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HW# 2 /Tutorial # 2 WWWR Chapter 17 ID Chapter 3

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## HW# 2 /Tutorial # 2 WWWR Chapter 17 ID Chapter 3

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**Tutorial #2**WWWR#17.4, 17.13, 17.2, 17.39. To be discussed on Jan. 28, 2014. By either volunteer or class list. HW# 2 /Tutorial # 2WWWR Chapter 17ID Chapter 3**Tutorial #2**WWWR #17.39: Line 2: The fins are made of aluminum, they are 0.3cm thick each. #17.2 The following correction should be made. (i) arithmetic mean area p(r0+ri) HW# 2 /Tutorial # 2Hints / Corrections**One-Dimensional Conduction**Steady-state conduction, no internal generation of energy For one-dimensional, steady-state transfer by conduction i = 0 rectangular coordinates i = 1 cylindrical coordinates i = 2 spherical coordinates**1/Rc=1/Ra+1/Rb**Ra Rb Equivalent resistance of the parallel resistors Ra and Rb is Rc**Adapted from Heat and Mass Transfer – A Practical**Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007. Thus, insulating the pipe may actually increase the rate of heat transfer instead of decreasing it.**For steady-state conduction in the x direction without**internal generation of energy, the equation which applies is Where k may be a function of T. In many cases the thermal conductivity may be a linear function temperature over a considerable range. The equation of such a straight-line function may be expressed by k = ko(1 + ßT) Where koand ß are constants for a particular material**One-Dimensional Conduction With Internal Generation of**Energy**. .**q = qL [ 1 + ß (T - T L)] Plane Wall with Variable Energy Generation The symmetry of the temperature distribution requires a zero temperature gradient at x = 0. The case of steady-state conduction in the x direction in a stationary solid with constant thermal conductivity becomes**Detailed derivation for the transformation**F = C + s q**Detailed Derivation for Equations 17-25**Courtesy by all CN5 Grace Mok, 2003-2004**Detailed Derivation for Equations 17-25**Courtesy by all CN5 Grace Mok, 2003-2004**Heat Transfer from Finned Surfaces**• Temperature gradient dT/dx, • Surface temperature, T, • Are expressed such that T is a function of x only. • Newton’s law of cooling • Two ways to increase the rate of heat transfer: • increasing the heat transfer coefficient, • increase the surface area fins • Fins are the topic of this section. Adapted from Heat and Mass Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.**(A)**For constant cross section and constant thermal conductivity Where • Equation (A) is a linear, homogeneous, second-order differential equation with constant coefficients. • The general solution of Eq. (A) is • C1 and C2 are constants whose values are to be determined from the boundary conditions at the base and at the tip of the fin. (B)**Boundary Conditions**Several boundary conditions are typically employed: • At the fin base • Specified temperatureboundary condition, expressed as: q(0)= qb=Tb-T∞ • At the fin tip • Specified temperature • Infinitely Long Fin • Adiabatic tip • Convection (and combined convection). Adapted from Heat and Mass Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.**How to derive the functional dependence of**for a straight fin with variable cross section area Ac = A = A(x)?**General Solution for Straight Fin with Three Different**Boundary Conditions**In set(a)**Known temperature at x = L In set(b) Temperature gradient is zero at x = L In set(c) Heat flow to the end of an extended surface by conduction be equal to that leaving this position by convection.**Detailed Derivation for Equations 17-36 (Case a).**Courtesy by CN3 Yeong Sai Hooi 2002-2003**Detailed Derivation for Equations 17-38 (Case b for extended**surface heat transfer). Courtesy by CN3 Yeong Sai Hooi, 2002-2003**Detailed Derivation for Equations 17-40 (Case c for extended**surface heat transfer). Courtesy by all CN4 students, presented by Loo Huiyun, 2002-2003**Detailed Derivation for Equations 17-46 (Case c for extended**surface heat transfer). Courtesy by all CN4 students, presented by Loo Huiyun, 2002-2003**Infinitely Long Fin (Tfintip=T) Adapted from Heat and Mass**Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007. • For a sufficiently long fin the temperature at the fin tip approaches the ambient temperature Boundary condition:q(L→∞)=T(L)-T∞=0 • When x→∞ so does emx→∞ C1=0 • @ x=0: emx=1 C2= qb • The temperature distribution: • heat transfer from the entire fin**Actual heat transfer rate from the fin**Ideal heat transfer rate from the fin if the entire fin were at base temperature Fin Efficiency Adapted from Heat and Mass Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007. • To maximize the heat transfer from a fin the temperature of the fin should be uniform (maximized) at the base value of Tb • In reality, the temperature drops along the fin, and thus the heat transfer from the fin is less • To account for the effect we define a fin efficiency or