Complex Heat Transfer Dimensional Analysis

Transcription

1 Lectues 4-5 CM30 Heat ansfe /8/06 CM30 anspot I Pat II: Heat ansfe Complex Heat ansfe Dimensional Analysis Pofesso Faith Moison Depatment of Chemical Engineeing Michigan echnological Uniesity (what hae we been up to?) Examples of (simple, D) Heat Conduction

2 Lectues 4-5 CM30 Heat ansfe /8/06 Examples of (simple, D) Heat Conduction But these ae highly simplified geometies 3 How do we handle complex geometies, complex flows, complex machiney? less hot cold less cold Q in W s, on hot Pocess scale 4

3 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis (Answe: Use the same techniques we hae been using in fluid mechanics) less hot Q in W s, on Engineeing Modeling cold less cold Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, e.g. add oughness lengthscale) Design additional expeiments Iteate until useful coelations esult hot Pocess scale 5 Complex Heat ansfe Dimensional Analysis Expeience with Dimensional Analysis thus fa: Flow in pipes at all flow ates (lamina and tubulent) Solution: Naie-Stokes, Re, F, /, dimensionless wall foce ; Re, / Rough pipes Non-cicula conduits Solution: add additional length scale; then nondimensionalie Solution: Use hydaulic diamete as the length scale of the flow to nondimensionalie Flow aound obstacles (sphees, othe complex shapes Solution: Naie-Stokes, Re, dimensionless dag ; Re Bounday layes Solution: wo components of elocity need independent lengthscales 6 3

4 Lectues 4-5 CM30 Heat ansfe /8/06 ubulent flow (smooth pipe) Rough pipe f Noncicula coss section Aound obstacles Sphees, disks, cylindes Re 7 ubulent flow (smooth pipe) Rough pipe f hese hae been exhilaating ictoies Noncicula coss section fo dimensional analysis Aound obstacles Sphees, disks, cylindes Re 8 4

5 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Now, moe to heat tansfe: Foced conection heat tansfe fom fluid to wall Solution:? Natual conection heat tansfe fom fluid to wall Solution:? Radiation heat tansfe fom solid to fluid Solution:? bulk fluid solid wall 9 Complex Heat ansfe Dimensional Analysis Now, moe to heat tansfe: Foced conection heat tansfe fom fluid to wall Solution:? Natual conection heat tansfe fom fluid to wall Solution:? Radiation heat tansfe fom solid to fluid Solution:? bulk fluid We hae aleady stated using the esults/techniques of dimensional analysis though defining the heat solid wall tansfe coefficient, 0 5

6 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Now, moe to heat tansfe: Foced conection heat tansfe fom fluid to wall Solution:? Natual conection heat tansfe fom fluid to wall Solution:? Radiation heat tansfe fom solid to fluid Solution:? bulk fluid We hae aleady stated using the esults/techniques of dimensional analysis though defining the heat solid wall tansfe coefficient, (ecall that we did this in fluids too: we used the Re coelation (Moody chat) long befoe we knew whee that all came fom) Handy tool: Heat ansfe Coefficient bulk fluid solid wall (x) bulk bulk wall in liquid wall x wall he tempeatue aiation nea-wall egion is caused by complex phenomena that ae lumped togethe into the heat tansfe coefficient, h x in solid 6

7 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis he flux at the wall is gien by the empiical expession known as Newton s Law of Cooling his expession sees as the definition of the heat tansfe coefficient. q x A h bulk wall depends on: geomety fluid elocity fluid popeties tempeatue diffeence 3 Complex Heat ansfe Dimensional Analysis he flux at the wall is gien by the empiical expession known as Newton s Law of Cooling his expession sees as the definition of the heat tansfe coefficient. q x A h bulk wall o get alues of fo aious situations, we need to measue data and ceate data coelations (dimensional analysis) depends on: geomety fluid elocity fluid popeties tempeatue diffeence 4 7

8 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Complex Heat tansfe Poblems to Sole: Foced conection heat tansfe fom fluid to wall Solution:? Natual conection heat tansfe fom fluid to wall Solution:? Radiation heat tansfe fom solid to fluid Solution:? he functional fom of will be diffeent fo these thee situations (diffeent physics) Inestigate simple poblems in each categoy, model them, take data, coelate 5 Complex Heat ansfe Dimensional Analysis Chosen poblem: Foced Conection Heat ansfe Solution: Dimensional Analysis Following pocedue familia fom pipe flow, What ae goening equations? Scale factos (dimensionless numbes)? Quantity of inteest? Heat flux at the wall 6 8

9 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Geneal Enegy anspot Equation (micoscopic enegy balance) As fo the deiation of the micoscopic momentum balance, the micoscopic enegy balance is deied on an abitay olume, V, enclosed by a suface, S. S ds nˆ ˆ Gibbs notation: C p t k S V see handout fo component notation 7 Complex Heat ansfe Dimensional Analysis Geneal Enegy anspot Equation (micoscopic enegy balance) ˆ C p t ate of change conection k S conduction (all diections) souce (enegy geneated pe unit olume pe time) elocity must satisfy equation of motion, equation of continuity see handout fo component notation 8 9

10 Lectues 4-5 CM30 Heat ansfe /8/06 0 Note: this handout is on the web: Equation of enegy fo Newtonian fluids of constant density,, and themal conductiity, k, with souce tem (souce could be iscous dissipation, electical enegy, chemical enegy, etc., with units of enegy/(olume time)). CM30 Fall 999 Faith Moison Souce: R. B. Bid, W. E. Stewat, and E. N. Lightfoot, anspot Pocesses, Wiley, NY, 960, page 39. Gibbs notation (ecto notation) p C p S C k t ˆ ˆ Catesian (xy) coodinates: p p y x C S y x C k y x t ˆ ˆ Cylindical () coodinates: p p C S C k t ˆ ˆ Spheical () coodinates: p C k t sin sin sin ˆ sin 9 R Example: Heat flux in a cylindical shell Assumptions: long pipe steady state k = themal conductiity of wall h, h = heat tansfe coefficients What is the steady state tempeatue pofile in a cylindical shell (pipe) if the fluid on the inside is at b and the fluid on the outside is at b? ( b > b ) Coole fluid at b Hot fluid at b R REVIEW REVIEW Foced-conection heat tansfe 0

11 Lectues 4-5 CM30 Heat ansfe /8/06 Now: How do deelop coelations fo h? Conside: Heat-tansfe to fom flowing fluid inside of a tube foced-conection heat tansfe = coe bulk tempeatue o = wall tempeatue (,,) = temp distibution in the fluid In pinciple, with the ight math/compute tools, we could calculate the complete tempeatue and elocity pofiles in the moing fluid. Complex Heat ansfe Dimensional Analysis What ae goening equations? Micoscopic enegy balance plus Naie-Stokes, continuity Scale factos? Re, F, L/D plus whatee comes fom the est of the analysis Quantity of inteest (like wall foce, dag)? Heat tansfe coefficient he quantity of inteest in foced-conection heat tansfe is h How is the heat tansfe coefficient elated to the full solution fo (,,) in the fluid?

12 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis fluid R 0 Assume: symmety Long tube Unknown function: is an unknown function (,, ) pipe wall 3 Complex Heat ansfe Dimensional Analysis At the bounday, (Newton s Law of Cooling is the bounday condition) otal heat flow though the wall in tems of h We can calculate the total heat tansfeed fom in the fluid: otal heat conducted to the wall fom the fluid We need in the fluid 4

13 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Equate these two: otal heat flow though the wall otal heat flow though the wall in tems of h otal heat conducted to the wall fom the fluid 5 Complex Heat ansfe Dimensional Analysis Equate these two: otal heat flow though the wall Now, non-dimensionalie this expession 6 3

14 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Non-dimensionalie non-dimensional aiables: position: D D tempeatue: o o 7 Complex Heat ansfe Dimensional Analysis h DL o L D 0 0 k o D D d d hd k L D L D 0 0 d d Nusselt numbe, Nu (dimensionless heattansfe coefficient) Nu Nu, L D one additional dimensionless goup 8 4

15 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis h DL o hd k L D 0 0 L D k L D 0 0 o D D d d d d his is a function of Re though Nusselt numbe, Nu (dimensionless heattansfe coefficient) Nu Nu, L D one additional dimensionless goup 9 Complex Heat ansfe Dimensional Analysis Non-dimensional Enegy Equation Pe Non-dimensional Naie-Stokes Equation D Dt P g Re F Cˆ p VD Pe PRe k Cˆ P p k Non-dimensional Continuity Equation x x y y 0 Quantity of inteest Nu L / D L D 0 0 d d 30 5

16 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Accoding to ou dimensional analysis calculations, the dimensionless heat tansfe coefficient should be found to be a function of fou dimensionless goups: Peclet numbe Pe Pandtl numbe P no fee sufaces Nu NuRe, P, F, Now, do the expeiments. L D 3 Complex Heat ansfe Dimensional Analysis Now, do the expeiments. Foced Conection Heat ansfe Build appaatus (seeal actually, with diffeent D, L) Run fluid though the inside (at diffeent V; fo diffeent fluids,,,) Measue on inside; on inside Measue ate of heat tansfe, Calculate : Repot alues in tems of dimensionless coelation: Nu Re, P, It should only be a function of these dimensionless numbes (if ou Dimensional Analysis is coect..) 3 6

17 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis 0000 Coelations fo Foced Conection Heat ansfe Coefficients 000 Nu 0.07 Re 0.8 P 3 b w 0.4 Nu 00 3 D b Nu.86 ReP L w Geankoplis, 4 th ed. eqn 4.5-4, page Re P = 8.07 (wate, 60 o F) iscosity atio =.00 L/D = Complex Heat ansfe Dimensional Analysis Coelations fo Foced Conection Heat ansfe Coefficients If dimensional 0000 analysis is ight, we should get a 0.4 single cue, not multiple b diffeent cues Nu 0.07 Re P w depending 000 on:,,,. Nu 00 3 D b Nu.86 ReP L w Geankoplis, 4 th ed. eqn 4.5-4, page Re P = 8.07 (wate, 60 o F) iscosity atio =.00 L/D =

18 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Coelations fo Foced Conection Heat ansfe Coefficients If dimensional 0000 analysis is ight, we should get a 0.4 single cue, not multiple b diffeent cues Nu 0.07 Re P w depending 000 on:,,,. Nu 00 3 D b Nu.86 ReP L w Geankoplis, 4 th ed. eqn 4.5-4, page Re Dimensional P = 8.07 (wate, 60 o F) iscosity atio =.00 L/D Analysis = 65 WINS AGAIN! 35 Complex Heat ansfe Dimensional Analysis Heat ansfe in Lamina flow in pipes: data coelation fo foced conection heat tansfe coefficients Nu a ha D k 3 D.86 Re P b L w 0.4 the subscipt a efes to the type of aeage tempeatue used in calculating the heat flow, q q h A a a w a bi w bo Geankoplis, 4 th ed. eqn 4.5-4, page 60 00, 00, hoiontal pipes; all physical popeties ealuated at the mean tempeatue of the bulk fluid except which is ealuated at the (constant) wall tempeatue. 36 8

19 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Foced conection Heat ansfe in Lamina flow in pipes N.86 ReP. Physical Popeties ealuated at:,, Foced conection Heat ansfe in ubulent flow in pipes N 0.07Re. P. May hae to be estimated,, all physical popeties (except ) ealuated at the bulk mean tempeatue Lamina o tubulent flow bulk mean tempeatue 37 Complex Heat ansfe Dimensional Analysis? In ou dimensional analysis, we assumed constant, k,, etc. heefoe we did not pedict a iscosity-tempeatue dependence. If iscosity is not assumed constant, the dimensionless goup shown below is pedicted to appea in coelations. Nu a ha D k 3 D.86 Re P b L w 0.4 (eminiscent of pipe wall oughness; needed to modify dimensional analysis to coelate on oughness) 38 9

20 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Viscous fluids with lage Δ heating cooling lowe iscosity fluid laye speeds flow nea the wall ==> highe h b w highe iscosity fluid laye etads flow nea the wall ==> lowe h b w empiical esult: b w 0.4 ef: McCabe, Smith, Haiott, 5th ed, p Complex Heat ansfe Dimensional Analysis Why does appea in lamina flow coelations and not in the tubulent flow coelations? LAMINAR h() h L Less lateal mixing in lamina flow means moe aiation in. 40 0

21 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis URBULEN h() 0.7 h L hl D h L In tubulent flow, good lateal mixing educes the aiation in along the pipe length. 4 Example of patial solution to Homewok lamina flow in pipes tubulent flow in smooth tubes ai at atm in tubulent flow in pipes wate in tubulent flow in pipes 3 h.86 Re P ad D b Nua k L w Nu h h lm lm lm hlmd k h h lm lm Re P b w 3.5V ( m / s) 0. D( m) 0.5V ( ft / s) 0. D( ft) o V m / s C D( m) o V ft / s 0.0 F 0. D( ft) Re<00, (RePD/L)>00, hoiontal pipes, eqn 4.5-4, page 38; all popeties ealuated at the tempeatue of the bulk fluid except w which is ealuated at the wall tempeatue. Re>6000, 0.7 <P <6,000, L/D>60, eqn 4.5-8, page 39; all popeties ealuated at the mean tempeatue of the bulk fluid except w which is ealuated at the wall tempeatue. he mean is the aeage of the inlet and outlet bulk tempeatues; not alid fo liquid metals. equation 4.5-9, page 39 4 < ( o C)<05, equation 4.5-0, page 39 4 ( mean)

22 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat tansfe Poblems to Sole: Foced conection heat tansfe fom fluid to wall Solution:? Natual conection heat tansfe fom fluid to wall Solution:? Radiation heat tansfe fom solid to fluid Solution:? We stated with a foced-conection pipe poblem, did dimensional analysis, and found the dimensionless numbes. o do a situation with diffeent physics, we must stat with a diffeent stating poblem. 43 Complex Heat ansfe Dimensional Analysis Fee Conection Fee Conection i.e. hot ai ises heat moes fom hot suface to cold ai (fluid) by adiation and conduction incease in fluid tempeatue deceases fluid density eciculation flow begins eciculation adds to the heat-tansfe fom conduction and adiation coupled heat and momentum tanspot 44

23 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection Fee Conection i.e. hot ai ises How can we sole eal poblems inoling fee (natual) conection? We ll ty this: Let s eiew how we appoached soling eal poblems in ealie cases, i.e. in fluid mechanics, foced conection. 45 Complex Heat ansfe Dimensional Analysis Fee Conection less hot Q in W s, on Engineeing Modeling cold less cold Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, e.g. add oughness lengthscale) Design additional expeiments Iteate until useful coelations esult hot Pocess scale 46 3

24 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection Example: Fee conection between long paallel plates o heat tansfe though double-pane glass windows (wam) b (cool) assumptions: long, wide slit steady state no souce tems iscosity constant density aies with y Calculate:, pofiles 47 Complex Heat ansfe Dimensional Analysis Fee Conection Example : Natual conection between etical plates Mass balance: 0 Momentum balance: (wam) b (cool) y 48 4

25 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection Example : Natual conection between etical plates You ty. (wam) b (cool) y 49 Complex Heat ansfe Dimensional Analysis Fee Conection (wam) b (cool) Mass balance: 0 y

26 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection (wam) b (cool) Mass balance: 0 y 0 steady tall, wide 5 Complex Heat ansfe Dimensional Analysis Fee Conection (wam) b (cool) Mass balance: 0 y 0 steady Conclusion: density must not ay with. tall, wide, 5 6

27 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection Momentum balance: 53 Complex Heat ansfe Dimensional Analysis Fee Conection Is Pessue a function of? YES, thee should be hydostatic pessue (due to weight of fluid) Pessue at the bottom of a column of fluid = pessue at top. at, aeage density Let at 0 y 54 7

28 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection (look up the physics in the liteatue) o account fo the tempeatue aiation of : mean density olumetic coefficient of expansion at 55 Complex Heat ansfe Dimensional Analysis Fee Conection (wam) b (cool) Enegy balance: y 56 8

29 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection (wam) b (cool) Enegy balance: y (sole) 57 Complex Heat ansfe Dimensional Analysis Fee Conection (wam) b (cool) Enegy balance: y (sole) 58 9

30 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection Enegy balance: Sole (wam) b y (cool) 59 Complex Heat ansfe Dimensional Analysis Fee Conection Final Result: (fee conection between two slabs) ( y) g b y b 3 y b (see next slide fo plot) 60 30

31 Lectues 4-5 CM30 Heat ansfe /8/06 Complex Heat ansfe Dimensional Analysis Fee Conection,max Velocity pofile fo fee conection between two wide, tall, paallel plates y b (Note that the tempeatue maxima ae not centeed) 6 Fee Conection i.e. hot ai ises Engineeing Modeling Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, e.g. add oughness lengthscale) Design additional expeiments Iteate until useful coelations esult 6 3

32 Lectues 4-5 CM30 Heat ansfe /8/06 Fee Conection i.e. hot ai ises (wam) b (cool) Mass balance: y 0 Momentum balance: Enegy balance: 63 Fee Conection i.e. hot ai ises Engineeing Modeling Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, e.g. add oughness lengthscale) Design additional expeiments Iteate until useful coelations esult 64 3

33 Lectues 4-5 CM30 Heat ansfe /8/06 Retun to Dimensional Analysis Nondimensionalie the goening equations; deduce dimensionless scale factos o nondimensionalied the Naie- Stokes fo fee conection poblems, we follow the simple poblem we just completed:, 0. density not constant P t thee was a tick fo this g diing the flow 65 How did we nondimensionalied the Naie-Stokes befoe? A coss-section A: FORCED CONVECION L EXAMPLE I: Pessuedien flow of a Newtonian fluid in a tube: steady state well deeloped long tube fluid R g hee was an aeage elocity used as the chaacteistic elocity 66 33

34 Lectues 4-5 CM30 Heat ansfe /8/06 FORCED CONVECION FORCED CONVECION FORCED CONVECION -component of the Naie-Stokes Equation: t P g Choose: = chaacteistic length = chaacteistic elocity / = chaacteistic time = chaacteistic pessue his elocity is an imposed (foced) aeage elocity We do not hae such an imposed elocity in natual conection 67 FORCED CONVECION FORCED CONVECION FORCED CONVECION non-dimensional aiables: time: position: elocity: tv t D D D V V V P g diing foce: P V g g 68 34

35 Lectues 4-5 CM30 Heat ansfe /8/06 FORCED CONVECION FORCED CONVECION FORCED CONVECION -component of the nondimensional Naie-Stokes Equation: Re F D Dt P VD gd g V D Dt t 69 FREE CONVECION FREE CONVECION We do not hae such FREE CONVECION an imposed elocity in natual conection Fo fee conection, what is the aeage elocity? fo foced conection we used:,max yp tall, paallel plates y b

36 Lectues 4-5 CM30 Heat ansfe /8/06 FREE CONVECION FREE CONVECION We do not hae such FREE CONVECION an imposed elocity in natual conection Fo fee conection, what is the aeage elocity? fo foced conection we used:,max yp tall, paallel plates.5 ZERO 0.5 y b FREE CONVECION FREE CONVECION We do not hae such FREE CONVECION an imposed elocity in natual conection Fo fee conection, what is the aeage elocity? Answe: eo! fo foced conection we used: Fo fee conection 0; what V should we use fo fee conection? Solution: use a Reynolds-numbe type expession so that no chaacteistic elocity imposes itself (we ll see now how that woks): 7 36

37 Lectues 4-5 CM30 Heat ansfe /8/06 FREE CONVECION FREE CONVECION FREE CONVECION When non-dimensionaliing the Naie-Stokes, what do I use fo? (answe fom idealied poblem) t P g hee we use because the issue is olumetic flow ate as befoe, fo pessue gadient we use hee we use because the issue is diing the flow by density diffeences affected by gaity 73 FREE CONVECION FREE CONVECION FREE CONVECION non-dimensional aiables: t time: position: elocity: t D D D D D D diing foce: 74 37

38 Lectues 4-5 CM30 Heat ansfe /8/ gd Dt D SOLUION: -component of the nondimensional Naie-Stokes Equation (fee conection): t Dt D Gashof numbe FREE CONVECION FREE CONVECION FREE CONVECION O any appopiate chaacteistic Δ 75 G Dt D P t Dimensionless Equation of Motion (fee conection) Dimensionless Enegy Equation (fee conection; Re = ) D L D L, P,G Nu Nu, Nu Nu FREE CONVECION FREE CONVECION FREE CONVECION G Δ 76 No Pe No Re

39 Lectues 4-5 CM30 Heat ansfe /8/06 Fee Conection i.e. hot ai ises Engineeing Modeling Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, e.g. add oughness lengthscale) Design additional expeiments Iteate until useful coelations esult 77 Fee Conection i.e. hot ai ises Engineeing Modeling Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, Done e.g. add oughness (see lengthscale) Design additional expeiments Iteate until useful coelations liteatue) esult 78 39

40 Lectues 4-5 CM30 Heat ansfe /8/06 Liteatue Results: G FREE CONVECION Δ Example: Natual conection fom etical planes and cylindes Nu hl k ag m P m a,m ae gien in able 4.7-, page 55 Geankoplis fo seeal cases L is the height of the plate all physical popeties ealuated at the film tempeatue, f Fee conection coelations use the film tempeatue fo calculating the physical popeties f w b Fee conection coelations use the film tempeatue fo calculating the physical popeties 79 Complex Heat ansfe Coelations fo Nu Natual conection Vetical planes and cylindes Nu G P all physical popeties ealuated at the film tempeatue, f Physical Popeties ealuated at: compae with: Foced conection Heat ansfe in Lamina flow in pipes N.86 ReP. Physical Popeties ealuated at:,, all physical popeties (except ) ealuated at the bulk mean tempeatue (tue also fo tubulent flow coelation) 80 40

41 Lectues 4-5 CM30 Heat ansfe /8/06 Fee Conection i.e. hot ai ises Engineeing Modeling Choose an idealied poblem and sole it Fom insight obtained fom ideal poblem, identify goening equations of eal poblem Nondimensionalie the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design expeiments to test modeling thus fa Reise modeling (stuctue of dimensional analysis, identity of scale factos, e.g. add oughness lengthscale) Design additional expeiments Iteate until Useful coelations esult Success! (Dimensional Analysis wins again) 8 Pactice Heat-ansfe Poblems: Foced Conection Fee Conection 8 4

42 Lectues 4-5 CM30 Heat ansfe /8/06 Pactice : A wide, deep ectangula oen (.0 tall) is used fo baking loaes of bead. Duing the baking pocess the tempeatue of the ai in the oen eaches a stable alue of 00. he oen side-wall tempeatue is measued at this time to be a stable 450. Please estimate the heat flux fom the wall pe unit width. Refeence: Geankoplis Ex page Pactice : A hydocabon oil entes a pipe ( inne diamete; 5.0 long) at a flow ate of 80 /. Steam condenses on the outside of the pipe, keeping the inside pipe suface at a constant 350. If the tempeatue of the enteing oil is 50, what is tempeatue of the oil at the outlet of the pipe? Hydocabon oil popeties: Mean heat capacity 0.50 hemal conductiity Viscosity 6.50, Refeence: Geankoplis Ex page

43 Lectues 4-5 CM30 Heat ansfe /8/06 Pactice 3: Ai flows though a tube 5.4 inside diamete, long tube) at 7.6 /. Steam condenses on the outside of the tube such that the inside suface tempeatue of the tube is If the ai pessue is 06.8 and the mean bulk tempeatue of the ai is / 477.6, what is the steady-state heat flux to the ai? Refeence: Geankoplis Ex page 6 85 Pactice 4: Had ubbe tubing inside adius 5.0; outside adius 0.0 is used as a cooling coil in a eaction bath. Cold wate is flowing apidly inside the tubing; the inside wall tempeatue is 74.9 and the outside wall tempeatue is 97.. o keep the eaction in the bath unde contol, the equied cooling ate is What is the minimum length of tubing needed to accomplish this cooling ate? What length would be needed if the coil wee coppe? Had ubbe popeties: Density 98 hemal conductiity Refeence: Geankoplis Ex. 4.- page 43, but don t do it his way follow class methods

44 Lectues 4-5 CM30 Heat ansfe /8/06 Pactice 5: A cold-stoage oom is constucted of an inne laye of pine (thickness.7 ), a middle laye of cok boad (thickness 0.6 ), and an oute laye of concete (thickness 76. ). he inside wall suface tempeatue is 55.4 and the outside wall suface tempeatue is 97.. What is the heat loss pe squae mete though the walls and what is the tempeatue at the inteface between the wood and the cok boad? Mateial popeties: hemal conductiity pine 0.5 hemal conductiity cok boad hemal conductiity concete 0.76 Refeence: Geankoplis Ex page 45, but don t do it his way follow class methods. 87 Pactice 6: A thick-walled tube (stainless steel; inne diamete; oute diamete; length ) is coeed with a thickness of insulation. he inside-wall tempeatue of the pipe is 8.0 and the outside suface tempeatue of the insulation is What is the heat loss and the tempeatue at the inteace between the steel and the insulation? Mateial popeties of stainless steel: hemal conductiity.63 Density 786 Heat Capacity 490 Mateial popeties of insulation: hemal conductiity 0.43 Refeence: Geankoplis Ex page 47, but don t do it his way follow class methods

45 Lectues 4-5 CM30 Heat ansfe /8/06 Expeience with Dimensional Analysis thus fa: Flow in pipes at all flow ates (lamina and tubulent) Solution: Naie-Stokes, Re, F, L/D, dimensionless wall foce = f; f=f(re, L/D) Flow aound obstacles (sphees, othe complex shapes Solution: Naie-Stokes, Re, dimensionless dag= C D ; C D = C D (Re) Foced conection heat tansfe fom fluid to wall Solution: Micoscopic enegy, Naie-Stokes, Re, P, L/D, heat tansfe coefficient=h; h = h(re,p,l/d) Natual conection heat tansfe fom fluid to wall Solution: Micoscopic enegy, Naie-Stokes, G, P, L/D, heat tansfe coefficient=h; h = h(g,p,l/d) 89 Expeience with Dimensional Analysis thus fa: Flow in pipes at all flow ates (lamina and tubulent) Solution: Naie-Stokes, Re, F, L/D, dimensionless wall foce = f; f=f(re, L/D) Flow aound obstacles (sphees, othe complex shapes Solution: Naie-Stokes, Re, dimensionless dag= C D ; C D = C D (Re) Foced conection heat tansfe fom fluid to wall Solution: Micoscopic enegy, Naie-Stokes, Re, P, L/D, heat tansfe coefficient=h; h = h(re,p,l/d) Natual conection heat tansfe fom fluid to wall Solution: Micoscopic enegy, Naie-Stokes, G, P, L/D, heat tansfe coefficient=h; h = h(g,p,l/d) Now, moe to last heat-tansfe mechanism: Radiation heat tansfe fom solid to fluid? Solution:? 90 45

46 Lectues 4-5 CM30 Heat ansfe /8/06 Expeience with Dimensional Analysis thus fa: Flow in pipes at all flow ates (lamina and tubulent) Solution: Naie-Stokes, Re, F, L/D, dimensionless wall foce = f; f=f(re, L/D) Actually, we ll hold off on Flow aound obstacles (sphees, othe complex shapes Solution: Naie-Stokes, Re, adiation and spend some dimensionless dag= C ; C = C (Re) D D D time on heat exchanges and othe pactical concens Foced conection heat tansfe fom fluid to wall Solution: Micoscopic enegy, Naie-Stokes, Re, P, L/D, heat tansfe coefficient=h; h = h(re,p,l/d) Natual conection heat tansfe fom fluid to wall Solution: Micoscopic enegy, Naie-Stokes, G, P, L/D, heat tansfe coefficient=h; h = h(g,p,l/d) Now, moe to last heat-tansfe mechanism: Radiation heat tansfe fom solid to fluid? Solution:? 9 Next: 9 46