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 \(20^{\circ} \mathrm{C}(1 \mathrm{~atm})\) 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

Question: Determine the surface temperature at the trailing edge of a silicon chip that dissipates power uniformly and is cooled by a flow of air. Answer: The surface temperature at the trailing edge of the silicon chip is approximately \(27.12^{\circ}\mathrm{C}\).

Step-by-step solution

Calculate Reynolds Number using given air properties and taking air property evaluation at \(50^{\circ} \mathrm{C}\).

To calculate the Reynolds number (\(Re\)), we have to use the given values for air velocity (\(V\)), temperature (\(T\)), pressure (P), and the unheated starting length (\(L\)). We can use the following correlation to estimate the kinematic viscosity (\(\nu\)) for air at \(50^{\circ}\mathrm{C}\): \(\nu = 16.97 \times 10^{-6}\,\mathrm{m^2 / s}\) Now, we can calculate the Reynolds number using this formula: \(Re = \frac{VL}{\nu}\) \(Re = \frac{(25\,\mathrm{m/s})(0.015\,\mathrm{m})}{16.97\times10^{-6}\,\mathrm{m^2 / s}} \approx 2.208 \times 10^{4}\)

Calculate the Nusselt number using the Reynolds number and the Prandtl number.

The Prandtl number (\(Pr\)) of air is given by: \(Pr = 0.89\) Since the Reynolds number is in the turbulent flow region, we can use the Hilpert's correlation to calculate the Nusselt number (Nu) as follows: \(Nu = 0.683\,Re^{0.466}\,Pr^{1/3}\) \(Nu = 0.683\,(2.208\times10^{4})^{0.466}\,(0.89)^{1/3} \approx 181.76\)

Calculate the heat transfer coefficient (h).

Using the calculated Nusselt number, we can find the heat transfer coefficient (\(h\)) as follows: \(h = \frac{Nu\,k}{L}\) Here, \(k\) is the thermal conductivity of air at \(50^{\circ}\mathrm{C}\), which is equal to: \(k = 0.0342\,\mathrm{W/m\,K}\) \(h = \frac{(181.76)(0.0342\,\mathrm{W/m\,K})}{0.015\,\mathrm{m}} \approx 349.74\,\mathrm{W/m^2 K}\)

Calculate the convective heat transfer from the chip to the air.

The convective heat transfer (\(q\)) can be calculated using the following formula: \(q = hA(T_{s}-T_{\infty})\) Here, \(A\) is the surface area of the chip, \(T_s\) is the surface temperature at the trailing edge of the chip, and \(T_{\infty}\) is the oncoming air temperature, which is given as \(20^{\circ}\mathrm{C}\). The chip dissipates \(1.4\,\mathrm{W}\) of power uniformly, so the heat transfer (\(q\)) can be expressed as \(q =1.4\,\mathrm{W}\)

Calculate the surface temperature at the trailing edge of the chip.

Now we can find the surface temperature (\(T_{s}\)) by isolating it in the convective heat transfer equation above: \(T_{s} = T_{\infty} + \frac{q}{hA}\) The surface area (\(A\)) of the chip is given by: \(A = 0.015\,\mathrm{m} \times 0.015\, \mathrm{m} = 2.25\times10^{-4}\,\mathrm{m^2}\) \(T_{s} = 20^{\circ}\mathrm{C} + \frac{1.4\,\mathrm{W}}{(349.74\,\mathrm{W/m^2\,K})(2.25\times10^{-4}\,\mathrm{m^2})} \approx 27.12^{\circ}\mathrm{C}\) The surface temperature at the trailing edge of the chip is approximately \(27.12^{\circ}\mathrm{C}\).

Key concepts

Reynolds Number Calculation

Understanding the Reynolds number calculation is crucial when analyzing fluid flow over objects, like a chip in our cooling analysis. The Reynolds number (Re) gauges the type of flow, laminar or turbulent, of a fluid over a surface. It can be calculated using the speed of the airflow (V), a characteristic length (L), such as the length of an unheated portion of a chip, and the kinematic viscosity of the fluid (\(u\)). The kinematic viscosity typically depends on the fluid's temperature and pressure. For air at a specific temperature, this value can be found in published data or through a correlation. The general equation for the Reynolds number is given by:
\[ Re = \frac{VL}{u} \]
In our exercise, with air flowing at 25 m/s over a chip with an unheated starting length of 15 mm, and assuming the kinematic viscosity at 50°C is \(16.97 \times 10^{-6} \text{ m}^2/\text{s}\), the Reynolds number will indicate the type of flow and help predict cooling performance. A higher Reynolds number typically points to turbulent flow, which can enhance the convective heat transfer rate away from the chip surface.

Nusselt Number Correlation

The Nusselt number (Nu) is a dimensionless quantity used in heat transfer calculations to relate the convective heat transfer to conductive heat transfer. It's a measure of the effectiveness of convective heat transfer relative to conductive heat transfer across the boundary layer. To determine the Nusselt number, we use correlations that take into account flow characteristics such as the Reynolds number (Re) and the Prandtl number (Pr), which is a dimensionless number relating the fluid's kinematic viscosity to its thermal diffusivity.
When dealing with turbulent flow, as in our problem, correlations such as Hilpert's can be used:
\[ Nu = 0.683 Re^{0.466} Pr^{1/3} \]
This correlation factors in the values of Re and Pr calculated earlier. In the provided example, a Prandtl number of 0.89 is used along with the Reynolds number to estimate the Nusselt number, which is then used to calculate the convective heat transfer coefficient.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient (h) is a pivotal factor in quantifying the heat transferred between a surface and a fluid moving over it. It is calculated using the Nusselt number, the characteristic length, and the thermal conductivity of the fluid (k):
\[ h = \frac{Nu \times k}{L} \]
High values of h indicate efficient transfer of heat, which is desirable in cooling applications. In chip cooling analysis, it's essential to maximize this coefficient to enhance cooling and prevent overheating. As shown in the exercise, by using the Nusselt number and thermal conductivity of air at the evaluated temperature, the heat transfer coefficient helps us determine how effectively the chip can release heat into the airflow.

Surface Temperature Determination

Determining the surface temperature of a cooling chip is a vital last step in understanding the thermal management of electronic components. This is achieved by using the convective heat transfer equation:
\[ q = hA(T_s - T_{\infty}) \]
Where q is the heat flux, A is the area of the heat transfer surface, Ts is the unknown surface temperature, and T∞ is the temperature of the fluid far from the surface. By rearranging the equation to solve for the surface temperature, we can find the trailing edge temperature of the chip:
\[ T_s = T_{\infty} + \frac{q}{hA} \]
This temperature is vital for ensuring the chip doesn't exceed critical operating temperatures. In our example, with the given heat dissipation, airflow properties, and calculated heat transfer coefficient, we're able to ascertain that the trailing edge of the chip remains within safe operating temperatures.