Numerical simulation of the heat transfer from a heated plate

Ja

b Schoo

a

ARRAccepted 9 April 2014

a swirl generator nozzle that, depending on its conguration, can produce jets with high or low swirlintensity levels. Different values of non-dimensional nozzle-to-plate distance, H=D, have been studied,as well as different values of the Reynolds number, Re. To know whether or not surface variations along

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Jet impingement on heated surfaces is frequently used as a toolto enhance heat transfer between them, especially at the stagnationpoint. Heat transfer has also a major effect at the region where thejet impinges on the surface, due to the development of the

through impinging jets is owing to its many engineeringapplications, such as the heat transfer of blades in gas turbines[1], the cooling of electronic devices [26], the heat transfer onwindshields of vehicles [7], or cooling in grinding processes [8],between others.

In all the industrial applications presented above, a jet impingeson a surface whose form depends on the particular applicationunder study. Normally, the surface is considered as a at plate with

Corresponding author. Tel.: +34 951952382.E-mail addresses: jortega@uma.es (J. Ortega-Casanova), F.J.GranadosOrtiz@

greenwich.ac.uk (F.J. Granados-Ortiz).

International Journal of Heat and Mass Transfer 76 (2014) 128143

Contents lists availab

International Journal of H

.e1. Introduction boundary layer along the surface, where heat exchange processestake place. The interest of studying this heat transfer mechanismhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.04.0220017-9310/ 2014 Elsevier Ltd. All rights reserved.5. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.1. Nozzle-to-plate distance effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.2. Surface variations effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.3. Nozzle-type effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Conflict of interest statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Keywords:Heat transferNumerical simulationsImpinging jetsTurbulent owsHeated platesSwirling jetsDimpled platesBumped plates

Contents

1. Introduction . . . . . . . . . . . . . . . . .2. Preliminary definitions . . . . . . . .3. The impinging jet . . . . . . . . . . . . .4. Numerical considerations . . . . . .the plate improve the heat transfer between the impinging jet and the plate, our results are comparedwith those obtained when a at plate is used.

2014 Elsevier Ltd. All rights reserved.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Available online 15 May 2014ment of two different types of axisymmetric turbulent jets on a non-at plate, located at a knowndistance H from the jet exit, have been conducted. The cylindrical jet used, of diameter D, is created byr t i c l e i n f o

rticle history:eceived 20 December 2013eceived in revised form 6 March 2014

a b s t r a c t

The study of heat transfer between impinging jets and non-uniform heated plates is presented here toanalyse if surface variations along the plates, (i.e. dimples, bumps, and bumps&dimples, as we studyhere), can improve the heat transfer phenomenon. To that end, numerical simulations of the impinge-l of Computing and Mathematical Sciences, University of Greenwich, Old Royal Naval College, Park Row, London SE10 9LS, UK. Ortega-Casanova a,, F.J. Granados-Ortiz bFluid Mechanics Group, E.T.S. Ingeniera Industrial, Universidad de Mlaga, C/Dr. Ortiz Ramos s/n, 29071 Mlaga, Spainwith surface variations to an impinging jetReview

journal homepage: wwwle at ScienceDirect

eat and Mass Transfer

l sevier .com/locate / i jhmt

l JouNomenclature

cp uid heat capacityCFD computational uid dynamicsds differential surface elementD diameter of the jete specic internal energyG production of turbulence due to mean velocity gradientsGCI grid convergence indexh enthalpyH distance from the nozzle to platek turbulent kinetic energyK thermal conductivity of the uidL swirl parameterLDA laser doppler anemometryn nth computational gridNr radial number of nodesNu Nusselt numberNu area-weighted average Nusselt numberNz axial number of nodes

J. Ortega-Casanova, F.J. Granados-Ortiz / Internationathe jet impinging perpendicular on it (see 9], for a review). How-ever, less studies have been carried out on non-uniform surfaces.In Ekkad and Kontrovitz [10], can be found an experimentallystudy of the effect of dimple location in a plate and the effect ofdimple depth, for different jets with different Reynold numbers.A similar study can be found in Kanokjaruvijit and MartinezBotas[11], where various jets impinge on a staggered array of hemi-spherical dimples with the consideration of various parametriceffects, such as Reynolds number, jet-to-plate distance, depth ofthe dimples, and curvature of the dimples for both impinging ondimples and impinging on at separation portions, showing thatthe variations on the shape of the plate (concretely with shallowdimples) are able to enhance the heat transfer up to a 70% withrespect to the plate one and showing that dimples are moreeffective when a strong crossow is present.

Other interesting studies of the shape variation, specically interms of seeing how one single dimple/bump is relevant, couldbe the one by Imbriale et al. [12], where the heat transfer betweena concave surface and a row of air jets impinging on it is studied byan experimental study by varying the inclination of the jets, pitch,impinging distance, Mach and Reynolds numbers. In ztekin et al.[13], an experimental and numerical study is carried out to

p pressurePIV particle image velocimetryPr Prandtl numberq heat uxQ ow rater; h; z radial, azimuthal and axial coordinatesR radius of the impinged surfaceRe Reynolds numberS impinged surfaceT temperatureu; v; w radial, azimuthal and axial velocitiesUDF user-dened functionv 0 velocity uctuations~V velocity vectorWc characteristic velocityy dimensionless wall distanceY turbulent dissipation

SymbolsN Nusselt coefcients ratioN area-weighted average Nusselt coefcients ratioR nozzle #1S2 nozzle #2

Greek symbolsd Diracs deltaD representative mesh sizeDzp position of the rst node from the plate turbulent kinetic energy dissipationc grid renement factorC effective diffusivityl uid viscositym uid kinematic viscosityx specic turbulent dissipation rateq uid densityr standard deviation. Heat transfer uniformitys computational timee relative error. observed order of accuracy

rnal of Heat and Mass Transfer 76 (2014) 128143 129investigate the turbulent slot jet impingement cooling characteris-tics on concave plates (big dimple) by varying the surfacecurvature and the Reynolds number (around ten times lowerReynolds numbers than those used in the present paper). The morerelevant outcomes of this research were that both the average andstagnation point Nusselt numbers decrease when the nozzle-to-surface distance increases, both the average and stagnation pointNusselt numbers increase when Reynolds number increases, andit is disclosed that the surface curvature increases the averageNusselt number from a depth value.

The analysis of a bump in a at plate can be seen in Zhang et al.[14], combining both PIV and numerical simulation. In this paper,the single jet impinges on the protrusion and the local Nusseltnumber increases with its presence, obtaining relevant conclusionssuch as the local Nusselt number increases when the depthincreases, and the average Nusselt number increases with bumprelative depth and the jet Reynolds number.

Another interesting variation on plate, that also certies howimportant the selection of its conguration is, in heat transferterms, is the inclination. This particular variation is not going tobe treated in the present research, but can be noticed in studiessuch as Beitelmal et al. [15], where an experimental analysis of

Subscript0 magnitude evaluated at r 0a approximatecoarse coarse gride Richardson extrapolation valuene ne gridit iterationseff effectiveext extrapolatei; j coordinate direction in compact tensor notationj jetp platet turbulent

Superscriptb bumped plated dimpled platedb dimpled&bumped platef at plate

of the jet. Once the effects of these parameters have been studied,

Usually, in heat transfer problems, the heat ux q between twomedia (the solid hot plate and the uid, in our case) is quantied,in a dimensionless way, by means of the Nusselt number Nu denedas

Nur qrDKTp T j ; 4

where Tp is the temperature of the heated plate (constant) and T j isthe temperature of the jet once it leaves the nozzle (constant, too).As can be seen, in (4), both the Nusselt number and q present radialdependence, so two values of Nu will be taken into account in orderto characterise the heat transfer process: the Nusselt number at thestagnation point Nu0 Nur 0, and the area-weighted averageNusselt number Nu, dened as

Nu 1S

ZSNurds; 5

where S represents the surface on which the jet impinges (with ds adifferential surface element on it). Nu0 is a local measure of the heattransfer at r 0 on the plate, while Nu is a global measure of theheat transfer taking into account the surface S of the plate.

Since the aim of this study is to know whether a plate withsurface variations works better than a at one, two more parame-ters will be dened to quantify the comparison: the stagnationNusselt coefcients ratio,

l Jouwe will be able to know how the heat transfer can be enhanced,comparing our results with those of a at plate under the sameconditions.

Regarding the plate surface, it will be modied by using smallaxisymmetric dimples, bumps, or both on it. With these changeson the plate surface, we will seek the conguration that providesthe highest heat exchange in comparison with a at plate, whenthe same type of jet and nozzle-to-plate distance are used.

Respecting to the jet, two kinds of nozzle congurations areused, so two families of jets can be generated, each of them withseven different ow rates, or Reynolds numbers. These jets havebeen previously experimentally characterised by the LDA (LaserDoppler Anemometry) technique and mathematically modelled(see [17]), and whose generator nozzle is shown in Fig. 1. Themathematical modelling of the jet is used in the numerical simula-tions as a boundary condition, such as it was also done in the heattransfer study by Ortega-Casanova [16].

Regarding the nozzle-to-plate distance, three different values ofthe non-dimensional parameter H=D were used: H=D 5;10;30.

Some preliminary denitions are given in the next section. Abrief description of the experimental device to generate the jet willbe given in Section 3, together with some details of its mathemat-ical model. Next, in Section 4, some details of the computationalconsiderations taken into account, will be described, as well as, adescription of the geometry under study. Section 5 will be dedi-cated to present and discuss the results. Finally, the conclusionswill be presented in Section 6.

2. Preliminary denitions

In this section, different parameters and variables appearingalong the document will be introduced, as well as different charac-teristic magnitudes and dimensionless variables.

The physical problem is going to be considered symmetric alongthe azimuthal direction h, that is, the problem will be axisymmet-ric. This kind of problems are usually described by cylindrical polarcoordinates r; h; z, with the velocity vector components dened as~Vr; h; z u;v ;w, where u;v and w are the velocities in radial,azimuthal and axial direction, respectively. As characteristiclength, the exit nozzle diameter D is used, while the characteristicvelocity Wc is based on the ow rate Q through the nozzle anddened as

Wc 4QpD2

: 1the effect of the inclination of an impinging two-dimensional airjet on the heat transfer from a uniformly heated at plate wasdeveloped, showing that the maximum Nusselt number decreaseswhen the inclination decreases; or the already mentioned Roy andPatel [7].

Due to the relevant impact of the shape on heat transfer byusing dimples/bumps noticed in the literature explained above,the study presented in this paper is considered by the authors asa relevant research in the heat transfer eld. This relevance issupported by the combination of the shape variation itself andthe addition of an impinging swirling jet to the test cases, whichwas identied as a type of jet that enhances heat transfer incomparison with the non-swirling one (see [16]).

In this article, we mainly focus our attention on how the heattransferred from the plate to the jet can be affected by four aspects:the shape variation of the plate surface, the type of jet created bythe nozzle, the nozzle-to-plate distance, and the Reynolds number

130 J. Ortega-Casanova, F.J. Granados-Ortiz / InternationaRegarding the uid, it will be considered incompressible with con-stant physical properties, such as the density q, the viscosity l orthe thermal conductivity K. All these magnitudes already denedwill allow us to introduce one of the dimensionless numbersgoverning the problem, the Reynolds number:

Re qWcDl

: 2

As in Ortega-Casanova et al. [17], for each nozzle blade congura-tion, seven ow rates were used, which also gives seven Reynoldsnumbers:

Re 0:7;0:9;1:1;1:3;1:5;1:7;1:9 104: 3

Fig. 1. Nozzle used to generate the jets (dimensions in mm).

rnal of Heat and Mass Transfer 76 (2014) 128143Nk0

Nuk0Nuf0

with k b; d; db; 6

and the area-weighted average Nusselt coefcients ratio,

N k Nuk

Nufwith k b; d; db; 7

where f stands for at plate and b, d, db for bumped, dimpled,dimpled&bumped plate. Therefore, values of these parametersabove unity mean that the heat transferred (locally or globally,depending on the parameter analysed) from the plate with surfacevariations is greater than from a at plate, whereas values underunity mean just the opposite.

In order to have comparable results, the radius of the surface Son which Nu was calculated by means of (5), was the same for all

line corresponds to a boundary condition where the ow isallowed to exit. Regarding the upper boundary, it is split in two

J. Ortega-Casanova, F.J. Granados-Ortiz / International JouNu calculated in the work. Obviously, when dimples and/or bumpsare used, the surface S is slightly higher than for a at plate. Inparticular, Sb Sd 1:056Sf and Sdb 1:123Sf .

Another aspect to take into account is that, due to the non-uni-form impinged surface, a bad performance in terms of heat transferuniformity is expected. This uniformity is typically measured (see[18], among others) by the evaluation of the standard deviationpercentage of the Nusselt number on the impinged area:

r 1001S

RS Nur Nur 2

dsq

Nur : 8

This characteristic will be discussed later for a particularconguration.

3. The impinging jet

As we have previously mentioned in the Introduction, the jetused in this work is exactly the same used to study the heat trans-fer from a at plate in Ortega-Casanova [16] (also used in [17], butin an underwater excavation study). The jet is generated by a noz-zle (see Fig. 1) and and it can also have azimuthal motion, which isgiven to the uid by means of eight rotatable blades located at noz-zle bottom and depending the jet swirl intensity on the blade rota-tion (see Fig. 2). Therefore, if the blades are radially oriented, asshown in Fig. 2(a), the swirl intensity is the lowest (nozzle R),while for their maximum rotation, see Fig. 2(b), the jet swirl inten-sity is the highest (nozzle S2). It depends on the Reynolds number,as can be seen in Fig. 3, where a swirl parameter L [see 19] based onthe nozzle exit axial and azimuthal velocity proles, and calculatedas

L R10 r

2wvdrD=2 R10 rw2 1=2v2dr 9

is depicted. Those two nozzle congurations are the ones used inthis study. Although the jet swirl intensity levels of the nozzle Rare not completely nulls, this type of jet will be referred hereinafteras the non-swirling jet.Fig. 2. Plan view of the blades: (a) radially oriented (R conguration); (b) with themaximum rotation (S2 conguration).parts, the left one, where the mathematical model of the jet is usedas boundary condition and ranging from r 0 to r D, and theright one (ranging from r D to the right side of the domain),where the same kind of boundary condition used at the right handside of the domain is imposed. It must be noted that the boundarycondition of the jet extends to D instead of D=2 since the velocityprole at the nozzle exit was measured until that radial position,where the uid velocities are practically nulls. According to theCFD software used to solve the problem, the commercial softwareFluentin our case, the different physical boundary conditions cor-respond to the next ones in Fluents terminology: the axis of sym-The swirling jet characteristics at the nozzle exit (i.e. thevelocity and turbulent magnitude proles), will be used in thenumerical simulations as an inlet boundary condition. They wereall previously experimentally measured using the LDA technique,and mathematically modelled both the velocity proles and itsturbulence (see [17], for details).

All we need to know is that, for each ow rate through thenozzle, i.e. for each Reynolds number, a mathematical function tomodel the jet emerging from the nozzle is totally known, and it willbe used in the numerical simulations as an inlet boundarycondition.

4. Numerical considerations

In this section, different considerations will be presented. Thoseconsiderations are mainly related to computational aspects of thestudy, such as computational geometries, boundary conditions,governing equations and a study of the grid convergence.

According to the azimuthal symmetry of the impinging jet, thephysical geometry will be considered also axisymmetric, so onlythe ow on a bi-dimensional plane is numerically simulated. Thisplane extends radially, from the symmetry axis of the jet to adimensionless distance R=D, far enough to not affect the develop-ment of the jet both from the nozzle and along the plate; and axi-ally, to a dimensionless distance H=D from the exit of the nozzle tothe plate. All these measurements are depicted in Fig. 4, where thegeometry of the at plate without surface variations is shown.

Once the computational plane have been commented, next thesurface variations on the uniform plate will be described. Thereader should keep in mind that the geometry under study is axi-symmetric, so a 3D image of it can be visualised if the 2D axisym-metric plane is rotated 2p radians. The diameter of both thedimples and bumps has been xed to 0:25D, while the separationbetween their centres was xed to D=2, as it is shown in Fig. 6.The joining between the at pieces and the dimples or bumpswas rounded with a radius of 0:05D. A total of 5 dimples, bumps,and dimples&bumps are made on the plate which means thatthe non-uniform zone of the plate extends radially to 2:5D. Theseall plate congurations with dimples, bumps and dimples&bumpsare shown in Fig. 6. The total length R of the plate is R 5D, actu-ally it is equals to 21D=4, when H=D 5;10, and R 15D whenH=D 30, because in this last case, the jet needs a higher radialdomain due to its spreading from the nozzle to the plate.

Fig. 4 can help us to introduce the different boundary conditionsused in the numerical simulations. Once the problem under studyis considered to be axisymmetric, the left line corresponds to theaxis of symmetry of the geometry; the lower boundary corre-sponds to the at/non-at heated plate, where a no-slip boundarycondition with a prescribed known temperature is used; the right

rnal of Heat and Mass Transfer 76 (2014) 128143 131metry is an axis boundary condition; the no-slip plate is a wallwith known temperature; the right boundary condition and theright part of the upper boundary are a pressure-outlet boundary

l Jou132 J. Ortega-Casanova, F.J. Granados-Ortiz / Internationacondition; and, nally, the left part of the upper boundary is avelocity inlet boundary condition, where the turbulent axisym-metric jet generated by the nozzle is imposed by means of a FluentUDF (User Dened Function), by which the velocity and turbulence

Fig. 3. Swirl parameter versus Re for the no

Fig. 4. Sketch of the geometry. Dimensions and boundary conditions.zzle congurations used in the study.

rnal of Heat and Mass Transfer 76 (2014) 128143proles of the jet are mathematically modelled for each Reynoldsnumber (or ow rate) and blade conguration, using the mathe-matical functions given in Ortega-Casanova et al. [17].

The governing equations, solved numerically by the CFDsoftware in order to get the evolution of the jet from the nozzleto the plate, are those corresponding to an axisymmetric incom-pressible turbulent swirling ow that, in Cartesian tensor notation,can be written as:

the continuity equation:

@Vi@xi

0; 10

the momentum equations:

@ViVj@xj

1q

@p@xi

m @@xj

@Vi@xj

@Vj@xi

23dij

@Vl@xl

@ v 0iv 0j @xj

; 11

and the energy equation:

@

@xiViqe p @

@xjKeff

@T@xj

; 12

where e, dened as

e h pq~V ~V2

; 13

is the internal energy per unit mass, T is the temperature, q is thedensity, p is the pressure, m is the kinematic viscosity, h is theenthalpy, d is the Diracs delta, K is the thermal conductivity andKeff K Kt is the effective thermal conductivity that takes intoaccount the turbulent thermal conductivity Kt : Kt cplt=Prt . cp isthe uid heat capacity, lt is the turbulent dynamic viscosity andPrt is the turbulent Prandtl number. However, due to the unknown

velocity uctuations v 0, the problem closure is achieved by using aturbulent model, such as the Shear Stress Transport (SST) kx inthis study. Its use is motivated by the work done by Sagot et al.[20], who found a good agreement between numerical and experi-mental heat transfer results by using the SST kx model. In addi-tion to that, it is also used a range of Reynold numbers close tothose employed in the referred paper, therefore the use of the cho-sen model is properly justied. Nevertheless, our own validationhas been carried out to have an idea about the goodness of the tur-bulent model SST kx is. To that end, we did a comparison of thesolution given by 3 different turbulent models with experimentaldata extracted from literature. In particular, we solved numerically

the impingement of an axial jet, with Reynolds numbers equals to23,000, against a at plate heated with a constant heat ux andlocated at a distance H (=2D) from the jet exit. The turbulent modelsunder study were the SST and the standard kx, and the enhancedk, while the experimental data were extracted from Lee et al. [21]and Baughn and Shimizu [22]. Fig. 5 shows all the proles of the Nu,together with a 10% deviation bands of the SST results. As one cansee, the experimental data are within these bands: the numericalvalues with the SST model agree very well with the experimentaldata both at the stagnation point and for large values of r=D, beingthe maximum discrepancies (of the order of 8%) around the secondpeak of the Nusselt prole; any other tested turbulent model gives

Fig. 5. Validation of the turbulent model.

J. Ortega-Casanova, F.J. Granados-Ortiz / International Journal of Heat and Mass Transfer 76 (2014) 128143 1330.125D

0.05D

0.125D

0.05D

0.125D

0.05D

D/2

D/2

D/2 D/2

0.125D

Fig. 6. Congurations and dimensions of plates.

worse results than the SST one. Due to all these reasons, the SSTmodel is the one used nally for the simulations. Despite of usingthe same turbulent model than Sagot et al. [20], their Nusselt evo-lution in the validation study they show is slightly different fromour. This is because the grid sed in both studies is not the same,

of our problem is the non-uniform plate, and that some magni-tudes evaluated on it will be used to choose the optimum mesh.So, the representative mesh size is chosen as D

pR2

p=Nr, thus

the grid renement factor is given by c Dcoarse=Dfine Nr =Nr . This last value, together with the area-weighted aver-

1;n

.6

134 J. Ortega-Casanova, F.J. Granados-Ortiz / International Journal of Heat and Mass Transfer 76 (2014) 128143although globally both Nusselt proles are quite similar.Therefore, two closure equations are required: one to know the

turbulent kinetic energy k and another one to know the specicturbulent dissipation ratex. In compact notation, they can be writ-ten as

q@

@xikVi @

@xjCk

@k@xj

Gk Yk; 14

q@

@xixVi @

@xjCx

@x@xj

Gx Yx: 15

Ck and Cx are the effective diffusivity of k and x, while Gk and Gxare the generation of k and x respectively, due to mean velocitygradients. Finally, Yk and Yx are the dissipation of k and x respec-tively. To know more about their denition and implementation inFluent, the reader is referred to Fluent 6.2 Users Guide [23].

In order to have an idea of the computational quality of thenumerical solutions, as well as, to chose the optimum grid, a gridconvergence study was carried out. In particular, we solved numer-ically the ow with different meshes for our shortest distance fromthe nozzle to the plate, H=D 5, the highest Reynolds number,Re 1:9 104, and a bumped plate. This conguration is thoughtto be the most unfavourable (with the highest velocities), so theconclusions of the grid convergence study, in terms of both theoptimum grid and its discretization error, can be used for the otherremaining congurations. This study of the grid convergence wasmainly focused on the number of grid points along the plate inr-direction, Nr, and how close the rst node must be from the platein z-direction, i.e. the value Dzp, since that region is, clearly, themore interesting of the domain and where the heat transfer takesplace. In that sense, the rst thing we looked for was the optimumvalue of Dzp, which also gives the value of the dimensionless turbu-lent distance y. It will tell us if the turbulent boundary layer alongthe plate is being solved properly. Thus, the study of y was carriedout for three different distances of the rst node from the plate:DzpD 0:001;0:002;0:003. Once the ow was solved and a steadystate reached, the values we obtained were y 1:1;2:2;3:3. Tosolve the turbulent boundary layer along the plate, values of y

of unity order are needed, so the rst node from the plate waschosen at 0:001D.

With the st row of nodes from the plates chosen, the next stepis to nd the optimum number of the grid nodes for the computa-tional domain. To that end, and as before, we have solved numer-ically the most unfavourable case (H=D 5; Re 1:9 104 and abumped plate) with three different mesh grids to choose the opti-mum one. The total number of grid nodes along radial and axialdirections was Nr 335;435;735 and Nz 80;110;150,respectively, with the nodes concentrated around the plate (withDzp 0:001D), the nozzle exit, and the transitions betweendimples/bumps and at regions. Following Celik et al. [24], wecan estimate the error of the discretization by using the gridspreviously commented. First, we have dened an average repre-sentative mesh size h taking into account that the region of interest

Table 1Discretization error results.

n Nr c hn1=hn Nu . Nune1 735 1.7 104.7 1.8 104

2 435 1.3 104.9 104.63 335 105.1 fine coarse

age Nusselt Nu on the plate obtained from the numerical solutions,are used to calculate the apparent (or observed) order of accuracy .of the method, the area-weighted average Nusselt extrapolatedvalue Nue (by means of the generalized Richardson extrapolation),the approximate relative error ea, the extrapolated relative erroreext and the grid convergence index GCI (for the expressions weused see [24]). They are all shown in Table 1, where the computa-tional time of the software to carry out 1000 iterations (s1000it) isalso shown. Hence, according to these results, the numerical uncer-tainty in the ne(medium)-grid solution for the area-weightedaverage Nusselt number can be reported as 0.4(1.0)%. Since bothare quite low, 1%, being the computational time more than threetimes greater with the ne grid than with the medium, we nallydecide to choose the grid having Nr 435 and Nz 110 nodes asthe optimum. Specically, this radial grid distribution has beenused in all the simulations when H=D 5;10 because of the radialdimension of the domain is the same. However, when H=D 30,the radial domain is higher and more nodes in the radial directionwere used. In particular, for H=D 30; Nr 510 were used, beingfrom node 1 until 435 in the same radial locations than whenH=D 5;10. Regarding the axial dimension for H=D 10;Nz 160, and for H=D 30; Nz 230. These grid points werenon-uniformly distributed along the axial direction having allmeshes not only Dzp 0:001D but also the same grid distributionfrom the plate to a distance equals to D=2, in order to assure thatthe uncertainty study should be valid for all nozzle-to-platedistances. In Fig. 7, one can see a detail of how the grid looksaround an arbitrary dimple.

Regarding the numerical methods used in the simulations, itmust be said that a typical simulation requires about 70 103iterations to converge, being around one fth of the total iterationsdone with rst-order error methods, while the remaining itera-tions were done with the second-order error schemes PRESTO(PREssure STaggering Option) and QUICK (Quadratic Upwind Inter-polation for Convective Kinematics), while the Pressure-VelocityCoupling were carried out with the SIMPLE (Semi-Implicit Methodfor Pressure-Linked Equations) scheme. Finally, the gravity effectswere not taken into account since the inertial forces are much big-ger than the gravitational ones, because of Froude number is muchgreater than one.

5. Results and discussion

Due to the big amount of data available once the simulationswere carried out, this section will be split in three subsections inorder to analyse independently the effect that the nozzle-to-platedistance, type of plate, and type of nozzle used to generate thejet, have on the heat transferred from the plate. All the results willbe shown in terms of N d;b;db0 and N

d;b;db, which help us to comparethe heat transfer between the at plate and non-at ones. In orderto have comparable results, the surface S used to calculate thearea-weighted average Nusselt number in (5) was the same forall geometries and it extends radially 5D.

en1;na (%) en1;next (%) GCI

n1;n (%) s1000it (s)

0.2 0.1 0.4 1,380

0.2 0.3 1.0 450 300

5.1. Nozzle-to-plate distance effect

All the information from the simulations is summarised inFigs. 8, 10 and 14, for H=D 5;10;30, respectively. Different curvescorrespond to different combinations of plate and nozzle-type, asindicated in the legend.

When the smallest distance between the nozzle and the plate isanalysed, see Fig. 8, at the stagnation point only the dimpled plate

impinged by a non-swirling jet (nozzle R) works slightly betterthan the at plate (N 0 > 1), whereas impinged by a swirling jet(nozzle S2) works as good as the at plate (N 0 1). For any othercombination of nozzle and surface type, the heat transferred at thestagnation point is lower than when a at plate is used (N 0 < 1),see Fig. 8(a). It is also remarkable that the lowest heat transfercoefcients ratio at the stagnation point occurs when the jet swirlintensity is the highest (nozzle S2 and Re 0:9 104, as shown in[17]), being the heat transfer at the stagnation point of a at platealmost twice that of a bumped plate, playing the bumps an impor-tant role at high jet swirl intensities. Regarding the global coef-cients ratio N , see Fig. 8(b), it is always greater than unity, sothat, any of the plates tested works better (higher heat transfer)than the at one. In addition to this, for this small separation, thebetter option to get the maximum heat transfer from the plate,both locally and globally, is by means of a dimpled plate (insteadof a at one) and a non-swirling jet (nozzle R) for any of theReynolds numbers under study. On the other hand, the worstbehaviour at the stagnation point is obtained when the bumpedplate is used, regardless of whether swirling or non-swirling jetsare used. This can be explained in terms of the velocity magnitudecontours when a at and a bumped plate are impinged by the samejet, as shown in Fig. 9: the presence of the rst bump creates a stag-nation region between it and the axis of symmetry greater than forthe case without the bump. Due to this, the ow has smaller veloc-ities around the bumped plate than around the at one. This meansthat both the heat transfer and the Nusselt number coefcient onthat small region are also smaller on the bumped plate than onthe at one.

For intermediate separations, in our case H=D 10, whoseresults are shown in Fig. 10, at the stagnation point none of the

Fig. 7. Grid detail around a dimple.

J. Ortega-Casanova, F.J. Granados-Ortiz / International Journal of Heat and Mass Transfer 76 (2014) 128143 135Fig. 8. (a) N 0; (b) N . When H=D 5 and for the pla(a)

(b)te-nozzle combination indicated in the legend.

thrad

l JouFig. 9. Contours of the dimensionless velocity magnitude. Comparison betweenH=D 5; Re 1:5 104 and nozzle R. The shown region extends 2D axially and 5D

136 J. Ortega-Casanova, F.J. Granados-Ortiz / Internationacongurations studied works better than the at plate, at the most,they behave as good as it, giving again the bumped plate the worstheat transfer regardless of the jet used, as shown in Fig. 10(a).Moreover, when the heat transfer is analysed globally by meansof N , as shown in Fig. 10(b), we nd two congurations workingworse than the at plate: the dimpled, and the dimpled&bumped,when a non-swirling jet impinges against them. It must be alsonoted that the impingement of a swirling jet against the bumpedplate gives globally the best results, whereas gives the worstresults locally at the stagnation point. As for H=D 5, the rstbump creates a stagnation region with low velocities, and thisgives rise to a smaller stagnation Nusselt numbers than for a at-plate. The reason why the bumped plate gives the highest heattransfer is because of the friction of the radial jet with the bumps,as can be seen in Fig. 11, where some streamlines are depicted.

Fig. 10. As in Fig. 8 be impingement on a at (left) and a axisymmetric bumped (right) plate forially. The velocity magnitude has been made dimensionless with Wc .

rnal of Heat and Mass Transfer 76 (2014) 128143Those impacts give rise to peaks in the radial distribution ofthe Nusselt number coefcient, as shown in Fig. 12, where it is alsosketched the radial distribution of bumps and dimples. Moreover,although stagnation regions appear between bumps, where theNusselt number diminishes, when the area-weighted average Nus-selt number is obtained, it is greater than the one of the at plate,that is, N > 1. In Fig. 12, it is also depicted the radial evolution ofNusselt number of the dimpled, and dimpled&bumped plate. Asone may observe, when the radial wall jet reaches the bumps,the Nusselt number increases, whereas when it gets to the dimples,the Nusselt number decreases. Despite of that behaviour, the join-ing of the right hand side of the dimple with the next at piece ofthe plate generates local peaks in the Nusselt number, as shown inFig. 12. These behaviours can be explained using Fig. 13, whichshows some of the streamlines of the ow when a dimpled or a

(a)

(b)

ut for H=D 10.

dimpled&bumped plate is located at the bottom. The friction of theradial jet with the bumps is observed again, giving rise to high localpeaks in the Nusselt number shown in Fig. 12, and it is alsoobserved how the radial jet owing on the dimples slightly hitthe dimple right side, giving rise to a peak in the Nusselt number,as shown in Fig. 12. In summary, for intermediate distances, theuse of a at plate always gives better heat transfer at the stagna-tion point than any other combination of nozzle and plate, beingremarkable that the heat transfer can be highly reduced, even

impingement against a bumped plate is better than against a atone. The impingement of swirling jets against any plate alwaysgive better results than impinging against a at one.

Fig. 14 shows the results when the largest nozzle-to-plate sep-aration is studied: H=D 30. Analysing the local heat transfer coef-cients ratio at the stagnation point, it is clear that only thedimpled plate works better than the at one, regardless of the typeof jet used. The remaining nozzle and plate combinations produceworse results at the stagnation point than the at plate, as can beseen in Fig. 19(a). Regarding the global heat transfer coefcientsratio, none of the nozzle and plate combinations is much betterthan the at plate, at the most, they work as good as it whenbumped plates are analysed, as can be seen in Fig. 14(b). InFig. 15, the radial Nusselt number prole, obtained when a swirlingjet (nozzle S2) impinges against the four different surfaces undercomparison, is depicted, and where a behaviour similar to theone described when H=D 10 is observed. In this gure it is alsoshown the radial heat transfer uniformity r, Fig. 15(c), where, asexpected, its high value for non-uniform plates shows a bad perfor-mance in comparison with a at plate. The reasons of this are theNusselt number oscillations associated with the dimples and/orbumps on the plate.

In summary, for large distances, the use of plates with surfacevariations has no benets in enhancing heat transfer in comparisonwith a at one, except at the stagnation point when dimpled platesare used, regardless of the type of jet used.

5.2. Surface variations effect

In the present section, results are plotted in a different way to

Fig. 11. Detail of the bumps region where some streamlines starting at the nozzleexit are shown. H=D 10; Re 1:5 104 and nozzle S2. The shown region extends2D axially and 3D radially.

J. Ortega-Casanova, F.J. Granados-Ortiz / International Journal of Heat and Mass Transfer 76 (2014) 128143 137halved, if bumped plates are used. In relation to global results,non-swirling jets (nozzle R) give the worst results, although theirFig. 12. (a) Sketch of the different plates; (b) Nu. When H=D 10; Re explain how surface variations on the plate affect the heattransferred from it, in comparison with a at one. The results are

(a)

(b)1:5 104, nozzle S2 and the plates are the indicated in the legend.

l Jou138 J. Ortega-Casanova, F.J. Granados-Ortiz / Internationadepicted in Figs. 1618 for the bumped, dimpled, and dim-pled&bumped plate, respectively.

Regarding the bumped plate (see Fig. 16), at the stagnationpoint the heat transfer coefcients ratio is always below unity,which means that any combination of nozzle and nozzle-to-platedistance gives worse heat transfer than a at plate, as shown inFig. 16(a): the rst bump generates a stagnation region with lowheat transfer at the stagnation point (see Fig. 9). However, inregard to the global heat transfer coefcients ratio, in Fig. 16(b)one can observed that its value is always greater (forH=D 5;10) or practically equal to (for H=D 30) unity. This isdue to the friction of the radial wall jet with the bumps (see

Fig. 13. Streamlines for H=D 10; Re 1:5 104 and nozzle S2, when the jet impingesthe shown region in each part are the same than in Fig. 11.

Fig. 14. As in Fig. 8 brnal of Heat and Mass Transfer 76 (2014) 128143Fig. 11), where high local values of heat transfer is obtained. Insummary, globally, a bumped plate will always work better than,or at the most as, a at plate, regardless of the type of jet used,whereas locally it will always work worse.

When the dimpled plate is analysed (see Fig. 17), with respectto the local heat transfer coefcients ratio N 0, it can be said thatsome congurations are slightly better than the at one, such asH=D 30 with any type of jet, and H=D 5 and nozzle R, whereasthe most unfavourable conguration is that one with H=D 10,regardless of the type of jet, for which the heat transfer is a littleworse than with a at plate (for most Reynolds numbers), as canbe seen in Fig. 17(a). With respect to the global heat transfer

against a dimpled&bumped (left part) and a dimpled plate (right part). The limits of

ut for H=D 30.

l Jou(a)

(b)

J. Ortega-Casanova, F.J. Granados-Ortiz / Internationacoefcients ratio, the separation H=D 5 with both types of jets isbetter than using a at plate, and also H=D 10 with a nozzle S2,whereas the worst results are with H=D 30 and any type of jet,and H=D 10 coupled with non-swirling jets for high Reynoldsnumbers, as can be seen in Fig. 17(b). To sum up, locally, few ben-ets can be obtained using dimpled plates whereas, globally, thebest congurations are H=D 5, and H=D 10 with nozzle S2.

For the dimpled&bumped plate, whose results are shown inFig. 18, it is easy to see that at the stagnation point the heat trans-fer coefcients ratio is always below unity, that is to say, the atplate gives always better results than the dimpled&bumped one,regardless of the separation H=D and the type of nozzle used, seeFig. 18(a): the rst bump works as in the bumped plate, creatinga low velocity region around the stagnation point. On the otherhand, with regards to the global coefcients ratio, it can be seenin Fig. 18(b) how with H=D 30 and any type of jet, and withH=D 10 and non-swirling jets, the heat transfer is worse thanusing at plates, whereas for the remaining combinations of sepa-ration and nozzle, the heat transfer coefcients ratio is slightly bet-ter than with the at plate. In summary, locally, none of thedimpled&bumped plates works better than the at one, whereasglobally, the best ones are obtained when H=D 5, and H=D 10with nozzle S2.

5.3. Nozzle-type effect

All the results used to explain the effect of the nozzle on theheat transfer are depicted in Figs. 19 and 20 for nozzles R and

(c)

Fig. 15. (a) Sketch of the different plates; (b) Nu; (c) r. When H=rnal of Heat and Mass Transfer 76 (2014) 128143 139S2, respectively. How the nozzle type affects the heat transfer coef-cients ratios is analysed in what follows.

Regarding the nozzle R and the local heat transfer coefcientsratio (see Fig. 19(a)), only the dimpled plates give better or similarresults than the at one, i.e. N d0J1. The remaining combinationsof separation and plate always give worse results than the at plateat the stagnation point. When the global heat transfer coefcientsratio is analysed, there are some situations where a non-at plateis more preferable than a at one. In particular, that is so for theseparation H=D 5 and any type of surface variations, and alsofor H=D 10 and a bumped plate. Nevertheless, the results withH=D 30 and a bumped plate, and H=D 10 and a dimpled plateare quite similar to those corresponding to a at plate. Thecombinations for which the global heat transfer is worse thanfor a at plate are H=D 30 together with a dimpled and a dim-pled&bumped plate, and H=D 10 with a dimpled&bumped plate.In summary, locally, the best results at the stagnation point areobtained with a dimpled plate, whereas globally, the best onesare obtained when H=D 5.

On the other hand, with respect to the nozzle S2 (see Fig. 20),with relation to the local heat transfer coefcients ratio, the dim-pled plate gives better results than the at one, when H=D 30,and quite similar to them, when H=D 5;10. When bumped ordimpled&bumped plates are used, the results are worse than withthe at one, as can be seen in Fig. 20(a). When the global heattransfer coefcients ratio is analysed, see Fig. 20(b), it is observedthat only when H=D 30, the results are worse than for a at plate,whereas any other separation works better than the at one. There

D 30; Re 1:5 104, nozzle S2 and the plates indicated.

l Jou140 J. Ortega-Casanova, F.J. Granados-Ortiz / Internationais, however, a special combination: the bumped plate and theseparation H=D 10, for which, globally, the results are the bestwhereas, locally, they are the worst. As it was said previously,the use of a bumped plate decreases the Nusselt number

Fig. 16. (a) N 0; (b) N . When a bumped plate is used and for the noz

Fig. 17. As in Fig. 16 but(a)

rnal of Heat and Mass Transfer 76 (2014) 128143coefcients ratio at the stagnation point. This can also beconrmed by having a look at Fig. 21, where the radial prole ofNusselt number is shown for bumped and at plates and all thenozzle-to-plate distances studied for an intermediate Reynolds

(b)

zle-to-plate distances and nozzle types indicated in the legend.

(a)

(b)

for a dimpled plate.

(a)

(b)

Fig. 18. As in Fig. 16 but for a dimpled&bumped plate.

(a)

(b)

Fig. 19. (a) N 0; (b) N . When a nozzle of type R is used and for the nozzle-to-plate distances and plates indicated in the legend.

J. Ortega-Casanova, F.J. Granados-Ortiz / International Journal of Heat and Mass Transfer 76 (2014) 128143 141

(a)

(b)

Fig. 20. As in Fig. 19 but for a nozzle S2.

(a)

(b)

Fig. 21. (a) Sketch of the bumped plate; (b) Nu. When Re 1:5 104, nozzle S2 and for the plates and H=D values indicated in the legend.

142 J. Ortega-Casanova, F.J. Granados-Ortiz / International Journal of Heat and Mass Transfer 76 (2014) 128143

number. It can be also seen how, at the stagnation point, the sep-aration H=D 10 is the worst: the highest difference existsbetween Nub0 and Nu

f0; whereas it is not so high for the other values

of H=D. The analysis of Fig. 21 also conrms that, when the radialjet ows on any bump, a peak always appears in the Nusselt num-ber, independently of the separation. Obviously, the farther theseparation, the lower the peak, being the area-weighted averageNusselt number for H=D 10 and the bumped plate the highest

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6. Conclusions

A numerical study has been conducted in order to identifywhether certain types of non-at plates (at plates with axisym-metric perturbations on them) can transfer more heat than a atone, to an impinging jet (generated by specic nozzle). Differentnon-at plates, nozzle-to-plate distances and Reynolds numbers(ranged from about 0:7 104 to 1:9 104), as well as two type ofjets, swirling and non-swirling have been analysed in the study.Although a general rule to improve the heat transfer from the platecan not be proposed, we are able to give some suggestions whichcould help to a better design of devices from which heat must betransferred by means of impinging jets. The suggestions willdepend on where the heat transfer must be increased, if at thestagnation point or on the whole plate:

At the stagnation point: In comparison with a at plate, the heattransfer can be increased, or at least be as good as the at plate,by using dimpled plates, regardless of the type of both the jetused to impinge the plate and the nozzle-to-plate separation.Bumped and dimpled&bumped plates always give worse heattransfer at the stagnation point than a at plate.On the whole plate:When the heat transfer must be increased inthe whole plate, it is recommended to use bumped plates forwhich the global heat transfer coefcient is greater than, or atleast as high as, that of a at plate. The same can be said fordimpled plates, although in this case there are some combina-tions of jets and nozzle-to-plate separations giving slightlyworse results than a bumped plate. The dimpled&bumped plategives better or clearly worse results than the at plate depend-ing on the jet and nozzle-to-plate separation used.

Obviously, all these suggestions are restricted to the ranges ofthe parameters used in this study.

The results and conclusions reported in this paper can be inter-esting, specically, for future research in cooling processes ofdevices where the temperature has an important role in theirworking behaviour, e.g. electronic components or industrialcomponents in thermodynamic cycles, between others.

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Numerical simulation of the heat transfer from a heated plate with surface variations to an impinging jet1 Introduction2 Preliminary definitions3 The impinging jet4 Numerical considerations5 Results and discussion5.1 Nozzle-to-plate distance effect5.2 Surface variations effect5.3 Nozzle-type effect

6 ConclusionsConflict of interest statementReferences