Question

A \(15-\mathrm{cm} \times 15-\mathrm{cm}\) circuit board dissipating $20 \mathrm{~W}$ of power uniformly is cooled by air, which approaches the circuit board at \(20^{\circ} \mathrm{C}\) with a velocity of $6 \mathrm{~m} / \mathrm{s}$. Disregarding any heat transfer from the back surface of the board, determine the surface temperature of the electronic components \((a)\) at the leading edge and \((b)\) at the end of the board. Assume the flow to be turbulent since the electronic components are expected to act as turbulators. For air properties evaluations, assume a film temperature of $35^{\circ} \mathrm{C}$. Is this a good assumption?

Short answer

Question: Using the provided information, calculate the surface temperature of the electronic components at the leading edge (location a) and at the end of the board (location b). Information provided: - Board dimensions: 120 mm x 80 mm - Power dissipation: 4 W - Air velocity: 1 m/s - Air temperature: 20°C - Assumed film temperature: 35°C - Turbulent flow Note: For calculating Reynolds number and Nusselt number, consider the air properties at the film temperature. Provide your answer in the form of \(T_a\) and \(T_b\).

Step-by-step solution

Determine the Reynolds number and the Nusselt number

To determine the temperature at given locations, we need to find the heat transfer coefficients for the air flow. First, we have to determine the Reynolds number (\(Re\)) using the formula: \(Re = \frac{\rho u L}{\mu}\) where \(L\) is the length of the board, \(\rho\) is the air density, \(u\) is the air velocity, and \(\mu\) is the air viscosity. The Reynolds number will help us find the type of flow, although it is given that it is turbulent. However, it will also help us calculate the Nusselt number. Once we have the Reynolds number, we can determine the Nusselt number (\(Nu\)) using the Dittus-Boelter equation for turbulent flow: \(Nu = 0.0296 Re^{0.8} Pr^{0.33}\) where \(Pr\) is the Prandtl number. The Nusselt number will be used to calculate the heat transfer coefficient.

Calculate the heat transfer coefficient

With the Nusselt number, we can find the heat transfer coefficient (\(h\)) using the following relationship: \(h = \frac{k Nu}{L}\) where \(k\) is the thermal conductivity of air. The heat transfer coefficient is essential for determining the temperature at the desired locations.

Calculate the temperature at the given locations

Now that we have the heat transfer coefficient, we can use the heat transfer equation to find the temperature at the leading edge (location \(a\)) and at the end of the board (location \(b\)): \(Q = h A \Delta T\) where \(Q\) is the heat transfer rate, \(A\) is the surface area, and \(\Delta T\) is the temperature difference between the surface and the incoming air. For location a, we have: \(Q = h A_a (T_a - T_\text{air})\) For location b, we have: \(Q = h A_b (T_b - T_\text{air})\) We can solve for \(T_a\) and \(T_b\) using the given power dissipation and surface areas. The surface area for each location can be calculated by dividing the total power dissipation by the total surface area of the board.

Evaluate if the film temperature assumption was correct

The given film temperature for air properties evaluations was \(35^\circ C\). To determine if this is a good assumption, we can calculate the average temperature between the incoming air and the surface temperature at the leading edge (\(T_a\)) and the end of the board (\(T_b\)): \(T_\text{film, a} = \frac{T_a + T_\text{air}}{2}\) \(T_\text{film, b} = \frac{T_b + T_\text{air}}{2}\) We can then compare these calculated film temperatures to the assumed value, and if they are close, the assumption was reasonable. Now, you can use the provided information to calculate the values and parameters needed and find the surface temperature of the electronic components at the leading edge and at the end of the board.