Forced and mixed convection heat transfer from an array of cylinders to a liquid submerged jet

Rev Gin Therm (1998) 37, 431-439 @ Elsevier, Paris

Forced and mixed convection heat transfer from an array of cylinders to a liquid submerged jet

Carlo Bartoli, Sergio Faggiani*, David Rossi Department of Energetic& University of Piss, via Diotisalvi 2, 56726 Pisa, /m/y

(Received 7 November 1997; accepted 28 January 1998)

Abstract-The results of an experimental investigation concerning the heat transfer from three cylindrical heaters to a water jet are reported in the form of correlating equations, which express the Nusselt number versus the Reynolds, Prandtl and Grashof numbers and some dimensionless ratios characterising the configuration. As the experienced range of the thermal flux is wide (2.104 5 Q 5 6.105 W.m-‘), the influence of the free convection, which was shown to be negligible in previous studies, is carefully investigated in the present one. This influence appears still negligible up to the maximum value of 4 for the heater impinged by the jet; on the contrary it is remarkable for the heaters lying in its wake. Another aspect which is carefully studied is the influence both of the ratios characterising the configuration and of the impingement direction: accordingly the values of these ratios and the kind of impingement which yield the maximum Nusselt number are clearly singled out. The investigation is completed by some visualization experiments which allow us to qualitatively clarify some aspects of the interaction between the dynamic and thermal fields. @ Elsevier, Paris

impingement heat transfer / forced and free convection / submerged jet

R&sum4 - Transfert thermique en convection for&e et mixte entre un groupe de cylindres et un jet submergi. Ce travail s’interesse au transfert thermique entre un groupe de trois cylindres rechauffes et un jet d’eau submerge. Les resultats permettent d’exprimer le nombre de Nusselt en fonction des nombres de Reynolds, de Prandtl et de Crashof, ainsi que d’autres parametres adimensionnalises. Dans cet article, a la difference des travaux precedents, on a dgalement etudie la contribution de la convection libre en augmentant le flux thermique jusque a 6.105 W.m- 2. En effet, la convection libre exerce une influence sur les deuxieme et troisieme cylindres : sur le premier, c’est-&dire celui qui est le plus proche du jet, cette influence est Cgalement negligeable pour les valeurs de flux thermique les plus &levees. On a aussi etudie I’influence du rapport qui caracterise la configuration geometrique, ainsi que celle de la direction d’impact, et on a mis en evidence les conditions necessaires pour atteindre les nombres de Nusselt les plus Clev&. Cette recherche a ete completee par plusieurs visualisations permettant de mettre en evidence les interactions entre les champs dynamique et thermique. 0 Elsevier, Paris

transfert thermique a I’impact / convection for&e et libre /jet submerge

Nomenclature

A slot area .............................

dc cylinder diameter. ....................

ds slot width.. ......................... G volumetric flow rate ..................

9 gravitational acceleration ............. h heat transfer coefficient ............... k thermal conductivity .................

* Correspondence and reprints. a volumetric thermal expansion coefficient K-l

11 spacing between the first and the second

m2 12 cylinder . . . . . . . . . . . . . , . . . . . m spacing between the second and the third

m cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m Q heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . w.m-f

m3.s-1 T temperature . . . . . . . . . . . . . . . . . . . . . . . . . K . -2

W.m-?;-’ z spacing between the slot exit and the

first cylinder. . . . . . . . . . . . . . . . . . m W.m-l.K-l

Greek symbols

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C. Bartoli et al.

p thermal diffusivity . . u kinematic viscosity . . . . . ,...........

Dimensionless numbers

Gr= PgIT,-Tbld:

Y2

h& Nu=- k

Pr = i

Ra= PrGr DsG

Re = - AU

Subscripts

E cylinder bulk

S slot F forced convection N natural convection

1. INTRODUCTION

Cooling by means of gas and liquid jets has been investigated by a number of researchers over the

m2.s-1 m2 .s- 1

last 30 years; these studies were motivated by the effectiveness of the method, by its versatility and by the considerable saving in refrigerant mass flow which can be attained with this cooling technique in comparison with the uniform flow and even with the duct flow. Also practical reasons spurred these studies: indeed cooling by means of a jet is largely employed in the metallurgic processes and in electronics, where the inexorable trend of an increasing number of circuits per chip and power per circuit imposes the use of more and more efficient cooling techniques. Most of the studies were concerned with impingement heat transfer on flat plates, while less attention was dedicated to the problem of heat transfer due to slot jet impinging on a circular cylinder. The latter configuration was studied by Schuh and Persson [l] , Kumada et al. 21, Miharaki and Sparrow [3], Sparrow and Alhomond I [4 , Kang and Greif [5], Bartoli et al. [6] and Faggiani and Bartoli [7]. Heat transfer from rows of horizontal cylinders in the crossflow in a water channel has been studied by Bennon and Incropera [8]; some aspects of heat transfer and fluid flow at an array of circular cylinders have been investigated by Kataoka et al. [9]. However, the configurations and the purposes of these studies were quite different from those of the above authors.

The configuration formed by a rectangular jet and a streamwise row of cylinders appears very interesting for the engineering applications, because further saving in refrigerant mass flow is allowed by the array for fixed total heat transfer or else larger total heat flux for fixed mass flow. Accordingly, in a previous paper

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[lo], the convective heat transfer from an array of three circular cylinders to a submerged slot jet of water has been studied. Very simple correlations based on known or easily measurable quantities have been searched: they expressed the Nusselt number versus the ratio between the diameter of the cylinders and the slot width, the Prandtl and the Reynolds numbers, with all properties evaluated at the bulk temperature which was nearly coincident with the temperature of the water at the slot exit. No influence of the buoyancy forces has been noted owing to the small range of the thermal fluxes (2.104 < Q < 5.104 W.m-‘) and the consequent temperature differences (AZ’ < 4 K). This limitation is avoided in the present investigation; furthermore the range of the distances between the cylinders, 11 and 12 are wider, consequently the influence of the free convection, some physical aspects and the configurations which allow the maximum heat transfer are carefully settled. Some flow visualization experiments complete the present study.

2. APPARATUS AND METHOD OF CAlCUlATiON

The experimental apparatus consists of a test section, a water thermostatic tank, a pump, a regulating valve, a turbine flowmeter and a rectangular duct which forms a slot jet. The test section is a stainless steel vessel, containing distilled water and three heaters in the middle. These are formed by hollow stainless cylinders supplied with dc current. The inner temperature of each heater is measured by means of two sliding thermocouples; the external one is calculated on the basis of the dissipated electric power, the thickness of the hollow cylinder and the inner temperature, under the hypothesis of axial symmetry. Because of this hypothesis the calculated temperatures represent circumferentially averaged values: accordingly also the Nusselt numbers or the heat transfer coefficients obtained from these values are averaged over the surface of the cylinder to which they refer. The array of heaters and the slot lie on the same plane (figure 1): consequently the submerged slot jet of water coming out from the duct impinges on the first cylinder. The diameter of the heaters varies from 2 to 6 mm, the slot width from 1 to 3 mm, the distance between the heaters from 3 to 30 mm and the slot exit-first cylinder spacing from 5 to 20 mm. The stainless steel vessel is 270 x 470 mm’. The temperature of the water coming out from the rectangular duct is maintained constant within f 0.3 K, therefore the value of Pr at the channel exit is affected by an error of f 1 %. For the temperature differences the precision is ItO. K, an error of about 4 % affects consequently T, - Tb, since 6 K is its reference value. The flowrate and the Reynolds number are affected by an error of about 3 % due to the flow oscillations induced by the pump and to the precision of the flowmeter.


Figure 1. Experimental configuration.

According to standard methods of evaluation and to the above uncertainties, the maximum error for h or NU is estimated to be less than 8 %.

The visualization experiments have been performed by means of a JVC KY 17 camera and a S-VHS tape recorder, using the hydrogen bubbles technique and/or injecting 65 or 100 j.rrn diameter polymeric microspheres in the upstream fluid.

A more detailed description of the experimental apparatus and of the test operations can be found in a previous paper [6].

3. EXPERIMENTAL RESULTS

The test runs have been performed within the following limits of variability of the parameters:

1.5 .103 <Re<3.104

2.7<Pr<7

2.104<4<6.105 W.rn-’

1 I h/d, I6

1<12/d,16

According to the previous results [lo], the maximum Nu for all the cylinders occurs in the range 3 5 T 5 7,

s

mostly at f x 5.5; a similar conclusion has recently

been obtainsed with air jets [ll]; therefore all the experiments described in the present paper, which are mainly aimed at the settlement of the condition allowing high heat transfer coefficients, have been performed at ;= 5.5. Concerning the influence of the ratio d,/d,,

&is problem has been carefully studied in the above mentioned previous paper, so the formerly proposed dependence is assumed here.

The trend of Nu in dependence of Re is quite complex, because of the occurrence of some irregularities in the form of sudden rises or sharp pits (figures 2 and 3); however, within limits of error usually accepted for practical purposes, the Nusselt number can be considered as regularly increasing with increasing Re for all the cylinders. In the case of the single heater, the above singularities were explained according to some flow visualizations, but in the present case of three heaters, an analogous explanation does not appear easy (see below).

The heat transfer coefficient of the first cylinder is independent of 11/d, and 12/d, for Re < 1.104; when the Reynolds number exceeds this value and Pr is less than 5, the influence of II/d, and 12/ds on Nu cannot be neglected. For the second cylinder, the heat transfer coefficient decreases with increasing II/d, and depends on the heat flux in the whole range of Re, when q exceeds 1.105. The influence of 4 on the Nus- selt number of the third cylinder is slightly larger than the influence regarding the second one; furthermore the heat transfer coefficient of the last cylinder exhibits an oscillatory trend in dependence of 12/d,: in particular it firstly decreases with increasing 12/d,, then suddenly

120 .-

f 80: OL

0. 3 60. 2 :

40 '-

.‘I ' mm

I 1000 5 :cKl 5coo 7!m 9cm 11000 1m 15000 17OKI

Re

Figure 2. Nusselt number in dependence of Pr and Re for upflow, Pr = 4.85 and configuration d, = d, = 3.0 mm, .z/ds = 5.3, II/d, = 12/d, = 3.3. Properties evaluated at the slot exit temperature. The arrows indicate the location of the singularities.

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C. Bartoli et al.

120

20

I 9000 13000 17000

Re

Figure 3. Nusselt number in dependence of Pr and Re for upflow, Pr = 3.52 and configuration d, = d, = 3.0 mm, z/ds = 5.3, II/d, = 12/d, = 3.3. Properties evaluated at the slot exit temperature. The arrows indicate the location of the singularities.

rises and, after this peak, it decreases monotonically. The value of 12/d, at which the above peak occurs is dependent on II/d,. The influence of the thermal flux is evidenced in table I, where the percent increases of the ratio Nu/PT’.~ caused by the change of 4 from 6.104 to 6.105 W.mP2 are reported for some configurations. It is worth nothing that these increases are always less than the expected error for the first heater.

Three types of impingement have been experienced: from below, from above and horizontal. In the cases of up and down flow the Nusselt number is practically the same, indeed the impingement from above yields increases of Nu which are less than 6 %, i.e. less than the expected error. In the last case (horizontal flow

Percent increase of Nu/PT-‘.~ caused by the change 01 1st 7 7 0

II/d, = 1.2 lz/ds = 1.2 2nd 23 17 13

3rd 22 17 14

II/d, = 1.2 12/d, = 3.2

II/d, = 3.2 12fds = 3.2

and horizontal array) for fixed values of IlId, and 12/d, the Nusselt numbers of the first, the second and the third cylinders are equal to, slightly larger and slightly less than the Nusselt numbers of the impingement from below; these differences increase with increasing 4 but remain less than 13 %. The just described behaviour is consistent with previous experiments [6], which have evidenced that the impingement direction exerts a negligible influence on the dynamic and thermal fields surrounding the impinged cylinder. Concerning the second one, the increase of the Nusselt number in the case of horizontal arrangement is probably due to the additional motion caused by the buoyancy forces: indeed, this motion can freely develop in this case because the third cylinder is no longer an obstacle. The decrease of the heat transfer coefficient of the third cylinder can be explained taking into account that, in a horizontal arrangement, the contribution to the buoyancy motion given by the second cylinder is absent. On the contrary this contribution is present in the case of vertical impingement.

Because of the very complex dependence of Nusselt numbers on II/d, and 12/d,, small variations of these parameters can cause remarkable changes in the temperature of the cylinders for fixed total heat flux, or remarkable changes in heat flux for fixed temperature distribution. This observation suggests that we should search for the best configuration from the viewpoint of heat transfer. Suitable experiments have been performed varying the parameters ll/cl, and 12/d, between 1 and 6: the configuration which allows the maximum Nusselt numbers resulted

II/d, = 1.2, l,/ds = 3.2

both for vertical and horizontal impingement. This configuration is markedly asymmetric. Such a characteristic can be unfit from the practical point of view, then symmetric configurations have been sought: Nusselt numbers slightly less than the former ones (the third Nu is about 10 % less, the other are unchanged) can be obtained with:

lx/6 = lz/ds = 1.2

Other configurations give less Nusselt numbers both on average and individually, nevertheless the configuration II/d, = 12/ds = 3.2 has been carefully studied, because it yields quite good heat transfer coefficients and larger spacing between the cylinders, which can be necessary in practice for maintenance and cleaning.

4. CORRELATiONS

The searched correlations have to take into account the influence of the parameters z/dsr d,/d,, II/d, and II/d,. Concerning the first two parameters, the

434


neighbourhood of the maximum heat transfer and the previously determined dependence, i.e. ; = 5.5 and

s dc

( >

0.15

d, respectively, are chosen for all the correlations

which follow. The ratios Ii/d, and lz/ds are taken into account giving a specific equation (i.e. specific exponents and constants) for each of the above discussed configurations.

According to table I, when the thermal flux, 4, increases beyond 6.104 W.rne2, the contribution of the natural convection must be taken into account up to Re = l&2.10* for the second and the third heater, but for the first cylinder it can be neglected in the whole range of the Reynolds number.

As the governing equations are non-linear, the dynamic field resulting from combined forced flow and buoyant motion cannot be described as the sum of two independent contributions. Since the thermal field is strongly dependent on the dynamic one, also the total Nusselt number cannot be described as the simple sum of two independent terms corresponding to forced and to free convection. The additive hypothesis is a convenient simplification which is acceptable when the free convection contribution to the combined Nusselt number is rather small [12, 131. This condition is well fulfilled in most of the situations of the present experimental study, therefore correlating equations having the form:

NU = (Nu)N +(Nu)F (1)

have been searched. For sake of completeness, however, also equations of the form

NuP = (NT& + (Nu);

have been studied [14].

(2)

These correlations contain a Reynolds number based on the slot exit velocity and width, G/A and d,. This Re is representative of the dynamic field around the first cylinder, which is placed close to the apex of the potential core of the jet; on the contrary it is not representative for the second and third cylinders which are placed in the wake of the first one, namely in a zone of reduced velocity. For these cylinders the Reynolds numbers ought to be based on the velocity which characterizes the wake, but the classical analytic expressions allowing this determination [15] refer to the uniform flow and are generally valid under the hypothesis of similarity, in other terms they are valid beyond distances from the trailing edge of the first cylinder larger than 80 d,; conversely the present investigation refers to jet flow and to distances which are less than 7 d,. Furthermore recent studies [16, 171 show that the complexity of the flow of the wake is far from being exhaustively described by the above mentioned expressions. Also the attribution to the second and third cylinder of an equivalent Re, defined as the Re which yields a value of Nu for the first cylinder which

equals the Nusselt numbers measured at the second and third cylinder, appears useless because the ratios Gr/Re2 based on this equivalent Re are less than 1, while they ought to be markedly larger than 2 [12]. Owing to these difficulties, it has been assumed that the Reynolds numbers of the second and third cylinder are proportional to the Reynolds number of the first one. All the following equations are based on this simplifying hypothesis; they clearly are empirical correlations.

According to these considerations, the proposed correlating equation is:

Nu - = Pro.*

0.15 Re” (3)

The values of Cl, C;? and n obtained by means of a least squares fit, are reported in table II for all the studied configurations. The percent maximum error corresponding to each case is reported too.

In equation (3) the exponent of the Grashof number has been taken as 0.25 because the ranges of Gr and Ra of the present experiments are: 1.102 < Gr < 2.105, 7.102 < Ra < 5.105. On the whole, however, the flow is fully turbulent, so also the exponent 0.33, which is specific for this dynamic condition, has been employed. The values of the constants and exponent of equation

NU - = Cl Gr”‘33 f C2 Pro.4

0.15 Re” (3’)

are reported in table III, where only the second and the third cylinder are considered, because for the first heater Cl is always zero like in table II. Obviously, for this heater, the previous equation and values of CZ and n hold.

The more complex correlations having the form of equation (2) does not yield better results, because the maximum errors are equal or slightly less than the previous ones. For the best configuration Ii/d, = 1.2, 12/ds = 3.2 and vertical impingement of such a correla- tion is:

Nuo.43

= (0.29 Ra 0.25)0.43 + o.016 (2)".15 pr~.4O Re~.85]o'43

for the second cylinder;

Nuo.57 = (0.29 Ra 0.25)0.57+ o,05g ($,""" prO.4O Re0.71

for the third cylinder. Concerning the maximum error, it is coincident with the previous one for the third cylinder, slightly less for the second one.

In all the above correlating equations the properties must be evaluated at the film temperature, 0.5(T, + Ti), T, being the circumferentially averaged temperature of the cylinder to which the Nusselt number refers.

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C. Bartoli et al.

r TABLE II Values of Cl, CZ and n for equation (3); properties evaluated at film temperature

Vertical

Horizontal

Configuration Cylinder Ci C2 n Max error, %

1st 0 0.23 0.63 11

11/d, = 1.2 12/d, = 1.2 2nd 1.9 0.025 0.84 9

3rd 1.3 0.102 0.65 13

1st 0 0.195 0.65 11

II/d, = 1.2 12/d, 3.2 2nd 1.9 0.021 0.85 8 =

If a larger error is accepted and the thermal flux is less than 6.104 W.m-2, the following very simple correlations can be used for the symmetric configurations:

NU 0.15

- =Alr dc ( >

d, Re” (4) pro.4 0.09

n = As

It holds for Re > 5. 103. The factor T is 1 for the first cylinder, while it is expressed by the equation

r=Azexp (Az+~$)~

for the second and the third cylinder. For equation (4) the properties are evaluated at the temperature of the water at the slot exit. The values of the constants and exponents of equation (4) are reported in table IV. These equations correlate 95 % of the experimental data

436


lee Re - 1.6 -1.16 4.15 0.82

- I I I ”

within 15 %. It can be useful for the determination of an initial value to be employed in iterative calculations for higher thermal fluxes.

5. VISUALIZATIONS

The motion of the hydrogen bubbles and/or reflecting spheres have been shot both in the direction perpen- dicular to the plane of the array and in the direction parallel to this plane, in order to evidence the flow patterns around and between cylinders. According to recent investigations [16, 171 for a detailed description of the flow patterns, frontal and side shootings are necessary.

Like the previous visualizations [7], the present ones have evidenced shear layer vortices, having the aspect and characteristics carefully described by Wu et al. [16], at the sides of the array beyond the first cylinder. In the two narrow regions contained between the first and the second, the second and the third cylinders and also in the wake of the last cylinder (figure 4, A, B, c) only intermittent vortices were present; no periodicity and, in particular, no alternate rotating and counter- rotating vortices have been observed. Concluding, the von Karman vortex street is not present in the case of jet impingement, at least in conditions of fully developed turbulence. The side shootings have also evidenced that the wake is wide and unstable when the Reynolds number is less than that one corresponding to the sudden rise or decrease of the Nusselt number (figures 2 and 3, first arrow). This instability is connected with the behaviour of the wake beyond the second cylinder: as schematically shown in figure 4, the streamlines of the flow around the last heater can bend on the right or on the left, so that the fluid laps one or the other side of this cylinder. This behaviour is intermittent and surely unperiodic. Some difficulties found in the approximation of the trend of Nu vs Re, and the larger error corresponding to the third heater (table II), are probably due to the just described behaviour. A more detailed conclusion, already proposed [S, 71 for a single

Figure 4. Sketch of observed bubble streaklines.

heater, do not seem possible in the present case of three heaters.

The frontal watching has evidenced that in the narrow regions between the cylinders (figure 4, A and B) the fluid velocity is markedly reduced: specifically the velocity between the first and the second heater is about l/4 of the velocity of the potential core of the jet, but in the next region it becomes l/10 of the above speed. At the array’s side the velocity slightly decreases in a streamwise direction and is comprised between 0.3 and 0.4 times the potential core velocity. From these approximate results, it can be deduced that around the second and the third cylinder the average velocity is 213 and l/3 of that around the first heater; about the same ratios characterise the Nusselt numbers of the second and the third cylinders in comparison with the Nu of the first one. It is worth nothing that these qualitative considerations refer to small and intermediate values of Reynolds numbers, namely 3.103 5 Re < 1.104; for larger values of Re the visualizations become progressively less clear. The complex flow around the cylinders is shown in figure 5; the flow is visualized by means of hydrogen bubbles and apears to be fully turbulent.

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C. Bartoli et al.

Figure 5. Still photographs of the flow around the heaters. Water velocity at the slot exit: 0.9 rn.s-l; shutter time: 0.1 s.

6. CONCLUSIONS

Three kind of correlating equations are proposed for the problem of heat transfer from an array of circular cylinders to a slot jet of water. The most simple correlations express the Nusselt number in dependence of Pr and Re, in which properties and characteristic velocity refer to the fluid at the slot exit, namely to a situation which is, in general, completely known. The costs of this simplicity are that the uncertainty is rather large (95 % of the experimental data within 15 %) and the thermal flux can not exceed 6.104 W.m-‘; in other words the simplest correlations hold when the contribution of the free convection is very small.

The second kind of correlating equations hold up to Q = fj.105 W.m-‘; they are based on the average film temperature and take into account the contribution of the forced and free convection under the simplifying additive hypothesis. All the experimental data are cor- related by these equations with an average uncertainty of 10 % and a maximum uncertainty of 13 %. It is

438

worth nothing that this maximum belongs to the Nus- selt number of the third cylinder, i.e. the heater with the minimum Nu.

The last kind of correlating equations are based on the film temperature but not based on the additive hypothesis. Notwithstanding the absence of this simplifying hypothesis they do not yield markedly larger accuracy, therefore their use seems of scarce interest for the present problems.

The contribution of the free convection is taken into account by means both of Gri/” and GT~/~, namely through the exponent recommended for the range of Gr of these experiments and the exponent recommended for fully turbulent conditions. The double choice is suggested by the presence of a forced mean flow in which the turbulence is fully developed, however for practical purposes both the exponents yield correlations of the same accuracy.

Most of the results reported and discussed in the present paper have been obtained with circular cylinders of 3 mm outer diameter, however some test runs have been performed with slightly smaller or larger heaters, namely with cylinders of 2 and 6 mm outer diameter. The results of these test runs are completely in agreement with those obtained with d, = 3 mm, consequently the proposed equations correlate, within the above specified limits of accuracy, these experimental results too. On the contrary the validity of the equations cannot be extended to other geometries, for instance to square cylinders, because the dynamic field markedly changes.

The present study is also intended to obtain useful information for electronics cooling, since some electronic components are cylindrical.

REFERENCES

[l] Schuh H., Persson B., Heat transfer on circular cylinders exposed to free-jet flow, Int. J. Heat Mass Trans. 7 (1964) 1257-1271.

[2] Kumada M., Mabuchi I., Kawashima Y., Mass transfer on a circular cylinder in the potential core region of a two-dimensional jet, Heat Transfer-Jpn. Res. 2 (1973) 53-66.

[3] Miyazaki H., Sparrow E.M., Potential flow solution for crossflow impingement of a slot jet on a circular cylinder, ASME J. Fluid Eng. 98 (1976) 249-255.

[4] Sparrow E.M., Alhomoud A., impingement heat transfer at a circular cylinder to an offset or non-offset slot

jet, Int. J. Heat Mass Trans. 27 (1984) 2297-2306.

[5] Kang SK., Creif R., Flow and heat transfer to a circular cylinder with a hot impinging air jet, Int. J. Heat Mass Trans. 35 (1992) 2 173-2 183.

[6] Bat-toli C., Di Marco P., Faggiani S., Impingement heat transfer at circular cylinder due to a slot jet of water, Exp. Therm. Fluid Sci. 7 (1993) 279-286.


[7] Faggiani S., Bartoli C., Impingement heat transfer from a circular cylinder to a liquid jet, in: Proceedings of the 2nd International Thermal Energy Congress, Agadir (Morocco), 1995, 2 l-26.

[8] Bennon W.B., lncropera F.P., Mixed convection heat transfer from horizontal cylinders in the crossflow of a finite water layer, J. Heat Tran.-T. ASME 103 (1981) 540- 545.

[9] Kataoka K., Hamano S., Minamiura K., Li C.Y., Lo- cal control of impinging jet heat transfer by an array of circular cylinders, in: Proceedings of the 9th Interna- tional Heat Transfer Conference, Jerusalem (Israel), 1990, pp. 203-208.

[lo] Bartoli C., Faggiani S., Impingement heat transfer at an array of circular cylinders, ExHFT4 Brussels 4 (1997) 2101-2106.

[12] Fand R.M., Keswani K.K., Combined natural and forced convection heat transfer from horizontal cylinders to water, Int. J. Heat Mass Trans. 16 (1973) 1175-l 189.

1131 Juma A.K.A., Richardson J.F., Heat transfer from horizontal cylinders to liquid, Chem. Eng. Sci. 37 (1 1) (1982) 1681-l 688.

[14] Churchill S.W., A comprehensive correlating equation for laminar, assisting, forced, and free convection, AlChEJ. 23 (1977) 10-16.

[15] Hinze J.O., Turbulence. MC Craw-Hill Inc., 1975.

[16] Wu J., Sheridan J., Hourigan K., Soria J., Shear layer vortices and longitudinal vortices in the near wake of a circular cylinder, Exp. Therm. Fluid Sci. 12 (1996) 169-l 74.

[l l] Marsili S., Convezione forzata di getti di aria su [17] Williamson C.H.K., Three-dimensional vortex dy una schiera di tre cilindri, Thesis, University of Rome, Tor namics in bluff body wakes, Exp. Therm. Fluid Sci. 12 Vergata, 1996. (1996) 150-l 68.

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Forced and mixed convection heat transfer from an array of cylinders to a liquid submerged jet