Consider a power transistor encapsulated in an aluminum case...

Consider a power transistor encapsulated in an aluminum case that is attached at its base to a square aluminum plate of thermal conductivity $k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, thickness $L=6 \mathrm{~mm}$, and width $W=20 \mathrm{~mm}$. The case is joined to the plate by screws that maintain a contact pressure of 1 bar, and the back surface of the plate transfers heat by natural convection and radiation to ambient air and large surroundings at $T_{\infty}=T_{\text {sur }}=$ $25^{\circ} \mathrm{C}$. The surface has an emissivity of $\varepsilon=0.9$, and the convection coefficient is $h=4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The case is completely enclosed such that heat transfer may be assumed to occur exclusively through the base plate. (a) If the air-filled aluminum-to-aluminum interface is characterized by an area of $A_{c}=2 \times 10^{-4} \mathrm{~m}^{2}$ and a roughness of $10 \mu \mathrm{m}$, what is the maximum allowable power dissipation if the surface temperature of the case, $T_{s, c}$, is not to exceed $85^{\circ} \mathrm{C}$ ? (b) The convection coefficient may be increased by subjecting the plate surface to a forced flow of air. Explore the effect of increasing the coefficient over the range $4 \leq h \leq 200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Consider a power transistor encapsulated in an aluminum case that is attached at its base to a square aluminum plate of thermal conductivity $k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, thickness $L=6 \mathrm{~mm}$, and width $W=20 \mathrm{~mm}$. The case is joined to the plate by screws that maintain a contact pressure of 1 bar, and the back surface of the plate transfers heat by natural convection and radiation to ambient air and large surroundings at $T_{\infty}=T_{\text {sur }}=$ $25^{\circ} \mathrm{C}$. The surface has an emissivity of $\varepsilon=0.9$, and the convection coefficient is $h=4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The case is completely enclosed such that heat transfer may be assumed to occur exclusively through the base plate.
(a) If the air-filled aluminum-to-aluminum interface is characterized by an area of $A_{c}=2 \times 10^{-4} \mathrm{~m}^{2}$ and a roughness of $10 \mu \mathrm{m}$, what is the maximum allowable power dissipation if the surface temperature of the case, $T_{s, c}$, is not to exceed $85^{\circ} \mathrm{C}$ ?
(b) The convection coefficient may be increased by subjecting the plate surface to a forced flow of air. Explore the effect of increasing the coefficient over the range $4 \leq h \leq 200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Key Concepts

Thermal Conduction and Material Properties

This concept involves understanding how heat is transferred through a solid material based on its thermal conductivity, geometry, and dimensions. It uses the material's conductivity along with the cross?sectional area and thickness to define the conduction thermal resistance, which is essential for calculating temperature gradients and heat flow within the solid parts of a system.

Thermal Contact Resistance

Thermal contact resistance arises at the interface between two surfaces where imperfect contact (due to surface roughness or insufficient contact pressure) causes an additional barrier to heat flow. Recognizing and quantifying this resistance is crucial in many thermal management applications because it can significantly affect the overall thermal performance of a device.

Convection and Radiation Heat Transfer

This concept focuses on the mechanisms by which heat is removed from a surface to the surroundings. Natural convection relies on fluid motion induced by temperature differences, while radiation involves the emission of thermal energy based on surface emissivity and temperature. Both processes contribute to the overall heat dissipation from a surface into the environment.

Forced Convection as an Enhanced Cooling Method

Forced convection applies external means, such as fans or blowers, to increase the convective heat transfer coefficient, thereby enhancing heat removal from a surface. This method is vital in scenarios where natural convection is insufficient, allowing for higher heat dissipation and improved thermal management in electronic devices and systems.

Thermal Circuit Analysis

Thermal circuit analysis is an analogy to electrical circuits where temperature differences are analogous to voltage differences and thermal resistances to electrical resistances. This approach simplifies the calculation of temperature drops and overall thermal performance in complex systems by modeling heat flow through a network of conduction, convection, and radiation pathways.

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