Chapter 5, Problem 5.40P

The objective of this problem is to develop thermal models for estimating the steady-state temperature and the transient temperature history of the electrical transformer shown.

The external transformer geometry is approximately cubical. with a length of 32 mm to a side. The combined mass of the iron and copper in the transformer is 0.28 kg, and its weighted-average specific heat is $400\text{J/kg}\cdot \text{K}\text{.}$ The transformer dissipates 4.0 W and is operating in ambient air at $T_{\infty }=20°\text{C,}$ with a convection coefficient of $10{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ List and justify the
assumptions made in your analysis, and discuss limitations of the models.

1. Beginning with a properly defined control volume, develop a model for estimating the steady-state temperature of the transformer, $T\left(\infty \right).$ Evaluate $T\left(\infty \right)$ for the prescribed operating conditions.
2. Develop a model for estimating the thermal response (temperature history) of the transformer if it is initially at a temperature of $T_i=T_{\infty }$ and power is suddenly applied. Determine the time required for the transformer to come within $5°\text{C}$ of its steady-state operating temperature.