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Numerical investigation of turbulent mixed convection in an open cavity: Effect of inlet and outlet openings
International Journal of Thermal Sciences 116 (2017) 103e117
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Numerical investigation of turbulent mixed convection in an open cavity: Effect of inlet and outlet openings Lounes Koufi a, b, c, Zohir Younsi b, c, Yassine Cherif a, c, Hassane Naji a, c, * Univ. Artois, Laboratoire de G enie Civil et g eo-Environnement (LGCgE - EA 4515), F-62400 B ethune, France FUPL, Hautes Etudes d'Ing enieur, LGCgE - EA 4515, 13 Rue de Toul, F-59000 Lille, France c Univ. Lille Nord de France, LGCgE - EA 4515, F-59000 Lille, France a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 September 2016 Received in revised form 21 December 2016 Accepted 11 February 2017
This paper deals with the numerical investigation of heat transfer by mixed convection inside ventilated cavities with supply and exhaust slots, and filled with air under a steady and turbulent flow regime. Four configurations, rated A, B, C and D are considered here, according to the position of the inlet and outlet air ports: A, the inlet is on the top of the left vertical wall, while the outlet is on the bottom of the right vertical wall; B, the inlet is on the bottom of the left vertical wall and the outlet at the top of the opposite wall; C, the two slots are on the same side, i.e. the inlet is at the bottom and the outlet at the top of the left vertical wall, and D, the inlet is at the top and the outlet at the bottom of the left vertical wall. The bottom of the cavity is kept at a temperature TH and other walls are fixed at a temperature TC , with TH > TC . The cavity is provided with two slots: an inlet slot for introducing fresh air, and an outlet slot to extract hot air. The main aim sought here is to analyze the ventilation efficiency for temperature distribution, and fix the best configuration providing the thermal comfort targeted. We also address the influence of heating on the behavior of flow and thermal comfort, while considering different Rayleigh numbers ranging from 6:4 108 to 3:2 109 . Numerical studies have been yet devoted to these configurations, using RANS simulations. The RNG k-ε turbulence model has been adopted for the turbulence closure, and the set of governing equations was then numerically solved via the finite volume method. The SIMPLEC algorithm was associated to ensure the pressure-velocity coupling. In terms of results achieved, the configuration D provides a better ventilation effectiveness for temperature distribution εT and ensures an even temperature in the occupied zone. As for configurations A and C, they maintain an acceptable level of heat and can be used in winter period to ensure good indoor air quality, while configuration B provides an efficiency close to unity and can be used to insure indoor air quality in temperate climate zones. © 2017 Elsevier Masson SAS. All rights reserved.
Keywords: Numerical simulation Convection Mixing ventilation Ventilation efficiency Thermal comfort Turbulence modeling
1. Introduction Today, many people spend more time indoors (homes, schools, offices, transports, stores, etc.) [1]. Thereby, the quality of indoor environment must ensure the occupants' requirements in terms of thermal comfort and indoor air quality that can adversely affect their health. In addition, the indoor environment is affected by rising energy costs. Besides, since the 1973 oil crisis, thermal insulation has been greatly increased to reduce heat loss and greenhouse emissions. To this, exchanges between the outside and
nie Civil et ge o-Environ* Corresponding author. Univ. Artois, Laboratoire de Ge thune, France. nement (LGCgE - EA 4515), F-62400 Be E-mail address: [email protected] (H. Naji). http://dx.doi.org/10.1016/j.ijthermalsci.2017.02.007 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.
inside buildings were greatly reduced. Such confinement led to discomfort of the occupants and caused damage to the building structure. To improve the quality of the environment, ventilation has proved to be one of the most promising solutions. Its principle is to renew sufficiently and permanently stale air by fresh air. For that, investigators and designers innovative offer different strategies of ventilation to guarantee the comfort for occupants. Note that a ventilation system can be used to ensure good indoor air quality and thermal comfort (heating or air conditioning). There are two mechanical ventilation modes that are widely used. These are based on the fresh air intake velocity, namely the mixing ventilation where fresh air is blown at high velocities (turbulent jet air), and the displacement ventilation where air is introduced at low velocities [2]. Convective motion can help to determine the quality of the indoor air circulation in rooms and passenger cabins (cars,
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Nomenclature Cp g h H, L k n Nu p Pr Q Ra Re T TC TH T0 T0 u, w u *, w * u0 , w0 U X, Z X*, Z*
Specific heat (Jkg1K1) Gravity acceleration (ms2) Height of the gap for the inlet or outlet air (m) Cavity's height and width (m) Turbulent kinetic energy (m2s2) Normal vector (m) Nusselt number Fluid pressure (Pa) Prandtl number, Pr ¼ no/ao Heat (W) Rayleigh number, Ra ¼ ðro g bDTL3 Þ=ðmo ao Þ Reynolds number, Re ¼ ðro Uin hin Þ=mo Fluid temperature (K) Cold temperature (K) Hot temperature (K) Reference temperature, To ¼ ðTH þ Tc Þ=2 ðKÞ Fluctuating temperature (K) Horizontal and vertical velocities (ms1) Dimensionless velocity components, pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðu* ; w* Þ ¼ ðu; wÞ= g bDTL Fluctuating velocity components (ms1) Velocity modulus (ms1) Horizontal and vertical coordinates (m) Dimensionless coordinates, ðX * ; Z * Þ ¼ ðX; ZÞ=L
planes, etc.). It also plays a significant role in heat exchangers and passive cooling devices for electronic equipments and computer chips. It should be noted that, in a ventilated room, natural convection (driven by buoyancy forces) and/or forced convection (due to the external forces) can take place. The combination of these modes gives rise to a mixed convection problem. To handle such a problem, numerical and experimental studies have been carried out to understand the behavior of airflow by mixed convection and the temperature distribution within ventilated rooms. In the present work, we consider a ventilated cavity often taken as a first model [3,4]. In the following, we briefly describe some works related to the subject dealt here. Blay et al. [5] studied numerically and experimentally mixed convection inside ventilated square cavity filled with air in turbulent and steady regime. This cavity is heated from below. The authors have performed simulations for different Froude number using two low-Reynolds number k-3 models [6,7]. Their results showed the existence of a critical Froude number for which the flow inside the cavity changes direction. Chen [8] evaluated the performance of five k-3 turbulence models on the heat transfer in a ventilated cavity. The author has found that the numerical predictions of average velocities are satisfactory. In addition, he noted that fluctuations are less estimated compared to available experimental results. Also, he noticed the RNG k-3 [9] model provides more accurate results than the standard k-3 model [10]. Raji and Hasnaoui [11e13] numerically analyzed the heat transfer by mixed convection inside ventilated cavities filled with air and subjected to heat flux conditions. They evaluated the influence of radiation on heat transfer by mixed convection in a rectangular cavity with an aspect ratio of 2. They found that radiation promotes the temperature's homogenization, and reduces the maximum temperature inside the cavity. Singh and Sharif [14] conducted a numerical study of mixed convection in a 2D rectangular cavity differentially heated
Greek symbols a Thermal diffusivity (m2s1) ta Local age of fluid (s) b Thermal expansion coefficient (K1) DT Characteristic temperature difference, DT ¼ ðTH TC ÞðKÞ 3 Turbulent energy dissipation (m2s3) 3T Ventilation effectiveness for temperature distribution l Thermal conductivity (Wm1K1) m Dynamic viscosity (kgm1s1) n Kinematic viscosity (m2s1) r Density (kgm3) d Boundary layer thickness (m) s Standard deviation Superscripts/subscripts C Cold H Hot in Inlet l Local m Mean max Maximum min Minimum out outlet T Thermal t Turbulent O Reference
in laminar and steady flow regime, and equipped with inlet and outlet slots. They have shown that the position of the slots can be crucial in terms of cooling efficiency. Considering the same regime, Rahman et al. [15] studied numerically the influence of key parameters as the Prandtl (Pr), Reynolds (Re) and Richardson (Ri) numbers on the mixed convection in a ventilated square cavity. The authors found that, for low values of Ri, the rate of heat transfer is minimum along the hot wall, and for high values of Pr, the Nusselt number is enhanced. Also, the increase of Ri avoids the separation of flow while exhibiting a linear behavior. They concluded that heat transfer is strongly influenced by high values of both Pr and Ri. Radhakrishnan et al. [16] performed experimental and numerical study of mixed convection in a ventilated cavity with a heat generator. They studied the influence of the size and position of heat sources on the heat transfer performance. They found that simulations corroborate experimental results. They stated that the size, position and inclination angle of the heat source influence greatly the thermal behavior inside the room. n et al. [3] numerically investigated conjugated heat Xama transfer in a ventilated cavity under turbulent flow regime. The authors sought to determine the optimal ventilation scenario in terms of air conditioning. They found that, for Reynolds numbers ranging from 5:103 to 104 , it is the cavity for which the exit is located at the top right which ensures temperature and velocity values close to those recommended by the standard ASHRAE 55 [17]. Ezzouhri et al. [18] took over the cavity considered by Blay et al. [5] to evaluate the LES model [19] and determine the exact value of the Froude number for which two different solutions exist. Comparisons with experimental data [5] have shown that such a model provides acceptable predictions. Hinojosa and Gortari [20] conducted a numerical study of heat transfer by laminar natural convection in an isothermal open cubic cavity. The authors presented the results for different Ra numbers (104 Ra 107 ).
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
Radhakrishnan et al. [21] performed a numerical and experimental study to optimize the turbulent heat transfer in a ventilated cavity with different heaters. They confronted and validated their results ~ oz et al. [22] invessimulations with experiments. Rodríguez-Mun tigated numerically the influence of Ra and Re numbers on the heat transfer by mixed convection under turbulent and steady flow regime inside a ventilated cavity. They observed that the fluid movement caused by a dominant natural convection in case of the considered boundary conditions. Also, they noted that increasing the inlet air velocity reduces the cavity temperature, and that a thermal stratification occurs in the upper part of the cavity for a particular value of the inlet velocity. Arce et al. [23] conducted a numerical study of the conjugate heat transfer by mixed convection and conduction in a two-dimensional ventilated cavity having one glass wall under laminar and steady flow. They scrutinized the influence of Ra and Re numbers and air outlet position. They have also found that it is the configuration with an air outlet near the glass panel which evacuates sufficient amount of heat entering the cavity. Di Liu et al. [24] performed a numerical study of heat and mass transfer by mixed convection in a ventilated cubic cavity under a turbulent regime. The cavity is provided with an air conditioner window having three slots for blowing, extraction and recirculation. They analyzed indoor air quality and thermal comfort as function of inlet air velocity, recirculation, fresh air ratio (r) and buoyancy forces. They stated that contaminant reduction rates can be achieved either by increasing r or Re, by increasing the supply air flow, or by decreasing the power of the heating source. Rodríguez~ oz et al. [25] examined the effect of heat generated by a human Mun being exposed to mixed turbulent convection with radiation and emitting a generation of CO2 through breathing in a ventilated room. They found that the ventilation reduces the average temperature room of about 4 C to 5:5 C, while radiation causes an increase in average temperature between 0:2 C and 0:4 C. Serrano-Arellano et al. [26] numerically analyzed conjugated heat and mass transfer of an air/CO2 mixture inside a ventilated cavity under turbulent flow regime. They scrutinized the ventilation effect on the thermal behavior and the indoor air quality inside the cavity as function of the outlet air location. In addition, they observed that the optimum location for situating the outlet flow of the mixture is ~ oz and Hinojosa [27] perclose to the heat source. Rodríguez-Mun formed a numerical study of heat transfer by mixed convection with radiation in a 3D ventilated room under a regime. They considered three ventilation strategies depending on the position of the air inlets. They found that the heat transfer due to radiation is approximately 50% for all configurations. As expected, the heat transfer coefficient and the temperature distribution efficiency vary notably depending on the position of the air inlets. Recently, ~ oz et al. [28] and Hinojosa et al. [29] studied Rodríguez-Mun numerically and experimentally the heat transfer by mixed convection inside a ventilation cubic cavity filled with air in turbulent and steady regime. The authors found that the boundary layer thickness in the vicinity of the heated and isothermal walls varies between 0:01m and 0:03m. Measured and computed temperatures exhibit discrepancies of 2% for Ra ¼ 2:7 108 and 3% for Ra ¼ 4:5 108 . Moreover, the predicted convective coefficients are closer to those obtained experimentally for Re ¼ 31466 with deviations of 0.8% and 0.3% for Ra ¼ 2:7 108 and Ra ¼ 4:5 108 , respectively. According to the above literature survey, there is little research on the ability of a ventilation system to ensure a better temperature distribution in a medium scale cavity with a weakly turbulent mixed convection, though the problem under consideration is of great practical interest. Thereby, it seems necessary to deepen our knowledge on gains and/or losses of thermal loads within buildings equipped with adequate ventilation. The aim of this study is
105
twofold: i) to carry out a numerical analysis of heat transfer by mixed convection inside ventilated square cavity heated by the floor, and ii) to analyze the influence of the heating intensity and position of the inlet and outlet air on the behavior of the flow and the temperature distribution to determine the best ventilation configuration in terms of temperature distribution efficiency. Note that another study has been carried out to test different mixing and displacement ventilation strategies. The main aim was to propose an effective strategy that takes less time to evacuate the contaminant and to ensure a good indoor air quality. For more details, see Ref. [30]. As possible applications, it can be stated that heating floors have been adopted over the last decades in residential or tertiary buildings, individual or collective houses, offices, etc. They have higher efficiency due to their ability to reduce the heat loss of distribution [31], making it possible to couple them with heat pumps, condensing boilers or thermal solar system [32]. Recall that the central ingredient to be modeled is a square cavity heated from below. Four ventilated cavities, with supply and exhaust slots differently placed have been studied here. These are denoted A, B, C and D, respectively. For the cavity A, the inlet of fresh air is placed at the top of the left vertical wall and the air stale outlet is installed at the bottom of the right vertical wall; for the cavity B, the air inlet is at the bottom of the left vertical wall and the air outlet is at the top of the opposite wall; for the cavity C, the air inlet is at the bottom of the left vertical wall and outlet at the top, while for the cavity D, the air inlet is at the top and the outlet at the bottom wall. Here, all these ventilated cavities have been studied with a CFD model in 2D, and the turbulence has been handled via the model RNG k-3 [10] using scStream commercial code [33]. The remainder of this article is organized as follows. The next section describes the physical model and the considered configurations. Section 3 presents the governing equations supplemented by the considered boundary conditions. In Section 4, key parameters of the heat transfer and thermal comfort are introduced. Section 5 is devoted to a brief description of the numerical approach while emphasizing its validation. In Section 6, results from simulations are presented and discussed. Finally, the major conclusions are outlined in Section 7.
2. Physical setup The physical model considered here is illustrated in Fig. 1 along with the Cartesian coordinate system. This is a ventilated square cavity of dimensions 1:04 1:04m2 filled with air. Such a
Fig. 1. Physical model of the ventilated cavity.
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configuration has been the subject of several numerical and experimental works [5,18,34,35] thanks to the availability of interesting and accurate experimental data that can validate the numerical simulations implemented. Initially, the air inside the cavity is at a constant temperature TC ; the bottom wall is maintained at a constant temperature TH (TH > TC ), while the other walls are maintained at a uniform temperature TC . Fresh air is introduced into the cavity via an inlet port at the temperature TC and at a velocity of 0:57ms1 and extracted naturally through an outlet port. It is the difference in temperature prevailing inside that induces natural convection, whereas the blowing of fresh air generates forced convection. It should be noted that the heating temperature was varied between 293K (Ra ¼ 6:4 108 ) and 313K (Ra ¼ 3:2 109 ) with an increment of 5K. The air inlet velocity is fixed at 0:57ms1 , which leads to Re ¼ 683. The four configurations considered according to the position of the inlet and outlet air gaps are shown in Fig. 2.
thermal diffusivity, Prt being equal to 0.9 [33]. Note that the eddy viscosity (mt ) is attached to the turbulent kinetic energy (k) and its dissipation rate (ε) by the Kolmorogov-Prandtl relationship:
.
mt ¼ Cm rk2 ε
(6)
Such a system (1e5) being open, it is needful to close it considering transport equations for the turbulent kinetic energy (k) and its dissipation rate (ε). These can be written, in their tensor formulation, as Eqs. (7) and (8):
! V rk U ¼ V$½ðm þ ðmt =sk ÞÞVk þ Gs þ Gt þ rε
(7)
! V rε U ¼ V$½ðm þ ðmt =sε ÞÞVε þ C1 ðGs þ Gt Þ 1 þ C3 Rf ðε=kÞ . C2 rε2 k (8)
3. Mathematical formulation The conservation equations governing a thermo-ventilation flow in a ventilated room are the continuity, momentum, and energy (heat) equations, which are based on the following conjectures: the fluid is Newtonian and incompressible, and the flow is turbulent and steady, and viscous. Dissipation, pressure forces' work, and radiation are assumed to be negligible. In addition, no heat source is inside the cavity, and physical properties of the fluid are constant except the buoyancy force term, which is modeled via the Boussinesq approximation. According to the aforementioned assumptions, these timeaveraged equations, in their tensor form, can be read hereafter.
V$ðrUÞ ¼ 0
(1)
! ! ! ! V r U 5 U ¼ Vp þ V$ mV U ru0i u0j þ rg bT ðT T0 Þ e3 (2) ! V rCp T U ¼ V$ lVT rCp u0i T 0
(3)
where ru0i u0j is the Reynolds stress tensor and ru0i T 0 is the turbulent heat flux, which are modeled as follows:
ru0i u0j ¼ mt Sij þ ð2=3Þrkdij
(4)
ru0i T 0 ¼ at VT
(5)
where k ¼ u0i u0i =2 is the turbulent kinetic energy, dij is the Kronecker tensor, mt is the eddy viscosity, and at ¼ mt =Prt is the turbulent
! with: Gs ¼ mt Sij V U is the producing shear rate of turbulent kinetic energy, Gt ¼ gi bT ðmt =Prt ÞVT is the buoyancy generation rate of turbulent kinetic energy, and sk , sε , C1 , C2 , C3 and Cm are the model constants whose values are given in Table 1. It should be noted that the RNG k-3 model can be derived from the instantaneous NaviereStokes equations, using the so-called ‘‘renormalization group’’ or RNG approach. It turned out that it is computationally efficient and stable compared to more complicated Reynolds stress models (RSM), which have several additional equations to be solved. The boundary conditions for the velocity are set at zero for all solid surfaces (u ¼ w ¼ 0). The velocity components at the air inlet are: u ¼ uin and win ¼ 0. At the air outlet, the boundary conditions are: vu=vn ¼ 0 and vw=vn ¼ 0, where n is the normal vector to the flow direction. The boundary conditions for the temperature are as follows: the temperature of air inlet is Tin ¼ 288K, the temperature of the floor is TH ¼ f ðRaÞ, the temperature of the other walls is TC ¼ 288K, and the temperature at the air outlet level is vT=vn ¼ 0. The boundary conditions for turbulent quantities (k, ε) are given in Ref. [5]: at the air inlet kin ¼ 1:25 103 m2 s2 and εin ¼ 0 m2 s3 , and at the air outlet vk=vn ¼ 0 and vε=vn ¼ 0. 4. Thermal parameters One of the main aim of the ventilation is to provide fresh air permanently to ensure a comfortable environment inside the building. In some cases, the ventilation is used as a means of warming, air conditioning and/or dilution of the level of pollution means. Therefore, a ventilation process results of physical phenomena of diffusion and/or convection. In this study, we are interested in heat transfer and thermal comfort induced by thermoventilation behavior of internal flows. This will allow us to identify
Fig. 2. Four study cases.
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
PPD ¼ 100 95 exp 0:033353ðPMVÞ4 þ 0:2179ðPMVÞ2
Table 1 Model constants of the RNG k-3 model.
sk
sε
C1
C2
C3
Cm
0.719
0.719
C1 ðhÞ
1.68
0
0.085
with C1 ðhÞ ¼ 1:42 ðhð1 h=4:38Þ=ð1 þ 0:012h3 ÞÞ, h ¼ kS=ε and S ¼ ðSij Sij Þ1=2 .
and analyze the processes of diffusion and convection. It should be noted that these processes are greatly influenced by the position, velocity and type of air supply, conservation of mass, and temperature gradients reining the area and the presence or absence of heat and/or mass sources. Several numerical and experimental parameters can be highlighted to quantify and evaluate the effectiveness of the ventilation and thermal behavior of a ventilated room. Hereinafter, we present the most relevant parameters in ventilation. 4.1. Local age of fluid
,
Z∞ Cp ðtÞdt
Cð0Þ
(12) where the PMV expresses the human perception, and the PPD describes the percentage of occupants that are dissatisfied with the given thermal conditions. It should be noted that the ISO 7730 standard advises a PPD lower than 10% for a comfortable thermal environment. This value corresponds to the range 0:5 PMV 0:5. Awbi and Gan [38] estimated that this model is limited to very specific conditions (sedentary, light clothes and well controlled environments). To quantify correctly the thermal comfort inside ventilated room, the authors have developed an expression for the thermal comfort number ðNT Þ that seems more suited to the thermal behavior of buildings, which can be given by:
NT ¼ εT =PPD
The age of air is the time taken by an air particle from its entry to a given point P. It can be simulated by using tracer gas techniques and computed by the following relationship [36]:
ta ¼
107
(13)
Recall that here, radiation is neglected. Thereby, we aim only to analyze the ventilation effectiveness for the temperature distribution εT . For details on radiation effect, the interested reader may refer to the works of Abraham and Sparrow [39e44].
(9) 5. Numerical formulation
0
where Cð0Þ is the initial concentration of the tracer gas, and Cp ðtÞ is the concentration at a given point in the room at time t. 4.2. Nusselt number To quantify the heat transfer inside the cavity, one often uses the mean Nusselt number (Num ). Here, it is defined as:
Z Num ¼
L 0
ðvT=vzÞz¼0 dx LðTH TC Þ
(10)
5.1. Discretization The abovementioned conservation equations governing thermo-ventilation flow are discretized by the finite volume method [45]. These equations can be represented by the following general equation:
! V r U f ¼ V$ðGVfÞ þ Sf
When such an equation is integrated over the corresponding control volume and substituted each term by the discrete values of f in the nodal points, we get the following algebraic equation for each nodal point:
ap fnþ1 ¼ p
4.3. Temperature distribution effectiveness
(14)
X
annb fnþ1 þS nb
(15)
nb¼E;O;N;S
The temperature distribution effectiveness (εT ) is a parameter that predicts the temperature distribution inside the cavity. Such a parameter can be defined as follows (see Ref. [2]):
εT ¼
Tout Tin Tm Tin
(11)
where Tout is the average air temperature at the outlet, Tin is the average air temperature at inlet, and Tm is the mean temperature inside the cavity. Note that an efficient ventilation system is one that ensures good thermal comfort in the occupied zone. Although values greater than unity of εT indicate high performance in terms of temperature distribution, this does not mean that thermal comfort is assured. Thereby, we present hereinafter another parameter to estimate correctly the thermal comfort. 4.4. Thermal comfort number Fanger [37] introduced a model to evaluate thermal comfort. Such a model has been developed to relate the Predicted Mean Vote (PMV) to the Predicted Percentage of Dissatisfied (PPD) via the following relationship:
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. The employed mesh is quadratic and structured with a geometric expansion coefficient of 1:05 to the center of the cavity. The convection terms are handled by the second-order upwind scheme, while diffusion terms are discretized using the central differencing scheme, which is also second-order accurate [46]. The well known SIMPLEC algorithm [47] was adopted for coupling pressure-velocity in the governing equations. Finally, the resulting algebraic system is solved by the Incomplete LU Conjugate Residual (ILUCR) method. The convergence criteria is set at 108 for the residual error of each variable. It should be noted that Qin ¼ Qout (see Table 2), thus demonstrating results' convergence. Prior to conduct the simulations mentioned, we have ensured the results independence with respect to the mesh. We put up several grids ranging from 156 156 to 216 216. The results of each grid were compared with those of the other grids. Table 3 shows the relative error of mean temperature Tm and maximum velocities umax (at Z * ¼ 0:5) and wmax (at X * ¼ 0:5) of the 196 196-grid. From the obtained results, we found that the 196 196-grid provides satisfactory results, with a maximum
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L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
Table 2 Inlets and outlets heat (W) for the last iteration of all the calculating. Ra
Config. A Qin
6.4 1.3 1.9 2.6 3.2
108 109 109 109 109
6.07 1.21 1.82 2.49 3.03
Config. B
Qout
103 102 102 102 102
6.07 1.21 1.82 2.49 3.03
Qin
103 102 102 102 102
3.47 6.94 1.04 1.42 1.74
Config. C
Qout
103 103 102 102 102
3.47 6.94 1.04 1.42 1.74
Qin
103 103 102 102 102
3.47 6.94 1.04 1.42 1.73
Qout
103 103 102 102 102
3.47 6.94 1.04 1.42 1.73
Config. D
Qin
103 103 102 102 102
4.56 9.13 1.37 1.87 2.28
Qout
103 103 102 102 102
4.56 9.13 1.37 1.87 2.28
103 103 102 102 102
Table 3 Relative errors between grids and 196 196 grid; configuration D with Ra ¼ 3:2 109 . Grid
Tm (K)
umax at X * ¼ 0:5 ðm:s1 Þ
wmax at Z * ¼ 0:5 ðm:s1 Þ
156 156 176 176 196 196 216 216
289.35 (0.01%) 289.50 (0.04%) 289.63 289.67 (0.01%)
0.35 (7.89%) 0.37 (2.63%) 0.38 0.384 (1.00%)
0.25 (7.41%) 0.26 (3.70%) 0.27 0.27 (0.00%)
deviation of 1% compared to 216 216-grid for maximum values of velocity components u and w, and average temperature Tm . Note that the number of grid nodes within the thermal boundary layer is 26. This value was calculated for the configuration D at X * ¼ 0:5 for Ra ¼ 3:2 109. The value of the thermal boundary layer thickness (dT ) found is of 1:91 102 m. Moreover, dT is linked to the momentum boundary layer thickness via the relationship (16).
dT ¼ 1:2d
(16)
d being given d ¼ 4:92b. Note that the mesh size has been selected such that wall coordinate yþ values remain less than 5 for all considered cases here.
5.2. Validation and verification It is now accepted that a numerical code requires verification and validation of the results with available data. To verify and
validate our approach, two different issues were considered: (a) heat transfer by natural convection inside closed square cavity in laminar and steady flow, and (b) heat transfer by mixed convection over ventilated cavity heated from below in turbulent and steady flow. The first problem has been the subject of several works of numerical validations [48e51]. The cavity is heated differentially with adiabatic horizontal walls. To perform this comparison, the following dimensionless parameters were used: Prandtl number ðPr ¼ 0:71Þ and Rayleigh number ð103 Ra 106 Þ. Table 4 gathers the values of Num , Numax , Numin at X * ¼ 0:, and the maximum u and w velocity at X * ¼ 0:5 and Z * ¼ 0:5, respectively. It demonstrates that our findings are in good agreement with various authors. The second problem was carried out by Blay et al. [5]. This is a ventilated cavity filled with air and heated from below. The bottom temperature wall is set at a constant temperature equal to 308:5 K. The other walls are kept at a uniform temperature of 288 K, and the air supplied (at the top of the left vertical wall) is at 288K and
Table 4 Comparison of Num , Numax , Numin at X * ¼ 0, u*max and w*max at ðX * ; Z * Þ ¼ ð0:5; 0:5Þ for 103 Ra 106, and Pr ¼ 0:71. Present work
[48]
[49]
[50]
[51]
Ra ¼ 103 Num Numax (at z=L) Numin (at z=L) u*max (at z=L) w*max (at x=L)
1.14 1.55 (0.09) 0.71 (0.99) 0.14 (0.81)
1.12 1.51 (0.09) 0.69 (1.00) 0.14 (0.81)
1.11 1.50 (0.08) 0.72 (0.99) (–) (0.83)
1.11 1.42 (0.08) 0.76 (1.00) 0.13 (0.83)
1.11 1.58 (0.10) 0.67 (0.99) 0.15 (0.81)
0.14 (0.18)
0.14 (0.18)
(–) (0.17)
0.13 (0.20)
0.16 (0.18)
Num Numax (at z=L) Numin (at z=L) u*max (at z=L)
2.17 3.53 0.58 0.19 0.23
2.24 3.53 0.57 0.19 0.23
2.20 3.48 (0.14) 0.64 (0.99) (–)(0.83) (–)(0.11)
2.30 3.65 0.61 0.20 0.23
2.25 3.54 0.58 0.19 0.23
Num Numax (at z=L) Numin (at z=L) u*max (at z=L) w*max (at x=L)
4.47 7.77 (0.08) 0.74 (1.00) 0.13 (0.85)
4.52 7.72 (0.08) 0.73 (1.00) 0.15 (0.86)
4.43 7.63 (0.08) 0.82 (0.99) (–) (0.86)
4.65 7.70 (0.08) 0.79 (1.00) 0.15 (0.86)
4.51 7.64 (0.08) 0.77 (1.00) 0.13 (0.86)
0.26 (0.07)
0.26 (0.07)
(–) (0.07)
0.25 (0.07)
0.26 (0.07)
Ra ¼ 104
w*max (at x=L)
(0.15) (1.00) (0.82) (0.12)
(0.14) (1.00) (0.82) (0.12)
(0.63) (1.00) (0.82) (0.12)
(0.14) (0.99) (0.82) (0.12)
Ra ¼ 105
Ra ¼ Num Numax (at z=L) Numin (at z=L) u*max (at z=L) w*max (at x=L)
106
9.36 17.88 (0.04) 1.03 (1.00) 0.08 (0.85)
8.80 17.92 (0.04) 0.99 (1.00) 0.08 (0.85)
8.75 17.87 (0.04) 1.23 (0.99) (–) (0.87)
9.01 17.67 (0.04) 1.26 (1.00) 0.08 (0.86)
8.81 17.44 (0.04) 1.00 (1.00) 0.08 (0.86)
0.26 (0.04)
0.26 (0.04)
(–) (0.04)
0.26 (0.03)
0.26 (0.04)
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
0:57 ms1 , giving a Rayleigh number Ra ¼ 2:6 109 and a Reynolds number Re ¼ 683; the latter is extracted naturally from the bottom of the opposite wall. Confrontations carried out are depicted in Fig. 3 inpterms of temperature (T), intensity of turbulent kiffiffiffi netic energy ð k Þ and velocity components (u and w) at the middle of the cavity (X * ¼ 0:5 or Z * ¼ 0:5). We observe that there is a slight difference between our predictions and experimental results. The temperature and velocities profiles are almost identical. As for pffiffiffi k-profiles, they are less well predicted by the RNG k-ε model, while comparing favorably with results of Blay et al. [5]. In light of these comparisons, we can conclude that the numerical approach opted to the case of this study provides satisfactory results.
temperature gradients reigning the domain. It seems obvious that these forces influence the behavior of the flow. It should be recalled that when the forced convection is low, the flow is governed by the natural convection, and if the buoyant forces are low, the flow is governed by the forced convection. However, in some cases, there is an interaction between the natural and forced convections. This interaction affects the flow with different intensities. To set this, we define the Richardson number (Ri ¼ ðRa=PrÞ=Re2 ) that represents the ratio between the natural and forced convections where Ra and Re being defined with the same scale L. Note that for Ri > > 1, the flow is only driven by the free convection, while for Ri < < 1, forced convection prevails. For Ri ¼ 1, it is the mixed convection where the two forces are equivalent. In other words, depending on the range wherein the problem is defined 0 < Ri < 1 or 1 < Ri < ∞, the natural and forced convection forces are present with a different interaction. In the present study, Richardson numbers considered are given in Table 5. To have a clear idea about the behavior of the flow and the temperature distribution inside the cavity, it is important to know how the fresh air is distributed in the cavity. For this, we present streamlines, isotherms and age of air at each point inside the cavity for the all configurations studied, namely A, B, C, and D. Fig. 4 illustrates effects of ventilation and Rayleigh number on streamlines. It appears that the increase of Rayleigh number has few effect on streamlines, whereas the ventilation clearly has an effect on these streamlines. In addition, we note that, for all configurations, the air flow remains parallel to the horizontal wall (top or bottom depending on configuration), and when it hits the right vertical wall, the flow change direction towards the exit giving rise of vortex structures. The primary structure is a large vortex located in the center of the cavity. As for secondary structures at the corners, they exhibit different sizes. Also, we note the emergence of a third structure close to the entrance for configurations A and D. This is due to the interaction between natural and forced convections. We observe that the increase of the Rayleigh number does not greatly influence the flow. Indeed, for configuration A and D, the secondary vortex near the inlet widens according to Ra. For
6. Results and discussion It is useful to recall that here we consider a square cavity heated from the floor of dimensions 1:04 1:04 m2 . This cavity is filled with air (Pr ¼ 0:71). The heights of the inlet and outlet air openings are hin ¼ 0:018m and hout ¼ 0:024m, respectively. The temperature at the air inlet is 288K. The velocity at the air inlet is set to 0:57ms1 , which corresponds to Reynolds number of 683 The vertical walls and upper horizontal wall are maintained at a uniform temperature of 288K. The lower horizontal wall is attached a constant temperature TH higher than 288K. To determine the best configuration of ventilation in terms of effectiveness for temperature distribution and to obtain an acceptable temperature of comfort, we performed an analysis based on the position of the air inlet and outlet openings (see Fig. 2). Another relevant parameter is the influence of the heating temperature, which has been scrutinized in increment of 5K leading to Rayleigh numbers between 6:4 108 and 3:2 109 . 6.1. Flow patterns (streamlines, isotherms and age of air) The air enters in the cavity with a certain velocity creating an additional forced convection flow to buoyancy forces caused by the
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Fig. 3. Temperature T (left), turbulent kinetic energy
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pffiffiffi k (middle), u and w(right) profiles; comparison with Ref. [5].
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L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
Table 5 Richardson numbers for the all cases; 6:4 108 Ra 3:2 109 . Ra
6:4 108
1:3 109
1:9 109
2:6 109
3:2 109
Ri
0.54
1.10
1.61
2.20
2.70
configuration C, the flow accelerates and splits into two vortex “top and bottom” at Ra ¼ 3:2 109 . Regarding isotherms (see Fig. 5), we found that the cold air is heated as it passes near the hot wall. Then, it branches to ascend to the upper part of the cavity. During this movement, the heat is supplied to the center of the cavity. Indeed, we observe a certain
Fig. 4. Streamlines for the all configurations for different Ra.
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
stratification of the temperature distribution in the heart of the cavity, thereby explaining the resting state of the fluid in this area. Furthermore, the increase of Rayleigh number causes a slight change of isothermal except in the case of the configuration C when Ra ¼ 3:2 109 . For the configurations A, B and C, the temperature in the middle of the cavity increases between 2 and 5K according to
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the heating temperature TH . However, the temperature in the middle of the configuration D does not change and remains substantially equal to 288K. This is due to the effect of the forced convection that wicks maximum heat before it is advected upward. To get an idea about the time what fresh air takes to sweep the cavity, the local age of air is illustrated in Fig. 6. We found that fresh
Fig. 5. Isotherms (K) for the all configurations for different Ra.
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L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
air takes less time to get close to walls due to the closure of the flow at these locations. In the middle of the cavity, air passes over time due to prevailing low velocities at the center of cavity. It appears that the increase of Rayleigh number from 6:4 108 to 3:2 109 causes a net decrease in the age of air: 60 sec for configuration A, 40 sec for configuration B and 100 sec for the configuration C. This
indicates an intensification of the flow based on Rayleigh number. Regarding the configuration D, the age of air does not change regardless of the Rayleigh number because of the non-contribution of the natural convection. This is because the hot air is caused to leave the cavity particularly quickly.
Fig. 6. Age of air (s) for the all configurations for different Ra.
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
6.2. Effect of the heating temperature To check the effect of the heating temperature imposed at the floor level on the flow behavior and the temperature distribution in the middle of the cavity, we plotted velocity profiles and temperature for all the cases considered. The Reynolds number is set to
683 and the Rayleigh number varies between 6:4 108 and 3:2 109 . The velocity components profiles along the vertical midplanes are depicted in Fig. 7. We observe that the horizontal and vertical velocities (u and w, respectively) have the same trend regardless of the Rayleigh number, for a given configuration, except C when
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Fig. 7. Velocity components for all configurations; horizontal velocity u (left) and vertical velocity w (right); 6:4 108 Ra 3:2 109 .
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Ra ¼ 3:2 109 . The increase of Ra, for all cases, induces an elevation of velocity. Note that it is clear that an elevation gradient of horizontal and vertical velocities occurs near walls. This indicates that the flow is accelerated when Ra increases. However, the fluid is almost at rest in the center of the cavity (velocities are practically zero). In the case of configurations A and D, the flow turns
counterclockwise, while for configurations B and C, the flow turns clockwise. In addition, configurations B and C exhibit the highest velocities, particularly at the floor and at the right levels. This allows us to state that natural convection effects increase with the Rayleigh number. The analysis of Fig. 8 shows that the natural convection causes
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Fig. 8. Temperature profiles for the all configurations; horizontal temperature (left) and vertical temperature (right); 6:4 108 Ra 3:2 109 .
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117 Table 6 Mean temperature inside the cavity and standard deviation (K). Ra 6.4 1.3 1.9 2.6 3.2
s
108 109 109 109 109
Config. A
Config. B
Config. C
Config. D
s
289.15 290.21 291.29 292.47 293.43 1.71
289.04 289.89 290.74 291.65 292.40 1.34
289.20 290.22 291.21 292.28 293.11 1.56
288.33 288.58 288.90 289.29 289.63 0.52
0.40 0.78 1.12 1.47 1.73 e
an increase of the temperature in the large central part of cavities. We observe that, for a given configuration, the temperature profiles are similar with Ra except for the configuration C when Ra ¼ 3:2 109 . These profiles are horizontal or vertical in the cavity center and all temperature gradients are located near walls. Also, we found that configurations B and C have almost identical temperatures for a given Ra. It should be noted that the configuration A provides highest temperatures, while the configuration D gives the lowest ones. Table 6 shows the mean temperature (Tm ) and its standard deviation s within the cavity according to the Ra for all the cases. It can be seen that Tm increases with Ra regardless of the configuration. It increases by 4K for cavity A, by 3K for cavities B and C, and only by 1:3K for cavity D. As expected, the configuration D has the lowest temperatures because of the low contribution of the natural convection. From the standard deviation (s) values, it can be concluded that when Ra increases, s becomes important. Indeed, it
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goes from 0:40K for Ra ¼ 6:4 108 to 1:73K for Ra ¼ 3:2 109. This is due to the increasing of the heating temperature. It is the configuration D, which presents the lowest value of s, while the configuration A exhibits a highest value (1:71K). As for configurations B and C, the deviation is of 1:34K and 1:56K, respectively. This indicates that the configuration D has the lowest mean temperature values. Note that the standard deviation s is obtained via Eq. (17).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX 2 . N s¼t fi f
(17)
i
where fi is the discrete variable, f is the average value, and N is the number of observations. From the results presented in this sub-section, we can conclude that the configuration A is better in terms of warming since it presents the highest temperatures. As for the configuration D, it can be used as a means of air conditioners since it serves to remove substantially all the heat from the bottom wall. Nevertheless, configurations B and C can also be used for heating as they have temperatures close to those of the configuration A. 6.3. Nusselt number on the floor The heat transfer effectiveness of the room is displayed in terms of local and average Nusselt numbers values which are calculated at the floor level. Fig. 9 shows the variation of the local Nusselt number (Nu) along the lower horizontal wall for all the considered
Fig. 9. Local Nusselt number for all configurations A, B, C and D; 6:4 108 Ra 3:2 109 .
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L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
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configurations. It is significantly larger than unity and decreases when Ra increases. This is due to the increasing of heating temperature, which was varied from 293K (Ra ¼ 6:4 108 ) to 313K (Ra ¼ 3:2 109 ). On the other hand, the heating temperatures which provide a better efficiency are less than 293 K, and that those which are higher than 293 K only contribute to rise the energy consumption without improving εT . It is configurations A and C that have provide an εT less than 1, while the configuration B provides a value close to unity From this figure, it looms up that it is the configuration D that provides the better εT and this whatever Ra. Based on the results obtained here, we find that configurations A and C uphold a higher temperature level. Thereby, they can be used in winter to insure indoor air quality. As for the B configuration, it can be used for sweep the outlet air without affecting the heating system, while the D configuration can be advised for the evacuation of the internal heat while contributing to refresh the room in summer.
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Fig. 10. Average Nusselt vs. Ra.
configurations. Rayleigh number varies between 6:4 108 and 3:2 108 . We observe that Nu has a similar evolution in the case of configurations B and C. Nu slightly increases with Ra only on the right side of the floor. As for configurations A and D, Nu increases with Ra along the bottom wall. Also, Nu is minimum near the left wall and increases to its maximum at the opposite wall. This is true for A and D cases. However, in B and C cases, Nu is maximum at the vicinity of the left wall and minimum at the opposite wall. This difference between the configurations B and C, and A and D is due to the location of the air inlet. In fact, the air inlet is located at the bottom of the left wall for B and C cases, and at the top of the left wall for A and D cases. In Fig. 10, we show the variations of the average Nusselt number (Num ) vs. Ra. We find that the heat transfer is greatly higher in the case of the cavity D with respect to cavities A, B and C. This is due to the existence of important temperature gradients between the floor and the inside air causing a maximum heat transfer for this case for which the interaction between floor and fluid is high compared to other configurations. Moreover, the heated air near the floor is caused to leave the field very quickly under the ventilation's effect. For configurations A, B and C, Num varies monotonically and decreases slightly at large Ra. 6.4. Temperature distribution effectiveness Fig. 11 illustrates temperature distribution effectiveness εT vs. Rayleigh number of all cavities. It demonstrates clear difference between the configuration D and the others. It is the configuration D that provides the higher value in comparison with the other
7. Conclusion In this work, a numerical study of heat transfer by mixed convection within a ventilated cavity, and under a turbulent and steady regime was carried out. This cavity is heated from below and is provided with two differently placed ports. The aim targeted here is to decide which optimum configuration ensures better efficiency of the temperature distribution and the internal flow velocities. The study was conducted for Pr ¼ 0:71, Re ¼ 683 and different Rayleigh numbers (6:4 108 Ra 3:2 109 ). The RNG k-3 model was associated with the conservation equations to handle turbulence, and pressure-velocity coupling is performed with the SIMPLEC algorithm. Based on the simulations findings, the main conclusions can be drawn as follows: The analysis of thermal comfort shows that configuration D has a much higher efficiency (>unit) than that of other configurations, with the lowest average temperatures. As a result, such a configuration is favorable for evacuating internal heat, and may be suitable for use in summer. As for configurations A, B and C, they have efficiency less than unity and exhibit high average temperature. Nonetheless, they can be used in winter for the discharge of pollutants seen that they present no significant losses. Through The heat transfer analysis, it is found that the configuration D provides a higher average Nusselt number than the other configurations due to the presence of strong temperature gradients. As for the other configurations, they provides almost the same Nusselt number. Finally, all configurations provide a flow velocity beneath that advised by ASHRAE 55 [17].
Conflict of interest The authors declare no potential conflicts of interest regarding authorship and/or publication of this article. Acknowledgements tudes d'inge nieur (HEI)” The authors are grateful to “Hautes e and the “Hauts de France” region, whose financial support made this work possible via the PhD thesis of the first author (Koufi).
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Fig. 11. Overall ventilation effectiveness vs. Ra.
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Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Numerical investigation of turbulent mixed convection in an open cavity: Effect of inlet and outlet openings Lounes Koufi a, b, c, Zohir Younsi b, c, Yassine Cherif a, c, Hassane Naji a, c, * Univ. Artois, Laboratoire de G enie Civil et g eo-Environnement (LGCgE - EA 4515), F-62400 B ethune, France FUPL, Hautes Etudes d'Ing enieur, LGCgE - EA 4515, 13 Rue de Toul, F-59000 Lille, France c Univ. Lille Nord de France, LGCgE - EA 4515, F-59000 Lille, France a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 September 2016 Received in revised form 21 December 2016 Accepted 11 February 2017
This paper deals with the numerical investigation of heat transfer by mixed convection inside ventilated cavities with supply and exhaust slots, and filled with air under a steady and turbulent flow regime. Four configurations, rated A, B, C and D are considered here, according to the position of the inlet and outlet air ports: A, the inlet is on the top of the left vertical wall, while the outlet is on the bottom of the right vertical wall; B, the inlet is on the bottom of the left vertical wall and the outlet at the top of the opposite wall; C, the two slots are on the same side, i.e. the inlet is at the bottom and the outlet at the top of the left vertical wall, and D, the inlet is at the top and the outlet at the bottom of the left vertical wall. The bottom of the cavity is kept at a temperature TH and other walls are fixed at a temperature TC , with TH > TC . The cavity is provided with two slots: an inlet slot for introducing fresh air, and an outlet slot to extract hot air. The main aim sought here is to analyze the ventilation efficiency for temperature distribution, and fix the best configuration providing the thermal comfort targeted. We also address the influence of heating on the behavior of flow and thermal comfort, while considering different Rayleigh numbers ranging from 6:4 108 to 3:2 109 . Numerical studies have been yet devoted to these configurations, using RANS simulations. The RNG k-ε turbulence model has been adopted for the turbulence closure, and the set of governing equations was then numerically solved via the finite volume method. The SIMPLEC algorithm was associated to ensure the pressure-velocity coupling. In terms of results achieved, the configuration D provides a better ventilation effectiveness for temperature distribution εT and ensures an even temperature in the occupied zone. As for configurations A and C, they maintain an acceptable level of heat and can be used in winter period to ensure good indoor air quality, while configuration B provides an efficiency close to unity and can be used to insure indoor air quality in temperate climate zones. © 2017 Elsevier Masson SAS. All rights reserved.
Keywords: Numerical simulation Convection Mixing ventilation Ventilation efficiency Thermal comfort Turbulence modeling
1. Introduction Today, many people spend more time indoors (homes, schools, offices, transports, stores, etc.) [1]. Thereby, the quality of indoor environment must ensure the occupants' requirements in terms of thermal comfort and indoor air quality that can adversely affect their health. In addition, the indoor environment is affected by rising energy costs. Besides, since the 1973 oil crisis, thermal insulation has been greatly increased to reduce heat loss and greenhouse emissions. To this, exchanges between the outside and
nie Civil et ge o-Environ* Corresponding author. Univ. Artois, Laboratoire de Ge thune, France. nement (LGCgE - EA 4515), F-62400 Be E-mail address: [email protected] (H. Naji). http://dx.doi.org/10.1016/j.ijthermalsci.2017.02.007 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.
inside buildings were greatly reduced. Such confinement led to discomfort of the occupants and caused damage to the building structure. To improve the quality of the environment, ventilation has proved to be one of the most promising solutions. Its principle is to renew sufficiently and permanently stale air by fresh air. For that, investigators and designers innovative offer different strategies of ventilation to guarantee the comfort for occupants. Note that a ventilation system can be used to ensure good indoor air quality and thermal comfort (heating or air conditioning). There are two mechanical ventilation modes that are widely used. These are based on the fresh air intake velocity, namely the mixing ventilation where fresh air is blown at high velocities (turbulent jet air), and the displacement ventilation where air is introduced at low velocities [2]. Convective motion can help to determine the quality of the indoor air circulation in rooms and passenger cabins (cars,
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Nomenclature Cp g h H, L k n Nu p Pr Q Ra Re T TC TH T0 T0 u, w u *, w * u0 , w0 U X, Z X*, Z*
Specific heat (Jkg1K1) Gravity acceleration (ms2) Height of the gap for the inlet or outlet air (m) Cavity's height and width (m) Turbulent kinetic energy (m2s2) Normal vector (m) Nusselt number Fluid pressure (Pa) Prandtl number, Pr ¼ no/ao Heat (W) Rayleigh number, Ra ¼ ðro g bDTL3 Þ=ðmo ao Þ Reynolds number, Re ¼ ðro Uin hin Þ=mo Fluid temperature (K) Cold temperature (K) Hot temperature (K) Reference temperature, To ¼ ðTH þ Tc Þ=2 ðKÞ Fluctuating temperature (K) Horizontal and vertical velocities (ms1) Dimensionless velocity components, pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðu* ; w* Þ ¼ ðu; wÞ= g bDTL Fluctuating velocity components (ms1) Velocity modulus (ms1) Horizontal and vertical coordinates (m) Dimensionless coordinates, ðX * ; Z * Þ ¼ ðX; ZÞ=L
planes, etc.). It also plays a significant role in heat exchangers and passive cooling devices for electronic equipments and computer chips. It should be noted that, in a ventilated room, natural convection (driven by buoyancy forces) and/or forced convection (due to the external forces) can take place. The combination of these modes gives rise to a mixed convection problem. To handle such a problem, numerical and experimental studies have been carried out to understand the behavior of airflow by mixed convection and the temperature distribution within ventilated rooms. In the present work, we consider a ventilated cavity often taken as a first model [3,4]. In the following, we briefly describe some works related to the subject dealt here. Blay et al. [5] studied numerically and experimentally mixed convection inside ventilated square cavity filled with air in turbulent and steady regime. This cavity is heated from below. The authors have performed simulations for different Froude number using two low-Reynolds number k-3 models [6,7]. Their results showed the existence of a critical Froude number for which the flow inside the cavity changes direction. Chen [8] evaluated the performance of five k-3 turbulence models on the heat transfer in a ventilated cavity. The author has found that the numerical predictions of average velocities are satisfactory. In addition, he noted that fluctuations are less estimated compared to available experimental results. Also, he noticed the RNG k-3 [9] model provides more accurate results than the standard k-3 model [10]. Raji and Hasnaoui [11e13] numerically analyzed the heat transfer by mixed convection inside ventilated cavities filled with air and subjected to heat flux conditions. They evaluated the influence of radiation on heat transfer by mixed convection in a rectangular cavity with an aspect ratio of 2. They found that radiation promotes the temperature's homogenization, and reduces the maximum temperature inside the cavity. Singh and Sharif [14] conducted a numerical study of mixed convection in a 2D rectangular cavity differentially heated
Greek symbols a Thermal diffusivity (m2s1) ta Local age of fluid (s) b Thermal expansion coefficient (K1) DT Characteristic temperature difference, DT ¼ ðTH TC ÞðKÞ 3 Turbulent energy dissipation (m2s3) 3T Ventilation effectiveness for temperature distribution l Thermal conductivity (Wm1K1) m Dynamic viscosity (kgm1s1) n Kinematic viscosity (m2s1) r Density (kgm3) d Boundary layer thickness (m) s Standard deviation Superscripts/subscripts C Cold H Hot in Inlet l Local m Mean max Maximum min Minimum out outlet T Thermal t Turbulent O Reference
in laminar and steady flow regime, and equipped with inlet and outlet slots. They have shown that the position of the slots can be crucial in terms of cooling efficiency. Considering the same regime, Rahman et al. [15] studied numerically the influence of key parameters as the Prandtl (Pr), Reynolds (Re) and Richardson (Ri) numbers on the mixed convection in a ventilated square cavity. The authors found that, for low values of Ri, the rate of heat transfer is minimum along the hot wall, and for high values of Pr, the Nusselt number is enhanced. Also, the increase of Ri avoids the separation of flow while exhibiting a linear behavior. They concluded that heat transfer is strongly influenced by high values of both Pr and Ri. Radhakrishnan et al. [16] performed experimental and numerical study of mixed convection in a ventilated cavity with a heat generator. They studied the influence of the size and position of heat sources on the heat transfer performance. They found that simulations corroborate experimental results. They stated that the size, position and inclination angle of the heat source influence greatly the thermal behavior inside the room. n et al. [3] numerically investigated conjugated heat Xama transfer in a ventilated cavity under turbulent flow regime. The authors sought to determine the optimal ventilation scenario in terms of air conditioning. They found that, for Reynolds numbers ranging from 5:103 to 104 , it is the cavity for which the exit is located at the top right which ensures temperature and velocity values close to those recommended by the standard ASHRAE 55 [17]. Ezzouhri et al. [18] took over the cavity considered by Blay et al. [5] to evaluate the LES model [19] and determine the exact value of the Froude number for which two different solutions exist. Comparisons with experimental data [5] have shown that such a model provides acceptable predictions. Hinojosa and Gortari [20] conducted a numerical study of heat transfer by laminar natural convection in an isothermal open cubic cavity. The authors presented the results for different Ra numbers (104 Ra 107 ).
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
Radhakrishnan et al. [21] performed a numerical and experimental study to optimize the turbulent heat transfer in a ventilated cavity with different heaters. They confronted and validated their results ~ oz et al. [22] invessimulations with experiments. Rodríguez-Mun tigated numerically the influence of Ra and Re numbers on the heat transfer by mixed convection under turbulent and steady flow regime inside a ventilated cavity. They observed that the fluid movement caused by a dominant natural convection in case of the considered boundary conditions. Also, they noted that increasing the inlet air velocity reduces the cavity temperature, and that a thermal stratification occurs in the upper part of the cavity for a particular value of the inlet velocity. Arce et al. [23] conducted a numerical study of the conjugate heat transfer by mixed convection and conduction in a two-dimensional ventilated cavity having one glass wall under laminar and steady flow. They scrutinized the influence of Ra and Re numbers and air outlet position. They have also found that it is the configuration with an air outlet near the glass panel which evacuates sufficient amount of heat entering the cavity. Di Liu et al. [24] performed a numerical study of heat and mass transfer by mixed convection in a ventilated cubic cavity under a turbulent regime. The cavity is provided with an air conditioner window having three slots for blowing, extraction and recirculation. They analyzed indoor air quality and thermal comfort as function of inlet air velocity, recirculation, fresh air ratio (r) and buoyancy forces. They stated that contaminant reduction rates can be achieved either by increasing r or Re, by increasing the supply air flow, or by decreasing the power of the heating source. Rodríguez~ oz et al. [25] examined the effect of heat generated by a human Mun being exposed to mixed turbulent convection with radiation and emitting a generation of CO2 through breathing in a ventilated room. They found that the ventilation reduces the average temperature room of about 4 C to 5:5 C, while radiation causes an increase in average temperature between 0:2 C and 0:4 C. Serrano-Arellano et al. [26] numerically analyzed conjugated heat and mass transfer of an air/CO2 mixture inside a ventilated cavity under turbulent flow regime. They scrutinized the ventilation effect on the thermal behavior and the indoor air quality inside the cavity as function of the outlet air location. In addition, they observed that the optimum location for situating the outlet flow of the mixture is ~ oz and Hinojosa [27] perclose to the heat source. Rodríguez-Mun formed a numerical study of heat transfer by mixed convection with radiation in a 3D ventilated room under a regime. They considered three ventilation strategies depending on the position of the air inlets. They found that the heat transfer due to radiation is approximately 50% for all configurations. As expected, the heat transfer coefficient and the temperature distribution efficiency vary notably depending on the position of the air inlets. Recently, ~ oz et al. [28] and Hinojosa et al. [29] studied Rodríguez-Mun numerically and experimentally the heat transfer by mixed convection inside a ventilation cubic cavity filled with air in turbulent and steady regime. The authors found that the boundary layer thickness in the vicinity of the heated and isothermal walls varies between 0:01m and 0:03m. Measured and computed temperatures exhibit discrepancies of 2% for Ra ¼ 2:7 108 and 3% for Ra ¼ 4:5 108 . Moreover, the predicted convective coefficients are closer to those obtained experimentally for Re ¼ 31466 with deviations of 0.8% and 0.3% for Ra ¼ 2:7 108 and Ra ¼ 4:5 108 , respectively. According to the above literature survey, there is little research on the ability of a ventilation system to ensure a better temperature distribution in a medium scale cavity with a weakly turbulent mixed convection, though the problem under consideration is of great practical interest. Thereby, it seems necessary to deepen our knowledge on gains and/or losses of thermal loads within buildings equipped with adequate ventilation. The aim of this study is
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twofold: i) to carry out a numerical analysis of heat transfer by mixed convection inside ventilated square cavity heated by the floor, and ii) to analyze the influence of the heating intensity and position of the inlet and outlet air on the behavior of the flow and the temperature distribution to determine the best ventilation configuration in terms of temperature distribution efficiency. Note that another study has been carried out to test different mixing and displacement ventilation strategies. The main aim was to propose an effective strategy that takes less time to evacuate the contaminant and to ensure a good indoor air quality. For more details, see Ref. [30]. As possible applications, it can be stated that heating floors have been adopted over the last decades in residential or tertiary buildings, individual or collective houses, offices, etc. They have higher efficiency due to their ability to reduce the heat loss of distribution [31], making it possible to couple them with heat pumps, condensing boilers or thermal solar system [32]. Recall that the central ingredient to be modeled is a square cavity heated from below. Four ventilated cavities, with supply and exhaust slots differently placed have been studied here. These are denoted A, B, C and D, respectively. For the cavity A, the inlet of fresh air is placed at the top of the left vertical wall and the air stale outlet is installed at the bottom of the right vertical wall; for the cavity B, the air inlet is at the bottom of the left vertical wall and the air outlet is at the top of the opposite wall; for the cavity C, the air inlet is at the bottom of the left vertical wall and outlet at the top, while for the cavity D, the air inlet is at the top and the outlet at the bottom wall. Here, all these ventilated cavities have been studied with a CFD model in 2D, and the turbulence has been handled via the model RNG k-3 [10] using scStream commercial code [33]. The remainder of this article is organized as follows. The next section describes the physical model and the considered configurations. Section 3 presents the governing equations supplemented by the considered boundary conditions. In Section 4, key parameters of the heat transfer and thermal comfort are introduced. Section 5 is devoted to a brief description of the numerical approach while emphasizing its validation. In Section 6, results from simulations are presented and discussed. Finally, the major conclusions are outlined in Section 7.
2. Physical setup The physical model considered here is illustrated in Fig. 1 along with the Cartesian coordinate system. This is a ventilated square cavity of dimensions 1:04 1:04m2 filled with air. Such a
Fig. 1. Physical model of the ventilated cavity.
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configuration has been the subject of several numerical and experimental works [5,18,34,35] thanks to the availability of interesting and accurate experimental data that can validate the numerical simulations implemented. Initially, the air inside the cavity is at a constant temperature TC ; the bottom wall is maintained at a constant temperature TH (TH > TC ), while the other walls are maintained at a uniform temperature TC . Fresh air is introduced into the cavity via an inlet port at the temperature TC and at a velocity of 0:57ms1 and extracted naturally through an outlet port. It is the difference in temperature prevailing inside that induces natural convection, whereas the blowing of fresh air generates forced convection. It should be noted that the heating temperature was varied between 293K (Ra ¼ 6:4 108 ) and 313K (Ra ¼ 3:2 109 ) with an increment of 5K. The air inlet velocity is fixed at 0:57ms1 , which leads to Re ¼ 683. The four configurations considered according to the position of the inlet and outlet air gaps are shown in Fig. 2.
thermal diffusivity, Prt being equal to 0.9 [33]. Note that the eddy viscosity (mt ) is attached to the turbulent kinetic energy (k) and its dissipation rate (ε) by the Kolmorogov-Prandtl relationship:
.
mt ¼ Cm rk2 ε
(6)
Such a system (1e5) being open, it is needful to close it considering transport equations for the turbulent kinetic energy (k) and its dissipation rate (ε). These can be written, in their tensor formulation, as Eqs. (7) and (8):
! V rk U ¼ V$½ðm þ ðmt =sk ÞÞVk þ Gs þ Gt þ rε
(7)
! V rε U ¼ V$½ðm þ ðmt =sε ÞÞVε þ C1 ðGs þ Gt Þ 1 þ C3 Rf ðε=kÞ . C2 rε2 k (8)
3. Mathematical formulation The conservation equations governing a thermo-ventilation flow in a ventilated room are the continuity, momentum, and energy (heat) equations, which are based on the following conjectures: the fluid is Newtonian and incompressible, and the flow is turbulent and steady, and viscous. Dissipation, pressure forces' work, and radiation are assumed to be negligible. In addition, no heat source is inside the cavity, and physical properties of the fluid are constant except the buoyancy force term, which is modeled via the Boussinesq approximation. According to the aforementioned assumptions, these timeaveraged equations, in their tensor form, can be read hereafter.
V$ðrUÞ ¼ 0
(1)
! ! ! ! V r U 5 U ¼ Vp þ V$ mV U ru0i u0j þ rg bT ðT T0 Þ e3 (2) ! V rCp T U ¼ V$ lVT rCp u0i T 0
(3)
where ru0i u0j is the Reynolds stress tensor and ru0i T 0 is the turbulent heat flux, which are modeled as follows:
ru0i u0j ¼ mt Sij þ ð2=3Þrkdij
(4)
ru0i T 0 ¼ at VT
(5)
where k ¼ u0i u0i =2 is the turbulent kinetic energy, dij is the Kronecker tensor, mt is the eddy viscosity, and at ¼ mt =Prt is the turbulent
! with: Gs ¼ mt Sij V U is the producing shear rate of turbulent kinetic energy, Gt ¼ gi bT ðmt =Prt ÞVT is the buoyancy generation rate of turbulent kinetic energy, and sk , sε , C1 , C2 , C3 and Cm are the model constants whose values are given in Table 1. It should be noted that the RNG k-3 model can be derived from the instantaneous NaviereStokes equations, using the so-called ‘‘renormalization group’’ or RNG approach. It turned out that it is computationally efficient and stable compared to more complicated Reynolds stress models (RSM), which have several additional equations to be solved. The boundary conditions for the velocity are set at zero for all solid surfaces (u ¼ w ¼ 0). The velocity components at the air inlet are: u ¼ uin and win ¼ 0. At the air outlet, the boundary conditions are: vu=vn ¼ 0 and vw=vn ¼ 0, where n is the normal vector to the flow direction. The boundary conditions for the temperature are as follows: the temperature of air inlet is Tin ¼ 288K, the temperature of the floor is TH ¼ f ðRaÞ, the temperature of the other walls is TC ¼ 288K, and the temperature at the air outlet level is vT=vn ¼ 0. The boundary conditions for turbulent quantities (k, ε) are given in Ref. [5]: at the air inlet kin ¼ 1:25 103 m2 s2 and εin ¼ 0 m2 s3 , and at the air outlet vk=vn ¼ 0 and vε=vn ¼ 0. 4. Thermal parameters One of the main aim of the ventilation is to provide fresh air permanently to ensure a comfortable environment inside the building. In some cases, the ventilation is used as a means of warming, air conditioning and/or dilution of the level of pollution means. Therefore, a ventilation process results of physical phenomena of diffusion and/or convection. In this study, we are interested in heat transfer and thermal comfort induced by thermoventilation behavior of internal flows. This will allow us to identify
Fig. 2. Four study cases.
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
PPD ¼ 100 95 exp 0:033353ðPMVÞ4 þ 0:2179ðPMVÞ2
Table 1 Model constants of the RNG k-3 model.
sk
sε
C1
C2
C3
Cm
0.719
0.719
C1 ðhÞ
1.68
0
0.085
with C1 ðhÞ ¼ 1:42 ðhð1 h=4:38Þ=ð1 þ 0:012h3 ÞÞ, h ¼ kS=ε and S ¼ ðSij Sij Þ1=2 .
and analyze the processes of diffusion and convection. It should be noted that these processes are greatly influenced by the position, velocity and type of air supply, conservation of mass, and temperature gradients reining the area and the presence or absence of heat and/or mass sources. Several numerical and experimental parameters can be highlighted to quantify and evaluate the effectiveness of the ventilation and thermal behavior of a ventilated room. Hereinafter, we present the most relevant parameters in ventilation. 4.1. Local age of fluid
,
Z∞ Cp ðtÞdt
Cð0Þ
(12) where the PMV expresses the human perception, and the PPD describes the percentage of occupants that are dissatisfied with the given thermal conditions. It should be noted that the ISO 7730 standard advises a PPD lower than 10% for a comfortable thermal environment. This value corresponds to the range 0:5 PMV 0:5. Awbi and Gan [38] estimated that this model is limited to very specific conditions (sedentary, light clothes and well controlled environments). To quantify correctly the thermal comfort inside ventilated room, the authors have developed an expression for the thermal comfort number ðNT Þ that seems more suited to the thermal behavior of buildings, which can be given by:
NT ¼ εT =PPD
The age of air is the time taken by an air particle from its entry to a given point P. It can be simulated by using tracer gas techniques and computed by the following relationship [36]:
ta ¼
107
(13)
Recall that here, radiation is neglected. Thereby, we aim only to analyze the ventilation effectiveness for the temperature distribution εT . For details on radiation effect, the interested reader may refer to the works of Abraham and Sparrow [39e44].
(9) 5. Numerical formulation
0
where Cð0Þ is the initial concentration of the tracer gas, and Cp ðtÞ is the concentration at a given point in the room at time t. 4.2. Nusselt number To quantify the heat transfer inside the cavity, one often uses the mean Nusselt number (Num ). Here, it is defined as:
Z Num ¼
L 0
ðvT=vzÞz¼0 dx LðTH TC Þ
(10)
5.1. Discretization The abovementioned conservation equations governing thermo-ventilation flow are discretized by the finite volume method [45]. These equations can be represented by the following general equation:
! V r U f ¼ V$ðGVfÞ þ Sf
When such an equation is integrated over the corresponding control volume and substituted each term by the discrete values of f in the nodal points, we get the following algebraic equation for each nodal point:
ap fnþ1 ¼ p
4.3. Temperature distribution effectiveness
(14)
X
annb fnþ1 þS nb
(15)
nb¼E;O;N;S
The temperature distribution effectiveness (εT ) is a parameter that predicts the temperature distribution inside the cavity. Such a parameter can be defined as follows (see Ref. [2]):
εT ¼
Tout Tin Tm Tin
(11)
where Tout is the average air temperature at the outlet, Tin is the average air temperature at inlet, and Tm is the mean temperature inside the cavity. Note that an efficient ventilation system is one that ensures good thermal comfort in the occupied zone. Although values greater than unity of εT indicate high performance in terms of temperature distribution, this does not mean that thermal comfort is assured. Thereby, we present hereinafter another parameter to estimate correctly the thermal comfort. 4.4. Thermal comfort number Fanger [37] introduced a model to evaluate thermal comfort. Such a model has been developed to relate the Predicted Mean Vote (PMV) to the Predicted Percentage of Dissatisfied (PPD) via the following relationship:
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. The employed mesh is quadratic and structured with a geometric expansion coefficient of 1:05 to the center of the cavity. The convection terms are handled by the second-order upwind scheme, while diffusion terms are discretized using the central differencing scheme, which is also second-order accurate [46]. The well known SIMPLEC algorithm [47] was adopted for coupling pressure-velocity in the governing equations. Finally, the resulting algebraic system is solved by the Incomplete LU Conjugate Residual (ILUCR) method. The convergence criteria is set at 108 for the residual error of each variable. It should be noted that Qin ¼ Qout (see Table 2), thus demonstrating results' convergence. Prior to conduct the simulations mentioned, we have ensured the results independence with respect to the mesh. We put up several grids ranging from 156 156 to 216 216. The results of each grid were compared with those of the other grids. Table 3 shows the relative error of mean temperature Tm and maximum velocities umax (at Z * ¼ 0:5) and wmax (at X * ¼ 0:5) of the 196 196-grid. From the obtained results, we found that the 196 196-grid provides satisfactory results, with a maximum
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L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
Table 2 Inlets and outlets heat (W) for the last iteration of all the calculating. Ra
Config. A Qin
6.4 1.3 1.9 2.6 3.2
108 109 109 109 109
6.07 1.21 1.82 2.49 3.03
Config. B
Qout
103 102 102 102 102
6.07 1.21 1.82 2.49 3.03
Qin
103 102 102 102 102
3.47 6.94 1.04 1.42 1.74
Config. C
Qout
103 103 102 102 102
3.47 6.94 1.04 1.42 1.74
Qin
103 103 102 102 102
3.47 6.94 1.04 1.42 1.73
Qout
103 103 102 102 102
3.47 6.94 1.04 1.42 1.73
Config. D
Qin
103 103 102 102 102
4.56 9.13 1.37 1.87 2.28
Qout
103 103 102 102 102
4.56 9.13 1.37 1.87 2.28
103 103 102 102 102
Table 3 Relative errors between grids and 196 196 grid; configuration D with Ra ¼ 3:2 109 . Grid
Tm (K)
umax at X * ¼ 0:5 ðm:s1 Þ
wmax at Z * ¼ 0:5 ðm:s1 Þ
156 156 176 176 196 196 216 216
289.35 (0.01%) 289.50 (0.04%) 289.63 289.67 (0.01%)
0.35 (7.89%) 0.37 (2.63%) 0.38 0.384 (1.00%)
0.25 (7.41%) 0.26 (3.70%) 0.27 0.27 (0.00%)
deviation of 1% compared to 216 216-grid for maximum values of velocity components u and w, and average temperature Tm . Note that the number of grid nodes within the thermal boundary layer is 26. This value was calculated for the configuration D at X * ¼ 0:5 for Ra ¼ 3:2 109. The value of the thermal boundary layer thickness (dT ) found is of 1:91 102 m. Moreover, dT is linked to the momentum boundary layer thickness via the relationship (16).
dT ¼ 1:2d
(16)
d being given d ¼ 4:92b. Note that the mesh size has been selected such that wall coordinate yþ values remain less than 5 for all considered cases here.
5.2. Validation and verification It is now accepted that a numerical code requires verification and validation of the results with available data. To verify and
validate our approach, two different issues were considered: (a) heat transfer by natural convection inside closed square cavity in laminar and steady flow, and (b) heat transfer by mixed convection over ventilated cavity heated from below in turbulent and steady flow. The first problem has been the subject of several works of numerical validations [48e51]. The cavity is heated differentially with adiabatic horizontal walls. To perform this comparison, the following dimensionless parameters were used: Prandtl number ðPr ¼ 0:71Þ and Rayleigh number ð103 Ra 106 Þ. Table 4 gathers the values of Num , Numax , Numin at X * ¼ 0:, and the maximum u and w velocity at X * ¼ 0:5 and Z * ¼ 0:5, respectively. It demonstrates that our findings are in good agreement with various authors. The second problem was carried out by Blay et al. [5]. This is a ventilated cavity filled with air and heated from below. The bottom temperature wall is set at a constant temperature equal to 308:5 K. The other walls are kept at a uniform temperature of 288 K, and the air supplied (at the top of the left vertical wall) is at 288K and
Table 4 Comparison of Num , Numax , Numin at X * ¼ 0, u*max and w*max at ðX * ; Z * Þ ¼ ð0:5; 0:5Þ for 103 Ra 106, and Pr ¼ 0:71. Present work
[48]
[49]
[50]
[51]
Ra ¼ 103 Num Numax (at z=L) Numin (at z=L) u*max (at z=L) w*max (at x=L)
1.14 1.55 (0.09) 0.71 (0.99) 0.14 (0.81)
1.12 1.51 (0.09) 0.69 (1.00) 0.14 (0.81)
1.11 1.50 (0.08) 0.72 (0.99) (–) (0.83)
1.11 1.42 (0.08) 0.76 (1.00) 0.13 (0.83)
1.11 1.58 (0.10) 0.67 (0.99) 0.15 (0.81)
0.14 (0.18)
0.14 (0.18)
(–) (0.17)
0.13 (0.20)
0.16 (0.18)
Num Numax (at z=L) Numin (at z=L) u*max (at z=L)
2.17 3.53 0.58 0.19 0.23
2.24 3.53 0.57 0.19 0.23
2.20 3.48 (0.14) 0.64 (0.99) (–)(0.83) (–)(0.11)
2.30 3.65 0.61 0.20 0.23
2.25 3.54 0.58 0.19 0.23
Num Numax (at z=L) Numin (at z=L) u*max (at z=L) w*max (at x=L)
4.47 7.77 (0.08) 0.74 (1.00) 0.13 (0.85)
4.52 7.72 (0.08) 0.73 (1.00) 0.15 (0.86)
4.43 7.63 (0.08) 0.82 (0.99) (–) (0.86)
4.65 7.70 (0.08) 0.79 (1.00) 0.15 (0.86)
4.51 7.64 (0.08) 0.77 (1.00) 0.13 (0.86)
0.26 (0.07)
0.26 (0.07)
(–) (0.07)
0.25 (0.07)
0.26 (0.07)
Ra ¼ 104
w*max (at x=L)
(0.15) (1.00) (0.82) (0.12)
(0.14) (1.00) (0.82) (0.12)
(0.63) (1.00) (0.82) (0.12)
(0.14) (0.99) (0.82) (0.12)
Ra ¼ 105
Ra ¼ Num Numax (at z=L) Numin (at z=L) u*max (at z=L) w*max (at x=L)
106
9.36 17.88 (0.04) 1.03 (1.00) 0.08 (0.85)
8.80 17.92 (0.04) 0.99 (1.00) 0.08 (0.85)
8.75 17.87 (0.04) 1.23 (0.99) (–) (0.87)
9.01 17.67 (0.04) 1.26 (1.00) 0.08 (0.86)
8.81 17.44 (0.04) 1.00 (1.00) 0.08 (0.86)
0.26 (0.04)
0.26 (0.04)
(–) (0.04)
0.26 (0.03)
0.26 (0.04)
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0:57 ms1 , giving a Rayleigh number Ra ¼ 2:6 109 and a Reynolds number Re ¼ 683; the latter is extracted naturally from the bottom of the opposite wall. Confrontations carried out are depicted in Fig. 3 inpterms of temperature (T), intensity of turbulent kiffiffiffi netic energy ð k Þ and velocity components (u and w) at the middle of the cavity (X * ¼ 0:5 or Z * ¼ 0:5). We observe that there is a slight difference between our predictions and experimental results. The temperature and velocities profiles are almost identical. As for pffiffiffi k-profiles, they are less well predicted by the RNG k-ε model, while comparing favorably with results of Blay et al. [5]. In light of these comparisons, we can conclude that the numerical approach opted to the case of this study provides satisfactory results.
temperature gradients reigning the domain. It seems obvious that these forces influence the behavior of the flow. It should be recalled that when the forced convection is low, the flow is governed by the natural convection, and if the buoyant forces are low, the flow is governed by the forced convection. However, in some cases, there is an interaction between the natural and forced convections. This interaction affects the flow with different intensities. To set this, we define the Richardson number (Ri ¼ ðRa=PrÞ=Re2 ) that represents the ratio between the natural and forced convections where Ra and Re being defined with the same scale L. Note that for Ri > > 1, the flow is only driven by the free convection, while for Ri < < 1, forced convection prevails. For Ri ¼ 1, it is the mixed convection where the two forces are equivalent. In other words, depending on the range wherein the problem is defined 0 < Ri < 1 or 1 < Ri < ∞, the natural and forced convection forces are present with a different interaction. In the present study, Richardson numbers considered are given in Table 5. To have a clear idea about the behavior of the flow and the temperature distribution inside the cavity, it is important to know how the fresh air is distributed in the cavity. For this, we present streamlines, isotherms and age of air at each point inside the cavity for the all configurations studied, namely A, B, C, and D. Fig. 4 illustrates effects of ventilation and Rayleigh number on streamlines. It appears that the increase of Rayleigh number has few effect on streamlines, whereas the ventilation clearly has an effect on these streamlines. In addition, we note that, for all configurations, the air flow remains parallel to the horizontal wall (top or bottom depending on configuration), and when it hits the right vertical wall, the flow change direction towards the exit giving rise of vortex structures. The primary structure is a large vortex located in the center of the cavity. As for secondary structures at the corners, they exhibit different sizes. Also, we note the emergence of a third structure close to the entrance for configurations A and D. This is due to the interaction between natural and forced convections. We observe that the increase of the Rayleigh number does not greatly influence the flow. Indeed, for configuration A and D, the secondary vortex near the inlet widens according to Ra. For
6. Results and discussion It is useful to recall that here we consider a square cavity heated from the floor of dimensions 1:04 1:04 m2 . This cavity is filled with air (Pr ¼ 0:71). The heights of the inlet and outlet air openings are hin ¼ 0:018m and hout ¼ 0:024m, respectively. The temperature at the air inlet is 288K. The velocity at the air inlet is set to 0:57ms1 , which corresponds to Reynolds number of 683 The vertical walls and upper horizontal wall are maintained at a uniform temperature of 288K. The lower horizontal wall is attached a constant temperature TH higher than 288K. To determine the best configuration of ventilation in terms of effectiveness for temperature distribution and to obtain an acceptable temperature of comfort, we performed an analysis based on the position of the air inlet and outlet openings (see Fig. 2). Another relevant parameter is the influence of the heating temperature, which has been scrutinized in increment of 5K leading to Rayleigh numbers between 6:4 108 and 3:2 109 . 6.1. Flow patterns (streamlines, isotherms and age of air) The air enters in the cavity with a certain velocity creating an additional forced convection flow to buoyancy forces caused by the
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Fig. 3. Temperature T (left), turbulent kinetic energy
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pffiffiffi k (middle), u and w(right) profiles; comparison with Ref. [5].
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Table 5 Richardson numbers for the all cases; 6:4 108 Ra 3:2 109 . Ra
6:4 108
1:3 109
1:9 109
2:6 109
3:2 109
Ri
0.54
1.10
1.61
2.20
2.70
configuration C, the flow accelerates and splits into two vortex “top and bottom” at Ra ¼ 3:2 109 . Regarding isotherms (see Fig. 5), we found that the cold air is heated as it passes near the hot wall. Then, it branches to ascend to the upper part of the cavity. During this movement, the heat is supplied to the center of the cavity. Indeed, we observe a certain
Fig. 4. Streamlines for the all configurations for different Ra.
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117
stratification of the temperature distribution in the heart of the cavity, thereby explaining the resting state of the fluid in this area. Furthermore, the increase of Rayleigh number causes a slight change of isothermal except in the case of the configuration C when Ra ¼ 3:2 109 . For the configurations A, B and C, the temperature in the middle of the cavity increases between 2 and 5K according to
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the heating temperature TH . However, the temperature in the middle of the configuration D does not change and remains substantially equal to 288K. This is due to the effect of the forced convection that wicks maximum heat before it is advected upward. To get an idea about the time what fresh air takes to sweep the cavity, the local age of air is illustrated in Fig. 6. We found that fresh
Fig. 5. Isotherms (K) for the all configurations for different Ra.
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air takes less time to get close to walls due to the closure of the flow at these locations. In the middle of the cavity, air passes over time due to prevailing low velocities at the center of cavity. It appears that the increase of Rayleigh number from 6:4 108 to 3:2 109 causes a net decrease in the age of air: 60 sec for configuration A, 40 sec for configuration B and 100 sec for the configuration C. This
indicates an intensification of the flow based on Rayleigh number. Regarding the configuration D, the age of air does not change regardless of the Rayleigh number because of the non-contribution of the natural convection. This is because the hot air is caused to leave the cavity particularly quickly.
Fig. 6. Age of air (s) for the all configurations for different Ra.
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6.2. Effect of the heating temperature To check the effect of the heating temperature imposed at the floor level on the flow behavior and the temperature distribution in the middle of the cavity, we plotted velocity profiles and temperature for all the cases considered. The Reynolds number is set to
683 and the Rayleigh number varies between 6:4 108 and 3:2 109 . The velocity components profiles along the vertical midplanes are depicted in Fig. 7. We observe that the horizontal and vertical velocities (u and w, respectively) have the same trend regardless of the Rayleigh number, for a given configuration, except C when
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Fig. 7. Velocity components for all configurations; horizontal velocity u (left) and vertical velocity w (right); 6:4 108 Ra 3:2 109 .
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Ra ¼ 3:2 109 . The increase of Ra, for all cases, induces an elevation of velocity. Note that it is clear that an elevation gradient of horizontal and vertical velocities occurs near walls. This indicates that the flow is accelerated when Ra increases. However, the fluid is almost at rest in the center of the cavity (velocities are practically zero). In the case of configurations A and D, the flow turns
counterclockwise, while for configurations B and C, the flow turns clockwise. In addition, configurations B and C exhibit the highest velocities, particularly at the floor and at the right levels. This allows us to state that natural convection effects increase with the Rayleigh number. The analysis of Fig. 8 shows that the natural convection causes
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Fig. 8. Temperature profiles for the all configurations; horizontal temperature (left) and vertical temperature (right); 6:4 108 Ra 3:2 109 .
L. Koufi et al. / International Journal of Thermal Sciences 116 (2017) 103e117 Table 6 Mean temperature inside the cavity and standard deviation (K). Ra 6.4 1.3 1.9 2.6 3.2
s
108 109 109 109 109
Config. A
Config. B
Config. C
Config. D
s
289.15 290.21 291.29 292.47 293.43 1.71
289.04 289.89 290.74 291.65 292.40 1.34
289.20 290.22 291.21 292.28 293.11 1.56
288.33 288.58 288.90 289.29 289.63 0.52
0.40 0.78 1.12 1.47 1.73 e
an increase of the temperature in the large central part of cavities. We observe that, for a given configuration, the temperature profiles are similar with Ra except for the configuration C when Ra ¼ 3:2 109 . These profiles are horizontal or vertical in the cavity center and all temperature gradients are located near walls. Also, we found that configurations B and C have almost identical temperatures for a given Ra. It should be noted that the configuration A provides highest temperatures, while the configuration D gives the lowest ones. Table 6 shows the mean temperature (Tm ) and its standard deviation s within the cavity according to the Ra for all the cases. It can be seen that Tm increases with Ra regardless of the configuration. It increases by 4K for cavity A, by 3K for cavities B and C, and only by 1:3K for cavity D. As expected, the configuration D has the lowest temperatures because of the low contribution of the natural convection. From the standard deviation (s) values, it can be concluded that when Ra increases, s becomes important. Indeed, it
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goes from 0:40K for Ra ¼ 6:4 108 to 1:73K for Ra ¼ 3:2 109. This is due to the increasing of the heating temperature. It is the configuration D, which presents the lowest value of s, while the configuration A exhibits a highest value (1:71K). As for configurations B and C, the deviation is of 1:34K and 1:56K, respectively. This indicates that the configuration D has the lowest mean temperature values. Note that the standard deviation s is obtained via Eq. (17).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX 2 . N s¼t fi f
(17)
i
where fi is the discrete variable, f is the average value, and N is the number of observations. From the results presented in this sub-section, we can conclude that the configuration A is better in terms of warming since it presents the highest temperatures. As for the configuration D, it can be used as a means of air conditioners since it serves to remove substantially all the heat from the bottom wall. Nevertheless, configurations B and C can also be used for heating as they have temperatures close to those of the configuration A. 6.3. Nusselt number on the floor The heat transfer effectiveness of the room is displayed in terms of local and average Nusselt numbers values which are calculated at the floor level. Fig. 9 shows the variation of the local Nusselt number (Nu) along the lower horizontal wall for all the considered
Fig. 9. Local Nusselt number for all configurations A, B, C and D; 6:4 108 Ra 3:2 109 .
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configurations. It is significantly larger than unity and decreases when Ra increases. This is due to the increasing of heating temperature, which was varied from 293K (Ra ¼ 6:4 108 ) to 313K (Ra ¼ 3:2 109 ). On the other hand, the heating temperatures which provide a better efficiency are less than 293 K, and that those which are higher than 293 K only contribute to rise the energy consumption without improving εT . It is configurations A and C that have provide an εT less than 1, while the configuration B provides a value close to unity From this figure, it looms up that it is the configuration D that provides the better εT and this whatever Ra. Based on the results obtained here, we find that configurations A and C uphold a higher temperature level. Thereby, they can be used in winter to insure indoor air quality. As for the B configuration, it can be used for sweep the outlet air without affecting the heating system, while the D configuration can be advised for the evacuation of the internal heat while contributing to refresh the room in summer.
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Fig. 10. Average Nusselt vs. Ra.
configurations. Rayleigh number varies between 6:4 108 and 3:2 108 . We observe that Nu has a similar evolution in the case of configurations B and C. Nu slightly increases with Ra only on the right side of the floor. As for configurations A and D, Nu increases with Ra along the bottom wall. Also, Nu is minimum near the left wall and increases to its maximum at the opposite wall. This is true for A and D cases. However, in B and C cases, Nu is maximum at the vicinity of the left wall and minimum at the opposite wall. This difference between the configurations B and C, and A and D is due to the location of the air inlet. In fact, the air inlet is located at the bottom of the left wall for B and C cases, and at the top of the left wall for A and D cases. In Fig. 10, we show the variations of the average Nusselt number (Num ) vs. Ra. We find that the heat transfer is greatly higher in the case of the cavity D with respect to cavities A, B and C. This is due to the existence of important temperature gradients between the floor and the inside air causing a maximum heat transfer for this case for which the interaction between floor and fluid is high compared to other configurations. Moreover, the heated air near the floor is caused to leave the field very quickly under the ventilation's effect. For configurations A, B and C, Num varies monotonically and decreases slightly at large Ra. 6.4. Temperature distribution effectiveness Fig. 11 illustrates temperature distribution effectiveness εT vs. Rayleigh number of all cavities. It demonstrates clear difference between the configuration D and the others. It is the configuration D that provides the higher value in comparison with the other
7. Conclusion In this work, a numerical study of heat transfer by mixed convection within a ventilated cavity, and under a turbulent and steady regime was carried out. This cavity is heated from below and is provided with two differently placed ports. The aim targeted here is to decide which optimum configuration ensures better efficiency of the temperature distribution and the internal flow velocities. The study was conducted for Pr ¼ 0:71, Re ¼ 683 and different Rayleigh numbers (6:4 108 Ra 3:2 109 ). The RNG k-3 model was associated with the conservation equations to handle turbulence, and pressure-velocity coupling is performed with the SIMPLEC algorithm. Based on the simulations findings, the main conclusions can be drawn as follows: The analysis of thermal comfort shows that configuration D has a much higher efficiency (>unit) than that of other configurations, with the lowest average temperatures. As a result, such a configuration is favorable for evacuating internal heat, and may be suitable for use in summer. As for configurations A, B and C, they have efficiency less than unity and exhibit high average temperature. Nonetheless, they can be used in winter for the discharge of pollutants seen that they present no significant losses. Through The heat transfer analysis, it is found that the configuration D provides a higher average Nusselt number than the other configurations due to the presence of strong temperature gradients. As for the other configurations, they provides almost the same Nusselt number. Finally, all configurations provide a flow velocity beneath that advised by ASHRAE 55 [17].
Conflict of interest The authors declare no potential conflicts of interest regarding authorship and/or publication of this article. Acknowledgements tudes d'inge nieur (HEI)” The authors are grateful to “Hautes e and the “Hauts de France” region, whose financial support made this work possible via the PhD thesis of the first author (Koufi).
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Fig. 11. Overall ventilation effectiveness vs. Ra.
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