Chapter 2: One-dimensional Steady State Conduction

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1 1 Chapter : One-imensional Steay State Conution.1 Eamples of One-imensional Conution Eample.1: Plate with Energy Generation an Variable Conutivity Sine k is variable it must remain insie the ifferentiation sign as shown in eq. (.1) T ( k ) + q = 0 (.1) This is a seon orer equation requiring two bounary onitions given in () an (). Diret integration of (.1) gives the solution T(). Eample.: Raial Conution in a Composite Cyliner with Interfae Frition. Sine no heat an be onute through the shaft, it must be at a uniform temperature. Shaft temperature is equal to sleeve temperature at the interfae r = Rs. Temperature istribution in the sleeve is obtaine by solving the heat equation T ( r ) = 0 (.) r r This is a seon orer equation requiring two bounary onitions: Speifie flu at r = Rs [equation (a)] an onvetion at r = Ro [equation (b)]. Diret integration of (.) gives the solution T(r). Eample.3: Composite Wall with Energy Generation For simpliity onsier symmetry an analyze half the system. Sine the two plates are ifferent an sine one has energy generation, two solutions are neee; one for eah plate. The heat equation for the two plates are: T1 q + = 0 k T These equations are seon orer requiring two bounary onitions eah. The four bounary onitions are given by equations ()-(f). = 0 (a) (b)

2 Diret integration of (a) an (b) gives the solutions T 1 () an T ().. Etene Surfaes: Fins..1 The Funtion of Fins Fins are use to: (i) Inrease the heat transfer rate from a surfae at a fie temperature, or (ii) Lower the temperature of a surfae having a fie heat transfer rate... Types of Fins Stuy the fins shown in Fig..5. The base is the en where the fin is attahe to a surfae. The tip is the opposite en. A variable area fin is one in whih the ross setion varies with istane from the base. Eamples are shown in Fig.,5 (b), () an (). Is it lear why () is a variable area fin?..3 Heat Transfer an Temperature Distribution in Fins Note the multi-imensional nature of heat transfer an temperature istribution in the fin of Fig..6. Uner ertain onitions the temperature istribution an be assume one-imensional...4 The Fin Approimation A major mathematial simplifiation in the analysis of fins is the assumption that fin temperature varies with aial istane only (from base to tip). This assumption is vali when the Biot number is small ompare to unity. That is hδ Biot number = Bi = << 1 k δ is a measure of the lateral istane along whih temperature variation is neglete...5 The Heat Equation: Convetion at Surfae Eah fin has its own heat equation epening on its geometry an what takes plae at its surfae. In this setion we present a general formulation for fins with surfae onvetion. The starting point is the seletion of an infinitesimal element in the iretion of temperature hange. In Fig..7 we selet an element of thikness. We apply onservation of energy to the element, as in equation (a). (.4)

3 3 Fourier s law is use to esribe heat ehange by onution, eq. (e). Newton s law is use to esribe heat ehange at the surfae of the element by onvetion, eq. (f). The resulting heat equation for this lass of fins is T 1 A A ( ) T h ( T T ka ( ) + As ) q + = 0 k (.5b) Note that A (), A / an A s / are geometri quantities whih are obtaine one the fin geometry is speifie (given). Unerstaning the physial meaning of A () an () is ruial...6 Determination of A s / A s is the surfae area of the element in Fig..7(b) through whih heat is ehange by onvetion. This area is A s A s = C( ) s (a) C() is the irumferene of the element in ontat with the ambient flui whih ehanges heat with the element by onvetion. From the geometry of Fig..7 (b) an () we obtain 1/ As y = ( ) 1 + s C ( ) (.6a) In many appliations the term y s / is usually small ompare to unity an thus an be neglete (unless it is equal to a onstant)...7 Bounary Conitions Eq. (.5b) is seon orer requiring two bounary onitions. Bounary onitions are base on what takes plae physially at two loations of a fin...8 Determination of Fin Heat Transfer Rate q f For steay state, heat transfer from a fin an be etermine by one of two methos: (1) Conution at the base (Fourier s law) () Convetion at the fin surfae ( Newton's law) q f T (0) = q( 0) = ka (0) (.7)

4 4 Note that it is easier to apply eq. (.7) than (.8). q = q = h[ T ( ) T ] A (.8).9 Appliations: onstant Area Fins with Surfae Convetion The heat equation for onstant area fins is obtaine from eq. (.5b). Note how eq. (.5b) simplifies to eq. (.10) Review the assumptions leaing to this equation. f s A s s θ m θ = 0 (.10) The solution to eq. (.10) is epresse in terms of hyperboli funtions: θ ) = C sinhm C oshm (.11b) ( 1 + This solution is vali for all onstant area fins with surfae onvetion. C1 an C are onstants of integration. They epen on speifi bounary onitions an hange aoringly: Case (i): Speifie base temperature, onvetion at tip. Solutions for θ () an q f are given by equations (.1) an (.13). Case (ii): Speifie base temperature, insulate tip. Solutions for θ () an q f are given by equations (.14) an (.15)...10 Correte Length L A small error in fin solution is introue if heat loss from the tip is neglete. To ompensate for this approimation the fin length is inrease by a small inrement Δ L...11 Fin Effiieny η f This imensionless fator ompares fin heat transfer rate with the maimum possible rate. Maimum fin heat transfer orrespons to a fin whose entire surfae is at the base temperature...1 Moving Fins Eample: A long wire moving through a furnae. The heat equation for a moving fin must take into onsieration the effet of fin veloity. We onsier a fin moving with onstant veloity an ehanging heat by onvetion an raiation. The starting point is the seletion of an infinitesimal element in the iretion of temperature hange (Fig..11) an applying onservation of energy.

5 5 Use Fourier s law, Newton s law an Stefan-Boltzmann raiation law to esribe heat ehange by onution, onvetion an raiation an aount for energy ehange ue to fin motion The resulting heat equation for this lass of fins is T ρ p U T hc ε σ C 4 4 ( T T ) ( T Tsur k k A k A Review the assumptions leaing to this result...13 Appliations of Moving fins Eample.4: Moving Fin with Surfae Convetion ) = 0 The heat equation for this fin is obtaine from eq. (.19) by negleting raiation to obtain T ρ p U k T hc k A ( T T ) = 0 (.19) (.19) This is a seon orer equation with onstant oeffiients. Two bounary onitions are neee. The solution base on bounary onitions (e) an (f) is..14 Variable Area Fins T ) T T T o pu pu h( W + t k k kw t = ep ρ ( ρ ) + (.) ( ) The area through whih heat is onute varies as one moves from the base towars the tip. We formulate the heat equation for two eamples of variable area fins. Both eamples represent speial ases of equation (.5b). The key fator in speializing eq. (.5b) is the etermination of orret epressions for the onution area A () an the onvetion irumferene C(). Case (i): The annular fin. This is a isk of onstant thikness whih is mounte on a tube. Eq. (.5b) reues to r T 1 T + (h / kt)( T T ) = 0 r r (.4) Case (ii): The Straight triangular fin. This is a wege like fin. Eq. (.5b) reues to T 1 T + 1 ( hl / kt) [(1 + ( t / L) ] 1/ ( T T ) = 0 (.5) Note that equations (.4) an (.5) are seon orer with variable oeffiients. Solutions to suh equations are isusse in the following setion..3 Bessel Differential Equations an Bessel Funtions

6 6.3.1 General Form of Bessel Equation The following is a general form of a seon orer equation with variable oeffiients y C [(1 A) B ] + [ C D + B B(1 A) + A C n ] = 0 y + y This equation is known as Bessel equation. Carefully stuy the general features of this equation (etaile on page 33). This equation represents many speial ases epening on the values of A, B, C, D an n. Certain variable area fin equations may be speial ases of this equation. In this ourse the epenent variable y in this equation represents temperature..3. Solutions: Bessel Funtions If a an equation you wish to solve is a speial ase of eq. (.6), proee as follows: (.6) (1) Rewrite your equation suh that the first term of your equation is iential to that of eq. (6.6) () Compare, term by term, your equation with eq. (.6) an etermine the values of the onstants A, B, C, D an n. (3) Depening on the values of D an n, selet one of the four solutions given by equations (.7)-(.30) Solutions to Bessel ifferential equations are epresse in terms of Bessel funtions Forms of Bessel Funtions Bessel funtions represent infinite power series. Equation (.31) is a typial eample Speial Close-form of Bessel Funtions: ointegral n = Suh funtions are epresse in terms of familiar funtions suh as sin, os, sinh an osh..3.5 Speial Relations for n = 1,,, Note relations between ertain Bessel funtions..3.6 Derivatives an Integrals of Bessel Funtions Equations (.39)-(.46) are formulas for the erivatives of Bessel funtions. Equations (.46) an (.47) an Appeni B give formulas for the integral of Bessel funtions..3.7 Tabulation an Graphial Representation of Selete Bessel Funtions

7 7 Reall that the foal in onution is the etermination of temperature istribution. In ertain problems the solution to temperature istribution may be epresse in terms of Bessel funtions. Thus it is helpful to know the general harateristis of some of these funtions. Stuy Table.1. It gives the values of si ommon Bessel funtions when the argument of the funtion = 0 an =. Fig..15 gives the general behavior of si ommon Bessel funtions..4 Equiimensional (Euler) Equation y y + a1 + a0 y = 0 (.49) Note the istint pattern of the oeffiients: multiplies the seon erivative, multiplies 0 the first erivative an multiplies the funtion y. Although this is a seon orer ifferential equation with variable oeffiient, it is not a Bessel equation. It is known as equiimensional or Euler equation. There are three possible solutions to this equation. They are given in equations (.51)- (.53)..5 Graphially Presente Solutions to Fin Heat Transfer Rate q f Solutions to ertain fins may be epresse in terms of Bessel funtions. In suh ases numerial etermination of fin heat transfer rate requires evaluation of infinite power series. Base on suh omputations, graphs of fin effiieny have been onstrute. Knowing fin effiieny the heat transfer rate an be easily etermine. Eamples are shown in Figs..16 an.17.

Math 225B: Differential Geometry, Homework 6

Math 225B: Differential Geometry, Homework 6 ath 225B: Differential Geometry, Homework 6 Ian Coley February 13, 214 Problem 8.7. Let ω be a 1-form on a manifol. Suppose that ω = for every lose urve in. Show that ω is exat. We laim that this onition