Radiation effects on natural convection in a vertical channel with an auxiliary plate

International Journal of Thermal Sciences 97 (2015) 41e55

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Radiation effects on natural convection in a vertical channel with an auxiliary plate Assunta Andreozzi a, *, Oronzio Manca b a b


degli Studi di Napoli Federico II, Piazzale Tecchio 80, 80125 Napoli, Italy Dipartimento di Ingegneria Industriale, Universita
degli Studi di Napoli, via Roma, 29, 81031 Aversa CE, Italy Dipartimento di Ingegneria Industriale e dell'Informazione, Seconda Universita

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 October 2014 Received in revised form 15 April 2015 Accepted 28 May 2015 Available online xxx

In this experimental study the effects of unheated auxiliary plate positions along the centerline of a vertical channel on combined natural convective in air and radiative heat transfer are investigated. The vertical channel is symmetrically heated and a uniform heat flux along the principal walls of the channel is assumed. Comparisons between experimental data carried out from different positions of the auxiliary plate and two surfaces emissivity values are given in terms of heated wall temperatures and radiative heat transfer. Average Nusselt numbers by means of experimental data are evaluated. Results are given for different thermal and geometrical dimensionless parameters. Results show that the lowest maximum channel wall temperatures are attained for the auxiliary plate placed at the channel exit for high channel Rayleigh numbers and heat flux values. Correlations for average Nusselt numbers, global average Nusselt numbers and dimensionless maximum wall temperatures in terms of channel Rayleigh numbers and auxiliary plate-to-channel height ratio, h/L, are evaluated in the ranges: 20 < Ra'< 1.5
105, 22 < Ra0 * < 1.7
105 and 0  h/L  0.5 for inlet and outlet positions and walls emissivity equal to 0.9. The proposed correlations are in good agreement with correlation for simple channel. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Natural convection Vertical channel Auxiliary plate Radiation effects Experimental investigation

1. Introduction Natural convection in air still remains an advantageous option in the thermal control of electronic systems and several other technical applications. In fact, it is very favorable in low heat generating electronic devices due to its low installation and maintenance cost and it is highly reliable and noise free. Moreover, natural convection between heated vertical parallel plates is receiving a new research interest in technological applications, such as thermal control in electronic equipments, nuclear reactors, solar collectors and chemical vapor deposition reactors. It has been extensively studied both experimentally and numerically as indicated in Refs. [1e7]. However, it should be underlined that the more significant limit in laminar natural convection is the heat transfer rate capacity and then the heat transfer improvement in natural convection explains the need to find new configurations or to analyze standard configurations to carry out optimal geometrical parameters for a higher heat transfer rate and behaviors for a suitable thermal design [4e6,8e24].

* Corresponding author. E-mail address: [email protected] (A. Andreozzi). http://dx.doi.org/10.1016/j.ijthermalsci.2015.05.013 1290-0729/© 2015 Elsevier Masson SAS. All rights reserved.

The added heat transfer mechanism of radiation can effectively improve the thermal performance of parallel plates vertical channels. As recently shown in Ref. [4], the thermal radiation can be considerable when added to the natural convection, especially with asymmetric heating conditions. The evaluation of the radiation effects on these configurations could be very challenging. In fact, the radiation in a vertical channel with an auxiliary plate placed on the centerline was investigated in Ref. [25]. The auxiliary plate was placed at three different locations and it was found that the best position in terms of heat transfer was the inlet. This geometric configuration was introduced by Aihara in 1963 [26]. In one of the first numerical studies on natural convection in vertical channels with an auxiliary plate along the centerline of the channel [27], the channel walls were isothermal and the auxiliary plate was either isothermal or isoflux. Results were obtained with the channel Rayleigh number in the 101e102 range and with the dimensionless length of auxiliary plate in the 0.005e0.2 range. It was found that a relatively long central plate shows more marked influence on the flow. An analysis of the configuration with the channel walls and the auxiliary plate at uniform temperature was carried out numerically in Ref. [28] and experimentally in Ref. [29]. The numerical study [28] showed that the heat transfer rate from the auxiliary plate can be significantly enhanced by the presence of

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A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

Nomenclature A b E g h J k L Nu Nu* Pr q R r Ra' Ra'* T x, y, z

surface, m2 channel spacing, m total emissive power, W m2 acceleration due to the gravity, m s2 height of the auxiliary plate, m radiosity, W m2 thermal conductivity, W m1 K1 height of the heated plates, m average Nusselt number, Eq. (6) global average Nusselt number, Eq. (6) Prandtl number heat flux, W m2 dummy variable regression coefficient channel Rayleigh number, Eq. (2) global channel Rayleigh number, Eq. (3) temperature, K coordinates, m

the channel walls. The highest average Nusselt number for complete system (channel and plate as a whole) was obtained by positioning the plate at the bottom of the channel. The experimental study [29] was obtained by means of a Mach-Zehender interferometer and the auxiliary plate was located at the bottom and top of the channel and its dimensionless height was 1/3. Experimental and numerical results were in good agreement even if the experimentally measured average Nusselt numbers were about 10% lower than the numerical results. Average Nusselt number correlation equations were proposed. The symmetrically heated configuration with uniform heat flux walls was numerically analyzed in Refs. [30e32]. Results were given for air and the channel Rayleigh number and the dimensionless auxiliary plate height ranged in 103e106 and 0.0e1.0, respectively, whereas the channel aspect ratio was equal to 10. It was shown in Ref. [30] that, at lower channel Rayleigh number values (103e104), the location of this auxiliary plate worsens the heat transfer performance of the channel, whereas at higher channel Rayleigh number values (105e106) an enhancement was detected and some optimal configurations were obtained either for insulated or heated auxiliary plate. The thermal and fluid dynamic behavior was analyzed in detail in Ref. [31] and correlation equations between channel Rayleigh number, dimensionless geometric parameters and average channel Nusselt number were proposed. The effect of the channel aspect ratio on the thermal behavior of the channel with auxiliary plate was investigated in Ref. [32]. It was found that the auxiliary plate worsened the local heat transfer and the larger both the height of the auxiliary plate and the channel aspect ratio the smaller the mass flow rate. At higher Rayleigh number the heat transfer improved with respect to the simple channel. Correlation equations for average Nusselt number and dimensionless maximum wall temperature were proposed. A concept for maximizing the heat transfer rate density in a fixed volume cooled by natural convection by means of the insertion of smaller plates at the entrance of channel was presented and evaluated in Ref. [13]. A numerical investigation of the interaction of surfaces radiation with developing laminar free convective heat transfer in a divided vertical channel was accomplished in Ref. [33]. The channel walls were at assigned temperature and the auxiliary plate was unheated. The auxiliary plate was placed in a central

Greek symbols volumetric coefficient of expansion, K1 ε emissivity coefficient n kinematic viscosity, m2 s1 * q dimensionless temperature

b

Subscripts ∞i inlet section ∞o outlet section avg average aux auxiliary plate c convective ch channel walls k conductive max maximum value o ambient r radiative ra reradiating ref reference value w wall U Ohmic dissipation

position inside the channel. The extension of the study given in Ref. [33] to a vented vertical channel was carried out in Ref. [34]. A combined extended channel-chimney system with an equidistant auxiliary plate inserted at the inlet was numerically investigated in Ref. [17]. Experimental and numerical investigations to study conjugate heat transfer by natural convection and surface radiation from a short planar heat generating element mounted freely between two thermally insulated vertical plates were accomplished in Ref. [35]. The results of combined experimental and numerical study of laminar conjugate natural convection heat transfer from four planar heat generating elements arranged in tandem and placed symmetrically in a vertical channel are reported in Ref. [36]. The importance of radiative heat transfer in natural convection in asymmetric heated vertical channel has been well reviewed and showed in Ref. [4]. A numerical study on the interaction of radiation with developing laminar natural convection between vertical parallel plates was carried out in Ref. [37]. The auxiliary plate was heated and the channel walls were unheated. The position of auxiliary plate was in the central part of the channel. The authors found that the maximum wall temperature for black surfaces was half the value attained in the case with no radiation. The radiative effects on a vertical channel with one plate adiabatic and the other at uniform temperature were investigated in Ref. [38]. It was found that if radiation heat transfer is accounted for, with the emissivity equal to one, the global heat transfer enhanced up to 75% for intermediate and large Grashof numbers. On the contrary, at low Grashof numbers the enhancement effect vanished. The effects of uniform heat flux at the wall, channel height, channel width, inlet air temperature and surface emissivity in a vertical channel with arbitrarily prescribed heat fluxes on each wall were studied numerically in Ref. [39]. The same configuration, with an absorbing and emitting medium, was studied analytically and experimentally in Ref. [40]. Results were in a fair agreement, showing a major effect of the wall emissivity on the combined heat transfer. The effect of the emissivity of the walls turned out to be higher in the asymmetric heating case was found in Ref. [41]. Larger heat transfer augmentation was obtained in asymmetric heating with high wall emissivity. Natural convection radiation cooling in a vented channel was numerically analyzed in Ref. [42]. An investigation on wall temperature distributions of vertical parallel plates channels

A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

heated uniformly and symmetrically and cooled by conduction, radiation, and natural convection in air was numerically and experimentally carried out in Ref. [43]. Laminar natural convection and surface radiation between vertical parallel plates, a central, highly emissive hot plate and two unheated polished plates, was investigated experimentally in Ref. [44] for various plate spacings. A correlation for the dimensionless average convective wall heat transfer was proposed. A numerical study on the heat transfer by natural convection and surface radiation in a two-dimensional vented enclosure in contact with a cold external ambient and a hot internal ambient was performed in Ref. [45]. Natural convection in a convergent channel with finite-thickness principal walls at uniform heat flux was numerically investigated in Ref. [46]. Heat conduction in the walls and the effects of walls emissivity were considered. The interaction of surface radiation with conduction and convection from a vertical channel with three flush-mounted discrete heat sources on wall with the other wall unheated was numerically studied in Ref. [47]. Numerical study by conjugate heat transfer by convection, conduction and radiation, in a solar chimney system for heating and ventilation of dwellings was presented in Ref. [48]. The effects of the radiative heat transfer on natural convection in air, in a convergent channel, uniformly heated at the principal walls, were experimentally investigated in Ref. [49]. The heat transfer and fluid flow in a vertical vented divided channel heated asymmetrically was carried out in Refs. [50,51]. In the numerical model also the surface radiation effect was taken into account. It seems that there is a lack of experimental information on the effects of radiation in a heated channel with unheated auxiliary plate though this configuration is employed in several application [6,10,13,25,26,28e37,52]. It should be underlined that, to the best authors' knowledge, there are not experimental investigations on heated channel with combined natural convection and radiation heat transfer. In fact, previous investigations, such as [33], are numerical and, in particular in Ref. [33] the channel walls are at assigned temperature. It is very interesting to analyze the effects of the plate position along the centerline of a heated channel on combined natural convection and radiation heat transfer, since these configurations are of main interest in the telecommunication systems and electronic cabinets with vertical cards. This paper investigates experimentally the effects of the auxiliary plate position along the centerline of a vertical channel with combined natural convective and radiative heat transfer in air. A comparison between the different positions of the plate in the channel is given in terms of wall temperature of the channel. A detailed analysis is carried out to find the optimal geometric configuration of the channel which improves the thermal performance in terms of maximum wall temperatures and overall heat transfer coefficients. Results are given in terms of heated wall temperature profiles and radiative heat transfer for different thermal and geometrical parameters. In addition, dimensionless maximum wall temperature, average Nusselt number and global Nusselt number are evaluated and useful correlation equations for these quantities are proposed in the ranges 20 < Ra'<1.5
105, 22 < Ra'*<1.7
105 with 0  h/ L  0.5 in terms of the channel Rayleigh number and dimensionless geometrical parameters. 2. Experimental apparatus The test sections of the investigated configurations are shown in Fig. 1(a)e(c). The three configurations of the vertical channel are: (a) channel with leading edge of the auxiliary plate at the channel inlet section, inlet position (inlet); (b) channel with the center point of the auxiliary plate at mid-height of the channel, central position

43

(central); (c) channel with the trailing edge of the auxiliary plate at the channel exit section, outlet position (outlet). The x and y axes are the axial and transversal coordinates, respectively, whereas the axis z is orthogonal to the picture plane and it is not shown in figure. The test section was made of two heated vertical plates, whose height is L ¼ 400 mm. The channel was 450 mm wide, the gap b between the channel walls equal to 20.0 or 40.0 mm. With these dimensions, the channel aspect ratio L/b was in the 10e20 range. The channel gap was measured by means of a caliper to within an accuracy of ±0.025 mm. Each of the principal channel walls was made of two plates as shown in Fig. 1(d). The wall facing the channel was a 3.2 mm thick and 530 mm wide phenolic fiberboard plate, with a typical thermal conductivity of 0.17 W m1 K1, and the surface adjacent to the inner part of the channel was coated with a 35 mm thick copper layer. The rear plate was 1.6 mm thick and its rear surface was a 17.5 mm thick copper layer, which was the heater (Fig. 1(d)). It was obtained by cutting the copper in a serpentine track of 19.6 mm wide and 9.0 m long. Its expected electric resistance was about 0.50 U. The two aforementioned plates were glued together with silicone rubber. A thick copper bar, bolted to the electric supply wire, was soldered to the ends of the heater. No electric resistance between the heater and the bars was measured during preliminary tests. A direct electrical current was passed through the copper and the dissipated heat flux was evaluated with an accuracy of ±2%, by measuring the voltage drop and the current through the electric resistance. A maximum variation of ±10% in the electrical resistivity of the copper was evaluated in the worst conditions, when the maximum difference in the wall temperatures was 30 K. Therefore, a uniform wall heat flux is assumed, with a ±5% maximum deviation from its average value. Heat losses from the rear surfaces of the channel were reduced by sticking a 120 mm polystyrene block on each wall. Side walls of the channel were made of Plexiglas rectangular rods, 4.0 mm thick, placed between the walls at their lateral edges. They were machined within an accuracy of ±0.03 mm. The distance between the principal walls is measured with an accuracy of ±0.25 mm using a dial-gauge caliper which can resolve ±0.025 mm. The channel was open to the environment along the top and the bottom edges. The walls are fastened together by a steel frame, which had been designed in such a way as not to obstruct the fluid flow in the proximity of the channel inlet. The channel was aligned vertically with horizontal leading edges using a plumb line and a level. The auxiliary unheated plate was a 0.1 mm thick stainless steel foil, coated with black varnish which resulted in a total emissivity of 0.9. If the plate was not coated with black varnish, its emissivity was 0.1. Its height varied between 0 and 200 mm, giving an auxiliary plate-to-wall height ratio, h/L, ranging from 0.0 to 0.5. It was fixed to the lateral walls of the channel by inserting and sticking it between them. The total normal emissivity of the heated walls was 0.9, obtained by coating the surfaces facing the channel with black varnish. Some experiments were carried out with the total normal emissivity of the heated walls equal to 0.1, obtained coating the surfaces with 0.025 mm thickness aluminum foils. The emissivity was evaluated by means of radiometric direct measurements by comparison with a black body with an accuracy of ±5%. Every week aluminum emissivity was measured by an infrared camera FLIR SC3000 in order to verify its variation due to oxidation. In case of differences more than 10% with respect to initial emissivity value, the aluminum foil was substituted. Electric insulation between the copper surfaces and the aluminum foil was ensured by uniformly spraying an electrically insulating varnish onto them before coating. Wall temperature measurements were carried out by 0.50 mm OD iron-constantan thermocouples (type J), embedded in the

A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

x

x

b

b h

L/2

h

b/2

L

h/2

x

L

L

b

h

44

b/2

b/2

y

y

y

Air

Air

Air

(a) Inlet Position

(c) Outlet Position

(b) Central Position 1.6 mm

3.2 mm

Phenolic fiberboard plates Polystyrene Copper Heater

Copper

Black Varnish

(d) Fig. 1. (a), (b) and (c) Sketch of the investigated configurations, (d) Cross view of the test section.

fiberboard close to the rear side of the copper layer facing the channel and bonded with a 3 M epoxy glue. Nine thermocouples in the centerline of each of the heated plates were placed at x ¼ 10, 30, 60, 100, 150, 250, 300, 350, 395 mm from the inlet section of the channel. At x ¼ 300 mm from the leading edge of one of the heated walls, eight additional thermocouples were located horizontally outward from the centreline at z ¼ ± 75.0 mm, ± 100.0 mm, ± 125.0 mm and ±150.0 mm, to provide indication on variation of the wall temperature along the horizontal z direction. Tests showed the wall temperature to be the same on two heated plates at the same x location, the differences being within ±0.2 K. Moreover, the wall temperature can be assumed to be independent of the z-coordinate within z ¼ ± 100 mm, since in this region the maximum deviation from the centreline temperature was found to be less than 1  C when the latter was 70.0  C for the heated plates. The ambient air temperature was measured by means of the same type of thermocouples located near the leading edge of the channel. The clearance of the bottom edges of the heated walls above the floor is 1.20 m and the minimum distance between the exit section of the channel and the ceiling is 1.80 m. The entire apparatus is located in an enclosed room, accurately sealed in order to eliminate extraneous air currents and air drafts are further reduced by vertical screens, 2.5 m high. A large fraction of the lower part of the screens is made up of a 0.20 m high mesh. The range of ambient temperature varied from 21 to 22  C during the experiments; the measured differences in the ambient air temperature in the proximity of the inlet and the exit sections of the apparatus were less than 0.8 K. Fifteen thermocouples were affixed to the rear surface of the plates and embedded in the polystyrene block in order to evaluate the conductive heat losses. Thermocouple voltages were recorded to 1 mV. Each thermocouple was calibrated in a 0.01 K thermostatic bath using a reference standard thermometer (Pt100). The calibration of the temperature measuring system showed an estimated precision of the thermocouple-readout system of ±0.2 K. A National Instruments SCXI module data acquisition

system and a personal computer were used for the data collection and reduction. The data acquisition was performed with the LabView™ software. 3. Data reduction The main geometrical parameters, with reference to the sketch of the investigated system (Fig. 1), used in this investigation are the channel aspect ratio, L/b, and the auxiliary plate-to-wall height ratio, h/L. The average convective heat flux of the channel walls, qc, is calculated as follows:

qc ¼

1 L

ZL qc ðxÞdx:

(1)

0

The channel Rayleigh number and the global channel Rayleigh number are defined as:

Ra0 ¼

g b qc b4 b Pr L n2 k

(2)

Ra0 ¼

g bðqc þ qr Þ b4 b Pr L n2 k

(3)

being qr the average radiative heat flux value over the heated walls, defined in an analogous way as the average convective heat flux, Eq. (1). The thermo-physical properties have been evaluated at the reference temperature:

Tref ¼

Tw þ To 2

with Tw the average heated wall temperature, defined as:

(4)

A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

Tw ¼

1 L

ZL Tw ðxÞdx

(5)

0

and To the ambient one. The average Nusselt number and the global average Nusselt number are:

Nu ¼

qc b ; kðTw  To Þ

Nu* ¼

ðqc þ qr Þ b kðTw  To Þ

(6)

The dimensionless maximum wall temperature is defined as:



q*w;max

 Tw;max  To k ¼ ðqc þ qr Þb

(7)

The local convective heat flux, qc(x), is not uniform because of radiative and conductive heat losses. Experimental data were reduced by first introducing, in the equations presented above, the local convective heat flux

qc ðxÞ ¼ qU ðxÞ  qk ðxÞ  qr ðxÞ

(8)

where qU(x) is the local heat flux due to the Ohmic dissipation and it is assumed to be uniform along the heated plates, qk(x) denotes the local conductive heat losses through the rear of the plates and qr(x) is the local radiative heat flux from the plate. For each run, the terms qk(x) were calculated by means of a numerical procedure, a threedimensional distribution of the temperature being assumed in the Polystyrene. The predicted numerical temperatures for some configurations of the system were previously compared with those measured by thermocouples embedded in the Polystyrene insulation and the agreement was very good, the maximum deviation being 3%. The maximum value of the conductive heat losses was about 9% of the Ohmic dissipated rate for the investigated wall heat fluxes.

h Fjra ¼

ðra  j þ 1Þ2 Dx2 þ b2

i1=2

45

The evaluation of qr was carried out considering the twodimensional cavity reported in Fig. 2. The inlet and outlet sections, ∞i and ∞o, were assumed black surfaces at ambient temperature, To. The heated channel wall, on the left in Fig. 2a, was assumed a grey surface whereas the symmetry surface, on the center of the heated channel, with and without the auxiliary plate was considered an adiabatic surface (q ¼ 0 and qra ¼ 0) and, consequently, a reradiating surface with a floating temperature distribution. The heated and reradiating surfaces were subdivided in k sub-surfaces and the hypothesis was that all sub-surface temperatures of heated wall and the inlet and outlet sections are known and the radiative heat fluxes of symmetry surface were equal to zero. Moreover, on each sub-surface and inlet and outlet surfaces the radiosity, J, is assumed uniform. The radiosity on each sub-surface on heated wall, Jj, was related to its temperature by the equations: k  X   Jj  1εj Fjra Jra ¼ εj Ebj þ 1εj Fjo Ebo ; for j ¼ 1;2;:::::;k ra¼1

(9) on the symmetry surface, with Ara ¼ Aj

Jra 

k X

Fjra Jj ¼ Frao Ebo ;

for

ra ¼ 1; 2; :::::; k

(10)

j¼1

where Eb ¼ sT4b for Eb ¼ Ebj or Eb ¼ Ebo Fjra ¼ Fraj Fjo ¼ Frao ¼ Fj∞o þ Fj∞i With reference to Fig. 2b the view factor Fj-ra is

i1=2 h i1=2 h þ ðra  j  1Þ2 Dx2 þ b2  ðra  jÞ2 Dx2 þ b2 2Dx

Fig. 2. Cavity for radiation heat transfer calculation with surface subdivision: (a) cavity; cross string for (b) Fj-ra; (c) Fj-∞i and Fj-∞o.

(11)

46

A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

With reference to Fig. 2c the view factors Fj-∞o and Fj-∞i are

Fj∞o ¼

Fj∞i ¼

i1=2 h i1=2 h Dx þ ðj  1Þ2 Dx2 þ b2  ðjDxÞ2 þ b2

(12)

2Dx

i1=2 h i1=2 h Dx þ ðk  jÞ2 Dx2 þ b2  ðk  j þ 1Þ2 Dx2 þ b2 2Dx (13)

The radiosities, Jj and Jra, were evaluated by Eqs. (9) and (10) and the flux on each sub-surface of the heated wall is

qj ¼

 εj  Ebj  Jj ; 1  εj

for

j ¼ 1; 2; :::::; k

(14)

and for the symmetry surface

qra ¼ 0;

for ra ¼ 1; 2; :::::; k

(15)

The radiative heat flux on the heated surface is

qr ¼

k  εj  Dx X Ebj  Jj L 1  εj

(16)

j¼1

4. Results and discussion Results are presented for the channel aspect ratio, L/b, equal to 10 and 20, h/L in the range [0.0, 0.5] and for four values of the Ohmic heat flux, qU, equal to 30, 60, 120 and 240 W m2. The corresponding channel Rayleigh number range is 20 < Ra'<1.5
105. The wall emissivity ε was equal to 0.9 and some comparisons were carried out with ε ¼ 0.1. The comparison between wall temperature rise values, with respect to the ambient temperature, for the three configurations (inlet, central, outlet) with h/L ¼ 0.5 and the simple channel wall temperatures is reported in Figs. 3 and 4. This comparison is reported in Fig. 3 for an Ohmic heat flux value equal to 60 W m2 and aspect ratio value equal to 20 (Fig. 3a, Ra'y1.4
103) and 10 (Fig. 3b, Ra'y4.5
104). It is observed that the wall temperature profiles for the simple channel (h/L ¼ 0.0) and the two aspect ratio values are very similar and the percent difference between the maximum wall temperature, referred to the value at L/b ¼ 20, is about 1.6%. The temperature profiles have their maximum value inside the channel; for both cases, Fig. 3a and b, the maximum value is located at x ¼ 0.75L ¼ 300 mm. The decrement of the wall temperature is due to the heat conduction through the insulated layer placed on the rear of the walls and to the radiative heat transfer with the cold surroundings, which are relevant in the upper part of the channel and can be identified as edge effects [41].

Half of the domain was considered to study the twodimensional radiative cavity, because of thermal and geometrical symmetry conditions. It was made of two plates, considered as diffuse-gray surfaces, which were the heated wall and the auxiliary adiabatic plate, two black edge sections at room temperature and a reradiating surface representing the symmetry plane. The qr(x) terms were calculated for each temperature distribution of the walls, ambient temperature and plate spacing by dividing the wall plate into ten equal length strips along its length and the reradiating surface and the auxiliary plate into other ten equal strips, This procedure is well described by Webb and Hill [53] and Manca and Nardini [54]. The qc and qc þ qr terms were obtained by the equations:

qc ¼ qU  qk  qr ;

qc þ qr ¼ qU  qk

(17)

where qk is the average values of the conductive heat fluxes along the plates. The uncertainty of the calculated quantities was determined according to the standard single sample analysis recommended in Refs. [55,56]. Accordingly, the uncertainty of a dependent variable R as a function of the uncertainties in the independent variables Xi is given by the following relation:

" dR ¼

n  X vR i¼1

vXi

2 # 1=2 (18)

dXi

The uncertainty in the values of the air thermo-physical properties can be assumed to be negligible. The maximum percent uncertainties in the values of the independent variables are reported in Table 1. These uncertainty values together with Eqs. (2) and (6) brought to maximum uncertainty of 7% both for the Rayleigh and Nusselt numbers.

Table 1 Maximum percent uncertainty values ((dXi/Xi)
100). Variable

Tw

T0

TwT0

b

qc

qr

qU

qk

qc þ qr

Uncertainty

0.50

0.93

1.1

1.2

5.0

4.7

2.0

4.0

2.5

Fig. 3. Wall temperature profiles vs the axial coordinate x for the investigated configurations with qU ¼ 60 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

Fig. 4. Wall temperature profiles vs the axial coordinate x for the investigated configurations with qU ¼ 240 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

The percentage decrease of the wall temperature at the outlet zone with respect to the maximum wall temperature is about 22% for both the L/b values. The edge effects are less relevant in the lower zone of the channel because of the lower temperature values. In Fig. 3, the configuration with the auxiliary plate placed in the lower part of the channel, inlet, shows lower wall temperature values in the lower channel zone, compared to the other configurations with auxiliary plate and the simple channel. For this configuration, the radiative heat transfer in the lower part of the channel is higher than in the other zones. The percent temperature difference with respect to the simple channel value at x ¼ 0.375 L ¼ 150 mm is about 17% for L/b ¼ 20 and 30% for L/ b ¼ 10. In Fig. 3a, it is observed that the presence of the auxiliary plate modifies the wall temperature profile slope in the upstream zone of the auxiliary plate position. The wall temperature decrease, for the configurations with the auxiliary plate and L/b ¼ 20, is not enough to reduce the maximum wall temperature value below the one of the simple channel. The presence of the auxiliary plate determines an increase of pressure drop and a consequent mass flow rate reduction which reduces the convective heat transfer on the heated walls and an increase in temperature walls is detected. In the upper zone of the channel the wall temperature profile of the simple channel, for this L/b value, is the lowest. For L/b ¼ 20, the pressure losses are greater and the radiative heat transfer toward

47

the external ambient is lower. The highest values of wall temperature profile are obtained for the configuration with the auxiliary plate in the central position, where the pressure losses are about the same but the radiative heat transfer toward the external ambient is lower with respect to the other two configurations. The percent differences between the maximum wall temperature for the channel with auxiliary plate with respect to the simple channel are about 12% for the central position, 5.8% for the outlet position and 3.7% for the inlet position. When L/b is smaller (L/b ¼ 10), Fig. 3b, the presence of the auxiliary plate allows a maximum wall temperature decrease and the lowest value is obtained for the configuration with the auxiliary plate in the inlet position. The pressure losses are lower and the radiative heat transfer toward the external ambient is greater with respect to the higher aspect ratio. The percent differences are about 4.8% for the central position, 13% for the outlet position and 14% for the inlet position. The temperature decrease for the inlet position could be explained as follows: the presence of the auxiliary plate inside the channel causes an increase of the localized pressure losses due to the wet surface increase; therefore, the mass flow rate could also be diminished with respect to the simple channel [32]. But the mass flow rate decrease is not high [30e32] and the velocity reduction in the channel central zone (zero on the symmetry plane) determines a velocity increase in the boundary layer adjacent to the channel walls; this determines a better local heat transfer between the fluid and the walls. Moreover, the presence of the auxiliary plate in the inlet zone reduces the “vena contracta” effects with a subsequent heat transfer improvement in this zone [30e32]. The radiation determines a wall temperature decrease due to the radiative heat transfer to the auxiliary plate, which implies a convective heat transfer to the air next to it. In this case, a thermal plume is present in the central zone of the channel, which contributes towards the local heat transfer increase. Clearly, the position of the auxiliary plate can be a decisive factor both with respect to the radiative heat transfer and to the thermal plume, also as function of the heat flux value and the channel aspect ratio. For the highest heat flux (qU ¼ 240 W m2) the wall temperature profiles, given in Fig. 4, show that the temperature decrease at the outlet zone is present for all cases and for the simple channel the percentage decreases with respect to the maximum temperature are again about 22% for the two considered L/b values. In the lower zone the configuration with the auxiliary plate in the inlet position presents the lowest temperature value even if for L/b ¼ 20 (Ra'y4.5
103) the maximum percent decrease with respect to the simple channel at x ¼ 0.375 L ¼ 150 mm is about 27% for the auxiliary plate in the central position and for the inlet position at x ¼ 0.25 L ¼ 100 mm (Fig. 4a). For L/b ¼ 10 (Ra'y1.4
105) the maximum wall decrease at x ¼ 0.375 L is attained for the central position (Fig. 4b) and the percent decrease is about 21%. The comparison shows that the configuration with the auxiliary plate at the outlet position has a lower value of the maximum wall temperature both at L/b ¼ 20 and 10. In this case the percent decreases with respect to the simple channel are about 19% and 21% for L/ b ¼ 10 and 20, respectively. This is due to the more efficient radiative heat transfer associated to the higher temperature values, and to the thinner boundary layer thickness; these two factors allow a more efficient thermal plume. The thermal plume is more efficient in the outlet position because it does not cause a channel narrowing which is possible for the other two configurations (central and inlet positions). Finally, it is observed that, with respect to the configurations considered in Figs. 3 and 4, the best configurations in terms of lower maximum wall temperature is the one with L/b ¼ 10, qU ¼ 60 W m2 and the auxiliary plate in the inlet position and the one with L/b ¼ 10, qU ¼ 240 W m2 and the auxiliary plate in the

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outlet position. Moreover, by comparing the three different configurations (inlet, central, outlet) shown in Figs. 3 and 4, it is more useful to analyze in detail the configuration with the auxiliary plate at the inlet and outlet positions. The effects of the height of the auxiliary plate are analyzed by means of Figs. 5e7 for the inlet position and Figs. 8e10 for the outlet position. In these figures the channel wall temperature profiles, pertinent to different heights of the auxiliary plate, h/ L ¼ 0.25, 0.33, 0.50, and to the simple channel, h/L ¼ 0.0, at qU ¼ 60, 120, 240 W m2 and L/b ¼ 20 and 10, are reported. In Figs. 5e7, it is observed that the auxiliary plate presence in the lower zone of the channel determines a reduction of wall temperature in this channel region independently of the auxiliary plate height, as already noted in Figs. 3 and 4. In particular, as observed in Fig. 5, for qU ¼ 60 W m2, the configuration with h/ L ¼ 0.33 presents in the lower channel zone a higher wall temperature decrease, which is about 31% with respect to the simple channel. The configuration with h/L ¼ 0.25 and L/b ¼ 20, Fig. 5a, shows the lowest maximum temperature value and the percent decrease with respect to the simple channel value is about 9.5%. But the configuration with h/L ¼ 0.5 has the highest maximum wall temperature value. The percentage increase with respect to the maximum wall temperature value pertinent to the simple channel is about 3.7%. As L/b decreases, Fig. 5b, all configurations of the channel with the auxiliary plate show a better thermal

Fig. 6. Wall temperature profiles vs the axial coordinate x for the configuration inlet with qU ¼ 120 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

Fig. 5. Wall temperature profiles vs the axial coordinate x for the configuration inlet with qU ¼ 60 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

performance than the simple channel also in the upper part of the channel. In this case, L/b ¼ 10, the configuration with h/L ¼ 0.5 exhibits the lowest wall temperature values and the best thermal performance in terms of maximum wall temperature. However, the configurations for a channel aspect ratio value equal to 10 show better thermal performance with respect to the configurations for L/b ¼ 20, for h/L ¼ 0.33 and h/L ¼ 0.5. Also increasing the Ohmic heat flux, Figs. 6 and 7, the lower part of the channel with auxiliary plate presents lower temperatures than those of the simple channel. In fact, for L/b ¼ 20 the maximum percent differences with respect to the simple channel are about 18% for qU ¼ 120 W m2 and 26% for qU ¼ 240 W m2, whereas for L/b ¼ 10 they are about 15% for qU ¼ 120 W m2 and 24% for qU ¼ 240 W m2. On the other hand, in Figs. 6a and 7a, L/b ¼ 20, it is noticed that the channel with auxiliary plate shows worse thermal performance in the upper part of the channel, in terms of maximum temperatures, than the simple channel. The maximum increase of the maximum wall temperature is for the configuration with h/L ¼ 0.5 both for qU ¼ 120 W m2 and 240 W m2. It is about 15% for qU ¼ 120 W m2 and 5.7% for qU ¼ 240 W m2 with respect to the simple channel values. The configuration with L/b ¼ 10 exhibits very small difference between the maximum wall temperatures, Figs. 6b and 7b. A better thermal performance, in terms of lower maximum wall temperatures, could be induced by a more efficient radiative heat transfer between the upper part of the channel and the auxiliary plate.

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49

Fig. 7. Wall temperature profiles vs the axial coordinate x for the configuration inlet with qU ¼ 240 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

Fig. 8. Wall temperature profiles vs the axial coordinate x for the configuration outlet with qU ¼ 60 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

In Fig. 8, outlet configuration, for qU ¼ 60 W m2 and L/b ¼ 20 and 10, the configuration with h/L ¼ 0.5 is the worst when L/b ¼ 20 whereas the configurations with h/L ¼ 0.25 and 0.33 show a wall temperature decrease with respect to the simple channel in the upper part of the channel. For L/b ¼ 20, in terms of maximum wall temperature, the percentage increase pertinent to the configuration with h/L ¼ 0.50 with respect to the simple channel is about 5.3% whereas the percentage maximum decrease obtained for h/ L ¼ 0.33 is equal to about 15%. Probably, for h/L ¼ 0.5, the thermal plume determines both a flow development from the half part of the channel, which decreases the mass flow rate for higher L/b, and a distributed pressure loss increase. When L/b diminishes, the simple channel configuration is the worst configuration in terms of maximum wall temperature, as shown in Fig. 8b. For L/b ¼ 10 the highest percentage decrease of the maximum wall temperature with respect to the simple channel is obtained for h/L ¼ 0.25 and it is about 17%. This is due to a better radiative heat transfer, to a greater influence of the thermal plume induced by the auxiliary plate on the chimney effect and to the pressure loss decrease pertinent to the b increase. When qU ¼ 120 W m2 and L/b ¼ 20, as shown in Fig. 9a, the channel thermal behavior improves for h/L ¼ 0.25. This improvement is evident in the upper part of the channel in terms of maximum wall temperature and the percent decrease with respect to the simple channel is about 7.6%. Also in this case, when L/b decreases, Fig. 9b, the configurations with the auxiliary plate are

better than the simple channel in terms of maximum wall temperatures, even if when L/b is equal to 10, the maximum wall temperature for h/L ¼ 0.25 is almost equal to the simple channel one. Moreover the insertion of the auxiliary plate for h/L ¼ 0.5 determines the lowest maximum temperature values and the percent decrease, with respect to the simple channel, is about 16%. Increasing the dissipated heat flux to 240 W m2 the simple channel always presents the worst thermal performance, as shown in Fig. 10a (L/b ¼ 20) and Fig. 10b (L/b ¼ 10). These configurations have the highest channel Rayleigh number for the same L/b value and, increasing qU, this implies a reduction of the boundary layer thickness along the channel walls and the auxiliary plate, with a minor interaction between each other. This occurrence also implies an improvement of the buoyancy. Further, as L/b decreases there is an increase of the buoyancy inside the simple channel whereas it worsens a little for the case with h/L ¼ 0.25. The maximum wall temperature for the simple channel (h/L ¼ 0.0) is 62.4  C for L/ b ¼ 20 and 57.8  C for L/b ¼ 10 and the percentage decrease with respect to the configuration with L/b ¼ 20 is about 7.4%. When L/ b ¼ 10 the profile of the h/L ¼ 0.25 case shows maximum temperature very close to that of the simple channel, whereas when the configuration has an auxiliary plate whose height is h/L ¼ 0.5, there is an augmentation of the heat transfer performance as far as the maximum wall temperatures are concerned. The percent decrease with respect to the simple channel is about 20% for both the L/b values. The two configurations with different L/b values show very

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Fig. 9. Wall temperature profiles vs the axial coordinate x for the configuration outlet with qU ¼ 120 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

Fig. 10. Wall temperature profiles vs the axial coordinate x for the configuration outlet with qU ¼ 240 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

similar profiles and it is interesting to observe that the percentage maximum wall temperature decrease between the configuration with L/b ¼ 20 and the one with L/b ¼ 10 is about 7.6% for h/L ¼ 0.5. The comparison of the wall temperature profiles for h/L ¼ 0.5 and outlet configuration between channels with different emissivity coefficients is reported in Fig. 11. In details the following three cases have been taken into account: (1) the channel walls and the auxiliary plate with εch ¼ εaux ¼ 0.1; (2) the channel walls with εch ¼ 0.9 and the auxiliary plate with εaux ¼ 0.1; (3) the channel walls and the auxiliary plate with εch ¼ εaux ¼ 0.9. The configuration with Ohmic wall heat flux of 120 W m2 is reported in Fig. 11a and b for L/b ¼ 20 and L/b ¼ 10, respectively. The case (1) shows the highest maximum wall temperature value with respect to the cases (2) and (3), for the two analyzed L/b values. It has to be observed that for L/b ¼ 20, Fig. 11a, the maximum wall temperature decrease of the cases (2) and (3) is about 5% with respect to the case (1). A greater emissivity value for the auxiliary plate, case (3), does not produce significant improvements with respect to the case (1) and moreover the thermal performance with respect to the case (2) is slightly worse. This fact shows that the auxiliary plate heating due to the radiative effects produces a boundary layer thickness increase, so that the interaction between the boundary layer adjacent to the channel plate and the one adjacent to the auxiliary plate is present upstream the outlet channel section. This determines a lower “chimney effect”. For L/ b ¼ 10, Fig. 11b, the best thermal performance is pertinent to the

case (3) and the percentage maximum wall temperature decrease with respect to the case (1) is about 18%. In this case not only the radiative exchange towards the environment is higher but even the “chimney effect” is higher with respect to the other analyzed cases. The profiles of the radiative heat transfer over the radiative and convective heat transfer, qr/(qr þ qc), as function of x/L are reported in Figs. 12 and 13. The numerical values of the radiative and convective heat fluxes have been determined through the numerical procedure previously described. These profiles allow the evaluation of the radiative heat losses from the channel walls. It is immediately clear in Fig. 12, inlet configuration, that the edge effects are very considerable at the channel exit because of the high temperature values and the better view factor between the walls and the ambient. Analogous trend is observed at the channel inlet, where this effect is less marked because of the lower wall temperature values. In addition, in this zone, the radiative heat transfer is as greater as bigger the h/L value. As L/b decreases the radiative heat transfer between the channel and the auxiliary plate enlarges; this trend is more marked when h/L ¼ 0.5. The increment with the L/b value decrease is explained by considering that the radiative heat transfer toward the outer environment is as lower as greater L/b. For h/L ¼ 0.5, the average radiative heat flux in the channel inlet zone is about 13% for L/b ¼ 20 and about 15% for L/ b ¼ 10 with respect to the average radiative and convective heat flux value in the same region.

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51

Fig. 11. Wall temperature profiles vs the axial coordinate x for different wall emissivity coefficients, outlet position, h/L ¼ 0.5, qU ¼ 120 W m2: (a) L/b ¼ 20; (b) L/b ¼ 10.

As far as the outlet configurations are concerned, Fig. 13, the profiles of qr/(qr þ qc) along x/L are analogous to the ones reported in Fig. 12, anyway showing higher values at the channel exit. Besides, this ratio is higher than the one for the previous configurations because the auxiliary plate is located opposite to the channel part with highest temperatures. This implies a greater radiative heat transfer between the surfaces. For this configuration the average radiative heat flux for h/L ¼ 0.5 in the channel outlet zone is about 28% for L/b ¼ 20 and about 23% for L/b ¼ 10 with respect to the average radiative and convective heat flux value in the same region. The global radiation fraction as a function of auxiliary plate height for inlet and outlet positions is given in Fig. 14 for two aspect ratio values, L/b ¼ 10 and 20 Fig. 14a and b, respectively. It is interesting to observe that for the lowest aspect ratio, in Fig. 14b, the global fraction is greater due to the higher view factor toward the external ambient with respect to the highest value, in Fig. 14a, and this determines a more efficient radiative heat transfer. Moreover, the outlet position of the auxiliary plate is more efficient with respect to the inlet position due to the higher heated wall temperatures in the outlet zone which determine a higher radiative heat transfer. Simple monomial equations are proposed to correlate the average channel Nusselt number to the channel Rayleigh number and the h/L ratio, for εch ¼ εaux ¼ 0.9. The average channel Nusselt numbers presented are the convective one, Nu, and the global or total one, Nu*, and the equations are sought in the form:

Fig. 12. Radiative heat flux over radiative and convective heat flux as function of x/L, inlet position at qU ¼ 120 W m2: a) L/b ¼ 20, b) L/b ¼ 10.

z ¼ a*xb *yc

(19)

Each correlation is estimated employing 58 experimental data points. The monomial correlation is obtained by the experimental data and the numerical coefficients have been calculated by means of the least square method. The correlation equations, together with their regression coefficient (RMS), r2, are:  Auxiliary plate at the channel inlet

Nu ¼ 0:334ðRa0 Þ

0:263

Nu* ¼ 0:341ðRa0 Þ

 1þ

0:268

 h 0:250 L

 1þ

h L

r2 ¼ 0:954

(20)

0:234 r2 ¼ 0:959

(21)

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A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

Fig. 13. Radiative heat flux over radiative and convective heat flux as function of x/L, outlet position at qU ¼ 120 W m2: a) L/b ¼ 20, b) L/b ¼ 10.

In Fig. 15(b) a comparison between the proposed correlation, Eq. (20), and the following

 Auxiliary plate at the channel outlet

Nu ¼ 0:456ðRa0 Þ

0:226

Nu* ¼ 0:462ðRa0 Þ

 1þ

0:233



 h 0:0163 L

 h 0:192 1þ L

r2 ¼ 0:956

r2 ¼ 0:950

(22) Nu ¼ 0:639ðRa0 Þ (23)

The channel Rayleigh number, the h/L ratio and the L/b values are in the 20e1.5 x 105, 0e0.5 and 10e20 ranges, respectively. It should be observed that the convective Nusselt number and the global Nusselt number values increase with increasing h/L values. In Fig. 15(a) a comparison between the proposed correlation, Eqs. (21) and (23), and the following

Nu* ¼

Fig. 14. Global radiation fraction as function of relative auxiliary plate height for qU ¼ 120 W m2 and for an aspect ratio: a) L/b ¼ 20 and b) L/b ¼ 10.

i i o nh h 0:50 2:0 0:18 2:0 1=2 0:288ðRa0 Þ þ 0:94ðRa0 Þ (24)

proposed in Ref. [57] is presented. The Eq. (24) is valid for εch  0.80. A good agreement is observed between the five curves, with those obtained by Eqs. (21) and (23) which are monomial.

0:223

 1þ

h L

0:140

r2 ¼ 0:976

(25)

proposed in Ref. [32] is presented. A good agreement is observed. Further, a comparison with the correlations evaluated in Refs. [58,59] for turbulent flow regime is also reported in Fig. 15b. It appears evident that the correlations present higher Nusselt number values than the ones estimated in the present experiments. This confirms that the flow can be considered laminar also for the highest Rayleigh number examined equal to about 1.5
105. Moreover, in Fig. 16, a comparison between the Nusselt number value from experimental data and the Nusselt number values calculated by the correlation in Eqs. (20) and (21) are reported together with an error level of ±10% and 20%. In addition, simple monomial equations are proposed to correlate the dimensionless maximum wall temperature to the global channel Rayleigh number and the h/L ratio. The correlation equations, together with their regression coefficient (RMS), r2, are:

A. Andreozzi, O. Manca / International Journal of Thermal Sciences 97 (2015) 41e55

7

10 9 8 Nu* 7 6

Nu (experimental)

6

5 4 3 Ref. [57] Eq.(21) with h/L=0 Eq.(21) with h/L=0.5 Eq.(23) with h/L=0 Eq.(23) with h/L=0.5

2

1

5 4 3 2

102

103

104

105

Ra'*

106

1

(a)

(a) 0

Ref. [32] h/L=0 Eq. (20) h/L=0 Ref. [32] h/L=0.5 Eq. (20) h/L=0.5 Ref. [58] h/L=0 Ref. [59] h/L=0

10

Nu

53

9 8 7 6

0

1

2

3

4

5

6

7

Nu (Eq.20) 8

Nu* (experimental)

5 4 3

2

1 101

102

103

104

105

106

Ra'

(b)

 Auxiliary plate at the channel inlet

0:315

 1þ

 h 0:346 L

0:256

 1þ

 h 0:281 L

5 4

2 1 0

r2 ¼ 0:979

(26)

(b) 0

1

2

3

4

5

6

7

8

Nu* (Eq.21) Fig. 16. Inlet Configuration: (a) comparison between experimental convective Nusselt numbers and the ones by correlation given by Eq. (20) with a percentage difference in the ±10% and ±20% range; (b) comparison between experimental global Nusselt numbers and the ones by correlation given by Eq. (21) with a percentage difference in the ±10% and ±20% range.

 Auxiliary plate at the channel outlet

q*w;max ¼ 3:35ðRa0 Þ

6

3

Fig. 15. Comparison between: (a) correlation equation of Ref. [57] and present Eqs. (21) and (23); (b) correlation equation of Refs. [32], [58] and [59] and present Eq. (20).

q*w;max ¼ 4:67ðRa0 Þ

7

r2 ¼ 0:975

(27)

5. Conclusions An experimental investigation on natural convection in air in a vertical parallel-plates channel with an auxiliary plate placed on the centerline of the channel was carried out. The investigation was focused on the impact that the placement of the auxiliary plate inside the channel involves on the combined convectioneradiation heat transfer inside the channel. The study was carried out by showing the channel wall temperature profiles for channel aspect

ratio, L/b, values of 10 and 20 and for the ratio of the auxiliary platechannel heights, h/L, equal to 0, 0.25, 0.33 and 0.5, with emissivity coefficients ε of the walls equal to 0.1 and 0.9 and three different auxiliary plate positions inside the channel. The channel Rayleigh number was in the range from 20 to 1.5
105. The comparison between the different positions of the auxiliary plate, for high radiative heat transfer, showed that the auxiliary plate placed at the channel exit gives the lowest maximum channel wall temperatures when high Rayleigh numbers and heat flux values are considered. Results showed that when the Rayleigh number is high the auxiliary plate always improves the thermal

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performance of the channel, by minimizing the maximum channel wall temperatures. A maximum reduction of maximum wall temperature rise of about 20%, with respect to the simple channel was detected for L/b ¼ 10 and 20. The h/L value which displayed nearly always the best thermal performance is equal to 0.5, unless when the lowest investigated heat flux was considered, qU ¼ 60 W/m2. When h/L ¼ 0.5 and configurations with the auxiliary plate placed at the channel exit were considered, the comparison between different superficial radiative coefficients of the walls has shown that the channel whose walls have emissivity equal to 0.9 displays the best thermal performance when the highest investigated channel gap b and heat flux are considered. The maximum wall temperature rise decreases of about 18% between εch ¼ 0.1 and 0.9 for L/b ¼ 10. The mean value of the radiative heat flux from the heated wall was about 28% for L/b ¼ 20 and 23% for L/b ¼ 10 of the average global heat flux (convective and radiative) with the auxiliary plate placed at the channel exit. For the configuration with the auxiliary plate at the channel inlet the average global heat flux was about 13% for L/b ¼ 20 and 15% for L/b ¼ 10. Correlation equations for average Nusselt numbers, Nu and Nu*, and dimensionless maximum wall temperature, q*w,max, in terms of channel Rayleigh numbers and auxiliary plate-channel height ratio for inlet and outlet position of the auxiliary plate were evaluated. Equations were proposed in the following ranges: 20 < Ra'<1.5
105, 22 < Ra'*<1.7
105, 0  h/L  0.50 and 10  L/ b  20. A good agreement between the proposed correlation for Nusselt numbers and correlation equations by experimental data for simple channel with and without radiative heat transfer [57] and by numerical data for channel with auxiliary plate without radiative heat transfer [32] was showed. Acknowledgments This work was supported by SUN and MIUR with a grant PRIN2009KSSKL3. Oronzio Manca thanks to UPEMLV for the support lisation et Simulation Multiduring his stay at Laboratoire Mode Echelle as Visiting Professor. References [1] O. Manca, B. Morrone, S. Nardini, V. Naso, Natural convection in open chann, G. Comini (Eds.), Computational Analysis of Convection nels, in: B. Sunde Heat Transfer, WIT Press, Southampton, 2000, pp. 235e278. [2] A.S. Krishnan, B. Premachandran, C. Balaji, S.P. Venkateshan, Combined experimental and numerical approaches to multi-mode heat transfer between vertical parallel plates, Exp. Therm. Fluid Sci. 29 (2004) 75e86. [3] D. Ryan, S.A.M. Burek, Experimental study of the influence of collector height on the steady state performance of a passive solar air heater, Sol. Energy 84 (2010) 1676e1684. nier, G. Lauriat, Effect of surface radiation on natural [4] R. Li, M. Bousetta, E. Che convective flows and onset of flow reversal in asymmetrically heated vertical channels, Int. J. Therm. Sci. 65 (2013) 9e27. [5] A. Auletta, O. Manca, Heat and fluid flow resulting from the chimney effect in a symmetrically heated vertical channel with adiabatic extensions, Int, J. Therm. Sci 41 (2002) 1101e1111. [6] S. Taieb, L. Ali Hatem, J. Balti, Natural convection in an asymmetrically heated vertical channel with an adiabatic auxiliary plate, Int. J. Therm. Sci. 74 (2013) 24e36. nier, A. Joulin, A. Bastide, B. Brangeon, J.P. Caltagirone, [7] G. Desrayaud, E. Che Y. Cherif, R. Eymard, C. Garnier, S. Giroux-Julien, Y. Harnane, P. Joubert, re , R. Li, D. Saury, A. Sergent, S. Xin, A. Zoubir, N. Laaroussi, S. Lassue, P. Le Que Benchmark solutions for natural convection flows in vertical channels submitted to different open boundary conditions, Int. J. Therm. Sci. 72 (2013) 18e33. [8] B. Morrone, A. Campo, O. Manca, Optimum plate separation in vertical parallel plate channels for natural convective flows: incorporation of large spaces at the channel extremes, Int. J. Heat Mass Transfer 40 (1997) 993e1000. [9] A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000. [10] S. Kazansky, V. Dubovsky, G. Ziskind, R. Letan, Chimney-enhanced natural convection from a vertical plate: experiments and numerical simulations, Int. J. Heat Mass Transfer 46 (2003) 497e512.

[11] A.K. da Silva, A. Bejan, S. Lorente, Maximal heat transfer density in vertical morphing channels with natural convection, Num. Heat Transfer A 45 (2004) 135e152. [12] G. Desrayaud, G. Lauriat, A numerical study of natural convection in partially open enclosures with a conducting side-wall, J. Heat Transfer 126 (2004) 76e83. [13] A.K. da Silva, A. Bejan, Constructal multi-scale structure for maximal heat transfer density in natural convection, Int. J. Heat Fluid Flow 26 (2005) 34e44. [14] A.K. da Silva, L. Gosselin, Optimal geometry of L and C-shaped channels with maximum heat transfer rate in natural convection, Int. J. Heat Mass Transfer 48 (2005) 609e620. [15] L.A. Florio, A. Harnoy, Use of a vibrating plate to enhance natural convection cooling of a discrete heat source in a vertical channel, Appl. Therm. Eng. 27 (2007) 2276e2293. [16] L. Langellotto, O. Manca, S. Nardini, Numerical investigation of transient natural convection in air in a convergent vertical channel symmetrically heated at uniform heat flux, Num. Heat Transfer A 51 (2007) 1065e1086. [17] A. Andreozzi, A. Campo, O. Manca, Compounded natural convection enhancement in a vertical parallel-plate channel, Int. J. Therm. Sci. 47 (2008) 742e748. [18] A. Andreozzi, B. Buonomo, O. Manca, Transient natural convection in vertical channels symmetrically heated at uniform heat flux, Num. Heat Transfer A 55 (2009) 409e431. gue , E. Bilgen, Heat transfer by convection, conduction and radi[19] H.F. Nouane ation in solar chimney systems for ventilation of dwellings, Int. J. Heat Fluid Flow 30 (2009) 150e157. [20] B. Zamora, A.S. Kaiser, Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation, App. Therm. Eng. 29 (2009) 762e769. [21] A. Andreozzi, B. Buonomo, O. 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