Power Transmitted Through Conductor Of Cross Sectional Area

Power Transmitted Through Conductor Of Cross Sectional Area A And Ther

The assignment focuses on analyzing thermal conduction, wave propagation, and related physical phenomena. It includes calculating the temperature distribution in a conductor based on thermal conductivity, determining the temperature of a copper and aluminum conductor, evaluating wave speed and wavelength, and understanding the implications of air’s kinetic energy and entropy. Additionally, the problem discusses the behavior of refrigerators with modern insulation, estimates joint movement in slabs due to thermal expansion, and derives characteristics of wave motion such as period, frequency, and wave speed.

Paper For Above instruction

Thermal conduction in conductors is governed by Fourier’s law, which states that the power transmitted through a conductor is proportional to its cross-sectional area, the temperature gradient, and its thermal conductivity. The fundamental relation is given by:

P = (K A ΔT) / L

where P is the power transmitted, K is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference, and L is the length of the conductor.

Applying this to copper and aluminum conductors, with their respective thermal conductivities (Kc for copper and Kal for aluminum), the temperature distribution can be derived. Given that:

P_copper = (K_c A (T_c - T)) / L

P_aluminium = (K_al A (T_al - T)) / L

Setting these equal allows solving for the conductor temperature:

T_c = [ (K_c - K_al) T + K_al T_al ] / K_c

Substituting known values, assuming K_c = 397 W/m·K (copper) and K_al = 238 W/m·K (aluminum), and temperatures T_al = 177°C, T_c can be calculated:

T_c = [(397 T + 238 177) ] / 397 = ( (397 - 238) T + 238 177 ) / 397

which simplifies to:

T_c ≈ 146.2°C

Wave propagation in air can be studied using the relation between the speed of sound, wavelength, and frequency. The speed of sound in air at room temperature is approximately 343.06 meters per second, and given the frequency f = 349.23 Hz, the wavelength λ is calculated as:

λ = c / f = 343.06 / 349.23 ≈ 0.98233 m (98.233 cm)

Analyzing the wave’s motion, if at t=0s, the center of the pulse is 1 meter from the origin and after 1.8 seconds it is 4 meters away, the wave has traveled 3 meters in 1.8 seconds, giving a velocity:

v = 3 m / 1.8 s ≈ 1.66 m/s

This indicates a wave speed of approximately 1.66 m/s, which is lower than the typical speed of sound, possibly indicating a different medium or wave type. The time period T of this wave is the reciprocal of the frequency, thus:

T = 1 / f = 0.667 seconds

The relation between wave speed, frequency, and wavelength is:

v = λ * f

Consequently, the wavelength can also be determined from the wave speed and frequency as:

λ = v / f = 1.00 m/s / 1.5 Hz ≈ 0.667 m

Understanding the statistical mechanics aspect, air molecules with greater kinetic energy are more likely to occupy higher gravitational potential states, increasing system entropy. This principle underscores the natural tendency of systems toward maximum entropy, favoring higher energy configurations statistically.

Modern refrigerator design involves advanced seals and insulation that minimize the influence of hot and cold air flow. These improvements lead to better thermal isolation, allowing freezers to be positioned on the top, bottom, or sides without affecting performance. Many units incorporate interconnecting passageways to enhance the cooling of specific compartments, such as vegetable and meat drawers, using cold air circulation effectively, and maintaining stable internal temperatures regardless of external temperature shifts.

Regarding structural thermal expansion, joint movement can be estimated using the formula:

ΔL = C L ΔT

where ΔL is the change in slab length, C is the coefficient of thermal expansion, L is the original length, and ΔT is the temperature change. For stabilized bases and granular bases, the coefficient C typically varies between 0.65 and 0.8, respectively. The temperature range from placement to the minimum daily temperature influences this expansion, which is crucial in construction and material engineering.

Finally, in wave mechanics, the period T relates to the wave’s time to complete one cycle, which in subsequent calculations shows a period of approximately 0.667 seconds. The wavelength corresponding to a wave speed of 1.0 m/s and frequency of 1.5 Hz is approximately 0.667 meters, reinforcing the fundamental relations among wave parameters.

References

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