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I want to calculate how long time it takes to heat an object with radiation. The object is the inner cylinder of two concentric cylinders. The inner cylinder is not hollow. If assuming black body:

[itex]\dot{Q}_E=\sigma A_1(T_1^4-T(t)^4)[/itex] [1]

Energy emitted must the same as the energy absorbed for the inner cylinder. I want the inner cylinder to reach temperature [itex]T_2[/itex] [itex](T_2<T_1)[/itex].

So the total energy needed for for inner cylinder is given by:

[itex]q=C_p(T_2-T(t))m[/itex] [2]

However, as can be seen, as the temperature of the inner cylinder increase, [itex]\dot{Q_E}[/itex] ([2]) will decrease.

So how do I calculate how long time it will take for it to reach [itex]T_2[/itex]?

Assumptions:

Black body

Inner cylinder has uniform temperature, i.e. surface temperature = core temperature at all times.