Question

A pipe is insulated such that the outer radius of the insulation is less than the critical radius. Now the insulation is taken off. Will the rate of heat transfer from the pipe increase or decrease for the same pipe surface temperature?

Short answer

Answer: The rate of heat transfer will decrease if the insulation is removed, provided that the outer radius of the insulation is less than the critical radius. This is because removing the insulation in this case reduces the effectiveness of conductive heat transfer, and the heat transfer rate is maximized when the outer radius is equal to the critical radius.

Step-by-step solution

Define the heat transfer rate equation

To determine the effect of removing the insulation, we need to use the heat transfer equation for cylindrical surfaces: \(Q = 2 * \pi * L * k * \frac{T_{pipe} - T_{infinity}}{ln(\frac{r_{outer}}{r_{pipe})}}\) Where: - \(Q\) is the heat transfer rate - \(L\) is the length of the pipe - \(k\) is the insulation's thermal conductivity - \(T_{pipe}\) is the temperature of the pipe surface - \(T_{infinity}\) is the surrounding temperature - \(r_{outer}\) is the outer radius of the insulation - \(r_{pipe}\) is the radius of the pipe

Analyze the critical radius

In the given problem, the outer radius of the insulation is less than the critical radius. The critical radius is given by the equation: \(r_c = \frac{k}{h}\) Where: - \(r_c\) is the critical radius - \(k\) is the thermal conductivity of the insulation - \(h\) is the convective heat transfer coefficient If the insulation's outer radius is less than the critical radius, adding insulation to the pipe will increase the rate of heat transfer. Conversely, if the insulation's outer radius is greater than the critical radius, adding insulation will decrease the rate of heat transfer.

Determine the effect of removing insulation

Since the outer radius of the insulation is less than the critical radius, removing it would cause the rate of heat transfer to decrease. This is because the smaller insulation layer allows for more effective conductive heat transfer and the overall heat transfer rate is maximized when the outer radius is equal to the critical radius. In conclusion, removing the insulation in this case will result in a decreased rate of heat transfer from the pipe for the same pipe surface temperature.