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L. LANGELLOTTO AND O. MANCA 441

Radiative effects are particularly interesting in convergent chan-nels, due to the large view factor toward the ambient [13, 14]. Inthe rst study, a numerical analysis was carried out in laminar,two-dimensional steady-state conditions, with the two principalat plates at uniform heat ux and taking into account wall con-ductivity and emissivity. Average Nusselt numbers were eval-

uated and simple monomial correlations for average Nusseltnumbers, in terms of channel Rayleigh numbers, were pro-posed. In the second study, an experimental investigation onnatural convection in air, in a convergent channel, with uniformheat ux at the walls, was carried out. Average Nusselt numberswere evaluated and simple monomial correlations for dimen-sionless maximum wall temperatures and average Nusselt num-bers were proposed in terms of channel Rayleigh numbers in thesame ranges given in [13]. Numerical results, obtained in [13],were in very good agreement with experimental results given in[14].

Design charts for the evaluation of thermal and geometricalparameters, for natural convection in air, were proposed for nat-ural convection in vertical convergent channels in [15]. Thermaldesign and optimization of a channel in stack of convergentchannels were obtained employing the correlations among themore signicant dimensionless thermal and geometrical param-eters.

Proposed correlations for natural convection in conver-gent channels, given in the already-mentioned papers, arereported in Table 1. In the present paper, a scale analysisis carried out following the procedure given in references[1620]. New correlations for convective heat transfer con-tribution in terms of Reynolds numbers, dimensionless walltemperature, and global Nusselt numbers are proposed. Moreaccurate new correlations for the ratio between radiative andglobal heat ux (radiative and convective heat uxes) areevaluated.

The new correlations extend the analysis presented in refer-ences [15] and [2023]. They are obtained by enlarging the re-sults given in [13] to large values of channel aspect ratioand lowRayleigh numbers. This also allows evaluation of the thermalbehavior of the convergent channels in a possible fully devel-oped ow. The analysis is proposed to evaluate the previouslymentioned variable for vertical convergent channel, with sur-face emissivity ranging from 0.10 to 0.90, for a single assignedwall thickness and thermal conductivity, for convergence anglefrom 0 to 10 , ratio between minimum and maximum channel

spacing, b min /bmax , in the range from 0.048 to 1.0, aspect ratio,Lw /bmin , in the range from 10 to 80, and global Rayleigh num-bers referred to b min , Ra bmin , in the range from 2.5 10 2 to2.3 105.

From a different point of view, the present study may beconceived as an effort to estimate the right balance betweenthe control of the maximum wall temperature and an ap-plied symmetrical wall heat ux. Moreover, this attempt canalso be viewed as the maximization of heat transfer for anassigned available total volume that is constrained by space

Figure 1 Sketch of the conguration: (a) physical domain; (b) computational

domain.

limitations. This goal has been studied in references [10],[16], [19], and [24], and reviewed lately in [25] and morerecently in [26]. The present geometry is important in elec-tronic cooling [9, 10, 27] and in solar energy components[28, 29].

MODEL DESCRIPTION AND NUMERICAL PROCEDURE

Model Description

The physical domain under investigation is shown inFigure 1a. It consists of two nonparallel plates that form a ver-tical convergent channel. Both plates are thermally conductive,gray, and heated at uniform heat ux. The imbalance betweenthe temperature of theambient air, T o , and the temperature of theheated plates draws air into the channel. The ow in the channelis assumed to be steady-state, two-dimensional, laminar, incom-pressible, with negligible viscous dissipation. All thermophys-

ical properties of the uid are assumed to be constant, exceptfor the dependence of density on the temperature (Boussinesqapproximation), which gives rise to the buoyancy forces. Thethermophysical properties of the uid are evaluated at the am-bient temperature, T o , which is assumed to be 300 K in allcases.

The ambient is assumed to be a black body at a temperatureof 300 K.

With the already mentioned assumptions, the governingequations in the conservative form and primitive variables

heat transfer engineering vol. 32 no. 6 2011

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442 L. LANGELLOTTO AND O. MANCA

are:

ux

+vy

= 0 (1)

uu

x

+ vu

y

= 1

f

p

x

+ 2u

x2 +

2u

y2 g (T To)

(2)

uvx

+ vvy

= 1f

py

+ 2vx2

+2vy2

(3)

uTx

+ vTy

= af 2Tx2

+ 2Ty2

(4)

where the pressure is referred to the ambient pressure, p o .A two-dimensional conduction model is employed. The heat

conduction equations in the steady-state regime with constantthermophysical properties is:

2Tsx2

+2Tsy2

= 0 (5)

The characteristic variables, for the investigated congura-tion in this paper, are the dimensionless maximum wall temper-ature, the channel Rayleigh number, the Reynolds number, andthe channel Nusselt number, dened as follows:

max =(Tmax To)k

qcb; max =

(Tmax To)k (qc + qr)b

(6)

Ra b = Gr Pr =gqcb

5

2kLwPr; Ra b =

g(qc + qr)b5

2kLwPr (7)

Re =uav,bmin bmin

(8)

and

Nu b =qcb

Tw,av To k Nu b =

(qc + qr)b

Tw,av To k (9)

where b is b min or b av or b max , and

qc =1

Lw Lw

0qc,x (xw )dx w (10)

qr =1

Lw Lw

0 qr,x(xw )dx w (11)

Tav =1

Lw Lw

0T(x w )dx w (12)

It is worth noticing that an evaluation of q c separate from q ris very difcult in practice. The value of (q c + qr) is not equalto the dissipated heat ux q due to the conductive heat lossestoward the ambient through the lower and upper edges of thewalls.

Table 2 Boundary conditions for the uid domain

Wall u v T

AH and FGuy

= 0vy

= 0 T = To

HGu

x

= 0v

x

= 0 T = To

AB, EF, IL, OP u = 0 v = 0Tx

= 0

BC, DE, LM, NO u = 0 v = 0 k f Tx

= k sTx

DN u = 0 v = 0 k f Tn

= k sTn

+ q + qr

CM u = 0 v = 0 k f Tn

= k sTn

q + qr

RQux

= 0vx

= 0Tx

= 0

IR, QPuy

= 0vy

= 0Ty

= 0

The letters in the column are in reference to Figure 1b.

Numerical Procedure

Since the two plates are placed in an innite medium, from anumerical point of view the problem has been solved with refer-ence to a computational domain of nite extension, as depictedin Figure 1b, by following the approach given in [1113]. Thiscomputational domain allows taking into account the diffusiveeffects peculiar to the elliptic model. The imposed boundaryconditions are reported in Table 2 for the uid domain and inTable 3 for the solid domain. The pressure defect is equal tozero at the inlet and outlet boundaries. The net radiative heatux from the surface is computed as a sum of the reectedfraction of the incident and emitted radiative heat uxes:

qr (xw) = (1 w ) qin (xw ) + wT4w (xw ) (13)

qin (xw) = sn> 0 Iin s nd (14)Table 3 Boundary conditions for the solid domain

Wall

DN k f Tn

= k sTn

+ q + qr

Tw = Tf

CM k f Tn

= k sTn

q + qr

Tw = Tf

BL, OETn

= 0

BC, DE, LM, NO k f Tx

= k sTx

+ qr

Tw = Tf

The letters in the column are in reference to Figure 1b.


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The computational uid dynamics code FLUENT [30] wasemployed to solve the present problem. The segregated methodwas chosen to solve the governing equations, which were lin-earized implicitly with respect to the equations dependent vari-able. The second-order upwind scheme was chosen for theenergy and momentum equations. The Semi Implicit Method

for Pressure-Linked Equations (SIMPLE)scheme waschosen tocouple pressure and velocity. Similar considerations were madefor choice of the discrete transfer radiation model (DTRM),which assumes all surfaces to be diffuse and grey. The conver-gence criteria of 10 6 for the residuals of the velocity compo-nents and of 10 8 for the residuals of the energy were assumed.

A grid dependence test is accomplished to realize the moreconvenient grid size and radiative subdivisions by monitoringthe induced dimensionless mass ow rate and the average Nus-selt number, referred to the minimum channel spacing for aconvergent channel system with L w /bmin = 40.6, = 10 atRa bmin = 30 and 220 and with L w /bmin = 10.2, = 10 , andRa bmin = 3.1 104 and 2.25 105 as reported in [13]. A moredetailed description on the numerical model is reported in [13].

A comparison between numerical and experimental [31] re-sults is reported in Figure 2. In Figure 2a wall temperatureproles, obtained for L w /bmin = 58.0, = 10 , and Ra bmin =37, are shown. The comparison between the numerical and ex-perimental data showed a good agreement with a maximumpercentage discrepancy of about 8%. In Figure 2b the compar-ison, in terms of average Nusselt number, is accomplished. Avery good accord between the numerical and experimental datais observed.

Since the numerical resultsand experimental data are in goodagreement, the assumptions of steady-state, two-dimensional,laminar, incompressible, with negligible viscous dissipation areconrmed, as well as the Boussinesq approximation.

SCALE ANALYSIS

For the convergent channel, the total volume (channel totalvolume) is:

V tot = Wb max Lw cos (15)

and it is greater than the channel volume as shown in Figure1a. The geometrical optimization of the convergent channel, interms of maximum or average wall temperature, should take

into account the channel total volume. The heat transfer rate inthe channel is:

Q = 2hWL w Tw (16)

In the optimization procedure, the channel total volume isconsidered:

Qb max = 2hWL w Twbmax (17)

For small convergent angles Eq. (17) becomes:

Qb max 2hV tot Tw (18)

xw [mm]0 100 200 300 400

T w

- T o [ K ]

0

10

20

30

40

50

60

Experimental Numerical

(a)Lw/bmin = 58.0 = 10q = 220 W/m 2

= 0.90

Ra'* bav

100 101 102 103 104 105 106 107 10

N u *

b a v

0.1

1

10

100

Present numerical dataExperimental data [14,31]

= 0.90 = 0 = 2

= 5 = 10

(b)

Figure2 Comparison between numerical and experimental data:(a) walltem-perature proles, (b) average Nusselt number.

Combining Eq. (18) with the average channel Nusselt num-ber, Eq. (9), we get:

V tot TwQb max b2kNu b

(19

for an assigned heat transfer rate. The optimal channel congu-ration that minimizes the product V tot Tw is the congurationthat maximizes the heat transfer as a function of the channeltotal volume in terms of b max .

In laminar, fully developed and two-dimensional natu-ral convection between parallel plates, heated at uniformheat ux, the maximum wall temperature is obtained at thechannel outlet section and the minimum Nusselt number is[20]:

Nu x= L =k

hb+ 48Rab

1

(20

The average wall temperature is approximately equal to thewall temperature at middle channel length and the average


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Nusselt number is estimated by [20]:

Nu x= L/ 2 =k

hb+ 12Ra b

1

(21)

The rst term on the right-hand side of the Eqs. (20) and

(21) is negligible with respect to the square root as given in[20]:

Nu x= L Ra b48 and Nu x= L/ 2 Rab12 (22)In vertical channels, at uniform heat ux, with small con-

vergence angle, the minimum and average Nusselt numbers,referred to the minimum channel spacing, can be evaluated asin [20]. It is obtained as:

Nu bminx w = Lw Rabmin48 bavbmin32

and

Nu bminx w = Lw / 2 Ra bmin12 bavbmin32

(23)

For fully developed ow, the comparison between theparallel-plate channel, Eqs. (20) and (21), and the convergentchannel, Eqs. (23), shows that, for the same b min , the convergentchannel has a higher Nusselt number value; i.e., the conver-gent channel, with the minimum channel spacing, equal to theparallel-plate channel spacing, presents lower maximum andaverage wall temperature values.

For the Nusselt number referred to the average channel spac-ing it is:

Nu bav x w = Lw Rabav

48bminbav

and

Nu bav x w = Lw / 2 Rabav12 bminbav (24)As shown in Eqs. (24), the Nusselt number for the con-

vergent channel, referred to b av , is lower than the one for theparallel-plate channel, i.e., the wall temperature in the conver-gent channel is greater than the one in the parallel-plate channel.Further, the convergence angle limit is:

bav Lw sin( ) = 0 = arcsinbavL

w

(25)

For the Nusselt number referred to the maximum channelspacing it is:

Nu bmaxx w = Lw Ra bav48 b2min b3avb5max andNu bmaxx w = Lw / 2 Rabav12 b2min b3avb5max (26)

Also in this case, the Nusselt number for the convergentchannel, referred to b max , is lower than the one for the parallel-plate channel. The convergence angle limit is equal to:

bmax 2Lw sin( ) = 0 = arcsinbmax2Lw

(27)

In laminar, developing, and two-dimensional natural convec-tion along an inclined single plate, heated at uniform heat ux,the average Nusselt number is [32]:

Nu Lw = 0.56 Ra Lw cos 0.2

(28)

For developing ow in convergent channels as limit condi-tion, Eq. (28) is employed in terms of b/L w with b equal to b minor b av or b max :

Nu b 0.56(Ra Lw cos )0.2b

Lw(29)

As suggested in [20], a composite relation is obtained bysumming the two expressions, the equation for fully developedlimit, indicatedwith Nu 0 [Eqs. (23)(25)], and single-plate limit,indicated with Nu [Eq. (29)]. The binomial correlation is:

Nu p = Nu p0 + Nu p (30)

as a rst approximation, the correlation exponent, p, is set equalto 2.

The term V tot Tw is evaluated by means of Eq. (30) usingbmin , bav , and bmax , respectively:

(V tot Tw )b minQb max bmin

2k

Rab min

12bavbmin

3 / 2 2

+ 0.56 Ra bmin cos 0.2 2

0.5

(31)

(V tot Tw )bavQb max bav

2k Ra bav12 bminbav 2

+ 0.56 Ra bav cos 0.2 2

0.5

(32)

(V tot Tw )b maxQb max bmax

2k Rabav12 b2min b3avb5max 2

+ 0.56 Ra bmax cos 0.2 2

0.5

(33)


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Figure 3 Wall temperature for channel total volume as a function of channelspacing and convergence angle with reference channel spacing equal to: (a)bmin ; (b) b av . (c) Optimal geometrical congurations, in terms of b min , bav andbmax values, as a function of convergence angle.

The values of V tot Tw , for qc equal to 30 W m 2, as afunction of the convergence angle and the minimum and av-erage channel spacing, are reported in Figures 3a and 3b. Thecontours of V tot Tw value in the (b, ) plane are also given. Itis noted, in Figure 3a, that (V tot Tw)bmin , Eq. (31), is alwaysdened except for b min equal to zero. The function shows thatthe absolute minimum value is obtained for = 0 . For the

considered convective heat ux the optimal channel spacing isbmin = 9.9 10 3 m. This value corresponds to the minimumvalue of V tot Tw , i.e., the minimum Tw with the minimumcompatible total volume V tot . For (V tot Tw )bav , Eq. (32), andFigure 3b, for assigned b av value, the convergence angle limit,limit , exists and a vertical asymptotic plane is detected for limit , according to Eq. (25). The optimal conguration in termsof (V tot Tw )bav , obtained by Eq. (32), is realized for = 0

and for the considered convective heat ux b av = 9.9 10 3 m.The same results are obtained for (V tot Tw )bmax , Eq. (33), butthe results are not reported here.

In Figure 3c, the optimal conguration, in terms of channelspacing, is given as a function of the channel convergence an-gle, for q c equal to 30 W m 2. The gure shows that the curvestend to an asymptotic value equal to 9.9 10 3 m, whichrepresents the optimal conguration for the parallel-plate chan-nel. Figure 3c shows that for (V tot Tw )bmin , increasing the

convergence angle, the minimum channel spacing decreases.For xed convergence angle, decreasing b min , the total volumedecreases and the Nusselt number increases, Eq. (31), and thenthe wall temperature decreases. In V tot Tw referred to b av anbmax , for xed convergence angle, the Nusselt number and thetotal volume increaseas the referencechannel spacing increases.

ANALYSIS AND PROCEDURES FOR CORRELATIONS

The results are obtained by the numerical procedure reportedin [13]. In this work, the analysis is focused on the radiativeeffects on natural convection in air, in a convergent channel,uniformly heated at the two principal walls. The wall thickness,t,is 3.2 mm, withtheratio t/b min varying in the range 0.0800.64.Its thermal conductivity is 0.198 W/m-K, with a solid-to-uidconductivity ratio k s /k f = 8.18. The input data are ranging from10 to 80 for aspect ratios, L w /bmin ; Rayleigh numbers, Ra bminranging from 2.5 10 2 to 2.3 105; convergent angles, ranging from 0 to 10 ; and wall emissivities, , ranging from0.1 to 0.9.

The percentage value of the conductive heat ux, q k , referredto the dissipated heat ux, q , for different Ra bmin values andfor the geometry here considered, are given in Table 4.

Mass Flow Rate

Mass ow rate, involved in the heat transfer, is an impor-tant parameter in design and control of electronic equipmentand solar energy in building. The following correlations formass ow rate, in a convergent uniformly heated vertical chan-nel, as a function of thermal and geometrical parameters areproposed. The mass ow rate for unit of width is dened asfollows:

m = uav,bmin bmin (34

where u av,bmin is the mean velocity at the minimum channelsection. From Eqs. (8) and (34):

m = Re (35

The Reynolds number, as a function of Ra bmax , is reportedin Figure 4. The gure shows that, when the Rayleigh num-ber increases, for xed aspect ratio and convergence angle, theReynolds number also increases. Decreasing the aspect ratio(increasing the spacing), the Reynolds number increases signif-icantly, whereas there is slight change in the mass ow rate inthe emissivity range 0.100.90. The maximum percent variation


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Table 4 Conductive heat losses

Ra bmax 30 Lw /bmin 58 0.048 bmin /bmax 1.0 q k /q 3%; q k /q 5%1.0 Ra bmax 30 58 Lw /bmin 80 0.048 bmin /bmax 1.0 q k /q 10%; q k /q 15%

Lw /bmin = 80 0.048 bmin /bmax 0.7Ra bmax 1.0 L w /bmin 80 0.048 bmin /bmax 1.0 q k /q 15%

between emissivity value equal to 0.90 and 0.10 is about 10%.Figure 4 shows that the Reynolds number is highly dependenton channel aspect ratio and convergence angle.

In order to reduce the Reynolds number scattering, the vari-

able Re bminbmax as a function of Ra bmaxbavLw

is considered:

Rebminbmax

= f Ra bmaxbavLw

(36)

A dependence on Lwbmin is observed and the best correlationsfor assigned Lwbmin are carried out employing = 1 and = 5/ 2. The plot of Eq. (36) is reported in Figure 5a. In thisgure, a greater dispersion is observed for high values of Ra bmax ( bavLw )

. The dispersion is due to the increase in relevanceof the aspect ratio in this zone. To obtain a monomial correlationfor the mass ow rate, in terms of geometrical and thermalvariables, the following relation is proposed in the form [15]:

Rebmaxb

min

= Lw

bmin

Ra bmaxbavL

w

2.5 Lw

bmin

(37)

Ra' bmax

100 101 102 103 104 10 5 106 107 10 8 10 9

R e

10

100

1000

= 0 = 2 = 5 = 10

= 0.10 = 0.50 = 0.90

Lw/bmin = 58.0

Lw/bmin = 40.6

Lw/bmin = 20.3

Lw/bmin = 12.6Lw/bmin = 10.2

Figure 4 Reynolds number versus Rayleigh number for various convergenceangles and wall emissivity values.

A new correlation, in terms of Ra bmax , is evaluated by re-gression analysis:

Re = 4.62 103bminLw

3

+ 752bminLw

2

+ 18 .6bminLw

Ra' bmax (bav/Lw)-2.5

104 105 106 10 7 108 109 1010 1011

R e (

b m a x

/ b m

i n )

101

102

103

104(a)

Re (numerical)0 100 200 300 400 500 600 700

R e

( E q .

3 8 )

0

100

200

300

400

500

600

700

(b)

Figure 5 (a) Re bminbmax as a function of Ra bmax (bavLw

) 2.5 . (b) Comparisonbetween numerical Reynolds numbers and Reynolds numbers by correlationgiven by Eq. (38) with a percentage difference in a 5% range.


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0.0248 ] Rab maxbavLw

2.50 0.249+ 0.233e 24 .6 bminLw

bminbmax

; r2 = 0.996 (38)

and a simpler expression is also proposed:

Re = 52 .9bminLw

0.436

Ra bmaxbavLw

2.50 0.249+ 0.233e 24 .6 bminLw

bminbmax

r2 = 0.990 (39)

They present a better regression coefcients than the corre-lation presented in [15] in terms of Ra bmax :

Re = 499bminLw

3

+ 72 .9bminLw

2

+ 54 .6bminLw

0.429 Ra bmax bavLw 2.50 0.242+ 0.230e

24.5 bminLw

bminbmax

; r2 = 0.989 (40)

Moreover, a new correlation in simplied form, in terms of Ra bmax , is proposed:

Re = 57 .4bminLw

0.454

RabmaxbavLw

2.50 0.242+ 0.230e 24 .5 bminLw

bminbmax

r2 = 0.988 (41)

In Figure 5b, a comparison between the Reynolds numbervalue from numerical data and the Reynolds number valuescalculated by the correlation in Eq. (38) are reported togetherwith an error level of 5%.

Radiative Heat Flux

Correlations to evaluate the ratio between radiative heat ux,qr , and total heat ux, q c + qr , for a convergent channel andsurface emissivity are proposed. The heat ux ratio, qr qc + qr ,

Ra'* bav

100 101 102 103 10 4 105 106 107 10

q r / ( q c + q r

)

0.01

0.1

1

= 0 = 2 = 5 = 10

= 0.10 = 0.50 = 0.90

Lw/bmin = 58.0

Lw/bmin = 40.6

Lw/bmin = 20.3

Lw/bmin = 12.6Lw/bmin = 10.2

Figure 6 Radiative heat ux ratio ( qr qc + qr ), as a function of Ra bav , for thredifferent wall emissivity values and different convergence angles.

is very useful when q c + qr is known, thereby allowing forestimation of radiative heat losses.

The ratios between radiative and total heat uxes, as a func-tion of Ra bav , are reported, for different angles and for threevalues, in Figure 6. It is observed that for xed L w /bmin valuesthe ratio qr qc + qr decreases when Ra bav increases, and the ratio

variationdecreases with increasingRa bav and decreasing aspectratio. The percentage reduction of heat ux ratio decreases withincreasing convergence angle. Furthermore, the ratio values in-creasewith decreasing aspect ratio value. Increase in the conver-gence angle produces a signicant increase in heat ux ratio. For = 10 , the radiative heat ux ranges between 20% and 40%of the total heat ux. Figure 6 shows that, for xed Rayleighnumber, the heat ux ratio decreases with decrease in the wallemissivity. The higher the Ra bav values, the higher is the vari-ation of the heat ux ratio. In fact, at Ra bav = 20, for = 0.10the heat ux ratio is 0.020, whereas for = 0.90, the ratio is0.032; at Ra bav = 2.4 107, the heat ux ratio is 0.11 and 0.27for = 0.10 and = 0.90, respectively. The percentage varia-tions referred to = 0.90 are about 38% and 59% for Ra baequal to 20 and 2.4 107, respectively.

For assigned wall length, Figure 6 allows to observe the de-pendence on the convergence angle and channel spacing whenthe wall heat ux is xed. Furthermore, the gure shows a data

scattering. In order to reduce the heat ux ratio scattering, thevariable qr+ qcqref is employed as suggested in [15]. A new com-posite correlation for = 0.90, obtained by means of regressionanalysis, is proposed:

qrqc + qr

= 0.153Ra 1.12x10 3

bav

4+ 0.0481Ra 0.186bav

4 1 / 4

qr + qc

qref

0.26

qref = 1 W/ m2; r2 = 0.991 (42


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Table 5 Coefcients and exponent of the Eq. (43)

m n p r2

0.10 0 .0478 0 .135 0.26 0 .9810.50 0 .0525 0 .173 0.26 0 .990

In Eq. (42), the value of reference heat ux is used to obtaina dimensionless equation. A good accord is observed betweenEq. (42) and the numerical data set.

For = 0.10 and 0.50, at high Rayleigh numbers, a monomialcorrelation is proposed:

qrqc + qr

= m Ra nbavqr + qc

qref

p

(43)

where the reference heat ux is equal to the values given inEq. (42). The coefcient m and exponents n and p, as well as

r2 values, are reported in Table 5 for the different values. Inthis case, simpler equations are proposed with respect to the onegiven in [15]. Moreover, a new global correlation is evaluatedusing all available data:

qrqc + qr

=oRa

obav

n+ Ra

bav

n 1 / n

1Ra1bav + f

bminbmax

m

1 + m

qr + qc

qref

p

(44)

With two different equations for f (bminbmax ):

f bminbmax

= a1lnbminbmax

+ a2 (45)

f bminbmax

= a1bminbmax

3

+ a2bminbmax

2

+ a3bminbmax

+ a4

(46)

The coefcients and exponents of Eq. (44) are reported inTable 6.

The comparison between the qr qc + qr

ratios obtained numer-ically and from Eq. (44), taking into account Eq. (46), arereported in Figure 7 together with an error level of 5%. Itis observed that the best agreement among the data and pro-posed correlation is obtained for high Ra bav value (Ra bav >104).

The main advantage of Eq. (44), with respect to the corre-lations in Eqs. (42) and (43) and ones in [15], is that it is asingle equation for all emissivity values. Furthermore, Eq. (44)provides a better estimation of qr qc + qr ratio with respect to theprevious correlations.

Table 6 Coefcients and exponent of the Eq. (44) withEqs. (45) and (46)

Eq. (44)

Eq. (45) Eq. (46)

o 0.1840 0 .1721

o 0.0746 0 .0237 0.0408 0 .0474 0.03569 0 .29681 0.0357 0 .03461 0.2257 0 .1743n 4 4m 1.075 1 .075 7.916 7 .916p 0.260 0.260qref 1.00 1 .00a1 0.0027 0 .5062a2 0.1501 0.7256a3 0.2325a4 0.1250r2 0.990 0 .991

Dimensionless Maximum Wall Temperature

Composite correlations between the dimensionless maxi-mum wall temperatures and Rayleigh numbers, referred to themaximum channel spacing b max , are evaluated for = 0.1, 0.5,and 0.9. The equations are obtained by means of the asymptoticrelations for the single tilted plates (large Rayleigh number,Ra > 104) and for the fully developed limit (small Rayleighnumber, Ra < 102), following the procedure suggested in [20].

q r /(q r +q c) (numerical)00.101.010.0

q r / ( q

r + q c

) E q .

( 4 4 ) w

i t h E q .

( 4 6 )

0.01

0.10

1.00

= 0.90 = 0.50 = 0.10

Figure 7 Comparison between the qr (qr + qc ) ratio obtained numericallyand that from the correlation Eq. (44) and Eq. (46), with a percentage differencein a 5% range.


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The composite correlation between bmax and Ra bmax , for = 0.9, is:

bmax = 6.13Ra 0.414bmax

4+ 1.88Ra 0.197bmax

4 1 / 4(47)

with r 2 = 0.997, in the ranges: 4.4 Ra bmax 2.9 103,10 Lw /bmin 80, 0 10 (0.048 bmin /bmax 1.0).Considering the variable

Ra mod = Ra bmaxbminbmax

0.10 (48)

a new correlation is proposed for = 0.1, 0.5, and 0.9:

bmax = 9.39Ra 0.628mod

2+ 2.48Ra 0.242mod

2 1 / 2(49)

with r 2 = 0.978, in the ranges 1.4 10 3 Ra 4.2 108,10 Lw /bmin 80, 0 10 (0.048 bmin /bmax 1.0).

A good agreement is observed in the comparison between thenumerical data and Eq. (49). The comparison shows that greaterdifferences are found for the highest Ra values. Equation (49)is dened in a larger Ra bmax and aspect ratio range, extendingits validity in the fully developed region with respect to theequation given in [15].

Nusselt Number Correlation

The average convective Nusselt numbers, dened in the Eq.(9), as a function of channel Rayleigh number referred to b min ,is given in the following for two asymptotic conditions: fullydeveloped ow Ra bmin < 50 and single plate limit Ra bmin >800. The correlation for fully developed ow depends on theconvergence angle. The following correlations are obtained bymeans of regression analysis:

Nu 0,bmin = 0.182Ra0.589bmin (50)

for = 0 and Ra bmin < 50 with r 2 = .992; and

Nu 0,bmin = 0.475Ra0.408bmin (51)

for 1 < < 10 and Ra bmin < 50 with r 2 = .991.For the single plate limit, the monomial correlation is:

Nu ,bmin = 0.660Ra0.200bmin (52)

for 0 < < 10 and Ra bmin > 800 with r 2 = .997.The composite correlations are obtained following the pro-

cedure suggested in [20]: N u p = N u p0 + N u

p

For p = 4, the two composite correlations are evaluated:

Nu bmin = 0.182Ra0.589bmin

4+ 0.660Ra 0.200bmin

4 0.25

(53)

for = 0 and 0.2 < Ra bmin < 1.9 105 with r 2 = .995; and

Nu bmin = 0.475Ra0.408bmin

4+ 0.660Ra 0.200bmin

4 0.25

(54

for 1 < < 10 and 0.02 < Ra bmin < 1.9 105 with r 2 = .999Numerical data and the two composite correlations are re-

ported in Figure 8a. A good accord between the numerical dataand the correlations is observed. A slightly better agreementis observed for higher Nusselt numbers corresponding to thehigher Rayleigh numbers.

In order to obtain a single composite correlation that takesinto account all convergence angles, a different monomial cor-relation for fully developed ow, Ra bmin < 50, is evaluated andemployed. The following monomial correlation is proposed interms of b av /bmin to take into account the convergence angles:

Nu 0,bmin = m0Ran0bmin

m0 = 0.468 0.29bav

bmin

3

(55

n0 = 0.40 + 0.142bavbmin

3

A composite correlation is obtained from the Eqs. (52) and(55), with p = 4:

Nu bmin = 0.468 0.29bavbmin

3

Ra0.40 0.142 bavbmin

3

bmin

4

+ 0.660Ra 0.200bmin 4 0.25

(56

for 0 < < 10 and 0.02 < Ra bmin < 1.9 105 with r 2 = .998In this case, a more noticeable difference is observed for

Nu bmin < 1.0, 0.0 < < 1.0 , and 0.02 < Ra bmin < 50. It iinteresting to observe that in Eq. (55), the coefcient m 0 anthe exponent n 0 are functions of b av /bmin , which conrm thescale analysis results. These functions are reported in Figure 8band a horizontal asymptotic value is observed in both functions.

Moreover, the critical values correspond to the zone where thefunctions change from vertical to horizontal asymptote.A similar analysis is given for average total Nusselt number,

dened in Eq. (9), which takes into account both radiation andconvective heat uxes. Average total Nusselt number, as func-tion of total channel Rayleigh number, referred to b min , is givenin the following for the two asymptotic conditions: fully de-veloped ow, Ra bmin < 100, and single plate limit, Ra bmin >1.0 103. The correlation for fully developed ow depends onthe convergence angle. In fact, by means of numerical data, thefollowing correlations are obtained employing the regression


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Ra' bmin10 -3 10 -2 10 -1 100 101 102 103 104 105 106

N u b

m i n

0.01

0.1

1

10

= 0.0

= 1.0 = 1.5 = 2.0 = 5.0 = 10Eq. (53)Eq. (54)

(a)

bav/bmin

0 2 4 6 8 10 12

m 0

a n d n 0 o

f E q .

( 5 5 )

0.1

0.2

0.3

0.4

0.5

0.6

n0 Eq. (55)

m0 Eq. (55)

n0m0

(b)

Ra'* bmin

10 -2 10 -1 100 101 102 103 104 105 106

N u *

b m i n

0.01

0.1

1

10

100

= 0.0 = 1.0 = 1.5 = 2.0 = 5.0 = 10Eq. (60)Eq. (61)

(c)

Figure 8 (a) Nusselt number versus Rayleigh numbers and correlations givenby Eqs. (53) and (54). (b) Coefcient m 0 and exponent n 0 in the Eq. (55). (c)Total Nusselt number vs total Rayleigh numbers and correlations given by Eqs.(60) and (61).

analysis:

Nu 0,bmin = 0.201Ra bmin0.545

(57)

for = 0 and Ra bmin < 100 with r 2 = .997; and

Nu 0,bmin = 0.492Ra bmin0.392

(58)

for 1 < < 10 and Ra bmin < 100 with r 2 = .992.For the single plate limit, the monomial correlation is:

Nu ,bmin = 0.725Ra bmin 0.210 (59)

for 0 < < 10 and Ra bmin > 1.0 103 with r 2 = .975.Two composite correlations are obtained, with p = 4, from

Eqs. (57) and (59) resulting in:

Nu bmin = 0.201Ra bmin0.545 4

+ 0.725Ra bmin0.210 4

0.25

(60)for = 0 and 0.08 < Ra bmin < 2.2 105 with r 2 = .988.

From Eqs. (58) and (63):

Nu bmin = 0.492Ra bmin0.392 4

+ 0.725Ra bmin0.210 4

0.25

(61)is carried out for 1 < < 10 and 0.02 < Ra bmin < 2.2 105

with r 2 = .992.Numerical data and the two composite correlations are re-

ported in Figure 8c. A good agreement between numerical dataand correlations is observed.

In the same way employed to obtain Eqs. (55), the followingmonomial correlation is carried out, in terms of b av /bmin , to takeinto account all convergence angles:

Nu 0,bmin = m0Rabmin

n0

m0 = 0.520 0.322bav

bmin

2

(62)

n0 = 0.288 + 0.340bav

bmin

2

For p = 4, a composite correlation is obtained from Eqs. (59)and (64):

Nu bmin = 0.520 0.322bavbmin

2

Rabmin 0.288 0.340bavbmin

2 4

+ 0.725Ra bmin 0.210 4 0.25

(63)

for 0 < < 10 and 0.02 < Ra bmin < 2.2 105 with r 2 =.991.

In Figure 9a, all proposed correlations are reported and theyare compared for = 0.10, with the correlation given in [12],


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Ra' bmin10 -1 100 10 1 102 103 104 105 106

N u b

m i n

0.1

1

10

= 0.10 [12] = 0.50 = 0.90 = 0.90 [13] = 0.90 = 0 = 0.90 1 < < 10

(a)

Ra'* bmin

10 -1 100 10 1 102 103 104 105 106

N u *

b m i n

0.1

1

10

= 0.10 [12] = 0.50 = 0.90 = 0.90 [13] = 0.90 = 0 = 0.90 1 < < 10

(b)

Figure 9 Nusselt number versus Rayleigh numbers for several wall emissivityvalues.

and for = 0.90, with the correlation proposed in [13]. In Figure9a, the correlations for convective average Nusselt number, asa function of convective channel Rayleigh number, referred tothe minimum channel spacing, are compared. The gure showsthat the greatest differences among the correlations are detectedfor = 0.1.

In Figure9b, thecorrelation fortotal average Nusselt number,as a function of total channel Rayleigh number, referred to theminimum channel spacing, for several wall emissivities, arereported. In this case as well, the greatest differences among thecorrelations are detected for = 0.1.

Channel Optimization

In the same manner of scale analysis, V tot Tw is obtainedby means of Eq. (56):

(V tot Tw)bmin =Qb max bmin

2k 0.468 0.29

bavbmin

3

Figure 10 (a) Wall temperature for channel total volume as function of min-imum wall spacing, b min , and channel convergence angle, . (b) Optimal geo-metrical congurationsin termsof channel spacing as a function of convergenceangle. (c) Optimal channel spacing as function of convective wall heat ux.

Ra0.40 0.142 bavbmin

3

bmin

4

+ 0.660Ra 0.200bmin 4

0.25

(64

In Figure 10a, the term V tot Tw , Eq. (64), evaluatedfor q c equal to 30 W m 2, as a function of b min and , ireported. In this gure the contours in the (b, ) plane are alsogiven. It is observed that the minimum value is obtained for = 0 and b min = 0.0104 m, which represents the optimalconguration for convergent channel. These values are almostequal to the ones estimated by the scale analysis. Moreover, thetwo diagrams in Figures 10a and 3a seem similar. This con-rms that the scale analysis provides a good estimation for thisconguration.


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In Figure 10b, the optimal channel spacing, evaluated bymeans of Eqs. (64), (31), (32), and (33), is given as a function of the convergence angle. The gure shows that the curves presentasymptotic values, corresponding to the optimal conguration,the parallel-plate channel. For all curves, the asymptotic valuesare almost equal. For the dimensionless quantities referred to

bmin and for small convergence angle, the gure shows that theoptimal congurations obtained by the numerical solution, Eq.(64), and scale analysis, Eq. (31), have a similar trend. In Figure10b, it is observed, for Eq. (64), that the minimum value of theoptimal minimum channel spacing value is about 8.1 10 3 mattained for = 0.44 .

By increasing the heat ux the optimal minimum channelspacing decreases as shown in Figure 10c. This result is inagreement with the results reported in [15] and [33].

CONCLUSIONS

Natural convection in air, in convergent channels, symmetri-cally heated at uniform heat ux, in a steady-state regime, wasstudied. A scale analysis allowed estimation of optimal geo-metrical congurations from Nusselt number correlations, forsingle plate and fully developed ow, in terms of the channelRayleigh numbers, the ratio b min /bmax , and the convergence an-gle. A new optimization procedure was obtained in terms of theminimum value of the product of average wall temperature andtotal volume, V tot Tw , as a function of convergence angle andminimum, average, and maximum wall spacing. It wasobservedthat in all cases, the best conguration is = 0 . These resultsare different from the ones given in previous papers [10, 12, 13,15].

Monomial and composite correlations were estimated by thenumerical results obtained by the numerical model proposed in[13].The correlation equationswereaccomplished formassowrate, radiative heat ux, and dimensionless maximum wall tem-perature in the emissivity range from 0.10 to 0.90, convergenceangle from 0 to 10 , ratio between minimum and maximumchannel spacing, b min /bmax , in the range of 0.048 to 1.0, 10 L/b min 58, and 5.0 Ra bmin 2.3 105. Average Nusseltnumber correlation was proposed for the emissivity value of 0.90, convergence angle ranging from 0 to 10 , bmin /bmax inthe range of 0.048 to 1.0, 10 L/b min 80, and 2.5 10 2

Ra bmin 2.3 105

. It was observed that all correlations werein very good accord with the numerical data.The analysis of Nusselt number, in fully developed ow,

in the channel, shows that the asymptotic value for = 0 isdifferent from the ones with > 0 . Moreover, for 1 ,all values were along a single asymptotic curve, Eqs. (56) and(63) for Nu and Nu , respectively. This asymptotic curve wasconsidered the border line between the fully developed ow andthe developing ow for low Ra values.

The new proposed optimization procedure was applied em-ploying the evaluated Nusselt number correlation. The optimal

conguration was obtained for = 0 . This conrms the resultobtained by means of the scale analysis. The optimal value of minimum channel spacing decreases with increase in the wallheat ux as shown in [15] and [33].

NOMENCLATURE

a thermal diffusivi ty (m 2 /s)b channel spacing (m)g acceleration of gravity (m/s 2)Gr Grashof numberGr Gr b / L w , Eq. (7)h Convective heat transfer coefcient (W/m 2-K)I radiation intensity (W/m 2)k thermal conductivity (W/m-K)L channel length (m)Lx , Ly reservoir dimensions (m)m mass ow rate for width unit (kg/s-m)

n normal to the wallNu average Nusselt number, Eq. (9)p reduced pressure referred to ambient pressure (N/m 2)Pr Prandtl numberq heat ux (W/m 2)Q heat transfer rate (W)r regression coefcientRa average Rayleigh numberRa Ra b/L w , Eq. (7)Re average Reynolds number, Eq. (8)s ray direction vectort wall thicknessT temperature (K)u velocity component along x axis (m/s)v velocity component along y axis (m/s)V volume (m 3)W plate weight (m)x, y coordinates (m)

Greek Symbols

volumetric coefcient of expansion (1/K) half angle from the vertical (degrees) emissivity dynamic viscosity (kg/m-s)

kinematic viscosity (m 2 /s) dimensionless temperature, Eq. (6) density (kg/m 3) StefanBoltzmann constant (W/m 2-K 4)

hemispherical solid angle (sr)

Subscripts

av averageb referred to the channel spacing


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[23] Olsson, C. O., Prediction of Nusselt Number and FlowRate of Buoyancy Driven Flow Between Vertical ParallelPlates, Journal of Heat Transfer , vol. 126, pp. 97104,2004.

[24] da Silva, A. K., and Bejan, A., Constructal Multi-ScaleStructure for Maximal Heat Transfer Density in Natural

Convection, International Journal of Heat and Fluid Flow ,vol. 26, pp. 3444, 2005.

[25] da Silva, A. K., Lorente, S., and Bejan, A., ConstructalMulti-ScaleStructuresfor Maximal Heat Transfer Density, Energy , vol. 31, pp. 620635, 2006.

[26] Andreozzi, A., Campo, A., and Manca, O., CompoundedNatural Convection Enhancement in a Vertical Parallel-Plate Channel, International Journal of Thermal Sciences ,vol. 47, no. 6, pp. 742748, 2008.

[27] Incropera, F. P., Convection Heat Transfer in ElectronicEquipment Cooling, ASME Journal of Heat Transfer , vol.110, pp. 10971110, 1988.

[28] Chena, Z. D., Bandopadhayay, P., Halldorsson, J.,Byrjalsen, C., Heiselberg, P., and Li, Y., An Experimen-tal Investigation of a Solar Chimney Model With UniformWall Heat Flux, Building and Environment , vol. 38, pp.893906, 2003.

[29] Harris, D. J., and Helwig, N., Solar Chimney and Build-ing Ventilation, Applied Energy , vol. 84, pp. 135146,2007.

[30] Fluent 6.2 User Manual , Fluent, Inc., Lebanon, NH, 2006.[31] Bianco, N., Langellotto, L., Manca, O., Nardini, S., and

Naso, V., Converging on New Cooling Technology, Fluent News, p. 28, summer, 2005.

[32] Fujii, T., and Imura, H., Natural Convection Heat Trans-fer from a Plate With Arbitrary Inclination, International Journal of Heat and Mass Transfer , vol. 15, p. 752,1972.

[33] Manca, O., and Nardini, S., Thermal Design of UniformlyHeated Inclined Channels in Natural Convection With andWithout Radiative Effects, Heat Transfer Engineering , vol.22, no. 2, pp. 116, 2001.

Luigi Langellotto is a researcher at Centro SviluppoMateriali S.p.A. (CSM), Rome Italy. He receivedhis Ph.D. in mechanical engineering from SecondaUniversit a degli Studi di Napoli (SUN). He is CSMproject leader in an RFCS project and several in-dustrial projects with TenarisDalmine S.p.A. or ABSS.p.A. His main scientic activities are on naturalconvection in an open-ended cavity; thermal controlof electronic equipment; solarsystems; analyticalandnumerical solutions in material processing such as

seamless pipe rolling, strip rolling, and ingot casting; and numerical analysisof austenite deformation and decomposition. He has co-authored more than 10refereed journal and conference publications.

Oronzio Manca is a professor of mechanical engi-

neering at Facolt a di Ingegneria della Seconda Uni-versit a degli Studi di Napoli (SUN), Naples, Italy. Hehas been coordinator of the Industrial EngineeringArea at SUN since January 2005. His main scien-tic activities are on active solar systems; passivesolar systems; refrigerant uids; natural and mixedconvection in an open-ended cavity with and withoutporous media; conduction in solids irradiatedby mov-ing heat sources; combined radiative and conductive

elds in multilayer thin lms; analytical and numerical solutions in materialprocessing; thermal control of electronic equipment and solar systems; and heattransfer augmentation by nanouids. He is a member of the ATA CampaniaCommittee. He is a member of the American Society of Mechanical Engi-neering, and Unione Italiana di Termouidodinamica UIT. He has co-authoredmore than 270 refereed journal and conference publications. He is currently a

member of the editorial advisory boards for The Open Thermodynamics Jour-nal , The Open Fuels & Energy Science Journal , and Advances in Mechanical Engineering .


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