Last time, Dimensional Analysis for Heat Transfer

Transcription

1 Last time, Dimensional Analsis fo Heat ansfe 0000 Coelations fo Foced Conection Heat ansfe Coefficients Re P b Nu w Nu D 3 b Nu.86 ReP L w 0 P = 8.07 (wate, 60 o F) iscosit atio =.00 L/D = Geanoplis, 4 th ed. eqn Re 4.5-4, page 60 Summa Dimensional analsis wos as well in heat tansfe as in momentum tansfe We should use it (and pobabl also in mass tansfe, but ) hese dimensionless numbes ae stacing up (and ) What do the eall mean? Nu Nu Re, P, L D

2 Dimensional Analsis hese numbes tell us about the elatie impotance of the tems the pecede. Dimensionless numbes fom the Equations of Change (micoscopic balances) mass eneg momentum Non-dimensional Naie-Stoes Equation Non-dimensional Eneg Equation Non-dimensional Continuit Equation (species A) ef: BSL, p58, 644 Re Renolds F Foude Pe Péclet ReP P Pandtl Pe Péclet ReSc Sc Schmidt 3 Dimensionless Numbes Dimensionless numbes fom the Equations of Change Re Renolds F Foude Pe Péclet ReP Pe Péclet ReSc P Pandtl Sc Schmidt LeP Le Lewis hese numbes tell us about the elatie impotance of the tems the pecede in the micoscopic balances (scenaio popeties). hese numbes compae the magnitudes of the diffusie tanspot coefficients ν, α, D (mateial popeties). themal diffusiit α 4

3 Dimensional Analsis Dimensionless numbes fom the Engineeing Quantities of Inteest hese numbes ae defined to help us build tanspot data coelations based on the fewest numbe of gouped (dimensionless) aiables (scenaio popet). momentum eneg mass fe Dimensionless Foce on the Wall (Dag) Newton s Law of Cooling Nu L / D L D 0 0 Dimensionless Mass ansfe Coefficient dz d f Fiction Facto Aspect Ratio (Fanning) Nu Nusselt Aspect Ratio Sh Shewood Aspect Ratio f F ρv A Nu hd Sh D D 5 momentum eneg mass Dimensionless Numbes Re Renolds F Foude Pe Péclet ReP Pe Péclet ReSc P Pandtl Sc Schmidt LeP Le Lewis f Fiction Facto Nu Nusselt Sh Shewood themal diffusiit α F hese numbes fom the goening equations tell us about the elatie impotance of the tems the pecede in the micoscopic balances (scenaio popeties). hese numbes compae the magnitudes of the diffusie tanspot coefficients ν, α, D (mateial popeties). hese numbes ae defined to help us build tanspot data coelations based on the fewest numbe of gouped (dimensionless) aiables (scenaio popeties). 6 3

4 Unstead State Heat ansfe: Dimensional Analsis NEW SUFF! Question: What now? Answe: Let s appl Dimensional Analsis to something new, unstead state heat tansfe, to sot out the aious effects. Engineeing Modeling (comple sstems) Choose an idealized poblem and sole it Fom insight obtained fom ideal poblem, identif goening equations of eal poblem Nondimensionalize the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design epeiments to test modeling thus fa Reise modeling (stuctue of dimensional analsis, identit of scale factos, e.g. add oughness lengthscale) Design additional epeiments Iteate until useful coelations esult 7 Unstead State Heat ansfe: Dimensional Analsis NEW SUFF! Question: What now? Answe: Let s appl Dimensional Analsis to something new, unstead state heat tansfe, to sot out the aious effects. Engineeing Modeling (comple sstems) Choose an idealized poblem and sole it Fom insight obtained fom ideal poblem, identif goening equations of eal poblem Nondimensionalize the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design epeiments to test modeling thus fa Reise modeling (stuctue of dimensional analsis, identit of scale factos, e.g. add oughness lengthscale) Design additional epeiments Iteate until useful coelations esult SPOILER ALER: hee ll be some new dimensionless numbes! 8 4

5 CM30 anspot/unit Opeations Moe comple Sstems: Unstead State Heat ansfe (Analtical Solutions) Pofesso Faith A. Moison Depatment of Chemical Engineeing Michigan echnological Uniesit 9 We model the dnamics of unstead state heat tansfe because thee ae e pactical poblems that we can sole with such models. 0 5

6 Eample: When will m pipes feeze? he tempeatue has been 35 o F fo a while now, sufficient to chill the gound to this tempeatue fo man tens of feet below the suface. Suddenl the tempeatue dops to 0 o F. How long will it tae fo feezing tempeatues (3 o F) to each m pipes, which ae 8 ft unde gound? /30/9 BU h.0 o h ft F soil soil ft 0.08 h BU 0.5 o h ft F Unstead State Heat ansfe: Dimensional Analsis Engineeing Modeling (comple sstems) Choose an idealized poblem and sole it. Fom insight obtained fom ideal poblem, identif goening equations of eal poblem Nondimensionalize the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design epeiments to test modeling thus fa Reise modeling (stuctue of dimensional analsis, identit of scale factos, e.g. add oughness lengthscale) Design additional epeiments Iteate until useful coelations esult SEP ONE: Idealized poblem: D heat tansfe in a semi infinite solid 6

7 Unstead State Heat ansfe: Dimensional Analsis Deelop a model: Eample : Unstead Heat Conduction in a Semi infinite solid A e long, e wide, e tall slab is initiall at a tempeatue. At time t 0, the left face of the slab is eposed to a igoousl mied gas at tempeatue. What is the timedependent tempeatue pofile in the slab? 3 Unstead State Heat ansfe Eample: Unstead Heat Conduction in a Semi infinite solid z H D H, D, e lage 4 7

8 D Heat ansfe: Unstead State Initial Condition: t 0 o t 0 o hen, t 0 t 0 (, t) 5 D Heat ansfe: Unstead State Geneal Eneg anspot Equation (micoscopic eneg balance) As fo the deiation of the micoscopic momentum balance, the micoscopic eneg balance is deied on an abita olume, V, enclosed b a suface, S. S ds nˆ ˆ Gibbs notation: C p t S V see handout fo component notation 6 8

9 9 Geneal Eneg anspot Equation (micoscopic eneg balance) see handout fo component notation ate of change conection conduction (all diections) souce elocit must satisf equation of motion, equation of continuit (eneg geneated pe unit olume pe time) S t C p ˆ 7 D Heat ansfe: Unstead State Equation of eneg fo Newtonian fluids of constant densit,, and themal conductiit,, with souce tem (souce could be iscous dissipation, electical eneg, chemical eneg, etc., with units of eneg/(olume time)). CM30 Fall 999 Faith Moison Souce: R. B. Bid, W. E. Stewat, and E. N. Lightfoot, anspot Pocesses, Wile, NY, 960, page 39. Gibbs notation (ecto notation) p C p S C t ˆ ˆ Catesian (z) coodinates: p p z C S z C z t ˆ ˆ Clindical (z) coodinates: p p z C S z C z t ˆ ˆ Spheical () coodinates: p C t sin sin sin ˆ sin 8 themal diffusiit α

10 D Heat ansfe: Unstead State Eample : Unstead Heat Conduction in a Semi infinite solid A e long, e wide, e tall slab is initiall at a tempeatue. At time t 0, the left face of the slab is eposed to a igoousl mied gas at tempeatue. What is the timedependent tempeatue pofile in the slab? t 0 o t 0 o Newton s law of cooling BC s: q ha t 0 t 0 (, t) 9 D Heat ansfe: Unstead State Micoscopic Eneg Equation in Catesian Coodinates t z z C S Cˆ p ˆ p z Cˆ p themal diffusiit what ae the bounda conditions? initial conditions? 0 0

11 D Heat ansfe: Unstead State Eample: Unstead Heat Conduction in a Semi infinite solid Initial Condition: t 0 o t 0 o t 0 t 0 (, t) You t. D Heat ansfe: Unstead State Unstead State Heat Conduction in a Semi Infinite Slab t 0 t 0 o o t 0 t 0 (, t) Initial condition: t Bounda conditions: ρc α t0 themal diffusiit α ρc 0 q A d d h t0 t

12 D Heat ansfe: Unstead State Unstead State Heat Conduction in a Semi Infinite Slab t 0 t 0 o o t 0 t 0 (, t) Initial condition: Bounda conditions: t α t0 fo all themal diffusiit α ρc 0 q A d d h t0 t fo all t 3 D Heat ansfe: Unstead State Unstead State Heat Conduction in a Semi Infinite Slab t 0 t 0 o o t 0 t 0 (, t) Initial condition: Bounda conditions: t0 See tet WRF p84 h t t 4

13 Unstead State Heat Conduction in a Semi Infinite Slab t 0 o t 0 o Solution: efcζe efc h t ζβ t t 0 t 0 (, t) Y Y complementa eo function of (a standad function in Ecel) eo function of efc ef ef e π d Geanoplis 4 th ed., eqn 5.3 7, page 363 WRF, eqn 8, page 86 themal diffusiit α 5 Unstead State Heat Conduction in a Semi Infinite Slab t 0 o t 0 o Solution: efcζe efc ζβ t 0 t 0 (, t) h t complementa eo function of eo function of themal diffusiit α efc ef t ef e π d o mae this solution easie to use, we can plot it. Y Y 6 3

14 Unstead State Heat Conduction in a Semi Infinite Slab his: efcζe efc ζβ t 0 t 0 o o t 0 t 0 (, t) Vesus this: At aious alues of this: themal diffusiit α h t t o mae this solution easie to use, we can plot it. Y Y 7 Unstead State Heat Conduction in a Semi Infinite Slab Y Y Y Plot design afte Geanoplis 4 th ed., Figue 5.3 3, page β 0.05 h t inceasing β β Zeta t

15 D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab With moden tools, we can plot the solution diectl (ealuated in Ecel) Unstead State Heat Conduction in a Semi Infinite Slab inceasing time, t time, hs , ft h.0 α.0.0 ft F themal diffusiit α 9 D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab With moden tools, we can plot the solution diectl (ealuated in Ecel) Unstead State Heat Conduction in a Semi Infinite Slab Notice the stead state effect of finite h inceasing time, t time, hs , ft h.0 α.0.0 ft F themal diffusiit α 30 5

16 Eample: When will m pipes feeze? he tempeatue has been 35 o F fo a while now, sufficient to chill the gound to this tempeatue fo man tens of feet below the suface. Suddenl the tempeatue dops to 0 o F. How long will it tae fo feezing tempeatues (3 o F) to each m pipes, which ae 8 ft unde gound? /30/9 BU h.0 o h ft F soil soil ft 0.08 h BU 0.5 o h ft F 3 D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab We need the appopiate phsical popet data fo the soil. BU h.0 o h ft F soil soil ft 0.08 h BU 0.5 o h ft F themal diffusiit α Geanoplis 4 th ed. 3 6

17 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab efcζe efc ζβ t 0 t 0 o o t 0 t 0 (, t) t h t Both ζ and β depend on time???? Y Y 33 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab Y β You t. Geanoplis 4 th ed., Figue 5.3 3, page 364 t 34 7

18 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab Y β Solution: Guess lage β (Inteatie solution) Geanoplis 4 th ed., Figue 5.3 3, page 364 t 35 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab t 0 t 0 o o t 0 t 0 (, t) efcζe efc ζβ Answe: t 480 hous 0 das Y Y 36 8

19 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab O, use Ecel. (How eactl?) t 0 t 0 o o t 0 t 0 (, t) efcζe efc t h t ζβ You t. 0= = = h= alpha= = = Answe: t. das β. pages.mtu.edu/~fmoiso/cm30/09pacticepoblemsinheatansfe%8geanoplis%9.pdf 37 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab With moden tools, we can plot the eolution of the model diectl (ealuated in Ecel) inceasing time, t h.0 α ft F 8ft 0. hous , ft, ft aa_solutions/unstead semi infinite solid plots.ls 38 9

20 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab With moden tools, we can plot the eolution of the model diectl (ealuated in Ecel), t 40 8ft o F inceasing time, t , ft, ft aa_solutions/unstead semi infinite solid plots.ls 0. hous F h.0 α ft F 39 Solution Summa:, t 40 8ft Answe: t 509 hous das o F inceasing time, t 0. hous F , ft, ft

21 We used unstead state heat tansfe modeling to sole one pactical poblem. What can we do to etend these methods to a wide class of poblems? 4 Bac to this: What is ou usual stateg fo comple phenomena? Answe: Dimensional Analsis Let s nondimensionalize the goening equations and BCs. Let s sot out the aious unstead cases. Engineeing Modeling (comple sstems) Choose an idealized poblem and sole it Fom insight obtained fom ideal poblem, identif goening equations of eal poblem Nondimensionalize the goening equations; deduce dimensionless scale factos (e.g. Re, F fo fluids) Design epeiments to test modeling thus fa Reise modeling (stuctue of dimensional analsis, identit of scale factos, e.g. add oughness lengthscale) Design additional epeiments Iteate until useful coelations esult 4

22 Let s nondimensionalize the goening equations and BCs. Let s sot out the aious cases. D Heat ansfe: Unstead State Unstead State Heat Conduction in a Semi Infinite Slab t 0 t 0 o o t 0 t 0 (, t) Initial condition: t0 Bounda conditions: themal diffusiit α ρc (Reiew: How did we do this befoe?) 43 Method: Identif the goening equation(s) Choose tpical alues (scale factos) Use them to scale the equations We ll modif ou solution fo Conectie Heat ansfe CM30 REVIEW Pipe flow Eneg Dimensional Analsis non-dimensional aiables: non-dimensional aiables: time: position: elocit: tv t D D z z D z z V V V diing foce: P P V g z g z g position: tempeatue: o o souce: 44

23 Dimensional Analsis We ll modif ou solution fo Conectie Heat ansfe Pipe flow Eneg Dimensional Analsis non-dimensional aiables: non-dimensional aiables: time: position: elocit: tv t D D z z D z z V V V diing foce: P P V g z g z g position: tempeatue: o o souce: Slight poblem: We need to nondimensionalize t fo the unstead case also, but thee is no chaacteistic elocit in themal conduction in a solid. 45 Choice: Fo the unstead case we ll choose a chaacteistic time based on the themal diffusiit, α. t αt D his dimensionless time is called Fouie numbe Fo. We need to nondimensionalize t fo the unstead case also, but thee is no chaacteistic elocit. Pipe flow non-dimensional aiables: time: position: elocit: tv z t z D D V z z D V V themal diffusiit α ρc P diing foce: g z P V g z g Conectie Heat ansfe Eneg non-dimensional aiables: position: tempeatue: o o souce: (Appeas in the eneg balance) themal diffusion time 46 3

24 , t, t F 40 Eample: When will m pipes feeze? D Heat ansfe: Unstead State Heat Conduction in a Semi Infinite Slab Eneg is diffusing down the tempeatue gadient o F h.0 α ft F inceasing time, t , ft, ft 0. hous themal diffusiit α ρc themal diffusion time 47 Dimensional Analsis, Unstead State Conection Non-dimensionalize (eqns, BCs) t α q ha t 0 t 0 o o t 0 t 0 (, t) non-dimensional aiables: position: D tempeatue: Y time: t αt D Fo Fouie Numbe αt D his dimensionless time is called Fouie numbe Fo. 48 4

25 D Heat ansfe: Unstead State Unstead State Heat Conduction in a Semi Infinite Slab Y t Y t 0 t 0 o o t 0 t 0 (, t) tempeatue: Initial condition: Bounda conditions: t 0 Y Y t 0 Y Bi Y t 0 Bi hd Bi Biot Numbe hd 49 In dimensionless fom, we see that this poblem educes to YY D,Fo,Bi D Heat ansfe: Unstead State Unstead State Heat Conduction in a Semi Infinite Slab Initial condition: Bounda conditions: t 0 o t 0 Y Y t 0 t 0 o t 0 t 0 (, t) tempeatue: Y Bi Y t 0 Dimensionless quantities: Y t Fo Bi Y (dimensionless tempeatue inteal) Fouie numbe (dimensionless time) Biot numbe (ponounced BEE OH) Ratio of heat tansfe esistance at the bounda to esistance in the solid. his is a tanspot issue. 50 5

26 Because we can sole this poblem analticall, we can confim that the dimensional analsis is coect: Solution: Unstead State Heat Conduction in a Semi Infinite Slab Solution: t 0 t 0 o o t 0 t 0 (, t) + h t t Y Y = Bi Biot Numbe hd Yefc D Fo Fouie Numbe αt D Fo ebi Bi Fo efc Fo Bi D Fo 5 Unstead State Heat ansfe in a Bod wo Additional Dimensionless Numbes Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) momentum eneg mass Re Renolds F Foude Pe Péclet ReP Pe Péclet ReSc Dimensionless Numbes P Pandtl Sc Schmidt LeP Le Lewis f Fiction Facto Nu Nusselt Sh Shewood F hese numbes fom the goening equations tell us about the elatie impotance of the tems the pecede in the micoscopic balances (scenaio popeties). hese numbes compae the magnitudes of the diffusie tanspot coefficients ν, α, D (mateial popeties). hese numbes ae defined to help us build tanspot data coelations based on the fewest numbe of gouped (dimensionless) aiables (scenaio popeties). Fo Fouie Numbe Scales the time eolution of the tempeatue pofile elatie to the mateial s themal popeties, α/ρc. 5 6

27 Dimensional Analsis in Unstead State Heat ansfe Note wo Diffeent Numbes with completel diffeent puposes and meanings but confusingl simila definitions Waning! Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) fo a bod in contact with a moing fluid. Nu Nusselt Numbe Dimensionless heat tansfe coefficient in conection. Quantifies the phsics in the moing fluid and how this esults in a esistance to heat tansfe, captued in the heat tansfe coefficient. 53 Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) At high Bi, the suface tempeatue equals the bul tempeatue; heat tansfe is limited b conduction in the bod. At modeate Bi, heat tansfe is affected b both conduction in the bod and the ate of heat tansfe to the suface. At low Bi, the tempeatue is unifom in a finite bod; heat tansfe is limited b ate of heat tansfe to the suface (h). High Bi: low, high h Modeate Bi: nethe pocess dominates Low Bi: high, low h 54 7

28 Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) High Bi: low, When the tempeatue is unifom in the high h bod, we can do a macoscopic eneg balance to sole man poblems Modeate of inteest. Bi: his is called a lumped paamete nethe pocess analsis. dominates At high Bi, the suface tempeatue equals the bul tempeatue; heat tansfe is limited b conduction in the bod. At modeate Bi, heat tansfe is affected b both conduction in the bod and the ate of heat tansfe to the suface. At low Bi, the tempeatue is unifom in a finite bod; heat tansfe is limited b ate of heat tansfe to the suface (h). Low Bi: high, low h 55 Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) At high Bi, the suface tempeatue equals the bul tempeatue; heat tansfe is limited b conduction in the bod. High Bi: low, high h Modeate Bi: When the wall tempeatue nethe pocess and the bul tempeatue ae equal, dominates the micoscopic eneg balance is easie to ca out (tempeatue bounda Low conditions). Bi: high, low h At modeate Bi, heat tansfe is affected b both conduction in the bod and the ate of heat tansfe to the suface. At low Bi, the tempeatue is unifom in a finite bod; heat tansfe is limited b ate of heat tansfe to the suface (h). 56 8

29 Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) When both pocesses affect the outcomes, the full At high Bi, the suface tempeatue solution ma be necessa. Fo High unifom Bi: stating equals the bul tempeatue; heat low, tansfe tempeatues, is limited b conduction the in solutions ae published. the bod. high h At modeate Bi, heat tansfe is affected b both conduction in the bod and the ate of heat tansfe to the suface. At low Bi, the tempeatue is unifom in a finite bod; heat tansfe is limited b ate of heat tansfe to the suface (h). Modeate Bi: nethe pocess dominates Low Bi: high, low h 57 Bi Biot Numbe Quantifies the tadeoffs between the ate of intenal heat flu (b conduction, ) and the ate of heat delie to the bounda (b conection, h) At high Bi, the suface tempeatue equals the bul tempeatue; heat tansfe is limited b conduction in the bod. At modeate Bi, heat tansfe is affected b both conduction in the bod and the ate of heat tansfe to the suface. At low Bi, the tempeatue is unifom in a finite bod; heat tansfe is limited b ate of heat tansfe to the suface (h). High Bi: low, high h Modeate Bi: nethe pocess dominates Low Bi: high, low h 58 9