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Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

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$**Heat Transfer from Extended Surfaces** ### Introduction This topic explores the heat transfer from extended surfaces, often referred to as fins. The analysis involves solving differential equations subject to specific boundary conditions. ### General Solution The temperature distribution along the fin is expressed as: \[ \theta = c_1 e^{mx} + c_2 e^{-mx} \] ### Case A: Boundary Conditions – Convection at the Tip - At $ x = 0 $: $ T = T_0 $ - At $ x = L $: \[ -k \left( \frac{dT}{dx} \right)_{x=L} = h(T_L - T_\infty) \] ### Temperature Profile Solution The temperature profile along the fin is given by: \[ \frac{\theta}{\theta_0} = \frac{\cosh[m(L-x)] + \frac{(h/mx)\sinh[m(L-x)]}}{\cosh mL + \frac{(h/mk)\sinh mL}} \] (Equation 17-40) ### Heat Transfer Rate The heat transfer rate from the fin is calculated using: \[ q = kA m \theta_0 \frac{\sinh mL + (h/mk)\cosh mL}{\cosh mL + (h/mk)\sinh mL} \] (Equation 17-46) ### Diagram Description The diagram includes schematic representations of two different types of extended surfaces or fins illustrating heat flow. The left image shows a cylindrical fin, while the right image displays a rectangular fin. These equations and concepts are crucial for understanding how fins enhance heat dissipation in various engineering applications.$

Transcribed Image Text:**Heat Transfer from Extended Surfaces** ### Introduction This topic explores the heat transfer from extended surfaces, often referred to as fins. The analysis involves solving differential equations subject to specific boundary conditions. ### General Solution The temperature distribution along the fin is expressed as: \[ \theta = c_1 e^{mx} + c_2 e^{-mx} \] ### Case A: Boundary Conditions – Convection at the Tip - At $ x = 0 $: $ T = T_0 $ - At $ x = L $: \[ -k \left( \frac{dT}{dx} \right)_{x=L} = h(T_L - T_\infty) \] ### Temperature Profile Solution The temperature profile along the fin is given by: \[ \frac{\theta}{\theta_0} = \frac{\cosh[m(L-x)] + \frac{(h/mx)\sinh[m(L-x)]}}{\cosh mL + \frac{(h/mk)\sinh mL}} \] (Equation 17-40) ### Heat Transfer Rate The heat transfer rate from the fin is calculated using: \[ q = kA m \theta_0 \frac{\sinh mL + (h/mk)\cosh mL}{\cosh mL + (h/mk)\sinh mL} \] (Equation 17-46) ### Diagram Description The diagram includes schematic representations of two different types of extended surfaces or fins illustrating heat flow. The left image shows a cylindrical fin, while the right image displays a rectangular fin. These equations and concepts are crucial for understanding how fins enhance heat dissipation in various engineering applications.

1. As more and more components are placed on a single integrated circuit (chip), the amount of heat that is dissipated continues to increase. However, this increase is limited by the maximum allowable chip operating temperature. Consider a 15 mm x 15 mm square chip. In order to improve the heat dissipation, a five by five array of copper pin fins are to be joined to the outer surface of the chip. The fins have a diameter of 1.5 mm and a length of 15 mm. Cooling air can be supplied at T∞ = 25°C, and the maximum allowable chip base temperature is T₀ = 75°C. The convective heat transfer coefficient is h = 250 W/m²·K, and the copper has a thermal conductivity of k = 401 W/m·K.

Consider two cases: Case A: Convecting Tip, and Case B: Adiabatic Tip

And for each calculate the following at steady state:

(a) Heat transfer rate from a single fin.
(b) Efficiency of a single fin.
(c) Total heat transfer rate of the array.

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