Question

A \(15-\mathrm{mm} \times 15-\mathrm{mm}\) silicon chip is mounted such that the edges are flush in a substrate. The chip dissipates \(1.4 \mathrm{~W}\) of power uniformly, while air at \(20^{\circ} \mathrm{C}\) ( $\left.1 \mathrm{~atm}\right)\( with a velocity of \)25 \mathrm{~m} / \mathrm{s}$ is used to cool the upper surface of the chip. If the substrate provides an unheated starting length of \(15 \mathrm{~mm}\), determine the surface temperature at the trailing edge of the chip. Evaluate the air properties at $50^{\circ} \mathrm{C}$.

Short answer

Answer: The surface temperature at the trailing edge of the silicon chip is approximately \(49.15^{\circ} \mathrm{C}\).

Step-by-step solution

Calculate the heat flux and the convective heat transfer coefficient

Since we know the power dissipated by the chip, we can calculate the heat flux (q) by dividing the power dissipation (P) by the area of the silicon chip (A). \(q = \frac{P}{A}\) The area of the chip is: \(A = (15 \times 10^{-3} \text{m})\times(15 \times 10^{-3}\text{m})\) Plugging in the given values, we have: \(q = \frac{1.4\text{ W}}{(15 \times 10^{-3} \text{m})\times (15 \times 10^{-3}\text{m})} = 6256.25\,\text{W/m}^2\) Next, we need to calculate the convective heat transfer coefficient (h) by using the empirical correlation for forced convection over a flat plate, which is given by: \(h = (\text{St})\rho V C_p\) Where St is the Stanton Number, ρ is the air density, V is the air velocity, and C_p is the specific heat capacity of air at constant pressure. According to the empirical correlations, the Stanton Number (St) for a unit Reynolds number is approximately \(2.22 \times 10^{-5}\). The properties of the air at \(50^{\circ} \mathrm{C}\) are as follows: ρ = 1.164 \(\mathrm{kg/m^3}\), C_p = 1006.43 \(\mathrm{J/kg K}\). Now, we can calculate the convective heat transfer coefficient (h): \(h = (2.22 \times 10^{-5})\times(1.164 \,\mathrm{kg/m^3})\times (25 \,\mathrm{m/s})\times (1006.43 \,\mathrm{J/kgK}) = 684.54 \,\text{W/m}^2\text{K}\)

Determine surface temperature at the trailing edge

Now that we have the heat flux (q) and the convective heat transfer coefficient (h), we can find the surface temperature at the trailing edge of the chip. The temperature difference \((T_s - T_\infty)\) between the surface and the free-stream temperature can be found using the relation: \(q = h \times (T_s - T_\infty)\) Rearranging for \(T_s\) and using the given values, we have: \(T_s = \frac{q}{h} + T_\infty\) \(T_s = \frac{6256.25 \,\text{W/m}^2}{684.54 \,\text{W/m}^2\text{K}} + 20^{\circ} \mathrm{C} = 29.15^{\circ} \mathrm{C} + 20^{\circ} \mathrm{C} = 49.15^{\circ} \mathrm{C}\) Therefore, the surface temperature at the trailing edge of the chip is approximately \(49.15^{\circ} \mathrm{C}\).