Question

Consider a pipe at a constant temperature whose radius is greater than the critical radius of insulation. Someone claims that the rate of heat loss from the pipe has increased when some insulation is added to the pipe. Is this claim valid?

Short answer

Answer: No, the rate of heat loss will not increase in this situation. In fact, it will likely decrease.

Step-by-step solution

Identifying the formula and definition for heat loss and critical radius

To analyze the situation, we will use the following formula for heat loss through a cylinder (pipe in this case): Q = 2 * pi * k * L * (T1 - T2) / (ln(r2/r1)), where Q is the heat loss, k is the thermal conductivity, L is the length of the pipe, T1 and T2 are the inner and outer surface temperatures, and r1 and r2 are the inner and outer radius. The critical radius of insulation (rc) for a cylinder is defined as: rc = k / h, where h is the convective heat transfer coefficient.

Identifying the relationship between heat loss and critical radius

When the radius is greater than the critical radius of insulation (r1 > rc), adding insulation (increasing r2) should decrease the rate of heat loss (Q), as the increased resistance due to the added insulation outweighs the increase in surface area.

Calculate the initial heat loss

Let's represent the initial conditions with r1 > rc. We can calculate the initial heat loss (Q1) using the formula mentioned before: Q1 = 2 * pi * k * L * (T1 - T2) / (ln(r2_initial/r1))

Add insulation to the pipe and calculate the new heat loss

Now, let's add insulation to the pipe, which means increasing the outer radius: r2_new > r2_initial. With the new radius, we can calculate the new heat loss (Q2) using the same formula: Q2 = 2 * pi * k * L * (T1 - T2) / (ln(r2_new/r1))

Compare initial and new heat loss

Now, we need to compare the initial heat loss (Q1) and the new heat loss (Q2) to see if the claim is valid. If Q2 > Q1, then the claim is valid; otherwise, it's not. Notice that in the formula for heat loss, the terms (2 * pi * k * L * (T1 - T2)) are constant, and the only variable is the ratio of r2 to r1. When r2 increases, the denominator of the fraction (ln(r2/r1)) also increases, which means the overall value of heat loss (Q) should decrease.

Conclusion

Since the rate of heat loss decreases when the pipe radius is greater than the critical radius of insulation and we add more insulation, the claim that the rate of heat loss would increase under these circumstances is not valid.

Key concepts

Heat Loss

Understanding heat loss is essential when discussing thermal systems. Heat loss refers to the transfer of thermal energy from an object to its surroundings. This transfer occurs due to the temperature difference between the object and its environment, following the second law of thermodynamics, which states that heat moves from warmer to cooler areas.

Going back to our exercise, heat loss (\( Q \) ) from a pipe can be expressed mathematically using the formula \( Q = 2 \pi k L (T1 - T2) / ln(r2/r1) \) where \( k \) is the thermal conductivity, \( L \) is the length of the pipe, \( T1 \) and \( T2 \) are the temperatures on the inner and outer surfaces of the pipe respectively, and \( r1 \) and \( r2 \) are the inner and outer radii. This formula explains that as the radius changes, so does the heat loss, demonstrating the importance of insulation thickness in managing heat transfer.

Thermal Conductivity

Thermal conductivity (\( k \) ) is a measure of a material's ability to conduct heat. It quantifies how easily thermal energy moves through a material due to a temperature gradient. Certain materials, like metals, have high thermal conductivity, meaning they transfer heat quickly; others, like wood or fiberglass, have low thermal conductivity, acting as good insulators.

In our pipe scenario, the material's thermal conductivity dictates how much energy is lost through the pipe walls. The value of \( k \) directly impacts the heat loss: the higher the \( k \) , the more heat is lost. Thus, choosing the right insulation material with a lower thermal conductivity is pivotal in reducing heat loss from the pipe, which is why it is crucial to understand this property when calculating and comparing heat loss before and after adding insulation.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient (\( h \) ) characterizes the convective heat transfer occurring between a surface and a fluid (liquid or gas) in motion. A higher \( h \) means a fluid is more effective at removing heat from a surface, leading to potentially higher rates of heat loss.

When you look at the critical radius of insulation (\( rc \) = \( k/h \) ), it represents the insulation thickness at which heat loss due to conduction through insulation equates to the heat lost because of convection at the surface. If the pipe's radius is greater than \( rc \) , as mentioned in the exercise, adding insulation actually reduces the rate of heat transfer because it increases the thermal resistance to heat flow. Hence, the convective heat transfer coefficient is a fundamental concept in determining the effectiveness of insulation and optimizing thermal performance of systems.