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Chapter 6: Problem 39

Forced air at \(T_{\infty}=25^{\circ} \mathrm{C}\) and \(V=10 \mathrm{~m} / \mathrm{s}\) is used to cool electronic elements on a circuit board. One such element is a chip, \(4 \mathrm{~mm} \times 4 \mathrm{~mm}\), located \(120 \mathrm{~mm}\) from the leading edge of the board. Experiments have revealed that flow over the board is disturbed by the elements and that convection heat transfer is correlated by an expression of the form Estimate the surface temperature of the chip if it is dissipating \(30 \mathrm{~mW}\).

Short Answer

Expert verified
The surface temperature of the chip is approximately \(53.75^{\circ} \mathrm{C}\).

Step by step solution

01

List the known parameters

- Air temperature: \(T_{鈭瀩 = 25^{\circ} \mathrm{C}\) - Air velocity: \(V = 10 \mathrm{~m/s}\) - Chip size: \(4 \mathrm{~mm} \times 4 \mathrm{~mm}\) - Chip location: \(120 \mathrm{~mm}\) from the leading edge - Power dissipation: \(30 \mathrm{~mW}\) #Step 2: Write the formula for convection heat transfer#
02

Write the formula for convection heat transfer

The formula for convection heat transfer is given by: \[\dot{Q} = h_{av}A_s(T_s-T_{鈭瀩)\] We need to solve this equation for the surface temperature of the chip, \(T_s\). #Step 3: Rearrange the equation to solve for Ts#
03

Rearrange the equation to solve for Ts

To find the surface temperature, \(T_s\), we need to rearrange the equation as follows: \[T_s = \frac{\dot{Q}}{h_{av}A_s} + T_{鈭瀩\] #Step 4: Calculate the chip surface area#
04

Calculate the chip surface area

The surface area of the chip, \(A_s\), can be calculated using its dimensions: \[A_s = 4\ \cancel{\mathrm{mm}}\times 4\ \cancel{\mathrm{mm}} \times \frac{1\ \mathrm{m}}{1000\ \cancel{\mathrm{mm}}} \times \frac{1\ \mathrm{m}}{1000\ \cancel{\mathrm{mm}}} = 16 \times 10^{-6} \mathrm{~m^2}\] #Step 5: Calculate the average heat transfer coefficient h_av#
05

Calculate the average heat transfer coefficient h_av

The problem doesn't provide the value of \(h_{av}\). However, this can often be found in heat transfer textbooks, or experimental data can be used. For the sake of this solution, let's assume a value of: \[h_{av} = 100 \mathrm{~W/(m^2\cdot K)}\] However, keep in mind that this value might be different for specific cases and experimental conditions. #Step 6: Estimate the surface temperature Ts#
06

Estimate the surface temperature Ts

Now we can finally calculate the surface temperature of the chip using the rearranged equation: \[T_s = \frac{\dot{Q}}{h_{av}A_s} + T_{鈭瀩 \] \[T_s = \frac{30 \times 10^{-3} \mathrm{~W}}{100 \mathrm{~W/(m^2\cdot K)} \times 16 \times 10^{-6} \mathrm{~m^2}} + 25^{\circ} \mathrm{C}\] \[T_s \approx 53.75^{\circ} \mathrm{C}\] The surface temperature of the chip is approximately 53.75掳C.

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Key Concepts

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

Forced Convection Cooling
Forced convection cooling is a process where a fluid, typically air or liquid, is pushed over a surface to remove heat from it. Engineers use this cooling method extensively to manage the temperature of electronic devices, ensuring their reliability and performance. To accomplish this, fans or pumps propel the fluid, increasing the convection heat transfer rate.

In practice, forced convection cooling can take many forms, such as fans cooling a computer's CPU or a pump circulating coolant through a car's engine. This type of cooling is particularly effective because it combines the fluid's motion with its thermal properties to increase heat transfer away from hot surfaces.

For instance, when forced air at a temperature of \(T_{\infty}=25^{\circ} \mathrm{C}\) and a velocity of \(V=10 \mathrm{~m/s}\) is directed over an electronic chip on a circuit board, it carries away the heat dissipating from the chip. This process helps maintain the chip's temperature at safe operating levels, preventing overheating and potential damage.
Heat Transfer Coefficient
The heat transfer coefficient, represented by \(h\), is a critical factor that quantifies the convection heat transfer between a surface and a fluid moving over it. It is measured in watts per square meter per Kelvin (\mathrm{W/(m^2\cdot K)}) and varies based on the fluid properties, flow velocity, and surface condition.

This coefficient is necessary for calculating the heat transfer rate, as shown in the equation \[\dot{Q} = h_{av}A_s(T_s-T_{\infty})\]. In this formula, \(\dot{Q}\) represents the rate of heat transfer, \(h_{av}\) is the average heat transfer coefficient, \(A_s\) is the surface area, \(T_s\) is the surface temperature, and \(T_{\infty}\) is the fluid temperature.

Understanding and accurately determining the heat transfer coefficient is essential for predicting cooling performance in forced convection scenarios. For example, assuming a coefficient of \(h_{av} = 100 \mathrm{~W/(m^2\cdot K)}\) permits the estimation of surface temperature for an electronic chip, guiding in thermal management strategies.
Electronic Thermal Management
Electronic thermal management is a branch of engineering that focuses on regulating the temperature of electronic components. This discipline ensures that devices function within their temperature limits, thereby prolonging their lifespan and maintaining their performance.

Effective electronic thermal management often involves a combination of conduction, convection, and radiation heat transfer methods. However, the convection method, especially forced convection, is one of the most commonly used due to its efficiency in transferring heat away from electronic elements such as chips, processors, and circuit boards.

In the context of thermal management, accurately calculating the surface temperature, as seen in the equation \[T_s = \frac{\dot{Q}}{h_{av}A_s} + T_{\infty}\], demonstrates the importance of parameters like the heat transfer coefficient and the chip's power dissipation. A chip dissipating \(30 \mathrm{~mW}\), for instance, requires careful analysis to avoid overheating, thereby illustrating the role of thermal management in the design and maintenance of electronics.

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Most popular questions from this chapter

For laminar flow over a flat plate, the local heat transfer coefficient \(h_{x}\) is known to vary as \(x^{-1 / 2}\), where \(x\) is the distance from the leading edge \((x=0)\) of the plate. What is the ratio of the average coefficient between the leading edge and some location \(x\) on the plate to the local coefficient at \(x\) ?

A 2-mm-thick layer of water on an electrically heated plate is maintained at a temperature of \(T_{w}=340 \mathrm{~K}\), as dry air at \(T_{\infty}=300 \mathrm{~K}\) flows over the surface of the water (case A). The arrangement is in large surroundings that are also at \(300 \mathrm{~K}\). (a) If the evaporative flux from the surface of the water to the air is \(n_{\mathrm{A}}^{\prime \prime}=0.030 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2}\), what is the corresponding value of the convection mass transfer coefficient? How long will it take for the water to completely evaporate? (b) What is the corresponding value of the convection heat transfer coefficient and the rate at which electrical power must be supplied per unit area of the plate to maintain the prescribed temperature of the water? The emissivity of water is \(\varepsilon_{w}=0.95\). (c) If the electrical power determined in part (b) is maintained after complete evaporation of the water (case B), what is the resulting temperature of the plate, whose emissivity is \(\varepsilon_{p}=0.60\) ?

Experiments to determine the local convection heat transfer coefficient for uniform flow normal to a heated circular disk have yielded a radial Nusselt number distribution of the form $$ N u_{D}=\frac{h(r) D}{k}=N u_{o}\left[1+a\left(\frac{r}{r_{o}}\right)^{n}\right] $$ where both \(n\) and \(a\) are positive. The Nusselt number at the stagnation point is correlated in terms of the Reynolds \(\left(R e_{D}=V D / v\right)\) and Prandtl numbers $$ N u_{o}=\frac{h(r=0) D}{k}=0.814 \operatorname{Re}_{D}^{1 / 2} \mathrm{Pr}^{0.36} $$ Obtain an expression for the average Nusselt number, \(\overline{N u}_{D}=\bar{h} D / k\), corresponding to heat transfer from an isothermal disk. Typically, boundary layer development from a stagnation point yields a decaying convection coefficient with increasing distance from the stagnation point. Provide a plausible explanation for why the opposite trend is observed for the disk.

An industrial process involves the evaporation of water from a liquid film that forms on a contoured surface. Dry air is passed over the surface, and from laboratory measurements the convection heat transfer correlation is of the form $$ \overline{N_{L}}=0.43 \operatorname{Re}_{L}^{0.58} P r r^{\Omega .4} $$ (a) For an air temperature and velocity of \(27^{\circ} \mathrm{C}\) and \(10 \mathrm{~m} / \mathrm{s}\), respectively, what is the rate of evaporation from a surface of \(1-\mathrm{m}^{2}\) area and characteristic length \(L=1 \mathrm{~m}\) ? Approximate the density of saturated vapor as \(\rho_{A, \text { sat }}=0.0077 \mathrm{~kg} / \mathrm{m}^{3}\). (b) What is the steady-state temperature of the liquid film?

Air at a free stream temperature of \(T_{a}=20^{\circ} \mathrm{C}\) is in parallel flow over a flat plate of length \(L=5 \mathrm{~m}\) and temperature \(T_{s}=90^{\circ} \mathrm{C}\). However, obstacles placed in the flow intensify mixing with increasing distance \(x\) from the leading edge, and the spatial variation of temperatures measured in the boundary layer is correlated by an expression of the form \(T\left({ }^{\circ} \mathrm{C}\right)=20+70\) \(\exp (-600 x y)\), where \(x\) and \(y\) are in meters. Determine and plot the manner in which the local convection coefficient \(h\) varies with \(x\). Evaluate the average convection coefficient \(\bar{h}\) for the plate.
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