### Background

Many heat transfer problems are time dependent. Transient conduction typically occurs when the boundary conditions of a system change. The system has an initial condition at time = 0 that changes to a different state thereafter. This change drives a thermal response in the body or system we are modeling until a steady-state temperature distribution is reached. For example, consider a small metal sphere initially heated in a furnace to an elevated temperature. At time 0, the sphere is removed from the furnace and exposed to a cool airstream. Energy is transferred via convection and radiation to surrounding environment. Energy transfer occurs via conduction from within the metal sphere body to external surface. The temperature of the sphere decreases until a steady-state temperature distribution is reached. We will use the lumped capacitance method to solve a simple transient conduction problem.

Lumped capacitance method is the simplest and most convenient method used to solve transient heating and cooling problems. The important assumption is the entire material temperature changes only with time. This means the temperature gradients within the solid are negligible. Therefore, we do not need to spatially discretize the components and solve 1-D, 2-D, or 3-D transient conduction equations. The transient temperature response is determined by formulating an overall energy balance on the solid.

### Problem

Steel balls 12 mm in diameter are annealed by heating to 1150 K and then exposed to an air environment for which the air temperature linearly increases with time, T_air = 325 K + a * ETime and convective heat transfer coefficient h = 20 W/(m2-K) and air temperature coefficient a = 0.1875 K/s.

Calculate the sphere temperature over a 1 hour period assuming the properties of the steel to have a thermal conductivity k = 40 W/(m-K), density rho = 7800 kg/m3, and specific heat c = 600 J/(kg-K).

NOTE: Due to the large initial sphere temperature, radiation effects are likely to be significant during the early portion of the transient. The sphere will cool faster if you include heat transfer via radiation which is not included in this example.

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