Question

Show that the thermal resistance of a rectangular enclosure can be expressed as \(R=L_{c} /(A k \mathrm{Nu})\), where \(k\) is the thermal conductivity of the fluid in the enclosure.

Short answer

Question: Show that the thermal resistance of a rectangular enclosure can be expressed as \(R = \frac{L_{c}}{Ak\mathrm{Nu}}\). Answer: The thermal resistance of a rectangular enclosure can be expressed as \(R = \frac{L_{c}}{Ak\mathrm{Nu}}\) by relating it to convective heat transfer coefficient and then substituting the Nusselt number definition.

Step-by-step solution

Define thermal resistance

Thermal resistance is a measure of the resistance offered by a material against the heat conduction. It is defined as the ratio of the temperature difference between the two surfaces of the enclosure to the rate of heat transfer (\(\dot{Q}\)) through the enclosure. Mathematically, it can be expressed as: $$R = \frac{\Delta T}{\dot{Q}}$$

Write the definition of Nusselt number

The Nusselt number (\(\mathrm{Nu}\)) is a dimensionless number used in heat transfer problems to determine the efficiency of convection relative to conduction. It is defined as the ratio of the convective heat transfer coefficient (\(h\)) multiplied by the characteristic length (\(L_{c}\)) to the thermal conductivity of the fluid (\(k\)). Mathematically, it can be expressed as: $$\mathrm{Nu} = \frac{hL_{c}}{k}$$

Express thermal resistance in terms of Nusselt number

Let's rewrite the definition of thermal resistance in terms of convective heat transfer coefficient, which is given by: $$R = \frac{\Delta T}{\dot{Q}} = \frac{1}{hA}$$ Now, we can substitute the definition of Nusselt number to relate thermal resistance with Nusselt number. Rearranging the Nusselt number definition for the convective heat transfer coefficient (\(h\)): $$h = \frac{k\mathrm{Nu}}{L_{c}}$$ Substitute this expression for \(h\) into the thermal resistance formula: $$R = \frac{1}{\frac{k\mathrm{Nu}}{L_{c}}A}$$

Provide the final expression for thermal resistance in the given form

Now, after substituting the convective heat transfer coefficient in terms of the Nusselt number, we get the required expression for the thermal resistance of a rectangular enclosure as: $$R = \frac{L_{c}}{Ak\mathrm{Nu}}$$ Hence, it has been shown that the thermal resistance of a rectangular enclosure can be expressed as \(R = \frac{L_{c}}{Ak\mathrm{Nu}}\).