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# COMPUTER AIDED DESIGN MECH 403

Rice University

GPA 3.92

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This 9 page Class Notes was uploaded by Shaina Lowe Jr. on Monday October 19, 2015. The Class Notes belongs to MECH 403 at Rice University taught by John Akin in Fall. Since its upload, it has received 29 views. For similar materials see /class/224973/mech-403-rice-university in Mechanical Engineering at Rice University.

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Date Created: 10/19/15

Heat Transfer Concepts Part 3 r ft l 93006 Classical Solutions There are afeW Well ml some he rlrst of these a planar wall wlth a ternperature dlffererlce on each slde Thls ls often appro rn te as lntlnlte wall whlch reduc ernper u h g a sernl s the problernt o e dlrnenslon l study The solutlon 2 shows that the at ret rou h wall ls llnear ln space Therefore the heat ux per unlt areal wlll be constant Ally rlnlte elernent rnodel should gwe the exact result everywhere Planar wall The heattransterthrough l d a slngle layer of elernents through the wall Here lt ls assurned that the analyth solutlon ls not known alloy steel The outer left slde ls kept atl r llll lll lllll lsldel a F a lled The otherfour faces of the body are planes of Symmetry and are automatlcally treated as ln shown ln Flgure l top along wlth the r sulated The rnesh ls esultlng urllform te ternperature dlstrloutlon ls rnore easlly seen wlth a graph o g n rnperature drop dlstrloutlon T e l n o eedge otthe rnesh Thellnear laei is39 legna Y i i3i aeae quot m e 332 eerie 3 1 a v we 3 v we a A 1 wet Eu a I Figure1 Temperatures ora homogeneous wall Pagelotg CopyrlghtJE Aker Allrlghtsreselved at OBOF tater bounds HSIedmtabt r H ta r h e ut r o 0134 BTU5 of powerto mamtam the outer temperature For a ptanarwatt made up of constant thtckness a erS of dtfference WM occur as hnear Changes from one nterface to the next Temn Fahmnh Favamelm Dmlance u nnnn menu D401 UEUB Benn HIDE Figure 2 Graph oftemperature through the wall HFMXN Emserum 1 aaaeeuuz 1 3352mm Vatue u Value ea mus umas75 tamSm mam awnSW2 umzam antterm nmam tamsum HM 7 n m 3375 BYUs m 2 u m 3575 Untx mm z Btuemu Figure 3 Constant heat flux through a wall Page2ot9 CowrtghtJE Akm AH rrgms reserved Cylindrical walls or pipes Another well knovm heat transfer problem with a simple analytic solution is that of radial conduction through an in nite pipe or curved wall In that case the temperature difference varies in a logarithmic mannerthrough the wall thickness 2 That means that the heat ux must also vary through the wall since it passes through more material as the radius increases The example here 4 will be for an alloy steel pipe with an inner radius of 1 0 inches and with a thickness of5 inches us it is very simi ar to the previous example having inner and outer temperatures of 100 F and 0 F respectively In this case each ofthose restraints are applied to cylindrical faces The other fourfaces are insulated and do not require speci c action The geometry a v 4 ery ne mesh and the resulting temperature contours are given In Figure v I 395 AV GEE AI w 5539274 1 221 was Figure 4 Conduction through a cylindrical wall The radial variation of the temperature in the contour plot of Figure 3 might appearto again be linear bu a graph ofthet d red t emperature along a ra ial edge see Figure 5 is actually logarithmic Compa to Figure 2 you see that at a distance of40 through the wall the temperature has dropped more Page 3 of 9 Copyright JE Akin All rights reserved lama Faluenheul 0 n nun n 2am H 4am I Bun I am mun Parametric Distance Figure 5 Radial temperature through a cylindrical wall 154924102 meme 1 mam 1 3192mm 1 zuaenuz 1 0392002 mm amem Figure 6 Heat flux contours through a cylindrical wall n 2am I Ann I EDD n Bun mun Pavamelm Distance Figure 7 Graph of radial heat flux Page 4 of9 Copyright JE Akin All rights reserved EH40 quot L 39 i seeninthe contour plot in Figure 6 and even more clearly in the radial edge heat ux graph of Figure 7 The last example wntch has one temperature unknown permesh node When the 5 degree soltd segment of thure 2 t t topts s ell dtrectton he mtddle of a plane of constant thicknesa Here the tn thure 8 ts generated tn a constant axial 2 plane c early t as on y a few percen as many equattons as the solt mes The otemperature restratnts are applte o the two ctrcularare edges The two s ratght edges and the shell face5 are tnsulated The temperature results agree very closelywlth the much more expensive soltd computattons That ts easily seen by examining the temperature results given tn Figure 10 se the heatfux eonto rs and radtal graph valueS tn thure M are also tn close agreement wtth tne soltd model and the analytic solutton Figure 9 Midsurface shell thermal mesh forpipe segment renp Fahrenhen t uuuem 2 t 22127EIEIE szp Fahrenheit El UUU I 2EU I 400 0600 I EDD 1 EIIJU Palametlm Didancs Figure 10 Pipe segment temperatures from midsurface shell mesh Page 5 of9 CopyrightJ E Akin AH nghts reserved HFluxN BTUJtSin QD 1 5498002 1 5395002 1 4233002 1 3198002 1 2095002 1 3398002 HFIUHH BTUsin Zl 39 UUUU 0200 DAUU 1500 0801 1000 Parametric Distance Figure 11 Midsurface shell heat flux result for the pipe Conducting rod with convection Most conduction problems also involve free convection That usually gives a steeper change in temperature over a region Here a segment of a circular rod Figure 12 is examined where the length is only two times the diameter That is near the lower limit where you might want to expect a one dimensional approximation to be accurate Convection occurs on the outer surface while one end is kept at 100 F The other three symmetry surfaces in the model are insulated Any wedge angle could have been used but a value of 30 degrees was picked to give good element aspect ratios A 4A Wz i 39mvm f e a mu r Lia Lg v mw g nm 4 mummy Figure 12 Circular rod segment with an end temperature and convection Page 6 of 9 Copyright JE Akin All rights reserved Myers 3 gives the onedimensional solution for a rod conducting heat along its interior and convection that heat away at its surface The temperature is shown to change with axial position X as a hyperbolic cosine of mx where m2 h P Lk A L is a ratio of convection strength to conduction strength It involves the surface convection coef cient h the perimeter P of the conducting area A over the length L and the material thermal conductivity k T ical temperature distributions for a low value of m are seen in Figure 13 The surrounding free convection iris assumed to be at F Comparing the centerline and surface grap s o the temperature there is very little difference and they both follow the onedimensional approximation given by Myers Notice that the far end plane temperature does not match that ofthe surrounding air A similar comparison of the heat ux magnitude is given in Figure 14 That gure shows a much larger difference between the centerline and surface heat ux But the average of the two graphs is still quite close to the analytic approximation given by Myers Temp Fahrenheit 39 39 l 0024002 animal V a 322mm 7 405mm 5 542mm 5 00mm 4 352mm A Imam 3 272mm 2 43mm l 392mm Temp Fahrenheit Temp lev fell 020 040 000 080 100 000 020 040 050 080 100 ParamemcDistance ParametricDistance Centerline Surface Figure 13 Rod temperature distributions for a small m value It is not uncommon for the user supplied convection coef cient to be in error due to measurement errors or errors occurring in a units conversion As an example the above stud was rerun with the 39 n oef cient increased by a factor of 10 That is the convection heat transfer mode was increased relative to the conduction mode m was approximately tripled The new temperatures in Figure 15 are signi cantly different from those of Figure 13 The surface and centerlinetemperature graphs are still about the same and still follow the hyperbolic cosine change given by Myers However the temperatures in the distal half ofthe bar have dropped to rapidly approach or match the temperature ofthe surrounding air Page 7 of9 Copyright JE Akin All rights reserved HFluxN ETUsln Z 39 v 7 Menu a 43mm 39 57DEUU2 A BENCH a 292mm 3 532mm 2 392mm 2 19mm 1 432mm 732am smmua SAW2 HFluxN Fmism HFluxN mu Parametric Distance Parametric Distance Centerline Surface Figure 14 Typical heat flux magnitude results for asmall m Limiting values of the convection coefficient The convection coef cient has lower and upper bounds of and N respectively They have different physical effects in a study A low value ofh causes the surface to a roach an insulated state while a high value causes the surface to approach a restraint of a speci ed temperature The latter state is what is seen in Figure 15 The distal end of the part is responding as if it had a restraint temperature of 0 F applied to it These two limits on h are also re ected in terms of the temperature contour lines The lower limit causes the contour lines to approach being perpendicular to the surface as they do for insulated boundaries Likewise the up er limit causes the temperature contours to approach being parallel to the surface as they would if it was subjected to a constant temperature restraint Page 8 of9 Copyright JE Akin All rights reserved Temp Fahrenhevi 1 E11102 a DUMJUT v v v E 0132mm 7 memm E Imam 5 DIe ZIUl 4 1112mm 3 Elle l 2 022mm 1 13251001 2 menu Tamp leveulvell Temp Falvreuhell 020 040 060 080 100 Pavametnc Dlstance Parametnc Dvslance Centerline Sur ce Figure 15 Temperatures when h is increased by a factor of 10 Closure These examples illustrate the value of using analytic approximations to estimate and validate the results from a nite element study The rst example also shows that if an analytic solution is not available for validating a solid w m i an 39 39 39 39 quot can be use ou aways should estimate the expected results before you start a study and to validate the study results when nished References Akin JE Finite Element Analysis with Error Estimates Elsevier 2005 Chapman A Fundamentals of eat Transfer Macmillan 1984 GE Myers Analytical Methods in Conduction Heat Transfer McGraw 1971 Tuba S Wright WB Eds Pressure Vessel and Piping Computer Programs Veri cation A ME 1972 ewwe Page 9 of9 Copyright JE Akin All rights reserved

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