7.0 Heat Transfer in an External Laminar Boundary Layer

Transcription

1 7.0 Heat ransfer in an Eternal Laminar Bundary Layer 7. Intrductin In this chapter, we will assume: ) hat the fluid prperties are cnstant and unaffected by temperature variatins. ) he thermal & mmentum bundary layers, which develp alng a given surface, are nt influenced by any ther adjacent surface. 3) Bdy frces are negligible, & the velcity is sufficiently lw such that the viscus dissipatin term, in the energy eq, can be neglected. 7. Flw Over a Flat Plate at Cnstant emperature Let a fluid with U = cnstant flw ver a cnstant temperature flat plate. A cnstant wall temperature ( = cnstant) indicates that the thermal and mmentum bundary layers will be initiated at the same lcatin (i.e., leading edge f the plate). Define the nn-dimensinal temperature as; he steady state energy eq fr a gaseus bundary layer; (7.) uc v c y y k u y y 0 (7.a) If k = cnstant, which is a gd assumptin fr liquids and a reasnable assumptin fr gases with small temperature gradients, eq 7.a becmes; u v k u y c y c y 0 ` (7.3a)

2 Pr c k Recall the thermal diffusivity, and ; k c u v y y Pr c u y 0 (7.3b) Nte the last term, which represents the dissipatin functin, depends nt nly n the velcity gradient, but als n the Pr. Viscus energy dissipatin can be imprtant fr high Prandlt fluids (i.e., ils) even under mderate velcity gradients. Fr gases, with Pr, the velcity must apprach the speed f sund befre this term is significant. If viscus dissipatin can be neglected; u v y y 0 Fr the current prblem the bundary cnditins are (7.3c) y 0 he hydrdynamic prtin f this prblem was slved using eqs 6. and 6., resulting in eq 6.8: c f 38. / Re Eq 7.3c is very similar t the crrespnding mmentum eq 6.. In fact, if the α = ν (i.e., Pr = ) the same functinal slutin satisfies bth equatins & b.c s. L

3 herefre, fr a fluid with Pr=, the nndimensinal velcity & temperature prfiles are similar and grw alng the plate at the same rate. he cnditin f similar V and prfiles leads t a relatin between the Stantn number (St) and the lcal skin frictin cefficient (cf); 3 St h c f Nu Nu and St and St CU Pe RePr p (7.4) Using a general similarity a slutin t eq 7.3c can be btained; ( ) where and y U u v U y F Substituting int eq 7.3c; Pr F 0 (7.5) where bth θ + and F are functins f η. he F functin was cmputed in chapter 6 frm slutin t the Blasius eq. Rewriting eq 7.5; d d Pr F 0 d Pr Fd 0

4 4 C ep Pr F d (7.6) C Pr ep F d d C 0 (7.7) Applying the b.c., ; gives C = 0 the secnd b.c, ; gives C C ep Pr Fdd (7.8) ( ) ep ep Pr Pr Fdd Fdd (7.9) Recall; h q and q k k( ) y y

5 q k( ) y 5 r in terms f Nu; Nu q k( ) h k U U ( 0) ( 0) but frm eq (7.6): Nu ( 0) U Re ( 0) / ( ) 0 C, Nu ep Re Pr / Fdd (7.0) F(η) was evaluated during the slutin t the mmentum eq.(chapt 6). herefre, the integral in eq (7.0) can be evaluated fr any Pr (see table belw). Values f {Nu Re -/ } fr a range f Pr, fr a laminar bundary layer w/cnstant prperties; U,, and = cnst. Nte: In eternal flw prblems the film temperature (f ) is used t evaluate thermphysical prperties

6 . f 6 Fr the Pr range: 0.5 < Pr < 5 Nu 033. Pr Re 3 / / (7.) Nte: - his crrelatin is gd fr Pr up t abut,000 t within %. - Fr Pr < 0.5 this errr nticeability increases. a. When the Pr = 0. the errr increases t 0%. b. Reducing the Pr t 0.0 the errr increases t 39 %. he fllwing temperature prfiles are displayed as a f(η) emperature prfiles fr a laminar cnstant prperty bundary layer, with U and cnstant Ө+ η Nte: i- Dependence f thermal bundary layer thickness ( ) n Pr. ii- Lw Pr ± > δ iii- High Pr ± < δ iv- Pr = ± = δ

7 7 When Pr << ( >> δ); Let s assume u = U, a slug flw prblem. herefre, F u U and Nu Pr Re / / If Pr >> ( << δ); he thermal bundary layer is entirely within the mmentum bundary layer where the velcity prfile is linear. Frm chapter 6, table 8-; and the final slutin; F ( 0) Nu Pr Re 3 / / Presenting this slutin in terms f the Stantn number, St h Nu CU Pe p Nu Re Pr St Pr 033. Re 3 / / (7.) St Pr 3 / c f Reynlds Clburn Analgy Nte, h decreases in the flw directin as the bundary layer becmes thicker, and as 0, h increases and becmes =0. his is due t an infinitely large temperature gradient at the surface, at = 0. Inspectin f eq 7. r 7. indicates, h = c -/

8 8 calculate the ttal heat transfer rate, integrate h ver the length f the plate (i.e., ) and btain a mean h, h m the mean Nusselt # (Num) c h d d h / (7.3) Nu m Pr Re 3 / / (7.4a) r divide by the prduct f (Re Pr) St m Pr Re 3 / / (7.4b) Eqs 7. & 4 are ften referred t as the Phlhausen slutins. Rewriting the Stantn number in terms f the enthalpy thickness ( ), rather than the distance () alng the plate. 43 / (7.5) Where the is cmputed frm eq (7.A) with i s, representing the freestream enthalpy w.r.t the wall () temperature, Fr a lw velcity, cnstant prperty flw f a perfect gas, and St Pr 005. Re u U uis u i s, (7.Aa) (7.Ab) a gd apprimatin, eq s (7.A) is als applicable t a cnstant prperty liquid flw. dy dy i c s

9 7. Flw Alng a Cnstant emperature Plate with U = c m 9 Since the similarity slutins were btained fr the mmentum eq, ne may epect a similarity slutin fr the energy eq. Using the same similarity variables as thse used fr the mmentum eq, the energy eq is transfrmed t an ODE similar t eq 7.5, with the Pr in the previus slutin (eq 7.9) replaced by Pr(m+). he slutin fr F is btained frm the wedge flw slutins (in terms f m). he fllwing table lists the lcal Nu fr varius values f m & Pr, Nu Re / Cnstant -/ Values f {Nu Re } are listed fr a range f Pr, fr a laminar bundary layer with/ cnstant prperties;, and = cnstant and U = C m m U du d Decelerating Flw Flat plate D Stagnatin Pint Flw Recall, m represents the varius types f wedge flw fields frm which, Or h k U / cnst. h cnst k U Fr wedge flw prblems where U = C m the heat transfer cefficient, / / / h cnst / k m ( )/ Nte: - Fr m =, indicates a D stagnatin pint flw where h = cnstant and des nt vary with.

10 a. therefre fr m = all temperature prfiles are similar b. a cnstant h can nly mean that the thermal bundary layer is f cnstant thickness. - Fr m <, including negative m values and zer (flat plate case, m=0) h starts infinitely = 0 and decreases alng the plate. 3- Fr m >, h zer and increases with. Cmputing the mean cnductance, hm which is defined frm 0 t as, 0 then, h m hd h m h m Ck / m ( m)/ Ck / d ( m)/ hus, h h m m (7.6) When m = and Pr ; Nu 057. Pr Re 04. / (7.7) 7.3 Flw Alng a Cnstant emperature Plate with Injectin r Suctin In chapter 6 a similarity slutin t the mmentum bundary layer eq fr the case V 0 and U =C m was btained. he slutin indicated that V (m-)/ Using eq 7.5 as the energy eq and b.c s, the slutin, eqs 7.9 & 0 are valid. Recall fr m 0, the Pr is replaced by Pr(m+) as nted in a previus sectin n wedge flw slutins.

11 herefre, it is nly necessary t emply the apprpriate values f F(η) frm the slutins t the mmentum eq fr the varius values f the blwing parameter; V U he fllwing tables prvide results fr Nu_, under varius blwing rates, Pr, and m values fr laminar bundary layer flw, respectively. Values f {Nu Re -/ } fr small range Pr and rates f blwing r suctin, fr a cnstant prperty laminar bundary layer n a flat plate (m=0); where,, U are all cnstants. U V / Re Re /

12 Values f {Nu Re -/ } fr varius rates f blwing fr an aisymmetric stagnatin pint (m>); fr a cnstant prperty laminar bundary layer; where the Pr = 0.7 and, = cnstant Values f {Nu Re -/ } fr varius rates f blwing r suctin, and varius values f m; fr a cnstant prperty laminar bundary layer; where,, are all cnstant and the Pr = 0.7. In general these tables indicate that blwing r injectin causes a decrease in the rate f heat transfer, while suctin results in increased heat transfer. Injectin shuld be applied when prtectin (i.e., cling) f surfaces epsed t ht gases is desired. Eamples are turbine blades, rcket engine nzzles, and reentry vehicle nse cnes.

13 Nte: i- Implicit in these slutins are the fact that the injected fluid is the same as the freestream fluid, and its temperature is equal t that f the surface it passes thrugh. ii- he same cnditin hlds fr the case f suctin. As injectin is increased, the Nu 0 which means the bundary layer is cmpletely blwn ff, & the fluid temperature the surface is zer. Injectin f a fluid, different frm the freestream fluid invlves bth the diffusin f heat and mass. If the suctin rate is increased t very large levels, the heat transfer rate becmes equal t the change in enthalpy f the fluid drawn in because the fluid appraches the surface temperature at the surface Flw Alng a Flat Plate w/ Cnstant U and an Unheated Starting Length Apply the energy integral t a laminar incmpressible bundary layer, grwing n a flat plate with a zer pressure gradient. U, X Assume the heating des nt start at the leading edge, = 0 where the hydrdynamic bundary layer begins, but at a psitin = ξ dwnstream frm the leading edge.

14 Assume: 4 - < δ u U 3 y y - Cubic velcity prfile; (7.4) 3 3- Cubic temperature prfile; 3 y y 3 (7.5) Where θ = ( - ) and θ = ( - ); and Substitute eqs 7.4 & 5 int the enthalpy thickness relatin (fr cnstant prperties) & integrate, u U dy (6.5) Nte: Only need t integrate t, thickness f the thermal bundary layer. he integratin yields, (7.6) he rati f thermal t hydrdynamic bundary layer thicknesses is defined, r r r (7.7) Nw substitute fr in eq 7.6;

15 r r 5 Differentiate w.r.t. d d r 0 70 r dr 3 r d 0 80 r d d If r <, the secnd term within each f the parentheses are small, and d 3r dr 3r d 0 d 0 d d using an alternate frm f the energy integral (eq 6.) d d q U c (7.8a) Epressing the heat flu, q in terms f temperature, d d k y U c (7.8b) Substituting this epressin fr d /d and evaluating the gradient at the surface, using eq 7.5, r dr d 0 r d d U 3 Recall, the slutins fr δ and δ dδ/d are available frm chapter 6 slutins f the hydrdynamic bundary layer, 464. U (7.9)

16 Substituting fr these terms, the fllwing ODE fr r is btained, r r r dr 3 4 Pr d 4 r 3 c 3 / 3 4 Pr Evaluating the cnstant c with the b.c., r = 0 at = ξ r / / / Pr 06. (7.30a) Fr ξ =0, heating begins at the leading edge, =0 (n unheated starting length). When Pr =, the tw bundary layers have the same thickness (δ = ). When Pr >, δ >. When Pr <, δ < ; r Pr / 3 (7.30b) Hwever, this slutin is nt applicable and therefre nt accurate, because it was assumed that < δ in the develpment f these slutins. hus, care must be eercised in using this slutin fr Pr <. Yet, fr gases in the range 0.5 < Pr < reasnable results are pssible. he heat transfer rate at the surface, using the assumed temperature prfile, q k y h Substituting eqs 7.5, 9, and 30a fr θ, δ, and r, and then slving fr h,

17 7 h U 033. k Pr 3 / ( / ) 34 / 3 / Determining the lcal Nu, Nu Pr Re ( / ) 3 / / 34 / 3 / (7.3) Fr the case f n unheated starting length (ξ = 0); Nu Pr Re 3 / / (7.3) which is identical t the eact slutin, eq Flw Alng a Flat Plate with a Cnstant U and an Arbitrary Specified Cnstant Surface emperature he fllwing sectin will eamine the heat transfer fr a cnstant prperty, cnstant U fluid passing ver a flat plate with a step change in the bundary cnditin, that is = fr < ξ and = cnstant (and ) fr > ξ. Slutin t eq 7.3c fllws, fr an arbitrary variatin in surface temperature, fr a cnstant U and ;,, y k d i (,, y ) d d y (,, ), i i (7.33)

18 8 Heat flu frm the surface, q k y q k (,, ) d k (,, ), d d y y i i i (7.34a) Nte fr the case f a single step, d d 0 and since q k ky(,, ) y y q h, eq 7.34b can be rewritten as, (7.34b) h( ) k (,, ) y y (,, ) h(, )/ k S fr an arbitrary wall temperature variatin, q h(, ) d k (, ), d d h i i i (7.34c) where h(ξ,) is the lcal cnvectin cefficient btained frm the single step functin slutin. he previusly derived slutin, eq 7.3 can be rewritten as,

19 9 h(, ) 0. 33k Pr Re ( / ) 3 / / 34 / 3 / (7.35) Substituting eq 7.35 int eq7.34c enables the heat transfer rate q fr a flat plate laminar bundary layer, with an aial wall temperature distributin t be determined. Nte: he change in as a f(ξ), and d d must be supplied t the integral eq 7.34c, as well as, any abrupt changes in in the summatin term. Eample: Cnsider a flat plate with a step change in at the leading edge, i.e., ξ = = 0, fllwed by a linear variatin in. 0,i 0, i b Find the wall heat flu? a Let the temperature prfile be written as: a b and the derivative, d d b d d Evaluate eq 7.34c with the step change in ccurring at ξ = 0.

20 0. 33k / / / / q Pr Re ( / ) bd a 0 (7.36) he abve integral can be put int the frm f a Beta functin, n, m z z dz m n (7.37) Where the Beta functin is defined as; mn, ( m) ( n) ( m n) (7.38) fr m > 0 and n < Simplfy by letting r W dw ( / ) / and 4 / d dw / d Rearrange and slve fr dξ, 4 d 3 4 / dw Nw substitute int the integral f eq 7.36 and integrate, b W 3 / d

21 3 / 4 b W 3 4 / dw b 3 / Using the riginal definitin f W, and rearranging, 4 3 W 4 / dw (7.39) W ( / ) / 3 4 aking the /3rd pwer, W 3 / 4 / ( / ) Substitute int eq b W W dw 0 / 3 / (7.40) Using the frmat f the Beta functin, eq 7.37, set the fllwing; m -= -/3 and n -= /3 Evaluate the Beta functin using apprpriate tables fr m = /3 and n = 4/3, mn, ( / 3) ( 4/ 3) ( 6/ 3) 09. S eq 7.40 becmes, q 0. 33k 3 / / 4 Pr Re b. a k 3 / / q Pr Re 6. b a

22 r in terms f the lcal Nu, Nu 3 / / Pr Re 6. b b a a (7.4) his relatin reduces t the cnstant wall temperature slutin when b = 0. Further if a = 0 (n step change in temperature): b 0, and Pr = 0.7 eq (7.4) reduces t, Nu Re / his slutin differs frm the eact slutin, btained using a similarity transfrmatin, by nly.%. --- he End Flw Alng a Flat Plate with a Cnstant U and with an Arbitrary Specified Surface Heat Flu In sme cases the heat transfer cefficient resulting frm a step functin slutin can be put in the fllwing frm; h(, ) f( ) Under this cnditin the wall temperature, as a functin f heat input can be btained. (7.4) Where q ( ) q( ) is an arbitrarily prescribed surface heat flu and g(ξ,) is a mdificatin f the functin h(ξ,) fund frm, ( ) q ( ) g(, ) d

23 g(, ) f ( ) (7.43) 3 Fr a laminar bundary layer grwing n a flat plate, the step functin slutin previusly btained is f the necessary frm, and thus g(ξ,) can be evaluated frm eq 7.3, g(, ) 3 / / Pr Re 6 (/) 4 3 (/) k ( / ) 34 / 3 / (7.44) Substituting eq 7.44 int 7.4 yields, / / 34 / 3 / ( ) Pr Re ( / ) q ( ) d k q If (7.45) = cnstant; the integral in eq 7.45 takes the frm f a Beta functin, and the Nu becmes, Nu (7.46) his result is 36% higher than that determined at the same lcatin, using eq 7., fr a flat plate with cnstant. Recall that the cnstant slutin was btained by a similarity analysis. N similarity slutin eists fr the cnstant heat flu bundary cnditin. Epressing this result in terms f a St, use the energy integral eq t determine the enthalpy thickness ( ) Pr Re 3 / / St Pr 43 / /Re (7.47) Nte the similarity with eq 7.5 fr the case f = cnstant. his suggests that when the lcal length scale is used, the past histry f the thermal bundary layer is nt as imprtant, and therefre eq 7.47 prvides a gd apprimatin.

24 Chapter 7 Addendum 4 Fr the case f a laminar flw ver an isthermal ( =cnstant) flat plate, the lcal Nusselt #, eq 7. applies, Nu 033. Pr Re fr Pr between: 0.6 < Pr < 50 r fr the average Nu # ver the plate L; Nu L 3 / / Pr Re 3 / / L (7.) Gd agreement is bserved in the lwer ReL regime (laminar flw) when cmparing eq 7.t eperimental data. Effects f urbulence (7.) Nu# data cmparisn between thery (eq 7.) and eperiments fr laminar flw ver a flat = cnstant Hwever, eq 7. des nt apply t fluids with very lw Pr, such as liquid

25 metals r fluids with high Pr, such as heavy ils and silicnes. 5 In 973 Churchill & Oze crrelated a large quantity f laminar flw flat plate data and develped the fllwing relatin: Nu C Pr C Pr Re ; fr / 3 / / 3 / 4 Re Pr00 (7.48) All thermphysical prperties are evaluated at the film temperature, he fllwing cnstants are t be applied t eq 7.48 Cnstants f = cnstant (isthermal) Cnstant heat flu C C Reynlds-Clburn Analgy: where St Pr / c 3 f St Nu Pr Re h cu p and c f represents the lcal skin frictin cefficient c f U

26 6 Aial Cnd=0 Aial Cnd=0 Mied B.L.

27 Bundary Layer Flw ver a Flat Plate with Viscus Dissipatin: 7 Effects f viscus dissipatin can be significant if the viscsity f the fluid is high (i.e., flw f lubricant thrugh a jurnal bearing) r if the fluid velcity is high (i.e., gas flw at relatively high M# s). If viscus dissipatin is imprtant thermphysical prperty variatins as f() must be cnsidered. his effect can be apprimated, by evaluating prperties at sme mean temperature. he D gverning eqs fr a flat plate with cnstant U and k are: u u u v 0 y u u p u v y u v k y c y c u y u y Recall, viscus dissipatin is represented by the last term in the energy eq. Velcity bcs are: u = v = y = 0 u = y δ hermal bcs are: q = y = 0 r y = 0 y δ he first thermal bc implies the wall is adiabatic, and frm Furiers the wall (y = 0), /y = 0 Since fluid prperties are assumed cnstant, viscus dissipatin has n effect n the cnservatin f mass and mmentum. herefre the similarity slutins fund in chapter 6 are valid.

28 Nw cnsider the Energy Eq. 8 Case : Adiabatic Wall Similarity temperature prfiles in the presence f viscus dissipatin include the wrk dne by the viscus frces, which results in an increased lcal fluid temperature. In the adiabatic case the temperature prfile is related t the fluids freestream kinetic energy, and has the fllwing nn-dimensinal frm: ( t ) a ( ) ( u / c p ) a ( ) Substituting this frm and the crrespnding velcity prfile int the energy eq, ne btains, after sme rearranging: Pr F a a Pr ( F") 0 Nte that bth θa + depends n bth Pr and η hermal bc=s are; a + = η = 0 a+ = η δ Slving the abve equatin: a ( 0) ep Pr FdPr( F") d ep Pr Fdd

29 9 he abve eq culd be numerically integrated r ne can apprimate θa + by, θ a + = Pr / and therefre r t t Where: aw = a is the adiabatic wall temperature t is the stagnatin temperature is the freestream static temperature r is the recvery factr a aw Fr a perfect gas: aw (0) a (u aw / c a (0) M p ) Where a+ (0) = r ; the recvery factr, therefre, r = Pr / he effect f M# n the recvery factr, fr a laminar flat plate bundary layer in air is presented.

30 30 Case : Cnstant Wall emperature () Since the energy eq is linear in, it was assumed that the cnstant case with viscus dissipatin can be thught f as a cmbinatin f : a) cnstant case w/ dissipatin, and b) cnstant heat flu case w/dissipatin. he resulting temperature prfile, dissipatin;, fr the case f cnstant 0 with viscus Where; = emperature prfile fr adiabatic wall w/ viscus dissipatin aw = emperature prfile fr 0 = cnstant w/ viscus dissipatin and, q 0 Re aw / k [ (0) 0.33 Pr aw / 3 (0)]( ) (, ad )

31 ( q 0 / / Re Pr, ad ) k 3 Defining the lcal Nu ; q0 (, ad ) k Nu ne can write, Nu 0.33 Re / Pr / 3 r integrating ver the plate length the average NuL can be determined, Nu L 0.33 Re L / Pr / 3

32 Eample: Air flws ver a flat plate@ cnstant (= 30 ºC). Determine if heat is being transferred t r frm the plate. 3 Assume a laminar bundary layer with Pr = 0.7 and γ =.4 Recall the recvery factr; r = Pr / =0.837 Slutin: aw r M aw r 0.M aw aw K 36.9C aw= 309.9K = 36.9EC herefre, heat is being transferred frm the air t the plate!!!! --- he End ---

33 33 Effect f Fluid Prperty Variatins w/ Viscus Dissipatin Since large temperature variatins can ccur when viscus dissipatin is present, changes in thermphysical prperties must be cnsidered. When viscus dissipatin is nt f cncern, a mean film temperature can be used t evaluate prperties: (w + )/ Hwever, when viscus dissipatin is imprtant there are 3 temperatures (, w, and aw) that affect fluid prperties and hence, the temperature distributin. he adjacent figure displays emperature distributins in a Laminar bundary layer with and w/ viscus dissipatin. N Viscus Dissipatin Adiabatic Wall with Viscus Dissipatin Actual aw Fluid prperty variatins can be accunted fr using the fllwing relatin, prp 0.5( 0 ) 0.( aw )