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International Communications in Heat and Mass Transfer
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A novel case study for thermal radiation through a nano fl uid as a semitransparent medium via discrete ordinates method to consider the absorption and scattering of nanoparticles along the radiation beams coupled with natural convection
Arash Karimipour
Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
A R T I C L E I N F O
Keywords:
Volumetric radiation Natural convection Nanofluid
Semitransparent medium
A B S T R A C T
Natural convection in a volumetric radiant enclosurefilled by a nanofluid is studied numerically for thefirst time by using discrete ordinates (DO) method to consider the absorption and scattering coefficients of nanoparticles on the radiation beams through the nanofluid as a semitransparent medium. Present nanofluid is a mixture of Al2O3nanoparticles suspended in water as the basefluid. The volume concentration percentages of nanoparticles are almost small to make a semitransparent medium which means the achieved results can be used in theflat plate solar collectors. Moreover the SIMPLE algorithm offinite volume method for Navier-Stokes continuity, momentum and energy equations are solved and coupled with DO to simulate the total radiation and natural convection in a shallow inclined rectangular 2-D enclosure. This shape of enclosure is chosen due to it might represent the usual configuration of a solar collector. The enclosure inner walls are the gray diffuse emitters and reflectors. The effects of various amounts of Rayleigh number and volume concentration at different values of wavelength are investigated. The positive effect of wave length on radiation heatflux and consequently total heatflux of radiation and natural convection is observed.
1. Introduction
A solar collector can be an example of the fluid flow and heat transfer in cavity or enclosure; however it can be a benchmark work at more complex geometries. Hence almost a great number of researches can be referred who they concernedfluidflow in cavities at different configurations and positions. Natural, force and mixed convection are three mechanisms of convection heat transfer which should be involved through them. For example the force convection is achieved while one of the walls moves or natural convection movement is generated when the suitable temperature gradient existences. The shallow cavity is called to one whose length is taller than its width. Now consider a shallow inclined enclosure which its hot lid moves in a constant velo- city; consequently the mixed convection heat transfer would be achieved. It should be mentioned that moving side walls also be able to generate the mixed convection movements. For the states which work only with the natural convection, Rayleigh number is used as a di- mensionless parameter to show the power of buoyancy forces[1–7].
The classic well known Navier-Stokes equations are applied for the wide range offlow regimes at macro scales level; however the particle based methods like direct simulation Monte Carlo, molecular dynamic
and lattice Boltzmann method should be used at micro and nano scales levels which represent the slipflow, transition and free molecularflow regimes. Several articles are introduced which concerned the recent methods in order to simulate theflow and heat transfer in macro and micro cavities and channels. Moreover the appropriate comparison between those of particle based methods and classic CFD ones were presented which proved their acceptable performances[8–14].
Available purefluids like water and oils have usually small thermal conductivity which corresponds neglecting the conduction heat transfer mechanism through them. It means conduction may be increased by adding the measured dispersed solid nanoparticles which leads to have a mixture called nanofluid. These suspended nanoparticles have very small masses which prevent from the sedimentation phenomenon. On the other side, the random movements of nanoparticles augment the convection heat transfer mechanism through the basefluid; therefore using nanofluid might be an innovative way to increase the heat transfer rate[15–17]. The investigation of nanofluid mixed convection composed of Cu/water in a lid driven cavity or inside a microchannel were reported by Karimipour et al.[18,19]. Moreover a large number of papers numerically studied the nanofluidflow inside the enclosures and pipes which they corresponded the positive effect of applying nanofluid
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on Nusselt number and its related heat transfer rate[20–25].
The radiant enclosures are called to the cavities which the thermal radiation effects are involved inside them; so that the influences of both free convection and thermal radiation should be considered at the same time. According to the medium substance and its thickness, two dif- ferent radiant approaches are used. Transparent media is usually ap- plied for the vacuum or pure simple gases like air which allows a wave to pass through it without any attenuation; that means the surface ra- diation should be coupled with natural convection only on the boundaries for the cavityflows. However the volumetric radiation must be applied for the medium with partial attenuation on the electro- magnetic wave which is named semitransparent media. In recent case, in addition of the effects of radiation on the boundaries, a more radiant transport equation must be solved to simulate the scattering and ab- sorption of medium elements along the radiation beams[26,27].
Several researchers reported the surface thermal radiation con- jugated natural convection in cavities. Their results versus experimental works implied that how important the radiation effects could be and should not be ignored in the numerical studies[28–31]. Alvarado et al.
[32]numerically investigated the interaction between natural convec- tion and surface radiation in a shallow cavity cooled from top and heated from bottom. They presented their results in the two ways of coupled and uncoupled between radiation and convection and devel- oped a correlation for Nusselt number.
2. Present work novelty
Several works can be addressed for the volumetric thermal radiation coupled with convection heat transfer in a cavityfilled by a nanofluid as a semitransparent media[33–35]. However they all used Rosseland approximation in order to compute the radiation source term in the energy equation. It should be mentioned that Rosseland approach is not be able to consider the scattering effects of medium particles on the radiation beam; so that the simulation of thermal radiation through a nanofluid which contains suspended nanoparticles via Rosseland method shows some detours in results versus the experimental ones.
Hence in present article, the natural convection in a volumetric radiant enclosurefilled by a nanofluid as a semitransparent media is studied by the discrete ordinates (DO) method for thefirst time in order to con- sider the nanoparticles scattering and absorption through the radiation beam.
Nomenclature AR aspect ratio
c specific heat (J/kg-K) d nanoparticles diameter (nm) B dimensionless emissive power g gravity (ms−2)
H enclosure height (m)
I intensity
k thermal conductivity (W/m-K) Ka absorption coefficient Ke extinction coefficient Ks scattering coefficient L enclosure length (m) m relative refractive index
np real part of the nanoparticles complex refractive index nm medium refractive index
p pressure (Pa)
P dimensionless pressure Pr Prandtl number Pl Planck number q heatflux (Wm−2)
qR radiation heatflux (Wm−2) QR dimensionless radiation heatflux Qe extinction efficiency
Qa absorption efficiency Qs scattering efficiency Ra Rayleigh number T temperature (K)
T0 reference temperature, = 0.5(TH+ TC) TC cold temperature (K)
TH hot temperature (K) u x-velocity (ms−1) U dimensionless x-velocity v y-velocity (ms−1)
V dimensionless y-velocity
x cartesian coordinate along the constant temperature walls (m)
X dimensionless form of x
y cartesian coordinate along the insulated walls (m) Y dimensionless form of y
Greek symbols
α thermal diffusivity (m2s−1)
ε emissivity
g inclination angle of enclosure
κp imaginary part of the nanoparticles complex refractive index
κbf extinction coefficient of the basefluid
σ Stefan Boltzmann constant (= 5.67 × 10−8Wm−2K−4) φ nanoparticles volume concentration
χ size factor λ wave length (nm)
μ dynamic viscosity (Nsm−2) μ,ξ direction cosines
θ dimensioless temperature
ϕ dimensionless temperature based on T0
ρ density (kgm−3)
υ kinematic viscosity (m2s−1) ω scattering albedo
τ optical thickness Subscripts
con convection
f fluid
nf nanofluid
rad radiation
s solid
Table 1
Thermo-physical properties of Al2O3and water[25].
cp(J/Kg-K) ρ(Kg/m3) k (W/m-K) μ(Pa s) β(1/K)
Water 4179 997.1 0.613 8.91 × 10−4 21 × 10−5
Al2O3 765 3970 40 – 0.85 × 10−5
3. Problem statement
Natural convection in a volumetric radiant enclosure filled by a nanofluid is studied numerically for thefirst time. Present Newtonian incompressible nanofluid is a mixture of spherical Al2O3nanoparticles suspended in water as the basefluid. The nanoparticles diameters are supposed as d = 13 nm which are at thermal equilibrium with the base fluid and move with the same velocity. Table 1shows the thermo- physical properties of Al2O3and water.
The volume concentration percentages of nanoparticles are almost small to make a semitransparent medium which means the achieved results can be used in the flat plate solar collectors. As a result the discrete ordinates (DO) approach is applied for the volumetric radiant transport equation to consider the absorption and scattering coefficients for nanofluid medium. Moreover the SIMPLE algorithm offinite volume method for Navier-Stokes continuity, momentum and energy equations are solved and coupled with DO to simulate the total radiation and natural convection in a shallow inclined rectangular 2-D enclosure. This shape of enclosure is chosen due to it might represent the typical configuration of a solar collector. According toFig. 1, the enclosure is made of two hot (at TH) and cold (at TC) walls and also two other in- sulated side walls. The inclination of enclosure is supposed asγ= 45°
which responses a usual solar collector. Aspect ratio of cavity is com- puted as the length over its height which is taken in to account as AR = L/H = 10. Boussinesq approximation is applied to simulate the buoyancy forces.
The enclosure inner walls are the gray diffuse emitters and reflectors
with emissivity ofε= 0.5. The effects of various amounts of Rayleigh number (Ra= 104 and Ra= 105) and volume concentration (φ= 0.1% andφ= 0.3%) at different values of wavelength of incident light (λ= 200 nm, λ= 300 nm, λ= 600 nm which illustrate the visible section of the thermal radiation wavelength) are investigated.
Fig. 1.The schematic configuration of the collector enclosure.
Table 2
Optical properties of water/Al2O3nanofluid[36].
Volume fraction λ= 200 (nm) λ= 300 (nm) λ= 600 (nm)
Κs(m−1) Κa(m−1) Κs(m−1) Κa(m−1) Κs(m−1) Κa(m−1)
φ= 0.1% 26 40 5 13 1 8
φ= 0.3% 80 40 15 13 1 8
Table 3
Grid independency study for the values of U andθat the point of X = 0.5 L and Y = 0.8H forφ= 0.1%,Ra= 105andλ= 200 nm.
650 × 65 700 × 70 750 × 75
U 217.2 217.9 218.2
θ 0.62 0.63 0.63
Table 4
Numalong the vertical wall and Umaxalong the vertical centerline versus the results of Tiwari and Das[17]atGr= 10,000.
φ= 0 φ= 8%
Present work
Tiwari & Das[17] Present work
Tiwari & Das[17]
Ri = 0.1 Umax 0.51 0.48 0.49 0.46
Num 32.02 31.64 43.97 43.37
Ri = 10 Umax 0.21 0.19 0.20 0.18
Num 1.41 1.38 1.91 1.85
Table 5
Validation with those of Alvarado et al.[32]for the interaction between natural con- vection and surface thermal radiation in a cavity for the case of coupled radiation &
natural convection at the state of aspect ratio = 16 and inclination angle = 15.
Ra qsouth(w/m2) qnorth(w/m2)
Alvarado et al.[32]
105 163.30 −162.45
106 351.77 −350.04
Present work
105 161.68 −160.93
106 348.58 −346.76
Fig. 2.Streamlines atRa= 104andφ= 0.1% for different values ofλ.λ= 200 nm (top),λ= 300 nm (middle),λ= 600 nm (bottom).
4. Governing equations
4.1. Optical properties of medium
Determine the scattering coefficient, absorption coefficient and re- fractive index of a nanofluid as a semitransparent medium is one of the main obstacles to simulate the volumetric radiation through it.
Scattering and extinction efficiencies are measured using Mie theory.
Moreover the relative refractive index should be used due to immersed nanoparticles through the basefluid.
= +
m n iκ n
p p
m (1)
where m, np,κpand nmindicate the relative refractive index, real part of the nanoparticles complex refractive index, imaginary part of the nanoparticles complex refractive index and medium refractive index, respectively.
The size factor is presented as follows according to the nanoparticle diameter and the wave length through the medium:
= χ πd
λ (2)
Now the extinction and scattering efficiencies of nanoparticle are achieved as:
∑
∑
= + +
= + +
=
∞
=
∞
Q χ n a b
Q χ n a b
2 (2 1) Re( )
2 (2 1)(| | | | )
e λ n
n n
s λ n
n n
, 2
1
, 2
1
2 2
(3) an and bn illustrate the Mia scattering coefficients[36,37]. The ex- tinction efficiency represents the absorption and scattering efficiencies,
= +
Qe λ, Qa λ, Qs λ, (4)
Eq.(3)corresponds to extinction coefficient of the medium:
= + −
K φQ
d φ πκ
λ
1.5 (1 )4
e λ e λ bf
, ,
(5) κbfshows the extinction coefficient of the basefluid. From the recent equation:
= +
Ke λ, Ka λ, Ks λ, (6)
Ka,λ,Ks,λpresent the absorption and scattering coefficients of the medium, respectively.
∑
∑
= + +
= + +
=
∞
=
∞
K λ
π n a b
K λ
π n a b
2 (2 1)(| | | | )
2 (2 1) Re{ }
s λ n
n n
e λ n
n n
, 2
0
2 2
, 2
0 (7)
which leads to determine the optical properties of water/Al2O3nano- fluid shown inTable 2. More details can be found in Refs[36,37].
4.2. Natural convection
The classic two dimensional Navier-Stokes equations of nanofluid are presented as follows,
Continuity:
∂
∂ + ∂
∂ = u x
v y 0
(8) x-momentum:
⎜ ⎟
∂
∂ + ∂
∂ = − ∂
∂ + ⎛
⎝
∂
∂ + ∂
∂
⎞
⎠
+ −
u u x v u
y ρ
p x
μ ρ
u x
u
y βg T T γ
1 ( ) sin
2 2
2
2 0
(9) y-momentum:
⎜ ⎟
∂
∂ + ∂
∂ = − ∂
∂ + ⎛
⎝
∂
∂ + ∂
∂
⎞
⎠
+ −
u v x v v
y ρ
p y
μ ρ
v x
v
y βg T T γ
1 ( ) cos
2 2
2
2 0
(10) Energy:
⎜ ⎟
∂
∂ + ∂
∂ = ⎛
⎝
∂
∂ +∂
∂
⎞
⎠
− ∇⋅
u T x v T
y α T
x T
y ρc1 q
R 2
2 2
2 (11)
The density, viscosity, heat capacity and thermal conductivity of nanofluid:
= + −
ρnf φρs (1 φ ρ) f (12)
Fig. 3.Isotherms atRa= 104andφ= 0.1% for different values ofλ.λ= 200 nm (top), λ= 300 nm (middle),λ= 600 nm (bottom).
Fig. 4.Streamlines atRa= 105andφ= 0.1% for different values ofλ.λ= 200 nm (top),λ= 300 nm (middle),λ= 600 nm (bottom).
Fig. 5.Isotherms atRa= 105andφ= 0.1% for different values ofλ.λ= 200 nm (top), λ= 300 nm (middle),λ= 600 nm (bottom).
= −
μnf μf (1 φ)2.5 (13)
= − +
ρc φ ρc φ ρc
( p nf) (1 )( p f) ( p s) (14)
= ⎡
⎣⎢
+ − −
+ + −
⎤
⎦⎥
k k k k φ k k
k k φ k k
( 2 ) 2 ( )
( 2 ) ( )
nf f
s f f s
s f f s (15)
Subscripts f and s indicate the base fluid and solid nanoparticles respectively; while φillustrates the nanoparticles volume concentra- tion.
The governing Navier-Stokes equations can be shown in di- mensionless forms according to the following variables:
= = = = =
= − − = − −
= = −
= = −
Y y H X x H V vH υ U uH υ P pH ρ υ
θ T T T T ϕ T T T T
υ α Gr gβH T T υ
Ra Gr gβH T T υα
, , , , ( )
( ) ( ), ( ) ( )
Pr , ( )
Pr ( ) ( )
C H C H C
H C
H C
2 0 2
0
3 2
3
(16)
∂
∂ +∂
∂ = U X
V Y 0
(17)
⎜ ⎟
∂
∂ + ∂
∂ = −∂
∂ + ⎛
⎝
∂
∂ +∂
∂
⎞
⎠ + U U
X V U Y
P X
U X
U Y
Raϕ γ Pr sin
2 2
2
2 (18)
U
-10 -8 -6 -4 -2 0 2 4 6 8 10
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =104, =0.1%, =200nm Ra =104, =0.1%, =300nm Ra =104, =0.1%, =600nm
U
-10 -8 -6 -4 -2 0 2 4 6 8 10
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =104, =0.3%, =200nm Ra =104, =0.3%, =300nm Ra =104, =0.3%, =600nm
Fig. 6.The profiles of U along the vertical centerline of enclosure atRa= 104.
⎜ ⎟
∂
∂ + ∂
∂ = −∂
∂ + ⎛
⎝
∂
∂ +∂
∂
⎞
⎠ + U V
X V V Y
P Y
V X
V Y
Raϕ γ Pr cos
2 2
2
2 (19)
⎜ ⎟
∂
∂ + ∂
∂ = ⎛
⎝
∂
∂ + ∂
∂
⎞
⎠
− ∇⋅
U ϕ X V ϕ
Y
ϕ X
ϕ Y
ϕ Pl Q 1
Pr Pr R
2 2
2 2
0
(20) where Pl= k ≈0.1
H σT4 03 and σare the Planck number and Stefan- Boltzmann constant. T0= 0.5(TH+ TC) is the mean reference tem- perature which corresponds toϕ0=T0/(TH−TC).
Based on the Cartesian coordinates shown inFig.1, dimensionless natural convection boundary conditions can be presented as:
= = = = ≤ ≤
= = = − = ≤ ≤
= = = + = ≤ ≤
U V at X and X Y
U V ϕ at Y X
U V ϕ atY X
0; 0 10, 0 1
0, 0.5; 0, 0 10
0, 0.5; 1, 0 10 (21)
4.3. Radiation
The source term of QRin energy Eq.(20), represents the radiative energy which can be calculated from the radiation transfer equation of discrete ordinates method; so that the steady state radiation intensity is achieved by the following equation for a medium with constant
U
-50 -40 -30 -20 -10 0 10 20 30 40 50
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =105, =0.1%, =200nm Ra =105, =0.1%, =300nm Ra =105, =0.1%, =600nm
U
-50 -40 -30 -20 -10 0 10 20 30 40 50
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =105, =0.3%, =200nm Ra =105, =0.3%, =300nm Ra =105, =0.3%, =600nm
Fig. 7.The profiles of U along the vertical centerline of enclosure at Ra= 105.
refractive index,
∂
∫
∂ + ∂
∂ + = − +
μ I X ξ I
Y τI τ
π ω B ω IdΩ
4 [(1 ) ]
π
4 (22)
= + = = +
ω K
K K B T
T τ K K H
, , ( )
s
a s
s a
04
(23) where I,ω, B andτshow the dimensionless intensity at the point of (X,Y) in a direction of Ω, single scattering albedo, dimensionless emissive power and optical thickness, respectively. In additionμ,ξare the direction cosines.
Planck number was chosen in Eq.(20)as the interaction variable rather than the well known parameter of N which illustrates the
conduction-radiation parameter:
= +
= ⋅
N k K K
σT Pl τ
( )
4
s a
03 (24)
The radiative heatflux vector is determined by using the intensity distribution,
∫ ∫
= IdΩ= μ +ξ IdΩ
QR Ω ( i j)
π π
4 4 (25)
Now integrating Eq.(20)in DO approach leads to[27,38]:
∫
∇⋅QR=(1−ω τ B) [ − IdΩ]
π
4 (26)
0.0 0.2 0.4 0.6 0.8 1.0
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =104, =0.1%, =200nm Ra =104, =0.1%, =300nm Ra =104, =0.1%, =600nm
0.0 0.2 0.4 0.6 0.8 1.0
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =104, =0.3%, =200nm Ra =104, =0.3%, =300nm Ra =104, =0.3%, =600nm
Fig. 8.The profiles ofθalong the vertical centerline of enclosure at Ra= 104.
which can be substituted in Eq.(20).
The boundary conditions for the pure natural condition was pre- sented in Eq. (21); however the following equation must be applied along the thermal insulated walls at X = 0 and X = 10 for 0≤Y≤1 in the presence of thermal radiation,
⎜ ⎟
⎛
⎝
−∇ + ⎞
⎠
⋅ = ϕ ϕ
Pl0QR i 0
(27)
5. Results and discussion
Natural convection in a volumetric radiant enclosure filled by
water/Al2O3nanofluid is numerically studied. The volume concentra- tion percentages of nanoparticles are almost small to make a semi- transparent medium which means the achieved results can be used in theflat plate solar collectors. As a result the discrete ordinates (DO) approach is applied for the volumetric radiant transport equation to consider the absorption and scattering coefficients for nanofluid medium.
Table 3indicates the grid independency study for the values of U andθat the point of X = 0.5 L and Y = 0.8H forφ= 0.1%,Ra= 105 andλ= 200 nm for three different grids of 650 × 65, 700 × 70 and 750 × 75. It is seen that the grids of 700 × 70 is suitable for the next computations.
Averaged Nusselt number on the vertical wall and maximum of
0.0 0.2 0.4 0.6 0.8 1.0
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =105, =0.1%, =200nm Ra =105, =0.1%, =300nm Ra =105, =0.1%, =600nm
0.0 0.2 0.4 0.6 0.8 1.0
Y
0.0 0.2 0.4 0.6 0.8 1.0
Ra =105, =0.3%, =200nm Ra =105, =0.3%, =300nm Ra =105, =0.3%, =600nm
Fig. 9.The profiles ofθalong the vertical centerline of enclosure at Ra= 105.
horizontal velocity on the vertical centerline from the present study versus those of Tiwari and Das [17] at Gr= 10,000 are shown in Table 4. They reported the heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. The acceptable agreements are found between the results.
More validation is presented in Table 5concerns the interaction between natural convection and surface thermal radiation in a cavity for the case of coupled radiation & natural convection for the state of aspect ratio = 16 and inclination angle = 15 which is reported by Al- varado et al.[32]. The appropriate accommodation is also observed at Ra= 105andRa= 106with those of Ref.[32].
Fig. 2andFig. 3show the streamlines and isotherms atRa= 104 and φ= 0.1% for different values of wave length as λ= 200 nm, λ= 300 nm andλ= 600 nm. Whole space of enclosure is dominated
by a strong large cell which illustrates the buoyancy forces existence and corresponds to the almost stratified isotherms at the middle parts of the enclosure and implies the cold and hotfluids existence adjacent to the cold and hot walls. However more curvature is observed in iso- therms near the adiabatic walls.Fig. 4andFig. 5indicate the stream- lines and isotherms atRa= 105andφ= 0.1% for different values ofλ asλ= 200 nm,λ= 300 nm andλ= 600 nm. Thesefigures say the same story as both previous ones; however various amounts of wave length do not have the significant effects on the shapes of streamlines and isotherms ofFig. 2toFig. 5.
The profiles of U along the vertical centerline of enclosure at Ra= 104is given byFig. 6at various amounts of nanoparticles volume concentration. It is observed that different amounts of wave length and volume concentration do not correspond to a noticeable change in U
X
0 2 4 6 8 10
Hea t flux (wm -2 )
0 20 40 60 80 100 120
140
Rad. (Ra =104, =0.1%, =200nm)
Rad.+ Con. (Ra =104, =0.1%, =200nm)
X
0 2 4 6 8 10
He at fl ux (wm -2 )
0 20 40 60 80 100 120
140
Rad. (Ra =104, =0.1%, =600nm)
Rad.+ Con. (Ra =104, =0.1%, =600nm)
Fig. 10.The local heatflux of radiation and the local heatflux of total radiation and natural convection along the hot surface of en- closure atRa= 104.
profiles so that U = 0 at Y = 0 and Y = 1 where the stationary cold and hot walls are being; However absolute value of Umaxwill be almost equal to 8 at Y≈0.2 and Y≈0.8. In the following,Fig. 7represents the profiles of U along the vertical centerline of enclosure atRa= 105 at differentλandφ.Ηere the volume concentration also has the neg- ligible effect on U however the effects of wave length are sensible; so that moreλcorresponds to higher U which implies the augmentation influence of radiation on the buoyancy forces.
The linear profiles ofθalong the vertical centerline of enclosure at Ra= 104at different values of wave length and volume concentration are presented inFig. 8which almost independent ofλandφ. However there is a constant temperaturefield in the core domain of enclosure at 0.3 < Y < 0.6 inFig. 9which concerns the state ofRa= 105. Wave length variation leads to small changes in the temperature profiles of this region. This fact means the influences of wave length would be
more important at higher amounts ofRa.
The local heatflux of radiation and the local heatflux of total ra- diation and natural convection along the hot surface of enclosure at Ra= 104 for various wave lengths is given byFig. 10. The positive effect ofλon radiation heatflux and consequently total heatflux of radiation and natural convection can be clearly observed in thisfigure.
More amount of heatflux is achieved around X < 2 while it decreases mildly with X. In order to show the influence of Ra on heatflux, the local heatflux of radiation and the local heatflux of total radiation and natural convection along the hot surface of enclosure at Ra = 105are presented inFig. 11. It might be supposed that higherRaonly corre- sponds to higher natural convection heat transfer; however a compar- ison between Fig. 10andFig. 11reveals that larger Raleads to in- vigorate the radiation heat transfer atfixed amount of wave length in addition of supporting the buoyancy forces.
X
0 2 4 6 8 10
Hea t flux (wm -2 )
0 50 100 150 200
250
Rad. (Ra =105, =0.1%, =200nm)
Rad.+ Con. (Ra =105, =0.1%, =200nm)
X
0 2 4 6 8 10
He at fl ux (wm -2 )
0 50 100 150 200
250
Rad. (Ra =105, =0.1%, =600nm)
Rad.+ Con. (Ra =105, =0.1%, =600nm)
Fig. 11.The local heatflux of radiation and the local heatflux of total radiation and natural convection along the hot surface of en- closure atRa= 105.
Fig. 12 illustrates the local heat flux of radiation along the hot surface of enclosure atRa= 104andRa= 105at differentλto show better the positive effect of wave length on radiation heat transfer.
Higher wave length corresponds to lower scattering and absorption coefficients which are the attenuators factors of the radiation beams through the medium; hence larger local heat flux of radiation is oc- curred at larger wave length. It should be mentioned that this process can be observed more severely at higher values ofRa.
Now it is tried to investigate the effects of volume concentration;
therefore the local heatflux of total radiation and natural convection along the hot surface of enclosure atRa= 104andRa= 105 at dif- ferentφandλare shown inFig. 13. The positive effect ofλand the great positive effect ofRaon total radiation and natural convection are also can be seen in thisfigure but the influence ofφshould be more
noticed. In the most previous works concerned nanofluid in enclosures, it was reported that more volume concentration led to larger heat transfer through the cavity. MoreoverTable 2implied that the using higher percentage of nanoparticles corresponded to more scattering coefficient and consequently less heat transfer of radiation through the medium. As a result it can be said that more volume concentration increases the natural convection and at the same time decreases the radiation heat transfer through the enclosure; so that the total radiation and natural convection heat transfer decreases with volume con- centration as shown inFig. 13for the state ofλ= 300 nm. However for the state ofλ= 600 nm which scattering coefficient is not significantly affected byφ(seeTable 2), the heatflux will be independent of volume concentration. This newfinding, which implies the negative effect of volume concentration on total heatflux, makes different the present
X
0 2 4 6 8 10
Hea t flux (wm -2 )
0 10 20 30 40 50 60
70
Rad. (Ra =104, =0.1%, =200nm)
Rad. (Ra =104, =0.1%, =300nm) Rad. (Ra =104, =0.1%, =600nm)
X
0 2 4 6 8 10
He at fl ux (wm -2 )
0 10 20 30 40 50 60
70
Rad. (Ra =105, =0.1%, =200nm)
Rad. (Ra =105, =0.1%, =300nm) Rad. (Ra =105, =0.1%, =600nm)
Fig. 12.The local heatflux of radiation along the hot surface of enclosure atRa= 104andRa= 105at differentλ.
work results from those of previous studies of nanofluid which ignore the radiation (or even consider it by inappropriate methods of radiation like Rosseland). More focus on this event, the local heatflux of radia- tion along the hot surface of enclosure atRa= 104andRa= 105at differentφandλis given byFig. 14which clearly shows the decreasing influence of volume concentration on the radiation heat transfer spe- cially atλ= 300 nm compared withλ= 600 nm.
Finally and in order to present a suitable visual aspect of all vari- ables, the heat transfer rate of radiation and the heat transfer rate of total radiation and natural convection over the hot surface of enclosure at different amounts ofRa,φandλare presented inFig. 15. Lowerφ corresponds to higher heat transfer rate at λ= 300 nm due to the scattering coefficient has the most growth withφat this state (almost 3 times more); however the effects of volume concentration is smaller at
λ= 200 nm and also its effect can be ignored atλ= 600 nm.
6. Conclusion
Natural convection in a volumetric radiant enclosure filled by water/Al2O3 nanofluid was numerically studied. The volume con- centration percentages of nanoparticles were supposed small to make a semitransparent medium which meant the achieved results could be used in the flat plate solar collectors. The following points can be provided according to the present work achievements:
1- Obtain the more accurate results for thermal radiation through a nanofluid as a semitransparent medium due to using discrete ordi- nates method to consider the absorption and scattering of
X
0 2 4 6 8 10
He at flux (wm -2 )
0 30 60 90 120 150 180 210 240 270
Rad.+ Con. (Ra =104, =0.1%, =300nm) Rad.+ Con. (Ra =104, =0.3%, =300nm) Rad.+ Con. (Ra =105, =0.1%, =300nm) Rad.+ Con. (Ra =105, =0.3%, =300nm)
X
0 2 4 6 8 10
He at fl ux (wm -2 )
0 30 60 90 120 150 180 210 240 270
Rad.+ Con. (Ra =104, =0.1%, =600nm) Rad.+ Con. (Ra =104, =0.3%, =600nm) Rad.+ Con. (Ra =105, =0.1%, =600nm) Rad.+ Con. (Ra =105, =0.3%, =600nm)
Fig. 13.The local heatflux of total radiation and natural convection along the hot surface of enclosure atRa= 104andRa= 105at differentφandλ.
nanoparticles along the radiation beams coupled with natural con- vection. In previous articles the Rosseland approximation was ap- plied which was not accurate enough for semitransparent mediums like nanofluid.
2- The volume concentration has the negligible effect on U however the effects of wave length are sensible; so that moreλcorresponds to higher U which implies the augmentation influence of radiation on the buoyancy forces. The influence of wave length on temperature profiles would be more important at higher amounts ofRa.
3- The positive effect ofλon radiation heatflux and consequently total heat flux of radiation and natural convection can be clearly ob- served. LargerRaleads to invigorate the radiation heat transfer at fixed amount of wave length in addition of supporting the buoyancy forces.
4- Higher wave length corresponds to lower scattering and absorption
coefficients which are the attenuators factors of the radiation beams through the medium; hence larger local heatflux of radiation is occurred at larger wave length. It should be mentioned that this process can be observed more severely at higher values ofRa.
5- Using higher percentage of nanoparticles corresponds to more scattering coefficient and consequently less heat transfer of radia- tion through the medium. As a result more volume concentration increases the natural convection and at the same time decreases the radiation heat transfer through the enclosure; so that the total ra- diation and natural convection heat transfer decreases with volume concentration specially at λ= 300 nm. This new finding, which implies the negative effect of volume concentration on total heat flux, makes different the present work results from those of previous studies of nanofluid which ignore the radiation or even consider it by the inappropriate methods of radiation.
X
0 2 4 6 8 10
Hea t flux (wm -2 )
0 10 20 30 40 50 60 70
Rad. (Ra =104, =0.1%, =300nm) Rad. (Ra =104, =0.3%, =300nm) Rad. (Ra =105, =0.1%, =300nm) Rad. (Ra =105, =0.3%, =300nm)
X
0 2 4 6 8 10
He at fl ux (wm -2 )
0 10 20 30 40 50 60
70
Rad. (Ra =104, =0.1%, =600nm)
Rad. (Ra =104, =0.3%, =600nm) Rad. (Ra =105, =0.1%, =600nm) Rad. (Ra =105, =0.3%, =600nm)
Fig. 14.The local heatflux of radiation along the hot surface of enclosure atRa= 104andRa= 105at differentφandλ.
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(nm)
200 300 400 500 600
Heat Tr ansf er Rate ( w )
0 100 200 300 400 500 600 700 800 900
1000
Rad.+ Con. (Ra =104, =0.1%)
Rad. (Ra =104, =0.1%) Rad.+ Con. (Ra =104, =0.3%) Rad. (Ra =104, =0.3%) Rad.+ Con. (Ra =105, =0.1%) Rad. (Ra =105, =0.1%) Rad.+ Con. (Ra =105, =0.3%) Rad. (Ra =105, =0.3%)
Fig. 15.The heat transfer rate of radiation and the heat transfer rate of total radiation and natural convection over the hot surface of enclosure at different amounts ofRa,φandλ.
