Question

A 10 -cm-high and 20-cm-wide circuit board houses on its surface 100 closely spaced chips, each generating heat at a rate of \(0.12 \mathrm{~W}\) and transferring it by convection and radiation to the surrounding medium at \(40^{\circ} \mathrm{C}\). Heat transfer from the back surface of the board is negligible. If the combined convection and radiation heat transfer coefficient on the surface of the board is \(22 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), the average surface temperature of the chips is (a) \(41^{\circ} \mathrm{C}\) (b) \(54^{\circ} \mathrm{C}\) (c) \(67^{\circ} \mathrm{C}\) (d) \(76^{\circ} \mathrm{C}\) (e) \(82^{\circ} \mathrm{C}\)

Short answer

The combined convection and radiation heat transfer coefficient on the surface of the board is 22 W/m² K and the circuit board is 10 cm high and 20 cm wide. The temperature of the surrounding medium is 40°C. a) 51°C b) 59°C c) 67°C d) 76°C Answer: c) 67°C

Step-by-step solution

Calculate the total heat generation

We are given that each of the 100 chips generates heat at a rate of \(0.12\mathrm{~W}\). Therefore, the total heat generation (\(q\)) for all the chips can be calculated as follows: \(q = N \times Q_{chip}\) where \(N\) is the total number of chips and \(Q_{chip}\) is the heat generation per chip. \(q = 100 \times 0.12\mathrm{~W} = 12\mathrm{~W}\) The total heat generation for all the chips is 12 W.

Determine the heat transfer area

The circuit board is 10 cm high and 20 cm wide. To calculate the area for heat transfer, we need to convert these measurements to meters and then multiply them together: \(A = H \times W\) where \(A\) is the heat transfer area, \(H\) is the height, and \(W\) is the width. \(A = 0.1\mathrm{~m} \times 0.2\mathrm{~m} = 0.02\mathrm{~m}^2\) The heat transfer area is 0.02 \(m^2\).

Use the heat transfer equation to find the temperature difference

The heat transfer equation for combined convection and radiation is: \(q = hA(T_s - T_{\infty})\) where \(q\) is the total heat generation, \(h\) is the heat transfer coefficient, \(A\) is the heat transfer area, \(T_s\) is the average surface temperature of the chips, and \(T_{\infty}\) is the surrounding medium temperature. We are given that the combined convection and radiation heat transfer coefficient is \(22\mathrm{~W}/\mathrm{m}^2 \cdot \mathrm{K}\). We can rearrange the heat transfer equation to find the temperature difference (\(\Delta{T}\)) between the chips and the surrounding medium: \(\Delta{T} = T_s - T_{\infty} = \frac{q}{hA}\) Substituting the given values: \(\Delta{T} = \frac{12\mathrm{~W}}{22\mathrm{~W}/\mathrm{m}^2 \cdot \mathrm{K} \times 0.02\mathrm{~m}^2} = 27.27\mathrm{~K}\) The temperature difference between the chips and the surrounding medium is 27.27 K.

Calculate the average surface temperature of the chips

We can now add the temperature difference to the surrounding medium temperature to find the average surface temperature of the chips: \(T_s = T_{\infty} + \Delta{T}\) We are given that the surrounding medium temperature is \(40^{\circ} \mathrm{C}\). \(T_s = 40^{\circ} \mathrm{C} + 27.27\mathrm{~K} \approx 67.27^{\circ}\mathrm{C}\) The average surface temperature of the chips is approximately 67.27 °C. Comparing this to the given options, we can see that the closest answer is: (c) \(67^{\circ} \mathrm{C}\)

Key concepts

Convection

When we talk about heat transfer, convection is one of the primary methods used to move heat between objects. It's particularly important in the example of our circuit board. Convection involves the transfer of heat through fluid movement—this could be air, a liquid, or any other fluid. Understanding convection is simple with a few key points:

• Fluid Movement: Heat is transferred by the flow of fluid, such as air, over a surface.
• Natural vs. Forced Convection: In natural convection, fluid movement is caused by temperature differences, whereas forced convection uses external means like fans to move the fluid.
• Heat Transfer Coefficient: This is a measure that indicates how effectively heat is transferred from the board to the fluid. In our example, it was combined with radiation in a single coefficient: 22 W/m²·K.

On the circuit board, convection helps carry away heat generated by the chips into the surrounding cooler air, preventing overheating. The combined heat transfer coefficient used in our problem also reflects how efficiently this process happens.

Radiation

Radiation forms another essential part of heat transfer. Unlike convection, it doesn't require a medium to move through. Heat can be transferred through a vacuum or empty space via electromagnetic waves. Here are a few insights into radiation:

• Electromagnetic Waves: Heat is radiated through infrared waves from one object to another.
• Direct Transfer: Heat energy moves directly from the hot surface of an object to whatever cooler object or space surrounds it.
• Surrounding Temperatures Matter: More heat will be radiated as the difference between the object temperature and surroundings increases.

For the circuit board case, radiation contributes to reducing heat by transferring energy away from the hot chips into the room. Because both convection and radiation were combined in the heat transfer coefficient, they collectively help to keep the board cooler.

Circuit Board Heat Management

Managing heat on a circuit board is crucial for ensuring the longevity and functionality of electronic devices. Overheating can lead to failures, so engineers work to keep temperatures in check using heat management strategies. Here's what you should know about managing heat in circuits:

• Materials and Design: Circuit board materials and chip arrangements are chosen to enhance heat dissipation. For example, materials with higher thermal conductivity are often used.
• Heatsinks and Fans: These are common tools that improve heat transfer. Heatsinks increase the surface area for heat transfer and fans enhance forced convection.
• Thermal Interfaces: Proper placement and contact between chips and heat-dissipating components ensure efficient heat flow.
• Combined Heat Transfer Mechanisms: As demonstrated in the example, using both convection and radiation helps maximize heat transfer efficiency.

By understanding these principles, engineers can create more reliable electronic systems that maintain their performance under various operating conditions. Efficient circuit board heat management is a balance of technology and design, optimizing each component's role in handling heat.