(vlachos)saris lekakis heatbelow

8/12/2019 (Vlachos)Saris Lekakis Heatbelow

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Natural convection in rectangular tanks heated locallyfrom below

I.E. Sarris, I. Lekakis, N.S. Vlachos *

Department of Mechanical and Industrial Engineering, University of Thessaly, Athens Avenue, 38334 Volos, Greece

Received 25 March 2003; received in revised form 4 December 2003

Abstract

The present study was undertaken in order to gain understanding of certain aspects of natural convection in a glass-melting tank heated locally from below. Based on numerical predictions, the effects of Rayleigh number, geometry of heated strip and tank on the ow patterns and heat transfer are investigated for Rayleigh numbers in the range 10 2 to107, strip width to tank length ratios 0.10.5 and half-tank aspect ratios 1.0, 3.0 and 7.0. The effect of the heated stripposition is studied by placing it at centre and off-centre positions at the tank bottom wall. The effect of Rayleigh numberon heat transfer is found to be signicant. The Nusselt number is obtained as a function of Rayleigh number, heatedstrip width to tank length ratio and tank aspect ratio. Augmentation of ow circulation intensity and uid temperatureresults when increasing the tank length and strip width. A scale analysis led to a behaviour which is conrmed by thenumerical results.

2004 Elsevier Ltd. All rights reserved.

1. Introduction

Natural convection above horizontal heated surfacesappears in a variety of physical circumstances (e.g.thermal plumes, meteorological and geophysical phe-nomena) as well as in industrial equipment (e.g. coolingof electronics, veneer). The melting of glass is anotherimportant industrial application in which horizontalheated surfaces could play a signicant role. Motivatedby the glass melting process, the present numerical studyexamines in detail the important parameters of thenatural convection above heated horizontal surfaceslocated on the bottom wall of open enclosures such asthose of glass melting tanks.

Although several studies have been performed onthe natural convection along vertical plates, a verylimited number of studies examine the problem of natural convection heat transfer above a horizontalplate. The heat transfer in a vertical cylindrical enclo-sure subjected to localized heating at the centre of its

bottom wall was studied numerically by Torrance andRockett [1] and experimentally by Torrance et al. [2].Their numerical and experimental results show verygood agreement in the laminar ow regime. Pretot et al.[3] reported a numerical and experimental study of natural convection in air above an upward-facing par-tially heated plate placed in a semi-innite medium.Sezai and Mohamad [4] also studied numerically nat-ural convection in air due to a discrete ush-mountedrectangular heat source on the bottom of a horizontalenclosure. Boehm and Kamyab [5] studied laminarnatural convection of air, due to stripwise heatingaccomplished by an array of alternatively heated andnon-heated strips placed on an innite horizontal sur-face in a uid of innite extent. Chu and Hickox [6]studied localized heating from below in a horizontalenclosure of square platform which contained a tem-perature-dependent property uid. In their work, whichwas complemented by experiments, a constant-temper-ature heated strip of xed width was placed on thebottom wall of the enclosure. Aydin and Yang [7]simulated numerically the natural convection heattransfer of air in a two-dimensional, square enclosure

with localized heating applied by a strip placed at the

*

Corresponding author.

E-mail address: [email protected] (N.S. Vlachos).0017-9310/$ - see front matter 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2003.12.022

International Journal of Heat and Mass Transfer 47 (2004) 35493563www.elsevier.com/locate/ijhmt
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bottom wall centre and symmetric cooling from the sidewalls. Their analysis included the inuence of the he-ated strip width and Rayleigh number on the uid owand heat transfer. Katsavos et al. [8] studied experi-mentally and numerically the natural convection of atemperature-dependent uid above a heated wire placedon the bottom of a rectangular tank with the use of anin-house PIV system. Emery and Lee [9] studied theeffect of property variations on the natural convectionin a square enclosure with different boundary condi-tions on the sidewalls. A comparison of the results for avariable property uid with those of a constant prop-erty showed that, although the uid ow and temper-ature elds seem to be different, the overall heattransfer is unaffected by the variation of the uidproperties.

The idea of using heated strips in glass-melting tanksseems to originate from Plumat [10]. He employed he-

ated strips placed at the bottom of the tank and normalto its main axis in physical model studies in order toenhance the melting procedure. The effect of horizontalheated strips at the bottom wall of an industrial glassmelting tank in two- and three-dimensions was studiednumerically by Sarris et al. [11]. Increasing the averagetemperature of the glass-melt, especially in the regionabove the cold bottom wall is of great practical impor-tance to the glass industry.

The heating of glass-melting tanks is mainly imple-mented through radiation from the ames and thecombustion chamber just above the tank. As a conse-

quence, the free surface temperature is controlled by the

radiation absorption of glass-melt. Cheong et al. [12]showed that the Rosseland (or diffusion) approximationis adequate to account for the radiation heat transfer inindustrial glass-melting tanks. This is achieved throughthe concept of the effective thermal conductivity. Thenatural convection due to the temperature variations atthe free surface is usually the driving force for thewhole ow of the glass-melt, considering the very lowpulling rates of the raw materials. Extensive discussionson the effects of the free surface temperature distribu-tion on the ow and heat transfer of the glass-melt canbe found, among others, in [13]. Modern glass-heatingfurnaces include a large number of closely spacedburners, resulting in almost uniform temperature dis-tribution at the glass-melt free surface and in consid-erable reduction of mixing. Therefore, in order toenhance mixing, electric boosting and/or air bubbles areusually introduced into the melt. The alternative

method of using heated strips at the bottom wall, whichhas the advantage of enhancing localized mixing at theplaces where it is needed most, has not been consideredin the open literature. The results presented here showfor which working parameters this method can beeffective.

In the present work, the natural convection in arectangular tank heated with a strip placed at the bot-tom wall of the tank is studied numerically, using a uidwith the same temperature-dependent properties as theglass-melt. The objective is to determine the inuence of the heated strip on the ow and heat transfer charac-

teristics of the glass-melt under uniform free surface

Nomenclature

A aspect ratio of half-tank, L= H As heated strip ratio, W s= L g gravitational acceleration H tank heightk thermal conductivity L tank half length Lt tank total length ( 2 L) Nufs local Nusselt number at the free surface Nu average Nusselt number at the heated strip p uid pressure P dimensionless pressure Pr Prandtl numberq00 heat ux rate from the strip Ra Rayleigh numberT uid temperature

t timeu, v velocity components in x and y directionsU , V dimensionless velocity components x, y spatial coordinates

X , Y dimensionless coordinatesW s strip width

Greek symbolsa thermal diffusivityb coefficient of thermal expansionH non-dimensional temperaturem uid kinematic viscosityq uid densitys non-dimensional time/ generalized variableW non-dimensional stream function

Subscriptsfs free surfaces strip

t totalmax maximumi, j coordinate indices0 reference value

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temperature distribution and to optimize the stripdimensions and heating conditions for efficient mixing of the melt.

2. Mathematical formulation

Consider a rectangular open tank of length Lt 2 Lwith a heated strip of width 2 W s at the bottom of thetank, as shown in Fig. 1, and lled up to height H withglass melt. The boundary conditions considered are: no-slip on the solid boundaries, zero shear stress and zerovertical velocity at the free surface. The temperature of the free surface is kept constant and the tank walls areconsidered adiabatic while a uniform heat lux q00 is as-sumed for the heated strip.

The glass-melt is considered as Newtonian uid, andthe ow incompressible and laminar. For the treatment

of the buoyant term in the momentum equation, theextended Oberbeck [14]Boussinesq [15] approximationis adopted, to account for the temperature variations of density, viscosity and thermal conductivity. Hofmann[16] provides a detailed discussion on the validity of theextended Boussinesq approximation as applied to thesimulation of glass melt in industrial tanks. The specicheat and the coefficient of thermal expansion of the glassmelt are considered temperature independent in thehigh-temperature ranges of the melt, Jian and Zhihao[17]. The thermo-physical properties of the glass-meltused are shown in Table 1. Based on these assumptions,the governing equations in dimensionless form for atwo-dimensional ow become

o U o X

o V o Y

0 1

o U o s

1 Pr 0

U o U o X V

oU o Y

o P o X

o

o X f H

o U o X oo Y f H o U o Y 2

o V o s

1 Pr 0

U oV o X V

oV o Y

o P o Y

o

o X f H

o V o X oo Y f H o V o Y

Ra0H 3

o Ho s

U oHo X

V o H

o Y

o

o X hH

o Ho X

o

o Y hH

o Ho Y 4

The coefficients f H and hH represent in dimension-less form the temperature-dependent viscosity and ther-mal conductivity, respectively. For a uid with constantproperties f H 1:0 and hH 1:0, and for the glassmelt 0 :42 6 f H 6 14:72 and 0 :57 6 hH 6 1:23 in thetemperature range 14001800 K. The dimensionlessvariables that appear in equations (1)(4), are

f H l H

l 0; hH

k H k 0

5

X x H

; Y y H

; U uH a0

; V vH a0

6

s t m0 H 2

; P pH 2

qa 0m0; H

T T 0q00 H k 0

7

where, P and H are the non-dimensional pressureand temperature, respectively, and U and V are the

Table 1Thermo-physical properties of a container glass-melt (afterMuschick and Muysenberg [32] and Jian and Zhihao [17])

Symbol Value ( T in K)

q , kg/m 3 2300.0c P , J/(kg K) 1300.0k eff , W/(m K) 5 :386 2:168 10 2T 2:058 10 5T 2

l , kg/(m s) 10 :0 2:58 4332T 521

b , K 1 6.0 10 5

v

u

y

x

g

q"q"

T 0T 0

adiabatic

a d i a b a t i c

adiabatic

a d i a b a t i c

L L

W sW s

H

Fig. 1. Flow conguration and boundary conditions.

I.E. Sarris et al. / International Journal of Heat and Mass Transfer 47 (2004) 35493563 3551


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non-dimensional velocity components in the x- and y -directions, respectively. The subscript 0 indicates thatthe reference values for the properties are evaluated atthe constant temperature of the free surface ( T 0 1400C).

The characteristic parameters of the ow are theRayleigh number Ra0 g bq00 H 4m0 a0 k 0 based on the tank height H and the strip heat ux q00, and the Prandtl number Pr 0 m0a0. All Rayleigh numbers studied here may beencountered in industrial glass melting tanks, throughcontrol of the externally supplied electric current. ThePrandtl number of the glass melt at the reference tem-perature was approximately 737.

The computations were performed using unsteadyformulation on the half domain for the cases of lowRayleigh number and with the heated strip centred atthe bottom wall. In all other cases, asymmetric heatingat the bottom wall or high Raleigh numbers, the com-putations were carried out in the entire domain. Theinitial and boundary conditions for the entire solutiondomain and with the heated strip centred at the bottomwall are given below:

s 0

U V H 0

s > 0o U o Y

V H 0 at Y 1

U V o H

o X 0 at X 0

U V o Ho X

0 at X 2 L H

2 A

U V o Ho Y

0 at Y 0 and0 6 X >>:U V 0

o Ho Y

19=;

at Y 0 and L W s

H 6 X 6

L W s H

8

The zero reference value of the streamfunction W (cal-culated from the velocity eld using the relationsV o Wo X or U

o Wo Y ) corresponds to the position

X ; Y 0; 0. The Nusselt number is calculated fromthe temperature eld, using the values at the boundaryand the next two interior grid nodes. The averageNusselt number at the heated strip is given by

Nu hH k 0

1H s

9

where, H s is the non-dimensional average temperature at

the heated strip, calculated as

H s T s T 0q00 H =k 0

10

and, the average dimensional temperature at the heatedstrip, T s, is calculated as

T s 1W s Z

L

L W sT x; 0d x 11

The local heat transfer at the free surface of the tank isdetermined by the local Nusselt number:

Nufs hH k

1H s

o Ho Y Y 1

12

and the average Nusselt number at the free surface isgiven as

Nufs W s L

1H s

W s L

Nu 13

indicating that the total heat supplied to the glass meltby the heated strip is equal to the total heat loss from thefree surface.

3. Numerical procedure

The governing equations together with the corre-sponding boundary conditions are solved numerically,employing a nite-volume method. The SIMPLEmethod of Patankar and Spalding [18] is used to couplethe momentum and continuity equations in a uniformstaggered grid. In order to minimize numerical diffusion,the convective terms in the momentum and energyequations are discretized using the QUICK scheme of Leonard [19], in the modied form proposed by Hayaseet al. [20]. The diffusion terms are discretized usingcentral differences, while a second order accurate im-plicit scheme is used for the transient terms. In all cal-culations presented here, under-relaxation factors withvalues of 0.5, 0.5, 0.7 and 0.3 were applied to U , V , Hand P , respectively.

Convergence within each time step is determinedthrough the sum of the absolute relative errors for eachdependent variable in the entire ow eld:

Xi; j/ k 1i; j /

k i; j

/ k i; j6 e 14

where, / represents the variables U , V or H , the super-script k refers to the iteration number and the subscriptsi and j refer to the space coordinates. The value chosenfor e was 10 5, for all calculations. Steady state isachieved when Pi; j j/

n 1i; j /

ni; j j 6 e, where n refers to the

time iteration. Time steps from 10 5 to 10 6 were used toinsure good accuracy in time and ability to capture

instabilities if they exist. All calculations are carried out



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on Intel CPU based personal computers using the in-house CFD code GLASS3D, [21].

The present numerical model was validated againstthe benchmark numerical solution of De Vahl Davis [22]for natural convection of air in a square cavity. Theseresults in the case of Ra 106 showed relative differ-ences of 0.1% and 0.06% for the streamfunction valuesat the centre of the cavity and the average Nusseltnumber, respectively. It was also validated against thenumerical results of Canzarolli and Milanez [23] for thecase of natural convection in a shallow enclosure

( L= H 7) heated from below at constant heat ux andcooled from the sidewalls. The relative differences ob-served in the maximum value of streamfunction and theaverage Nusselt number, at the highest Rayleigh numberof 10 6 studied, were 0.5% and 0.04%, respectively. Fi-nally, the model was tested against the work of Emeryand Lee [9] for natural convection in a square enclosurewith both temperature-dependent viscosity and thermalconductivity, for the case of Ra0 105. At the dimen-sionless height of Y 0:743, the maximum relative dif-ference was found to be 0.01% for the temperature andless than 0.4% for the vertical velocity component.

Prior to the nal computations, grid independencetests were performed for every tank aspect ratio, A,studied. For the square half-tank, the representative caseof As 0:2 was tested for Ra0 104 and 10 6, usinguniform grid sizes of 30 30 to 100 100, in both x- and y -directions of the half tank. As shown in Fig. 2, the

variation of the maximum local Nusselt number at thefree surface appears to be negligible, and for grid sizeslarger than 80 80 the variation of the maximum streamfunction are less than 0.1%. For these reasons, the80 80 uniform grid was selected for all calculations inthe present study. For the shallow tank, a uniform gridof 80 nodes was adopted in the y -direction for both tankaspect ratios ( A 3 and 7) with 140 nodes for A 3 and200 nodes for A 7 in the x-direction. These selectedgrids resulted in differences less than 0.8% for the max-imum local Nusselt number at the free surface andmaximum value of streamfunction from ner grids.

In order to determine possible three-dimensionaleffects, computations were performed for a three-dimensional tank with width equal to the height. In the

x (or y) grid nodes

N u

20 30 40 50 60 70 80 90 100 110

5

10

15

0

5

10

15

20

25

30

Ra 0 = 104

Ra 0 = 106

Ra 0 = 104

Ra 0 = 106

Nusselt

Streamfunction m a x

Fig. 2. Grid independence tests with respect to maximum Nu fsat the free surface and maximum streamfunction value.

(a)

(b)

Fig. 3. Temperature and vertical velocity distributions for Ra0 106 , A 1 and As 0:1 along horizontal planes: (a) Y 0:045, and

(b) Y 0:365.



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z -direction, symmetry conditions were considered andthe grid employed was uniform of size 160 80 40.Indicative results for the case of Ra0 106, A 1 and As 0:1 are presented in Fig. 3, where temperature andvertical velocity distributions along planes parallel to thebottom wall are shown. These results show clearly thatthe heated strip does not produce signicant three-dimensional effects in the range of parameters studied.This behaviour applies to all ow properties and is inagreement with the stability analysis of the glass-meltingtank by Lim et al. [24]. Comparison of their resultsbased on two-dimensional steady-state computationswith those from two- and three-dimensional unsteadycomputations shows no observable differences. For thesereasons, two-dimensional computations were performedin the present study. Numerical experiments also con-ducted in a tank with rigid sides showed no three-dimensional effects for the Rayleigh numbers used in the

present simulation.

4. Results

The present numerical results were obtained for arange of Rayleigh numbers from 10 2 to 10 7, tank aspectratios A 1:0, 3.0 and 7.0 and strip width to tank lengthratios As 0:1, 0.2, 0.3, 0.4 and 0.5. The ow andtemperature elds in the tank are shown in the form of streamlines and isotherms, with 15 equally spaced con-tour levels. The streamline levels are between therespective maximum value and zero and the temperaturelevels range between the zero value at the free surfaceand the highest value corresponding to the heated stripregion.

4.1. Square half-tank with symmetric heating

Figs. 49 show the results corresponding to thesquare half-tank A 1 where the heated strip is cen-tred on the bottom wall of the tank. The ow just abovethe strip ascends towards the cold free surface, thenmoves horizontally towards the corner of the tank,descends to the bottom of the tank and nally returns to

the heated strip region. Fig. 4 shows the streamlines andisotherms for different Rayleigh numbers and a heatedstrip width to tank length ratio As 0:1. When Ra0 102, the isotherms are almost concentric circlesaround the heated strip, because heat is transferred tothe uid body by pure conduction. As the Rayleighnumber increases, and consequently the uid starts tocirculate, the convective mode of heat transfer starts todominate over that of conduction. This increase in thecirculation intensity results in a decrease of the uidtemperature. The free surface, the sidewall and the partof the bottom wall not covered by the heated strip have

almost the same low temperature ( H 0) whereas the

temperature gradients are concentrated just abovethe heated strip, where a thermal plume is formed. In thehighest Rayleigh number cases studied (i.e. Ra0 106,107) the isotherms show signicant curvature just abovethe heated strip.

Fig. 5 shows the dimensionless temperature distri-bution, for strip width ratio As 0:1 along the bottomwall (a) and the symmetry plane (b). The almost similartemperature distributions corresponding to the lowerRayleigh numbers of 10 2 and 10 3 indicate that heattransfer is mainly due to conduction. The maximumtemperature of approximately 0.24 occurs at the centreof the heated strip and corresponds to the pure con-duction limit. With increasing Rayleigh number, thetemperature at the bottom wall not covered by the he-ated strip, decreases and, at Ra0 107, it becomes al-most zero. From the temperature distribution along thesymmetry plane for all cases studied, a conduction

thermal boundary layer is shown to exist adjacent to thefree surface. This layer covers approximately 30% of theuid depth for Ra0 104, 10% for Ra0 105 and lessthan 5% for Ra0 106 and 10 7 . Chu and Hickox [6]observed also this surface conduction layer to have athickness of approximately 15% for Rayleigh numbersof the order of 10 4.

The distribution of local Nusselt number at the freesurface, for the case As 0:1 and for all the Rayleighnumbers studied, is shown in Fig. 6. The positive signindicates that heat is lost from the uid. For the lowerRaleigh numbers, no signicant variations across thesurface are observed due to the pure conduction heattransfer. For higher Rayleigh numbers, the free surfaceheat transfer rate is maximum near the symmetry planedue to the rising plume and minimum at the sidewalls.Increasing the Rayleigh number increases the maximumvalues of the local Nusselt number at the free surfaceabove the heated strip.

The effect of heated strip width to tank length ratio Ason the ow and heat transfer is illustrated in Fig. 7, forthe convection-dominated case of Ra0 106 and therange of As 0:20.5 (the case As 0:1 was shown inFig. 4). Increasing As, does not result in signicantchanges of the ow patterns but intensies the circula-

tion zones and increases the maximum uid tempera-ture. The strip width affects strongly the temperaturedistribution within the body of the uid, but causesinsignicant changes near the sidewall.

The inuence of the Rayleigh number on the presentnatural convection ow can be explained qualitativelyon pure scaling arguments (see [23,25,26]). In the fol-lowing analysis, all quantities are estimated at the ref-erence temperature. As shown in Fig. 4, at high enoughRayleigh numbers the ow along and above the heatedstrip forms two boundary layers. A thermal boundarylayer is formed along the heated strip which turns into a

buoyant thermal plume centred on top of the strip. The



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initiation of this ow can be understood by consideringthe simpler problem of a heated horizontal plate in auid of innite extent. This problem, for the case of

constant uniform temperature, has been studied exten-sively by, for example, Goldstein and Lau [27], Rotemand Claassen [28] and Lewandowski [29], and relativelyless for constant heat ux by Pera and Gebhart [30] andChen et al. [31]. In this kind of heating, the ow is drawninwards from the two edges of the heated strip formingboundary layers along its surface which may meet at thecentre of the strip before they turn through a right angleand form a thermal plume. If the strip width were suf-ciently wide and its heating rate very high, the localbuoyancy will cause the boundary layers to separateprior to the strip centre. The latter is unlikely to occur in

the present ow conguration. The ow inwards is

caused entirely by the lower pressure levels towards thestrip centre which are induced entirely by the buoyancyforce away from the surface.

Using the boundary layer momentum and energyequations for a Prandtl number value of 1 or greater andpure scaling arguments (according to which the pressuregradient term balances the buoyant term in the y -momentum equation, the pressure gradient term bal-ances the viscous term in the x-momentum equation,and the inertial term balances the diffusion term in theenergy equation) leads to a NusseltRayleigh number

dependence: RaW s 1=5

for an isothermal plate, and

(a)

(d)

(c)

(b)

Fig. 4. Streamlines and isotherms for A 1, As 0:1: (a) Ra0 102 (Wmax 0:0, H max 0:24); (b) Ra0 105 (6.48, 0.14);(c) Ra 0 106 (18.77, 0.094); (d) Ra 0 107 (52.208, 0.06).

0 0.25 0.5 0.75 1

X

0

0.05

0.1

0.15

0.2

0.25

Ra 0 = 102

Ra 0 = 103

Ra 0 = 104

Ra 0 = 105

Ra 0 = 106

Ra 0 = 107

A = 1.0, A s = 0.1

Y = 0

0 0.1 0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

Ra 0 = 102

Ra 0 = 103

Ra 0 = 104

Ra 0 = 105

Ra 0 = 106

Ra 0 = 107

A = 1.0, A s = 0.1

X = 0

(a)

(b)

Fig. 5. Temperature distribution in the square half-tank fordifferent values of Ra0 and As 0:1 along: (a) the bottom wall,and (b) the symmetry plane.



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RaW s 1=6 for a constant heat ux plate (present study).

The results of this remarkably simple analysis, agree inform with the analytic results of Chen et al. [31]. Theabove modied Rayleigh number is based on the stripwidth W s as characteristic length.

Because, in the present problem the uid is of niteextent, the rising uid in the thermal plume will have toreturn to the bottom to replace that already risen,resulting in a recirculation pattern. The sense of therecirculation is such that the ow along the bottomwall is in the same direction as that induced by thebuoyancy forces corresponding to the innite uid,when the uid starts moving from rest. This uid mo-tion is expected to result in a thinner thermal bound-ary, which in turn is expected to increase the rate of heat transfer.

Using pure scaling arguments again as in [25] forboth the thermal boundary layer and the plume and with Pr 1 or greater leads to the following scaling laws:

(a) For the energy equation in the thermal boundarylayer of thickness d t over the strip having a temper-ature difference DT :

Longitudinal convection uD T

W s

Transverse diffusion aD T d2t

15

(b) For the u-momentum equation within the thermalboundary layer over the strip:

Pressure gradient D P

W sViscous forces

l ud2

t

16

In the lower region of the plume, where the temperatureis high, the viscosity of the glass-melt could be approx-imately 35 times smaller than that away from the heatedstrip and thus, the viscous forces are not expected to bevery important. In this region, the plume is subjected tostrong acceleration as was observed experimentally byKatsavos et al. [8] using a glycerol solution (a uid withsimilar viscosity behaviour to the glass-melt). Based onthis discussion, it is expected, that in the lower region of the plume, the pressure gradient in the vertical directionis of the same order with the buoyancy forces and theinertial forces, i.e.:

D P

H

qv2

H

q g b D T 17

(b)

(a)

(c)

(d)

Fig. 7. Streamlines and isotherms for Ra0 106 , A 1: (a) As 0:2 (Wmax 25:468, H max 0:117), (b) As 0:3 (30.59,0.132), (c) As 0:4 (34.98, 0.144), (d) As 0:5 (38.86, 0.154).

X

N u

f s

0 0.25 0.5 0.75 10

1

2

3

4

5

6

Ra 0 = 102

Ra 0 = 103

Ra 0 = 104

Ra 0 = 105

Ra 0 = 106

Ra 0 = 107

Fig. 6. Distribution of Nufs along the free surface for A 1, As 0:1, and different values of Ra .



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The pressure rise at the centre of the heated strip on thewall region where the plume stagnates is

D P qv2 18

and from mass continuity

vdp udt 19

where dp is the width of the plume.

Eqs. (15)(19) are sufficient for determining the un-known scales. Of particular interest are the resultingscaling laws for the thickness dt , and the velocity u in thethermal boundary layer along the width of the strip:

dt H W s

H 1=2

Ra 1=4T

20

u a H

Ra1=2T 21

for a strip at uniform temperature with the Rayleigh

number dened as Ra T g bD TH 3

m0 a0, where D T the difference

between the temperature of the strip and T 0, and

dt H W s

H 2=5

Ra 1=50 22

u a

H

W s

H 1=5

Ra2=50

23

for a heated strip of uniform ux, as in the presentstudy.

The maximum value of the streamfunction wmax is of the order of ud, where d is the velocity boundary layerthickness along the heated strip. For the case of Pr 1,the velocity boundary layer thickness d is of the order:d dt Pr 1=2.

Thus, we have

For constant temperature :

wmax a W s

H 1=2

Ra1=4T Pr

1=2

24

and for constant flux :

wmax a W s

H 3=5

Ra1=50 Pr 1=2 25

Finally, from the total heat rate ( Q q00W s) entering thetank bottom from the heated strip, the average Nusseltnumber over the heated strip can be expressed as

Constant temperature :

Nu W s

H 1=2

Ra1=4T As A1=2 Ra1=4T 26

Constant flux :

Nu W s

H 2=5

Ra1=50 As A2=5 Ra1=50 27

Owing to the fact that dt W s, the domain in which thescale analysis is valid can be determined from Eqs. (20)and (22):

Constant temperature :

W s H

1=2

Ra1=4T As A1=2 Ra1=4T 1 28

Rayleigh

N u s s e l t

10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 810 0

10 1

10 2

As = 0.1As = 0.5Scale Analysis

Nu Ra 1/5

Pure conductionlimits:

Fig. 8. Variation of Nu at the heated strip as a function of Ra0for A 1 ( As 0:1 and 0.5).

W s /L

N u s s e l t

0.1 0.2 0.3 0.4 0.5 0.610 -1

10 0

10 1

Ra 0 = 102

Ra 0 = 103

Ra 0 = 104

Ra 0 = 105

Ra 0 = 106

Ra 0 = 107

Scale Analysis

Nu A s-2/5

Fig. 9. Variation of Nu at the heated strip for A 1 as afunction of As at different R0 .



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shrinks and this process continues until the Ra 0 reachesthe value of 10 4. Further increase in the Rayleighnumber causes convection to dominate over conductionand as a result the circulation cell expands to occupy theentire tank.

The temperature elds show the existence of two welldened regions, a cold uniform temperature region anda hot region. A thermal penetration length can be de-ned which represents the horizontal distance from thevertical plane passing through the centre of the heatedstrip which is affected by the thermal plume. For com-parison purposes, this distance is determined by thefurthest point of the lowest value isotherm (level value is1/15). A thermal penetration due to convection appearsat approximately Ra0 104 , while at Ra0 105 , thethermal penetration occupies more than one-third of thetank length; at Ra0 106, occupies approximately two-thirds of the tank length; and at Ra0 107, reaches al-

most the sidewall of the tank. This thermal penetrationlength concept has also been studied, among others, byPoulikakos [26] and Ganzarolli and Milanez [23], butwas based on an isotherm value of 1/10. Increase of theRayleigh number makes the thermal plume stronger andslender, as in the square half-tank case.

The inuence of the heated strip width ratio As on theow and heat transfer characteristics for A 3 and Ra0 106 is illustrated in Fig. 11 (the value of As 0:1was already illustrated in Fig. 10). The circulation pat-terns are not affected signicantly by As, but the maxi-mum streamfunction value, which is a measure of theirintensity, is increased by a factor of 2 between As 0:1and 0.5. The temperature distribution inside the uid

body is strongly affected by As, especially near the heatedstrip, as in the corresponding square half-tank case.

The effect of Rayleigh number and heated strip widthratio As on the Nusselt number for a tank with A 3is shown in Fig. 12. Increasing As, the Nusselt numberdecreases as in the square half-tank case. The Nusseltnumber corresponding to a given As increases alsowith increasing Rayleigh number. For the two smaller Ra0 studied, heat transfer is dominated mainly by

(d)

(c)

(b)

(a)

Fig. 11. Streamlines and isotherms for Ra 0 106, A 3: (a) As 0:2 (Wmax 38:335, H max 0:148), (b) As 0:3 (45.756, 0.167), (c)

As 0:4 (51.609, 0.183), (d) As 0:5 (56.585, 0.197).

W s /L

N u s s e l t

0.1 0.2 0.3 0 .4 0.5 0.610 -1

10 0

10 1

Ra 0 = 102

Ra 0 = 103

Ra 0 = 104

Ra 0 = 105

Ra 0 = 106

Ra 0 = 107

Scale Analysis

Nu A s-2/5

Fig. 12. Variation of Nu at the heated strip for A

3, as afunction of As at different Ra 0 .



12/15

conduction and the Nu vs. As curves are relatively closeto each other, especially for small As. However, for Ra0 103, as As increases, a weak contribution fromconvection causes the small differences observed.

The ow and temperature elds for the most shallowtank studied ( A 7) with As 0:1 and different Ra0values are shown in Fig. 13. The region where the tem-perature is affected by the strip is large at low Ra0 dueto conduction and shrinks initially with the Ra0 until athermal plume is formed at approximately Ra0 104.This plume becomes more slender by further increas-ing the Rayleigh number and simultaneously the tem-perature penetration length is increased as a result of the inuence of the free surface. At the highest Ray-leigh number the thermal penetration approaches thetank sidewall. As previously observed (case A 3) theextent of the circulation cell increases monotonicallywith increasing Ra0 in the pure convective regime of

Ra0 > 104.

The ow and temperature elds for strip width values As 0:2 to 0.5 and for Ra 0 106 are shown in Fig. 14.The circulation cell seems to be of larger extent andmore intense with increasing As, as previously observedfor the other tank aspect ratios. The thermal penetrationlength increases also with increasing As and, for thelargest value it reaches almost the sidewall. In principle,both parameters Ra0 and As can increase the heattransfer in a tank of specied aspect ratio A, in accor-dance with Eq. (27). However, an important aspect of glass production process, which affects the quality of thenal product, is homogenization of the glass melt. Thisbecomes possible through the existence of circulationcurrents in the tank and uniform temperature elds,avoiding the formation of cold regions. Comparing theow and temperature elds in Figs. 10 and 13, whichcorrespond to xed As with variable Ra0 , with therespective Figs. 11 and 14 which correspond to xed Ra0

with variable As, we observe by increasing As that: (a) the

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 13. Streamlines and isotherms for A 7, As 0:1: (a) Ra0 102 (Wmax 0:0, H max 0:734), (b) Ra0 103 (1.131, 0.715), (c) Ra0 104 (5.875, 0.406), (d) Ra 0 105 (15.933, 0.244), (e) Ra 0 106 (40.247, 0.149), (f) Ra 0 107 (111.475, 0.089).

(a)

(b)

(c)

(d)

Fig. 14. Streamlines and isotherms for Ra 0 106 , A 7: (a) As 0:2 (Wmax 53:546, H max 0:183), (b) As 0:3 (62.898, 0.208), (c)

As 0:4 (70.309, 0.229), (d) As 0:5 (76.503, 0.248).



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circulation current is stronger as indicated by the peakstreamline values and of larger extent covering the entirehalf of the tank, which in turn promotes mixing andhomogenization of the glass melt, and (b) the isothermsshow that the increase of As raises the temperature in theentire tank, thus avoiding the presence of cold regionsand the possibility of local solidication of the melt.This temperature eld contributes further to thehomogenization of the glass melt. The increase in Ray-leigh number (through increasing the heat ux rate q00

and consequently the operational costs) increases mainlythe temperature locally above the heated strip. Based onthe above arguments, it seems that the most effectiveway of increasing ow circulation currents and temper-ature of the glass melt is by increasing As for all tankaspect ratios studied.

The dependence of the Nusselt number on the Ray-leigh number for different tank aspect ratios and the

xed value of As 0:1 is shown in Fig. 15. The Nu / Ra1=5 relation describes well the heat transfer byconvection, for all the tank aspect ratios and for allRayleigh numbers studied. For low Rayleigh numbersthe heat transfer is due to conduction and, thus, theNusselt numbers remain unchanged. The Nusselt num-bers corresponding to pure conduction are shown in Fig.15 by the horizontal dashed lines. For this range of Rayleigh numbers the criterion of validity of the scaleanalysis, Eq. (29), is not satised anymore. For A 1, 3and 7, this criterion gives that Eq. (27) is approximatelyvalid for Ra 0 102, 11 and 2, respectively.

Finally, the effect of tank aspect ratio on the Nusseltnumber for different Rayleigh numbers and a xed value As 0:1 is shown in Fig. 16. The Nu decreases almostlinearly on a loglog plot with the tank aspect ratio andwith approximately the same slope for Ra0 P 104 (con-

vective regime) as is indicated by Eq. (27). As shown inFig. 16, for Ra 0 102 and all aspect ratios studied, theNusselt number approximates the conductive regime,whereas for Ra0 103 at some aspect ratios heat transferis due to pure conduction and at some others to a mixedmode. For Ra0 103 , a mixed mode behaviour whenincreasing the tank aspect ratio occurs for A > 2. Thisdemonstrates the inuence of A on the initiation of convective currents, as implied by Eq. (27). The slope of the curve Nusselt number vs. A curve is different than 2

5as expected, because heat transfer is not yet due mainlyto the convection.

4.3. Non-symmetric heating

The heated strip is placed successively at the tankcentre (i.e. 12 Lt ) and to the right of the centre in incre-ments of 112 Lt . The cases studied include all Ra0 , ant the Aand As ranges as in the symmetric heating cases. Therecirculation patterns and temperature elds for therepresentative case of Ra 0 106 , A 3 and As 0:1 areshown in Fig. 17 with the heating strip centred at posi-

tions 12 Lt , 812 Lt and 1112 Lt . When the strip position is closeto the tank centre, the familiar double cell ow patterndoes appear. The extent of each of the two cells, as wellas the resulting temperature penetration length arecompletely controlled by the strip position. As the stripapproaches the adiabatic sidewall, the smaller circula-tion cell disappears while the other cell covers the entiretank, resulting in a small increase of the maximum uidtemperature.

The effect of the heated strip position on the heattransfer at the free surface of the tank (expressed in theform of local Nusselt number distribution) is shown in

Fig. 18. The peak value of the local Nu fs is considerably

L/H

N u s s e l t

1 2 3 4 5 6 7 810 -1

10 0

10 1

Ra 0 = 102

Ra 0 = 103

Ra 0 = 104

Ra 0 = 105

Ra 0 = 106

Ra 0 = 107

Scale Analysis

Nu A -2/5

Fig. 16. Variation of Nu at the heated strip with As 0:1 as afunction of A at different Ra .

Rayleigh

N u s s e l t

10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 810 0

10 1

10 2

A = 1.0A = 3.0A = 7.0Scale Analysis

Nu Ra 1/5

As = 0.1

Pure conductionlimits:

Fig. 15. Variation of Nu at the heated strip with As 0:1 as a

function of Ra 0 at different aspect ratios A.



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