When the operating condition of a gas turbine engine changes from one steady-state to another, the cooling must ensure that the solid's temperatures never exceed the maximum allowable throughout the transient process. Exceeding the maximum allowable temperature is possible even though cooling is increased to compensate for the increase in heating because there is a time lag in how the solid responds to changes in its convective heating and cooling environments. In this paper, a closed-form solution (referred to as the 1D model) is derived to estimate the over temperature and its duration in a flat plate subjected to sudden changes in heating and cooling rates. For a given change in heating rate, the 1D model can also be used to estimate the minimum cooling needed to ensure that the new steady-state temperature will not exceed the maximum allowable. In addition, this model can estimate the temperature the material must be cooled to before imposing a sudden increase in heat load to ensure no over temperature throughout the transient process. Comparisons with the exact solutions show the 1D model to be accurate within 0.1%. This 1D model was generalized for application to problems in multidimensions. The generalized model was used to estimate the duration of over temperature in a two-dimensional problem involving variable heat transfer coefficient (HTC) on the cooled side of a flat plate and provided results that match the exact solution within 5%.