On thermal boundary layers on a flat plate subjected to a variable heat flux

Transcription

1 Souce: Intenational Jounal of Heat and Fluid Flo Vol. 9 No. pp ; DOI:.6/S-77X(97)6- On theal bounda laes on a flat plate subjected to a vaiable heat flu Jian-Jun SHU and Ioan O School of Mechanical & Aeospace Engineeing Nanang Technological Univesit 5 Nanang Avenue Singapoe jjshu@ntu.edu.sg Abstact The poble of a stead foced convection theal bounda-lae past a flat plate ith a pescibed suface heat flu is investigated both analticall and nueicall. In vie of the pesent foulation the govening equations educe to the ell-kno Blasius siilait equation and to the full boundalae eneg equation ith to paaetes: andtl nube and suface heat flu paaete. The ange of eistence of solutions is consideed. The asptotic solutions ae deived and copaed ith the nueical solutions of the full bounda-lae equation. A ve good ageeent beteen these asptotic solutions and nueical siulations ae found in the ange of andtl nubes consideed. Keods-coponent; foced convection; bounda lae; vaiable all heat flu; eistence of siilait solutions; continuous tansfoations I. INTRODUCTION Histoicall the theoetical desciption of the theal bounda-lae flo began ith the analsis of andtl [] ho applied the bounda-lae concept to the heat tansfe pobles. The ok of andtl as especiall noteoth as it fist intoduced the atheatical technique of bounda-lae theo into the subject of heat tansfe. Subsequentl ohlhausen [] identified siilait solutions fo the heat tansfe pat of the foced convection flo past a flat plate b intoducing the diensionless siilait pofile fo the bounda-lae eneg equation. ohlhausen s analsis has been ve uch efined and genealized since then. It is ell-established that convective heat tansfe depends on the fo of the theal bounda conditions iposed ith it being usual to take eithe a pescibed tepeatue o a pescibed heat flu on the bounda suface. Hoeve in an pobles paticulal those involving the cooling of electical and nuclea coponents the all heat flu is knon. In such pobles oveheating bunout and eltdon ae ve ipotant issues; theefoe one of the objects of heat tansfe theo is the pediction of the all tepeatue as all heat flu vaies. The design objective is to contol this all tepeatue distibution [3]. Hoeve the situation of a pescibed heat flu ate at a suface is often appoiated in pactical applications and is easie to easue in a laboato than the case of a suface ith pescibed all tepeatue. The ain objective of this pape is to coplete the eisting solutions in the open liteatue of the foced convection theal bounda-lae on a flat plate b consideing the case hen a pescibed all heat flu is given and is of the fo () hee and ae the non-diensional Catesian coodinates along and noal to the plate espectivel is the nondiensional tepeatue and is a constant. The analogous poble of fee convection bounda-lae flo on a vetical plate iesed in a viscous (Netonian) fluid ith the pescibed all heat flu given b () as discussed b Mekin and Mahood []. In hat follos the solution of the pesent classical poble ith the case given b () ill be pesented and copaed ith the coesponding esults fo the pescibed unifo all heat flu case. It as shon that as a ust fo a solution of the eneg equation to eist. Hoeve it is found that this equation has a phsicall acceptable asptotic solution fo lage hen. The behaviou of this asptotic solution fo is full discussed. Finall the analtical solutions fo the vaious values of the paaete and andtl nube ae copaed ith the esults obtained using nueical techniques. The nueical esults confi the heat tansfe featues anticipated b the asptotic solutions. It is oth entioning to this end that the pecise fo that the suface heat flu takes is not ipotant onl that it has the functional fos fo sall and lage given b (). II. BASIC EQUATIONS Conside the laina foced convection bounda-lae flo of a viscous incopessible fluid past a flat plate ith a

2 Souce: Intenational Jounal of Heat and Fluid Flo Vol. 9 No. pp ; DOI:.6/S-77X(97)6- unifo fee stea velocit U and constant tepeatue T. The bounda-lae equations in tes of the stea function and tepeatue can be itten in non-diensional fo as 3 () 3 (3) hee is the andtl nube. Equations () and (3) ae subject to the bounda conditions : : () hee is a eal constant. Fo () it is seen that fo sall hile fo lage so that though it is possible to ite don siilait equations fo both sall and lage in the latte case these possess a solution onl if. A -dependenc fo is intoduced into continuous tansfoation as follos f g. (5) Without loss of genealit it can pescibe v ln (6) hee z z v z and z (7) z z ae called the altenative unit-step function and the Konecke delta function espectivel [5]. Substituting (5) into () and (3) e get 3 d f 3 d d f f d (8) g g d g f ln g d d d (9) along ith the bounda conditions g : f d : g. () d The all tepeatue distibution is given no b g. () A standad Kelle bo ethod can be adapted to solve (8) and (9) nueicall [6-6]. The siilait solutions of (9) ae possible fo and ae given b g dg d g f g d () g (3) hich coespond to the theal bounda-lae on a flat plate ith a pescibed unifo heat flu [3]. On the othe hand on letting in (9) e get g g f g d () dg g d. (5) These equations coespond to a pescibed all heat flu ([7] fo ). B integating () and using () fo f and (5) fo g e obtain gd d. (6) Hence e ust have (7) fo a solution of (9) to eist fo both sall and lage. Hoeve thee ae thee sepaate cases to conside fo the pesent poble fo lage nael and espectivel. III. ASYMTOTIC SOLUTION FOR The all tepeatue distibution fo lage is given b g O (8) hee the nueical esults of g ae shon in Table fo the diffeent values of the paaete and andtl nube. The vaiation of the all tepeatue is shon in Fig. fo. It can be seen that the nueical solution attains its asptotic condition as given b (8) fo the case hich coesponds to a unifo all tepeatue fo lage.

3 Souce: Intenational Jounal of Heat and Fluid Flo Vol. 9 No. pp ; DOI:.6/S-77X(97)6- TABLE I. VALUES OF g OBTAINED FROM (9) g We no look fo a solution of () subject to () b epanding h in the fo of seies h h O () ln hee h satisfies the equation dh fh ith h d. (3) We integate (3) and appl () to get d ln ln ln O. () The asptotic epession fo the all tepeatue distibution fo lage and can then be epessed as ln O (5) J hee the values of J ae listed in Table fo diffeent andtl nube. TABLE II. VALUES OF J J Figue. Gaph of calculated fo the nueical solution of (9) (full line) and fo (8) (boken line) fo and IV. ASYMTOTIC SOLUTION FOR To find an asptotic solution of (9) hich is valid fo lage hen e ake the tansfoation ln f h (9) hich is the one that gives () fo the citical case. Using (9) (3) and () becoe h h h f h () ln d subject to the bounda conditions h h ln. () Figue. Gaph of calculated fo the nueical solution of (9) (full line) and fo (5) (boken line) fo and

4 Souce: Intenational Jounal of Heat and Fluid Flo Vol. 9 No. pp ; DOI:.6/S-77X(97)6- The vaiation of the all tepeatue is shon in Fig. fo this case and. V. ASYMTOTIC SOLUTION FOR To find an asptotic solution of (9) hich is valid fo lage hen e ake the tansfoation f G Using (6) (3) and () becoe G G f G d subject to the bounda conditions G d. (6) G (7) G. (8) We look fo a solution of (7) subject to (8) b epanding G in poes of G ( G O ) of the fo (9) hee G satisfies the equation dg fg ith G d. (3) We integate (3) and appl () to get Γ 3 d ; ; F O Γ (3) hee Γ z and F ab;c; z ae the Gaa function and the Gauss function espectivel [5]. The asptotic epession fo the all tepeatue distibution fo lage and can then be epessed as Γ O. (3) Γ J A gaph of (3) is shon in Fig. 3 fo hee e can see that it is in ve good ageeent ith the value obtained fo the nueical solution of (9). The case of as teated b Mekin and Mahood [] as an eception due to the eistence of eigensolutions. No such eigensolution appeas in ou case because the oentu equation (Blasius equation) is decoupled fo the eneg equation. Figue 3. Gaph of calculated fo the nueical solution of (9) (full line) and fo (3) (boken line) fo and CONCLUSION The behaviou of the solution of the equations has been consideed fo the foced convection theal bounda-lae on a flat plate ith a pescibed heating ate popotional to. B solving the govening eneg equation both analticall and nueicall using the Kelle bo schee in cobination ith the continuous tansfoation ethod it has been possible to povide a detailed desciption of the solutions fo lage in the ange and the diffeent values of the andtl nube. Asptotic analsis esults in an altogethe siple foulation giving equations hose analtical solution poves to be oe easil tactable than the oiginal full bounda-lae equation. Ageeent beteen the asptotic and nueical solutions fo the thee egies and poved to be ve good leading us to believe that the asptotic appoach although siple in natue as successful in captuing the essential featues of the heat tansfe chaacteistics. The eponent is theefoe obseved to influence the heat tansfe chaacteistics significantl. REFERENCES [] L. andtl Eine beziehung zischen äeaustausch und stöungsidestand de flüssigkeiten hsikalische Zeitschift vol. pp [] E. ohlhausen De äeaustausch zischen festen köpen und flüssigkeiten it kleine eibung und kleine äeleitung Zeitschift fü Angeandte Matheatik und Mechanik vol. pp [3] A. Bejan Convection Heat Tansfe nd ed. Wile Ne Yok 995. [] J. H. Mekin and T. Mahood On the fee convection bounda lae on a vetical plate ith pescibed suface heat flu Jounal of Engineeing Matheatics vol. no. pp [5] J. Spanie and K. B. Oldha An Atlas of Functions Heisphee Washington DC 987.

5 Souce: Intenational Jounal of Heat and Fluid Flo Vol. 9 No. pp ; DOI:.6/S-77X(97)6- [6] J.-J. Shu and G. Wilks An accuate nueical ethod fo sstes of diffeentio-integal equations associated ith ultiphase flo Coputes & Fluids vol. no. 6 pp [7] J.-J. Shu and G. Wilks Mied-convection laina fil condensation on a sei-infinite vetical plate Jounal of Fluid Mechanics vol. 3 pp [8] J.-J. Shu and G. Wilks Heat tansfe in the flo of a cold todiensional vetical liquid jet against a hot hoizontal plate Intenational Jounal of Heat and Mass Tansfe vol. 39 no. 6 pp [9] J.-J. Shu and I. op Inclined all plues in poous edia Fluid Dnaics Reseach vol. no. pp [] J.-J. Shu and I. op Tansient conjugate fee convection fo a vetical flat plate in a poous ediu subjected to a sudden change in suface heat flu Intenational Jounal of Engineeing Science vol. 36 no. pp [] J.-J. Shu and I. op Theal inteaction beteen fee convection and foced convection along a vetical conducting all Heat and Mass Tansfe vol. 35 no. pp [] J.-J. Shu Micoscale heat tansfe in a fee jet against a plane suface Supelattices and Micostuctues vol. 35 no. 3-6 pp [3] J.-J. Shu and G. Wilks Heat tansfe in the flo of a cold aisetic vetical liquid jet against a hot hoizontal plate Jounal of Heat Tansfe-Tansactions of the ASME vol. 3 no. pp. 8. [] J.-J. Shu and G. Wilks Heat tansfe in the flo of a cold todiensional daining sheet ove a hot hoizontal clinde Euopean Jounal of Mechanics B-Fluids vol. 8 no. pp [5] J.-J. Shu Laina fil condensation heat tansfe on a vetical nonisotheal sei-infinite plate Aabian Jounal fo Science and Engineeing vol. 37 no. 6 pp [6] J.-J. Shu and G. Wilks Heat tansfe in the flo of a cold aisetic jet ove a hot sphee Jounal of Heat Tansfe- Tansactions of the ASME vol. 35 no. 3 pp [7] E. M. Spao and S. H. Lin Bounda laes ith pescibed heat flu application to siultaneous convection and adiation Intenational Jounal of Heat and Mass Tansfe vol. 8 pp