The roof of a house consists of a 15 -cm-thick concrete slab (k=2 W / m ·K) that is 15 m wide

step 1 We have got area equals 7 multiplied by 10. Length equals 0 .25 meter. Heat transfer coefficient

The roof of a house consists of a 15 -cm-thick concrete slab...

The roof of a house consists of a 15 -cm-thick concrete slab $(k=2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ that is $15 \mathrm{~m}$ wide and $20 \mathrm{~m}$ long. The convection heat transfer coefficients on the inner and outer surfaces of the roof are 5 and $12 \mathrm{~W} / \mathrm{m}^{2}$. , respectively. On a clear winter night, the ambient air is reported to be at $10^{\circ} \mathrm{C}$, while the night sky temperature is $100 \mathrm{~K}$. The house and the interior surfaces of the wall are maintained at a constant temperature of $20^{\circ} \mathrm{C}$. The emissivity of both surfaces of the concrete roof is $0.9$. Considering both radiation and convection heat transfers, determine the rate of heat transfer through the roof, and the inner surface temperature of the roof. If the house is heated by a furnace burning natural gas with an efficiency of 80 percent, and the price of natural gas is $\$ 1.20 /$ therm ( 1 therm $=105,500 \mathrm{~kJ}$ of energy content), determine the money lost through the roof that night during a 14-h period.

The roof of a house consists of a 15 -cm-thick concrete slab $(k=2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ that is $15 \mathrm{~m}$ wide and $20 \mathrm{~m}$ long. The convection heat transfer coefficients on the inner and outer surfaces of the roof are 5 and $12 \mathrm{~W} / \mathrm{m}^{2}$. , respectively. On a clear winter night, the ambient air is reported to be at $10^{\circ} \mathrm{C}$, while the night sky temperature is $100 \mathrm{~K}$. The house and the interior surfaces of the wall are maintained at a constant temperature of $20^{\circ} \mathrm{C}$. The emissivity of both surfaces of the concrete roof is $0.9$. Considering both radiation and convection heat transfers, determine the rate of heat transfer through the roof, and the inner surface temperature of the roof.
If the house is heated by a furnace burning natural gas with an efficiency of 80 percent, and the price of natural gas is $\$ 1.20 /$ therm ( 1 therm $=105,500 \mathrm{~kJ}$ of energy content), determine the money lost through the roof that night during a 14-h period.

Key Concepts

Thermal Resistance Network

A thermal resistance network models the combined effects of different heat transfer mechanisms—conduction, convection, and radiation—by representing each as a thermal resistance. This approach simplifies the analysis by allowing the determination of overall heat transfer rates and temperature distributions across the layered system.

Conduction

Conduction is the process by which heat is transmitted through a material due to a temperature gradient, with energy being transferred from the hot side to the cold side. It is characterized by the material's thermal conductivity and the geometry (thickness and area) of the medium through which the heat flows.

Convection

Convection involves the transfer of heat between a surface and a fluid (such as air) in motion. This mode of heat transfer is quantified by the convection heat transfer coefficient and accounts for the effect of fluid movement on the rate of heat transfer from the concrete roof surfaces to the air inside or outside.

Radiative Heat Transfer

Radiative heat transfer is the emission or absorption of energy in the form of electromagnetic radiation, typically infrared radiation in thermal applications. This mode of heat transfer depends on the surface emissivity and the temperature difference between the surfaces, such as between the roof and the night sky.

Energy Efficiency and Cost Analysis

Energy efficiency and cost analysis involve evaluating the real-world economic impact of energy losses. This includes determining how inefficient heating (such as energy losses through a roof) translates into additional fuel consumption and cost, accounting for factors like furnace efficiency and unit fuel costs.

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