Consider the composite wall of Problem 3.13 under conditions...

Consider the composite wall of Problem $3.13$ under conditions for which the inside air is still characterized by $T_{\infty, i}=20^{\circ} \mathrm{C}$ and $h_{i}=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. However, use the more realistic conditions for which the outside air is characterized by a diurnal (time) varying temperature of the form $$ \begin{array}{ll} T_{\infty, o}(\mathrm{~K})=273+5 \sin \left(\frac{2 \pi}{24} t\right) & 0 \leq t \leq 12 \mathrm{~h} \\ T_{\infty, a}(\mathrm{~K})=273+11 \sin \left(\frac{2 \pi}{24} t\right) & 12 \leq t \leq 24 \mathrm{~h} \end{array} $$ with $h_{o}=60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming quasi-steady conditions for which changes in energy storage within the wall may be neglected, estimate the daily heat loss through the wall if its total surface area is $200 \mathrm{~m}^{2}$.

Consider the composite wall of Problem $3.13$ under conditions for which the inside air is still characterized by $T_{\infty, i}=20^{\circ} \mathrm{C}$ and $h_{i}=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. However, use the more realistic conditions for which the outside air is characterized by a diurnal (time) varying temperature of the form
$$
\begin{array}{ll}
T_{\infty, o}(\mathrm{~K})=273+5 \sin \left(\frac{2 \pi}{24} t\right) & 0 \leq t \leq 12 \mathrm{~h} \\
T_{\infty, a}(\mathrm{~K})=273+11 \sin \left(\frac{2 \pi}{24} t\right) & 12 \leq t \leq 24 \mathrm{~h}
\end{array}
$$
with $h_{o}=60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming quasi-steady conditions for which changes in energy storage within the wall may be neglected, estimate the daily heat loss through the wall if its total surface area is $200 \mathrm{~m}^{2}$.

Key Concepts

Thermal Resistance and Energy Loss Calculation

Thermal resistance is a measure of a material's opposition to heat flow and is used to model heat transfer in composite systems. By summing the resistances of individual layers and the convective interfaces, one can determine the overall thermal resistance, which in turn allows for the estimation of the heat flux and total energy loss. In problems involving varying external conditions, integration over time of the heat transfer rate yields the total energy loss over a given period.

Diurnal Temperature Variation

Diurnal temperature variation refers to the regular fluctuation of the ambient temperature over a 24?hour period. In thermal analysis, accounting for these time?varying external conditions is crucial as they affect the driving temperature difference for heat transfer through the wall, and hence influence the transient or quasi-steady calculations of heat loss.

Composite Wall

A composite wall consists of multiple layers of materials with different thermal properties, typically arranged in series. Understanding composite walls involves analyzing how heat is conducted through each layer via thermal resistance, and how the combined effect of these layers impacts the overall heat transfer between the two environments.

Quasi-Steady Thermal Analysis

In quasi-steady thermal analysis, the system is assumed to reach a steady state quickly compared to the time scale of external temperature changes, allowing one to neglect energy storage within the wall. This assumption simplifies the analysis by enabling the use of steady-state heat transfer equations despite the presence of time-varying boundary conditions.

Convection Heat Transfer

Convection heat transfer occurs at the surfaces of a wall where a fluid, such as air, is in contact with it. The rate of heat transfer is governed by convection heat transfer coefficients, which relate the temperature difference between the wall surface and the ambient air to the resulting heat flux. Both interior and exterior surfaces have their own convection coefficients that must be considered in the overall energy balance.

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