A bonding operation utilizes a laser to provide a constant...

A bonding operation utilizes a laser to provide a constant heat flux, $q_{o}^{\prime \prime}$, across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness $d=1.25 \mathrm{~mm}$, and its width is large relative to that of the film. The thermophysical properties of the strip are $\rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=435 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The thermal resistance of the plastic film of width $w_{1}=40 \mathrm{~mm}$ is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at $25^{\circ} \mathrm{C}$ and a convection coefficient of $10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature $\left(T_{\infty}\right)$. (a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film $\left(-w_{1} / 2 \leq x \leq+w_{1} / 2\right)$. (b) If the heat flux provided by the laser is 10,000 $\mathrm{W} / \mathrm{m}^{2}$, determine the temperature of the plastic film at the center $(x=0)$ and its edges $\left(x=\pm w_{1} / 2\right)$. (c) Plot the temperature distribution for the entire strip and point out its special features.

A bonding operation utilizes a laser to provide a constant heat flux, $q_{o}^{\prime \prime}$, across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness $d=1.25 \mathrm{~mm}$, and its width is large relative to that of the film. The thermophysical properties of the strip are $\rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=435 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The thermal resistance of the plastic film of width $w_{1}=40 \mathrm{~mm}$ is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at $25^{\circ} \mathrm{C}$ and a convection coefficient of $10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature $\left(T_{\infty}\right)$.
(a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film $\left(-w_{1} / 2 \leq x \leq+w_{1} / 2\right)$.
(b) If the heat flux provided by the laser is 10,000 $\mathrm{W} / \mathrm{m}^{2}$, determine the temperature of the plastic film at the center $(x=0)$ and its edges $\left(x=\pm w_{1} / 2\right)$.
(c) Plot the temperature distribution for the entire strip and point out its special features.

Key Concepts

One-Dimensional Steady-State Heat Conduction

This concept deals with the transfer of thermal energy through a material in one direction under steady?state conditions, meaning that the temperature profile does not change with time. The governing equation is typically a form of Fourier’s law of conduction, and its solution involves balancing the internal conduction with any applied heat flux or generation. This analysis is fundamental in determining temperature distributions in solid bodies when affected by external heat sources and sinks.

Convection Heat Transfer

Convection refers to the heat transfer between a solid surface and a fluid (often air or liquid) in motion. It is characterized by the convective heat transfer coefficient which quantifies how efficiently heat is removed from (or added to) the surface. In problems involving conduction within a solid, convection boundary conditions are applied at surfaces to account for heat loss (or gain) to the surrounding environment, affecting the overall temperature distribution.

Boundary Conditions in Heat Transfer

Boundary conditions specify the behavior of the temperature or heat flux at the limits of the domain under consideration. They are critical for solving differential equations governing heat conduction. In many practical problems, certain boundaries are held at a prescribed temperature (Dirichlet conditions), while others may have a specified heat flux (Neumann conditions) or convective heat transfer (Robin conditions). These conditions ensure that the mathematical model accurately represents the physical situation.

Thermal Resistance

Thermal resistance is a concept used to quantify the opposition to heat flow through a material or across an interface. In composite systems or layered structures, different materials present different resistances to heat flow. When one component’s thermal resistance is negligible relative to another’s, it simplifies the analysis since its effect on the overall temperature drop can often be omitted. This concept helps in modeling complex systems by reducing them to simpler equivalent thermal circuits.

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