Question

The temperature distribution across a wall 0.4-------------

Answer 1

The question requires applying thermodynamics principles to find the rates of heat conduction through different window constructions and an air gap, with a comparison between them, followed by an economic analysis of insulation payback time.

Step-by-step explanation:

The student's question pertains to the rate of heat conduction through a double-paned window and a comparison with a single-pane window given specific dimensions, materials, and temperature differences.

This is a classic problem in thermodynamics, specifically relating to Fourier's law of heat conduction.

To solve part (a) of the question, we need to consider the temperature difference across each pane of glass and the air gap.

Since the air gap is also part of the window assembly, its contribution to the overall heat transfer rate is calculated separately and then combined with that of the glass panes.

The formula to use is Q = kA(T1-T2)/d, where Q is the heat transfer per unit time, k is the thermal conductivity, A is the area, T1 and T2 are the temperatures on either side of the material, and d is the thickness of the material.

For part (b), one must calculate the rate of heat conduction through a single 1.60-cm-thick glass pane.

Again, the same formula is used, but the calculations are simpler because there is only one layer involved.

After finding the rate of heat conduction, comparisons can be made between the two scenarios.

For the payback period problem, the calculation involves economic analysis comparing the energy savings over a heating season to the cost of insulation.

The formula used for estimating payback time is the total cost of insulation divided by the savings per season.

Answer 2

The temperature distribution across a wall 0.4 meters thick can be expressed as T(x) = T1 + (T2 - T1) * (x / d), where T1 and T2 are the temperatures at the two surfaces of the wall, x is the distance from one surface, and d is the thickness of the wall.

Step-by-step explanation:

In heat conduction through a solid, the temperature distribution across the wall can be modeled using the one-dimensional heat conduction equation:

`\[ T(x) = T_1 + (T_2 - T_1)/(d) \cdot x \]`

Where:

`\( T(x) \)` is the temperature at a distance
`\( x \)` from one surface,

`\( T_1 \)` is the temperature at one surface,

`\( T_2 \)` is the temperature at the other surface,

`\( d \)` is the thickness of the wall.

This equation represents a linear temperature gradient across the wall, assuming steady-state conditions and constant thermal conductivity. The term
`\( (T_2 - T_1)/(d) \)` is the slope of the temperature profile, indicating how rapidly temperature changes with distance.

For instance, if the temperatures at the two surfaces
`(\( T_1 \)` and
`\( T_2 \))` are known to be 100°C and 50°C, respectively, and the thickness of the wall
`(\( d \))` is 0.4 meters, the temperature distribution
`(\( T(x) \))` at any point within the wall can be determined using the formula provided.

This mathematical model is fundamental in analyzing heat transfer through walls and helps engineers and scientists understand how temperature varies within different materials, guiding the design and optimization of thermal systems.