A house has well-insulated walls 19.5 cm thick (assume...

A house has well-insulated walls $19.5 \mathrm{~cm}$ thick (assume conductivity of air) and area $410 \mathrm{~m}^{2},$ a roof of wood $5.5 \mathrm{~cm}$ thick and area $280 \mathrm{~m}^{2}$, and uncovered windows $0.65 \mathrm{~cm}$ thick and total area $33 \mathrm{~m}^{2} .$ ( $a$ ) Assuming that heat is lost only by conduction, calculate the rate at which heat must be supplied to this house to maintain its inside temperature at $23^{\circ} \mathrm{C}$ if the outside temperature is $-15^{\circ} \mathrm{C}$. $(b)$ If the house is initially at $12^{\circ} \mathrm{C}$, estimate how much heat must be supplied to raise the temperature to $23^{\circ} \mathrm{C}$ within 30 min. Assume that only the air needs to be heated and that its volume is $750 \mathrm{~m}^{3}$. ( $c$ ) If natural gas costs $$\$ 0.080$$ per kilogram and its heat of combustion is $5.4 \times 10^{7} \mathrm{~J} / \mathrm{kg},$ how much is the monthly cost to maintain the house as in part $(a)$ for $24 \mathrm{~h}$ each day, assuming $90 \%$ of the heat produced is used to heat the house? Take the specific heat of air to be $0.24 \mathrm{kcal} / \mathrm{kg} \cdot \mathrm{C}^{\circ}$

Key Concepts

Fourier's Law of Heat Conduction

This fundamental law states that the rate of heat transfer through a material is proportional to the negative gradient in the temperature and the area, and inversely proportional to the material's thickness. It forms the basis for calculating how much energy is lost or gained by conduction, and is used to model how insulating barriers, like walls or roofs, slow down heat flow.

Thermal Resistance in Conduction

Thermal resistance quantifies a material's opposition to heat flow. It is determined by the material's thickness, its thermal conductivity, and the area through which heat is transferred. In building physics, thermal resistance is key for assessing the insulating capability of walls, roofs, and windows, allowing for the determination of the overall heat loss or gain.

Specific Heat Capacity

Specific heat capacity is a property that indicates the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius. It is crucial when calculating the energy needed to heat a space, such as a room or an entire house, and is used to estimate the heating requirements of materials like air.

Heat Energy Calculation for Temperature Change

This concept involves computing the total energy needed to increase the temperature of a substance by a given amount, using the specific heat capacity and the mass of the substance. It is fundamental in thermal management and energy budgeting, particularly when determining the energy needed for short-term heating processes.

Fuel Energy Content and Efficiency

When comparing energy costs, the energy content of a fuel (heat of combustion) and the efficiency of its energy conversion must be considered. This concept connects the theoretical energy needed (calculated through conduction or specific heat considerations) to the practical fuel consumption, accounting for real-world inefficiencies in the heating system.

Transcript

Please give Ace some feedback