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An array of electronic chips is mounted within a sealed rectangular enclosure, and cooling is implemented by attaching an aluminum heat \(\operatorname{sink}(k=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The base of the heat sink has dimensions of \(w_{1}=w_{2}=\) \(100 \mathrm{~mm}\), while the 6 fins are of thickness \(t=10 \mathrm{~mm}\) and pitch \(S=18 \mathrm{~mm}\). The fin length is \(L_{f}=50 \mathrm{~mm}\), and the base of the heat sink has a thickness of \(L_{b}=10 \mathrm{~mm}\). If cooling is implemented by water flow through the heat sink, with \(u_{\infty}=3 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=17^{\circ} \mathrm{C}\), what is the base temperature \(T_{b}\) of the heat sink when power dissipation by the chips is \(P_{\text {elec }}=1800 \mathrm{~W}\) ? The average convection coefficient for surfaces of the fins and the exposed base may be estimated by assuming parallel flow over a flat plate. Properties of the water may be approximated as \(k=0.62 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=995 \mathrm{~kg} / \mathrm{m}^{3}\), \(c_{p}=4178 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \nu=7.73 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\), and \(\operatorname{Pr}=5.2\).

Step by step solution

01

Calculate the Fin Efficiency

We start by calculating the fin efficiency. We can use the formula for the fin efficiency of a rectangular fin as follows: \( \eta = \frac{\tanh \left(\frac{h SK}{2 k_{al} t^2}\right)}{\frac{h SK}{2 k_{al} t^2}} \) where \(h\) is the convection coefficient, \(S\) is the fin spacing, \(K_{al}\) is the thermal conductivity of the aluminum heat sink, and \(t\) is the fin thickness. But before calculating the fin efficiency, we need to first estimate the average convection coefficient.

02

Estimate the Average Convection Coefficient

To estimate the average convection coefficient, we can use the correlation for flow over a flat plate: \( h = \frac{k_{w}}{L_{b}} \cdot 0.037 Re^{4 / 5} Pr^{1/3} \) where, \(k_{w}\) is the thermal conductivity of water, \(L_{b}\) is the base length of the heat sink, and \(Re\) and \(Pr\) are the Reynolds and Prandtl numbers for water flow, respectively. The Reynolds number is calculated as: \( Re = \frac{u_{\infty} L_{b}}{\nu} \) Substitute the given values to calculate the Reynolds number and convection coefficient.

03

Calculate Fin Efficiency

Now that we have estimated the average convection coefficient, we can calculate the fin efficiency using the formula from Step 1. Substitute the given values and the calculated convection coefficient to find the fin efficiency:

04

Determine the Overall Heat Transfer Area

With the fin efficiency, we can determine the overall heat transfer area of the fins and the exposed base: \( A_{total} = A_{base} + (N \cdot \eta \cdot A_{fin}) \) where \(N\) is the number of fins and \(A_{base}\) and \(A_{fin}\) represent the exposed base area and the single fin surface area, respectively.

05

Calculate Total Heat Transfer

The total heat transfer from the heat sink can be calculated using the formula: \( Q = h A_{total} (T_{b} - T_{\infty}) \) Substitute the given values, calculated convection coefficient, and overall heat transfer area into the equation.

06

Determine Base Temperature

Now, we can determine the base temperature \(T_{b}\) based on the given power dissipation: \( P_{elec} = Q \) Solve for the base temperature \(T_{b}\) using the total heat transfer value. This will give you the base temperature of the heat sink when the power dissipation of the chips is \(1800 W\).

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Key Concepts

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

Fin Efficiency

Fin efficiency plays a critical role in the design and analysis of heat sinks in thermal management systems. It represents the ratio of actual heat transfer to the ideal heat transfer from a fin, assuming that the entire fin is at the base temperature. The efficiency depends on several factors, including the material's thermal conductivity, the fin's geometry, and the convection coefficient.

The mathematical expression to calculate this efficiency considers the physical and thermal properties, allowing us to evaluate the effectiveness of the fins in dissipating heat. In practice, higher fin efficiency means that the fin material and design are highly effective in transferring heat away from the component such as electronic chips.

Convection Coefficient

The convection coefficient, represented as 'h' in thermal analysis, measures the convective heat transfer per unit area and per degree of temperature difference between the object's surface and the surrounding fluid. It's a crucial variable when assessing the cooling performance of heat sinks.

Estimation of the average convection coefficient for a heat sink involves correlations derived for particular flow conditions, like those over a flat plate. The value of 'h' contrasts across different parts of the heat sink, influenced by both the velocity of the fluid flow and its properties.

Reynolds Number

Reynolds number is a dimensionless value used to predict the flow regime—laminar, transitional, or turbulent flow—in fluid mechanics. It's a ratio that compares inertial forces to viscous forces within the fluid flow. For heat sinks cooled by fluid flow, like water, a higher Reynolds number generally indicates a turbulent flow, which enhances the convection coefficient and thereby improves the cooling performance.

In our case, using the provided flow velocity, fluid properties, and characteristic length, the Reynolds number was calculated to determine the average convection coefficient. Understanding and calculating the Reynolds number helps in predicting the nature of the flow and, thus, the performance of the heat sink.

Heat Transfer

Heat transfer is a fundamental concept in thermal systems, encompassing the movement of thermal energy from one place to another. It occurs through three main mechanisms: conduction, convection, and radiation. In the context of a heat sink, conduction occurs within the solid material, while convection is responsible for heat removal from the heat sink to the surrounding fluid.

To determine the thermal performance of a heat sink, we calculate the total heat transfer, which considers the area for heat dissipation (taking into account the fin efficiency and the number of fins) along with the temperature difference between the base of the heat sink and the flowing fluid. The given power dissipation by the electronic chips must match the heat transferred away by the heat sink to maintain the components at a safe operating temperature.

Thermal Conductivity

Thermal conductivity, denoted as 'k', is a material property that indicates how well a material can conduct heat. High thermal conductivity materials are often chosen for heat sink construction as they can quickly spread heat away from heat sources like electronic components.

In our exercise, the aluminum's high thermal conductivity is essential for distributing the heat efficiently through the fins and into the surrounding water. It's a key parameter when calculating both the fin efficiency and the convection coefficient, ultimately affecting how well the heat sink can keep the electronic chips at the desired temperature.