Small-diameter electrical heating elements dissipating \(50 \mathrm{~W} / \mathrm{m}\) (length normal to the sketch) are used to heat a ceramic plate of thermal conductivity \(2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The upper surface of the plate is exposed to ambient air at \(30^{\circ} \mathrm{C}\) with a convection coefficient of \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the lower surface is well insulated.

Initially, the ceramic plate \(\left(\alpha=1.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) is at a uniform temperature of \(30^{\circ} \mathrm{C}\), and suddenly the electrical heating elements are energized. Using the implicit finite-difference method, estimate the time required for the difference between the surface immediately above a heating element and initial temperatures to reach \(95 \%\) of the difference for steady-state conditions. Use a grid spacing of \(\Delta x=6 \mathrm{~mm}\) and \(\Delta y=2 \mathrm{~mm}\) and a time increment of \(1 \mathrm{~s}\).

Transcribed Image Text:

6 mm Ceramic plate Air Th Heating element- -x 2 mm 24 mm- -24 mm-