Question

A \(15-\mathrm{cm} \times 20\)-cm circuit board is being cooled by forced convection of air at \(1 \mathrm{~atm}\). The heat from the circuit board is estimated to be \(1000 \mathrm{~W} / \mathrm{m}^{2}\). If the airstream velocity is \(3 \mathrm{~m} / \mathrm{s}\) and the shear stress of the circuit board surface is \(0.075 \mathrm{~N} / \mathrm{m}^{2}\), determine the temperature difference between the circuit board surface temperature and the airstream temperature. Evaluate the air properties at \(40^{\circ} \mathrm{C}\) and $1 \mathrm{~atm}$.

Short answer

Based on the step-by-step solution, complete the following sentence: The temperature difference between the circuit board surface and the airstream temperature can be determined by evaluating the convective heat transfer coefficient using the Reynolds number, Prandtl number, and air properties at the given conditions, calculating the heat transfer rate from the given heat flux, and then applying Newton's Law of Cooling to find the temperature difference.

Step-by-step solution

Write down the given quantities and find missing properties of the air at the given conditions

The given quantities are: - Circuit board dimensions: \(15\,\mathrm{cm}\times 20\,\mathrm{cm}\) - Heat flux from circuit board: \(q'' = 1000\,\mathrm{W}/\mathrm{m}^2\) - Air velocity: \(V = 3\,\mathrm{m/s}\) - Surface shear stress: \(\tau_0 = 0.075 \,\mathrm{N/m}^2\) - Air conditions: \(40^\circ\mathrm{C}\) and \(1\,\mathrm{atm}\) At first, we need to find the air properties at the given conditions. These can be retrieved from air property tables or online tools/calculators. In particular, we need the air density \(\rho\), specific heat \(c_p\), and dynamic viscosity \(\mu\).

Calculate the Reynolds number and Prandtl number

To determine the convective heat transfer coefficient, we first need to find the Reynolds Number (Re) and the Prandtl Number (Pr). The Reynolds Number is expressed as: $$\text{Re} = \frac{\rho V L}{\mu}$$ Here, \(L\) is the characteristic length, which we can choose as the smaller dimension of the circuit board (\(0.15\,\mathrm{m}\)). The Prandtl Number is given by: $$\text{Pr} = \frac{c_p\mu}{k}$$ Where \(k\) is the thermal conductivity of the air. Now, calculate Re and Pr using the air properties obtained in Step 1.

Calculate the convective heat transfer coefficient (h)

To find the convective heat transfer coefficient (h), we will use the following correlation: $$\text{h} L = 0.664 \,k \sqrt{\text{Re}}\,\text{Pr}^{1/3}$$ Calculate the convective heat transfer coefficient by substituting the values obtained in the previous steps.

Determine the heat transfer rate (Q) from the heat flux (q'')

To determine the heat transfer rate, we need to multiply the heat flux by the surface area of the circuit board: $$Q = q''A$$ Here, \(A = 0.15\,\mathrm{m} \times 0.20\,\mathrm{m}\). Calculate the heat transfer rate by substituting the given heat flux and the surface area.

Calculate the temperature difference between the circuit board surface and airstream temperature

Finally, we can determine the temperature difference by applying the Newton's Law of Cooling: $$Q = hA\Delta T$$ Rearrange for \(\Delta T\) and substitute the values obtained earlier to find the temperature difference between the surfaces. $$\Delta T = \frac{Q}{hA}$$ Following these steps will give you the desired temperature difference between the circuit board surface temperature and the airstream temperature.