Question

Can the one-term approximate solutions for a plane wall exposed to convection on both sides be used for a plane wall with one side exposed to convection while the other side is insulated? Explain.

Short answer

Answer: No, the one-term approximate solutions cannot accurately predict the behavior of a plane wall with one side exposed to convection while the other side is insulated due to differences in temperature distribution and the boundary conditions imposed by the insulated side.

Step-by-step solution

Understand One-Term Approximate Solutions

In heat transfer analysis, one-term approximate solutions are used to solve the temperature distribution in a plane wall when there is convection on both sides. This method simplifies the solution by assuming that the temperature distribution depends primarily on the first term of the infinite series solution (hence, "one term"). It is useful when the temperature distribution is relatively uniform, and when convective conditions dominate.

Analyze the Scenario with One Side Insulated

Now, consider a plane wall with one side exposed to convection while the other side is insulated. In this case, heat conduction occurs in the wall, while convective heat transfer occurs on the exposed surface. The insulated side has no heat transfer by definition, which imposes a specific boundary condition in our analysis.

Check the Applicability of One-Term Approximate Solutions

The one-term approximate solution assumes that temperature distribution is relatively uniform in the wall and is mainly governed by convection. However, for a plane wall with one side insulated, the temperature distribution is not uniform. In fact, it is affected by both conduction within the wall and convection on the exposed surface. As a consequence, the one-term approximation may not be accurate for such a scenario.

Conclusion

In conclusion, the one-term approximate solutions for a plane wall exposed to convection on both sides cannot accurately predict the behavior of a plane wall with one side exposed to convection while the other side is insulated. This is due to the differences in temperature distribution and the boundary conditions imposed by the insulated side. Therefore, a more appropriate method should be used to analyze such cases.