A thermopane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm

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A thermopane window consists of two pieces of glass 7 mm...

A thermopane window consists of two pieces of glass $7 \mathrm{~mm}$ thick that enclose an air space $7 \mathrm{~mm}$ thick. The window separates room air at $20^{\circ} \mathrm{C}$ from outside ambient air at $-10^{\circ} \mathrm{C}$. The convection coefficient associated with the inner (room-side) surface is $10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) If the convection coefficient associated with the outer (ambient) air is $h_{o}=80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, what is the heat loss through a window that is $0.8 \mathrm{~m}$ long by $0.5 \mathrm{~m}$ wide? Neglect radiation, and assume the air enclosed between the panes to be stagnant. (b) Compute and plot the effect of $h_{o}$ on the heat loss for $10 \leq h_{o} \leq 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Repeat this calculation for a triple-pane construction in which a third pane and a second air space of equivalent thickness are added.

A thermopane window consists of two pieces of glass $7 \mathrm{~mm}$ thick that enclose an air space $7 \mathrm{~mm}$ thick. The window separates room air at $20^{\circ} \mathrm{C}$ from outside ambient air at $-10^{\circ} \mathrm{C}$. The convection coefficient associated with the inner (room-side) surface is $10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) If the convection coefficient associated with the outer (ambient) air is $h_{o}=80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, what is the heat loss through a window that is $0.8 \mathrm{~m}$ long by $0.5 \mathrm{~m}$ wide? Neglect radiation, and assume the air enclosed between the panes to be stagnant.
(b) Compute and plot the effect of $h_{o}$ on the heat loss for $10 \leq h_{o} \leq 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Repeat this calculation for a triple-pane construction in which a third pane and a second air space of equivalent thickness are added.

Key Concepts

Effect of Convection Coefficient Variation

Analyzing the effect of varying the convection heat transfer coefficient on heat loss helps in understanding the sensitivity of the system’s thermal performance to changes in ambient conditions. By varying the convective heat transfer coefficient over a range, one can determine how much the overall heat loss rate changes, and this analysis can be extended to similar systems, such as triple-pane constructions. This approach is critical when designing energy-efficient systems or optimizing insulation performance.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient is an important parameter that quantifies the total rate of heat transfer per unit area per unit temperature difference across a composite assembly. It is derived from the sum of the individual thermal resistances and is used to simplify complex systems into a single effective resistance, which then facilitates the calculation of the overall heat transfer rate.

Composite Wall Analysis

Composite wall analysis involves treating a multilayered structure, such as a window constructed with multiple panes and air gaps, as a series of thermal resistances. Each layer—whether a solid layer with conduction resistance or an air gap/interfacial layer with convective resistance—adds to the overall thermal resistance. This method allows for the determination of heat loss across the entire assembly by combining the effects of conduction and convection in a systematic manner.

Conduction Heat Transfer

Conduction heat transfer occurs in solids due to temperature gradients and is governed by Fourier’s law. In the context of composite layers (such as glass panes), the conduction resistance is determined by the material’s thermal conductivity, the thickness of the layer, and the cross-sectional area through which the heat flows. This concept is essential for determining the heat loss through each solid component of the system.

Thermal Resistance

Thermal resistance is a measure of a material’s opposition to heat flow. In heat transfer problems, each layer – whether it is a solid or a boundary condition such as convection – contributes a resistance to heat flow, analogous to resistors in an electrical circuit. The overall resistance in a system with multiple layers is obtained by summing the individual resistances in series, which then allows for the calculation of the heat transfer rate based on the temperature difference across the composite system.

Convection Heat Transfer

Convection heat transfer involves the transport of heat between a surface and a fluid in motion, and it is characterized by a convection heat transfer coefficient. On the surfaces of the composite system, such as the room side and the ambient air side, convective resistances account for the heat transferred from the solid surfaces to the air. This effect is modeled using Newton’s law of cooling and significantly influences the overall heat transfer when the air’s motion is considered.

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