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Test of turbulence models for heat transfer within a ventilated cavity with and without an internal heat source
International Communications in Heat and Mass Transfer 94 (2018) 106–114
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Test of turbulence models for heat transfer within a ventilated cavity with and without an internal heat source
T
⁎
A. Piña-Ortiza, J.F. Hinojosaa, , J.P. Xamánb, J.M.A. Navarroa a b
Departamento de Ingeniería Química y Metalurgia, Universidad de Sonora, Hermosillo 83000, Sonora, Mexico Centro Nacional de Investigación y Desarrollo Tecnológico, CENIDET-DGEST-SEP, prol. Av. Palmira S/N. Col. Palmira, Cuernavaca, Morelos 62490, Mexico
A R T I C LE I N FO
A B S T R A C T
Keywords: Ventilated cavity Internal heat source Turbulence models
In this study, six of most frequently used turbulence models in computational fluid dynamics were compared with experimental data. The experimental cavity is a cube whose edge is 1 m long. The left vertical wall receives a constant and uniform heat flux, while the opposite wall is kept at a constant temperature. The rest of the walls are adiabatic. The heat source is a rectangular parallelepiped with square base 0.3 m on a side and height 0.61 m. The cavity represents a ventilated room in a 1:3 scale with multiple inlets and outlets of air, considering ventilation by ducts of an air-conditioning system. The experimental setup was built to obtain temperature profiles and heat transfer coefficients. Temperature profiles were obtained at six different depths and heights consisting of 14 thermocouples each. It was found that the lowest average of percentage differences for the case with the heat source turned on corresponds to realizable k-ε turbulence model (rkε) with 0.84%.
1. Introduction At recent years, there has been a sustained increase in energy consumption at arid regions around the world also known as desert climate. In this climate, there are large diurnal temperature variations. Therefore, due to the design of existing buildings, artificial air conditioning systems are required to achieve comfort conditions, resulting in high electricity consumption. The consequence is an ecological footprint, since most of primary electricity production around the world comes from fossil fuels that are responsible of the discharge into the atmosphere of important amounts of greenhouse effect gases, mainly carbon dioxide. Therefore, the study of buildings ventilation has become an important need to reduce the energy consumption. On the other hand, the interaction between electronic equipment or people's heat fluxes at the interior may cause different flow and temperature patterns, therefore it must be analyzed to optimize and to reduce the electricity consumption of the ventilation systems. To understand the effect of heat transfer on airflow and temperature distribution, theoretical studies may be used considering a room as a cavity, to get a better control of the studied parameters. With the mathematical model, it is possible to predict what will happen in the system, but it is necessary to establish its predictive capacity by comparing with experimental data of the thermal system. The investigations of heat transfer in ventilated cavities are numerous [1–32], therefore, will be described only the studies with
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Corresponding author. E-mail address: [email protected] (J.F. Hinojosa).
https://doi.org/10.1016/j.icheatmasstransfer.2018.03.021
0735-1933/ © 2018 Elsevier Ltd. All rights reserved.
internal heat generation. Those studies may be organized as: (a) Numerical, (b) Experimental and (c) Numerical-Experimental. (a) Numerical studies in ventilated cavities with internal heat generation. Papanicolaou and Jaluria [18] carried out a numerical study of the mixed convection in a rectangular ventilated cavity with a discrete heat source in it. The effects of the Reynolds number, Richardson number, position of the heat source and position of the outflow on heat transfer and temperature distribution were observed for a Reynolds number from 50 to 2000. The obtained results were used to study heat removal mechanisms in practical systems. In another article, these authors [19] studied numerically a problem of mixed convection considering a cavity with two ventilation ports, conductive walls and a discrete heat source on a wall. The Reynolds number was fixed at 100 and Richardson number was varying between 0 and 10. The results show that the configuration in which the two types of convection are assisted presents higher heat transfer and lower sources temperatures. Hsu and Wang [20] performed a numerical study about mixed convection in a ventilated cavity with discrete heat sources embedded on a vertical board which is situated on the bottom wall of the enclosure. The Reynolds number was studied from 100 to 1000. They found that when the source is located on the right surface of the board the Nusselt number is not depending neither on the variation of the
International Communications in Heat and Mass Transfer 94 (2018) 106–114
A. Piña-Ortiz et al.
location of the source nor of the board. Ghasemi and Aminossadati [22] studied numerically the mixed convection in a two-dimensional ventilated cavity with discrete heat sources. They examined the effects of the number and position of the sources, the Rayleigh number from 0 to 107 at a Reynolds number of 100. Results show that increasing significantly Rayleigh number improves the heat transfer process in the cavity. The arrangement of sources also has a great contribution on the cooling performance but when the Rayleigh number is increased this contribution decreases. Bilgen and Muftuoglu [23] numerically investigated a cooling strategy in a square cavity with adiabatic walls and a heat source on the left wall. The Rayleigh number was studied from 103 to 107 and the Reynolds number from 102 to 103. The authors observed that the optimal position of the source is almost insensitive to variation in Rayleigh and Reynolds numbers, but it is strongly affected by the arrangements on the ventilation ports. It was found that the highest cooling performance is given by placing the air outlet on the upper left part of the cavity. Rodriguez-Muñoz et al. [24] studied numerically the combined effect of heat generation produced by a human being and the mixed turbulent convection with thermal radiation, as well as the CO2 production from respiration in a rectangular ventilated room. The results showed that the ventilation reduces the average temperatures in the room between 4 °C and 5.5 °C, while the thermal radiation increases the average temperature between 0.2 °C and 0.4 °C. Biswas et al. [25] numerically studied thermal management in a ventilated enclosure undergoing mixed convection by dividing the entire heating element into multiple equal segments and by positioning them appropriately on vertical side walls, namely at bottom, middle or top positions. Analysis of segmental heating and whole heating are conducted for different Richardson number (0.01–100) and Reynolds number (50–200). Nine positional configurations of bi-segmental heating reveal the possibility of significant enhancement in heat transfer.
Fig. 1. Physical model of the studied cavity.
ventilated cavity without and with an internal heat source (turned on and turned off). The cavity has multiple inlets and outlets of air, considering ventilation by ducts of an air-conditioning system. In addition, the cavity receives a heat flux in one vertical wall (representing incoming heat from the exterior). The experimental setup was built to obtain temperature profiles and heat transfer coefficients. The numerical results obtained with six turbulence models, are compared with experimental results and percentage differences are computed. 2. Physical and mathematical model 2.1. Physical model The study of turbulent mixed convection was performed in the cubic cavity shown in Fig. 1. The cavity represents a ventilated room in a 1:3 scale. The dimensions of the system are as follows: Lx = Ly = Lz = 1.0 m and it consists of one vertical wall (x = 0) receiving a constant and uniform heat flux, while the vertical facing wall (x = Lx) was kept at a constant temperature Tc (298 K). The remaining walls were assumed as adiabatic. Every wall of the cavity was covered with a film of polished aluminum to minimize the thermal radiation exchange. The thermal fluid is air. The dimensions of the inlet and outlet are lx = ly = 0.08 m and their positions are described in Table 1. The air at constant temperature enters the cavity with velocities of 0.5 or 1.3 m/s. At the center of the cavity, there is a parallelepiped with a height of 0.61 m and a depth and length of 0.30 m with electrical heaters in every surface to represent the generated heat by a person in a room. The fluid flow was assumed as turbulent and because the temperature differences are < 30 K, the Boussinesq approach was considered valid.
(b) Experimental studies in ventilated cavities with internal heat generation. Ajmera and Mahur [26] performed an experimental investigation of mixed convection in multiple ventilated enclosures with discrete heat sources. The flow velocity and applied heat flux were varied. The experimental investigations were executed for a range of Reynolds number and Grashof number of 270 ≤ Re ≤ 6274 and 7.2 × 106 ≤ Gr ≤ 5.5 × 107, whereas the Richardson number lied in the range of 0.201–571. Different correlations were proposed for Nusselt number within the range of considered parameters. (c) Experimental and numerical studies in ventilated cavities with internal heat generation. Radhakrishnan et al. [21] reported a numerical and experimental work about turbulent mixed convection in a ventilated cavity with adiabatic walls and a discrete heat source inside. The k–ε (RNG) model is adopted for the turbulence closure in the two-dimensional numerical study. Correlations were developed for the average Nusselt number and the maximum dimensionless temperature occurring in the heat source, in these parameter ranges: 1200 ≤ Re ≤ 10,000 and 0.003 ≤ Ri ≤ 0.2. The authors concluded that a combined experimental and numerical investigation would significantly reduce the effort required to optimize the cooling of electronic equipment. The literature review indicates very scarce numerical and experimental studies analyzing the effect of an internal heat source on turbulent mixed convection in a ventilated cavity. Nevertheless, the study of turbulent heat transfer in a ventilated cavity with an internal heat source is relevant for an adequate thermal design of a building. Considering the above, this paper presents an experimental and numerical study of three-dimensional turbulent mixed convection in a
2.2. Mathematical model The considerations for the mathematical model are: steady state, Newtonian fluid, turbulent flow regime, negligible viscosity dissipation and the use of the Boussinesq approximation. The time averaged Table 1 Inlet and outlet positions. Position (x, y, z) Inlet 1 Inlet 2 Inlet 3 Outlet 1 Outlet 2 Outlet 3
107
0.46 m ≤ x ≤ 0.54 m 0.46 m ≤ x ≤ 0.54 m 0.46 m ≤ x ≤ 0.54 m 0.71 m ≤ x ≤ 0.79 m 0.71 m ≤ x ≤ 0.79 m 0.71 m ≤ x ≤ 0.79 m
y = 1.0 m y = 1.0 m y = 1.0 m y = 1.0 m y = 1.0 m y = 1.0 m
0.21 m ≤ z ≤ 0.29 m 0.46 m ≤ z ≤ 0.54 m 0.71 m ≤ z ≤ 0.79 m 0.21 m ≤ z ≤ 0.29 m 0.46 m ≤ z ≤ 0.54 m 0.71 m ≤ z ≤ 0.79 m
International Communications in Heat and Mass Transfer 94 (2018) 106–114
A. Piña-Ortiz et al.
that is based on the omega equation [31]. This model is ideal for modeling flows over curved surfaces and swirling flows and it is good for predictions over a wide range of turbulent flows. In this model, the closure coefficients are identical to the skω model; however, there are additional closure coefficients. The hydrodynamic boundary conditions are non-slip conditions on the walls; therefore, the velocity components are equal to zero. Since the air was considered to enter perpendicularly to the opening, the ycomponent of the velocity had a constant value whereas the remaining components are equal to zero. The pressure outlet boundary condition is used for the outgoing air. The turbulent kinetic energy and the dissipation of the turbulent kinetic energy for the incoming air were obtained by applying empirical correlations [33].
governing equations in tensor notation, are as follow: Continuity:
∂ui =0 ∂x i
(1)
Momentum:
ρuj
∂ui ∂P ∂ ⎡ ∂ui = + μ − ρui′ u′j ⎤ + ρgi β (T − T0) ⎥ ∂x j ∂x i ∂x j ⎢ ⎣ ∂x j ⎦
(2)
Energy:
ρuj
∂T 1 ∂ ⎡ ∂T = λ − ρCP T ′u′ j ⎤ ⎥ ∂x j Cp ∂x j ⎢ ⎦ ⎣ ∂x j
(3)
kin = 1.5(0.04Uin )2
where xi and xj are the Cartesian coordinates of the system (i = x,y,z and j = x,y,z), ū is the mean velocity, P is the mean dynamic pressure, T is the mean temperature, g is the gravitational acceleration; λ, ρ, Cp are the thermal conductivity, density, and the specific heat at constant pressure, respectively. The reference temperature (T0) is the ambient temperature. However, the above set of equations is not complete due to the presence of the Reynolds stress tensor ( ρui′ u′j ) in the momentum equation and the turbulent heat flux vector ( ρCP T ′u′ j ) in the energy equation. To close the turbulence mathematical problem, a turbulence model has to be considered.
εt , in =
(4)
(kin ) 1.5 ly
(5)
The thermal boundary conditions are uniform and constant heat flux on the hot wall and internal heat source, constant temperature on the cold wall and heat flux equal to zero on the remaining walls. 2.3. Heat transfer parameters With the purpose of generalize the results, the non-dimensional Rayleigh and Reynolds numbers were defined as:
2.2.1. Turbulence models In this work, six of most frequently used turbulence models in computational fluid dynamics were considered: the standard k-ε model (skε), the realizable k-ε model (rkε), the renormalized k-ε model (rngkε), the linear pressure strain Reynolds model (RS-LPS), the low Reynolds stress omega model (RS-LRSO) and the standard k-ω model (skω). The standard k-ε model [28] is a semi-empirical two-equation eddy viscosity model, which is based on the Boussinesq hypothesis that assumes that the Reynolds stresses can be expressed in terms of mean velocity gradients and that the turbulent eddy viscosity is related to the turbulent kinetic energy (k) and the dissipation rate of turbulent kinetic energy (ε). Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. The realizable k-ε model [29] is a modification of the skε model and differs in two important ways: it contains a new formulation for the turbulent viscosity and a new transport equation for the dissipation rate has been derived from an exact equation of the transport of the meansquare vorticity fluctuation. The renormalization group theory developed a modified k-ε model [30], which is like the skε but includes the following refinements: it has an additional term in its ε equation that improves the accuracy for rapidly strained flows, the effect of swirl on turbulence is included and provides an analytical formula for turbulent Prandtl numbers, while skε uses a user-specified constant values. The standard k-ω model [31] incorporates modifications for lowReynolds-number effects, compressibility and shear flow spreading. It predicts free shear flow spreading rates that are in close agreement with measurements for far wakes, mixing layers, and plane, round, and radial jets, and is thus applicable to wall-bounded flows and free shear flows. The linear pressure strain Reynolds model [32] is the most elaborate turbulence model among the RANS based models. It does not use the Boussinesq hypothesis and rather than assuming isotropic turbulent viscosity, the RSM closes the RANS equations by solving individual transport equations for the Reynolds stresses, together with an equation for the dissipation rate. This means that five additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D flows. The low Reynolds stress omega model is a stress-transport model
Ra =
Re =
gβq″L4 ναλ
(6)
Uin L y (7)
ν ″
where ν is the kinematic viscosity, q is the heat flux on the heated wall, α is the thermal diffusivity, Ly is the height of the cavity, Uin is the air inlet velocity and λ is the thermal conductivity of the fluid. The Nusselt number is defined as the ratio between the heat flux at the hot wall in the presence of natural convection and the heat flux due to conduction only, i.e.,
Nu =
qconvective qconductive
=
hL y λ
(8)
where h is the convective heat transfer coefficient. 3. Experimental system The experimental cavity is shown in Fig. 2. The walls of the cavity were built of medium density fiberboard (MDF) with a 0.05 m thick core of polystyrene. The hot wall has an electrical heater covered by silicon which has dimensions of 0.91 m × 1.01 m. The electrical heater is in contact with an aluminum plate for a better distribution of the heat across the hot face of the cavity wall and it is supported in a box of MDF with 0.1 m of mineral wool and 0.1 m of polystyrene as thermal insulation. The electrical heater is connected to a Powerstat AC variable autotransformer model 3PN136B, allowing regulate the electrical tension and obtaining the desired thermal power. To keep the right wall of the cavity isothermal, a heat exchanger from the TEMP-PLATE brand, is connected to Cole-Parmer thermostatic bath with water as thermal fluid. The internal heat source is a parallelepiped (base of 0.3048 m × 0.3048 m and height of 0.6096 m) filled with mineral wool. The walls are built with MDF and covered with electrical heaters from the Omega brand. Three thermocouples are placed at each vertical surface and one at the top surface. The electrical heaters in the vertical surfaces are connected in parallel to a BK PRECISION 1790 high current DC regulating power supply and the top surface to a BK PRECISION 1786B single output programmable DC power supply. All the surfaces inside the cavity are covered with polished aluminum sheets 108
International Communications in Heat and Mass Transfer 94 (2018) 106–114
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Fig. 2. Experimental temperature profile (y = z = 0.5 m) for Rawall = 5.5 × 1011, Rasource = 2.3 × 1011 and Re = 32,467.
q″ = hAr (Th − Tc )
Table 2 Independence mesh study. Total nodes
Nusselt number
% difference
20 × 30 × 30 30 × 30 × 30 40 × 30 × 30 50 × 30 × 30
146.9 148.6 148.4 149.2
– 1.14 0.13 0.55
(9)
where q″ is the heat supplied to the surface, h is the convective heat transfer coefficient, Ar is the heat transfer surface, Th is the average temperature of the surface (K) and Tc is the average temperature of the cold wall (K). Considering q = VA, hence:
h=
VA Ar (Th − Tc )
(10)
where V and A, are the electric tension and current, applied to the electrical heater, respectively. The thermocouples were calibrated providing an uncertainty of ± 0.20% of the reading. The equipment for air velocity measurement has an uncertainty of ± 1%. The uncertainty of the experimental heat transfer coefficients was calculated from the following expression [34]:
(εr ≈ 0.03). The data acquisition system is composed of three Agilent data acquisition system model 34970A, with three multiplexor cards and twenty thermocouples capacity each. To monitor the temperature inside the cavity, 84 K type thermocouples with a diameter of 0.079 mm (40 AWG) are used. The thermocouples formed an array of six temperature profiles, on the following heights: y = 0.25 m, 0.50 m, 0.75 m and 0.9 m and at the following depths z = 0.25 m, 0.50 m and 0.75 m. At every temperature profile, 14 thermocouples are placed at the following positions on the x-axis: 0 m, 0.004 m, 0.008 m, 0.012 m, 0.016 m, 0.02 m, 0.03 m, 0.97 m, 0.98 m, 0.984 m, 0.988 m, 0.992 m, 0.996 m and 1.0 m. Besides, 12 thermocouples are located on aluminum film added to the hot surface to determine the heat losses. Three thermocouples measuring the air temperature at the outlets and one thermocouple monitoring the milieu temperature. The experiment initiates setting the heat power at the hot wall and the heat source, and adjusting the wanted air velocity at the inlets. The monitoring interval time for temperature data was every 10 s for a total of 12 h. The behavior of experimental temperature data is illustrated with the results obtained for Rawall = 5.5 × 1011, Rasource = 2.3 × 1011 and Re = 32,467 (Fig. 2). The experimental data are plotted from the beginning to the end of the experiment. The Fig. 2 show that for every position the temperature varies with time, indicating the presence of a turbulent flow regime. The fluctuations of temperature in time are higher inside the thermal boundary layer adjacent to the heated wall. The temperature of heated wall (x = 0 m) reach its maximum values after 6 h and remain almost constant along the experiment. To compare experimental and numerical results a time average treatment is performed on the last 3 h of experimental values, obtaining the average temperatures and its standard deviations. The experimental values of the convective coefficient (h) are computed from the Newton law:
2
2
uh =
2 2 ⎡ ∂h uV ⎤ + ⎡ ∂h uI ⎤ + ⎡ ∂h uTH ⎤ + ⎡ ∂h uTC ⎤ ⎥ ⎢ ⎥ ⎢ ⎣ ∂V ⎦ ⎣ ∂I ⎦ ⎦ ⎣ ∂TH ⎦ ⎣ ∂TC 2
2
2
∂h ∂h ∂h +⎡ u LY ⎤ + ⎡ u LZ ⎤ + ⎡ uθσ ⎤ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎦ ⎦ ⎣ ∂LZ ⎣ ∂LY ⎣ ∂θσ ⎦
(16)
where the first term of the right side of the equation is the uncertainty of the heat transfer coefficient with respect to the voltage. The next terms are the uncertainties with respect to the amperage, the hot surface temperature, the cold surface temperature, the width measure of the cavity wall, height measure and finally the uncertainty associated to the random errors of the experimental procedure. The values of: uV, uI, uTH , uTc, uLy and uLz, were obtained from technical specifications of the measurement devices. However, to obtain the value of uθσ, the experimental procedure was repeated three times for each considered case. The uncertainty percentage values (uncertainty/heat transfer coefficient) varied between 2.1% and 3.9%. 4. Numerical procedure The numerical results are obtained using the CFD commercial package ANSYS FLUENT 15.0, which is based on the finite volume method to solve the governing equations of the fluid motion. For the coupling of momentum and continuity equations, the SIMPLE algorithm is used. The convective terms are discretized applying the MUSCL scheme [35]. The criteria for convergence is that the weighted residue of each of the governing equations is < 10−4. The appropriate mesh 109
International Communications in Heat and Mass Transfer 94 (2018) 106–114
A. Piña-Ortiz et al.
Fig. 3. Experimental-numerical temperature profiles for the case with heat source on for z = 0.5 m.
mesh with 30 nodes at each direction, giving 27,000 computational nodes.
size for cavity is obtained with grid independence study performed with the following conditions: heated wall receives a heat flux of 150 W, the heat source produces a total heat flux of 128 W and the velocity at the inlets is 1.3 m/s. It is shown in Table 2, that the average Nusselt number of the heated wall becomes independent using a structured non-uniform 110
International Communications in Heat and Mass Transfer 94 (2018) 106–114
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Fig. 4. Experimental-numerical temperature profiles for the case with heat source on for y = 0.5 m.
5. Discussion of results
an air velocity at the three inlets of 1.3 m/s). Fig. 3 presents a qualitative comparison between experimental and numerical temperature profiles for the case with the heat source turned on at z = 0.5 m. The y = 0.25 m profile exhibits that the prediction of six turbulence models has an agreement with the experimental thermal boundary layer at the heated wall (x = 0). However, differ significantly
The results are obtained for a Rayleigh numbers at the heated wall of 5.5 × 1011 (corresponding to heat power of 150 W), a Rayleigh number of the heat source of 4.6 × 1011 (corresponding to a total heat power of 128 W), and a Reynolds number of 77,922 (corresponding to 111
International Communications in Heat and Mass Transfer 94 (2018) 106–114
A. Piña-Ortiz et al.
Table 3 Experimental and numerical temperature profile comparison at y = 0.5 m, z = 0.5 m. x (m)
Exp.
rkε
% dif
rngkε
% dif
skε
% dif
RS-LRSO
% dif
RS-LPS
% dif
skω
% dif
0.3 1.8 1.3 1.1 0.9 0.9 0.9 0.7 0.7 0.7 0.7 0.6 0.6 0.1
306.0 304.4 303.3 302.6 302.1 301.9 301.7 301.4 301.1 300.9 300.5 300.0 299.1 297.9
6.2 2.0 0.2 0.5 0.7 0.8 1.0 1.2 1.2 1.1 0.9 0.7 0.2 0.1
329.3 303.7 303.3 303.2 303.3 303.3 303.1 300.4 300.0 299.8 299.5 299.1 298.6 297.9
0.9 2.2 0.2 0.7 1.1 1.3 1.5 0.9 0.8 0.7 0.5 0.4 0.0 0.1
326.3 317.3 311.4 307.6 305.2 303.9 302.7 300.9 300.4 300.1 299.6 299.1 298.5 297.9
0.0 2.2 2.4 2.1 1.8 1.5 1.3 1.1 0.9 0.8 0.6 0.4 0.0 0.1
321.5 314.0 309.0 305.8 303.9 302.8 301.9 301.3 300.9 300.6 300.2 299.6 298.9 297.9
1.5 1.1 1.7 1.5 1.3 1.1 1.1 1.2 1.1 0.9 0.8 0.5 0.1 0.1
327.3 317.7 311.3 307.2 304.7 303.3 302.0 299.8 299.5 299.3 299.0 298.7 298.3 297.8
0.3 2.3 2.4 2.0 1.6 1.3 1.1 0.7 0.6 0.5 0.4 0.2 0.0 0.1
309.3 306.6 304.8 303.7 303.0 302.6 302.3 302.1 301.8 301.5 301.1 300.4 299.3 297.9
5.2 1.3 0.3 0.8 1.0 1.1 1.2 1.5 1.4 1.3 1.1 0.8 0.3 0.1
Heat source 0 0.004 0.008 0.012 0.016 0.02 0.03 0.97 0.98 0.984 0.988 0.992 0.996 1
turned off 327.0 ± 0.4 312.7 ± 1.5 304.5 ± 1.2 301.4 ± 1.0 299.8 ± 0.8 299.0 ± 0.7 298.1 ± 0.7 295.6 ± 0.6 295.6 ± 0.6 295.6 ± 0.6 295.6 ± 0.6 295.8 ± 0.6 296.2 ± 0.6 296.7 ± 0.1
321.3 310.6 304.4 301.0 299.1 298.2 297.5 295.4 292.8 291.5 290.5 290.1 291.7 296.6
1.8 0.6 0.0 0.1 0.2 0.3 0.2 0.1 0.9 1.4 1.7 1.9 1.5 0.0
317.9 309.4 304.1 300.9 299.0 298.0 297.2 295.6 296.4 296.8 297.2 297.4 297.3 296.5
2.8 1.0 0.1 0.2 0.3 0.3 0.3 0.0 0.3 0.4 0.5 0.5 0.3 0.0
317.0 307.0 302.2 299.9 298.6 298.0 297.1 295.6 295.6 295.6 295.7 295.8 296.0 296.6
3.1 1.8 0.7 0.5 0.4 0.4 0.3 0.0 0.0 0.0 0.0 0.0 0.1 0.0
325.6 312.1 304.9 301.8 300.2 299.2 298.0 295.7 295.8 295.9 295.9 296.1 296.2 296.6
0.4 0.2 0.1 0.1 0.1 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.0 0.0
316.9 307.2 302.1 299.6 298.6 298.2 297.9 295.0 290.3 287.9 286.0 285.4 288.1 296.6
3.1 1.7 0.8 0.6 0.4 0.3 0.1 0.2 1.8 2.6 3.3 3.5 2.7 0.0
327.0 312.6 304.8 301.4 299.8 299.0 298.1 296.3 296.3 296.3 296.4 296.4 296.4 296.5
0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.2 0.3 0.3 0.3 0.2 0.1 0.0
Heat source 0 0.004 0.008 0.012 0.016 0.02 0.03 0.97 0.98 0.984 0.988 0.992 0.996 1
turned on 329.1 ± 0.3 315.3 ± 1.6 308.3 ± 1.2 305.7 ± 1.0 304.3 ± 0.9 303.6 ± 0.8 302.8 ± 0.8 300.1 ± 0.7 300.1 ± 0.7 300.0 ± 0.7 300.1 ± 0.7 300.2 ± 0.7 300.8 ± 0.6 299.4 ± 0.1
325.1 313.6 307.7 304.8 303.5 302.9 302.1 299.5 298.9 298.1 297.1 296.1 296.2 298.8
0.6 0.3 1.1 1.2 1.3 1.3 1.3 1.3 1.1 0.9 0.5 0.1 0.0 0.7
321.8 313.2 307.5 304.6 303.0 302.1 300.8 299.5 299.4 299.3 299.3 299.1 299.0 298.9
1.6 0.2 1.0 1.1 1.1 1.0 0.9 1.3 1.3 1.3 1.2 1.1 0.9 0.7
323.5 313.3 308.1 305.6 304.5 303.9 303.0 299.4 296.8 295.2 293.4 292.1 293.1 298.8
1.1 0.2 1.2 1.4 1.6 1.6 1.6 1.3 0.4 0.1 0.8 1.2 1.0 0.7
329.5 314.6 307.1 303.8 302.5 302.1 301.6 299.9 296.3 294.2 291.9 290.2 291.7 298.8
0.7 0.6 0.9 0.8 0.9 1.0 1.2 1.4 0.2 0.5 1.3 1.9 1.5 0.7
321.9 311.4 306.5 304.5 303.7 303.5 302.8 299.6 294.9 292.2 289.4 287.7 289.9 298.9
1.6 0.4 0.7 1.0 1.3 1.5 1.6 1.4 0.2 1.1 2.1 2.7 2.2 0.7
329.9 315.2 307.7 304.3 303.0 302.6 302.2 299.9 293.9 290.2 286.8 284.8 287.9 298.9
0.9 0.8 1.1 1.0 1.1 1.2 1.4 1.5 0.6 1.8 3.0 3.7 2.8 0.8
No heat source 0 326.4 0.004 310.6 0.008 303.9 0.012 301.2 0.016 299.9 0.02 299.4 0.03 298.7 0.97 297.6 0.98 297.6 0.984 297.8 0.988 297.9 0.992 298.0 0.996 298.5 1 298.2
± ± ± ± ± ± ± ± ± ± ± ± ± ±
Table 4 Comparison of experimental and numerical average convective heat transfer coefficients of the heated wall. Exp
rkε
No heat source 3.9 3.3
Dif (%)
rngkε
Dif (%)
RS-LPS
Dif (%)
RS-LRSO
Dif (%)
skε
Dif (%)
skω
Dif (%)
15.4
3.3
15.4
3.3
15.4
3.5
10.3
3.2
17.9
3.3
15.4
Heat source turned off 3.8 5.3 39.5
5.5
44.7
5.4
42.1
4.2
10.5
5.4
42.1
3.6
5.3
Heat source turned on 3.8 4.3 14.2
4.8
27.9
4.6
21.3
3.6
4.7
4.5
18.9
3.2
14.2
Table 5 Comparison of experimental and numerical Nusselt numbers of the heated wall. Exp
rkε
No heat source 147 127
Dif (%)
rngkε
Dif (%)
RS-LPS
Dif (%)
RS-LRSO
Dif (%)
skε
Dif (%)
skω
Dif (%)
13.6
126
14.3
127
13.6
136
7.5
124
15.6
126
14.3
Heat source turned off 144 202 40.3
211
46.5
209
45.1
161
11.8
209
45.1
138
4.2
Heat source turned on 146 166 13.7
186
27.4
176
20.5
138
5.5
173
18.5
125
14.4
112
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values of average convective heat transfer coefficient and Nusselt number of the heated wall are minimally modified by the presence of the heat source (even if it is turned on or turned off), since the first varies from 3.8 to 3.9 W/m2 K and the last oscillate from 144 till 147. The previous behavior is predicted by the skω turbulence model.
around the isothermal wall (x = 1), the highest deviation corresponds to the RS-LRSO turbulence model, whereas the lowest deviation is obtained with RS-LPS turbulence model. For y = 0.75 m, it is observed that the turbulence models under predict the boundary layers adjacent to the heated and cold walls, the better agreement corresponds to rkε and RS-LPS turbulence models. The experimental temperature profile at y = 0.9 m, is better described for rkε and RS-LPS turbulence models, however all turbulence models under predict the temperature of the heated wall and the highest deviation is observed for RS-LRSO turbulence model near the cold wall. The comparison between experimental and numerical temperature profiles for the case with heat source turned on for y = 0.5 m, is shown in Fig. 4. The experimental temperature profiles are very similar; however, turbulence models predict differences between profiles. At z = 0.25 m is observed the better agreement correspond to the rkε turbulence model, whereas the RS-LRSO differ considerably from the experimental temperatures near the cold wall. For z = 0.5 m, most of turbulence models predict well the thermal boundary layer adjacent to the heated wall (except the RS-LRSO turbulence model) but differ near the cold wall. Finally, for z = 0.75 m all turbulence models are very close to the thermal boundary layer of the heated wall, but again deviate near the cold wall. Therefore, for the three profiles the numerical values under predict the temperatures near the cold wall with the highest deviation observed in the z = 0.5 m profile. A quantitative comparison between an experimental and numerical temperature profile (y = 0.5 m, z = 0.5 m) is presented at Table 3. The comparison is made for three cases (a) ventilated cavity without heat source, (b) ventilated cavity with the heat source turned off (it acts like an obstacle to the fluid flow) and (c) ventilated cavity with the heat source turned on. The case without heat source presents a difference of 6.2% with rkε model in x = 0 m and no difference with several models (rngkε, skε, RS-LPS) at x = 0.996 m, however the average of percentage differences indicates a minimum 0.81% with the rngkε turbulence model and a maximum of 1.24% with skω turbulence model. On the other hand, the case with the heat source turned off indicates no difference with several models at various positions of the profile, whilst the highest difference is 3.1% at x = 0 m with skε and RS-LPS turbulence models; in this case the average of percentage differences shows a minimum of 0.1% with the LR-LRSO turbulence model and a maximum of 1.51% with the RS-LPS turbulence model. Finally, the case with the heat source turned on exhibit the highest difference at the heated and cold wall boundary layer points with a 3.7% for the skω at x = 0.992 m and the lowest percentage difference is at x = 0.996 m with rkε turbulence model with zero difference. However, the average of temperature differences indicates that the best turbulence model was rkε (0.84%) and the worst was the LR-LRSO turbulence model (1.55%). The comparison between experimental and numerical average convective heat transfer coefficients of the heated wall is presented in Table 4. For the case without heat source, the lowest and highest percentage differences were for the RS-LRSO and skε models with 10.3 and 17.9%, respectively. Instead, for the case with heat source turned off the highest and lowest percentage differences are 44.7 and 5.3 with the rngkε and skω turbulence models, respectively. At last, the case with heat source turned on presents a minimum difference of 4.7% with RSLRSO model and a maximum difference with the rngkε model with 27.9%. In Table 5, is presented the comparison between experimental and numerical Nusselt numbers of the heated wall. The minimum and maximum percentage difference for the case with heat source turned on, were obtained with RS-LRSO and rngkε models with a 5.5 and 27.4%, respectively. The case with the heat source turned off, displays a maximum and minimum difference of 46.5 and 4.2% with the rngkε and skω model, in the order given. Finally, the case without heat source exhibits a maximum difference of 15.6% with the skε model and the minimum difference is for the RS-LRSO model with 7.5%. From the experimental results presented in Tables 4 and 5, is observed that the
6. Conclusions In this paper, a comparison between experimental data and numerical results for the turbulent heat transfer in a ventilated cavity, with and without an internal heat source, was presented. From the results, we can conclude the following:
• The comparison between experimental and numerical temperature •
•
values indicates that, the lowest average of percentage differences for the case with the heat source turned on corresponds to realizable k-ε turbulence model (rkε) with 0.84%. The comparison between the average convective heat transfer coefficient and Nusselt number indicates that the best predictions are obtained with: (a) RS-LRSO turbulence model for the case without heat source, (b) skω turbulence model for the case with heat source turned off and (c) RS-LRSO for the case with heat source turned on. The experimental average convective heat transfer coefficient and Nusselt number of the heated wall, are minimally influenced by the presence of a heat source (regardless of whether this on or off) within the cavity. The previous behavior is predicted by the skω turbulence model.
Acknowledgments The first author would like to acknowledge to the National Science and Technology Council of the Mexican Republic (291113), for the support granted through its program of postdoctoral fellowships. References [1] S. Singh, M.A.R. Sharif, Mixed convective cooling of a rectangular cavity with inlets and exit openings on differentially heated side walls, Numer. Heat Transf. A 44 (2002) 233–253. [2] J.D. Posner, C.R. Buchanan, D. Dunn-Rankin, Measurement and prediction of indoor air flow in a model room, Energy Build. 35 (2003) 515–526. [3] N.O. Moraga, S.E. Lopez, Numerical simulation of three-dimensional mixed convection in an air-cooled cavity, Numer. Heat Transf. A 45 (2004) 811–824. [4] V. Haslavsky, J. Tanny, M. Teitel, Interaction between the mixing and displacement modes in a naturally ventilated enclosure, Build. Environ. 41 (2006) 1755–1761. [5] M.M. Rahman, M.A. Alim, Numerical study of opposing mixed convection in a vented enclosure, ARPN J. Eng. Appl. Sci. 2 (2007) 25–36. [6] R. Daghigh, N.M. Adam, B.B. Sahari, Influences of air exchange effectiveness and its rate on thermal comfort: naturally ventilated office, J. Build. Phys. 32 (2008) 175–194. [7] A. Raji, M. Hasnaoui, A. Bahlaoui, Numerical study of natural convection dominated heat transfer in a ventilated cavity: case of forced flow playing simultaneous assisting and opposing roles, Int. J. Heat Fluid Flow 29 (2008) 1174–1181. [8] J. Tanny, V. Haslavsky, M. Teitel, Airflow and heat flux through the vertical opening of buoyancy-induced naturally ventilated enclosures, Energy Build. 40 (2008) 637–646. [9] S. Saha, A.H. Mamun, Z. Hossain, S. Islam, Mixed convection in an enclosure with different inlet and exit configurations, J. Appl. Fluid Mech. 1 (2008) 78–93. [10] G.M. Stavrakakis, M.K. Koukou, M.G. Vrachopoulos, N.C. Markato, Natural crossventilation in buildings: building-scale experiments, numerical simulation and thermal comfort evaluation, Energy Build. 40 (2008) 1666–1681. [11] L. Susanti, H. Homma, H. Matsumoto, Y. Suzuki, M. Shimizu, A laboratory experiment on natural ventilation through a roof cavity for reduction of solar heat gain, Energy Build. 40 (2008) 2196–2206. [12] R. Ezzouhri, P. Joubert, F. Penot, S. Mergui, Large Eddy simulation of turbulent mixed convection in a 3D ventilated cavity: comparison with existing data, Int. J. Therm. Sci. 48 (2009) 2017–2024. [13] A. Lariani, H. Nesreddine, N. Galanis, Numerical and experimental study of 3D turbulent airflow in a full scale heated ventilated room, Eng. Appl. Comput. Fluid Mech. 3-1 (2009) 1–14. [14] P. Karava, T. Stathopoulos, A.K. Athienitis, Airflow assessment in cross-ventilated buildings with operable façade elements, Build. Environ. 46 (2011) 266–279. [15] T.S. Larsen, N. Nikolopoulos, A. Nikolopoulos, G. Strotos, K.S. Nikas,
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Characterization and prediction of the volume flow rate aerating a cross ventilated building by means of experimental techniques and numerical approaches, Energy Build. 43 (2011) 1371–1381. N.A. Rodríguez, J.F. Hinojosa, Numerical study of airflow and heat transfer in an air-cooled room whit different inlet positions, J. Build. Phys. 37 (3) (2014) 246–268. J.F. Hinojosa, N.A. Rodriguez, J. Xaman, Heat transfer and airflow study of turbulent mixed convection in a ventilated cavity, J. Build. Phys. 40 (3) (2016) 204–234. E. Papanicolaou, Y. Jaluria, Mixed convection from an isolated heat source in a rectangular enclosure, Numer. Heat Transf. A 18 (1990) 427–461. E. Papanicolaou, Y. Jaluria, Mixed convection from localized heat source in a cavity with conducting walls, Numer. Heat Transf. A 23 (1993) 463–484. T. Hsu, S. Wang, Mixed convection in a rectangular enclosure with discrete heat sources, Numer. Heat Transf. A 38 (2000) 627–652. T. Radhakrishnan, A. Verma, C. Balaji, S.P. Venkateshan, An experimental and numerical investigation of mixed convection from a heat generating element in a ventilated cavity, Exp. Thermal Fluid Sci. 32 (2007) 502–520. B. Ghasemi, S.M. Aminossadati, Numerical simulation of mixed convection in a rectangular enclosure with different numbers and arrangements of discrete heat sources, Arab. J. Sci. Eng. 33 (2008) 189–207. E. Bilgen, A. Muftuoglu, Cooling strategy by mixed convection of a discrete heater at its optimum position in a square cavity with ventilation ports, Int. Commun. Heat Mass Transf. 35 (2008) 545–550. N.A. Rodriguez-Muñoz, Z.C. Briceño-Ahumada, J.F. Hinojosa-Palafox, Numerical study of heat transfer by convection and thermal radiation in a ventilated room with human heat generation and CO2 production, Lat. Am. Appl. Res. 43 (2013) 353–361. N. Biswas, P.S. Mahapatra, N.K. Manna, Thermal management of heating element in a ventilated enclosure, Int. Commun. Heat Mass Transf. 66 (2015) 84–92. S.K. Ajmera, A.N. Mathur, Experimental investigation of mixed convection in multiple ventilated enclosure with discrete heat sources, Exp. Thermal Fluid Sci. 68 (2015) 402–411. M.M. Gibson, B.E. Launder, Ground effects on pressure fluctuations in the atmospheric boundary layer, J. Fluid Mech. 86 (1978) 491–511. T.H. Shih, W.W. Liou, A. Shabbir, Z. Yang, J. Zhu, A new k-ε eddy-viscosity model for high Reynolds number turbulent flows-model development and validation, Comput. Fluids 24 (3) (1995) 227–238. D. Choudhury, Introduction to the renormalization group method and turbulence modeling, Technical Memorandum, Fluent Inc., 1993(TM-107). D.C. Wilcox, Turbulence Modeling for CFD, DCW Industries Inc., La Canada, California, 1998. S. Fu, B.E. Launder, M.A. Leschziner, Modelling strongly swirling recirculating jet flow with Reynolds-stress transport closures, Sixth Symposium on Turbulent Shear
Flows, Touluose, France, 1987. [32] P. Nielsen, Specification of a two-dimensional test case, Energy Conservation in Building and Community System. Annex 20, Vol. 20 1990. [33] J.P. Holman, Experimental Methods for Engineers, Sixth edition, McGraw-Hill, 1994. [34] B. Van Leer, Towards the ultimate conservative difference scheme, J. Comput. Phys. 32 (1979) 101–136.
Nomenclature A: electric current, (A) Ar: area, m2 Cp: specific heat at constant pressure, (J/kg K) g: gravitational acceleration, (m/s2) Gr: GRASHOF number, nondimensional h: average convective heat transfer coefficient, (W/m2 K) k: turbulent kinetic energy, (m2/s2) L: cavity wall length, (m) li: inlet length, (m) Nu: Nusselt number, nondimensional q: heat flux, (W/m2) Ra: modified Rayleigh number, nondimensional Re: Reynolds number, nondimensional Ri: Richardson number, nondimensional Th : average temperature of the hot wall, (K) Tc: temperature of the cold wall, (K) uh: experimental uncertainty, nondimensional V: voltage, (V) x, y, z: coordinate system, (m)
Greek symbols α: thermal diffusivity, (m2/s) β: thermal expansion coefficient, (1/K) εr: emissivity, nondimensional ε: turbulent kinetic energy dissipation, (J/kg) λ: thermal conductivity, (W/m K) μt: turbulent viscosity, (kg/m s) ν: kinematic viscosity, (m2/s) ρ: density, (kg/m3) ω: turbulent specific dissipation rate, nondimensional
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Test of turbulence models for heat transfer within a ventilated cavity with and without an internal heat source
T
⁎
A. Piña-Ortiza, J.F. Hinojosaa, , J.P. Xamánb, J.M.A. Navarroa a b
Departamento de Ingeniería Química y Metalurgia, Universidad de Sonora, Hermosillo 83000, Sonora, Mexico Centro Nacional de Investigación y Desarrollo Tecnológico, CENIDET-DGEST-SEP, prol. Av. Palmira S/N. Col. Palmira, Cuernavaca, Morelos 62490, Mexico
A R T I C LE I N FO
A B S T R A C T
Keywords: Ventilated cavity Internal heat source Turbulence models
In this study, six of most frequently used turbulence models in computational fluid dynamics were compared with experimental data. The experimental cavity is a cube whose edge is 1 m long. The left vertical wall receives a constant and uniform heat flux, while the opposite wall is kept at a constant temperature. The rest of the walls are adiabatic. The heat source is a rectangular parallelepiped with square base 0.3 m on a side and height 0.61 m. The cavity represents a ventilated room in a 1:3 scale with multiple inlets and outlets of air, considering ventilation by ducts of an air-conditioning system. The experimental setup was built to obtain temperature profiles and heat transfer coefficients. Temperature profiles were obtained at six different depths and heights consisting of 14 thermocouples each. It was found that the lowest average of percentage differences for the case with the heat source turned on corresponds to realizable k-ε turbulence model (rkε) with 0.84%.
1. Introduction At recent years, there has been a sustained increase in energy consumption at arid regions around the world also known as desert climate. In this climate, there are large diurnal temperature variations. Therefore, due to the design of existing buildings, artificial air conditioning systems are required to achieve comfort conditions, resulting in high electricity consumption. The consequence is an ecological footprint, since most of primary electricity production around the world comes from fossil fuels that are responsible of the discharge into the atmosphere of important amounts of greenhouse effect gases, mainly carbon dioxide. Therefore, the study of buildings ventilation has become an important need to reduce the energy consumption. On the other hand, the interaction between electronic equipment or people's heat fluxes at the interior may cause different flow and temperature patterns, therefore it must be analyzed to optimize and to reduce the electricity consumption of the ventilation systems. To understand the effect of heat transfer on airflow and temperature distribution, theoretical studies may be used considering a room as a cavity, to get a better control of the studied parameters. With the mathematical model, it is possible to predict what will happen in the system, but it is necessary to establish its predictive capacity by comparing with experimental data of the thermal system. The investigations of heat transfer in ventilated cavities are numerous [1–32], therefore, will be described only the studies with
⁎
Corresponding author. E-mail address: [email protected] (J.F. Hinojosa).
https://doi.org/10.1016/j.icheatmasstransfer.2018.03.021
0735-1933/ © 2018 Elsevier Ltd. All rights reserved.
internal heat generation. Those studies may be organized as: (a) Numerical, (b) Experimental and (c) Numerical-Experimental. (a) Numerical studies in ventilated cavities with internal heat generation. Papanicolaou and Jaluria [18] carried out a numerical study of the mixed convection in a rectangular ventilated cavity with a discrete heat source in it. The effects of the Reynolds number, Richardson number, position of the heat source and position of the outflow on heat transfer and temperature distribution were observed for a Reynolds number from 50 to 2000. The obtained results were used to study heat removal mechanisms in practical systems. In another article, these authors [19] studied numerically a problem of mixed convection considering a cavity with two ventilation ports, conductive walls and a discrete heat source on a wall. The Reynolds number was fixed at 100 and Richardson number was varying between 0 and 10. The results show that the configuration in which the two types of convection are assisted presents higher heat transfer and lower sources temperatures. Hsu and Wang [20] performed a numerical study about mixed convection in a ventilated cavity with discrete heat sources embedded on a vertical board which is situated on the bottom wall of the enclosure. The Reynolds number was studied from 100 to 1000. They found that when the source is located on the right surface of the board the Nusselt number is not depending neither on the variation of the
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location of the source nor of the board. Ghasemi and Aminossadati [22] studied numerically the mixed convection in a two-dimensional ventilated cavity with discrete heat sources. They examined the effects of the number and position of the sources, the Rayleigh number from 0 to 107 at a Reynolds number of 100. Results show that increasing significantly Rayleigh number improves the heat transfer process in the cavity. The arrangement of sources also has a great contribution on the cooling performance but when the Rayleigh number is increased this contribution decreases. Bilgen and Muftuoglu [23] numerically investigated a cooling strategy in a square cavity with adiabatic walls and a heat source on the left wall. The Rayleigh number was studied from 103 to 107 and the Reynolds number from 102 to 103. The authors observed that the optimal position of the source is almost insensitive to variation in Rayleigh and Reynolds numbers, but it is strongly affected by the arrangements on the ventilation ports. It was found that the highest cooling performance is given by placing the air outlet on the upper left part of the cavity. Rodriguez-Muñoz et al. [24] studied numerically the combined effect of heat generation produced by a human being and the mixed turbulent convection with thermal radiation, as well as the CO2 production from respiration in a rectangular ventilated room. The results showed that the ventilation reduces the average temperatures in the room between 4 °C and 5.5 °C, while the thermal radiation increases the average temperature between 0.2 °C and 0.4 °C. Biswas et al. [25] numerically studied thermal management in a ventilated enclosure undergoing mixed convection by dividing the entire heating element into multiple equal segments and by positioning them appropriately on vertical side walls, namely at bottom, middle or top positions. Analysis of segmental heating and whole heating are conducted for different Richardson number (0.01–100) and Reynolds number (50–200). Nine positional configurations of bi-segmental heating reveal the possibility of significant enhancement in heat transfer.
Fig. 1. Physical model of the studied cavity.
ventilated cavity without and with an internal heat source (turned on and turned off). The cavity has multiple inlets and outlets of air, considering ventilation by ducts of an air-conditioning system. In addition, the cavity receives a heat flux in one vertical wall (representing incoming heat from the exterior). The experimental setup was built to obtain temperature profiles and heat transfer coefficients. The numerical results obtained with six turbulence models, are compared with experimental results and percentage differences are computed. 2. Physical and mathematical model 2.1. Physical model The study of turbulent mixed convection was performed in the cubic cavity shown in Fig. 1. The cavity represents a ventilated room in a 1:3 scale. The dimensions of the system are as follows: Lx = Ly = Lz = 1.0 m and it consists of one vertical wall (x = 0) receiving a constant and uniform heat flux, while the vertical facing wall (x = Lx) was kept at a constant temperature Tc (298 K). The remaining walls were assumed as adiabatic. Every wall of the cavity was covered with a film of polished aluminum to minimize the thermal radiation exchange. The thermal fluid is air. The dimensions of the inlet and outlet are lx = ly = 0.08 m and their positions are described in Table 1. The air at constant temperature enters the cavity with velocities of 0.5 or 1.3 m/s. At the center of the cavity, there is a parallelepiped with a height of 0.61 m and a depth and length of 0.30 m with electrical heaters in every surface to represent the generated heat by a person in a room. The fluid flow was assumed as turbulent and because the temperature differences are < 30 K, the Boussinesq approach was considered valid.
(b) Experimental studies in ventilated cavities with internal heat generation. Ajmera and Mahur [26] performed an experimental investigation of mixed convection in multiple ventilated enclosures with discrete heat sources. The flow velocity and applied heat flux were varied. The experimental investigations were executed for a range of Reynolds number and Grashof number of 270 ≤ Re ≤ 6274 and 7.2 × 106 ≤ Gr ≤ 5.5 × 107, whereas the Richardson number lied in the range of 0.201–571. Different correlations were proposed for Nusselt number within the range of considered parameters. (c) Experimental and numerical studies in ventilated cavities with internal heat generation. Radhakrishnan et al. [21] reported a numerical and experimental work about turbulent mixed convection in a ventilated cavity with adiabatic walls and a discrete heat source inside. The k–ε (RNG) model is adopted for the turbulence closure in the two-dimensional numerical study. Correlations were developed for the average Nusselt number and the maximum dimensionless temperature occurring in the heat source, in these parameter ranges: 1200 ≤ Re ≤ 10,000 and 0.003 ≤ Ri ≤ 0.2. The authors concluded that a combined experimental and numerical investigation would significantly reduce the effort required to optimize the cooling of electronic equipment. The literature review indicates very scarce numerical and experimental studies analyzing the effect of an internal heat source on turbulent mixed convection in a ventilated cavity. Nevertheless, the study of turbulent heat transfer in a ventilated cavity with an internal heat source is relevant for an adequate thermal design of a building. Considering the above, this paper presents an experimental and numerical study of three-dimensional turbulent mixed convection in a
2.2. Mathematical model The considerations for the mathematical model are: steady state, Newtonian fluid, turbulent flow regime, negligible viscosity dissipation and the use of the Boussinesq approximation. The time averaged Table 1 Inlet and outlet positions. Position (x, y, z) Inlet 1 Inlet 2 Inlet 3 Outlet 1 Outlet 2 Outlet 3
107
0.46 m ≤ x ≤ 0.54 m 0.46 m ≤ x ≤ 0.54 m 0.46 m ≤ x ≤ 0.54 m 0.71 m ≤ x ≤ 0.79 m 0.71 m ≤ x ≤ 0.79 m 0.71 m ≤ x ≤ 0.79 m
y = 1.0 m y = 1.0 m y = 1.0 m y = 1.0 m y = 1.0 m y = 1.0 m
0.21 m ≤ z ≤ 0.29 m 0.46 m ≤ z ≤ 0.54 m 0.71 m ≤ z ≤ 0.79 m 0.21 m ≤ z ≤ 0.29 m 0.46 m ≤ z ≤ 0.54 m 0.71 m ≤ z ≤ 0.79 m
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that is based on the omega equation [31]. This model is ideal for modeling flows over curved surfaces and swirling flows and it is good for predictions over a wide range of turbulent flows. In this model, the closure coefficients are identical to the skω model; however, there are additional closure coefficients. The hydrodynamic boundary conditions are non-slip conditions on the walls; therefore, the velocity components are equal to zero. Since the air was considered to enter perpendicularly to the opening, the ycomponent of the velocity had a constant value whereas the remaining components are equal to zero. The pressure outlet boundary condition is used for the outgoing air. The turbulent kinetic energy and the dissipation of the turbulent kinetic energy for the incoming air were obtained by applying empirical correlations [33].
governing equations in tensor notation, are as follow: Continuity:
∂ui =0 ∂x i
(1)
Momentum:
ρuj
∂ui ∂P ∂ ⎡ ∂ui = + μ − ρui′ u′j ⎤ + ρgi β (T − T0) ⎥ ∂x j ∂x i ∂x j ⎢ ⎣ ∂x j ⎦
(2)
Energy:
ρuj
∂T 1 ∂ ⎡ ∂T = λ − ρCP T ′u′ j ⎤ ⎥ ∂x j Cp ∂x j ⎢ ⎦ ⎣ ∂x j
(3)
kin = 1.5(0.04Uin )2
where xi and xj are the Cartesian coordinates of the system (i = x,y,z and j = x,y,z), ū is the mean velocity, P is the mean dynamic pressure, T is the mean temperature, g is the gravitational acceleration; λ, ρ, Cp are the thermal conductivity, density, and the specific heat at constant pressure, respectively. The reference temperature (T0) is the ambient temperature. However, the above set of equations is not complete due to the presence of the Reynolds stress tensor ( ρui′ u′j ) in the momentum equation and the turbulent heat flux vector ( ρCP T ′u′ j ) in the energy equation. To close the turbulence mathematical problem, a turbulence model has to be considered.
εt , in =
(4)
(kin ) 1.5 ly
(5)
The thermal boundary conditions are uniform and constant heat flux on the hot wall and internal heat source, constant temperature on the cold wall and heat flux equal to zero on the remaining walls. 2.3. Heat transfer parameters With the purpose of generalize the results, the non-dimensional Rayleigh and Reynolds numbers were defined as:
2.2.1. Turbulence models In this work, six of most frequently used turbulence models in computational fluid dynamics were considered: the standard k-ε model (skε), the realizable k-ε model (rkε), the renormalized k-ε model (rngkε), the linear pressure strain Reynolds model (RS-LPS), the low Reynolds stress omega model (RS-LRSO) and the standard k-ω model (skω). The standard k-ε model [28] is a semi-empirical two-equation eddy viscosity model, which is based on the Boussinesq hypothesis that assumes that the Reynolds stresses can be expressed in terms of mean velocity gradients and that the turbulent eddy viscosity is related to the turbulent kinetic energy (k) and the dissipation rate of turbulent kinetic energy (ε). Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. The realizable k-ε model [29] is a modification of the skε model and differs in two important ways: it contains a new formulation for the turbulent viscosity and a new transport equation for the dissipation rate has been derived from an exact equation of the transport of the meansquare vorticity fluctuation. The renormalization group theory developed a modified k-ε model [30], which is like the skε but includes the following refinements: it has an additional term in its ε equation that improves the accuracy for rapidly strained flows, the effect of swirl on turbulence is included and provides an analytical formula for turbulent Prandtl numbers, while skε uses a user-specified constant values. The standard k-ω model [31] incorporates modifications for lowReynolds-number effects, compressibility and shear flow spreading. It predicts free shear flow spreading rates that are in close agreement with measurements for far wakes, mixing layers, and plane, round, and radial jets, and is thus applicable to wall-bounded flows and free shear flows. The linear pressure strain Reynolds model [32] is the most elaborate turbulence model among the RANS based models. It does not use the Boussinesq hypothesis and rather than assuming isotropic turbulent viscosity, the RSM closes the RANS equations by solving individual transport equations for the Reynolds stresses, together with an equation for the dissipation rate. This means that five additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D flows. The low Reynolds stress omega model is a stress-transport model
Ra =
Re =
gβq″L4 ναλ
(6)
Uin L y (7)
ν ″
where ν is the kinematic viscosity, q is the heat flux on the heated wall, α is the thermal diffusivity, Ly is the height of the cavity, Uin is the air inlet velocity and λ is the thermal conductivity of the fluid. The Nusselt number is defined as the ratio between the heat flux at the hot wall in the presence of natural convection and the heat flux due to conduction only, i.e.,
Nu =
qconvective qconductive
=
hL y λ
(8)
where h is the convective heat transfer coefficient. 3. Experimental system The experimental cavity is shown in Fig. 2. The walls of the cavity were built of medium density fiberboard (MDF) with a 0.05 m thick core of polystyrene. The hot wall has an electrical heater covered by silicon which has dimensions of 0.91 m × 1.01 m. The electrical heater is in contact with an aluminum plate for a better distribution of the heat across the hot face of the cavity wall and it is supported in a box of MDF with 0.1 m of mineral wool and 0.1 m of polystyrene as thermal insulation. The electrical heater is connected to a Powerstat AC variable autotransformer model 3PN136B, allowing regulate the electrical tension and obtaining the desired thermal power. To keep the right wall of the cavity isothermal, a heat exchanger from the TEMP-PLATE brand, is connected to Cole-Parmer thermostatic bath with water as thermal fluid. The internal heat source is a parallelepiped (base of 0.3048 m × 0.3048 m and height of 0.6096 m) filled with mineral wool. The walls are built with MDF and covered with electrical heaters from the Omega brand. Three thermocouples are placed at each vertical surface and one at the top surface. The electrical heaters in the vertical surfaces are connected in parallel to a BK PRECISION 1790 high current DC regulating power supply and the top surface to a BK PRECISION 1786B single output programmable DC power supply. All the surfaces inside the cavity are covered with polished aluminum sheets 108
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Fig. 2. Experimental temperature profile (y = z = 0.5 m) for Rawall = 5.5 × 1011, Rasource = 2.3 × 1011 and Re = 32,467.
q″ = hAr (Th − Tc )
Table 2 Independence mesh study. Total nodes
Nusselt number
% difference
20 × 30 × 30 30 × 30 × 30 40 × 30 × 30 50 × 30 × 30
146.9 148.6 148.4 149.2
– 1.14 0.13 0.55
(9)
where q″ is the heat supplied to the surface, h is the convective heat transfer coefficient, Ar is the heat transfer surface, Th is the average temperature of the surface (K) and Tc is the average temperature of the cold wall (K). Considering q = VA, hence:
h=
VA Ar (Th − Tc )
(10)
where V and A, are the electric tension and current, applied to the electrical heater, respectively. The thermocouples were calibrated providing an uncertainty of ± 0.20% of the reading. The equipment for air velocity measurement has an uncertainty of ± 1%. The uncertainty of the experimental heat transfer coefficients was calculated from the following expression [34]:
(εr ≈ 0.03). The data acquisition system is composed of three Agilent data acquisition system model 34970A, with three multiplexor cards and twenty thermocouples capacity each. To monitor the temperature inside the cavity, 84 K type thermocouples with a diameter of 0.079 mm (40 AWG) are used. The thermocouples formed an array of six temperature profiles, on the following heights: y = 0.25 m, 0.50 m, 0.75 m and 0.9 m and at the following depths z = 0.25 m, 0.50 m and 0.75 m. At every temperature profile, 14 thermocouples are placed at the following positions on the x-axis: 0 m, 0.004 m, 0.008 m, 0.012 m, 0.016 m, 0.02 m, 0.03 m, 0.97 m, 0.98 m, 0.984 m, 0.988 m, 0.992 m, 0.996 m and 1.0 m. Besides, 12 thermocouples are located on aluminum film added to the hot surface to determine the heat losses. Three thermocouples measuring the air temperature at the outlets and one thermocouple monitoring the milieu temperature. The experiment initiates setting the heat power at the hot wall and the heat source, and adjusting the wanted air velocity at the inlets. The monitoring interval time for temperature data was every 10 s for a total of 12 h. The behavior of experimental temperature data is illustrated with the results obtained for Rawall = 5.5 × 1011, Rasource = 2.3 × 1011 and Re = 32,467 (Fig. 2). The experimental data are plotted from the beginning to the end of the experiment. The Fig. 2 show that for every position the temperature varies with time, indicating the presence of a turbulent flow regime. The fluctuations of temperature in time are higher inside the thermal boundary layer adjacent to the heated wall. The temperature of heated wall (x = 0 m) reach its maximum values after 6 h and remain almost constant along the experiment. To compare experimental and numerical results a time average treatment is performed on the last 3 h of experimental values, obtaining the average temperatures and its standard deviations. The experimental values of the convective coefficient (h) are computed from the Newton law:
2
2
uh =
2 2 ⎡ ∂h uV ⎤ + ⎡ ∂h uI ⎤ + ⎡ ∂h uTH ⎤ + ⎡ ∂h uTC ⎤ ⎥ ⎢ ⎥ ⎢ ⎣ ∂V ⎦ ⎣ ∂I ⎦ ⎦ ⎣ ∂TH ⎦ ⎣ ∂TC 2
2
2
∂h ∂h ∂h +⎡ u LY ⎤ + ⎡ u LZ ⎤ + ⎡ uθσ ⎤ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎦ ⎦ ⎣ ∂LZ ⎣ ∂LY ⎣ ∂θσ ⎦
(16)
where the first term of the right side of the equation is the uncertainty of the heat transfer coefficient with respect to the voltage. The next terms are the uncertainties with respect to the amperage, the hot surface temperature, the cold surface temperature, the width measure of the cavity wall, height measure and finally the uncertainty associated to the random errors of the experimental procedure. The values of: uV, uI, uTH , uTc, uLy and uLz, were obtained from technical specifications of the measurement devices. However, to obtain the value of uθσ, the experimental procedure was repeated three times for each considered case. The uncertainty percentage values (uncertainty/heat transfer coefficient) varied between 2.1% and 3.9%. 4. Numerical procedure The numerical results are obtained using the CFD commercial package ANSYS FLUENT 15.0, which is based on the finite volume method to solve the governing equations of the fluid motion. For the coupling of momentum and continuity equations, the SIMPLE algorithm is used. The convective terms are discretized applying the MUSCL scheme [35]. The criteria for convergence is that the weighted residue of each of the governing equations is < 10−4. The appropriate mesh 109
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Fig. 3. Experimental-numerical temperature profiles for the case with heat source on for z = 0.5 m.
mesh with 30 nodes at each direction, giving 27,000 computational nodes.
size for cavity is obtained with grid independence study performed with the following conditions: heated wall receives a heat flux of 150 W, the heat source produces a total heat flux of 128 W and the velocity at the inlets is 1.3 m/s. It is shown in Table 2, that the average Nusselt number of the heated wall becomes independent using a structured non-uniform 110
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Fig. 4. Experimental-numerical temperature profiles for the case with heat source on for y = 0.5 m.
5. Discussion of results
an air velocity at the three inlets of 1.3 m/s). Fig. 3 presents a qualitative comparison between experimental and numerical temperature profiles for the case with the heat source turned on at z = 0.5 m. The y = 0.25 m profile exhibits that the prediction of six turbulence models has an agreement with the experimental thermal boundary layer at the heated wall (x = 0). However, differ significantly
The results are obtained for a Rayleigh numbers at the heated wall of 5.5 × 1011 (corresponding to heat power of 150 W), a Rayleigh number of the heat source of 4.6 × 1011 (corresponding to a total heat power of 128 W), and a Reynolds number of 77,922 (corresponding to 111
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Table 3 Experimental and numerical temperature profile comparison at y = 0.5 m, z = 0.5 m. x (m)
Exp.
rkε
% dif
rngkε
% dif
skε
% dif
RS-LRSO
% dif
RS-LPS
% dif
skω
% dif
0.3 1.8 1.3 1.1 0.9 0.9 0.9 0.7 0.7 0.7 0.7 0.6 0.6 0.1
306.0 304.4 303.3 302.6 302.1 301.9 301.7 301.4 301.1 300.9 300.5 300.0 299.1 297.9
6.2 2.0 0.2 0.5 0.7 0.8 1.0 1.2 1.2 1.1 0.9 0.7 0.2 0.1
329.3 303.7 303.3 303.2 303.3 303.3 303.1 300.4 300.0 299.8 299.5 299.1 298.6 297.9
0.9 2.2 0.2 0.7 1.1 1.3 1.5 0.9 0.8 0.7 0.5 0.4 0.0 0.1
326.3 317.3 311.4 307.6 305.2 303.9 302.7 300.9 300.4 300.1 299.6 299.1 298.5 297.9
0.0 2.2 2.4 2.1 1.8 1.5 1.3 1.1 0.9 0.8 0.6 0.4 0.0 0.1
321.5 314.0 309.0 305.8 303.9 302.8 301.9 301.3 300.9 300.6 300.2 299.6 298.9 297.9
1.5 1.1 1.7 1.5 1.3 1.1 1.1 1.2 1.1 0.9 0.8 0.5 0.1 0.1
327.3 317.7 311.3 307.2 304.7 303.3 302.0 299.8 299.5 299.3 299.0 298.7 298.3 297.8
0.3 2.3 2.4 2.0 1.6 1.3 1.1 0.7 0.6 0.5 0.4 0.2 0.0 0.1
309.3 306.6 304.8 303.7 303.0 302.6 302.3 302.1 301.8 301.5 301.1 300.4 299.3 297.9
5.2 1.3 0.3 0.8 1.0 1.1 1.2 1.5 1.4 1.3 1.1 0.8 0.3 0.1
Heat source 0 0.004 0.008 0.012 0.016 0.02 0.03 0.97 0.98 0.984 0.988 0.992 0.996 1
turned off 327.0 ± 0.4 312.7 ± 1.5 304.5 ± 1.2 301.4 ± 1.0 299.8 ± 0.8 299.0 ± 0.7 298.1 ± 0.7 295.6 ± 0.6 295.6 ± 0.6 295.6 ± 0.6 295.6 ± 0.6 295.8 ± 0.6 296.2 ± 0.6 296.7 ± 0.1
321.3 310.6 304.4 301.0 299.1 298.2 297.5 295.4 292.8 291.5 290.5 290.1 291.7 296.6
1.8 0.6 0.0 0.1 0.2 0.3 0.2 0.1 0.9 1.4 1.7 1.9 1.5 0.0
317.9 309.4 304.1 300.9 299.0 298.0 297.2 295.6 296.4 296.8 297.2 297.4 297.3 296.5
2.8 1.0 0.1 0.2 0.3 0.3 0.3 0.0 0.3 0.4 0.5 0.5 0.3 0.0
317.0 307.0 302.2 299.9 298.6 298.0 297.1 295.6 295.6 295.6 295.7 295.8 296.0 296.6
3.1 1.8 0.7 0.5 0.4 0.4 0.3 0.0 0.0 0.0 0.0 0.0 0.1 0.0
325.6 312.1 304.9 301.8 300.2 299.2 298.0 295.7 295.8 295.9 295.9 296.1 296.2 296.6
0.4 0.2 0.1 0.1 0.1 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.0 0.0
316.9 307.2 302.1 299.6 298.6 298.2 297.9 295.0 290.3 287.9 286.0 285.4 288.1 296.6
3.1 1.7 0.8 0.6 0.4 0.3 0.1 0.2 1.8 2.6 3.3 3.5 2.7 0.0
327.0 312.6 304.8 301.4 299.8 299.0 298.1 296.3 296.3 296.3 296.4 296.4 296.4 296.5
0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.2 0.3 0.3 0.3 0.2 0.1 0.0
Heat source 0 0.004 0.008 0.012 0.016 0.02 0.03 0.97 0.98 0.984 0.988 0.992 0.996 1
turned on 329.1 ± 0.3 315.3 ± 1.6 308.3 ± 1.2 305.7 ± 1.0 304.3 ± 0.9 303.6 ± 0.8 302.8 ± 0.8 300.1 ± 0.7 300.1 ± 0.7 300.0 ± 0.7 300.1 ± 0.7 300.2 ± 0.7 300.8 ± 0.6 299.4 ± 0.1
325.1 313.6 307.7 304.8 303.5 302.9 302.1 299.5 298.9 298.1 297.1 296.1 296.2 298.8
0.6 0.3 1.1 1.2 1.3 1.3 1.3 1.3 1.1 0.9 0.5 0.1 0.0 0.7
321.8 313.2 307.5 304.6 303.0 302.1 300.8 299.5 299.4 299.3 299.3 299.1 299.0 298.9
1.6 0.2 1.0 1.1 1.1 1.0 0.9 1.3 1.3 1.3 1.2 1.1 0.9 0.7
323.5 313.3 308.1 305.6 304.5 303.9 303.0 299.4 296.8 295.2 293.4 292.1 293.1 298.8
1.1 0.2 1.2 1.4 1.6 1.6 1.6 1.3 0.4 0.1 0.8 1.2 1.0 0.7
329.5 314.6 307.1 303.8 302.5 302.1 301.6 299.9 296.3 294.2 291.9 290.2 291.7 298.8
0.7 0.6 0.9 0.8 0.9 1.0 1.2 1.4 0.2 0.5 1.3 1.9 1.5 0.7
321.9 311.4 306.5 304.5 303.7 303.5 302.8 299.6 294.9 292.2 289.4 287.7 289.9 298.9
1.6 0.4 0.7 1.0 1.3 1.5 1.6 1.4 0.2 1.1 2.1 2.7 2.2 0.7
329.9 315.2 307.7 304.3 303.0 302.6 302.2 299.9 293.9 290.2 286.8 284.8 287.9 298.9
0.9 0.8 1.1 1.0 1.1 1.2 1.4 1.5 0.6 1.8 3.0 3.7 2.8 0.8
No heat source 0 326.4 0.004 310.6 0.008 303.9 0.012 301.2 0.016 299.9 0.02 299.4 0.03 298.7 0.97 297.6 0.98 297.6 0.984 297.8 0.988 297.9 0.992 298.0 0.996 298.5 1 298.2
± ± ± ± ± ± ± ± ± ± ± ± ± ±
Table 4 Comparison of experimental and numerical average convective heat transfer coefficients of the heated wall. Exp
rkε
No heat source 3.9 3.3
Dif (%)
rngkε
Dif (%)
RS-LPS
Dif (%)
RS-LRSO
Dif (%)
skε
Dif (%)
skω
Dif (%)
15.4
3.3
15.4
3.3
15.4
3.5
10.3
3.2
17.9
3.3
15.4
Heat source turned off 3.8 5.3 39.5
5.5
44.7
5.4
42.1
4.2
10.5
5.4
42.1
3.6
5.3
Heat source turned on 3.8 4.3 14.2
4.8
27.9
4.6
21.3
3.6
4.7
4.5
18.9
3.2
14.2
Table 5 Comparison of experimental and numerical Nusselt numbers of the heated wall. Exp
rkε
No heat source 147 127
Dif (%)
rngkε
Dif (%)
RS-LPS
Dif (%)
RS-LRSO
Dif (%)
skε
Dif (%)
skω
Dif (%)
13.6
126
14.3
127
13.6
136
7.5
124
15.6
126
14.3
Heat source turned off 144 202 40.3
211
46.5
209
45.1
161
11.8
209
45.1
138
4.2
Heat source turned on 146 166 13.7
186
27.4
176
20.5
138
5.5
173
18.5
125
14.4
112
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values of average convective heat transfer coefficient and Nusselt number of the heated wall are minimally modified by the presence of the heat source (even if it is turned on or turned off), since the first varies from 3.8 to 3.9 W/m2 K and the last oscillate from 144 till 147. The previous behavior is predicted by the skω turbulence model.
around the isothermal wall (x = 1), the highest deviation corresponds to the RS-LRSO turbulence model, whereas the lowest deviation is obtained with RS-LPS turbulence model. For y = 0.75 m, it is observed that the turbulence models under predict the boundary layers adjacent to the heated and cold walls, the better agreement corresponds to rkε and RS-LPS turbulence models. The experimental temperature profile at y = 0.9 m, is better described for rkε and RS-LPS turbulence models, however all turbulence models under predict the temperature of the heated wall and the highest deviation is observed for RS-LRSO turbulence model near the cold wall. The comparison between experimental and numerical temperature profiles for the case with heat source turned on for y = 0.5 m, is shown in Fig. 4. The experimental temperature profiles are very similar; however, turbulence models predict differences between profiles. At z = 0.25 m is observed the better agreement correspond to the rkε turbulence model, whereas the RS-LRSO differ considerably from the experimental temperatures near the cold wall. For z = 0.5 m, most of turbulence models predict well the thermal boundary layer adjacent to the heated wall (except the RS-LRSO turbulence model) but differ near the cold wall. Finally, for z = 0.75 m all turbulence models are very close to the thermal boundary layer of the heated wall, but again deviate near the cold wall. Therefore, for the three profiles the numerical values under predict the temperatures near the cold wall with the highest deviation observed in the z = 0.5 m profile. A quantitative comparison between an experimental and numerical temperature profile (y = 0.5 m, z = 0.5 m) is presented at Table 3. The comparison is made for three cases (a) ventilated cavity without heat source, (b) ventilated cavity with the heat source turned off (it acts like an obstacle to the fluid flow) and (c) ventilated cavity with the heat source turned on. The case without heat source presents a difference of 6.2% with rkε model in x = 0 m and no difference with several models (rngkε, skε, RS-LPS) at x = 0.996 m, however the average of percentage differences indicates a minimum 0.81% with the rngkε turbulence model and a maximum of 1.24% with skω turbulence model. On the other hand, the case with the heat source turned off indicates no difference with several models at various positions of the profile, whilst the highest difference is 3.1% at x = 0 m with skε and RS-LPS turbulence models; in this case the average of percentage differences shows a minimum of 0.1% with the LR-LRSO turbulence model and a maximum of 1.51% with the RS-LPS turbulence model. Finally, the case with the heat source turned on exhibit the highest difference at the heated and cold wall boundary layer points with a 3.7% for the skω at x = 0.992 m and the lowest percentage difference is at x = 0.996 m with rkε turbulence model with zero difference. However, the average of temperature differences indicates that the best turbulence model was rkε (0.84%) and the worst was the LR-LRSO turbulence model (1.55%). The comparison between experimental and numerical average convective heat transfer coefficients of the heated wall is presented in Table 4. For the case without heat source, the lowest and highest percentage differences were for the RS-LRSO and skε models with 10.3 and 17.9%, respectively. Instead, for the case with heat source turned off the highest and lowest percentage differences are 44.7 and 5.3 with the rngkε and skω turbulence models, respectively. At last, the case with heat source turned on presents a minimum difference of 4.7% with RSLRSO model and a maximum difference with the rngkε model with 27.9%. In Table 5, is presented the comparison between experimental and numerical Nusselt numbers of the heated wall. The minimum and maximum percentage difference for the case with heat source turned on, were obtained with RS-LRSO and rngkε models with a 5.5 and 27.4%, respectively. The case with the heat source turned off, displays a maximum and minimum difference of 46.5 and 4.2% with the rngkε and skω model, in the order given. Finally, the case without heat source exhibits a maximum difference of 15.6% with the skε model and the minimum difference is for the RS-LRSO model with 7.5%. From the experimental results presented in Tables 4 and 5, is observed that the
6. Conclusions In this paper, a comparison between experimental data and numerical results for the turbulent heat transfer in a ventilated cavity, with and without an internal heat source, was presented. From the results, we can conclude the following:
• The comparison between experimental and numerical temperature •
•
values indicates that, the lowest average of percentage differences for the case with the heat source turned on corresponds to realizable k-ε turbulence model (rkε) with 0.84%. The comparison between the average convective heat transfer coefficient and Nusselt number indicates that the best predictions are obtained with: (a) RS-LRSO turbulence model for the case without heat source, (b) skω turbulence model for the case with heat source turned off and (c) RS-LRSO for the case with heat source turned on. The experimental average convective heat transfer coefficient and Nusselt number of the heated wall, are minimally influenced by the presence of a heat source (regardless of whether this on or off) within the cavity. The previous behavior is predicted by the skω turbulence model.
Acknowledgments The first author would like to acknowledge to the National Science and Technology Council of the Mexican Republic (291113), for the support granted through its program of postdoctoral fellowships. References [1] S. Singh, M.A.R. Sharif, Mixed convective cooling of a rectangular cavity with inlets and exit openings on differentially heated side walls, Numer. Heat Transf. A 44 (2002) 233–253. [2] J.D. Posner, C.R. Buchanan, D. Dunn-Rankin, Measurement and prediction of indoor air flow in a model room, Energy Build. 35 (2003) 515–526. [3] N.O. Moraga, S.E. Lopez, Numerical simulation of three-dimensional mixed convection in an air-cooled cavity, Numer. Heat Transf. A 45 (2004) 811–824. [4] V. Haslavsky, J. Tanny, M. Teitel, Interaction between the mixing and displacement modes in a naturally ventilated enclosure, Build. Environ. 41 (2006) 1755–1761. [5] M.M. Rahman, M.A. Alim, Numerical study of opposing mixed convection in a vented enclosure, ARPN J. Eng. Appl. Sci. 2 (2007) 25–36. [6] R. Daghigh, N.M. Adam, B.B. Sahari, Influences of air exchange effectiveness and its rate on thermal comfort: naturally ventilated office, J. Build. Phys. 32 (2008) 175–194. [7] A. Raji, M. Hasnaoui, A. Bahlaoui, Numerical study of natural convection dominated heat transfer in a ventilated cavity: case of forced flow playing simultaneous assisting and opposing roles, Int. J. Heat Fluid Flow 29 (2008) 1174–1181. [8] J. Tanny, V. Haslavsky, M. Teitel, Airflow and heat flux through the vertical opening of buoyancy-induced naturally ventilated enclosures, Energy Build. 40 (2008) 637–646. [9] S. Saha, A.H. Mamun, Z. Hossain, S. Islam, Mixed convection in an enclosure with different inlet and exit configurations, J. Appl. Fluid Mech. 1 (2008) 78–93. [10] G.M. Stavrakakis, M.K. Koukou, M.G. Vrachopoulos, N.C. Markato, Natural crossventilation in buildings: building-scale experiments, numerical simulation and thermal comfort evaluation, Energy Build. 40 (2008) 1666–1681. [11] L. Susanti, H. Homma, H. Matsumoto, Y. Suzuki, M. Shimizu, A laboratory experiment on natural ventilation through a roof cavity for reduction of solar heat gain, Energy Build. 40 (2008) 2196–2206. [12] R. Ezzouhri, P. Joubert, F. Penot, S. Mergui, Large Eddy simulation of turbulent mixed convection in a 3D ventilated cavity: comparison with existing data, Int. J. Therm. Sci. 48 (2009) 2017–2024. [13] A. Lariani, H. Nesreddine, N. Galanis, Numerical and experimental study of 3D turbulent airflow in a full scale heated ventilated room, Eng. Appl. Comput. Fluid Mech. 3-1 (2009) 1–14. [14] P. Karava, T. Stathopoulos, A.K. Athienitis, Airflow assessment in cross-ventilated buildings with operable façade elements, Build. Environ. 46 (2011) 266–279. [15] T.S. Larsen, N. Nikolopoulos, A. Nikolopoulos, G. Strotos, K.S. Nikas,
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Nomenclature A: electric current, (A) Ar: area, m2 Cp: specific heat at constant pressure, (J/kg K) g: gravitational acceleration, (m/s2) Gr: GRASHOF number, nondimensional h: average convective heat transfer coefficient, (W/m2 K) k: turbulent kinetic energy, (m2/s2) L: cavity wall length, (m) li: inlet length, (m) Nu: Nusselt number, nondimensional q: heat flux, (W/m2) Ra: modified Rayleigh number, nondimensional Re: Reynolds number, nondimensional Ri: Richardson number, nondimensional Th : average temperature of the hot wall, (K) Tc: temperature of the cold wall, (K) uh: experimental uncertainty, nondimensional V: voltage, (V) x, y, z: coordinate system, (m)
Greek symbols α: thermal diffusivity, (m2/s) β: thermal expansion coefficient, (1/K) εr: emissivity, nondimensional ε: turbulent kinetic energy dissipation, (J/kg) λ: thermal conductivity, (W/m K) μt: turbulent viscosity, (kg/m s) ν: kinematic viscosity, (m2/s) ρ: density, (kg/m3) ω: turbulent specific dissipation rate, nondimensional
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