An iron bar 2.00 cm x 3.00 cm x 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 Q (J/min) = UA (Tb – Tw) where U [= 0.050 J/ (min∙cm2∙°C)] is a hear transfer coefficient, A (cm2) is the exposed surface area of the bar, and Tb (°C) and Tw (°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). 1-feat conduction in iron is rapid enough for the temperature Tb (t) to be considered uniform throughout the bar.

(a) Write an energy balance on the bar, assuming that all six sides are exposed. Your result should be an expression for dTb/dt and an initial condition.

(b) Without integrating the equation, sketch the expected plot of Tb versus r, labeling the values of Tb at t = 0 and t → ∞.

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