Question

Consider heat transfer between two identical hot solid bodies and the air surrounding them. The first solid is being cooled by a fan while the second one is allowed to cool naturally. For which solid is the lumped system analysis more likely to be applicable? Why?

Short answer

Answer: The lumped system analysis is more likely to be applicable to the solid cooled by the fan. This is because the increased convective heat transfer coefficient provided by the fan results in a smaller Biot number, indicating that the internal temperature gradients within the solid are negligible compared to the cooling effect provided by the surrounding air.

Step-by-step solution

Identify the assumptions of the lumped system analysis

The lumped system analysis assumes that the temperature within the solid is uniform and that any temperature gradients within the body are negligible. This assumption is valid when the solid has much smaller thermal resistance than the fluid surrounding it. The lumped system analysis relies on the Biot number (Bi), which is the ratio of the internal thermal resistance to the convective resistance. If the Biot number is much smaller than one (Bi << 1), the lumped system analysis is applicable.

Determine the effect of the cooling methods on the Biot number

In this exercise, one solid is being cooled by a fan, while the other cools naturally. The fan increases the convective heat transfer coefficient, which reduces the convective resistance surrounding the solid. This results in a lower Biot number for the solid cooled by the fan. In contrast, the solid that cools naturally has a higher convective resistance due to the natural convection, resulting in a higher Biot number.

Compare the Biot numbers and lumped system applicability

Since both solids are identical, they have the same material properties and geometries, meaning their internal thermal resistances are the same. The difference in the cooling methods results in different Biot numbers. As mentioned earlier, the Biot number is smaller for the solid cooled by the fan due to the increased convective heat transfer coefficient. A smaller Biot number means that the lumped system analysis is more applicable to the solid cooled by the fan.

Explain the reasoning

The lumped system analysis is more applicable to the solid cooled by the fan because the internal temperature gradients in the solid are negligible compared to the cooling effect provided by the surrounding air. The increased convective heat transfer coefficient from the fan ensures that the thermal resistance of the solid is much smaller than the thermal resistance of the fluid surrounding the solid, making the lumped system analysis more suitable for this condition.