Determine the conduction heat flux through various plane...

Determine the conduction heat flux through various plane layers that are subjected to boundary temperatures of $T_{s, 1}=301 \mathrm{~K}$ and $T_{s, 2}=299 \mathrm{~K}$ at atmospheric pressure. Hint: Do not account for micro- or nanoscale effects within the solid, and assume the thermal accommodation coefficient for an aluminum-air interface is $\alpha_{t}=0.92$. (a) Case A: The plane layer is aluminum. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{100}=600 \mu \mathrm{m}$ and $L_{10 \mathrm{t}}=600 \mathrm{~nm}$. (b) Case B: Conduction occurs through an air layer. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{\mathrm{tor}}=600 \mu \mathrm{m}$ and $L_{\mathrm{tot}}=600 \mathrm{~nm}$. (c) Case C: The composite wall is composed of air held between two aluminum sheets. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{\text {ot }}=600 \mu \mathrm{m}$ (with aluminum sheet thicknesses of $\delta=40 \mu \mathrm{m}$ ) and $L_{\text {tot }}=600 \mathrm{~nm}$ (with aluminum sheet thicknesses of $\delta=40 \mathrm{~nm}$ ). (d) Case D: The composite wall is composed of 7 air layers interspersed between 8 aluminum sheets. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{\text {lot }}=600 \mu \mathrm{m}$ (with aluminum sheet and air layer thicknesses of $\delta=40 \mu \mathrm{m}$ ) and $L_{\mathrm{lot}}=600 \mathrm{~nm}$ (with aluminum sheet and air layer thicknesses of $\delta=40 \mathrm{~nm}$ ).

Determine the conduction heat flux through various plane layers that are subjected to boundary temperatures of $T_{s, 1}=301 \mathrm{~K}$ and $T_{s, 2}=299 \mathrm{~K}$ at atmospheric pressure. Hint: Do not account for micro- or nanoscale effects within the solid, and assume the thermal accommodation coefficient for an aluminum-air interface is $\alpha_{t}=0.92$.
(a) Case A: The plane layer is aluminum. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{100}=600 \mu \mathrm{m}$ and $L_{10 \mathrm{t}}=600 \mathrm{~nm}$.
(b) Case B: Conduction occurs through an air layer. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{\mathrm{tor}}=600 \mu \mathrm{m}$ and $L_{\mathrm{tot}}=600 \mathrm{~nm}$.
(c) Case C: The composite wall is composed of air held between two aluminum sheets. Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{\text {ot }}=600 \mu \mathrm{m}$ (with aluminum sheet thicknesses of $\delta=40 \mu \mathrm{m}$ ) and $L_{\text {tot }}=600 \mathrm{~nm}$ (with aluminum sheet thicknesses of $\delta=40 \mathrm{~nm}$ ).
(d) Case D: The composite wall is composed of 7 air layers interspersed between 8 aluminum sheets.
Determine the heat flux $q_{x}^{\prime \prime}$ for $L_{\text {lot }}=600 \mu \mathrm{m}$ (with aluminum sheet and air layer thicknesses of $\delta=40 \mu \mathrm{m}$ ) and $L_{\mathrm{lot}}=600 \mathrm{~nm}$ (with aluminum sheet and air layer thicknesses of $\delta=40 \mathrm{~nm}$ ).

Key Concepts

Continuum Assumption and Scale Considerations

The continuum assumption treats materials as continuous media, neglecting atomic-scale and nanoscale effects. This is acceptable when the characteristic dimensions of the system are much larger than the mean free path of the molecules. In problems involving micro- or nanoscale layers, special effects might become important, but if these scales are neglected, standard continuum theories like Fourier’s law remain applicable.

Thermal Accommodation Coefficient

The thermal accommodation coefficient measures the efficiency with which a gas molecule exchanges energy with a surface upon collision. It is particularly significant when analyzing heat transfer processes at interfaces, especially between gases and solids, where it helps account for non-equilibrium effects that might otherwise alter the effective thermal conductivity across the interface.

Thermal Resistance and Composite Wall Analysis

Thermal resistance is an analog to electrical resistance in heat conduction problems. In composite systems, the overall resistance to heat flow is the sum of the individual resistances of each layer arranged in series. This concept allows for the analysis of complex structures – such as layers of aluminum and air – by calculating the effective heat transfer rate based on the sum of individual thermal resistances.

Thermal Conductivity

Thermal conductivity is a material property that measures a material's ability to conduct heat. It plays a crucial role in conduction calculations since it determines how rapidly heat can diffuse through a material. In multilayer or composite systems, differing thermal conductivities lead to varied temperature drops across each layer.

Fourier’s Law

Fourier’s law is the fundamental law governing conduction heat transfer. It states that the heat flux, or the rate of heat transfer per unit area, is proportional to the negative of the temperature gradient. This relationship is commonly expressed as q = -k dT/dx, where k is the thermal conductivity. This law forms the basis for calculating heat flux through uniform and composite plane layers.

Conduction Heat Transfer

This concept refers to the energy transfer within a material or between materials that are in direct contact, driven by a temperature gradient. It is governed by the principles of energy conservation and the microscopic motion of particles, and it is typically analyzed using Fourier’s law, which relates the heat flux to the temperature gradient and the material's thermal conductivity.

Transcript

Please give Ace some feedback