Question

The Biot number during a heat transfer process between a sphere and its surroundings is determined to be \(0.02\). Would you use lumped system analysis or the one-term approximate solutions when determining the midpoint temperature of the sphere? Why?

Short answer

Answer: Lumped system analysis should be used to determine the midpoint temperature of the sphere, as the given Biot number (Bi = 0.02) indicates that the internal temperature gradients are negligible.

Step-by-step solution

Understanding the Biot Number

The Biot number (Bi) is a dimensionless quantity that helps to determine whether internal temperature gradients in an object are significant or not. It is defined as: Bi = \(hL_c/k\) where: - h = heat transfer coefficient \((W/m^2K)\), - \(L_c\) = the characteristic length of the object (in this case, the sphere) \((m)\), and - k = thermal conductivity of the sphere \((W/mK)\). The given Biot number is \(Bi = 0.02\).

Criteria for Lumped System Analysis

Lumped system analysis assumes that internal temperature gradients in the object are negligible. This implies that the object has uniform temperature throughout its volume. Lumped system analysis is valid when the Biot number (Bi) is less than or equal to 0.1 (Bi ≤ 0.1). This criterion needs to be checked against the given Biot number.

Criteria for One-Term Approximate Solution

The one-term approximate solution is more accurate than the lumped system analysis and accounts for internal temperature gradients within the object. It is valid when the Biot number is between 0.1 and 10, i.e., 0.1 < Bi < 10.

Comparing the Biot Number to Criteria

Given that \(Bi = 0.02\), let's compare it to the criteria for both cases: 1. Lumped System Analysis: \(Bi \leq 0.1\), and 2. One-Term Approximate Solution: \(0.1 \lt Bi \lt 10\)

Selecting the Appropriate Method

Since the given Biot number (\(Bi = 0.02\)) is less than the criterion for lumped system analysis (Bi ≤ 0.1), we can use the lumped system analysis method to determine the midpoint temperature of the sphere, as the internal temperature gradients are considered negligible. In conclusion, we should use the lumped system analysis to determine the midpoint temperature of the sphere as the given Biot number indicates that the internal temperature gradients are negligible.