CHAPTER 8 CONVECTION IN EXTERNAL TURBULENT FLOW

Transcription

1 CHAPTER 8 CONVECTION IN EXTERNAL TURBULENT FLOW 8.1 Introdction Common phsical phenomenon, bt comple Still relies on empirical data and rdimentar conceptal drawings Tremendos growth in research over last 30 ears V o () trblent jet V o trblent wake behind bod trblent wake behind smokestack 1

2 8.1.1 Eamples of Trblent Flows (i) Miing Processes (ii) Free Shear Flows (iii) Wall-Bonded Flows Varing shape of instantaneos velocit profile Instantaneos velocit flctation (,t) (,t) (,t) velocit profiles Instantaneos velocit profile (t) Instantaneos velocit flctation t flctating component 2

3 Trblent velocit profile vs. laminar Time averaged () trblent δ trb δ lam laminar Trblent flows can enhance performance Trblators Dimpled golf balls 3

4 8.1.2 The Renolds Nmber and the Onset of Trblence Osborne Renolds (1883) first identifies laminar and trblent regimes Renolds nmber: Re D D ν = (8.1) Internal flow: critical flow nmber is Rec = D / ν 2300 Flow over semi-infinite flat plate is Re = V / ν 500, 000 Wh the Renolds nmber predicts the onset of trblence Renolds nmber represents the ratio of inertial to viscos forces o Inertial forces accelerate a flid particle o Viscos forces slow or damp the motion of the particle At low velocit, viscos forces dominate o Infinitesimal distrbances damped ot o Flow remains laminar c t 4

5 At high enogh flid velocit, inertial forces dominate o Viscos forces cannot prevent a waward particle from motion o Chaotic flow enses Trblence near wall For wall-bonded flows, trblence initiates near the wall 5

6 8.1.3 Eddies and Vorticit An edd is a particle of vorticit, ω, r ω = V r (8.2) Eddies tpicall form in regions of velocit gradient. Vorticit can be fond from Eqn. (8.2) to be v ωz = A Common View of Edd Formation Edd begins as a distrbance near the wall Vorte filament forms Stretched into horseshoe or hairpin vorte Lifting phenomenon 6

7 7

8 8.1.4 Scales of Trblence Largest eddies break p de to inertial forces Smallest eddies dissipate de to viscos forces Richardson Energ Cascade (1922) 8

9 Kolgomorov Microscales (1942) Attempt to estimate size of smallest eddies η / l Re v / Re τ / t Re 3/4 1/4 1/2 (8.3a) (8.3b) (8.3c) Important impacts: o There is a vast range of edd sizes, velocities, and time scales in a trblent flow. This cold make modeling difficlt. o The smallest eddies small, bt not infinitesimall small. Viscosit dissipates them into heat before the can become too small. o Scale of the smallest eddies are determined b the scale of the largest eddies throgh the Renolds nmber. Generating smaller eddies is how the viscos dissipation is increased to compensate for the increased prodction of trblence. 9

10 8.1.5 Characteristics of Trblence Trblence is comprised of irreglar, chaotic, three-dimensional flid motion, bt containing coherent strctres. Trblence occrs at high Renolds nmbers, where instabilities give wa to chaotic motion. Trblence is comprised of man scales of eddies, which dissipate energ and momentm throgh a series of scale ranges. The largest eddies contain the blk of the kinetic energ, and break p b inertial forces. The smallest eddies contain the blk of the vorticit, and dissipate b viscosit into heat. Trblent flows are not onl dissipative, bt also dispersive throgh the advection mechanism Analtical Approaches Considering small eddies, is continm hpothesis still valid? o The smallest eddies: approimatel m 10

11 o Mean free path of air at atmospheric pressre is on the order 8 of 10 m three orders of magnitde smaller o Continm hpothesis OK Are nmerical simlations possible? o Direct Nmerical Simlation (DNS) a widespread topic of research o However, short time scales and size range of trblence a problem o Still have to rel on more traditional analtical techniqes Two Common Idealizations Homogeneos Trblence: Trblence, whose microscale motion, on average, does not change from location to location and time to time. Isotropic Trblence: Trblence, whose microscale motion, on average, does not change as the coordinate aes are rotated. 11

12 8.2 Conservation Eqations for Trblent Flow Renolds Decomposition Trblent flow seems well-behaved on average. Renolds Decomposition: Separate velocit, properties into timeaveraged and flctating components: Time-averaged component is determined b: g g = g g (8.4) τ 1 = g( t) dt (8.5) τ Time average of flctating component is zero: τ g = g ( t) dt = 0 τ (8.6) 12

13 Average Identities: For two variables a = a a and b = b b a = a (8.7a) ab = ab (8.7b) 2 2 ( a ) ( a ) = (8.7c) aa = 0 ab = ab a b = (8.7e) a = ( a ) ( a ) (8.7d) = (8.7f) a a a b = a b (8.7g) = a = 0 t a = t (8.7i) 0 (8.7h) (8.7j) Conservation of Mass 13

14 B identities (8.7a) and (8.6): Epanding, ( ρ ) ( ρ v) ( ρ w) ρ = t z (2.2a) Assme incompressible, two-dimensional flow. Sbstitting the Renolds-decomposed velocities = and v = v v, ( ) ( v v ) (a) Time-average the eqation: = v v = v v = Then, simplif each term b invoking identit (8.7h): (b) (c) 14

15 B identities (8.7a) and (8.6), v v = 0 v = Conservation of Momentms The and momentm eqations are given b: (d) (8.8) p ρ v w = ρ g µ t z z (2.10) v v v v p v v v ρ v w = ρ g µ t z z (2.10) 15

16 Simplifing for stead, 2D flow, no bod forces: 2 2 p ρ v = µ 2 2 t 2 2 v v v p v v ρ v = µ 2 2 t (8.10) (8.10) For the -momentm eqation, the terms ( / ) and v( / ) can be replaced b the following relations, derived from the prodct rle of derivation: 2 = (a) ( v) v v = Sbstitte (a) into the -momentm eqation (8.10): ( v) v p ρ = µ 2 2 t { { a b (b) (c) 16

17 Note that terms marked and in the above can be combined as: v = 0 b continit Ths, the -momentm eqation redces to: ( v) p ρ = µ 2 2 t Following Renolds decomposition and averaging, p ( ) v ρ v = µ ρ ρ 2 2 v v p v v v ( v ) ρ v = µ ρ ρ (d) (8.11) (8.12) (8.12) Conservation of Energ 17

18 For incompressible flow, negligible heat generation, constant properties, the energ eqation is given b T T T T T T T ρ c p v w = k µ Φ t z z The energ eqation redces to: 2 2 T T T T T ρc p v = k 2 2 t (2.19b) (8.13) Following Renolds decomposition and time averaging, Eqn. (8.13) becomes: ( T ) ( v T ) 2 2 T T T T ρ c p v = k ρ c 2 2 p ρ c p NOTE TWO NEW TERMS (8.14) 18

19 8.2.5 Smmar of Governing Eqations for Trblent Flow Continit: v = 0 (8.8) -momentm: p ( ) v ρ v = µ ρ ρ 2 2 -momentm: v v p v v v ( v ) ρ v = µ ρ ρ 2 2 Energ: (8.12) (8.12) ( T ) ( v T ) 2 2 T T T T ρ c p v = k ρ c 2 2 p ρ c p (8.14) 19

20 8.3 Analsis of Eternal Trblent Flow Trblent Bondar Laer Eqations (i) Trblent Momentm Bondar Laer Eqation Consider a flat plate in trblent flow. Assme bondar laer is thin: δ 1 L (8.15) Following the same argments as for the laminar bondar laer, the following scalar argments are made: V L δ (8.16a) (8.16b) (8.16c) 20

21 It can be shown that the viscos dissipation terms in (8.12) compare as follows: 2 2 (8.17) 2 2 Also, the pressre gradient in the -direction is negligible: p The pressre gradient in the -direction can be epressed as: Simplifing the Flctation Terms: Flctation Terms: 0 p p dp dp = = d d ( ) 2 and v (8.18) (8.19) If flctation terms are the reslt of eddies, one cold arge that there is no preferred direction to the flctations: 21

22 or v ( ) 2 (8.20) v (8.21) Using scale analsis: First Flctation Term: Second Flctation Term: δ / L 1 ( ) ( ) Since, we conclde that: ( ) L ( ) 2 (a) v v (b) δ δ v (8.22) The -momentm eqation for the trblent bondar laer redces to: 22

23 2 dp v ρ v = µ ρ 2 d (ii) Trblent Energ Eqation Scaling argments for the thermal bondar is: L (8.20) (8.16b) Then: Flctation terms: ρ c p δ t (8.24) T Ts T T (8.25) T 2 2 (8.26) 2 2 ( T ) ( v T ) and ρ c p Assming there is no preferred direction to the flctations: v (8.20) 23

24 or We can then show that: T v T (8.27) ( T ) ( v T ) The energ eqation then redces to: (8.28) ρ c p v k c ( v T ) 2 T T T = ρ 2 p (8.29) Renolds Stress and Heat Fl Can write the -momentm and energ bondar laer eqations as: dp ρ v = µ ρ v d T T T ρ c p v = k ρ c pv T (8.30) (8.31) 24

25 Flctating term in (8.30) looks like a shear stress Flctating term in (8.31) looks like a heat fl Consider a particle flctation imposed on some average velocit profile ρ v is called the trblent shear stress or the Renolds stress ρ c v T is called the trblent heat fl or the Renolds heat fl p The Closre Problem of Trblence Trblent bondar laer eqations: Continit: v = 0 (8.8) 25

26 -momentm: dp ρ v = µ ρ v d T T T ρ c v = k ρ c v T Energ: p p Bondar conditions: Also have, otside the bondar laer: (8.30) (8.31) (,0) = 0 (8.31a) v (,0) = 0 (8.31b) (, ) = V (8.31c) (0, ) = V (8.31d) T (,0) = Ts (8.31e) T (, ) = T (8.31f) T (0, ) = T (8.31g) dp dp = d d (8.32) dv 1 dp V = d ρ d (8.33) 26

27 Leaves s with three eqations (8.8), (8.30) and (8.31), bt five nknowns:, v, T, v and v T This is the closre problem of trblence Edd Diffsivit Cstomar to model the Renolds stress as ρ v = ρε M (8.34) ε is called the momentm edd diffsivit. ρε M M is often referred to as edd viscosit. Similarl, we can model the Renolds heat fl as ρ c v T = ρ c ε p p H ε H is called the thermal edd diffsivit. ρ ε c p ε H T (8.35) is often referred to as edd condctivit. We can then write the bondar laer momentm and energ eqations as 27

28 v = ( ν ε M ) T T T v = ( α ε H ) (8.38) (8.39) The terms in brackets represent the apparent shear stress and apparent heat fl, respectivel: τ ( ν ε ) app = M ρ q app T = ( α ε H ) ρc p (8.40) (8.41) 8.4 Momentm Transfer in Eternal Trblent Flow Modeling Edd Diffsivit: Prandtl s Miing Length Theor Simplest model b Bossinesq: constant ε M o does not allow v to approach zero at the wall 28

29 Prandtl (1925): sed analog to kinetic theor of gases Define the miing length l as the distance the particle travels towards the wall as the reslt of a flctation. The velocit flctation that reslts can be approimated from a Talor series as initial d = (a) = final initial (a) Ths, l (b) 29

30 If we assme, as we have before, that flctations have no preferred direction, then, and so v v l v One cold arge, then, that the trblent stress term is of the following scale: Finall, we can solve Eqation (8.34), for the edd viscosit: 2 (c) 2 v ( )( v ) l (d) ε M v 2 = l / Prandtl proposed the following model for the miing length, Leading to Prandtl s miing-length model: ε M l = κ (8.42) = κ (8.43) 2 2 (8.44) 30

31 8.4.2 Universal Trblent Velocit Profile One wa to solve momentm is to assme a velocit profile, then se approimate methods to solve integral momentm (like in Chap. 5) (i) Large-Scale Velocit Distribtion: Velocit Defect Law / V / δ First Step: normalize variables: vs. Doesn t collapse crves with varing friction Velocit Defect Law Introdce coefficient of friction: C f τ = (8.45) o ρ 2 ( 1 / 2) V Second, define a friction velocit as: * τ o / ρ (8.46) * V C f = / 2 (8.47) 31

32 Define velocit defect: This works, bt doesn t provide enogh detail near the wall. (ii) Wall Coordinates * V Dimensional analsis sggests the following wall coordinates: (8.49) (iii) Near-Wall Profile: Coette Flow Assmption Ver close to the wall, scaling analsis sggests: Near the Wall: τ app ρ * ( ν ε ) 0 (8.48) 32 (8.51) M (8.50) ( ν ε ) constant = M

33 This reslt is similar to Coette flow: Coette Flow Assmption Reslting crve: 33

34 What do we do with this? We can se (8.51) to develop an epression for the velocit profile. First, we need to epress (8.51) in terms of the wall coordinates and. Sbstitting their definitions, it can be shown that: ε M 1 = 1 ν And after rearranging and integrating, (iv) Viscos Sblaer = d ( ε ν ) 1 0 M / Ver close to the wall, viscos forces dominate, Coette Flow Assmption (8.52) redces to: Integrating, with bondar condition at (8.52) (8.53) ν = 1 = 0 = 0 =, (0 7) (8.54) ε M 34

35 This relation compares well to eperimental data from which we call the viscos sblaer. (v) Fll Trblent Region: Law of the Wall Frther awa from the wall, trblent flctations dominate, Coette Flow Assmption (8.52) becomes ε M ν Sbstitte Prandtl s miing length, and wall coordinates: Sbstitte into Eqation (8.55), = ( ) ε M = κ ν 2 κ 0 to 7 ε M (8.55) = (8.56) ( ) Solve for the velocit gradient, 2 = 1 κ 2 = 1 (8.57) ν 35

36 Finall, integrate the above to obtain 1 ln = B (8.58) κ This is sometimes referred to as the Law of the Wall. The constant κ is called von Karman s constant, and eperimental measrements show that κ The constant of integration B can be estimated b noting that the viscos sblaer and the Law of the Wall region appear to intersect at roghl = Using this as a bondar condition, the integration constant is fond to be β 5.0. Ths, an approimation for the Law of the Wall region is: = 2.44 ln 5.0 (50 < < 1500) (8.59) (vi) Other Models van Driest s continos law of the wall o van Driest proposed a miing length model of this form: ( / A 1 ) l = κ e (8.60) o van Driest sed this eqation with (8.42) and (8.51) to obtain: 36

37 τ Spalding s Law: app 2 2 / A ρ = ν κ ( 1 e ) 2 Transforming (8.61) into wall coordinates, and solving for one can obtain: = κ 1 ( ) A e / / (8.61) (8.62) ( ) 2 ( ) 3 κ κ κβ κ = e e 1 κ (8.63) 2 6 o Works for flat plate and pipe flow Reichardt s Law, applied freqentl to pipe flow: 1 ln ( 1 ) 1 / X 0.33 = κ C e e κ X (vii) Effect of Pressre Gradient (8.64) 37

38 In the presence of an adverse pressre gradient, the velocit profile beond 350 deviates from the Law of the Wall model. > 350 The deviation is referred to as a wake, and the region is commonl referred to as the wake region, where the velocit profile deviates even from the overlap region. Law of the Wall-tpe models developed earlier model flat plate flow reasonabl well in the presence of zero pressre gradient. A favorable pressre gradient is approimatel what we enconter in pipe flow, which helps eplain wh the models developed here appl as well to pipe flow. 38

39 8.4.3 Approimate Soltion for Momentm Transfer: Momentm Integral Method (i) Prandtl - von Karman Model Consider a flat, impermeable plate eposed to incompressible, zero-pressre-gradient flow The integral momentm eqation redces to eqation (5.5), δ ( ) δ ( ) 2 V d d 0 0 (,0) d d ν = d d (5.5) Applies to trblent flows as well withot modification if we look at the behavior of the flow on average, and we interpret the flow properties as time-averaged vales. Estimate of Velocit Profile The integral method reqires an estimate for the velocit profile in the bondar laer. Prandtl and von Kármán both sed a crde bt simple model for the velocit profile sing prior knowledge abot pipe flow. 39

40 Using the Blasis model for the shear at the wall of a circlar pipe, Prandtl [18] and von Kármán [16] each showed that the velocit profile in the pipe cold be modeled as CL = ro This is the well-known 1/7th Law velocit profile, discssed frther in Chapter 9. Wh base a velocit profile for flat plate on pipe flow? The velocit data for pipe flow and flat plate flow (at zero or favorable pressre gradient) have essentiall the same shape, so the se of this model to describe flow over a flat plate is not nreasonable. To appl the 1/7th law to flat plate, we approimate as the edge of the bondar laer, and approimate as. Then, Model for Wall Shear δ V = δ LHS of Integral Momentm is an epression for wall shear; ses assmed velocit profile. 1/7 1/7 CL V r o (8.65) 40

41 Problem: or assmed profile goes to infinit as approaches zero. To avoid this dilemma, Prandtl and von Kármán again looked to pipe flow knowledge The adapted the Blasis correlation for pipe flow friction factor to find an epression for the wall shear on a flat plate Recasting the Blasis correlation terms of the wall shear and the tbe radis, the obtained C f τ o V δ = = ρ V ν 1/4 This is sed in the LHS of the momentm integral relation. Eample 8.2: Integral Soltion for Trblent Bondar Laer Flow over a Flat Plate Consider trblent flow over a flat plate, depicted in Fig Using the 1/7th law velocit profile (8.65) and the epression for friction factor (8.67), obtain epressions for the bondar laer thickness and friction factor along the plate. (8.67) 41

42 (1) Observations. The soltion parallels that of Chapter 5 for laminar flow over a flat plate. (2) Problem Definition. Determine epressions for the bondar laer thickness and friction factor as a fnction of. (3) Soltion Plan. Start with the integral Energ Eqation (5.5), sbstitte the power law velocit profile (8.65) and friction factor (8.67), and solve. (4) Plan Eection. (i) Assmptions. (1) Bondar laer simplifications hold, (2) constant properties, (3) incompressible flow, (4) impermeable flat plate. (ii) Analsis. Sbstitte 1/7th power law velocit profile into the Mom. Int. Eqation: δ ( ) 1/7 δ ( ) 2/7 2 d d V δ ( ) 1/ 7 2/7 τ o d d = V V d V d ρ δ δ (a) Dividing the epression b, and collecting terms, τ o d = 2 ρ V d 0 δ δ d (b) 42

43 After integrating, τ o 2 ρv 7 dδ 72 d = (8.68) Now sbstitting (a) into the wall shear epression (8.67), V δ ν 1/4 = 7 72 Then, separating variables and integrating, dδ d 1/4 4 5/4 72 V δ = C 45 7 ν (c) (8.69) To complete the soltion, a bondar condition is needed. Can assme that δ ( ) is zero at = 0, which ignores the initial laminar bondar laer region However crde the assmption, we find that the reslts of this analsis compare well to eperimental data. 43

44 1/5 ( ) V δ ν = (d) Finall, solve for the friction factor. Sbstitting (8.70) into (8.67), With the bondar condition established, the integration constant C eqals zero. Then solving (8.69) for, ( ) δ 1/ Re δ = (8.70) or 1/4 1/ V V C ν 44 Which redces to f C ν ν = (e) 1/ f C Re = (8.71) (5) Checking. Eqations (8.70) and (8.71) are both dimensionless, as epected.

45 (6) Comments. Note that, according to this model, the trblent 1/5 bondar laer δ/ varies as Re, as does the friction factor C f. This 1/2 is contrast to laminar flow, in which δ/ and var as Re. (ii) Newer Models One limitation of the Prandtl-von Kármán model is that the approimation for the wall shear, Eqn. (8.66), is based on limited eperimental data, and considered to be of limited applicabilit even for pipe flow. White s Model White [14] ses the Law of the Wall velocit profile (8.59) to model the wall shear. First, sbstitting the definitions of and, as well as *, into the Law of the Wall epression (8.59), C f 1/4 2 V C f = 2.44 ln 5.0 V C f ν 2 In theor, an vale within the wall law laer wold satisf this epression, bt a sefl vale to choose is the edge of the bondar laer, where ( = δ ) = V. Then, the above can be epressed as 45

46 C 1 f / 2 Re δ 10 to 10 C C f = 2.44 ln Re δ f (8.72) Still a difficlt relation to se, bt a simpler crve fit over a range of 4 7 vales from gives 1/ (8.73) Re δ We can now se this epression to estimate the wall shear in the integral method. For the velocit profile, the 1/7th power law is still sed. It can be shown that the soltion to the momentm integral eqation in this case becomes and δ 0.16 = 1/7 Re δ (8.74) = 1/7 Re δ (8.75) C f 2 Eqations (8.74) and (8.75) replace the less accrate Prandtl-von Kármán correlations, and White recommends these epressions for general se. 46

47 Kestin and Persen s Model Perhaps a more accrate correlation wold reslt if we se one of the more advanced velocit profiles to estimate the wall shear, as well as to replace the crde 1/7th power law profile. Kestin and Persen sed Spalding s law of the wall for the velocit profile and shear stress. The reslting model is etremel accrate, bt cmbersome. White [20] modified the reslt to obtain the simpler relation C f ln 0.06 = (8.76) ( Re ) White reports that this epression is accrate to within 1% of Kestin and Persen s model. (iii) Total Drag The total drag is fond b integrating the wall shear along the entire plate. Assming the presence of an initial laminar flow region, crit L D = ( τ o ) ( τ o ) (8.76) lam trb 0 F wd wd crit 47

48 ρ V A ρ V wl 2 2 crit L 1 C D = C f, lamd C f, trbd L 0 crit Dividing b, the drag coefficient C is: = D (8.78) Sbstitting Eqn. (4.48) for laminar flow and sing White s model (8.75) for trblent flow, we obtain with some maniplation, Assme = 5 10 crit C 5 D = (8.79) Re Re 1/7 L Effect of Srface Roghness on Friction Factor The interaction between the alread comple trblent flow and the comple, random geometric featres of a rogh wall is the sbject of advanced std and nmerical modeling. However, with crde modeling and some eperimental std we can gain at least some phsical insight. Define k as the average height of roghness elements on the srface. In wall coordinates: k = k * / ν L 48

49 Eperiments show that for small vales of k (less than approimatel 5), the velocit profile and friction factor are naffected b roghness For k > 10 or so, however, the roghness etends beond the viscos sblaer, and the viscos sblaer begins to disappear, likel de to the enhanced miing in the roghness provided. Beond k > 70 viscos effects are virtall eliminated, and the flow is referred to as fll rogh. Beond this vale of roghness, the shape of the velocit profile changes ver little. Conseqentl, we might epect that once the srface is fll rogh, increasing the roghness wold not change the friction factor. 49

50 8.5 Energ Transfer in Eternal Trblent Flow Not srprisingl, energ transfer is also greatl complicated nder trblent flow. We fond in Chapter 2 that the heat transfer for flow over a geometricall similar bod like a flat plate (neglecting both boanc and viscos dissipation) cold be correlated throgh dimensionless analsis b * N = f (, Re, Pr) (2.52) Trblence introdces two new variables into the analsis: the momentm and thermal edd diffsivities, ε and ε. One wa to deal with these new terms is to introdce a new dimensionless parameter: Trblent Prandtl Nmber Pr t ε ε M = (8.81) Approaches to Analzing Trblent Heat Transfer Find a mathematical analog between heat and mass transfer Develop a niversal temperatre profile, similar to how we developed a niversal velocit profile. H M H 50

51 o Then attempt to obtain an approimate soltion for heat transfer sing the integral method The niversal temperatre profile ma also lend itself to a simple algebraic method for evalating the heat transfer. There are more advanced methods, like nmerical soltions to the bondar laer flow, which we will forgo in this tet. We will instead remain focsed on some of the more traditional methods, which are the basis of the correlations commonl in se Momentm and Heat Transfer Analogies Osborne Renolds first discovered a link between momentm and heat transfer in 1874 while stding boilers. He theorized that the heat transfer and the frictional resistance in a pipe are proportional to each other. This is a significant and bold assertion! If we can measre or predict the friction along a wall or pipe, we can determine the heat transfer simpl b sing a mltipling factor. This approach wold allow s to solve for the heat transfer directl, avoiding the difficlt of solving the energ eqation. 51

52 (i) Renolds Analog Consider parallel flow over a flat plate. The pressre gradient dp/d is zero, and the bondar laer momentm and energ eqations (8.38) and (8.39) redce to The bondar conditions are v = ( ν ε M ) T T T v = ( α ε H ) (8.82a) (8.82b) ( = 0) = 0, T ( = 0) = Ts (8.83a) ( ) = V, T ( ) = T (8.83b) Notice that eqations (8.82a&b) and their respective bondar conditions are ver similar; if the were identical, their soltions the velocit and temperatre profiles wold be the same. Normalizing the Variables Select the following variables, 52

53 v T Ts U =, V =, θ =, X = and Y = V V T T L L The bondar laer eqations become U U 1 U U V = ( ν ε M ) X Y V L Y Y θ 1 U θ V θ ( H ) = α ε X Y V L Y With bondar conditions: s (8.84a) (8.84b) U( Y = 0) = 0, θ ( Y = 0) = 0 (8.85a) U( Y ) = 1, θ ( Y ) = 1 (8.85b) Normalizing the variables has made the bondar conditions identical. The bondar laer eqations (8.84) can then be made identical if which is possible nder two conditions. ν ε = α ε M H 1. The kinematic viscosit and thermal diffsivit are eqal: 2. The edd diffsivities are eqal: ν = α ( Pr = 1) (8.86) 53

54 ε = ε ( Pr = 1) (8.87) M H t We can provide some jstification for this assmption b arging that the same trblent mechanism the motion and interaction of flid particles is responsible for both momentm and heat transfer. Renolds made essentiall the same argment, and so Eqation (8.87) b itself is sometimes referred to as Renolds analog. The analog is now complete, meaning that the normalized velocit and temperatre profiles, and are eqal. U ( X, Y ) V ( X, Y ) Developing the Analog Begin b writing the ratio of the apparent heat fl and shear stress (eqations and 8.41), 54

55 ( α ε ) ( ) q app / ρc p α ε H T / = τ app / ρ ν ε M / ν = α ε = ν Imposing the two conditions (8.86) and M H (8.87), sbstitting the dimensionless variables ields app α ( ) θ / q app c p Ts T Y = τ V U / Y (8.88) Since the dimensionless velocit and temperatre profiles are identical, their derivatives cancel. ε (8.89) Another important implication of (8.89) is that the ratio app app is constant throghot the bondar laer. This means we can represent this ratio b the same ratio at the wall. Eqation (8.89) then becomes q o = τ o p ( ) c T T can recast this into a more convenient form b sbstitting 2 and into the above, and rearranging, ( ) = qo = h Ts T τ o = 0.5C h V c ρ s V f ρ V p = C 2 f q / τ 55

56 The terms on the left side can also be written in terms of the Renolds, Nsselt and Prandtl nmbers, St This is the Renolds Analog. St is called the Stanton Nmber. N C f = = ( Pr = 1) (8.90) Re Pr 2 Note: The same analog can also be derived for laminar flow over a flat plate (for Pr =1). Limitations to the Renolds Analog It is limited to Pr = 1 flids. o A reasonable approimation for man gases, bt for most liqids the Prandtl nmbers are mch greater than nit vales of p to 700 are possible. Therefore the Renolds analog is not appropriate for liqids. Also doesn t accont for the varing intensit of moleclar and trblent diffsion in the bondar laer. (ii) Prandtl-Talor Analog 56

57 In independent works, Prandtl [9] and Talor [10] modified the Renolds analog b dividing the bondar laer into two regions o a viscos sblaer where moleclar effects dominate: Analog for the Viscos Laer ν ε and α ε Define the viscos sblaer from = 0 to = 1 The bondar conditions for this region are M o a trblent oter laer, where trblent effects dominate: ν ε and α ε M o Notice that neither of these conditions restricts s to Pr = 1 flids. Define the following normalized variables: H H (0) = 0, T (0) = Ts ( ) =, T ( ) = T v T Ts U =, V =, θ =, X = and Y = T T s 1 1 Then, for the viscos sblaer, the ratio of the apparent heat fl and apparent shear stress (Eqn. 8.86) leads to the following: 57

58 Where we have again noted that Analog for the Oter Laer q T T = Pr (8.93) s o 1 1 τ oc p q / τ = q / τ = constant app app o o Closel resembles the Renolds analog, with M H bt this time we assme that the trblent effects otweigh the moleclar effects, eqation (8.92). The bondar conditions for this region are ε ( 1 ) = 1, T ( 1 ) = T 1 ( ) = V, T ( ) = T = ε The following normalized variables will make the analog valid in this region: v T T = = θ = = = V V T T L L U, V,, X and Y Then, for the oter region, the ratio of the apparent heat fl and apparent shear stress (eqation 8.86) leads to q T T = V (8.94) ( ) o 1 1 τ oc p 58

59 q / τ As before, the ratio app app is constant, so we have chosen the vale at = 1 (which, as we fond for the viscos sblaer, can be represented b q / τ ). o o Adding (8.93) and (8.94) gives q o 1 Ts T = V ( Pr 1) 1 τ oc p V 1 2 τ o = C f ρ V 2 q o C f / 2 St = = ρv c p ( Ts T ) 1 ( Pr 1) 1 V Sbstitting into the above ields The velocit at the edge of the viscos sblaer,, is still nknown 1 Estimate sing the niversal velocit profile. Approimate the edge of the viscos region = 5 Then, from the definition of = 5 = V 1 2 C 1 f 59

60 1 5 Ths, the Prandtl-Talor analog is St (iii) von Kármán Analog V C f = (8.95) N C / 2 f = = Re Pr C f 5 ( Pr 1) (8.96) Theodore von Kármán etended the Renolds analog even frther to inclde a third laer a bffer laer between the viscos sblaer and oter laer. The reslt, developed in Appendi D, is St (iv) Colbrn Analog N C / 2 f = = RePr C f 5Pr ( Pr 1) ln 2 6 (8.97) Colbrn [24] proposed a prel empirical modification to the Renolds analog that acconts for flids with varing Prandtl nmber. 60

61 He proposed the following correlation throgh an empirical fit of available eperimental data: C 2/3 f St Pr = (8.98) The eponent (2/3) on the Prandtl nmber is entirel empirical, and does not contain an theoretical basis. The Colbrn analog is considered to ield acceptable reslts for 7 Re < 10 (inclding the laminar flow regime) and Prandtl nmber ranging from abot 0.5 to 60. Eample 8.3: Average Nsselt Nmber on a Flat Plate Determine the average Nsselt nmber for heat transfer along a flat plate of length L with constant srface temperatre. Use White s model (8.75) for trblent friction factor, and assme a laminar region eists along the initial portion of the plate. (1) Observations. This is a mied-flow tpe problem, with the initial portion of the plate eperiencing laminar flow. (2) Problem Definition. Determine an epression for the average Nsselt nmber for a flat plate of length L. 2 61

62 (3) Soltion Plan. Start with an epression for average heat transfer coefficient, eqation (2.50), and split the integral p between laminar and trblent regions. (4) Plan Eection. (i) Assmptions. (1) Bondar laer assmptions appl, (2) mied (laminar and trblent) flow, (3) constant properties, (4) incompressible flow, (5) impermeable flat plate, (6) niform srface temperatre. (7) transition occrs at c = 5*10 5. (ii) Analsis. The average heat transfer coefficient is fond from: hl Splits this into laminar and trblent regions: L 1 = h( ) d (2.50) L 0 1 c L hl = hlam ( ) d htrb ( ) d L 0 c (8.99) From the definition of Nsselt nmber, we can write the above as: c L hl L 1 1 L = =, lam, trb k 0 (a) N N d N d c 62

63 To find epressions for local Nsselt nmber, we will se the friction factors for laminar flow, and White s model for trblent flow (8.75), and appl them to Colbrn s analog (8.98). The reslts are N, lam 0.332Pr Re N, trb Pr Re = (b) 1/3 1/2 = (c) 1/3 6/7 Sbstitting these epressions into (a) gives: c 1/ 2 L 6/7 hl L V d V N L = = Pr Pr d 1/3 1/3 ( ) ( ) 1/7 = = k ν 0 Which ields c ν 1/3 1/2 7 1/3 ( ) ( 6/7 6/7 N ) L = Pr Re Pr Re Re (d) c c 6 L Finall, since Re c = 5*10 5, (d) redces to: ( ) 6/7 1/3 NL = ReL Pr = (8.100) (iii) Checking. The reslting Nsselt nmber correlation is dimensionless. 63

64 (5) Checking. If the laminar length had been neglected, the reslting correlation wold be 6/7 1/ 3 NL = ReL Pr (8.101) This reslt also makes sense when eamining the mied-flow correlation (8.100). If the plate is ver long, sch that the majorit of the plate is in trblent flow, the second term in the parentheses becomes negligible, leading to (8.101) Validit of Analogies Generall valid for slender bodies, where pressre gradient does not var greatl from zero. Approimatel valid for internal flows in circlar pipes as well, althogh other analogies have been developed specificall for internal flow. Althogh the are derived assming constant wall temperatre, the above correlations work reasonabl well even for constant heat fl. To address propert variation with temperatre, evalate properties at the film temperatre: Ts T T = (8.102) f 2 64

65 Effect of the Trblent Prandtl Nmber Analogies also that Pr t = 1. Valid? Pr t as high as 3 near wall, bt otside the viscos sblaer Pr t seems to be affected slightl b pressre gradient, thogh largel naffected b srface roghness or the presence of bondar laer sction or blowing. A vale of Pr t 0.85 is considered reasonable for most flows. This sggests that the analogies shold be approimatel valid for real flows. Validit of the Colbrn Analog Argabl, the most poplar analog is that of Colbrn. The analog is as primitive as the Renolds analog, adds no new theoretical insight, and is in fact merel a crve-fit of eperimental data. Wh has this method maintained its seflness over the decades? Eas to se More advanced models are based on theoretical assmptions that are, at best, approimations. 65

66 o Prandtl-Talor and von Kármán analogies assme that the viscos sblaer and condction sblaer are the same thickness. Colbrn analog backed b eperimental data over a range of conditions and flids. Empiricism is sometimes better than pre theoretical argments; the test is the eperimental data. Colbrn analog does have critics. Chrchill and Zajic [29] demonstrated that that the Colbrn analog nder-predicts the Nsselt nmber b 30-40% for flids with Prandtl nmbers greater than 7. Despite their shortcomings, analogies are fairl straightforward, and facilitate the development of empirical correlations that are often reasonabl accrate and eas to se. Nmerical soltions, on the other hand, are still difficlt to obtain and are limited in applicabilit. For these reasons, heat and mass transfer analogies remain in widespread se, and new correlations are still being developed often based on this techniqe Universal Trblent Temperatre Profile 66

67 (i) Near-Wall Profile Begin with the trblent energ eqation, (8.39). Akin to the Coette flow assmption, we assme that, near the wall, the velocit component v ~ 0, as is the temperatre gradient T /. Ths the left-hand-side of (8.39) approaches zero. Then, Near the Wall: ( α ε ) 0 H This implies that the apparent heat fl is approimatel constant with respect to, T q app T = ( α ε H ) constant ρc (8.103) p The idea here is the same as we developed for the niversal velocit profile: we can solve the above relation for the temperatre profile. q / ρc First, recognize that, since is constant throghot this region, app p we can replace q with. Then, sbstitting wall coordinates app q o and, (8.103) can be rearranged to * T ρc p ν = q α ε o H (8.104) 67

68 Define a a temperatre wall coordinate as, Then (8.104) becomes, ( s ) T T T T ν = α ε ρc p q We can now integrate the above epression: T = H ν d α ε o * (8.105) (8.106) (8.107) Ver close to the wall, we epect moleclar effects to dominate the heat transfer; that is, α ε H 0 H We will divide the bondar laer into two regions in order to evalate this epression. (ii) Condction Sblaer Invoking this approimation, (8.107) redces to: T Prd Pr = = C 68

69 The constant of integration, C, can be fond b appling the bondar condition that T ( = 0) = 0. This condition ields C = 0, so the temperatre profile in the condction sblaer is T = Pr, ( < ) (8.108) 1 In the above, 1 is the dividing point between the condction and oter laers. (iii) Fll Trblent Region Otside the condction-dominated region close to the wall, we epect that trblent effects dominate: α ε ν ε Rather than develop some new model for ε H, we invoke the trblent Prandtl nmber: ε H = 1 Pr ε t M H T = T d = (8.109) 1 H We alread have a model for ε M, and will assme a constant vale for Pr t Prandtl s miing length theor was ε M = κ 2 2 (8.44) 69

70 In terms of wall coordinates, we can write (8.44) as = / The partial derivative can be fond from the Law of the Wall, Eqation (8.58). Sbstitting the above and (8.58) into (8.109), we obtain T ( ) 2 2 ε M κ ν Pr d κ = (8.111) 1 Finall, if we assme Pr t and κ are constants, then (8.110) becomes: Pr t T = ln, ( > 1 ) κ 1 Kas et al. [30] assmed Pr t = 0.85 and κ = 0.41, bt fond that the thickness of the condction sblaer ( ) 1 varies b flid. (8.110) (8.112) White [14] reports a correlation that can be sed for an flid with Pr 0.7: Pr 2/3 T t ln = 13Pr 7 (8.113) κ Pr t is assmed to be approimatel 0.9 or

71 (iv) A 1/7th Law for Temperatre As with the velocit profile, a simpler 1/7th power law relation is sometimes sed for the temperatre profile: T Ts = T T δ s 1/7 (8.114) Algebraic Method for Heat Transfer Coefficient The eistence of a niversal temperatre and velocit profile makes for a fairl simple, algebraic method to estimate the heat transfer. 71

72 Begin with the definition of the Nsselt nmber, which can be epressed sing Newton s law of cooling as: N h q o = k k T T ( ) s (8.115) Invoke the niversal temperatre profile, T : Using the definition of T, eqation 8.105, we can define the free stream temperatre as follows, * ρc / 2 p ρc pv C f T = ( Ts T ) = ( Ts T ) q q o o (8.116) Sbstitting this epression into (8.115) for (T s T ) and rearranging, N = ρc V C / 2 p kt Then, mltipling the nmerator and denominator bν, N Re Pr T C f f / 2 = (8.117) We can now se the niversal temperatre profile, Eqn. (8.113), to evalate T, 72

73 Pr 2/3 T t ln = 13Pr 7 (8.118) κ A precise vale for is not eas to determine. However, we can make a clever sbstittion sing the Law of the Wall velocit profile, Eqn. (8.58). In the free stream, we can evalate (8.58) as N = 1 ln = B (8.119) κ Sbstitting (8.119) into (8.118) for ln, the Nsselt nmber relation then becomes Re Pr C / 2 ( ) 2/3 Prt B 13 Pr 7 We can simplif this epression frther. Using the definition of Stanton nmber, St = N / ( RePr ),selecting B = 5.0 and Pr t = 0.9, and noting that the definition of leads to = 2 / C f, we can rearrange the relation above to arrive at the final reslt: St = C f / 2 ( 2/3 Pr ) C / 2 Note the similarit to the more advanced momentm-heat transfer analogies of Prandtl and Talor (8.96) and von Kármán (8.97). f f (8.120) 73

74 8.5.5 Integral Methods for Heat Transfer Coefficient One se for the niversal temperatre profile is to model the heat transfer sing the integral energ eqation. Consider trblent flow over a flat plate, where a portion of the leading srface is nheated Can assme a 1/7th power law profile for both the velocit and temperatre (eqations 8.65 and 8.114), and sbstitte them into the Energ Integral Eqation Even with the simplest of assmed profiles, the development is mathematicall cmbersome. A detailed development appears in Appendi E; the reslt of the analsis is: 74

75 St N C f o = = 1 Re Pr 2 Applies to trblent flow over a flat plate with nheated starting length o. Note that (8.121) redces to the Renolds analog when o = 0. This is becase the Prandtl nmber was assmed to be 1 as part of the derivation. The model has been sed to approimate heat transfer for other flids as follows. Eqation (8.121) can be epressed as N N o = 0 = 9/10 1/9 1 ( o / ) 9/10 1/9 (8.121) (8.122) In this form, other models for heat transfer, like von Kármán s analog, cold be sed to approimate for Pr 1 flids. N = Effect of Srface Roghness on Heat Transfer o We wold epect roghness to increase the heat transfer, like it did for the friction factor. However, the mechanisms for momentm and heat transfer are different. 0 75

76 As roghness increases, the viscos sblaer diminishes, to sch an etent that for a fll rogh srface the viscos sblaer disappears altogether. o The trblent flid elements are echanging momentm with srface directl (like profile or pressre drag), and the role of moleclar diffsion (i.e., skin friction) is diminished. Heat transfer, on the other hand, relies on moleclar condction at the srface, no matter how rogh the srface, or how trblent the flow. o There is no pressre drag eqivalent in heat transfer. o Moreover, flid in the spaces between roghness elements is largel stagnant, and transfers heat entirel b moleclar condction. o The condction sblaer, then, can be viewed as the average height of the roghness elements. o The stagnant regions between roghness elements effectivel create a resistance to heat transfer, and is the major sorce of resistance to heat transfer [27]. Bottom line: we can not epect roghness to improve heat transfer as mch as it increases friction. 76

77 This is also means that we can not predict the heat transfer b simpl sing a friction factor for rogh plates along with one of the momentm-heat transfer analogies. What Inflences Heat Transfer on a Rogh Plate? The roghness size k o Epect that roghness size has no inflence ntil it etends beond the viscos and condction sblaers. o Its inflence reaches a maimm beond some roghness size (the fll rogh limit). The Prandtl nmber o Since moleclar condction is important. o Flids with higher Prandtl nmber (lower condctivit) wold be affected more b roghness. Wh? The lower-condctivit flid trapped between the roghness elements will have a higher resistance to heat transfer. Also, the condction sblaer is shorter for these flids, so roghness elements penetrate relativel frther into the thermal bondar laer. o In contrast, for a liqid metal, the condction sblaer ma fll englf the roghness elements, virtall eliminating their inflence 77 on the heat transfer.

78 Kas et al. [30] develop a correlation for rogh plate, which is eqivalent to C f 0.44 (8.123) ( ) St = Pr C k Pr C 2 k = k * / ν t s f / 2 where s s is based on the eqivalent sand-grain roghness k s and C is a constant that depends on roghness geometr. Bogard et al. [31] showed that this model compared well with eperimental data from roghened trbine blades. o Showed a 50% increase in heat transfer over smooth plates. o Demonstrated that increasing roghness beond some vale showed little increase in the heat transfer. 78