A chip that is of length L = 5 mm on a side and thickness t = 1 mm is encased in a ceramic substrate, and its exposed surface is convectively cooled by a dielectric liquid for which h = 150 W/m² ∙ K and T_∞ = 20°C.

In the off-mode the chip is in thermal equilibrium with the coolant (T_i = T_∞). When the chip is energized, however, its temperature increases until a new steady-state is established. For purposes of analysis, the energized chip is characterized by uniform volumetric heating with q = 9 x 10⁶ W/m³ ∙ Assuming an infinite contact resistance between the chip and substrate and negligible conduction resistance within the chip, determine the steady-state chip temperature Tf' Following activation of the chip, how long does it take to come within 1°C of this temperature? The chip density and specific heat are p = 2000 kg/m³ and c = 700 J/kg ∙ K, respectively.

Chip, q, Tj, P. Cp Substrate