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A chip that is of length \(L=5 \mathrm{~mm}\) on a side and thickness \(t=1 \mathrm{~mm}\) is encased in a ceramic substrate, and its exposed surface is convectively cooled by a dielectric liquid for which \(h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(T_{\infty}=20^{\circ} \mathrm{C}\). In the off-mode the chip is in thermal equilibrium with the coolant \(\left(T_{i}=T_{\infty}\right)\). When the chip is energized, however, its temperature increases until a new steady state is established. For purposes of analysis, the energized chip is characterized by uniform volumetric heating with \(\dot{q}=9 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\). Assuming an infinite contact resistance between the chip and substrate and negligible conduction resistance within the chip, determine the steady-state chip temperature \(T_{f}\). Following activation of the chip, how long does it take to come within \(1^{\circ} \mathrm{C}\) of this temperature? The chip density and specific heat are \(\rho=2000 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively.

Step by step solution

01

Calculate heat generation in the chip

Since the energized chip is characterized by a uniform volumetric heat generation \(\dot{q}=9\times10^6~W/m^3\). We can calculate the heat generated in the chip, \(Q_g\), using the following formula: $$Q_g = \dot{q} \cdot V$$ where \(V\) is the volume of the chip. We are given the dimensions of the chip (\(L=5~mm\), \(t=1~mm\)) so we can calculate the volume as: $$V = L^2 \cdot t$$

02

Calculate the convection heat transfer

Once we calculate the heat generation in the chip, we can determine the convection heat transfer at the exposed surface, \(Q_h\), using $$Q_h = hA_s(T_f-T_\infty)$$ where \(h = 150~W/m^2\cdot K\) is the convection heat transfer coefficient, \(A_s = L^2\) is the area of the exposed surface, \(T_f\) is the final steady-state temperature of the chip, and \(T_\infty=20°C\) is the temperature of the coolant.

03

Set up the energy balance equation

Since it will ultimately reach steady-state, the heat generation within the chip will equal the heat transfer to the coolant. $$Q_g = Q_h$$ Now we can substitute the expressions for \(Q_g\) and \(Q_h\) from steps 1 and 2 and solve for the temperature \(T_f\).

04

Calculate the time to reach \(1^{\circ}C\) of the final temperature

Once we find the steady-state temperature \(T_f\), we can calculate the time it takes for the chip to come within \(1^{\circ}C\) of this temperature. The time constant \(\tau\) for the transient heat conduction is given by: $$\tau = \frac{\rho c V}{hA_s}$$ where \(\rho=2000~kg/m^3\) is the density and \(c=700~J/kg\cdot K\) is the specific heat of the chip. Using the first approximate solution for lumped capacitance method, we can calculate the time \(t\) as: $$\frac{T_f-T_\infty}{T_f-T_\infty-1} = e^{-t/\tau}$$ By rearranging and substituting the known values, we can solve for the time \(t\).

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Key Concepts

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

Convection

Convection is a fundamental concept in heat transfer which involves the movement of heat through a fluid (like a liquid or gas) by the combined action of conduction and fluid flow. In the problem, the chip is cooled by a dielectric liquid using convection. The convection heat transfer is characterized by the convection heat transfer coefficient, denoted as \(h\). This coefficient represents how effectively heat is being transferred between the chip's surface and the surrounding fluid.

For the chip, the convection heat flow can be calculated using the formula:

• \( Q_h = hA_s(T_f-T_\infty) \)

Here, \(A_s\) is the area of the surface where convection occurs, \(T_f\) is the steady-state temperature of the chip, and \(T_\infty\) is the ambient fluid temperature. This process continues until the rate of heat loss by convection equals the rate of heat being generated within the chip. This is what helps to achieve the steady-state temperature the chip stabilizes at during operation.

Steady-State Temperature

Steady-state temperature is a critical concept in thermal analysis where the system reaches a condition where temperatures remain constant over time despite ongoing heat inputs or losses. In this exercise, steady-state temperature is the ultimate chip temperature when the heat generated inside the chip equals the heat being transferred to the dielectric coolant.

For this situation, an energy balance equation is used:

• \( Q_g = Q_h \)

This indicates that the heat generated internally within the chip \(Q_g\) is equal to the heat convected away \(Q_h\). By solving this equation using known variables — including the heat transfer coefficient and ambient temperatures — the steady-state temperature \(T_f\) can be determined, providing a crucial understanding of how effectively heat management is achieved.

Transient Heat Conduction

Transient heat conduction refers to the change of temperature distribution within an object over time, until the system eventually reaches a new state of equilibrium or a steady-state condition. In this exercise, transient heat conduction is important in determining the time it takes for the chip to nearly reach the steady-state temperature after it is activated.

The amount of time it takes for a system to reach a certain degree of its steady-state temperature is influenced by the system's time constant, \(\tau\), given by:

• \( \tau = \frac{\rho c V}{hA_s} \)

This formula incorporates several material properties such as density \(\rho\), specific heat \(c\), and geometric attributes like volume \(V\) and surface area \(A_s\). These factors jointly define how quickly the chip responds to changes in heat generation and how efficiently it moves towards achieving thermal balance.

Volumetric Heat Generation

Volumetric heat generation is the process by which heat is generated per unit volume within a system. In the chip example, the uniform volumetric heat generation is specified as \( \dot{q} = 9 \times 10^6 \) W/m³, implying that every cubic meter of the chip produces a constant amount of heat.

This concept allows engineers to predict how much heat is generated inside the chip, which impacts how effectively it can transfer this heat to the surrounding environment. This heat generation parameter is crucial for materials that generate heat internally, as it defines the total energy produced:

• \( Q_g = \dot{q} \cdot V \)

The evaluation of this heat is necessary to determine whether the heat dissipation by convection is sufficient to maintain safe operational temperatures, emphasizing its critical role in thermal management of electronic devices.