LECTURER: PM DR MAZLAN ABDUL WAHID PM Dr Mazlan Abdul Wahid
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1 M 445 LU: M D MZL BDUL WID hapte teady-tate tate One Dimensional eat onduction M bdul Wahid UM aculty of Mechanical ngineeing Univesiti eknologi Malaysia
2 One-Dimensional teady-tate onduction onduction poblems may involve multiple diections and timedependent conditions Inheently complex Difficult to detemine tempeatue distibutions One-dimensional steady-state models can epesent accuately numeous engineeing systems In this chapte we will: Lean how to obtain tempeatue pofiles fo common geometies with and without heat geneation. Intoduce the concept of themal esistance and themal cicuits Intoduce to the analysis of one dimensional conduction analysis on extended sufaces ssumptions - dimensional heat tansfe - Isothemal sufaces - teady state
3 ouie s law hemal esistance concept - onduction hemal wold lectical wold
4 hemal esistance concept - onvection he lane Wall onside a simple case of onedimensional conduction in a plane wall, sepaating two fluids of diffeent tempeatue, without enegy geneation empeatue is a function of x eat is tansfeed in the x-diection Must conside onvection fom hot fluid to wall onduction though wall onvection fom wall to cold fluid Begin by detemining tempeatue distibution within the wall,,,h s, q x ot fluid x0 xl x old fluid,,h s,, 4
5 k x x + empeatue Distibution k y y + d d k 0 dx dx k y z eat diffusion equation in the x-diection fo steady-state conditions, with no enegy geneation: Bounday onditions: + q ρc p ( 0 s,, ( L s, empeatue pofile, assuming constant k: x + L ( x ( s, s, s, t q x is constant (. empeatue vaies linealy with x hemal esistance Based on the pevious solution, the conduction heat tansfe ate can be calculated: q d k dx k L ( ( s, s, (.a x s, s, L / k imilaly fo heat convection, ewton s law of cooling applies: q x ( h( / h nd fo adiation heat tansfe: q (.b ( s su h ( s su (.c / h ad ecall electic cicuit theoy - Ohm s law fo electical esistance: otential Diffeence lectic cuent esistance 5
6 hemal esistance We can use this electical analogy to epesent heat tansfe poblems using the concept of a themal cicuit (equivalent to an electical cicuit. q Oveall Diving oce esistance oveall ompae with equations.a-.c he tempeatue diffeence is the potential o diving foce fo the heat flow and the combinations of themal conductivity, convection coefficient, thickness and aea of mateial act as a esistance to this flow: L, t, conv, t, k h t, cond ad h,,,h hemal esistance fo lane Wall s, q x old fluid,,h s, ot fluid x0 xl x, q x, s / h s, s L / k s, / h, s,,, In tems of oveall tempeatue diffeence:,, q x tot tot + h L k + h 6
7 7 omposite Walls xpess the following geomety in tems of a an equivalent themal cicuit. omposite Walls What is the heat tansfe ate fo this system? ltenatively U q U q t tot x whee U is the oveall heat tansfe coefficient and the oveall tempeatue diffeence. ] (/ / ( / ( / ( [(/ h 4 k L k L k L h U B B tot
8 8 hemal esistance concept - onvection & adiation
9 9 xample single laye wall xample multi laye wall
10 0 xample single laye window
11 xample two laye window omposite Walls with paallel esistances o esistances in seies: tot n o esistances in paallel: / tot / +/ + +/ n (a ufaces nomal to the x- diection ae isothemal (b ufaces paallel to x- diection ae adiabatic
12 omposite Walls with paallel esistances o esistances in seies: tot n o esistances in paallel: / tot / +/ + +/ n omposite Walls with paallel esistances
13 aallel heat conduction ttention he esult obtained will be somewhat appoximate, since the sufaces of the thid laye will pobably not be isothemal, and heat tansfe between the fist two layes is likely to occu.
14 4 eat loss though a composite wall ontact esistance
15 q x hc ontact esistance int he tempeatue dop acoss the inteface between mateials may be appeciable, due to suface oughness effects, leading to ai pockets. We can define themal contact esistance: c int / h c 5
16 hemal contact esistance is obseved to decease with deceasing suface oughness and inceasing inteface pessue, as expected. Most expeimentally detemined values of the themal contact esistance fall between and m /W (the coesponding ange of themal contact conductance is 000 to 00,000 W/m. 6
17 ontact esistance of tansistos 7
18 8 omposite Walls with contact esistances adial ystems-ylindical oodinates onside a hollow cylinde, whose inne and oute sufaces ae exposed to fluids at diffeent tempeatues empeatue distibution
19 empeatue Distibution - hemal esistance fo cylinde k + k + k + q ρc φ φ z z p t fo constant k d d d k 0 d 0 d d d d eat diffusion equation (eq..5 in the -diection fo steady- state conditions, with no enegy geneation: d d d d ouie s law: q k k L const ( π Bounday onditions: ( s,, ( s, empeatue pofile, assuming constant k: ( + s, s, ( ln s, ln( / Logaithmic tempeatue distibution theefoe empeatue Distibution - themal esistance fo cylinde ( + s, s, ( ln s, ln( / q q q d k k( πl d π kl ( ln( / ln( / / π kl cyl d d ln( / / π kl 9
20 empeatue Distibution - themal esistance fo cylinde hemal esistance fo cylinde Based on the pevious solution, the conduction hea tansfe ate can be calculated: ouie s law: d d q k k ( π L const d d q x ( ( ( πlk s, s, s, s, s, ln( / ln( / /(πlk In tems of equivalent themal cicuit: q x tot,, ln( / + + h (π L πkl h tot (π L t, cond s, 0
21 omposite Walls xpess the following geomety in tems of a an equivalent a an equivalent themal cicuit. omposite Walls What is the heat tansfe ate? whee U is the oveall heat tansfe coefficient. If π L: ln ln ln h k k k h U B altenatively we can use π L, π L etc. In all cases: t U U U U 4 4
22 pheical oodinates ouie s law: d q sph k d d k(4π d tating fom ouie s law, acknowledging that q is constant, independent of, and assuming that k is constant, deive the equation descibing the conduction heat tansfe ate. What is the themal esistance? empeatue Distibution - themal esistance fo sphee q sph q sph / 4 π sph d / 4π k ( s ( / 4π k k s ( / d (
23 onduction with Geneation hemal enegy may be geneated o consumed due to convesion fom some othe enegy fom. If themal enegy is geneated in the mateial at the expense of some othe enegy fom, we have a souce: is +ve Deceleation and absoption of neutons in a nuclea eacto xothemic eactions onvesion of electical to themal enegy:. q & V g I e V q. whee I is the cuent, e the electical esistance, V the volume of the medium If themal enegy is consumed we have a sink: ndothemic eactions. q is -ve 4 he lane Wall onside one-dimensional, steady-state conduction in a plane wall of constant k, with unifom geneation, and asymmetic suface conditions: eat diffusion equation (eq.. : d q. dx q + 0 k Geneal olution:. q kk x + x + Bounday onditions: ( L s,, ( L s, 5
24 . empeatue ofile ql x ( s, s, x s, + s, ( x + k L + (. L ofile is paabolic. eat flux not independent of x What happens when: q... 0, q inceases, q < 0? 6 ymmetical Distibution When both sufaces ae maintained at a common tempeatue, s, s, s. ql x ( x + k L ( (.4a s What is the location of the maximum tempeatue? ( x s max max x L (.4b 7 4
25 ymmetical Distibution ote that at the plane of symmety: d dx x 0 0 q" x 0 quivalent to adiabatic suface 0 8 alculation of suface tempeatue s In equations (.4a and (.4b the suface tempeatue, s is needed. Bounday condition at the wall: k d dx x L h( ubstituting (d/dx xl fom equation (.4a:. s ql (.5 h s + 9 5
26 xample he steady-state tempeatue distibution in a composite plane wall of thee diffeent mateials, each of constant themal conductivity, is shown in the schematic below. a Does heat geneation occu in any of sections,b, o? b Based on the schematic, what is the bounday condition at location (4? c omment on the elative magnitudes of q and q. d omment on the elative magnitudes of k and k B. 0 xample plane wall of thickness 0. m and themal conductivity 5 W/m.K having unifom volumetic heat geneation of 0. MW/m is insulated on one side, while the othe side is exposed to a fluid at 9. he convection heat tansfe coefficient between the wall and the fluid is 500 W/m.K. Detemine the maximum tempeatue in the wall. 6
27 adial ystems ylindical (ube Wall olid ylinde (icula od pheical Wall (hell olid phee eat diffusion equation in the -diection fo steadystate conditions: Geneal olution: adial ystems. d d q k + d d k. q + ln + k d Bounday onditions: 0, ( o s d 0 empeatue pofile:. q o ( + k 4 o q( π o L h(π L( o s s (.5 alculation of suface tempeatue: and s 0 qo + h,h L 7
28 xample cylindical shell of inne and oute adii i and o, espectively, is filled with a heat-geneating mateial that povides a unifom volumetic geneation ate. he inne suface is insulated, while the oute suface of the shell is exposed to a fluid with a convection coefficient h. a Obtain an expession fo the steady-state tempeatue distibution ( in the shell. b Detemine an expession fo the heat ate q ( o at the oute adius of the shell in tems of the heat geneation ate and the shell dimensions 4 8
29 9 itical adius of insulation Optimum thickness of insulation
30 xtended ufaces (ins n extended suface (also know as a combined conduction-convection system o a fin is a solid within which heat tansfe by conduction is assumed to be one dimensional, while heat is also tansfeed by convection (and/o adiation fom the suface in a diection tansvese to that of conduction 5 xtended ufaces (ins xtended sufaces may exist in many situations but ae commonly used as fins to enhance heat tansfe by inceasing the suface aea available fo convection (and/o adiation. hey ae paticulaly beneficial when h is small, as fo a gas and natual convection. olutions fo vaious fin geometies can be found in the liteatue (see fo example in textbook. 6 0
31 xtended sufaces- ins hee ae two ways to incease the ate of heat tansfe: to incease the convection heat tansfe coefficient h o to incease the suface aea. ins enhance heat tansfe fom a suface by exposing a lage suface aea to convection and adiation. in equation o
32 Infinitely long fin egligible heat loss fom the fin tip
33 in efficiency fficiency of fins (I
34 4 fficiency of fins (II in effectiveness (I
35 5 in effectiveness (II ope length of a fin
36 hemal esistance 6
Chapter 2 ONE DIMENSIONAL STEADY STATE CONDUCTION. Chapter 2 Chee 318 1
hapte ONE DIMENSIONAL SEADY SAE ONDUION hapte hee 38 HEA ONDUION HOUGH OMPOSIE EANGULA WALLS empeatue pofile A B X X 3 X 3 4 X 4 Χ A Χ B Χ hapte hee 38 hemal conductivity Fouie s law ( is constant) A A
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