acceptable temperatures on the inner and outer surfaces. Incorporating these limits into the analysis ensures that the system design is both safe and functional under expected operating conditions.

Heat Transfer Enhancement with Fins and Thermal Contact Resistance

Adding fins to the outer surface is a common technique to enhance gas?side heat transfer by increasing the effective surface area. However, in this application, the fins are attached using a dielectric adhesive which introduces an additional thermal contact resistance. Understanding the influence of this contact resistance is essential, as it affects the overall thermal performance and must be incorporated into the thermal resistance network to accurately predict the outer surface temperatures.

Cylindrical Heat Conduction with Volumetric Generation

This concept deals with solving the energy (heat conduction) equation in cylindrical coordinates when there is a uniform generation of heat within a cylindrical wall. The analysis involves setting up and solving the differential equation for the radial temperature profile, applying appropriate boundary conditions at the inner and outer surfaces to relate the generated heat to the surface temperatures and geometry, and deriving expressions for the heat rate per unit length.

Thermal Resistance Networks

Thermal resistance networks simplify complex heat transfer problems by modeling the conductive and convective processes as resistances in series or parallel. In this context, the tube wall conduction and the convection from the inner water side and the outer air (and fins) side can be treated as separate thermal resistances. This method provides a systematic way to relate the surface temperatures to the overall heat load and to identify the temperature drop contributions from each segment of the heat transfer pathway.

Convection Heat Transfer and Boundary Conditions

Convection plays a crucial role in determining the surface temperatures of the tube by facilitating heat exchange with the surrounding fluids. The use of convection heat transfer coefficients at the water and air sides sets the boundary conditions for the conduction problem. These coefficients bridge the temperature difference between the tube surfaces and the bulk fluid temperatures, thereby allowing the formulation of expressions to calculate surface temperatures based on the convective heat losses and gains.

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