Experimental and numerical natural convection in an asymmetrically heated double vertical facade

International Journal of Thermal Sciences 152 (2020) 106288

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts

Experimental and numerical natural convection in an asymmetrically heated double vertical facade Yassine Cherif a, **, Emilio Sassine b, *, Stephane Lassue a, Laurent Zalewski a a b

Univ. Artois, ULR 4515, Laboratoire de G�enie Civil et g�eo-Environnement (LGCgE), B�ethune, F-62400, France Laboratory of Applied Physics (LPA), Lebanese University, Faculty of Sciences, Fanar Campus, Lebanon

A R T I C L E I N F O

A B S T R A C T

Keywords: Natural convection Double vertical channel Radiative heat flux Convective heat flux

The present work addresses the numerical and experimental study of natural convection inside an asymmetri­ cally heated open double vertical facade. Two heating cases were considered independently, the constant heat flux (Neumann condition) and the constant temperature (Dirichlet condition). The double facade has been modeled using a vertical two-dimensional channel with one wall being maintained at the heating condition and the other one insulated. The boundary conditions at the inlet and outlet were controlled through the addition of adiabatic walls upstream and downstream of the studied area. The airflow is assumed to be laminar and per­ manent. This study is conducted for several modified Rayleigh numbers ranging from 102 � Ram �107 and different aspect ratios (A ¼ 25, 12.5, 8.34, 6.25, and 5). Various parameters have been evaluated and high­ lighted, namely velocity and temperature profiles. In the first part of this study, the radiative heat transfer is not considered, comparison results give excellent agreement with the experimental work of Webb and Hill [18]. Similarly, the streamline results show a return flow through the outlet of the channel starting from a modified Raleigh value of Ram ¼ 104. In the second case of this study and for isothermal conditions, the radiative transfer is taken into account with mutual radiative heat exchanges between the surfaces and for a transparent not participating medium. The comparison results between the experimental and numerical heat fluxes along the heated plate gives a very good agreement, as well as for the mean Nusselt values for 2.28 � 102�Ram�8.22 � 105.

1. Introduction Convection is the most common heat transfer mode adopted in miscellaneous practical engineering fields systems due to its simplicity, reliability, and cost effectiveness. Natural convection heat transfers and fluid flow in vertical parallel-plate channels are relevant to a wide range of heat exchange applications especially in buildings where passive solar heating and ventilation of buildings is gaining more and more impor­ tance [1,2]. Natural convection in vertical parallel-plate channels may take place under laminar or turbulent flow regimes depending on the geometrical size and thermal parameters. Laminar natural convection in vertical parallel-plate channels has been studied using experimental, analytical, and numerical techniques is many research works due to its wide applicability and [3–11]. Work of Elenbaas 1942 [12] appears among the first in this field, it presents an experimental device of vertical plane plates heated using a

density flux of constant heat (square plates of 12 cm side). It determines that the parameters of this flow are the numbers of Nusselt, Grashof and Prandtl and proposes a correlation connecting them. This one is ob­ tained in an analytical way for an infinitely long channel and is compared with the experimental results, which are corrected to be valid with the assumption infinite length. These results sweep a range of modified Rayleigh (Ram) going from 0.1 to 105. Lastly, Elenbaas deter­ mined the optimal spacing making it possible to maximize the transfer of heat in the channel. In 1980, Sparrow and Bahrami [13] presented an original experimental device to confirm the study of Elenbaas [12]. They made the analogy between mass transfer (via Sherwood num­ ber) and heat transfer (Nusselt number) by using the sublimation of naphthalene. Their device is almost the same one as that of Elenbaas except that the vertical walls were covered with naphthalene, which quantity was given before and after the experiment according to the quantity of matter which evaporated. They assessed the influence of the

* Corresponding author. Laboratoire de Physique Appliqu� ee (LPA), Universit�e Libanaise, Facult�e des Sciences, Campus Fanar, Lebanon. ** Corresponding author. Univ. Artois, ULR 4515, Laboratoire de G�enie Civil et g�eo-Environnement (LGCgE), B�ethune, F-62400, France. E-mail addresses: [email protected] (Y. Cherif), [email protected] (E. Sassine). https://doi.org/10.1016/j.ijthermalsci.2020.106288 Received 2 May 2019; Received in revised form 4 January 2020; Accepted 23 January 2020 Available online 21 February 2020 1290-0729/© 2020 Elsevier Masson SAS. All rights reserved.

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

side walls by closing the sides of the channel. They noted that without side walls (experiment of Elenbaas), for low modified Rayleigh numbers (Ram), the 2D approximation was not valid anymore and that new parameter must be introduced in order to account the 3D aspect. Moreover, they underlined the importance of the choice of the reference temperature being used to calculate the thermophysical properties of the fluid. Their results differed about 15% compared to those of Elenbaas for high modified Rayleigh numbers of (Ram > 100). It was explained by the fact that Elenbaas used the wall temperature (300 � C) for the calculation of the thermophysical properties instead of using a fluid temperature which can be estimated by the average between the wall temperature and the temperature at the entry. Sparrow et al. [14,15] presented in 1984 a numerical and experimental study of natural convection flow in a heated channel with one only side having an imposed temperature. The experiment was made in water with a 14.5 cm height channel and they observed a recirculation at the exit of the unheated wall. By neglecting the recirculation in simulations, the calculation of heat exchange remained adequate because the recirculation took place on the unheated wall. They gave numerical correlations for Prandtl numbers varying between 0.7 and 10. The same year, Bar-Cohen and Rohsenow [16] presented an analytical study on the correlations between the Nusselt number and the modified Rayleigh number. They were interested in the four heating cases: temperature or heat flux density imposed on one (asymmetrical) or the two walls (symmetrical). They start from solutions for vertical plane plate and channel diffusive flow regimes and propose, for each case, a correlation linking these two asymptotes using the method developed by Churchill and Usagi [17]. However, the majority of the correlations of the literature are of the same type as that of Sparrow et al. [14] and thus cannot take into account the change of exhibitor corresponding to the change of mode. In 1989, Webb and Hill [18] worked on an experimental device similar to that of Wirtz and Stutzman [19]. It was a 30 cm height air channel having a heated wall with imposed flux heat density and the other wall adiabatic. The heated zone corresponded to the half-height so that the entry and the exit are adiabatic; and they swept a range of modified Rayleigh Ram going from 102 to 107. The originality of their study came from the fact that the radiative heat transfers were estimated separately. Thus, they attribute the gap with the vertical plane plate solution to these radiative transfers as well as conduction losses in the insulation placed on the back of the heated wall. They have, in addition to conventional correlations on heat transfer, a correlation giving the maximum temperature in the wall. Many researchers analyzed thermal and fluid dynamic behaviors of natural convection in vertical channel systems heated symmetrically with parallel plates [20–26] and also convergent channels [27,28]. Due to asymmetric heating the opposing wall is at a different tem­ perature which causes an additional radiation transfer across the channel to the cooler surface. In this case, the radiation has an important effect and cannot be neglected [29]. This radiative effect has been detailed and highlighted in the work of Lauriat et al. [30]. Asymmetric heating in vertical channels is done by heating one side with constant heat flux and insulating the other. Heating can be done by either imposing a considered a uniform constant heat flux [31–35] or a uni­ form constant temperature [36,37]. Parallel vertical plates heated from below were also addressed in the work of Huang et al. [38]. Other configurations and shapes were also investigated such as in­ clined adiabatic channel wall [39], tilted channels [40,41], and multiple discrete heat sources [42]. One of the most important applications for free convection between vertical flat plates is the conventional solar chimney (SC) consisting of a black storage wall and a glazing cover that form an asymmetrically heated channel [43–45]. Natural convective heat transfers from an array of vertical parallel plates, forming open channels containing heated protruding elements attached to one of the walls, was analyzed both numerically and

experimentally by Avelar et al. [46]. Numerical investigation of combined surface radiation and natural convection in a differentially heated vertical square cavity were re­ ported by Balaji and Venkateshan [47]. They showed that radiation contributes to the overall heat transfer and decreases the convective component. They underlined that simple formulas that account for ra­ diation in an additive way are not adequate. In a subsequent study, Balaji and Venkateshan [48] proposed correlations for both convective and radiative heat transfers for air as the working fluid. These correla­ tions were found in fair agreement with the results of an experimental study conducted by Ramesh and Venkateshan [49]. Ridouane et al. [50] studied the impact of surface radiation inside a square two-dimensional cavity heated from below. The emissivity values of the isothermal hor­ izontal walls were different from those of the vertical adiabatic walls and were set to ε ¼ 0.05 or to ε ¼ 0.85. Most experimental investigations studying natural convection in heated open channels are based on the determination of global or local correlations linking thermal or kinematical quantities to the character­ istic Rayleigh numbers. Most of the studies were limited to heat transfer phenomena and neglected the study of the instantaneous kinematical structure of the flow [51,52]. Olsson [53] referenced different correla­ tions for the Nusselt number in an open channel heated at a uniform wall temperature (UWT) or uniform wall heat flux (UWF) by considering a laminar steady flow in both boundary conditions. The laminar regime is actually the most documented for the two main wall boundary condi­ tions: the uniform wall temperature (UWT) and the uniform wall heat flux (UWF). Similarly, Aung et al. [54] realized many numerical and experimental laminar flows studies for UWT and UWF thermal boundary conditions. Among the very few experimental studies dedicated to transient and turbulent regimes, Wirtz and Stutzman [45] conducted experiments on free convection for uniform and symmetric heat fluxes ranging from 50 to 150 W/m2 with different aspect ratios. Also, Miya­ moto [55] studied the natural convection in a 5 m vertical air channel with a UWF (104 and 208 W/m2) imposed on one wall, the other being adiabatic with aspect ratios ranging between 1/25 and 1/100. All these studies showed that the wall temperature profile along the heated wall presented an inflexion point. When the width of the channel is suffi­ ciently large, both boundary layers develops independently and the location of the inflexion point corresponds to that obtained for a single vertical heated plate. The location of the inflexion point is shifted up­ ward in the channel as the width of the channel decreases. Subsequently, Katoh et al. [56] investigated the influence of the inlet on the flow development. A quasi-flat velocity profile was measured by LDA (Laser Doppler Anemometer) at this level. Yilmaz and Fraser [57] and Yilmaz and Gilchrist [58] also used an LDA system to study a 3 m high vertical channel with one wall at imposed temperature, facing a glass with an injected heat flux density of 344 W/m2 and an aspect ratio of 1/25. They also performed instantaneous simultaneous velocity and temperature measurements and presented correlations between the Nusselt number and the Reynolds number. Moreover, they showed that the turbulent kinetic energy was relatively high at the channel inlet; it decreased until the mid-height of the channel corresponding to a relaminarization zone of the flow and increased in the upper half of the channel. Desrayaud et al. [59]. brings this diversity of works and results to the fact that the choice of the boundary conditions in entry and exit remains a debatable question. In spite of the diversity of the numerical studies carried out on the thermosiphons in free convection, the numerical resolution of this type of problem remains particularly delicate [59]. Within this framework, the obtained solutions for this kind of problems are primarily related to the boundary conditions adopted at the inlet and outlet of the channel. In real contexts, these conditions are very difficult to control by the experimenter. Indeed, the inner flow inside the channel influences the outer flow outside the channel and vice versa [60]. Few authors [62] proposed to solve this problem numerically by using a computational domain including the environment in which the channel is positioned. The boundary conditions defining this 2

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

environment replace then the boundary conditions at the conditions at the inlet and outlet of the channel when taken alone. Conversely, Uygur and Edrican [60] were satisfied to consider only the channel. In this article, two studies were developed for studying the natural convection in an asymmetrically heated double vertical facade, which was assimilated to a vertical channel. The first part concerns the natural convection in an asymmetrically heated double vertical channel with a constant heat flux were a numerical study has been developed and compared to the experimental work of the literature. In this part, the influence of the Neumann condition on the flow structure has also been demonstrated for different modified Rayleigh values. The second part concerns the study of natural convection in a vertical channel asym­ metrically heated at a constant temperature; an experimental device was realized in our laboratory allowing the thermal and aeraulic character­ ization of natural convection in a vertical air channel. Different pa­ rameters were studied and highlighted. The mean Nusselt number which is the parameter influencing the convective transfer behavior in the channel, was evaluated and compared experimentally, numerically, and with correlations available in the literature. Through these results of numerical and experimental benchmark, this article shows the interest of the means developed in our laboratory LGCgE using an original instrumentation to show how it is possible to access an interesting physical quantity which is the density of heat flow, this information makes it possible to go directly back to the other dimensionless quan­ tities available in the literature. On the other hand, this benchmark exercise complements and expands the fields of dynamic and thermal exploration in the case of natural convection in an asymmetrically heated vertical channel.

Fig. 1. Geometrical configuration.

2. Physical problem and numerical setup

channel are those of a fluid at room temperature T0 and at the driving pressure determined according to the following equation:

2.1. Problem formulation

1 P ¼ P0 þ ρU 20 2

The aim of this paper is to make a contribution to the study of natural convection in an asymmetrically heated vertical channel. In the first case where the heating is of Neumann type (constant flux), in this first case radiative heat exchanges are not considered, a numerical study is developed and confronted with the experimental results of Webb and Hill [18]. In the second case where the heating is of Dirichlet type (constant temperature), numerical and experimental studies are per­ formed, the radiative exchanges between the walls of the channel are taken into account and the air is considered transparent. The experi­ mental setup will be exposed and detailed later on in this study. Both studies are based on the Webb and Hill configuration [18]. This configuration was also selected by the French convection community research groups [62] to perform a benchmark exercise comparing the different natural convection calculation codes used by research teams. Fig. 1 shows the geometric and physical configuration of a twodimensional system that Webb and Hill have used. It is a vertical channel composed of two parallel plates; the air can flow freely through the channel via the inlet and the outlet. The two plates are composed of three parts, a first part (0
(1)

According to the Boussinesq approximation, the density ρ is a func­ tion of the temperature variation and a mean or reference density ρ0 :

ρ ¼ ρ0 ½1

βðT

(2)

T0 Þ�

On the other hand, we conjecture that (1) the air is incompressible and Newtonian, and (2) the variations of the density are taken into ac­ count only in the volume forces [1–16]. In this first case (Newman condition) the radiative exchanges in the channel are not taken into account. The dimensionless equations gov­ erning the natural convection flow in the channel with the above as­ sumptions are: 8 !!
3

∂θ ¼ ∂x

1 and u ¼ v ¼ 0

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

� ∂θ ¼ NRC Qrd H 2 � y � 0 and H � y � 1:5H : u ¼ v ¼ 0; ∂x ∂θ x ¼ 0; 0 � y � H : u ¼ v ¼ 0; ¼ 1 ∂x � ∂θ x ¼ d; H 2 � y � 1:5H : u ¼ v ¼ 0; ¼ NRC Qrd ∂x � ∂u G2 y ¼ H 2; 0 � x � d : v ¼ ¼ 0; θ ¼ 0; Pm ¼ ∂y 2

� At the inlet and the outlet ð0 � x � d and y ¼ H=2 or y ¼ 1:5HÞ :

x ¼ 0;

If !! U: n < 0 ; v ¼ ∂∂uy ¼ 0; θ ¼ 0; Pm ¼ Else,

v¼

G2 2

;! n is the unit outward normal.

∂u ∂θ ¼ ¼ 0 and P ¼ 0 ∂y ∂y

y ¼ 1:5H; 0 � x � d : v ¼

G is the inward mass flow rate defined as: R! R n :ds ¼ U:dX ; s is the inlet section through which the air G ¼ V :! s

The dimensionless parameters appearing above:

s

NRC ¼

enters the channel. The local Nusselt local number of the heated wall can be evaluated from the following equation: Nuy ¼

∂u ∂θ ¼ ¼ 0; P ¼ 0 ∂y ∂y

σT 4h H λΔT

; Qrd ¼

qrd

σ T 4h

The radiative exchanges amongst the surfaces were calculated by considering the channel as a two-dimensional enclosure consisting of 4 Gray-diffuse, vertical surfaces (the three parts of the left-hand side wall and the adiabatic right-hand side wall) and two horizontal surfaces regarded as black radiators at an effective temperature T0 and emissiv­ ity, ε ¼ 1. Since the vertical walls have the same emissivity, ε ¼ 0.98. The net radiative flux leaving the surface is given by Ref. [67]: Z Iin ! s :! n dΩ; ! s :! n <0 qrd; in ¼

hðyÞ:d 1 ¼ λ θU ðYÞ

The dimensionless overall heat transfer rate is represented by average Nusselt number and defined as: Z 1 1:5A Num ¼ NuðyÞdY A 0:5A Local Nusselt numbers will be determined for NuY¼0:5 ; NuY¼0:90 ; NuY¼0:97 , these values represent the local Nusselt numbers in the middle of the heated plate (y ¼ 0.5H), and at the limit of the heated plate (y ¼ 0.9H) and (y ¼ 0.97H). In the second study of this work, (paragraph 3.2), the radiative transfer in the channel was considered in order to study. The fluid is assumed transparent to thermal radiation leaving the channel walls. The energy equation is coupled with the equation dealing with radiant in­ terchanges amongst surfaces through the thermal boundary conditions. The experimental channel made in the laboratory was covered by a thin layer of black matt paint to control the emissivities of the plates which are estimated at 0.98 according to Refs. [63,64]. The radiative exchanges in the channel were modeled using the model of discrete ordinates. The radiative transfer equation (RTE) for an absorbing, emitting, and scattering at position ! r in the direction ! s is: Z ! ! 4 dIð r ; s Þ σT σ s 4π ! ! ! ! 0 r ;! s Þ ¼ an2 þ ða þ σ s ÞIð! þ Ið r ; s ÞΦð s : s ÞdΩ (4) ds π 4π 0

4π

qrd; out ¼ ð1

εw Þqrd; in þ n2 εw σT 4w

2.2. Numerical resolution The above conservation equations governing thermo-aeraulic flow are discretized by the finite volume method [61]. These equations can be represented by the following general equation: ! rðρ U φÞ ¼ r:ðΓ φ rφÞ þ Sφ

(5)

When such an equation is integrated over the corresponding control volume and substituted each term by the discrete values of φ in the nodal points, we get the following algebraic equation for each nodal point: X aP φnþ1 ¼ annb φnþ1 (6) P nb þ S nb ¼ E;O;N;S

where nb denotes the neighbor nodes of the point P, n indicates the number of iteration, S is a source term and f stands for the discrete value of the dependent variable over the control volume. Numerical simulations for solving the above equations are conducted via the Fluent Computational Fluid Dynamics (CFD) Software tool. The employed mesh is quadratic and structured. The convergence is checked by sevral criteria. The boundary conditions mentioned above are imposed at the different boundaries of the channel. The resolution is based on a steady 2D model using various schemes such as the upwind scheme of second order for convective terms of momentum equation, the energy equation. A SIMPLE (semi-implicit method for pressure linked equations) method proposed by Patankar and Spalding [61] was employed to solve coupling pressure and veloc­ ity. Finally, the convergence criteria is set at 10 6 for the residual error of each variable. To carry out this numerical work and verify the insensitivity of the numerical results with respect to the meshing, various grid densities were developed and tested compared to the dimensionless values of local and average Nusselt and mass flow. Table 1 presents the different

The Discrete Ordinates Method ‘DOM’ consists in evaluating such integral terms by means of a Gaussian quadrature over the solid angle. Therefore, the DOM correctly evaluates angular integrals of intensity radiation field, but it is poorly suited to follow the track of a particular photo. The discrete ordinates radiation model solves the radiative transfer equation (4) for a finite number of discrete solid angles, each associated with a vector direction fixed in the global Cartesian system (x,y). One can control the fineness of the angular discretization. Instead, the DO model transforms Equation (4) into a transport equation for radiation intensity in the spatial coordinates (x,y). Each octant of the angular space 4π at any spatial location is discretized into Nθ � Nϕ solid angles, called control angles. The angles θ and ϕ are the polar and azimuthal angles respectively, and are measured with respect to the global Carte­ sian system ðx;yÞ. In two-dimensional calculations, only four octants are solved due to symmetry, making a total of 4Nθ Nϕ directions. In this second case the boundary conditions can be written as below:

4

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

This laminar flow is fully developed beyond Ram ¼ 105, the vertical velocity profile at the entrance of the heated zone takes a more accel­ erated pace towards the heated wall than the adiabatic wall, this is well seen on the graph for Ram ¼ 107. In the middle of the heated zone (Fig. 2b), the velocity profiles at this level show a flow shape of a boundary layer type, the speeds are accelerated near the heated wall and in the boundary limit layer, then they take a very weak or nil tendency in the rest of profiles at this level (Y ¼ H). Fig. 2c has a similar tendency to Fig. 2b, dimensionless velocity values are very important near the heated wall. On the same level (Y ¼ 1.5H), velocity profiles record visible negative values for modified Rayleigh values ranging from Ram ¼ 106 to Ram ¼ 107. These negative values are the results of feeding the channel from the top; this remark will be detailed later in this work. Concerning the horizontal velocity profiles (Fig. 2d, 2e, 2f), they show comparable values at the three levels (inlet of the heated zone, middle of the heated zone and outlet of the heated zone), for modified Rayleigh values lower than 104, the hori­ zontal speeds are almost nil. On the other hand, for Ram values beyond 105, the profiles are more developed and proportional as a function of the modified Rayleigh numbers. Akbari et al. [65] stated that the vertical velocity component U, which represents the movement of fluid across the channel gap, is caused by a combination of friction and heating. At high flows, an initial rapid flow towards the center of the channel develops. Soon after the initial adjustment, the movement of the fluid to the center is greatly diminished, since the parabolic character of the upward velocity is already quite well established. As the heating becomes asymmetric, the fluid flow becomes noticeably greater at the heated plate than that at the cold plate which represents a temperature distribution close the outside fluid temperature. The comparison of the temperature results with respect to the three levels of the heated zone (Fig. 3a, 3b, and 3c) confirms the development of the boundary layer of the heated wall. This profile becomes very important in terms of temperature and thickness at Y ¼ 1.5H, while at Y ¼ 0.5H its thickness and value are small; this explains the beginning of the boundary layer. Similarly, it is noted that the temperature profiles with respect to the modified Rayleigh number, and for each level of the heated zone, dimensionless temperatures decrease when the modified Rayleigh values increase. Fig. 4a shows the temperature profiles along the left wall including the heated wall according to the different modified Rayleigh values, this profile is composed of three distinct zones: on the first adiabatic zone (0
Table 1 Grid density in function of local and mean Nusselt numbers and dimensionless mass flow. Grid

Num

Nuy¼0.5

Nuy¼0.90

Nuy¼0.97

G

Error %

25 � 150 50 � 150 50 � 200 50 � 300 50 � 400 100 � 400 200 � 400 [62] [18]

7.1179 7.0939 7.0938 7.0737 6.978 6.925 6.924 6.93 7.65

6.3816 6.3694 6.3694 6.3694 6.3694 6.3654 6.3653 6.25 6.21

5.5804 5.5710 5.5710 5.5710 5.5710 5.5710 5.5710 5.45 6.46

5.4824 5.4735 5.4735 5.4735 5.4735 5.4735 5.4735 5.37 6.99

73.3996 73.5418 73.5456 73.5605 73.6083 73.6325 73.6353 73.35 –

0,07 0,26 0,27 0,29 0,35 0,38 0,38 – –

values in function of the densities of meshes (with x high), in the same way, these values were compared with those of references [18,62] for an aspect ratio A ¼ 5 and a modified Rayleigh number Ram ¼ 105. This table shows that the local Nusselt numbers (Nuy¼0.5, Nuy¼0.90, Nuy¼0.97) give stability from a density of 50 � 150 while for the average Nusselt number (Num) the stability of the results is obtained from a density of 100 � 400. Also, the mass flow values were compared with reference [62], we note that the relative errors remain very small and <0.5%. For a better estimation of mass flow and good stability of simulation results we used a grid of density 100 � 400. 3. Results and discussion The experimental work developed in 1982 by Wirtz and Stutzman [19] and in 1989 by Webb and Hill [18] on an open vertical channel heated with constant heat flux, made it possible to make a contribution to the expression of the average Nusselt number expressed by a corre­ lation reflecting the convective transfer rate as a function of the modi­ fied Rayleigh number. However, no data on the dynamic flow was provided during the experimental study. Similarly, in 2013 research teams began to compare the solving methods implied in several com­ puter codes for modeling and simulating the problem of Webb and Hill; however, the choice of the boundary condition at the input of the channel remains a debatable question for converging the different equations governing the system [62]. In this context, at first, we wanted to make a contribution on the numerical simulation of the Webb and Hill model in a constant heat flux condition and without radiative heat transfer covering a wide range of modified Rayleigh numbers varying between 102 -107. The dynamic flow in the channel will be highlighted. Different velocity and temper­ ature profiles will be analyzed according to the modified Rayleigh number. Finally, the results of the mean Nusselt number resulting from the numerical simulations will be compared to the correlations of Webb and Hill for the verification and validation of our numerical model. In the second part, we will study the natural convection in an open vertical channel heated at constant temperature taking into account the mutual radiative exchanges present in the channel. An experimental device developed in our laboratory to meet this need will be presented and detailed in paragraph (3.2). 3.1. Study of the convective heat transfer in the channel: Neumann boundary condition Figs. 2 and 3 show the results of the velocity and temperature profiles at the inlet of the heated zone, in the middle of the heated zone and at the outlet of the heated zone, for different modified Rayleigh values. Comparing the results of vertical velocity profiles, at the entrance to the heated zone (Fig. 2a), and as a function of the modified Rayleigh number varying between 102 and 104, the velocity profile is parabolic and symmetrical with respect to the vertical axis of the channel with zero velocities near the walls. 5

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 2. Vertical velocity component profile U (left), horizontal velocity component profile V (right).

6

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 3. Temperature profiles: (a) inlet to the heated zone; (b) middle of the heated zone; and (c) outlet from the heated zone.

remain higher than the local values recorded from the middle of the heated wall to the top of the wall. Regarding the dimensionless mass flow, the values obtained are proportional to the modified Rayleigh values Ram, when Ram ¼ 107 the flow becoming more than 10 times higher than that obtained for Ram ¼ 102, which shows the intensity of the acceleration of the flow and the heating rate. These Notes are consistent with the speed profile obtained in Fig. 2 (c). In order to describe the flow sructure, streamlines for the different modified Rayleigh numbers from Ram ¼ 102 to Ram ¼ 107 are reported in Fig. 5. When the modified Rayleigh values are sufficiently low (Ram ¼ 102 to 103) the flow is in a fully developed state. The streamlines are � 107) parallel to the walls. For modified Rayleigh numbers (Ram ¼ 104 a the flow becomes boundary layer type. Air near the heated wall is accelerated along the boundary layer; it can be noted that on the upper side of the cold wall of the channel, the appearance of a return flow in the form of a supply of the channel by the outlet cold wall side (right). For a modified Rayleigh number Ram ¼ 104, a small recirculation zone adjacent to the cold wall appears at the channel exit; its size grows when the Rayleigh number increases. When Ram becomes very important from 105 to 107, the return flow feeds the channel in depth and size and

reaches the heated zone (Fig. 5d, 5e, and 5f). In order to explain the appearance of this return current, Sparrow et al. [14,15] have proposed the following description: under the effect of buoyancy, the fluid heated in the vicinity of the hot plate moves along it and affects the fluid upstream. Knowing that on each section of the channel, the constraint of conservation of the mass must be satisfied, one arrives at a level where the fluid in the channel is not enough anymore, at the same time, to feed the boundary layer which develops along the hot plate and to keep the flow. The channel feeds, then through the outlet, in the region where there is little thrust force (close to the cold plate) to maintain the flow. The threshold would correspond to Rayleigh number values high enough for the boundary layer to no longer affect the cold plate. When the critical Rayleigh number is crossed, the inversion zone develops in a V-shaped pattern. The fluid enters from the top of the channel, descends along the cold plate and then goes up, on the hot plate side, carried away by the upflow caused by the presence of the convective boundary layer. The characteristic quantities of this phenomenon are the fluid height (from the outlet) affected by the reversal and the width of this zone. This simulation campaign conducted to study the free natural 7

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 4. (a) Wall temperature T (X ¼ 0), (b) local Nusselt number along the heated plate, (c) the mean Nusselt number compared with [18].

Table 2 Local and mean Nusselt numbers and dimensionless flow in terms of Ram. Ram

Num

Nux¼0.5

Nux¼0.90

Nux¼0.97

G

102 103 104 105 106 107

1.966 3.165 4.718 6.625 10.80 17.50

1.655 2.733 4.248 6.365 9.875 18.215

1.230 2.213 3.541 5.571 8.615 16

1.212 2.157 3.465 5.475 8.475 15.723

9.72 26.58 55.16 73.63 85.30 117.08

average Nusselt numbers with the experimental work of Webb and Hill gave an excellent concordance. This validates our results and our nu­ merical models. 3.2. Study of the convective and radiative heat transfer in the channel: Dirichlet boundary condition 3.2.1. Experimental setup Before presenting the results of the second case study (vertical channel heated asymmetrically at constant temperature), we will describe the experimental device that was developed in our laboratory for this case study. Fig. 6a shows the configuration of the experimental device which is the same as the one studied in the first case but with a Dirichlet heating condition. Fig. 6b shows the experimental channel developed in the laboratory. The channel is composed of two main vertical walls. In order to control experimental conditions, the channel is introduced in a protected large volume protected from external thermal and airflow disturbances. The distances between the ends of the parallel vertical walls and the floor and/or the ceiling are such that the distur­ bances of the air flows at the inlet and at the outlet of the channel are very small [63,66]. The spacing between the two plates is adjustable

convection without radiative transfer, in an open vertical channel (Pr ¼ 0.71, Ram ¼ 102 - 107 and A ¼ 5) made it possible to highlight the different vertical and horizontal velocity profiles and temperature pro­ files in the heated zone, and thus the evolution of the flow structure in the channel. We have noticed that for a modified Rayleigh value lower than 103 the flow in the channel has a fully developed nature. When the number of modified Rayleigh value approaches 104 or more, the channel con­ tains two different flows, a bottom flow of boundary layer type and a second flow downwards from the cold wall side. For very high modified Rayleigh values, we found that the return flow develops in size and depth in the channel. Finally, the results of comparisons of numerical 8

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 5. Streamlines with different modified Rayleigh number.

Fig. 6. Vertical channel in Dirichlet condition.

with a precision of the order of a millimeter. One of the two walls is adiabatic, while the other is maintained at uniform temperature. The geometric configuration follows that used by Webb and Hill [18] under Neumann conditions in which the thermally controlled wall consisted of three parts, an adiabatic inlet zone of length H=2 ¼ 25 cm, a plate maintained at a constant temperature of length H ¼ 50 cm and finally an

adiabatic outlet zone of length H=2 ¼ 25 cm (see Fig. 6). The adiabatic wall opposite the hot wall consists of a 9 cm thick polystyrene plate, while the adiabatic inlet and outlet walls of this same hot wall are made of polystyrene of a 7 cm thickness; the thermally regulated plate is made of aluminum, 2.5 cm thick. On the heated plate, 19 heat flux sensors (fluxmeter with tangential 9

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

gradient) of dimension 2 � 10 cm2 were placed in order to obtain heat flux density measurements over the central part of the heated zone of the channel. These fluxmeters are rigorously integrated with the heated wall in order to limit the effects of wall roughness and the boundary layer separation zone (see Fig. 6b and 6c). Then, all the walls of the channel were covered with a black matte paint film which makes it possible to control the various radiative reflections. Radiative and convective heat transfers coexist in the channel. In order not to disturb the air flow between the two plates, we positioned laterally two plexiglass panes of 5 mm thick. The edge effects are negligible in the measurement zone. The gap d for the experiments presented in this second part is fixed at d ¼ 2, 4, 6, 8 and 10 cm. Therefore, the aspect ratios are respectively A ¼ 25; 12:5; 8:34; 6:25 and 5. Type T thermocouples have been regularly inserted at various points of the channel (especially on the isothermal plate) and provide temperature measurements. Several measurement campaigns have been conducted with the purpose of highlighting the evolution of the local surface heat flux as a function of the heating temperature. These measurements were compared with the numerical results ob­ tained using the Ansys-Fluent calculation code [67]. To overcome the aforementioned problem, the use of heat flux sen­ sors is suggested in this experiment. Tangential gradient fluxmeters [68] intrinsically produce temperature variations in the measurement plane. Then, these temperature gradients are detected by a large number of thermoelectric junctions connected in series. However, the sensor pro­ duces an electro-motive force (EMF) called Seebeck effect, this force is proportional to the heat flux. These fluxmeters consist of highly conductive copper surfaces which make it possible to homogenize the surface temperature and thus to provide average values of heat flux. The sensors used have a surface 10 � 2 cm2 each and their sensitivities are of the order 12 μv/W.m 2. In this experiment, 19 flux sensors were indi­ vidually calibrated using the zero flux method detailed in Ref. [69]. Then, these sensors were rigorously inserted on the heated plate and this heat exchange plate was placed in the vertical channel to form the heated wall (see Fig. 6a), these heat flux sensors were presented and detailed by the author in several configurations (refer to Ref. [66]).

Fig. 8. Total heat flux density as a function of heated height for A ¼ 12:5 with various modified Rayleigh numbers Ram

Fig. 9. Total heat flux density as a function of heated height for A ¼ 8:34 with various modified Rayleigh numbers Ram

3.2.2. Heat flux measurements Heat flux sensors have been regularly improved and used in our laboratory in particular for the study of the thermophysical properties of building materials where conductive heat transfer is the most important. In this study, we are particularly interested in thermal exchanges be­ tween a wall and its microclimatic environment. In this context, the fluxmeters proposed are used to locally measure the radiative and convective heat transfer with respect to an isothermal vertical wall

Fig. 10. Total heat flux density as a function of heated height for A ¼ 6:25 with various modified Rayleigh numbers Ram

(exchange plate Fig. 6b) forming part of the open vertical channel. In what follows, we present the results obtained experimentally and numerically for the heat flux for aspect ratios A ¼ 5; 6.25; 8.34; 12.5 and 25. The unheated walls are kept at zero flux; however, for each aspect ratio A, the isothermal wall is held at constant temperatures (θ) during each test, 4 values were selected and applied to the heated plate: 303 K, 313 K, 323 K and 342 K, which corresponds to the following modified Rayleigh number range:

Fig. 7. Total heat flux density as a function of heated height for A ¼ 25 with various modified Rayleigh numbers Ram

10

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 13. Radiative heat flux according to the height heated for ratio A ¼ 12:5 with various modified Rayleigh numbers Ram

Fig. 11. Total heat flux density as a function of heated height for A ¼ 5 with various modified Rayleigh numbers Ram

are almost stable and very close, this for each aspect ratio and for each test. In fact, at the end of the isothermal wall (Y/H � 0.8) the air be­ comes hot and its temperature approaches the temperature of the isothermal wall; that is to say that beyond (Y/H � 0.8) the temperature difference is practically low and stable. The confrontation with the nu­ merical simulations is very interesting; the numerical model of CFD developed under Fluent effectively describes the tendencies obtained experimentally. As the temperature of the isothermal plate increases, the relative difference between the measured heat flux densities and the numerically obtained heat flux densities is very small and remains less than 5%. This shows that fluxmetric, non-intrusive measurements pro­ duce excellent results. To control the radiative exchanges in the channel, we have covered the two vertical walls by a thin film of matte black paint having an emissivity of 0.98 [64]; this application allows a mutual heat exchange between the channel surfaces. The results obtained numerically show that the isothermal wall is very affected by the radiative exchanges in the channel. Figs. 12–16 present radiative heat flux densities at each fluxmeter. Note that this density is very important at the beginning of the isothermal wall and then gradually decreases along the height of the heated plate, this variation is substantially parabolic; this can be explained by relying on the Stephan-Boltzman law, since the tempera­ ture of the heated wall is constant, only the temperature of the adiabatic walls varies, that is to say, that the temperature difference between the isothermal wall and the other walls varies substantially.

2:28x102 � Ram � 8:22x105 Figs. 7–11 present the total heat flux densities (convection þ radiation) along the heated wall, according to the different modified Rayleigh numbers Ram. It should be noted that the heat flux density relative to the surface of the fluxmeter corresponds to the total heat flux, that is to say the sum of the convective and radiative experimental fluxes: φexp T

¼ φexp þ φexp φexp T R C Through Figs. 7–11, it is noted that for each aspect ratio, the measured heat flux density increases as a function of the modified Rayleigh number. It is also noted that measured heat flux densities change proportionally with respect to the modified Rayleigh number in the range of 0–475 W/m2. In each test and for each modified Rayleigh value the isothermal wall is kept at a constant temperature, this heating develops a natural boundary layer in the channel. The heat flux values measured along the heated wall inform us that the total heat flux is maximum at the beginning of the heated wall (Y/H ¼ 0.12) and then it decreases grad­ ually as a function of the height of the plate; this can explained, using Newton law on convective heat exchange. At the beginning of the heated wall (Y/H ¼ 0.12) the temperature difference between the heated wall and the fluid is very important, so the air did not have enough time to warm up and the flow becomes very important. Afterwards, this gap decreases gradually and the air warms up while moving upwards in the channel. Similarly, when (Y/H � 0.8) the measured heat flux densities

Fig. 14. Radiative heat flux according to the height heated for ratio A ¼ 8:34 with various modified Rayleigh numbers Ram

Fig. 12. Radiative heat flux according to the height heated for ratio A ¼ 25 with various modified Rayleigh numbers Ram 11

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

agreement between the different studies. 3.2.3. Flow structure To visualize the effect of the evolution of the dynamic and thermal fields on the flow structure, streamlines and isotherms are plotted over the entire computation domain as a function of the modified Rayleigh number (Figs. 18–20). The modified Rayleigh values are set both as a function of the values of the imposed heating temperature (323 K) and as a function of the aspect ratio A ¼ 25, 12.5 and 6.25. Fig. 18, shows the evolution of isotherms along the channel for a modified Rayleigh number 7:56x102 � Ram � 1:97x105 . For an aspect ratio range (5 � A � 25), the thermal field evolves significantly Fig. 18(a–c). It can be seen that the air temperatures occupy the entire central part of the channel. The profiles of the isotherms Fig. 18 (a) show curves forming a parabolic jet in full vertical axis, likewise, the boundary layers develop on both sides on the two walls of the channel. This can be explained by the simultaneous effect of the convective and radiative exchanges generated by the heating of the right wall. When the aspect ratio decreases from 12.5 to 6.25 (Fig. 19b and 19c), the central isotherms widen and occupy the channel, this gives rise to a thermosiphon that draws fresh air through these two side walls. Fig. 19 (a, b and c) present the vertical velocity field lines U for aspect ratios range A ¼ 25; 12:5; and 6:25 as a function of the modi­ fied Rayleigh number varying between 7:56x102 � Ram � 1:97x105 . For an aspect ratio A ¼ 25 (Fig. 19a), the structure of the dynamic flow at the inlet of the channel up to the entrance of the heated zone repre­ sents a zone of high vertical velocity with zero vertical velocities to­ wards the walls. These lines of vertical velocity fields deform and give an accelerated pace more towards the heated walls. When the aspect ratio decreases, that is to say the spacing increases proportionally with the modified Rayleigh, the lines of the flow field take an acceleration more toward the edges of the walls than at the center of the channel, (Fig. 19b and c), this effect can be explained by the creation of the layers that develop near the walls of right and left and reflecting the nature of the convective and radiative exchanges in the channel. The walls of the channel have been coated with a matte black color, this characteristic favors mutual radiative exchanges between the walls of the channel. This remark is further developed in figures (12)– (16) dealing with radiative flux densities. Fig. 19 (c) shows that when the aspect ratio equals A ¼ 6.25 the air in the middle of the vertical axis

Fig. 15. Radiative heat flux according to the height heated for aspect ratio A ¼ 6:25 with various modified Rayleigh numbers Ram

Similarly, we notice that the difference in temperature is very important at the beginning of the isothermal plate (Y/H ¼ 0.12) and then decreases gradually with respect of the height of the heated plate up to (YH � 0.8), which explains the parabolic curve of radiative flux densities. Also, we note that for an aspect ratio equal to 8.34; 6.25 and 5 (Figs. 14–16) when (Y/H � 0.7) the radiative flux densities increase again. Indeed, once the spacing d between the walls increases, the temperature difference between the isothermal wall and the adiabatic walls becomes weak, that is to say that the increase of the temperature of the adiabatic walls is sensitive to the spacing d. The first significant correlations of the average Nusselt number are made by Elenbaas in 1942 [12] and Bar-Cohen [16]. Their channel consisted of two vertical flat plates asymmetrically heated at constant temperature. They showed that the average Nusselt number (denoted Num) is a function of a single parameter: the modified Rayleigh number (Ram). � � Ra A; aspect ratio Num ¼ f ðRam Þ and Ram ¼ A In addition to this, we determined the average Nusselt number, based on measured flux densities. The radiative heat flux obtained numerically from the total flux is then deduced. Fig. 17 illustrates the mean Nusselt numbers in comparison with those given by by Elenbaas [12] and Bar-Cohen [16] for an open vertical channel differentially heated at constant temperature. An examination of this figure shows a perfect

Fig. 17. The average Nusselt number as a function of the modified Ray­ leigh number.Ram

Fig. 16. Radiative heat flux according to the height heated for aspect ratio A ¼ 5 with various modified Rayleigh numbers Ram

12

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 18. Isotherms for the all configurations for different Ram.

is weakly heated, from a dynamic point of view, this translates into a reduction or a braking with very low or negligible vertical speeds; that is to say that when the aspect ratio becomes very low (very large spacing d) and the modified Rayleigh becomes very important, the flow could be trapped by a return flow and zones of depression. Fig. 20 shows the streamlines for aspect ratios A ¼ 25; 12:5; and 6:25 as a function of the modified Rayleigh numbers. It can be seen that in all three cases, the channel is transformed into a thermosiphon. For the very important aspect ratio A ¼ 25 (d ¼ 2 cm) the streamlines are tight; on the other hand, when the aspect ratio becomes weak, A ¼ 6.25, the streamlines are very close to the walls of the channel and dispersed in the vicinity of the vertical central axis of the channel.

taking place in miscellaneous building engineering applications such as Trombe walls, and parietodynamic windows, and to enrich the scientific database by quantitative and qualitative results. The main findings of the first part can be summarized as follows: a) Different dimensionless temperature profiles and vertical and hori­ zontal velocities in the channel were used to identify the flow structure. We found that the temperature profile values in the channel vary inversely with respect to the modified Rayleigh num­ ber: when Ram ¼ 102, the values of dimensionless temperature pro­ file are higher than the other values of other temperature profiles (Ram> 102) in the different sections (heated inlet zone, heated middle zone and heated outlet zone). On the other hand, the evo­ lution of the temperature curve remains parabolic for all profiles. As for the dynamic results, the velocity profiles are proportional to the modified Rayleigh numbers. We found that in the heated outlet section, the values and shapes of the velocity profiles change and remain very sensitive to the modified Rayleigh number. b) For modified Rayleigh values lower than 103, it was found that the flow in the channel has a fully developed nature. Beyond Ram ¼ 104, the channel contains two different flows, the first one is a boundary layer flow type at the heated wall while the second is a downward

4. Conclusion This paper provides a contribution in natural convection in an asymmetric double open vertical façade. In the first part of this work, a numerical study was developed for constant heat flux asymmetric heating (Neumann condition) and for modified Rayleigh values ranging from 102 to 107 and an aspect ratio A ¼ 5. The major interest of this study is to develop numerical and experimental models for describing accurately natural free convection in asymmetric heating processes 13

International Journal of Thermal Sciences 152 (2020) 106288

Y. Cherif et al.

Fig. 19. U velocity for the all configurations for different Ram.

flow from the cold wall side. The average Nusselt results were compared successfully with those of Webb and Hill [18]. c) The comparison of the numerical results with those obtained by Refs. [18,62] and for a Ram ¼ 105 show a very good agreement for the local and average Nusselt numbers, same for the mass flow, these dimensionless values are proportional to the modified Rayleigh numbers. The relative mass flow error compared to the works of the literature is very small and lower than 0.5%.

In the light of the results thus obtained, we can say that: a) The effect of the Rayleigh number on the aerodynamic and thermal fields at the numerical level is observed especially at the entry of the channel and it fades along the channel; b) In this second case study, the radiative exchanges in the channel contributed in increasing the surface temperature of the adiabatic walls which avoided the return flow of the fluid by the channel exit from the side of the adiabatic walls. Likewise, we found that the radiative exchanges are proportional to the modified Rayleigh number; c) The use of the radiation model based on the discrete ordinate method makes it possible to directly obtain the radiative flux on the surface of the channel and to obtain dimensionless numbers Num. These values have been compared with excellent agreement experimentally and numerically with those available in the literature. The Nusselt number decreases as the channel aspect ratio increases; d) Finally, the flow structure obtained numerically through isotherms and streamlines have shown boundary layers developed near the heated surface and adiabatic surfaces on the two walls of the chan­ nel. We explained this phenomenon by the simultaneous effect of the convective and radiative exchanges generated by the heating of the adiabatic wall.

In the second part of this article, we carried out an experimental and numerical study on ascending natural convection in an open channel whose wall is heated asymmetrically at a constant temperature (Dirichlet condition) and the other is adiabatic. In this case, the radiative exchanges are mutual between the different surfaces of the channel. The fluid is considered as not participating in the radiation. To ensure a fully developed flow at the entrance to the heated zone, an adiabatic inlet and outlet representing the half height of the heated zone have been developed. An original instrumentation was developed in our laboratory under the same conditions of Webb and Hill to determine different quantitative results for temperature and heat flow in the channel. This has direct access to the dimensionless quantity (Num, Nuy). The results were presented in terms of heat flux along the heated wall and the average Nusselt number for a modified Rayleigh number varying be­ tween 2 � 102 and 8 � 105 and for different aspect ratios.

14

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288

Fig. 20. Streamlines for the different Ram and aspect ratios.

Nomenclature A a ! d d g G H Ir n NCR Nu Nuy Num p P Pr q_ qrd Qrd ! r Ra Ram ! s’

s T Th T0 U; V u; v x, y

aspect ratio, A ¼ Hd absorption coefficient direction vector channel with (m) gravity acceleration (m.s 2) dimensionless flow height of the heated wall (m) total irradiation intensity, W/sr refractive index radiation conduction number Nusselt number local Nusselt number average Nusselt number pressure (Pa) d2 dimensionless pressureP ¼ ðpþρραgyÞ 2 ν Prandtl number Pr ¼ α heat flux (W.m 2) net radiative heat flux (W.m 2) dimensionless net radiative flux position vector 3 Rayleigh number, Ra ¼ gβΔTd νa modified Rayleigh number, Ram ¼ Ra A

scattering direction vector path length local temperature (K) heat temperature plate (K) inlet temperature (K) vd dimensionless vertical and horizontal velocity, U ¼ ud α; V ¼ α vertical and horizontal velocity (m.s 1) horizontal and vertical coordinates (m) 15

Y. Cherif et al.

X, Y

dimensionless Cartesian coordinates, X ¼ xd; Y ¼ dy

Greek symbols α thermal diffusivity, m2.s 1 β thermal expansion coefficient, K ΔT

International Journal of Thermal Sciences 152 (2020) 106288

1

difference temperature, K, ΔT ¼ qd λ _

ε

dimensionless temperature, θ ¼ TΔTT0 thermal conductivity, W.m 1.K 1 density, kg.m 1 phrase function heat flux density, W.m 2 scattering coefficient Stefan-Boltzmann constant (5.669 � 10 solid angle emissivity of the radiative surface

Subscripts 0 m y exp C R T w

and Superscripts reference value average local experimental convective component radiative component global component wall

θ λ

ρ Φ φ

σS σ Ω’

8

W/m2.K4)

References

[20] B. Andreozzi, O. Manca Buonomo, Thermal and fluid dynamic behaviors of natural convection in symmetrical heated channel – chimney systems, Int. J. Numer. Methods Heat Fluid Flow 20 (7) (2010) 811–833, https://doi.org/10.1108/ 09615531011065584. [21] O. Manca Andreozzi, Radiative effects on natural convection in a vertical channel with an auxiliary parallel plates, Int. J. Therm. Sci. 97 (2015) 41–55, https://doi. org/10.1016/j.ijthermalsci.2015.05.013. [22] O. Manca Auletta, Heat and fluid flow resulting from the chimney effect in a symmetrically heated vertical channel with adiabatic extensions, Int. J. Therm. Sci. 41 (2002) 1101–1111, https://doi.org/10.1016/S1290-0729(02)01396-0. [23] C.F. Kettleborough, Transient laminar free convection between heated vertical plates including entrance effects, Int. J. Heat Mass Tran. 15 (1972) 883–896. [24] W. Lewandowski, M. Ryms, H. Denda, Infrared techniques for natural convection investigations in channels between two vertical, parallel, isothermal and symmetrically heated plates, Int. J. Heat Mass Tran. 114 (2017) 958–969. [25] W. Lewandowski, M. Ryms, H. Denda, Natural convection in symmetrically heated vertical channels, Int. J. Therm. Sci. 134 (2018) 530–540. [26] A. Campo Morrone, O. Manca, Optimum plate separation in a vertical parallelplate channel for natural convection flows: incorporation of large spaces at the channel extremes, Int. J. Heat Mass Tran. 40 (1997) 993–1000, https://doi.org/ 10.1016/0017-9310(96)00197-4. [27] N. Bianco, L. Langellotto, O. Manca, V. Naso, Numerical analysis of radiative effects on natural convection in vertical convergent and symmetrically heated channels, Numer. Heat Tran. Part A: Appl. 49 (4) (2006) 369–391. [28] N. Bianco, L. Langellotto, O. Manca, S. Nardini, Radiative effects on natural convection in air in symmetrically heated convergent channels, Int. J. Heat Mass Tran. 53 (2010) 3513–3524, https://doi.org/10.1016/j. ijheatmasstransfer.2010.04.012. [29] A.A. Dehghan, M. Behnia, Combined natural convection- conduction and radiation heat transfer in a discretely heated open cavity, J. Heat Tran. 118 (February 1996). [30] G. Lauriat, G. Desrayaud, Effect of surface radiation on conjugate natural convection in partially open enclosures, Int. J. Therm. Sci. 45 (2006) 335–346. [31] J. Hernandez, B. Zamora, Effects of variable properties and non-uniform heating on natural convection flows in vertical channels, Int. J. Heat Mass Tran. 48 (2005) 793–807, https://doi.org/10.1016/j.ijheatmasstransfer.2004.09.024. [32] J.R. Carpenter, D.G. Briggs, V. Sernas, Combined radiation and developing laminar free convection between vertical flat plates with asymmetric heating, J. Heat Tran. (1976) 95–100. [33] G.C. Vliet Hall, T.L. Bergman, Natural convection cooling of vertical rectangular channels in air considering radiation and wall conduction, J. Electron. Packag. 121 (1999) 75–84. [34] Y.H. Wong Moutosoglou, Convection-radiation interaction in buoyancy induced channel flow, J. Thermophys. Heat Tran. 3 (2) (1989) 175–181. [35] H.F. Nouan� egu� e, E. Bilgen, Heat transfer by convection, conduction and radiation in solar chimney systems for ventilation of dwellings, Int. J. Heat Fluid Flow 30 (1) (2009) 150–157. [36] W.-S. Fu, W.-S. Chao, T.-E. Pen, C.-G. Li, Flow downward penetration of vertical parallel plates natural convection with an asymmetrically heated wall, Int. Commun. Heat Mass Tran. 74 (2016) 55–62.

[1] A. Bejan, Convection Heat Transfer, Wiley-InterSciences, 1994. [2] F.P. Incropera, D.P. Dewitt, Fundamentals of Heat and Mass Transfer, fourth ed., J. Wiley & Sons, New York, 1996. [3] 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) 1101–1111. [4] O. Manca, M. Musto, V. Naso, Experimental analysis of asymmetrical isoflux channel-chimney systems, Int. J. Therm. Sci. 42 (2003) 837–846. [5] O. Manca, M. Musto, V. Naso, Experimental investigation of natural convection in an asymmetrically heated vertical channel with an asymmetric chimney, J. Heat Tran. 127 (2005) 888–896. [6] A. Auletta, O. Manca, B. Morrone, V. Naso, Heat transfer enhancement by the chimney effect in a vertical isoflux channel, Int. J. Heat Mass Tran. 44 (2001) 4345–4357. [7] J. Hernandez, B. Zamora, Effects of variable properties and non-uniform heating on natural convection flows in vertical channels, Int. J. Heat Mass Tran. 48 (2005) 793–807. [8] B. Zamora, L. Hern� andez, Influence of variable property effects on natural convection flows in asymmetrically-heated vertical channels, Int. Commun. Heat Mass Tran. 24 (1997) 1153–1162. [9] B. Zamora, A.S. Kaiser, A. Viedma, On the effects of Rayleigh number and inlet turbulence intensity upon the buoyancy-induced mass flow rate in sloping and convergent channels, Int. J. Heat Mass Tran. 51 (2008) 4985–5000. [10] A. Andreozzi, B. Buonomo, O. Manca, Transient natural convection in a symmetrically heated vertical channel at uniform wall heat flux, Numer. Heat Tran. A 55 (2009) 409–431. [11] A.K. Da Silva, L. Gosselin, Optimal geometry of L and C-shaped channels for maximum heat transfer rate in natural convection, Int. J. Heat Mass Tran. 48 (2005) 609–620. [12] W. Elenbaas, Heat dissipation of parallel plates by free convection, Physica IX 39 (1) (1942) 1–28. [13] E.M. Sparrow, P.A. Bahrami, Experiments on natural convection from vertical parallel plates with either open or closed edges, J. Heat Tran. 102 (1980) 221–227. [14] E.M. Sparrow, N. Cur, Turbulent heat transfer in a symmetrically or asymmetrically heated flat rectangular duct with flow separation at inlet, J. Heat Tran. 104 (1982) 82–89. [15] E. M Sparrow, G. M Chrysler, L. F Azevedo, Observed flow reversals and measuredpredicted Nusselt numbers for natural convection in a one-sided heated vertical channel, J. Heat Tran. 106–325 (1984). [16] A. Bar-Cohen, W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled, parallel plates, J. Heat Tran. 106 (1984) 116–123. [17] S.W. Churchill, R. Usagi, A general expression for the correlation of rates of transfer and other phenomena, AIChE J. 18 (6) (1972) 1121–1128. [18] B.W. Webb, D.P. Hill, High Rayleigh number laminar natural convection in an asymmetrically heated vertical channel, J. Heat Tran. 111 (1989) 649–656. [19] R.A. Wirtz, R.J. Stutzman, Experiments on free convection between vertical plates with symmetric heating, J. Heat Tran. 104 (1982) 501–507.

16

Y. Cherif et al.

International Journal of Thermal Sciences 152 (2020) 106288 [52] J. Vareilles, Etude des transferts de chaleur dans un canal vertical diff� erentiellement chauff�e: application aux enveloppes photovoltaïques-thermique, Lyon 1 University, 2007. [53] C.O. Olsson, Prediction of Nusselt number and flow rate of buoyancy driven flow between vertical parallel plates, ASME J. Heat Transf. 126 (2004) 97e104. [54] W. Aung, L.S. Fletcher, V. Sernas, Developing laminar free convection between vertical flat plates with asymmetric heating, Int. J. Heat Mass Tran. 15 (1972) 2293e2308. [55] M. Miyamoto, Turbulent free convection heat transfer from vertical parallel plates, J. Heat Tran. 4 (1986) 1593e1598. [56] Y. Katoh, M. Miyamoto, J. Kurina, S. Kaneyasu, Turbulent free convection heat transfer from vertical parallel plates, effect of entrance bell-mouth shape, JSME Int. J. 34 (1991) 496e501. [57] T. Yilmaz, S.M. Fraser, Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating, Int. J. Heat Mass Tran. 50 (2007) 2612e2623. [58] T. Yilmaz, A. Gilchrist, Temperature and velocity field characteristics of turbulent natural convection in a vertical parallel-plate channel with asymmetric heat flux, Heat Mass Tran. 43 (2007) 707e719. [59] G. Desrayaud, R. Bennacer, J.P. Caltagirone, E. Chenier, A. Joulin, N. Laaroussi, K. et Mojtabi, VIII� eme Colloque Interuniversitaire Franco-Qu�ebecois sur la thermique des syst� emes, 2007, pp. 389–394. [60] S. Uygur, N. Egrican, Energy conversion and management 37 (5) (1996) 505–520. [61] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, 1980. [62] G. Desrayaud, E. Ch� enier, A. Joulin, A. Bastide, B. Brangeon, J.P. Caltagirone, Y. Cherif, R. Eymard, C. Garnier, S. Giroux-Julien, Y. Harnane, P. Joubert, N. Laaroussi, S. Lassue, P. Le Qu� er� e, R. Li, D. Saury, A. Sergent, S. Xin, A. Zoubir, Benchmark solutions for natural convection flows in vertical channels submitted to different open boundary conditions, Int. J. Therm. Sci. 72 (2013) 18–33. [63] Y. Cherif, A. Joulin, L. Zalewski, S. Lassue, Contribution � a l’� etude exp�erimentale de la convection naturelle entre deux plaques planes verticals diff� erentiellement chauff�ees, in: VIII�eme Colloque Interuniversitaire Franco Qu�ebecois sur la thermique des syst� emes, 2007, pp. 131–136. [64] Y. Cherif, A. Joulin, L. Zalewski, S. Lassue, Superficial heat transfer by forced convection and radiation in a horizontal channel, Int. J. Therm. Sci. 48 (2009) 1696–1706. [65] H. Akbari, T.R. Borger, Free convective laminar flow within the trombe wall channel, in: Solar Energy, vol. 22, Pergamon Press Ltd., 1979, 165-17. [66] Y. Cherif, PhD, Universit�e d’Artois, France, 2007. [67] Ansys-Fluent, Documentation, Fluent Inc, 2015. [68] D. Leclercq, P. Thery, Apparatus for simultaneous temperature and heat flow measurement under transient conditions, Rev. Sci. Instrum. 54 (1983) 374–380. [69] D.L. Marinoski, S. Guths, F.O.R. Pereira, R. Lamberts, Improvement of a measurement system for solar heat gain through fenestrations, Energy Build. 39 (2007) 478–487.

[37] T.S. Chang, T.F. Lin, On the reversed flow and oscillating wake in an asymmetrically heated channel, Int. J. Numer. Methods Fluid. 10 (1990) 443–459. [38] M. Huang, J. P Yeh, H.J. Shaw, Problem of conjugate conduction and convection between finitely vertical plates heated from below, Comput. Struct. 37 (5) (1990) 823–883. [39] N. Kimouche, Z. Mahri, A. Abidi-Saad, C. Popa, G. Polidori, C. Maalouf, Effect of inclination angle of the adiabatic wall in asymmetrically heated channel on natural convection: application to double-skin façade design, J. Build. Eng. 12 (2017) 171–177. [40] O. Manca, S. Nardini, Composite correlation for air natural convection in tilted channels, Heat Tran. Eng. 20 (3) (1999) 64–72, https://doi.org/10.1080/ 014576399271439. [41] O. Manca, S. Nardini, Thermal design of uniformly heated inclined channels in natural convection with and without radiative effects, Heat Tran. Eng. 22 (2) (2001) 13–28, https://doi.org/10.1080/01457630118178. [42] C.G. Rao, Interaction of surface radiation with conduction and convection from a vertical channel with multiple discrete heat sources in the left wall, Numer. Heat Tran. Part A: Appl. 52 (2007) 831–848. [43] I. Zavala-Guill�en, J. Xam� an, I. Hern� andez-P�erez, et al., Ventilation potential of an absorber-partitioned air channel solar chimney for diurnal use under Mexican climate conditions, Appl. Therm. Eng. 149 (2019) 807–821. [44] I. Zavala-Guill�en, J. Xam� an, I. Hern� andez-P� erez, et al., Numerical study of the optimum width of 2a double air channel solar chimney, Energy 147 (2018) 403–417. [45] I. Zavala-Guillen, J. Xaman, G. Alvarez, et al., Computational fluid dynamics for modeling the turbulent natural convection in a double air-channel solar chimney system”, Int. J. Mod. Phys. C 27 (2016) 1650095. [46] Averlar Ana Cristina, Marcelo Moreira Ganzarolli, Natural convection in an array of vertical channels with two-dimensional heat sources: uniform and non-uniform plate heating, Heat Tran. Eng. 25 (7) (2004) 46–56. [47] C. Balaji, S. Venkateshan, Interaction of surface radiation with free convection in a square cavity, Int. J. Heat Fluid Flow 14 (3) (1993) 260–267. [48] C. Balaji, S. Venkateshan, Correlations for free convection and surface radiation in a square cavity, Int. J. Heat Fluid Flow 15 (3) (1994) 249–251. [49] N. Ramesh, S. Venkateshan, Effect of surface radiation on natural convection in a square enclosure, J. Thermophys. Heat Tran. 13 (3) (1999). [50] E. Ridouane, M. Hasnaoui, A. Campo, Effects of surface radiation on natural convection in a Rayleigh-benard square enclosure: steady and unsteady conditions, Heat Mass Tran. 42 (3) (2006) 214e225. [51] M. Fossa, C. Menezo, E. Leonardi, Experimental natural convection on vertical surfaces for building integrated photovoltaic (BIPV) applications, Exp. Therm. Fluid Sci. 32 (2008), 980e990.

17