Chapter 10, Problem 10.28P

An iron bar 2.00 cm × 3.00 cm × 10.0 cm at a temperature of 95°C is dropped into a barrel of water at 25°C. The barrel is large enough so that the water temperature rises negligibly as the bar cools. The rate at which heat is transferred from the bar to the water is given by the expression

   $\stackrel{\dot{}}{Q}\left(J/\min \right)=UA\left(T_b-T_w\right)$

where U[= 0.050 J/(min?cm²?°C)] is a heat transfer coefficient, A(cm²) is the exposed surface area of the bar, and T_b(°C) and T_W(°C) are the surface temperature of the bar and the water temperature, respectively. The heat capacity of the bar is 0.460 J/(g? °C). Heat conduction in iron is rapid enough for the temperature T_b(t) to be considered uniform throughout the bar.

1. Write an energy balance on the bar, assuming that all six sides are exposed. Your result should be an expression for dT_b/dt and an initial condition.
2. Without integrating the equation, sketch the expected plot of T_bversus t, labeling the values of T_bat t = 0 and t ? 8.

Derive an expression for T_b(t) and check it three ways. How long will it take for the bar to cool to 30°C?