Unsteady state heat conduction

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Our discussions thus far have been limited to the case that the temperature is not a function of time. For many applications, it is necessary to consider the variation of temperature with time. In this case, the energy equation for classical heat conduction, eq. \rho {c_p}\frac{{\partial T}}{{\partial t}} = \nabla  \cdot (k\nabla T) + q''' from Basics of heat conduction, should be solved. If the thermal conductivity is independent from the temperature, the energy equation is reduced to eq. \frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} = {\nabla ^2}T + \frac{{q'''}}{k} from Basics of heat conduction. Analysis of transient heat conduction using lumped (zero-dimensional), one-dimensional, and multidimensional models will be presented in this section.

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