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An array of 10 silicon chips, each of length \(L=10 \mathrm{~mm}\) on a side, is insulated on one surface and cooled on the opposite surface by atmospheric air in parallel flow with \(T_{\infty}=24^{\circ} \mathrm{C}\) and \(u_{\infty}=40 \mathrm{~m} / \mathrm{s}\). When in use, the same electrical power is dissipated in each chip, maintaining a uniform heat flux over the entire cooled surface. If the temperature of each chip may not exceed \(80^{\circ} \mathrm{C}\), what is the maximum allowable power per chip? What is the maximum allowable power if a turbulence promoter is used to trip the boundary layer at the leading edge? Would it be preferable to orient the array normal, instead of parallel, to the airflow?

Step by step solution

01

Find properties of air

First, we need to find the properties of air at the mean temperature, which is the average of the chip's surface temperature and the free stream air temperature. For this exercise, we will consider air properties at \(\left(24+80\right)/2 = 52^{\circ}\mathrm{C}\) or \(325\mathrm{K}\). Using air property tables at the given temperature and around atmospheric pressure, we get: - Dynamic Viscosity \(\mu = 2.0 \times 10^{-5} \mathrm{~kg/m.s}\) - Thermal Conductivity, \(k = 0.029 \mathrm{~W/m.K}\) - Specific Heat, \(c_p = 1005 \mathrm{~J/kg.K}\) - Density, \(\rho = 1.10 \mathrm{~kg/m^3}\)

02

Calculate the Reynolds number

The Reynolds number (\(\mathrm{Re}\)) for airflow over the chip is: \(\mathrm{Re} = \frac{\rho u_{\infty} L}{\mu}\) Plugging in the known values, we get: \(\mathrm{Re} = \frac{1.10\times 40 \times 0.01}{2.0\times 10^{-5}} = 2.2 \times 10^5\)

03

Calculate the Nusselt number for laminar flow

Since the Reynolds number is greater than \(5\times 10^4\), the flow is turbulent. However, we will first calculate the Nusselt number for laminar flow as follows: \(Nu = 0.664 \times \mathrm{Re}^{0.5} \times Pr^{1/3}\) In case of air, the Prandtl number (\(Pr\)) is close to \(0.71\). Thus, we get: \(Nu = 0.664 \times \sqrt{2.2 \times 10^5} \times {0.71}^{1/3} \approx 184.6\)

04

Calculate the heat transfer coefficient for laminar flow

The heat transfer coefficient (\(h\)) can be found using the Nusselt number and thermal conductivity: \(h = \frac{k \times Nu}{L}\) Plugging in the values, we get: \(h = \frac{0.029 \times 184.6}{0.01} = 537.34 \mathrm{~W/m^2.K}\)

05

Calculate the maximum allowable power for laminar flow

The maximum allowable power can be found using the following formula: \(P = h \times A \times (T_{s} - T_{\infty})\) Here, \(A = L^2\) is the area of one chip. Rearranging the formula, we get: \(P = \frac{h \times L^2}{1} \times (80 - 24)\) Plugging in the values, we get: \(P = 537.34 \times 0.01^2 \times (56) \approx 0.301 \mathrm{W}\)

06

Check the maximum allowable power with turbulence promoter

If a turbulence promoter is used, the Nusselt number for turbulent flow needs to be considered. The Nusselt number formula for turbulent flow is: \(Nu = 0.037 \times \mathrm{Re}^{0.8} \times Pr^{1/3}\) Replacing the values, we get: \(Nu = 0.037 \times (2.2 \times 10^5)^{0.8} \times 0.71^{1/3} \approx 501.0\) Now calculate the heat transfer coefficient for turbulent flow: \(h = \frac{0.029 \times 501.0}{0.01} = 1452.9 \mathrm{~W/m^2.K}\) Finally, we calculate the maximum allowable power for turbulent flow: \(P = \frac{1452.9 \times 0.01^2}{1} \times (80 - 24) \approx 0.819 \mathrm{W}\) Thus, the maximum power per chip is \(0.819 \mathrm{W}\) if a turbulence promoter is used.

07

Discussion on the array orientation

Changing the orientation of the chips so that they are normal to the airflow would increase the exposed surface area and potentially lowering the temperature since a higher convection rate occurs. However, placing the chips normal to the airflow would create flow disturbance and increase the resistance to the flow, reducing the efficiency of the case. Determining if this arrangement is preferable would require a more in-depth analysis considering the trade-offs between higher convection rates and flow disturbances.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reynolds number

The Reynolds number (\( \mathrm{Re} \)) is key in determining the flow characteristics of a fluid. It is defined as the ratio between the inertial forces and the viscous forces within a fluid flow. It's important because it helps predict whether the flow will be laminar or turbulent.

• A higher Reynolds number indicates that inertial forces dominate, which is usually associated with turbulent flow.
• A lower Reynolds number suggests that viscous forces dominate, indicating a laminar flow.

The formula to calculate the Reynolds number in a flow situation, such as airflow over a chip, is given by:\[\mathrm{Re} = \frac{\rho u_{\infty} L}{\mu}\]Where:

• \( \rho \) is the fluid density.
• \( u_{\infty} \) is the velocity of the fluid.
• \( L \) is the characteristic length, often the length of the object the fluid flows over.
• \( \mu \) is the dynamic viscosity of the fluid.

For our exercise, the calculation confirmed a Reynolds number of \( 2.2 \times 10^5 \), indicating a turbulent flow. Understanding the Reynolds number is crucial for assessing flow behaviors such as the transition from laminar to turbulent flow.

Nusselt number

The Nusselt number (\( Nu \)) is an essential dimensionless number in heat transfer. It correlates the convective heat transfer to the conductive heat transfer across a boundary. Essentially, it helps determine the efficiency of heat transfer in a fluid flow over a surface.

• A higher Nusselt number suggests more efficient convective heat transfer, which is desirable in cooling applications.
• For laminar flow, the Nusselt number is typically calculated with empirical correlations like \( Nu = 0.664 \times \mathrm{Re}^{0.5} \times Pr^{1/3} \)
• For turbulent flow, a different correlation is used, such as \( Nu = 0.037 \times \mathrm{Re}^{0.8} \times Pr^{1/3} \)

The Nusselt number allows the determination of the heat transfer coefficient (\( h \)) using the formula:\[h = \frac{k \times Nu}{L}\]where \( k \)is the thermal conductivity of the fluid. In our example, the Nusselt number for laminar flow was found to be approximately 184.6, but with turbulent flow, it increased to approximately 501.0, indicating a substantial improvement in heat transfer.

Turbulent flow

Turbulent flow refers to a type of fluid flow characterized by chaotic and irregular fluctuations. Unlike laminar flow, turbulent flow does not follow a streamlined path and is more complex and less predictable.

• This flow type occurs when the Reynolds number exceeds a critical threshold, typically around \( 5 \times 10^4 \).
• In turbulent flow, eddies, vortices, and rapid changes in pressure and flow velocity dominate.

In the context of the problem, when the inflow surpassed \( \mathrm{Re} = 2.2 \times 10^5 \), turbulent flow was determined. This increased the heat transfer coefficient significantly. Using a turbulence promoter can trigger turbulent flow at lower Reynolds numbers or earlier in the flow over a surface. This property was leveraged to achieve a more substantial cooling effect on the chips by raising the Nusselt number, hence enhancing the power dissipation capability of each chip.

Laminar flow

Laminar flow is often described as smooth, streamlined, or regular flow. In laminar flow, fluid particles move in parallel layers and there is minimal mixing between them. It occurs at lower Reynolds numbers, typically less than \( 2 \times 10^3 \).

• Laminar flow is more stable and predictable than turbulent flow.
• It has lower momentum transfer across layers compared to turbulent flow, which results in lower heat transfer efficiency.

In our specific setting, we initially calculated parameters as if the flow might be laminar, even though the problem context indicated turbulent flow. Despite the smooth nature of laminar flow, in many engineering applications, especially cooling or heating, turbulent flow is preferred due to its higher heat transfer capabilities. However, laminar flow can be advantageous in situations where low friction and high accuracy are necessary.