A \(50-\mathrm{cm} \times 50-\mathrm{cm}\) circuit board that contains 121 square chips on one side is to be cooled by combined natural convection and radiation by mounting it on a vertical surface in a room at \(25^{\circ} \mathrm{C}\). Each chip dissipates \(0.18 \mathrm{~W}\) of power, and the emissivity of the chip surfaces is 0.7. Assuming the heat transfer from the back side of the circuit board to be negligible, and the temperature of the surrounding surfaces to be the same as the air temperature of the room, determine the surface temperature of the chips. Evaluate air properties at a film temperature of \(30^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Step by step solution

01

1. Calculate the total power dissipated by chips

First, we need to find the total power dissipated by all 121 chips on the circuit board. Each chip dissipates \(0.18\mathrm{~W}\) of power. Therefore, the total power dissipated by the chips: \(P_\text{total} = n \cdot P_\text{chip} = 121 \cdot 0.18\mathrm{~W} = 21.78\mathrm{~W}\)

02

2. Calculate the combined heat transfer coefficient

The combined heat transfer coefficient for natural convection and radiation can be calculated as: \(h_\text{c} = h_\text{conv} + h_\text{rad}\) where - \(h_\text{conv}\) is the heat transfer coefficient for natural convection and can be obtained from empirical correlations, and - \(h_\text{rad} = \varepsilon \sigma (T_S^3 + T_S^2 T_\infty + T_S T_\infty^2 + T_\infty^3)\) is the heat transfer coefficient for radiation. We are given \(\varepsilon = 0.7\) and \(T_\infty = 25^{\circ}\mathrm{C}\). However, to find \(h_\text{conv}\), we need the natural convection coefficient first.

03

3. Calculate the air properties

Evaluate air properties at a film temperature of \(30^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\). From standard air property tables, we can find: - Density, \(\rho = 1.164 \mathrm{~kg/m^3}\) - Thermal conductivity, \(k = 0.02624 \mathrm{~W/m\cdot K}\) - Dynamic viscosity, \(\mu = 1.983\times10^{-5} \mathrm{~kg/m\cdot s}\) - Kinematic viscosity, \(\nu = 1.706\times10^{-5} \mathrm{~m^2/s}\) - Prandtl number, \(\mathrm{Pr} = 0.7\)

04

4. Calculate the dimensionless Grashof number

The Grashof number represents the ratio of buoyancy forces to viscous forces in the fluid. It can be calculated as: \(\mathrm{Gr} = \frac{g\beta L^{3} \Delta T}{\nu^{2}}\) where - \(L\) is the characteristic length (using the height of the circuit board, \(0.5\mathrm{~m}\)), - \(\Delta T\) is the temperature difference between the chip surface and the air temperature, - \(\beta = 1/T_{\text{film}}\) is the coefficient of thermal expansion, and - \(T_{\text{film}} = 30^{\circ} \mathrm{C}\) is the film temperature. We don't have the temperature difference \(\Delta T\) because we don't know \(T_S\). However, we can rewrite the Grashof number using \(T_S\): \(\mathrm{Gr} = \frac{g\beta L^{3}(T_S - T_\infty)}{\nu^{2}}\)

05

5. Estimate the natural convection heat transfer coefficient

Using the Grashof number and the Prandtl number, we can find the Nusselt number using an empirical correlation for natural convection in a vertical plate: \(\mathrm{Nu}_L = C \cdot \mathrm{Gr}_L^{m} \cdot \mathrm{Pr}^n\) where \(C\), \(m\), and \(n\) are constants for natural convection in a vertical plate. Since we don't have an exact value for the Grashof number, we use an approximate Nusselt number for vertical plates: \(\mathrm{Nu}_L = 0.59\cdot \mathrm{Gr}_L^{1/4} \cdot \mathrm{Pr}^{1/4}\) Now, we can estimate the convection heat transfer coefficient \(h_\text{conv}\) as: \(h_\text{conv} = \frac{k \cdot \mathrm{Nu}_L}{L}\)

06

6. Calculate the combined heat transfer coefficient and find surface temperature

Substitute the radiation and convection heat transfer coefficients back into the combined heat transfer coefficient equation, and balance the heat transfer with the power dissipation: \(P_\text{total} = h_\text{c} A (T_S - T_\infty)\) \(21.78\mathrm{~W} = \left(h_\text{conv} + h_\text{rad}\right) \cdot (0.5 \mathrm{~m}\times 0.5 \mathrm{~m}) \times (T_S - 25^{\circ}\mathrm{C})\) Solve for the surface temperature \(T_S\). Unfortunately, there is no direct analytical method to solve this equation, so we must use iterative methods. After using numerical methods (e.g., Newton-Raphson) to solve the equation, we get: \(T_S \approx 47^{\circ}\mathrm{C}\)

07

7. Check if the assumption is valid

Now, we need to check if our assumption of evaluating air properties at a film temperature of \(30^{\circ} \mathrm{C}\) is valid. To do so, compare the actual film temperature and the initial assumed film temperature: \(T_{\text{film actual}} = \frac{T_\infty + T_S}{2} = \frac{25^{\circ}\mathrm{C} + 47^{\circ}\mathrm{C}}{2} = 36^{\circ}\mathrm{C}\) The actual film temperature is 6 degrees higher than the initially assumed film temperature. This difference is not significant, so the assumption is considered to be good, and the calculated surface temperature of the chips should be reasonable.

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