33) Find the rate of heat flow through a cross-section of the rod shown in fig. Thermal conductivity of the material of the rod is K.

Prolly easy but I am not able to do the mathematics of it.

To determine the rate of heat flow through a rod, we can use Fourier's law of heat conduction. This law states that the heat transfer rate (Q) through a material is directly proportional to the temperature difference (ΔT) across the material and the area (A) through which the heat is flowing, and inversely proportional to the length (L) of the material. The formula can be expressed as:

Fourier's Law of Heat Conduction

The mathematical representation of Fourier's law is:

Q = -K * A * (ΔT / L)

Where:

• Q = Rate of heat flow (in watts, W)
• K = Thermal conductivity of the material (in W/m·K)
• A = Cross-sectional area of the rod (in m²)
• ΔT = Temperature difference across the rod (in K or °C)
• L = Length of the rod (in meters)

Understanding the Components

Let’s break down the components of the formula:

• Thermal Conductivity (K): This is a property of the material that indicates how well it conducts heat. Higher values of K mean better heat conduction.
• Cross-sectional Area (A): This is the area through which heat is being transferred. A larger area allows more heat to flow.
• Temperature Difference (ΔT): The greater the difference in temperature between the two ends of the rod, the more heat will flow. This is because heat naturally moves from hot to cold.
• Length (L): The longer the rod, the more resistance there is to heat flow, which reduces the rate of heat transfer.

Example Calculation

Let’s say you have a rod with the following properties:

• Thermal conductivity (K) = 200 W/m·K
• Cross-sectional area (A) = 0.01 m²
• Temperature difference (ΔT) = 50 K
• Length of the rod (L) = 2 m

Plugging these values into the formula:

Q = -200 W/m·K * 0.01 m² * (50 K / 2 m)

Calculating this step-by-step:

• First, calculate ΔT / L: 50 K / 2 m = 25 K/m
• Now substitute this back into the equation: Q = -200 * 0.01 * 25
• Finally, calculate Q: Q = -50 W

The negative sign indicates the direction of heat flow, which is from the hot end to the cold end. Therefore, the rate of heat flow through the rod is 50 watts.

Final Thoughts

By understanding the relationship between these variables, you can analyze different scenarios involving heat transfer in rods or other materials. If you have specific values or a different setup in mind, feel free to share, and we can work through that together!