Arod is initially at a uniform temperature of 0^∘ C throughout. One end is kept at 0^∘ C , and t

step 1 First, writing the given data, given that L is equal to 12070 meter and the temperature gradient

Arod is initially at a uniform temperature of 0^∘ C...

Arod is initially at a uniform temperature of $0^{\circ} \mathrm{C}$ throughout. One end is kept at $0^{\circ} \mathrm{C}$ , and the other is brought into contact with a steam bath at $100^{\circ} \mathrm{C}$ . The surface of the rod is insulated so that heat can flow only lengthwise along the rod. The cross-sectional area of the rod is $2.50 \mathrm{cm}^{2},$ its length is 120 $\mathrm{cm}$ , its thermal conductivity is $380 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},$ its density is $1.00 \times 10^{4} \mathrm{kg} / \mathrm{m}^{3},$ and its specific heat is 520 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ . Consider a short cylindrical element of the rod 1.00 $\mathrm{cm}$ in length. (a) If the temperature gradient at the cooler end of this element is 140 $\mathrm{C}^{\circ} / \mathrm{m}$ , how many joules of heat energy flow across this end per second? (b) If the average temperature of the element is increasing at the rate of 0.250 $\mathrm{C} \%$ /s, what is the temperature gradient at the other end of the element?

Key Concepts

Conservation of Energy in Heat Transfer

This principle states that the energy entering a system, minus the energy leaving it, must equal the change in energy stored within the system. In the context of heat conduction problems, this involves balancing the heat entering and leaving a small element of the material to account for any change in internal energy due to temperature changes.

Temperature Gradient

The temperature gradient is the rate of change of temperature with respect to distance inside a material. This gradient is key in determining the direction and magnitude of heat flow according to Fourier’s law. Understanding the temperature gradient is also essential when assessing how changes in temperature affect the internal energy of a system.

Heat Capacity

Heat capacity describes the amount of energy required to change the temperature of a substance by a given amount. In problems involving heating or cooling of a material, the specific heat capacity is used along with the material's mass to determine how much energy is stored or released during a temperature change. This concept is crucial when linking the conduction of heat with the actual temperature rise or fall in the material.

Fourier's Law of Heat Conduction

This law states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient and the area perpendicular to that gradient. It is fundamental to understanding how heat flows in solids and is mathematically expressed by q = -kA(?T/?x), where q is the heat flux, k is the thermal conductivity, A is the cross-sectional area, and ?T/?x represents the temperature gradient.

Thermal Conductivity

Thermal conductivity is a material property that quantifies a substance's ability to conduct heat. It appears in Fourier’s law and is critical in problems involving heat transfer through solids. High thermal conductivity indicates that a material can transfer heat efficiently, while low thermal conductivity corresponds to poor heat conduction.

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