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Optical thickness effect on natural convection in a vertical channel containing a gray gas
International Journal of Heat and Mass Transfer 107 (2017) 510–519
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Optical thickness effect on natural convection in a vertical channel containing a gray gas I. Zavala-Guillén a,⇑, J. Xamán a, C. Salinas b, K.A.R. Ismail b, I. Hernández-Pérez c, I. Hernández-López d a
Centro Nacional de Investigación y Desarrollo Tecnológico CENIDET-TecNM-SEP, Prol. Av. Palmira S/N. Col. Palmira, Cuernavaca, Morelos CP 62490, Mexico Faculty of Mechanical Engineering, University of Campinas-UNICAMP, CP 6066, 13083-970 Campinas, SP, Brazil c Universidad Juárez Autónoma de Tabasco, División Académica de Ingeniería y Arquitectura (DAIA-UJAT), Carretera Cunduacán-Jalpa de Méndez km. 1, Cunduacán, Tabasco CP 86690, Mexico d Cátedras Conacyt, Universidad de Quintana Roo, CONACYT-UQRoo, Boulevard Bahía S/N, Col. Del Bosque, Chetumal, Quintana Roo CP 77019, Mexico b
a r t i c l e
i n f o
Article history: Received 20 June 2016 Received in revised form 17 October 2016 Accepted 21 November 2016
Keywords: Natural convection DOM Participating media Vertical channel
a b s t r a c t The effect of radiation on natural convection heat transfer in a vertical parallel-plate channel with asymmetric heating, considering the radiation effects for both walls and participating air is presented. The channel is formed by one vertical wall heated by a uniform heat flux and by a vertical adiabatic plate. The governing equations of laminar natural convection and radiative transfer are solved by the finite volume method (FVM) and by the discrete ordinates method (DOM), respectively. The code was validated and verified with data reported in the literature. The effect of optical thickness (s), channel width (b) and wall emissivity (eh ) on the heat transfer and mass flow are investigated. The mass flow of the channel for s ¼ 0:1 is up to 42% greater than that obtained for a transparent medium (s ¼ 0:0). When s ¼ 0:1, the average temperature difference between the air at the inlet and air at the outlet of the channel decreases up to 75% due to the increase of b from 0.02 to 0.10 m. Varying eh from 0.1 to 0.9 increases the radiative heat flux at the heated wall up to 72% and the mass flow rate increases up to 29%. A set of correlations were obtained for the mass flow, average convective Nusselt number and average radiative Nusselt number. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The combined natural convection and radiation in a vertical channel is representative of several applications such as solar chimneys, solar panels, double skin facades or Trombe wall [1]. The natural convection flow is induced between vertical parallel-plates due to the buoyancy generated when the fluid is heated. Thus, it occurs when at least one of the two walls is heated. The resulting regime flow can be laminar or turbulent depending on the channel geometry, the fluid properties and the temperature of boundary conditions. Natural convection between vertical parallel-plates symmetrically heated and non-symmetrically heated has been extensively studied in the last years, both experimentally and numerically [2–9]. In most of these applications represented by a vertical parallel-plate channel the radiative heat transfer is significant, nevertheless in numerical studies this mechanism is often ⇑ Corresponding author. E-mail addresses: [email protected] (I. Zavala-Guillén), jxaman@cenidet. edu.mx (J. Xamán), [email protected] (C. Salinas), [email protected] (K.A.R. Ismail), [email protected] (I. Hernández-Pérez), irving.hernandez@ uqroo.edu.mx (I. Hernández-López). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.068 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
neglected. The problem of combined radiation and natural convection has been studied for closed cavities with surface radiation, and with both surface and gas radiation [10–14]. However, there are few studies that consider natural convection combined with surface and gas radiation in a vertical parallel-plate channel. One of the first experimental and analytical studies was reported by Yamada [15]. He reported the combined heat transfer of convection and radiation in a vertical channel asymmetrically heated with an absorbing and emitter medium. The author considered the natural convection with laminar flow regime in two dimensions, and he used an exponential wide-band model and gray gas model in one dimension. He concluded that even at intervals of low temperature between ambient temperature and 150 °C, the surface radiation is important in the studied system. He also concluded that an increase of the emissivity of the surfaces from 0.18 to 0.82 increases the heat transfer of the heated surface by 30%. Webb and Hill [16] reported the effect of surface radiation on the heat transfer in a vertical parallel-plate channel, one wall heated with uniform heat flux and the other thermally insulated. Local temperatures along both walls were measured for a range of Rayleigh number regime within 5036Ra61.75 107. The temperatures were used to determine
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Nomenclature _ m b Cp L q T u; v wk x; y Greek
j k
l
mass flow rate, kg/ms channel width, m specific heat, J/kgK channel height, m heat flux, W/m2 temperature, °C components of velocity, m/s weight of angular quadrature coordinates, m
absorption coefficient m1 thermal conductivity, W/mK dynamic viscosity, kg/ms
the local radiative heat flux, by solving the radiation exchange for the channel with gray-diffuse walls considering an emissivity equal to 0.1. Manca et al. [17] studied experimentally the thermal radiation and natural convection in vertical parallel-plate channels. A correlation for Nusselt number in terms of Rayleigh number (Ra) was proposed for Ra up to 106. The measurements showed that the effect of surface radiation is more important for asymmetric heating than for symmetric heating. Experimental and semiexperimental research of laminar natural convection and surface radiation between three parallel vertical plates, a central, highly emissive (e = 0.85) hot plate and two unheated polished plates (e = 0.05) was investigated by Krishnan et al. [18]. The radiative heat transfer rate at the hot surface was computed by the radiosity-irradiation method and it was calculated from the power input to the heater. The convective heat transfer rates were obtained from the measured temperatures. Krishnan et al. [18] concluded that the radiation heat transfer rate is significant even at temperatures below 37 °C. Li et al. [19] also studied numerically the influence of surface radiation on the laminar air flow induced by natural convection in vertical asymmetrically-heated channels. They observed that the effect of surface radiation delayed the onset of recirculation at the top part of the channel due to the increased temperature of the adiabatic wall, even for a wall emissivity as small as 0.1. The studies available in the literature show that radiation heat transfer has a significant effect in the system of vertical parallelplates with asymmetric heating; hence the radiative effects cannot be neglected. However, most of the research only takes into account the radiative exchange between the walls of the channel (transparent medium), solved by the radiosity-irradiation method. Therefore, the aim of this work is to report the effect of radiation (optical thickness) on natural convection heat transfer in a vertical parallel-plate channel with asymmetric heating, considering the radiation effects for both walls and participating medium. In addition, the effect of emissivity of the heated wall and the channel width on the heat transfer and mass flow rate is analyzed.
2. Physical and mathematical model The geometry of the vertical channel is shown in Fig. 1. The channel is formed by one vertical wall heated by a uniform heat flux, and by a vertical adiabatic plate. The height of the channel is L = 1 m and the width of the channel b ranges from 0.02 to 0.10 m. The uniform heat flux at the wall is equal to 250 W/m2. The physical properties of the fluid are assumed constant and they were evaluated at ambient temperature (T1 = 24 °C), with exception of the density, which was
q r rs s e eh
nm ; lm
density, kg/m3 Stefan–Boltzmann constant (5.67 108 W/m2K4) scattering coefficient m1 optical thickness, s ¼ jb emissivity heated wall emissivity direction cosines
Subscripts 1 ambient c convective g glazing r radiative
Toutlet y
0
g qc
q' ' qw
qr
0
L
b
y
x
Tinlet
T
Fig. 1. Physical model of the vertical channel with asymmetric heating.
considered with the Boussinesq approximation. The fluid inside the channel is air (Pr = 0.71) and it is considered in steady laminar regime. The radiation heat transfer is assumed to be twodimensional, the participating medium is considered as a gray gas with an optical thickness of s ¼ 0:1. The vertical walls are considered as gray-diffusely reflecting surfaces, the emissivity of the adiabatic wall is 0.9 and the emissivity of the heated wall has different values ranging from 0.1 to 0.9. The inlet and outlet of the channel are assumed to be black body surfaces. 2.1. Mathematical model The governing conservation equations for a two-dimensional incompressible flow in steady state are:
@ qu @ qv þ ¼0 @x @y
ð1Þ
@ ðquuÞ @ ðquv Þ @P @ þ ¼ þ @x @y @x @x
@ ðqv uÞ @ ðqvv Þ @P @ þ ¼ þ @x @y @y @x
l l
@u @ @u þ l @x @y @y
ð2Þ
@v @ @v þ l @y @x @y
qgbðT T 1 Þ
ð3Þ
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@ ðquT Þ @ ðqv T Þ @ k @T @ k @T 1 þ ¼ þ r qr @x @y @x C p @x @y C p @y Cp
ð4Þ
No-slip boundary conditions on the velocity components are imposed in the vertical surfaces. At the inlet of the channel, the total pressure depression imposed is p p0 þ qv 2 =2 ¼ 0, hence p0 ¼ qv 2 =2 where p0 is the ‘reduced’ static pressure (the difference between the pressure inside the channel and the pressure outside at the same elevation) and p0 is the ambient pressure. The inflow is assumed to be normal to the inlet. The temperature of incoming air (Tinlet ) is equal to the ambient temperature, T1 . The fluid pressure at the outlet is assumed to be the same as the ambient pressure. The streamwise variations of velocity components and temperature are neglected. The temperature boundary conditions are shown in the Fig. 1. The left wall is exposed to a uniform heat flux (qw ), and the right wall is considered adiabatic (qc þ qr ¼ 0). The local divergence of radiative flux (radiative source term) r qr appearing in the energy equation accounts for gas radiation, and is related to the local intensities by:
r qr ¼ j 4pIb ðrÞ
Z 4p
Iðr; XÞdX
ð5Þ
To obtain the radiation intensity field and calculate r qr , it is necessary to solve the radiative transfer equation (RTE). This equation for an absorbing, emitting and scattering gray medium can be written as [20]:
dI rs ð r Þ ¼ ðjðr Þ þ rðrÞÞIðr; ^sÞ þ ds 4p
Z
Iðr; ^sÞdX þ jðr ÞIb ðr Þ 0
4p
ð6Þ
and the radiative boundary condition for diffusely reflecting surfaces in Eq. (6) is:
Iðr w ; ^sÞ ¼ eðrw ÞIb ðr w Þ þ
qðrw Þ p
Z ^ ^s0 <0 n
^ ^s0 jdX0 Iðr w ; ^s0 Þjn
ð7Þ
surfaces, in discrete ordinates for cartesian coordinates can be written as:
I m ¼ eI b þ
qX wm0 l0m Im0 ; lm > 0 in x 2 C p m0
ð12Þ
q X 0 w 0 n I 0 ; nm > 0 in y 2 C p m0 m m m
ð13Þ
lm0 <0
I m ¼ eI b þ
nm0 <0
2.3. Interpolation schemes for the spatial discretization of RTE In this work the genuinely multidimensional (GM) scheme is used. The stencil used in this scheme depends on the characteristics directions [21]. It is a TVD scheme and the Van Albada limiter is used, in the form:
Limiter ½x; y ¼
xy2 þ yx2 x2 þ y 2
ð14Þ
The following relations are used to define the scheme (see Fig. 2): when lm P 0 ; nm P 0 ; Dlxm 6 Dnym i;j
l
m m Iw
¼l "
i;j
m m Ii1;j1 þ 0:5Limiter
l
m m m Ii;j I i1;j1
# ! n m Dxi m m Ii1;j Ii1;j1 lm Dyj
Dxi Sm i;j ;
ð15Þ and m nm Im s ¼ nm I i;j1
when
ð16Þ
lm P 0 ; nm P 0 ; Dlxmi;j P Dnymi;j
lm Imw ¼ lm Imi1;j1
ð17Þ
and 2.2. Radiative transport equation The radiative transfer equation (Eq. (6)) is solved using the discrete ordinates method (DOM). In this method, the RTE is substituted by a set of M discrete equations for a finite number of directions Xm , and each integral is substituted by a quadrature series of the form: M rs X ðXm rÞIðr; Xm Þ ¼ bIðr; Xm Þ þ jIb ðr Þ þ wk Iðr; Xk Þ 4p k¼1
ð8Þ
subject to the boundary condition:
Iðr w ; Xm Þ ¼ eðr w ÞIb ðrw Þ þ
qX w I r ; X0 jn Xk j p nXk 0 k w k
ð9Þ
where wk are the ordinates weight, b is the coefficient extinction (b ¼ j þ rs ) and rs is the scattering coefficient. This angular approximation transforms the original equation into a set of coupled differential equations. In cartesian coordinates, Eq. (9) becomes:
@I @I lm m þ nm m ¼ bIm þ jIb þ Sm @x @y
M rs X wk Iðr; Xk Þ 4p k¼1
ð18Þ where m m Sm i;j ¼ ðji;j þ rsi;j ÞIi;j þ ji;j Ibi;j
Ii,j+1 In Ii-1,j Iw
and lm y nm are the directional cosines of Xm and Sm is the source radiative term. And the boundary conditions for diffusely reflecting
Ii,j Is
Ii-1,j-1
ð11Þ
ð19Þ
The above expressions give us the fluxes when the characteristics point into the quadrant of the xy-plane. When the characteristics point into other quadrants of the xy-plane, the fluxes can be obtained by applying symmetries to Eqs. (15)–(18).
ð10Þ
where
Sm ¼
m nm Im s ¼ nm I i1;j1 þ 0:5Limiter
l m Dy j m m m Ii;j1 Im nm Im i;j I i1;j1 Dyj Si;j ; nm i1;j1 Dxi
Ie
Ii+1,j
Δyi,j
Ii,j-1
Δxi,j
Fig. 2. Interpolation stencil in GM scheme.
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3. Numerical procedure and verification The Navier–Stokes and energy equations are numerically solved using the finite volume method [22]. The numerical code was developed using Fortran computational language. The coupling between the momentum and continuity equations is made with the SIMPLEC algorithm, and a false-transient technique is used. The convective terms were discretized applying the hybrid scheme and the diffusive terms were discretized applying the central difference scheme. The system of algebraic equations obtained in the discretization is solved applying the line by line method (LBL) with the alternating direction implicit scheme (ADI). The radiation source term in energy equation is calculated by solving RTE using DOM. We used an angular quadrature set as LC-11 [23]. The convergence criterion for RTE and the others governing equations are about 1007. A study of spatial grid independence considering the highest value of emissivity was performed for each channel width. From this study, the computational grid that gives grid independent solutions were 31 201, 26 201, 21 201, 16 201 and 11 201 for the channel widths of 0.10, 0.08, 0.06, 0.04 and 0.02 m, respectively; which show non-significant differences (2.0%) for the components of velocity and for the temperature profile at the outlet of the channel. To validate the accuracy of our numerical code, three comparisons were made: free convection in antisymmetrically heated vertical channel, turbulent mixed convection and surface thermal radiation in a vertical channel and natural convection and radiation in a differentially-heated cavity with participating gases. First, numerical code was validated comparing our results against the results for free convection in non-symmetrically heated channel with a hot wall heated 7.5 °C above the ambient temperature and a cold wall was cooled 7.5 °C below the ambient temperature. The Table 1 shows comparison of the average Nusselt numbers at the hot wall of the experimental and numerical data reported by Roeleveld et al. [24] and the solution obtained for three values of Ra as 1:48 103 , 4:79 103 and 1:18 104 corresponding to aspect ratios of 26.4, 17.6 and 13.2. It can be seen that the present results agree with those presented by Roeleveld et al. [24]. The overall deviations are within 7% of the experimental data for the three Rayleigh numbers. The numerical code was also verified comparing our results against the results for a turbulent mixed convection and thermal radiation in a vertical channel reported by Barhaghi and Davidson [25] and Wang et al.[26]. The boundary conditions of the vertical channel were a constant heat flux (qw = 449 W/m2) in one channel wall while the other wall is insulated. The comparison of the maximum velocity (v max ) and average Nusselt number (Nuav e ) at the hot wall show a good agreement with the data from literature. Table 2 shows the comparison of results for Re = 5080 and e = 0.125, for v max , the percentage difference respect to the experimental data was 8.2%; and for Nuav e , the percentage difference respect to the experimental data was 4.3%. To validate the coupling of natural convection with radiation, we present a comparison of combined heat transfer in a
Table 2 Comparison of the hot wall average Nusselt number (Nuav e ) and maximum velocity (v max ) for vertical channel. Parameter
Wang et al. [26]
Barhaghi and Davidson [25]
Present study
v max (m/s)
1.22 39.5
1.46 33.4
1.33 (8.2)⁄ 40.7 (4.3)⁄
Nuav e ⁄
Absolute differences in percentage respect to the experimental data.
differentially-heated cavity. Table 3 shows the overall Nusselt number at the hot wall for two values of the optical thickness (s = 1 and s = 5) and two Rayleigh numbers. The results are compared with the values reported by [27] and shown a maximum difference of 1.1%. According to the table, the results obtained in the present work have a good agreement with the results of Lari et al. [27]. 4. Results To observe the effect that the gray gas radiation has on heat transfer and fluid flow in a vertical parallel-plate channel with asymmetric heating, in the first section we compare results of the channel with gray gas with the results of the channel with transparent gas under the same boundary conditions. The following section shows the effect that the channel width has on the mass flow and the heat transfer. 4.1. Optical thickness effect To show the effect of gray gas radiation on natural convection in the channel, two optical thicknesses s = 0.0 and s = 0.1 were considered. The results shown in this section are for a channel width b = 0.1 m and emissivity e = 0.9 in both channel walls. Fig. 3 presents the air vertical velocity component and its temperature with (a) s = 0.0 and (b) s = 0.1. The velocity field is qualitatively similar in both cases, higher velocity gradients occur in areas near the walls, with higher values near the heated wall (left wall). In both cases the lowest velocity occurs in the central zone of the channel; however, when s = 0.0 the velocity in the central zone of the channel is about 0.1 m/s, but when s = 0.1 it is approximately 0.25 m/s. The temperature behavior of the channel in both cases shows higher gradients in zones close to the walls; due to heat flux imposed on the left wall, the air close to this wall has the highest temperature values. On the other hand, for s = 0.0 the air temperature in the central zone along the channel has no variation from its inlet temperature; but when s = 0.1 approximately from two-thirds of the height of the channel, the air in the central zone has higher temperature than its inlet temperature. This effect occurs because in the case of surface thermal radiation, the energy is distributed on the walls; however, when considering radiative participant gas, part of the energy is absorbed by the air distributed across the channel width. Therefore, the air increases its temperature even in the central zone of the channel and decreases the heat transfer of the walls.
Table 1 Comparison of the hot wall average Nusselt number (Nuov e;h ) for an antisymmetrically heated vertical channels. Aspect ratio A
Rayleigh number Ra (103)
26.4 17.6 13.2 ⁄
Absolute differences in percentage.
1.48 4.79 1.18
Experimental comparison
Numerical comparison
Roeleveld et al.[24]
Present study
Roeleveld et al. [24]
Present study
0.97 1.03 1.32
1.03 (6.1)⁄ 1.10 (6.7)⁄ 1.22 (7.5)⁄
1.02 1.11 1.40
1.03 (0.9)⁄ 1.10 (0.9)⁄ 1.22 (12.8)⁄
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Table 3 Comparison of average overall Nusselt number (Nuov e;h ) at different optical thickness and Rayleigh numbers.
s=1
Rayleigh number Ra Lari et al. [27]
Present study
Lari et al. [27]
8.367
8.277 (1.1)⁄
6.811
6.814 (0.1)⁄
17.086
16.965 (0.7)⁄
14.514
14.580 (0.5)⁄
105 6
10 ⁄
s=5 Present study
Absolute differences in percentage.
(a) τ =0.0
(b) τ =0.1
Fig. 3. Vertical velocity (m/s) and temperature (°C) when b = 0.1 m with (a)
The above is observed in Fig. 4 that show the temperature at the heating wall of the vertical channel (T w ). In general, T w increases as the channel height increases. For s = 0.0, the T w shows a maximum difference of 32 °C between y = 0 m and y = 1.0 m. The T w for s = 0.0 reaches up 4.8 °C more than for s = 0.1 and shows a maximum difference of 7.7%.
1.0
τ 0.8
0.0 0.1
y(m)
0.6
0.4
0.2
0.0 35
40
45
50
55
60
65
70
Tw (°C) Fig. 4. Temperature of the heated wall of the vertical channel (T w ).
s = 0.0 y (b) s = 0.1.
The previous observations can also be seen in Fig. 5 where the (a) velocity profile and (b) temperature of the air at the outlet are shown; in this figure the x coordinate was dimensionless with the channel width b. In general, a higher the temperature the velocity is higher in the channel output. The air velocity and its temperature near the channel walls are greater when s = 0.0 than s = 0.1, with a difference of up to 0.70 m/s and 4.1 °C, respectively; while variables in the central zone of the channel when s = 0.1 are greater when s = 0.0, with a difference of up to 0.19 m/s and 1.7 °C, respectively. The velocity and temperature of the air in the channel show such behavior because the ability of the air to absorb energy depends on its optical thickness; therefore, the higher the optical thickness of the gas, the greater the increase in its internal energy and hence its buoyancy force. _ the convective heat flux Table 4 presents the mass flow rate (m), (qc;h ) and radiative heat flux (qr;h ) at the heated wall of the channel, _ = 18.74 g/ms when for s = 0.0 and s = 0.1. We can observe that m s = 0.0 and m_ = 32.32 g/ms when s = 0.1, for such values of m_ there is a percentage difference of 42%. The table also shows that qc;h is 142.6 and 127.7 W/m2 for s = 0.0 and s = 0.1, respectively; with a percentage difference between them of 10%. The qc;h presents this behavior because when the fluid is radiatively transparent (s = 0.0) the energy is concentrated on the walls, thus, the temperature difference between the wall and the adjacent air increases and therefore the convective heat flux increases. When s = 0.1, the temperature difference between the wall and the adjacent air is smaller than that of s = 0.0 because the radiative energy is absorbed by the gas increasing its temperature along the channel.
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0.7
50
τ 0.0 0.1
τ
40
0.5
30
0.4
20
T-Tinlet (°C)
v(m/s)
0.6
0.3 0.2
0.0 0.1
10
4 3
0.1
2 1
0.0
0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
x/b
x/b
(a)
(b)
0.6
0.8
1.0
Fig. 5. (a) Velocity profile and (b) temperature of the air at the outlet.
Table 4 _ the convective heat flux (qc;h ) and radiative heat flux (qr;h ) of the Mass flow rate (m), heated wall for a vertical channel.
s
_ (g/ms) m
qc;h (W/m2)
qr;h (W/m2)
0.0 0.1
18.7 32.3
142.6 127.7
107.4 122.3
Furthermore, when s = 0.0 the qr;h is 13.8% lower than that obtained for s = 0.1. This effect occurs because when s = 0.0 the energy from the other walls directly incides on the heated wall. However, when s = 0.1 the energy is attenuated by the gas before reaching the heated wall, which increases the net radiative heat flux because the amount of energy emitted is greater than the incident one. Therefore, from the energy balance between energy that is emitted and the energy that incident on the heated wall we obtained values for qr;h of 107.4 and 122.3 W/m2 for s = 0.0 and s = 0.1, respectively. The behavior described shows that gas radiation significantly affects the behavior of natural convection, for air with an optical thickness about 0.1. 4.2. Channel width effect This section presents the results obtained for b = 0.02, 0.04, 0.06, 0.08 and 0.10 m, considering an optical thickness s = 0.1 and emissivity e = 0.9 in both channel walls. Fig. 6 shows the contour of vertical velocity, for different values of b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10. When b P0.04 m higher values and velocity gradients occur in the region near to the heated wall. On the other hand, when b = 0.02 m the highest value of velocity is present in the central zone of the channel because such a channel width allows the air to reach similar conditions to those of a developed flow, therefore, the air velocity shows a nearly parabolic profile during its route through the channel. In addition, when b P0.06 m the velocity profile is almost constant in the central region of the channel, and when increasing the channel width, the velocity value decreases in the same region. When b = 0.04 m the air is accelerated in the region the near walls and with greater intensity near the heated wall, however in the central zone of the channel, the velocity neither presents a constant behavior nor parabolic profile as the other cases. Fig. 7 presents the temperature contour for different values of b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10. In general, the
(a)
(b)
(c)
(d)
(e)
Fig. 6. Contour of vertical velocity, for different b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10.
highest values and temperature gradients are present in the region near to the left wall, where the constant heat flux is received. On the other hand, when b P0.06 m the temperature of the air at the inlet remains almost constant in the central region of the channel up to a certain height, such height increases as the channel width increases; for instance, when b = 0.06 m that behavior is presented up to half of the height and when b = 0.10 m up to approximately two thirds of the channel. For b 60.04 m the air temperature in the central region of the channel increases as the channel height increases, in such a way that the air average temperature at the outlet of the channel reaches up to 28.1 °C more than the inlet temperature. This effects occurs because the thickness of the thermal boundary layers developed on the walls of the channels covers the width of the channel, eliminating the almost constant behavior of the temperature in the central region of the channel. Fig. 8 shows (a) the velocity profile at the exit of channel (v) and (b) the temperature difference of the air between the inlet and the outlet (T T inlet ) for all channel widths considered. In this figure the x coordinate was made dimensionless with the channel width b. Fig. 8a shows that the maximum velocity in the channel output
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(a)
(b)
(c)
(d)
central area of the channel, with a minimum value of up to 1.7 °C for b = 0.10 m. The average of T T inlet is 28.1 and 6.8 °C for b = 0.10 m and b = 0.02 m respectively; therefore, the average of T T inlet decreases up to 75% due to the increase of b from 0.02 to 0.10 m. _ when varying the Fig. 9 shows the behavior of the mass flow (m) channel width b. As the channel width increases the mass flow _ are in a range of 10.15 to 32.32 g/ms. increases, the values of m The mass flow rate increases up to 68% when b changes from 0.02 m to 0.10 m. Furthermore, when increasing channel width _ decreases, obtaining a minimum of the slope of the curve of m 12% percentage difference between the results for b = 0.08 and b = 0.10 m. This effect occurs because as the air volume increases the amount of energy required to move it is greater; in this case, the volume of air increases by increasing the channel width, however, the imposed heat flux on the wall remains constant. There_ is not proportional to the fore, we observe that the increase of m _ tends to decrease increase of the b and the slope of the curve of m with increasing b.
(e)
Fig. 7. Contour of temperature, for different values of b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10.
35
0.7
0.5
20
15
10 0.02
0.04
0.06
0.08
0.10
b (m) Fig. 9. Mass flow rate behavior with respect to the channel width.
b (m) 0.10 0.08 0.06 0.04 0.02
40
30
T-T inlet (°C)
0.4
v (m/s)
25
50
b (m) 0.10 0.08 0.06 0.04 0.02
0.6
30
m (g/m.s)
is about 0.59 m/s for all channel widths considered and this velocity is present in the region near the heated wall of the channel, except for b = 0.02 m that presents an almost parabolic (near to developed flow conditions) profile where the maximum speed on the channel output is presented in x=b 0.35. When b P0.06 m the velocity has an almost constant behavior in the central area of the channel, in addition, as the channel width decreases the velocity in the central region and has a minimum value up 0.21 m/s when b = 0.10 m. When b = 0.04 m the velocity decreases as moving away from the walls up to a value 0.34 m/s in the center of the channel (x=b=0.5). Fig. 8b shows that highest values T T inlet are present in the zone near the heated wall with values up to 49 °C for b = 0.02 m. As b increases it decreases the value of T T inlet , therefore for b = 0.10 m there is a maximum of 41 °C. Furthermore, the T T inlet of air adjacent to the insulated wall, decreases as the channel width increases. For b 60.04 m one can observe that in the central region of the channel T T inlet decreases as it moves away from the solid walls, with minimum values of 4.7 and 17.8 °C for b = 0.02 m and b = 0.04 m respectively. For b 60.06 m an almost constant behavior of T T inlet occurs in the
0.3
20
0.2 10 0.1
0.0
0 0.0
0.2
0.4
x/b
(a)
0.6
0.8
1.0
0.0
0.2
0.4
x/b
0.6
0.8
1.0
(b)
Fig. 8. (a) Velocity profile at the exit of channel and (b) the temperature difference of the air between the inlet and the outlet, for all channel widths.
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517
4.3. Wall emissivity effect 32
To observe the effect of the heated wall emissivity eh on the _ and the effect of eh on the conbehavior of the mass flow rate m, vective and radiative heat flux (qc;h ,qr;h ), this section presents the results obtained for eh = 0.1, 0.3, 0.5, 0.7 and 0.9, considering an optical thickness s = 0.1 and b = 0.10 m. Fig. 10 shows the behavior of the mass flow when varying the emissivity of the heated wall of the vertical channel. As the value _ also increases, to a maximum value of of eh increases the m _ shows this behavior because the 32.3 g/ms for an eh = 0.9. The m radiative energy emitted by the heated wall increases with the increment of walls emissivity. Therefore, the gas in the channel absorbs more energy. The radiative energy absorbed by the gas is proportional to the divergence of radiative flux r qr ðW=m3 Þ obtained in the gas, as this is the source term in the energy equation. Therefore, Fig. 11 shows the divergence of radiative flow of the gas inside the channel when b = 0.1 and for different values of eh : a) 0.1, b) 0.3, c) 0.5, d) 0.7 and e) 0.9. This figure also shows the profile of radiative heat
m (g/m.s)
30
28
26
24
0.1
0.3
0.5
0.7
0.9
εh Fig. 10. Mass flow rate behavior with respect to the
eh .
(a)
(b)
(d)
(c)
(e)
Fig. 11. Profile of qr;h and r qr ðW=m Þ of the air inside the channel for different values of 3
eh : a) 0.1, b) 0.3, c) 0.5, d) 0.7 and e) 0.9.
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qr,h
200
qc,h
q (W/m2)
150
100
50
0.1
0.3
0.5
0.7
0.9
εh Fig. 12. Convective and radiative heat fluxes for different values of b = 0.10 m.
eh and
4.4. Numerical correlations
Table 5 _ correlation. Coefficients for m
s
bðmÞ A
B
0.0
0.10 0.08 0.06 0.04 0.02
0.5246 0.7127 0.9874 0.3914 0.4725
18.968 19.703 19.975 17.463 10.429
0.10 0.08 0.06 0.04 0.02
3.411 2.8003 2.1343 1.4519 0.3771
32.966 28.972 24.449 18.850 10.225
0.1
increasing values qr;h along the channel with higher gradients than the other cases. Fig. 12 shows the behavior of the convective and radiative heat fluxes when the emissivity of the heated wall varies as eh = 0.1, 0.3, 0.5, 0.7 and 0.9 and channel width b = 0.10 m. The figure indicates that as the emissivity of the heated wall increases, the value of qr;h increases and the value of qc;h decreases. The values of qr;h are in the interval 34.16qr;h 6122.3 W/m2, moreover the values have a maximum percentage difference of 72% between the results for eh = 0.1 and eh = 0.9. The behavior of qr;h is because this amount is the result of an energy balance between the energy emitted by the wall and the energy that strikes it, where the amount of energy emitted by the wall is proportional to the emissivity. The values of qc;h are in the interval 215.66qc;h 6127.3 W/m2, moreover the values have a maximum percentage difference of 40% between the results for eh = 0.1 and eh = 0.9. The values of qc;h present such behavior because as the radiative heat flux (qr;h ) emitted by the wall to the air inside the channel increases, the temperature of the gas increases and the temperature difference between the heated wall and the adjacent air decreases, therefore, the value of qc;h also decreases.
_ was obtained. The corA correlation for the mass flow rate (m) relation was established as a function of the emissivity of the heated wall (eh ) for each channel width. This correlation can be expressed as:
Coefficient
_ ¼ Alnðeh Þ þ B m
flux from the heated wall (qr;h ) for each field of r qr . The negative sign of r qr means that the radiative volume flow is absorbed/ attenuated by the gas, therefore, it is decreasing the radiative energy beam. Fig. 11 shows that the divergence of radiative heat flow and the qr;h increase when increasing the emissivity of the heated wall. The behavior of qr;h significantly affects the distribution of r qr . When eh = 0.1, r qr does not present considerable variations along the channel, because the profile qr;h is nearly constant, with a maximum variation of 33.5 W/m2. On the contrary, when eh = 0.9 the profile qr;h presents a steeper slope, with a maximum variation of 102.7 W/m2, while values of r qr increase with
ð20Þ
Table 5 shows the coefficients for each channel width. The maximum deviation from the numerical results is presented by the channel width b = 0.04 m. Such deviation is 1.7 and 1.0% for s = 0.0 and s = 0.1, respectively. In addition, correlations for average convective Nusselt number (Nuc ) and average radiative Nusselt number (Nur ) were also obtained. The correlations were established depending on the emissivity of the heated wall (eh ) for each channel width. Such correlations are expressed as:
Nuc ¼ Aðe2h Þ þ Bðeh Þ þ C
ð21Þ
Nur ¼ Aðe2h Þ þ Bðeh Þ þ C
ð22Þ
Table 6 shows the correlation coefficients for Nuc , for each channel width. The maximum deviation from the numerical results are presented by a channel width b = 0.06 m. Such deviation is 0.5 and 3.0% for s = 0.0 and s = 0.1, respectively. Table 7 shows coefficients for the correlation of Nur for each channel width. The maximum deviation with respect to numerical
Table 6 Coefficients for Nuc correlation.
s
bðmÞ
B
C
0.0
0.10 0.08 0.06 0.04 0.02
1.9173 1.4313 0.9542 0.4902 0.2569
3.762 2.7846 1.8613 0.9898 0.4979
16.171 13.112 9.9635 6.7919 3.5246
0.1
0.10 0.08 0.06 0.04 0.02
1.684 1.3042 2.7501 0.6111 0.3837
3.8607 2.9937 0.4623 1.4269 0.8566
16.554 13.397 9.8415 6.9335 3.5681
Coefficient A
519
I. Zavala-Guillén et al. / International Journal of Heat and Mass Transfer 107 (2017) 510–519 Table 7 Coefficients for Nur correlation.
s
bðmÞ
0.0
0.10 0.08 0.06 0.04 0.02
8.7602 6.9543 5.1806 3.4344 1.6977
19.019 14.825 10.853 7.1183 3.4957
0.7457 0.616 0.4759 0.3203 0.1553
0.1
0.10 0.08 0.06 0.04 0.02
8.3427 6.6905 1.3055 3.315 1.5906
3.8607 17.611 10.255 8.3561 3.834
0.4821 0.3945 0.6225 0.2032 0.1007
Coefficient A
results for s = 0.0 is 3.0% and is presented by a channel width b = 0.04 m. When s = 0.1 the maximum deviation is 4% and is presented by a channel width b = 0.06 m. 5. Conclusions Based on the results of natural convection and radiation in a vertical parallel-plate channel with asymmetric heating, considering the radiation effects for both walls and participating medium, it can be concluded that: The mass flow of the channel for s = 0.1 is up to 42% greater than that obtained for a transparent medium (s = 0.0). In addition, it was observed that considering radiation in both the walls and in the medium (s = 0.1) increases the radiative heat flux at the heated wall up to 13.8% compared to the case with transparent medium (s = 0.0). On the other hand, convective heat flux at the heated wall for s = 0.0 is up to 10% greater than for s = 0.1. When b changes from 0.02 m to 0.10 m it increases the mass flow rate up to 68%. However, the average temperature difference between the air at the inlet and air at the outlet of the channel decreases up to 75% due to the increase of b from 0.02 m to 0.10 m. The mass flow rate increases up to 29% due to an increment of emissivity of the heated wall from 0.1 to 0.9. Varying eh from 0.1 to 0.9 increases the radiative heat flux up to 72% in the heated wall and it decreases the convective heat flux in the heated wall up to 40%.
Acknowledgements The authors are grateful to Consejo Nacional de Ciencia y Tecnología (CONACYT) whose financial support made this work possible. I. Zavala-Guillén would like to thanks TWAS-CNPQ program for the financial support during her stay at UNICAMP. C. Salinas wishes to thanks for the financial support grant #2016/01493-9, São Paulo Research Foundation (FAPESP). References [1] G. Gan, General expressions for the calculation of air flow and heat transfer rates in tall ventilation cavities, Build. Environ. 46 (2011) 2069–2080. [2] B. Zamora, J. Hernández, Influence of variable property effects on natural convection flows in asymmetrically-heated vertical channels, Int. Commun. Heat Mass Transfer 24 (1997) 1153–1162. [3] G. Desrayaud, A. Fichera, Laminar natural convection in a vertical isothermal channel with symmetric surface-mounted rectangular ribs, Int. J. Heat Mass Transfer 23 (2002) 519–529.
B
C
[4] B. Zamora, J. Hernández, Effects of variable properties and non-uniform heating on natural convection flows in vertical channels, Int. J. Heat Mass Transfer 48 (2005) 793–807. [5] Q. Lu, S. Qiu, S. Qiu, W. Tian, Z. Ye, Experimental research on heat transfer of natural convection in vertical rectangular channels with large aspect ratio, Exp. Therm. Fluid Sci. 34 (2010) 73–80. [6] D. Ospir, C. Popa, C. Chereches, G. Polidori, S. Fohanno, Flow visualization of natural convection in a vertical channel with asymmetric heating, Int. Commun. Heat Mass Transfer 39 (2012) 486–493. [7] H. Sun, R. Li, E. Chénier, G. Lauriat, J. Padet, Optimal plate spacing for mixed convection from an array of vertical isothermal plates, Int. J. Therm. Sci. 55 (2012) 16–30. [8] G. Desrayaud, E. Chénier, A. Joulin, A. Bastide, B. Brangeon, J.P. Caltagirone, Y. Cherif, R. Eymard, C. Garnier, S. Giroux-Julien, Y. Harnane, P. Joubert, N. Laaroussi, S. Lassue, P. Le Quéré, R. Li, D. Saury, A. Sergent, S. Xin, A. Zoubir, Benchmark solutions for natural convection flows in vertical channels submitted to different open boundary conditions, Int. J. Therm. Sci. 72 (2013) 18–33. [9] D. Roeleveld, D. Naylor, W.H. Leong, Free convection in asymmetrically heated vertical channels with opposing buoyancy forces, J. Heat Transfer 136 (2014) 062502. [10] C. Balaji, S. Venkateshan, Interaction of surface radiation with free convection in a square cavity, Int. J. Heat Fluid Flow 14 (1993) 260–267. [11] N. Ramesh, S. Venkateshan, Effect of surface radiation on natural convection in a square enclosure, J. Thermophys. Heat Transfer 13 (1993) 299–301. [12] G. Colomer, M. Costa, R. Consul, A. Oliva, Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method, Int. J. Heat Mass Transfer 47 (2004) 257–269. [13] O. Manca, S. Nardini, Experimental investigation of radiation effects on natural convection in horizontal channels heated from above, J. Heat Transfer 131 (2009) 062503. [14] A. Ibrahim, D. Saury, D. Lemonnier, Coupling of turbulent natural convection with radiation in an air-filled differentially-heated cavity at Ra=1.5 109 , Comput. Fluids 88 (2013) 115–125. [15] Y. Yamada, Combined radiation and free convection heat transfer in a vertical channel with arbitrary wall emissivities, Int. J. Heat Mass Transfer 31 (1988) 429–440. [16] B. Webb, D. Hill, High rayleigh number laminar natural convection in an asymmetrically heated vertical channel, J. Heat Transfer 111 (1989) 649–656. [17] O. Manca, S. Nardini, V. Naso, G. Ruocco, Experiments on natural convection and radiation in asymmetrical heated vertical channels with discrete heat source, ASME-HTD 317 (1995) 309–319. [18] A.S. Krishnan, B. Premachandran, C. Balaji, S.P. Venkateshan, Combined experimental and numerical approaches to multi-mode heat transfer between vertical parallel plates, Exp. Therm. Fluid Sci. 29 (2004) 75–86. [19] R. Li, M. Bousetta, E. Chénier, G. Lauriat, Effect of surface radiation on natural convective flows and onset of flow reversal in asymmetrically heated vertical channels, Int. J. Therm. Sci. 65 (2013) 9–27. [20] M.F. Modest, Radiative Heat Transfer, McGraw-Hill, 1993. [21] D. Balsara, Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods, J. Quant. Spectrosc. Radiat. Transfer 69 (2001) 671–707. [22] S. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Co. McGraw-Hill Co., 1980. [23] R. Koch, R. Becker, Evaluation of quadrature schemes for the discrete ordinates method, J. Quant. Spectrosc. Radiat. Transfer 84 (2004) 423–435. [24] D. Roeleveld, D. Naylor, W.H. Leong, Free convection in antisymmetrically heated vertical channels, J. Heat Transfer 136 (2014) 012502. [25] D.G. Barhaghi, L. Davidson, Large-eddy simulation of mixed convection radiation heat transfer in a vertical channel, Int. J. Heat Mass Transfer 52 (2009) 3918–3928. [26] J. Wang, J. Li, J.D. Jackson, A study of the influence of buoyancy on turbulent flow in a vertical plane passage, Int. J. Heat Fluid Flow 25 (2004) 420–430. [27] K. Lari, M. Baneshi, S.A. Gandjalikhan Nassab, A. Komiya, S. Maruyama, Combined heat transfer of radiation and natural convection in a square cavity containing participating gases, Int. J. Heat Mass Transfer 54 (2011) 5087–5099.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Optical thickness effect on natural convection in a vertical channel containing a gray gas I. Zavala-Guillén a,⇑, J. Xamán a, C. Salinas b, K.A.R. Ismail b, I. Hernández-Pérez c, I. Hernández-López d a
Centro Nacional de Investigación y Desarrollo Tecnológico CENIDET-TecNM-SEP, Prol. Av. Palmira S/N. Col. Palmira, Cuernavaca, Morelos CP 62490, Mexico Faculty of Mechanical Engineering, University of Campinas-UNICAMP, CP 6066, 13083-970 Campinas, SP, Brazil c Universidad Juárez Autónoma de Tabasco, División Académica de Ingeniería y Arquitectura (DAIA-UJAT), Carretera Cunduacán-Jalpa de Méndez km. 1, Cunduacán, Tabasco CP 86690, Mexico d Cátedras Conacyt, Universidad de Quintana Roo, CONACYT-UQRoo, Boulevard Bahía S/N, Col. Del Bosque, Chetumal, Quintana Roo CP 77019, Mexico b
a r t i c l e
i n f o
Article history: Received 20 June 2016 Received in revised form 17 October 2016 Accepted 21 November 2016
Keywords: Natural convection DOM Participating media Vertical channel
a b s t r a c t The effect of radiation on natural convection heat transfer in a vertical parallel-plate channel with asymmetric heating, considering the radiation effects for both walls and participating air is presented. The channel is formed by one vertical wall heated by a uniform heat flux and by a vertical adiabatic plate. The governing equations of laminar natural convection and radiative transfer are solved by the finite volume method (FVM) and by the discrete ordinates method (DOM), respectively. The code was validated and verified with data reported in the literature. The effect of optical thickness (s), channel width (b) and wall emissivity (eh ) on the heat transfer and mass flow are investigated. The mass flow of the channel for s ¼ 0:1 is up to 42% greater than that obtained for a transparent medium (s ¼ 0:0). When s ¼ 0:1, the average temperature difference between the air at the inlet and air at the outlet of the channel decreases up to 75% due to the increase of b from 0.02 to 0.10 m. Varying eh from 0.1 to 0.9 increases the radiative heat flux at the heated wall up to 72% and the mass flow rate increases up to 29%. A set of correlations were obtained for the mass flow, average convective Nusselt number and average radiative Nusselt number. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The combined natural convection and radiation in a vertical channel is representative of several applications such as solar chimneys, solar panels, double skin facades or Trombe wall [1]. The natural convection flow is induced between vertical parallel-plates due to the buoyancy generated when the fluid is heated. Thus, it occurs when at least one of the two walls is heated. The resulting regime flow can be laminar or turbulent depending on the channel geometry, the fluid properties and the temperature of boundary conditions. Natural convection between vertical parallel-plates symmetrically heated and non-symmetrically heated has been extensively studied in the last years, both experimentally and numerically [2–9]. In most of these applications represented by a vertical parallel-plate channel the radiative heat transfer is significant, nevertheless in numerical studies this mechanism is often ⇑ Corresponding author. E-mail addresses: [email protected] (I. Zavala-Guillén), jxaman@cenidet. edu.mx (J. Xamán), [email protected] (C. Salinas), [email protected] (K.A.R. Ismail), [email protected] (I. Hernández-Pérez), irving.hernandez@ uqroo.edu.mx (I. Hernández-López). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.068 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
neglected. The problem of combined radiation and natural convection has been studied for closed cavities with surface radiation, and with both surface and gas radiation [10–14]. However, there are few studies that consider natural convection combined with surface and gas radiation in a vertical parallel-plate channel. One of the first experimental and analytical studies was reported by Yamada [15]. He reported the combined heat transfer of convection and radiation in a vertical channel asymmetrically heated with an absorbing and emitter medium. The author considered the natural convection with laminar flow regime in two dimensions, and he used an exponential wide-band model and gray gas model in one dimension. He concluded that even at intervals of low temperature between ambient temperature and 150 °C, the surface radiation is important in the studied system. He also concluded that an increase of the emissivity of the surfaces from 0.18 to 0.82 increases the heat transfer of the heated surface by 30%. Webb and Hill [16] reported the effect of surface radiation on the heat transfer in a vertical parallel-plate channel, one wall heated with uniform heat flux and the other thermally insulated. Local temperatures along both walls were measured for a range of Rayleigh number regime within 5036Ra61.75 107. The temperatures were used to determine
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Nomenclature _ m b Cp L q T u; v wk x; y Greek
j k
l
mass flow rate, kg/ms channel width, m specific heat, J/kgK channel height, m heat flux, W/m2 temperature, °C components of velocity, m/s weight of angular quadrature coordinates, m
absorption coefficient m1 thermal conductivity, W/mK dynamic viscosity, kg/ms
the local radiative heat flux, by solving the radiation exchange for the channel with gray-diffuse walls considering an emissivity equal to 0.1. Manca et al. [17] studied experimentally the thermal radiation and natural convection in vertical parallel-plate channels. A correlation for Nusselt number in terms of Rayleigh number (Ra) was proposed for Ra up to 106. The measurements showed that the effect of surface radiation is more important for asymmetric heating than for symmetric heating. Experimental and semiexperimental research of laminar natural convection and surface radiation between three parallel vertical plates, a central, highly emissive (e = 0.85) hot plate and two unheated polished plates (e = 0.05) was investigated by Krishnan et al. [18]. The radiative heat transfer rate at the hot surface was computed by the radiosity-irradiation method and it was calculated from the power input to the heater. The convective heat transfer rates were obtained from the measured temperatures. Krishnan et al. [18] concluded that the radiation heat transfer rate is significant even at temperatures below 37 °C. Li et al. [19] also studied numerically the influence of surface radiation on the laminar air flow induced by natural convection in vertical asymmetrically-heated channels. They observed that the effect of surface radiation delayed the onset of recirculation at the top part of the channel due to the increased temperature of the adiabatic wall, even for a wall emissivity as small as 0.1. The studies available in the literature show that radiation heat transfer has a significant effect in the system of vertical parallelplates with asymmetric heating; hence the radiative effects cannot be neglected. However, most of the research only takes into account the radiative exchange between the walls of the channel (transparent medium), solved by the radiosity-irradiation method. Therefore, the aim of this work is to report the effect of radiation (optical thickness) on natural convection heat transfer in a vertical parallel-plate channel with asymmetric heating, considering the radiation effects for both walls and participating medium. In addition, the effect of emissivity of the heated wall and the channel width on the heat transfer and mass flow rate is analyzed.
2. Physical and mathematical model The geometry of the vertical channel is shown in Fig. 1. The channel is formed by one vertical wall heated by a uniform heat flux, and by a vertical adiabatic plate. The height of the channel is L = 1 m and the width of the channel b ranges from 0.02 to 0.10 m. The uniform heat flux at the wall is equal to 250 W/m2. The physical properties of the fluid are assumed constant and they were evaluated at ambient temperature (T1 = 24 °C), with exception of the density, which was
q r rs s e eh
nm ; lm
density, kg/m3 Stefan–Boltzmann constant (5.67 108 W/m2K4) scattering coefficient m1 optical thickness, s ¼ jb emissivity heated wall emissivity direction cosines
Subscripts 1 ambient c convective g glazing r radiative
Toutlet y
0
g qc
q' ' qw
qr
0
L
b
y
x
Tinlet
T
Fig. 1. Physical model of the vertical channel with asymmetric heating.
considered with the Boussinesq approximation. The fluid inside the channel is air (Pr = 0.71) and it is considered in steady laminar regime. The radiation heat transfer is assumed to be twodimensional, the participating medium is considered as a gray gas with an optical thickness of s ¼ 0:1. The vertical walls are considered as gray-diffusely reflecting surfaces, the emissivity of the adiabatic wall is 0.9 and the emissivity of the heated wall has different values ranging from 0.1 to 0.9. The inlet and outlet of the channel are assumed to be black body surfaces. 2.1. Mathematical model The governing conservation equations for a two-dimensional incompressible flow in steady state are:
@ qu @ qv þ ¼0 @x @y
ð1Þ
@ ðquuÞ @ ðquv Þ @P @ þ ¼ þ @x @y @x @x
@ ðqv uÞ @ ðqvv Þ @P @ þ ¼ þ @x @y @y @x
l l
@u @ @u þ l @x @y @y
ð2Þ
@v @ @v þ l @y @x @y
qgbðT T 1 Þ
ð3Þ
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@ ðquT Þ @ ðqv T Þ @ k @T @ k @T 1 þ ¼ þ r qr @x @y @x C p @x @y C p @y Cp
ð4Þ
No-slip boundary conditions on the velocity components are imposed in the vertical surfaces. At the inlet of the channel, the total pressure depression imposed is p p0 þ qv 2 =2 ¼ 0, hence p0 ¼ qv 2 =2 where p0 is the ‘reduced’ static pressure (the difference between the pressure inside the channel and the pressure outside at the same elevation) and p0 is the ambient pressure. The inflow is assumed to be normal to the inlet. The temperature of incoming air (Tinlet ) is equal to the ambient temperature, T1 . The fluid pressure at the outlet is assumed to be the same as the ambient pressure. The streamwise variations of velocity components and temperature are neglected. The temperature boundary conditions are shown in the Fig. 1. The left wall is exposed to a uniform heat flux (qw ), and the right wall is considered adiabatic (qc þ qr ¼ 0). The local divergence of radiative flux (radiative source term) r qr appearing in the energy equation accounts for gas radiation, and is related to the local intensities by:
r qr ¼ j 4pIb ðrÞ
Z 4p
Iðr; XÞdX
ð5Þ
To obtain the radiation intensity field and calculate r qr , it is necessary to solve the radiative transfer equation (RTE). This equation for an absorbing, emitting and scattering gray medium can be written as [20]:
dI rs ð r Þ ¼ ðjðr Þ þ rðrÞÞIðr; ^sÞ þ ds 4p
Z
Iðr; ^sÞdX þ jðr ÞIb ðr Þ 0
4p
ð6Þ
and the radiative boundary condition for diffusely reflecting surfaces in Eq. (6) is:
Iðr w ; ^sÞ ¼ eðrw ÞIb ðr w Þ þ
qðrw Þ p
Z ^ ^s0 <0 n
^ ^s0 jdX0 Iðr w ; ^s0 Þjn
ð7Þ
surfaces, in discrete ordinates for cartesian coordinates can be written as:
I m ¼ eI b þ
qX wm0 l0m Im0 ; lm > 0 in x 2 C p m0
ð12Þ
q X 0 w 0 n I 0 ; nm > 0 in y 2 C p m0 m m m
ð13Þ
lm0 <0
I m ¼ eI b þ
nm0 <0
2.3. Interpolation schemes for the spatial discretization of RTE In this work the genuinely multidimensional (GM) scheme is used. The stencil used in this scheme depends on the characteristics directions [21]. It is a TVD scheme and the Van Albada limiter is used, in the form:
Limiter ½x; y ¼
xy2 þ yx2 x2 þ y 2
ð14Þ
The following relations are used to define the scheme (see Fig. 2): when lm P 0 ; nm P 0 ; Dlxm 6 Dnym i;j
l
m m Iw
¼l "
i;j
m m Ii1;j1 þ 0:5Limiter
l
m m m Ii;j I i1;j1
# ! n m Dxi m m Ii1;j Ii1;j1 lm Dyj
Dxi Sm i;j ;
ð15Þ and m nm Im s ¼ nm I i;j1
when
ð16Þ
lm P 0 ; nm P 0 ; Dlxmi;j P Dnymi;j
lm Imw ¼ lm Imi1;j1
ð17Þ
and 2.2. Radiative transport equation The radiative transfer equation (Eq. (6)) is solved using the discrete ordinates method (DOM). In this method, the RTE is substituted by a set of M discrete equations for a finite number of directions Xm , and each integral is substituted by a quadrature series of the form: M rs X ðXm rÞIðr; Xm Þ ¼ bIðr; Xm Þ þ jIb ðr Þ þ wk Iðr; Xk Þ 4p k¼1
ð8Þ
subject to the boundary condition:
Iðr w ; Xm Þ ¼ eðr w ÞIb ðrw Þ þ
qX w I r ; X0 jn Xk j p nXk 0 k w k
ð9Þ
where wk are the ordinates weight, b is the coefficient extinction (b ¼ j þ rs ) and rs is the scattering coefficient. This angular approximation transforms the original equation into a set of coupled differential equations. In cartesian coordinates, Eq. (9) becomes:
@I @I lm m þ nm m ¼ bIm þ jIb þ Sm @x @y
M rs X wk Iðr; Xk Þ 4p k¼1
ð18Þ where m m Sm i;j ¼ ðji;j þ rsi;j ÞIi;j þ ji;j Ibi;j
Ii,j+1 In Ii-1,j Iw
and lm y nm are the directional cosines of Xm and Sm is the source radiative term. And the boundary conditions for diffusely reflecting
Ii,j Is
Ii-1,j-1
ð11Þ
ð19Þ
The above expressions give us the fluxes when the characteristics point into the quadrant of the xy-plane. When the characteristics point into other quadrants of the xy-plane, the fluxes can be obtained by applying symmetries to Eqs. (15)–(18).
ð10Þ
where
Sm ¼
m nm Im s ¼ nm I i1;j1 þ 0:5Limiter
l m Dy j m m m Ii;j1 Im nm Im i;j I i1;j1 Dyj Si;j ; nm i1;j1 Dxi
Ie
Ii+1,j
Δyi,j
Ii,j-1
Δxi,j
Fig. 2. Interpolation stencil in GM scheme.
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3. Numerical procedure and verification The Navier–Stokes and energy equations are numerically solved using the finite volume method [22]. The numerical code was developed using Fortran computational language. The coupling between the momentum and continuity equations is made with the SIMPLEC algorithm, and a false-transient technique is used. The convective terms were discretized applying the hybrid scheme and the diffusive terms were discretized applying the central difference scheme. The system of algebraic equations obtained in the discretization is solved applying the line by line method (LBL) with the alternating direction implicit scheme (ADI). The radiation source term in energy equation is calculated by solving RTE using DOM. We used an angular quadrature set as LC-11 [23]. The convergence criterion for RTE and the others governing equations are about 1007. A study of spatial grid independence considering the highest value of emissivity was performed for each channel width. From this study, the computational grid that gives grid independent solutions were 31 201, 26 201, 21 201, 16 201 and 11 201 for the channel widths of 0.10, 0.08, 0.06, 0.04 and 0.02 m, respectively; which show non-significant differences (2.0%) for the components of velocity and for the temperature profile at the outlet of the channel. To validate the accuracy of our numerical code, three comparisons were made: free convection in antisymmetrically heated vertical channel, turbulent mixed convection and surface thermal radiation in a vertical channel and natural convection and radiation in a differentially-heated cavity with participating gases. First, numerical code was validated comparing our results against the results for free convection in non-symmetrically heated channel with a hot wall heated 7.5 °C above the ambient temperature and a cold wall was cooled 7.5 °C below the ambient temperature. The Table 1 shows comparison of the average Nusselt numbers at the hot wall of the experimental and numerical data reported by Roeleveld et al. [24] and the solution obtained for three values of Ra as 1:48 103 , 4:79 103 and 1:18 104 corresponding to aspect ratios of 26.4, 17.6 and 13.2. It can be seen that the present results agree with those presented by Roeleveld et al. [24]. The overall deviations are within 7% of the experimental data for the three Rayleigh numbers. The numerical code was also verified comparing our results against the results for a turbulent mixed convection and thermal radiation in a vertical channel reported by Barhaghi and Davidson [25] and Wang et al.[26]. The boundary conditions of the vertical channel were a constant heat flux (qw = 449 W/m2) in one channel wall while the other wall is insulated. The comparison of the maximum velocity (v max ) and average Nusselt number (Nuav e ) at the hot wall show a good agreement with the data from literature. Table 2 shows the comparison of results for Re = 5080 and e = 0.125, for v max , the percentage difference respect to the experimental data was 8.2%; and for Nuav e , the percentage difference respect to the experimental data was 4.3%. To validate the coupling of natural convection with radiation, we present a comparison of combined heat transfer in a
Table 2 Comparison of the hot wall average Nusselt number (Nuav e ) and maximum velocity (v max ) for vertical channel. Parameter
Wang et al. [26]
Barhaghi and Davidson [25]
Present study
v max (m/s)
1.22 39.5
1.46 33.4
1.33 (8.2)⁄ 40.7 (4.3)⁄
Nuav e ⁄
Absolute differences in percentage respect to the experimental data.
differentially-heated cavity. Table 3 shows the overall Nusselt number at the hot wall for two values of the optical thickness (s = 1 and s = 5) and two Rayleigh numbers. The results are compared with the values reported by [27] and shown a maximum difference of 1.1%. According to the table, the results obtained in the present work have a good agreement with the results of Lari et al. [27]. 4. Results To observe the effect that the gray gas radiation has on heat transfer and fluid flow in a vertical parallel-plate channel with asymmetric heating, in the first section we compare results of the channel with gray gas with the results of the channel with transparent gas under the same boundary conditions. The following section shows the effect that the channel width has on the mass flow and the heat transfer. 4.1. Optical thickness effect To show the effect of gray gas radiation on natural convection in the channel, two optical thicknesses s = 0.0 and s = 0.1 were considered. The results shown in this section are for a channel width b = 0.1 m and emissivity e = 0.9 in both channel walls. Fig. 3 presents the air vertical velocity component and its temperature with (a) s = 0.0 and (b) s = 0.1. The velocity field is qualitatively similar in both cases, higher velocity gradients occur in areas near the walls, with higher values near the heated wall (left wall). In both cases the lowest velocity occurs in the central zone of the channel; however, when s = 0.0 the velocity in the central zone of the channel is about 0.1 m/s, but when s = 0.1 it is approximately 0.25 m/s. The temperature behavior of the channel in both cases shows higher gradients in zones close to the walls; due to heat flux imposed on the left wall, the air close to this wall has the highest temperature values. On the other hand, for s = 0.0 the air temperature in the central zone along the channel has no variation from its inlet temperature; but when s = 0.1 approximately from two-thirds of the height of the channel, the air in the central zone has higher temperature than its inlet temperature. This effect occurs because in the case of surface thermal radiation, the energy is distributed on the walls; however, when considering radiative participant gas, part of the energy is absorbed by the air distributed across the channel width. Therefore, the air increases its temperature even in the central zone of the channel and decreases the heat transfer of the walls.
Table 1 Comparison of the hot wall average Nusselt number (Nuov e;h ) for an antisymmetrically heated vertical channels. Aspect ratio A
Rayleigh number Ra (103)
26.4 17.6 13.2 ⁄
Absolute differences in percentage.
1.48 4.79 1.18
Experimental comparison
Numerical comparison
Roeleveld et al.[24]
Present study
Roeleveld et al. [24]
Present study
0.97 1.03 1.32
1.03 (6.1)⁄ 1.10 (6.7)⁄ 1.22 (7.5)⁄
1.02 1.11 1.40
1.03 (0.9)⁄ 1.10 (0.9)⁄ 1.22 (12.8)⁄
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Table 3 Comparison of average overall Nusselt number (Nuov e;h ) at different optical thickness and Rayleigh numbers.
s=1
Rayleigh number Ra Lari et al. [27]
Present study
Lari et al. [27]
8.367
8.277 (1.1)⁄
6.811
6.814 (0.1)⁄
17.086
16.965 (0.7)⁄
14.514
14.580 (0.5)⁄
105 6
10 ⁄
s=5 Present study
Absolute differences in percentage.
(a) τ =0.0
(b) τ =0.1
Fig. 3. Vertical velocity (m/s) and temperature (°C) when b = 0.1 m with (a)
The above is observed in Fig. 4 that show the temperature at the heating wall of the vertical channel (T w ). In general, T w increases as the channel height increases. For s = 0.0, the T w shows a maximum difference of 32 °C between y = 0 m and y = 1.0 m. The T w for s = 0.0 reaches up 4.8 °C more than for s = 0.1 and shows a maximum difference of 7.7%.
1.0
τ 0.8
0.0 0.1
y(m)
0.6
0.4
0.2
0.0 35
40
45
50
55
60
65
70
Tw (°C) Fig. 4. Temperature of the heated wall of the vertical channel (T w ).
s = 0.0 y (b) s = 0.1.
The previous observations can also be seen in Fig. 5 where the (a) velocity profile and (b) temperature of the air at the outlet are shown; in this figure the x coordinate was dimensionless with the channel width b. In general, a higher the temperature the velocity is higher in the channel output. The air velocity and its temperature near the channel walls are greater when s = 0.0 than s = 0.1, with a difference of up to 0.70 m/s and 4.1 °C, respectively; while variables in the central zone of the channel when s = 0.1 are greater when s = 0.0, with a difference of up to 0.19 m/s and 1.7 °C, respectively. The velocity and temperature of the air in the channel show such behavior because the ability of the air to absorb energy depends on its optical thickness; therefore, the higher the optical thickness of the gas, the greater the increase in its internal energy and hence its buoyancy force. _ the convective heat flux Table 4 presents the mass flow rate (m), (qc;h ) and radiative heat flux (qr;h ) at the heated wall of the channel, _ = 18.74 g/ms when for s = 0.0 and s = 0.1. We can observe that m s = 0.0 and m_ = 32.32 g/ms when s = 0.1, for such values of m_ there is a percentage difference of 42%. The table also shows that qc;h is 142.6 and 127.7 W/m2 for s = 0.0 and s = 0.1, respectively; with a percentage difference between them of 10%. The qc;h presents this behavior because when the fluid is radiatively transparent (s = 0.0) the energy is concentrated on the walls, thus, the temperature difference between the wall and the adjacent air increases and therefore the convective heat flux increases. When s = 0.1, the temperature difference between the wall and the adjacent air is smaller than that of s = 0.0 because the radiative energy is absorbed by the gas increasing its temperature along the channel.
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0.7
50
τ 0.0 0.1
τ
40
0.5
30
0.4
20
T-Tinlet (°C)
v(m/s)
0.6
0.3 0.2
0.0 0.1
10
4 3
0.1
2 1
0.0
0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
x/b
x/b
(a)
(b)
0.6
0.8
1.0
Fig. 5. (a) Velocity profile and (b) temperature of the air at the outlet.
Table 4 _ the convective heat flux (qc;h ) and radiative heat flux (qr;h ) of the Mass flow rate (m), heated wall for a vertical channel.
s
_ (g/ms) m
qc;h (W/m2)
qr;h (W/m2)
0.0 0.1
18.7 32.3
142.6 127.7
107.4 122.3
Furthermore, when s = 0.0 the qr;h is 13.8% lower than that obtained for s = 0.1. This effect occurs because when s = 0.0 the energy from the other walls directly incides on the heated wall. However, when s = 0.1 the energy is attenuated by the gas before reaching the heated wall, which increases the net radiative heat flux because the amount of energy emitted is greater than the incident one. Therefore, from the energy balance between energy that is emitted and the energy that incident on the heated wall we obtained values for qr;h of 107.4 and 122.3 W/m2 for s = 0.0 and s = 0.1, respectively. The behavior described shows that gas radiation significantly affects the behavior of natural convection, for air with an optical thickness about 0.1. 4.2. Channel width effect This section presents the results obtained for b = 0.02, 0.04, 0.06, 0.08 and 0.10 m, considering an optical thickness s = 0.1 and emissivity e = 0.9 in both channel walls. Fig. 6 shows the contour of vertical velocity, for different values of b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10. When b P0.04 m higher values and velocity gradients occur in the region near to the heated wall. On the other hand, when b = 0.02 m the highest value of velocity is present in the central zone of the channel because such a channel width allows the air to reach similar conditions to those of a developed flow, therefore, the air velocity shows a nearly parabolic profile during its route through the channel. In addition, when b P0.06 m the velocity profile is almost constant in the central region of the channel, and when increasing the channel width, the velocity value decreases in the same region. When b = 0.04 m the air is accelerated in the region the near walls and with greater intensity near the heated wall, however in the central zone of the channel, the velocity neither presents a constant behavior nor parabolic profile as the other cases. Fig. 7 presents the temperature contour for different values of b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10. In general, the
(a)
(b)
(c)
(d)
(e)
Fig. 6. Contour of vertical velocity, for different b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10.
highest values and temperature gradients are present in the region near to the left wall, where the constant heat flux is received. On the other hand, when b P0.06 m the temperature of the air at the inlet remains almost constant in the central region of the channel up to a certain height, such height increases as the channel width increases; for instance, when b = 0.06 m that behavior is presented up to half of the height and when b = 0.10 m up to approximately two thirds of the channel. For b 60.04 m the air temperature in the central region of the channel increases as the channel height increases, in such a way that the air average temperature at the outlet of the channel reaches up to 28.1 °C more than the inlet temperature. This effects occurs because the thickness of the thermal boundary layers developed on the walls of the channels covers the width of the channel, eliminating the almost constant behavior of the temperature in the central region of the channel. Fig. 8 shows (a) the velocity profile at the exit of channel (v) and (b) the temperature difference of the air between the inlet and the outlet (T T inlet ) for all channel widths considered. In this figure the x coordinate was made dimensionless with the channel width b. Fig. 8a shows that the maximum velocity in the channel output
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(a)
(b)
(c)
(d)
central area of the channel, with a minimum value of up to 1.7 °C for b = 0.10 m. The average of T T inlet is 28.1 and 6.8 °C for b = 0.10 m and b = 0.02 m respectively; therefore, the average of T T inlet decreases up to 75% due to the increase of b from 0.02 to 0.10 m. _ when varying the Fig. 9 shows the behavior of the mass flow (m) channel width b. As the channel width increases the mass flow _ are in a range of 10.15 to 32.32 g/ms. increases, the values of m The mass flow rate increases up to 68% when b changes from 0.02 m to 0.10 m. Furthermore, when increasing channel width _ decreases, obtaining a minimum of the slope of the curve of m 12% percentage difference between the results for b = 0.08 and b = 0.10 m. This effect occurs because as the air volume increases the amount of energy required to move it is greater; in this case, the volume of air increases by increasing the channel width, however, the imposed heat flux on the wall remains constant. There_ is not proportional to the fore, we observe that the increase of m _ tends to decrease increase of the b and the slope of the curve of m with increasing b.
(e)
Fig. 7. Contour of temperature, for different values of b (m): (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10.
35
0.7
0.5
20
15
10 0.02
0.04
0.06
0.08
0.10
b (m) Fig. 9. Mass flow rate behavior with respect to the channel width.
b (m) 0.10 0.08 0.06 0.04 0.02
40
30
T-T inlet (°C)
0.4
v (m/s)
25
50
b (m) 0.10 0.08 0.06 0.04 0.02
0.6
30
m (g/m.s)
is about 0.59 m/s for all channel widths considered and this velocity is present in the region near the heated wall of the channel, except for b = 0.02 m that presents an almost parabolic (near to developed flow conditions) profile where the maximum speed on the channel output is presented in x=b 0.35. When b P0.06 m the velocity has an almost constant behavior in the central area of the channel, in addition, as the channel width decreases the velocity in the central region and has a minimum value up 0.21 m/s when b = 0.10 m. When b = 0.04 m the velocity decreases as moving away from the walls up to a value 0.34 m/s in the center of the channel (x=b=0.5). Fig. 8b shows that highest values T T inlet are present in the zone near the heated wall with values up to 49 °C for b = 0.02 m. As b increases it decreases the value of T T inlet , therefore for b = 0.10 m there is a maximum of 41 °C. Furthermore, the T T inlet of air adjacent to the insulated wall, decreases as the channel width increases. For b 60.04 m one can observe that in the central region of the channel T T inlet decreases as it moves away from the solid walls, with minimum values of 4.7 and 17.8 °C for b = 0.02 m and b = 0.04 m respectively. For b 60.06 m an almost constant behavior of T T inlet occurs in the
0.3
20
0.2 10 0.1
0.0
0 0.0
0.2
0.4
x/b
(a)
0.6
0.8
1.0
0.0
0.2
0.4
x/b
0.6
0.8
1.0
(b)
Fig. 8. (a) Velocity profile at the exit of channel and (b) the temperature difference of the air between the inlet and the outlet, for all channel widths.
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4.3. Wall emissivity effect 32
To observe the effect of the heated wall emissivity eh on the _ and the effect of eh on the conbehavior of the mass flow rate m, vective and radiative heat flux (qc;h ,qr;h ), this section presents the results obtained for eh = 0.1, 0.3, 0.5, 0.7 and 0.9, considering an optical thickness s = 0.1 and b = 0.10 m. Fig. 10 shows the behavior of the mass flow when varying the emissivity of the heated wall of the vertical channel. As the value _ also increases, to a maximum value of of eh increases the m _ shows this behavior because the 32.3 g/ms for an eh = 0.9. The m radiative energy emitted by the heated wall increases with the increment of walls emissivity. Therefore, the gas in the channel absorbs more energy. The radiative energy absorbed by the gas is proportional to the divergence of radiative flux r qr ðW=m3 Þ obtained in the gas, as this is the source term in the energy equation. Therefore, Fig. 11 shows the divergence of radiative flow of the gas inside the channel when b = 0.1 and for different values of eh : a) 0.1, b) 0.3, c) 0.5, d) 0.7 and e) 0.9. This figure also shows the profile of radiative heat
m (g/m.s)
30
28
26
24
0.1
0.3
0.5
0.7
0.9
εh Fig. 10. Mass flow rate behavior with respect to the
eh .
(a)
(b)
(d)
(c)
(e)
Fig. 11. Profile of qr;h and r qr ðW=m Þ of the air inside the channel for different values of 3
eh : a) 0.1, b) 0.3, c) 0.5, d) 0.7 and e) 0.9.
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qr,h
200
qc,h
q (W/m2)
150
100
50
0.1
0.3
0.5
0.7
0.9
εh Fig. 12. Convective and radiative heat fluxes for different values of b = 0.10 m.
eh and
4.4. Numerical correlations
Table 5 _ correlation. Coefficients for m
s
bðmÞ A
B
0.0
0.10 0.08 0.06 0.04 0.02
0.5246 0.7127 0.9874 0.3914 0.4725
18.968 19.703 19.975 17.463 10.429
0.10 0.08 0.06 0.04 0.02
3.411 2.8003 2.1343 1.4519 0.3771
32.966 28.972 24.449 18.850 10.225
0.1
increasing values qr;h along the channel with higher gradients than the other cases. Fig. 12 shows the behavior of the convective and radiative heat fluxes when the emissivity of the heated wall varies as eh = 0.1, 0.3, 0.5, 0.7 and 0.9 and channel width b = 0.10 m. The figure indicates that as the emissivity of the heated wall increases, the value of qr;h increases and the value of qc;h decreases. The values of qr;h are in the interval 34.16qr;h 6122.3 W/m2, moreover the values have a maximum percentage difference of 72% between the results for eh = 0.1 and eh = 0.9. The behavior of qr;h is because this amount is the result of an energy balance between the energy emitted by the wall and the energy that strikes it, where the amount of energy emitted by the wall is proportional to the emissivity. The values of qc;h are in the interval 215.66qc;h 6127.3 W/m2, moreover the values have a maximum percentage difference of 40% between the results for eh = 0.1 and eh = 0.9. The values of qc;h present such behavior because as the radiative heat flux (qr;h ) emitted by the wall to the air inside the channel increases, the temperature of the gas increases and the temperature difference between the heated wall and the adjacent air decreases, therefore, the value of qc;h also decreases.
_ was obtained. The corA correlation for the mass flow rate (m) relation was established as a function of the emissivity of the heated wall (eh ) for each channel width. This correlation can be expressed as:
Coefficient
_ ¼ Alnðeh Þ þ B m
flux from the heated wall (qr;h ) for each field of r qr . The negative sign of r qr means that the radiative volume flow is absorbed/ attenuated by the gas, therefore, it is decreasing the radiative energy beam. Fig. 11 shows that the divergence of radiative heat flow and the qr;h increase when increasing the emissivity of the heated wall. The behavior of qr;h significantly affects the distribution of r qr . When eh = 0.1, r qr does not present considerable variations along the channel, because the profile qr;h is nearly constant, with a maximum variation of 33.5 W/m2. On the contrary, when eh = 0.9 the profile qr;h presents a steeper slope, with a maximum variation of 102.7 W/m2, while values of r qr increase with
ð20Þ
Table 5 shows the coefficients for each channel width. The maximum deviation from the numerical results is presented by the channel width b = 0.04 m. Such deviation is 1.7 and 1.0% for s = 0.0 and s = 0.1, respectively. In addition, correlations for average convective Nusselt number (Nuc ) and average radiative Nusselt number (Nur ) were also obtained. The correlations were established depending on the emissivity of the heated wall (eh ) for each channel width. Such correlations are expressed as:
Nuc ¼ Aðe2h Þ þ Bðeh Þ þ C
ð21Þ
Nur ¼ Aðe2h Þ þ Bðeh Þ þ C
ð22Þ
Table 6 shows the correlation coefficients for Nuc , for each channel width. The maximum deviation from the numerical results are presented by a channel width b = 0.06 m. Such deviation is 0.5 and 3.0% for s = 0.0 and s = 0.1, respectively. Table 7 shows coefficients for the correlation of Nur for each channel width. The maximum deviation with respect to numerical
Table 6 Coefficients for Nuc correlation.
s
bðmÞ
B
C
0.0
0.10 0.08 0.06 0.04 0.02
1.9173 1.4313 0.9542 0.4902 0.2569
3.762 2.7846 1.8613 0.9898 0.4979
16.171 13.112 9.9635 6.7919 3.5246
0.1
0.10 0.08 0.06 0.04 0.02
1.684 1.3042 2.7501 0.6111 0.3837
3.8607 2.9937 0.4623 1.4269 0.8566
16.554 13.397 9.8415 6.9335 3.5681
Coefficient A
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I. Zavala-Guillén et al. / International Journal of Heat and Mass Transfer 107 (2017) 510–519 Table 7 Coefficients for Nur correlation.
s
bðmÞ
0.0
0.10 0.08 0.06 0.04 0.02
8.7602 6.9543 5.1806 3.4344 1.6977
19.019 14.825 10.853 7.1183 3.4957
0.7457 0.616 0.4759 0.3203 0.1553
0.1
0.10 0.08 0.06 0.04 0.02
8.3427 6.6905 1.3055 3.315 1.5906
3.8607 17.611 10.255 8.3561 3.834
0.4821 0.3945 0.6225 0.2032 0.1007
Coefficient A
results for s = 0.0 is 3.0% and is presented by a channel width b = 0.04 m. When s = 0.1 the maximum deviation is 4% and is presented by a channel width b = 0.06 m. 5. Conclusions Based on the results of natural convection and radiation in a vertical parallel-plate channel with asymmetric heating, considering the radiation effects for both walls and participating medium, it can be concluded that: The mass flow of the channel for s = 0.1 is up to 42% greater than that obtained for a transparent medium (s = 0.0). In addition, it was observed that considering radiation in both the walls and in the medium (s = 0.1) increases the radiative heat flux at the heated wall up to 13.8% compared to the case with transparent medium (s = 0.0). On the other hand, convective heat flux at the heated wall for s = 0.0 is up to 10% greater than for s = 0.1. When b changes from 0.02 m to 0.10 m it increases the mass flow rate up to 68%. However, the average temperature difference between the air at the inlet and air at the outlet of the channel decreases up to 75% due to the increase of b from 0.02 m to 0.10 m. The mass flow rate increases up to 29% due to an increment of emissivity of the heated wall from 0.1 to 0.9. Varying eh from 0.1 to 0.9 increases the radiative heat flux up to 72% in the heated wall and it decreases the convective heat flux in the heated wall up to 40%.
Acknowledgements The authors are grateful to Consejo Nacional de Ciencia y Tecnología (CONACYT) whose financial support made this work possible. I. Zavala-Guillén would like to thanks TWAS-CNPQ program for the financial support during her stay at UNICAMP. C. Salinas wishes to thanks for the financial support grant #2016/01493-9, São Paulo Research Foundation (FAPESP). References [1] G. Gan, General expressions for the calculation of air flow and heat transfer rates in tall ventilation cavities, Build. Environ. 46 (2011) 2069–2080. [2] B. Zamora, J. Hernández, Influence of variable property effects on natural convection flows in asymmetrically-heated vertical channels, Int. Commun. Heat Mass Transfer 24 (1997) 1153–1162. [3] G. Desrayaud, A. Fichera, Laminar natural convection in a vertical isothermal channel with symmetric surface-mounted rectangular ribs, Int. J. Heat Mass Transfer 23 (2002) 519–529.
B
C
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