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Many laptop computers are equipped with thermal management systems that involve liquid cooling of the central processing unit (CPU), transfer of the heated liquid to the back of the laptop screen assembly, and dissipation of heat from the back of the screen assembly by way of a flat, isothermal heat spreader. The cooled liquid is recirculated to the CPU and the process continues. Consider an aluminum heat spreader that is of width \(w=275 \mathrm{~mm}\) and height \(L=175 \mathrm{~mm}\). The screen assembly is oriented at an angle \(\theta=30^{\circ}\) from the vertical direction, and the heat spreader is attached to the \(t=3\)-mm-thick plastic housing with a thermally conducting adhesive. The plastic housing has a thermal conductivity of \(k=0.21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and emissivity of \(\varepsilon=0.85\). The contact resistance associated with the heat spreaderhousing interface is \(R_{t, c}^{\prime \prime}=2.0 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). If the CPU generates, on average, \(15 \mathrm{~W}\) of thermal energy, what is the temperature of the heat spreader when \(T_{\infty}=T_{\text {sur }}=23^{\circ} \mathrm{C}\) ? Which thermal resistance (contact, conduction, radiation, or free convection) is the largest?

Step by step solution

01

Calculate the contact thermal resistance.

To calculate the contact thermal resistance, we will use the given contact resistance \(R_{t, c}^{\prime \prime}\): \[R_{t, c} = R_{t, c}^{\prime \prime} \times A\] where \(A = w \times L\) is the surface area of the heat spreader.

02

Calculate the conduction thermal resistance.

To calculate the conduction thermal resistance, we will use the formula: \[R_{t, k} = \frac{t}{k \times A}\] where \(t\) is the thickness of the plastic housing and \(k\) is its thermal conductivity.

03

Calculate the radiation thermal resistance.

To calculate the radiation thermal resistance, we will use the formula: \[R_{t, rad} = \frac{1}{\varepsilon \sigma A (T_{\infty}+T_{hs})}\] where \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, \(T_{\infty}\) is the surrounding temperature and \(T_{hs}\) is the heat spreader temperature. Since we don't know \(T_{hs}\) yet, we will leave this expression in terms of \(T_{hs}\).

04

Calculate the free convection thermal resistance.

To calculate the free convection thermal resistance, we will use the formula: \[R_{t, conv} = \frac{1}{h_{c} \times A}\] where \(h_{c}\) is the convection heat transfer coefficient. We will find \(h_c\) from the correlating Nusselt number for natural convection of a flat plate inclined at angle \(\theta\) (this expression can be found in heat transfer literature): \[Nu = 0.56 Ra_L^{\frac{1}{4}}\] where \(Ra_L = \frac{g \beta (T_{hs} - T_{\infty}) L^3}{\nu \alpha}\) is the Rayleigh number, \(g\) is the acceleration due to gravity, \(\beta\) is the thermal expansion coefficient, \(\nu\) is the kinematic viscosity, and \(\alpha\) is the thermal diffusivity. We will use the properties of air at the film temperature \({(T_{hs} + T_{\infty})}/2\). Once we have the Nusselt number, we can find \(h_c = Nu \times k_f / L\), where \(k_f\) is the thermal conductivity of the fluid (air). Since we don't know \(T_{hs}\) yet, we will leave this expression in terms of \(T_{hs}\).

05

Calculate the total thermal resistance, \(R_t\).

We can calculate the total thermal resistance by summing the individual thermal resistances: \[R_t = R_{t, c} + R_{t, k} + R_{t, rad} + R_{t, conv}\]

06

Calculate the heat spreader temperature, \(T_{hs}\).

We will use the heat transfer equation, which states that the heat generation (\(Q\)) is equal to the temperature difference across the total thermal resistance: \[Q = (T_{hs} - T_{\infty}) R_t\] Now, we will solve for \(T_{hs}\): \[T_{hs} = Q R_t + T_{\infty}\] Once we have found the heat spreader temperature \(T_{hs}\), we can evaluate the thermal resistances and identify the largest one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Conduction

Thermal conduction is a fundamental process where heat energy is transferred through a material without the physical movement of the substance itself. In a scenario like the one with the laptop's aluminum heat spreader, thermal conduction is significant. However, it's important to recognize that the conduction happens not just within the aluminum, but predominantly through the attached plastic housing.
This occurs because the plastic housing acts as a bridge for thermal energy transfer between the aluminum and external environments, having its own thermal conductivity denoted by the letter 'k'.

• The formula used for calculating thermal resistance due to conduction is: \[R_{t, k} = \frac{t}{k \times A}\]
  Where:
  • R_{t, k} is the conduction resistance
  • t is the thickness of the material
  • k is the thermal conductivity, which is key for efficient thermal conduction
  • A is the area over which the heat is conducted, in this instance, the area of the heat spreader

The lower the thermal conductivity, the higher the thermal resistance, resulting in poorer heat transfer. Understanding these concepts can help in selecting materials with better thermal properties for improved thermal management.

Thermal Radiation

Thermal radiation is a mode of heat transfer that doesn't require contact or a medium. Instead, it relies on electromagnetic waves. This type of heat transfer occurs as energy emitted by bodies due to their temperature.
In the context of the aluminum heat spreader dispersing heat from a laptop, thermal radiation plays a crucial role. It functions in conjunction with the heat spreader's properties, such as emissivity \((\varepsilon)\), to determine how effective the radiation process is.

• Formula for thermal radiation resistance:\[R_{t, rad} = \frac{1}{\varepsilon \cdot \sigma \cdot A (T_{\infty}+T_{hs})}\]
• \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8} \text{W/m}^2\cdot\text{K}^4\)), and\(A\) represents the surface area
• The temperatures involved include the surrounding \(T_{\infty}\) and the heat spreader \(T_{hs}\)

Emissivity is essential here because a high emissivity value means the material is more effective at radiating heat.
By understanding and manipulating these parameters, efficiency in heat dispersal can be maximized, crucial for electronic devices to prevent overheating.

Free Convection

Free convection refers to a type of heat transfer that occurs when a fluid (such as air or liquid) moves due to temperature differences, without being induced by external forces like fans or pumps. It is an essential mechanism for dissipating heat in electronic devices.
In our aluminum heat spreader scenario, free convection helps remove heat from the system into the air.

• In the calculation of free convection resistance, the convection heat transfer coefficient, denoted as \(h_c\), comes into play:\[R_{t, conv} = \frac{1}{h_{c} \times A}\]
• This is heavily dependent on the orientation of the surface, described by the angle \(\theta\), as well as temperature differences.
• The Nusselt number \(Nu\) relates to this process for an inclined surface and is calculated using:\[Nu = 0.56 Ra_L^{\frac{1}{4}}\]Where \(Ra_L\) is the Rayleigh number, a dimensionless value that considers the thermal expansion, viscosity, and thermal properties at the film temperature.

Understanding and optimizing these variables enables effective heat management, consequently lifting the performance and longevity of devices such as laptops.

Thermal Management Systems

Thermal management systems are critical components in electronic devices, especially in maintaining operational efficiency and longevity. Such systems are designed to control and dissipate heat effectively, which is why they are implemented in laptops for cooling the CPU.
The goal is to prevent overheating by ensuring the produced heat is efficiently transferred away from critical components like CPUs. The aluminum heat spreader, in this scenario, is a core part of the thermal management system, effectively distributing and dissipating heat.

• Key elements of thermal management systems include:
  • Heat conduction components like heat spreaders and heat sinks.
  • Liquid cooling mechanisms, enhancing thermal conductivity and providing uniform temperature distribution.
  • Passive heat dissipation via free convection and radiation, using materials with high emissivity.

The efficiency of these systems relies on a synergistic approach where conduction, radiation, and convection work in parallel. High-performing systems also include sensors and feedback loops to adapt to changes in thermal loads, ensuring optimal cooling strategies in real-time. Understanding and designing these systems effectively can greatly improve device durability and user experience, making them indispensable in modern electronic design.