Consider a \(15-\mathrm{cm} \times 20\)-cm printed circuit board \((\mathrm{PCB})\) that has electronic components on one side. The board is placed in a room at \(20^{\circ} \mathrm{C}\). The heat loss from the back surface of the board is negligible. If the circuit board is dissipating \(8 \mathrm{~W}\) of power in steady operation, determine the average temperature of the hot surface of the board, assuming the board is \((a)\) vertical, \((b)\) horizontal with hot surface facing up, and (c) horizontal with hot surface facing down. Take the emissivity of the surface of the board to be \(0.8\) and assume the surrounding surfaces to be at the same temperature as the air in the room. Evaluate air properties at a film temperature of \(32.5^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Step by step solution

01

Calculate the area of the PCB and the heat loss rate per unit area

First, we determine the area of the PCB and the heat loss rate per unit area: Area: \(A = width \times height = 15 \times 20 = 300\,\mathrm{cm^2}\) Heat loss rate per unit area: \(q'' = \dfrac{8\,\mathrm{W}}{A} = \dfrac{8\,\mathrm{W}}{300\,\mathrm{cm^2}} \cdot \dfrac{10^4\,\mathrm{cm^2}}{1\,\mathrm{m^2}} = \dfrac{8}{0.03}\,\mathrm{W/m^2} = 266.67\,\mathrm{W/m^2}\)

02

Obtain convection correlations for different orientations

We will use the following correlations for natural convection heat transfer over a flat surface: a) Vertical orientation (correlation for vertical plates): \(Nu_{L} = 0.825 + \dfrac{0.387Ra_{L}^{1/6}}{[1+(0.492/Pr)^{9/16}]^{8/27}}\) b) Horizontal orientation with hot surface facing up (correlation for horizontal plates facing upward): \(Nu_{L} = 0.54Ra_{L}^{1/4}\) c) Horizontal orientation with hot surface facing down (correlation for horizontal plates facing downward): \(Nu_{L} = 0.27Ra_{L}^{1/4}\) Here, \(Nu_{L}\) is the Nusselt number based on length, \(Ra_{L}\) is the Rayleigh number based on length, and \(Pr\) is the Prandtl number.

03

Calculate the air properties at the film temperature

We will calculate the air properties at the film temperature of \(32.5^{\circ} \mathrm{C}\) and \(1\,\mathrm{atm}\) pressure. We can use any thermodynamic property table or online resource to obtain the properties: \(T_f = 32.5^{\circ} \mathrm{C}\) Properties at film temperature: - Kinematic viscosity: \(\nu = 1.61 \times 10^{-5}\,\mathrm{m^2/s}\) - Thermal conductivity: \(k = 0.026\,\mathrm{W/(m\cdot K)}\) - Prandtl number: \(Pr = 0.7\) - Coefficient of thermal expansion: \(\beta = 1/\left(273.15 + T_f\right) = 1/305.65\,\mathrm{K^{-1}} = 3.27 \times 10^{-3}\,\mathrm{K^{-1}}\)

04

Calculate the Rayleigh number for each orientation and find the Nusselt number

Using the length-based Grashof number (\(Gr_{L}\)), we can calculate the Rayleigh number. \(Gr_{L} = \dfrac{g\beta\Delta T L^3}{\nu^2}\) \(Ra_{L} = Gr_L \cdot Pr\) For each orientation (a, b, and c), calculate \(Gr_{L}\) and \(Ra_{L}\), and use the appropriate \(Nu_{L}\) correlations (from Step 2) to find their values.

05

Calculate the average convective heat transfer coefficients

Calculate the average convective heat transfer coefficient (\(h\)) for each orientation: \(h = \dfrac{k}{L}\cdot Nu_{L}\)

06

Determine the hot surface temperature

Now that we have the convective heat transfer coefficient, we express the total heat transfer from the PCB (through convection and radiation) as: \(q_{conv} + q_{rad} = q\) Use the following formulas: \(q_{conv} = h A \Delta T = hA(T_s - T_{\infty})\) \(q_{rad} = \sigma\varepsilon A \left(T_s^4 - T_{\infty}^4\right)\) where \(\sigma = 5.67 \times 10^{-8}\,\mathrm{W/(m^2\cdot K^4)}\) is the Stefan-Boltzmann constant, \(\varepsilon = 0.8\) is the emissivity, and \(T_{\infty} = 20^{\circ} \mathrm{C}\). Combine the two equations and solve for the hot surface temperature \(T_s\) for each orientation.

07

Reflect on the assumption of film temperature for air properties

After finding the hot surface temperature, evaluate if the assumption of the film temperature as \(32.5^{\circ}\,\mathrm{C}\) for calculating air properties is reasonable. If the calculated values of \(T_s\) are close to this temperature, then the assumption is justified. Otherwise, air properties might need to be recalculated at a new film temperature, and the calculations repeated.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!