Discrete ordinates simulation of radiative participating nanofluid natural convection in an enclosure

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Journal of Thermal Analysis and Calorimetry

An International Forum for Thermal Studies

ISSN 1388-6150 Volume 134 Number 3

J Therm Anal Calorim (2018) 134:2183-2195

DOI 10.1007/s10973-018-7233-8

Discrete ordinates simulation of radiative participating nanofluid natural convection in an enclosure

Mohamadali Hassani & Arash

Karimipour

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Discrete ordinates simulation of radiative participating nanofluid natural convection in an enclosure

Mohamadali Hassani¹^• Arash Karimipour¹

Received: 8 November 2017 / Accepted: 31 March 2018 / Published online: 17 April 2018 Akade´miai Kiado´, Budapest, Hungary 2018

Abstract

Natural convection was studied by discrete ordinates approach within a radiation enclosure in which nanoparticles absorption and scattering were taken into consideration. Nanofluid was prepared in deionized water as the base fluid and Al₂O₃nanoparticles. In order to reach a semitransparent medium suspension, nanofluid was prepared by adding small amount of Al₂O₃ nanoparticles volume concentration. In addition, the Navier–Stokes equations were solved based on SIMPLE algorithm of finite volume method to simulate the natural convection and then coupling to radiation by discrete ordinates approach. Two-dimensional inclined rectangular enclosure was assumed, while the inner emissive walls were found to be diffuse-gray. The various values of Ra number and volume concentration of nanoparticles at different wavelengths were studied. The results declared that the wavelength had positive impact on the radiation heat flux. More accurate results were obtained in comparison with the previous researches used Roseland approximation, due to using the discrete ordinates method to consider the adsorption and scattering coefficients. With increasing nanoparticles volume concentration, the scattering coefficient increased; hence, the heat transfer of radiation declined. On the other hand, it enhanced the natural convective heat transfer. By coupling these two conclusions, it was observed that more volume concentration corresponded to less total heat transfer which implied the main outcome of present work and made it different from the present works used nanofluid.

Keywords Volumetric radiationSemitransparentDiscrete ordinatesNanofluidParticipating media

List of symbols

AR = 3 Enclosure aspect ratio

B Emissive power in non-dimensional form

H Enclosure height (m)

I Intensity

K_a,K_eand K_s

Absorption, extinction and scattering coefficients

L Enclosure length (m)

P Pressure (Pa)

Pl Planck number

q_R Heat flux of radiation (W m^-2) Q_e,Q_aand

Q_s

Extinction, absorption and scattering efficiency

T Temperature (K)

Greek symbols e Emissivity c Inclination angle

r Stefan Boltzmann constant (= 5.67910^-8W m^-2K^-4) k Wave length (nm)

x Scattering albedo s Optical thickness

Subscripts Con Convection

f Fluid

Introduction

One of the methods was used in previous effort for increasing the heat transfer efficiency was concentrated on application of particles in micro- or nano-scales. The results of experimentations exhibit that by decreasing the size of particles more heat transfer efficiency can be

& Arash Karimipour

[email protected];

[email protected]

1 Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

Journal of Thermal Analysis and Calorimetry (2018) 134:2183–2195 https://doi.org/10.1007/s10973-018-7233-8(0123456789().,-volV)(0123456789().,-volV)

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obtained, and this is due to the fact that by decreasing the particles size more solid surface is obtained [1–5]. Thus, implementation of nanoparticles can enhance the heat transfer within the fluid due to the smaller size of nanoparticles and more available solid–liquid interface.

Nanoparticles are made in nanometer dimension that their mean diameters are at the range of 10–100 nm. There are wide ranges of studies through the literature on the effect of nanoparticles on heat transfer efficiency within the base fluid [6–10].

Das et al. [11] studied numerically transient natural convection within a vertical channel containing nanofluid in the presence of heat radiation. According to their findings, the nanofluid velocity in PST was much higher than PHF and by increasing the nanoparticle volume concentration, the velocity and nanofluid temperature decreased.

Moreover, their results showed that heat radiation had major impact on nanofluid velocity and temperature.

Ho et al. [12] studied the natural convective heat transfer within a square-shaped enclosure containing water/Al₂O₃ nanofluid by means of finite element method. In their simulation, upper and lower walls were chosen to be insulated and different temperatures were chosen for two other walls. According to their findings, different models for viscosity estimated different Nu numbers during solving partial differential equations numerically.

Bourantas et al. [13] studied numerically natural convective heat transfer in nanofluid with constant physical properties within a square enclosure with high porosity. In their experimentation, the lower wall was chosen for pro- ducing constant heat flux and other walls were kept at low temperature. Their results showed that at constant nanoparticle volume fraction, with the increase in Ra number, the averaged Nu number increased significantly.

Oztop et al. [14] presented a research in order to investigate the natural convective heat transfer within a highly porous square-shaped enclosure. Their results declared that by increasing Darcy number at constant Ra number, the averaged Nu number increased significantly. There are wide range of studies concerned heat transfer and fluid flow within the porous media inside the micro- or macrochan- nels [15–21].

In previous efforts, it is mentioned that by adding nanoparticles to base fluid, the heat transfer efficiency increases which is due to the thermal conductivity of nanoparticles. Adding nanoparticles to the base fluid, the overall thermal conductivity of nanofluid increases and leads to higher heat transfer rate through the fluid. Several other works can be addressed to predict the nanofluid thermo-physical properties [22–27]. Karimipour et al.

[28,29] studied heat transfer within a microchannel, and they concluded that heat transfer in nanofluid is much higher than heat transfer in the base fluid. Also other

numerical and experimental results can be referred in order to prove the more Nusselt number because of the higher values of nanoparticles concentration [30–40]. Some of the researchers have investigated heat radiation and convective natural heat transfer together. They found that increasing radiation on the boundaries led to much higher impact on radiation equations as well as the emission and absorption coefficients. The significant effects of radiation on natural convective heat transfer were reported in the several works [41–52]. Alvarado et al. [53] studied the effect of natural convection and surficial radiation and showed by increasing the inclination angle, and the total heat transfer rate increased. Sheikholeslami et al. [54] studied numerically the effect of heat radiation on heat transfer enhancement within a square enclosure in which water/Al₂O₃was used as the nanofluid. According to their results, the heat transfer enhancement had direct relation to Ha number, but it had reverse relation with Ra number. Ghalambaz et al.

[55] investigated the impact of viscous dissipation and radiation on natural convection heat transfer within a porous square enclosure containing nanofluid. Their experimentation declared that the values of Nu number were not the same as the condition near to the hot and cold walls due to the dissipation viscous impacts. In addition, they concluded that by increasing Ec number, Nu number decreased near to the hot wall and increased near to the cold wall.

Naghib et al. [56] investigated experimentally the induced natural convective heat transfer produced by means of direct radiation to the square-shaped enclosure containing pure water. Akbar et al. [57] studied numerically the impact of various thermal conductivity and heat radiation on carbon nanotubes flow with slippery boundary conditions. Based on their results, velocity profiles and boundary layer thickness increased for single-wall carbon nanotubes as well as multiwall carbon nanotubes at constant Gr number. Several other works reported the nanofluid applications at various conditions [58–60].

It was seen that the most of the previous researches reported their results while they all used Roseland approximation to obtain the heat radiation. However, the Roseland method did not take the scattering of particles into account through a radiation beam. The comparison between the experimental method and the Roseland approach showed that the results were slightly different.

Thus, in order to present the more accurate numerical results in this study, discrete ordinates (DO) approach was used to simulate the natural convection and heat radiation in a cavity filled with nanofluid to consider the scattering and absorption of nanoparticles.

2184 M. Hassani, A. Karimipour

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Problem statement

In this study, natural convection together with thermal radiation was studied numerically in an enclosure by using commercial software of FLUENT 6.3.2. The nanofluid used in this research was water/Al2O3 which was in heat equilibrium and velocity with water. Table1 implies the properties of Al₂O₃and water. Small amount of nanoparticles was dispersed within the base fluid to generate a semitransparent environment. In this research, the shape of the enclosure was like a solar collector panel which was located with inclination angle of 45, (c= 45). According to Fig.1, the enclosure was made of two cold (T_C) and hot (T_H) walls and two insulated vertical walls. The cavity aspect was chosen as AR =L/H= 3. Navier–Stokes equations were solved by using SIMPLE algorithm with finite volume method coupled by DO model. Discrete ordinates approach was used due to the fact that the absorption and scattering coefficients of nanoparticles in the nanofluid could be taken into consideration. In order to take into account the floating forces, Boussinesq approximation was implemented.

The inner side of walls was chosen to be gray with emissivity of e= 0.5. The effects of different values of Rayleigh number (Ra= 10⁴andRa= 10⁵) and nanoparticles volume concentration (u= 0.1% andu= 0.3%) were investigated at various amounts of radiation wavelengths (k= 200 nm, k= 300 nm and k= 600 nm). It was well known that the radiation effects were more important at the large values of surfaces temperatures. However, the dimensionless approach was used at present work to show the effects of surfaces radiations according to the non-dimensional physical conditions of the supposed problem.

Governing equations

Optical properties

One of the important issues for simulating thermal radiation in a cavity was obtaining the absorption coefficient, scattering coefficient and refractive index of nanofluid.

Therefore, by using Mie theory, the coefficients of scattering, extinction and refractive index of nanofluid were estimated by lowest deviation.

m¼npþiKp

nm

ð1Þ

wherem,np,kp andnm were the relative refractive index, the actual refractive index of the nanoparticle, the virtual refractive index of the nanoparticles, and the refractive index of the medium, respectively. Also, the size factor was calculated by following equation:

v¼pd

k ð2Þ

Also, the scattering and extinction coefficients of the nanoparticles were obtained as follows:

Q_s;k¼ 2 X²

X¹

n¼1

2nþ1

ð ÞRe aj jn²þj jbn²

Q_e;k¼ 2 X²

X¹

n¼1

2nþ1

ð ÞReðanþbnÞ

ð3Þ

an andbn showed the Mie scattering coefficients [50,51].

The coefficients of absorption, scattering and extinction were related to each other by the following relationship:

Q_e;k ¼Q_a;kþQ_s;k ð4Þ

Equation3 indicates the extinction coefficient of medium:

Table 1 Properties of Al₂O₃

and water [35] c_p/J Kg^-1K^-1 q/Kg m^-3 k/W m^-1K^-1 l/Pa s bð1=KÞ

Water 4179 997.1 0.613 8.91910^-4 21910^-5

Al₂O₃ 765 3970 40 – 0.85910^-5

L g

T_H T_C

H

Nanofluid

y x γ

Fig. 1 The geometric configuration of the enclosure

Discrete ordinates simulation of radiative participating nanoﬂuid natural convection in an… 2185

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K_e;k¼1:5uQe;k

d þð1uÞ4pKbf

k ð5Þ

Kbfwas the extinction coefficient of base fluid that could be calculated by following relation:

K_e;k¼K_a;kþK_s;k ð6Þ

K_s;k andK_a;k were absorption and scattering coefficient of medium which could be calculated by the following relation by means of physical properties of water/Al₂O₃ nanofluid (Table2).

K_s;k¼k² 2p

X¹

n¼0

2nþ1

ð Þj jan²þj jbn²

K_e;k¼k² 2p

X¹

n¼0

2nþ1

ð ÞRe af nþb_ng

ð7Þ

More details can be found in Refs. [50,51].

Natural convection

The Navier–Stokes equation was presented as follows for a two-dimensional radiant enclosure [52]:Continuity:

ou oxþov

oy¼0 ð8Þ

x-momentum:

uov oxþvou

oy¼ 1 q

op oxþl

q o²u ox²þo²u

oy²

þbg Tð T0Þsinc ð9Þ y-momentum:

uov oxþvou

oy¼ 1 q

op oyþl

q o²v ox²þo²v

oy²

þbg Tð T0Þcosc ð10Þ Energy:

uoT oxþvoT

oy¼a o²T ox²þo²T

oy²

1

qcr qR ð11Þ The density, viscosity, heat capacity, thermal conductivity and thermal expansion coefficient were calculated according to the following equations [61]:

q_nf¼uq_sþð1uÞq_f ð12Þ

l_nf ¼l_f=ð1uÞ^2:5 ð13Þ qcp

nf¼ð1uÞqcp

fþu qc p

s ð14Þ

knf¼kf

ksþ2kf

ð Þ 2uðkf ksÞ ksþ2kf

ð Þ þuðkfksÞ

ð15Þ

ðqbÞ_nf¼ð1uÞðqbÞ_fþu qbð Þ_s ð16Þ Equations (13,15) were used according to Ref. [61] by Aminossadati and Ghasemi who investigated the natural convection cooling of a localized heat source at the bottom of an enclosure filled by a nanofluid composed of Al₂O₃/ water. That article has been considered as a one of benchmarks studies of nanofluid in the cavities. However, more new experimental and theoretical correlations have been presented for the thermo-physical properties of that nanofluid. It should be mentioned that noticeable differ- ences were not observed between the values of nanofluid’s thermal conductivity and viscosity from Eqs. (13, 15) of Ref. [61] with those of other available articles. In this equations, f and sreferred to the base fluid and nanoparticles and u was volume concentration of nanoparticles.

Navier–Stokes equation could be rewritten in the forms of dimensionless parameter by using the following equations [52]:

Y¼y=H;X¼x=H;V¼vH=t;U¼uH=t;P¼pH²=q₀t² h¼ðTTCÞ=ðTHTCÞ;/¼ðTT0Þ=ðTHTCÞ

Pr¼t=a;Gr¼gbH³ðT_HT_CÞ=t² Ra¼Gr:Pr¼gbH³ðTHTCÞ=ð Þta

ð17Þ oU

oXþoV

oY ¼0 ð18Þ

Table 2 Optical properties of

water/Al₂O₃nanofluid [50] Volume fraction k¼200=nm k¼300=nm k¼600=nm

Ks=m¹ Ka=m¹ Ks=m¹ Ka=m¹ Ks=m¹ Ka=m¹

u= 0.1% 26 40 5 13 1 8

u= 0.3% 80 40 15 13 1 8

Table 3 Grid study at X= 0.5L and Y= 0.8H for /= 0.1%, Ra= 10⁵andk= 200 nm

255985 270990 285995

U 7.31 7.32 7.325

h 0.66 0.67 0.67

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UoU oXþVoU

oY ¼ oP

oXþ o²U oX²þo²U

oY²

þRa Pr/sinc

ð19Þ

UoV oXþVoV

oY ¼ oP

oYþ o²V oX²þo²V

oY²

þRa Pr/cosc

ð20Þ

Uo/

oXþVo/

oY ¼ 1 Pr

o²/ oX²þo²/

oY²

/₀

PlPrr QR ð21Þ Pl¼_H4rT^k 3

0¼0:1 andrwere Planck number and Stefan–

Boltzmann constant, respectively. In addition, the averaged

reference temperature could be calculated according to the following equation:

T0 ¼0:5ðTHþTCÞ ð22Þ which was equal to /₀¼T0=ðTHTCÞ. According to Fig.1, the natural convection boundaries were estimated using the following equations,

U¼V ¼0: at X¼0;X¼3;0Y1 U¼V ¼0;/¼ 0:5: atY ¼0;0X3 U¼V ¼0;/¼ þ0:5: atY ¼1;0X3

ð23Þ Table 4 Validation versus those

of Tiwari and Das [62] at Gr= 10000

u= 0 u= 8%

Present work Tiwari and Das [62] Present work Tiwari and Das [62]

Ri= 0.1 U_max 0.51 0.48 0.49 0.46

Nu_m 32.02 31.64 43.97 43.37

Ri= 10 U_max 0.21 0.19 0.20 0.18

Nu_m 1.41 1.38 1.91 1.85

Table 5 Alvarado et al. results

[53] versus present work ones Ra q_south q_north

Present work Alvarado et al. [53] Present work Alvarado et al. [53]

10⁵ 161.68 163.30 -160.93 -162.45

10⁶ 348.58 351.77 346.76 -350.04

Fig. 2 Streamlines forRa= 10⁴and/= 0.1% atk= 200 nm (top), k= 300 nm (middle),k= 600 nm (bottom)

Fig. 3 Isotherms forRa= 10⁴and /= 0.1% at k= 200 nm (top), k= 300 nm (middle),k= 600 nm (bottom)

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Radiation

According to Eq.21, the term of QR in energy equation could be calculated by using discrete ordinates (DO) approach and the radiation intensity for the medium with

constant refractive index was calculated by using the following correlations [41,42]:

loI oXþnoI

oYþsI¼ s

4p ð1xÞB Z

4p

IdX 2

4

3

5 ð24Þ

x¼ Ks

K_aþK_s;B¼ T

T₀⁴;s¼ðK_sþK_aÞH ð25Þ where I, x, B, and s were dimensionless intensity at coordinate (X,Y) in one path on X, the unit reflect of the scattering, the strength of dimensionless scattering and visual thickness, respectively. Plank number in Eq.21was related to N parameter, (introducing convection-radiation), by means of following relation:

N¼k Kð sKaÞ

4rT₀³ ¼Pls ð26Þ

The vector of radiation heat flux was obtained by, Fig. 4 Streamlines forRa= 10⁵and/= 0.1% atk= 200 nm (top),

k= 300 nm (middle),k= 600 nm (bottom)

Fig. 5 Isotherms for Ra= 10⁵ and /= 0.1% atk= 200 nm (top), k= 300 nm (middle),k= 600 nm (bottom)

–10 –8 –6 –4 –2 U0 2 4 6 8 10

Y

0.0 0.2 0.4 0.6 0.8 1.0

Ra = 10⁴, = 0.1%, = 200 nm Ra = 10⁴, = 0.1%, = 300 nm Ra = 10⁴, = 0.1%, = 600 nm

–10 –8 –6 –4 –2 U0 2 4 6 8 10

Y

0.0 0.2 0.4 0.6 0.8 1.0

Ra = 10⁴, = 0.3%, = 200 nm Ra = 10⁴, = 0.3%, = 300 nm Ra = 10⁴, = 0.3%, = 600 nm

Fig. 6 Horizontal dimensionless velocity profiles at vertical central line atRa= 10⁴

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Q_R¼ Z

4p

XIdX¼ Z

4p

liþnj

ð ÞIdX ð27Þ

With integrating over Eq. (21) by means of DO, the following relation was obtained [42,52]:

r QR ¼ð1xÞs B Z

4p

IdX 2

4

3

5 ð28Þ

Due to the heat radiation, the following equation should be taken into consideration as a boundary condition over the insulated walls atX= 0 andX = 3 for 0B YB1.

r/þ/₀ PlQ_R

i¼0 ð29Þ

Results and discussion

In the present study, the natural convection and volumetric heat radiation were investigated numerically in a rectangular enclosure filled with water/Al₂O₃ nanofluid as a semitransparent medium. Discrete ordinates approach was used to simulate the radiation heat transfer considering the scattering and absorption of nanoparticles.

In order to reach an independent mesh, U and h for u= 0.1%, Ra= 10⁴ and k= 200 nm at X= 0.5L and Y= 0.8H were studied and the results were reported in three different meshes of 85 9255, 909270 and 959285 in Table3. As it was concluded from the results, it was more appropriate to use mesh of 90 9270 due to the better results were obtained.

In Table4, a comparison was made on averaged Nu number over the vertical wall and maximum horizontal velocity over the central vertical line with the results obtained by Tiwari and Das [62] atGr = 10000. As it could be seen, the results were same and no significant deviations U

–40 –30 –20 –10 0 10 20 30 40

Y

0.0 0.2 0.4 0.6 0.8 1.0

Ra = 10⁵, = 0.1%, = 200 nm Ra = 10⁵, = 0.1%, = 300 nm Ra = 10⁵, = 0.1%, = 600 nm

U

–40 –30 –20 –10 0 10 20 30 40

Y

0.0 0.2 0.4 0.6 0.8 1.0

Ra = 10⁵, = 0.3%, = 200 nm Ra = 10⁵, = 0.3%, = 300 nm Ra = 10⁵, = 0.3%, = 600 nm

Fig. 7 Horizontal dimensionless velocity profiles at vertical central line atRa= 10⁵

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%, = 200 nm Ra = 10⁴, = 0.1%, = 300 nm Ra = 10⁴, = 0.1%, = 600 nm

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%, = 200 nm Ra = 10⁴, = 0.3%, = 300 nm Ra = 10⁴, = 0.3%, = 600 nm

Fig. 8 Dimensionless temperature profiles at vertical central line at Ra= 10⁴

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were observed. In order to validate the results of this study with other studies, more data were extracted from the article published by Alvarado et al. [53] which is presented in Table5. The results also showed that the deviation of data obtained in this research and the literature was less than 5%.

Figures2 and 3 show the streamlines and isotherms at Ra= 10⁴ and u= 0.1% at k = 200 nm, k = 300 nm and k= 600 nm. As it can be seen, a large cell surrounds the enclosure’s space indicating that there is a floating force in the enclosure. The presence of hot and cold liquids helps to form isotherms within the middle of the enclosure. On the other hand, the isotherms have specific and same curvature near the insulating walls. Figures4 and5 show the flow and isotherms are obtained at the same conditions and Ra= 10⁵. The results of Figs.2 and 3 are the same for Figs.4and5. As it is observed, no significant changes are seen on streamlines and isotherms at various wavelengths.

Figure6 shows the U profiles along the vertical center line of the enclosure for u= 0.1% and u= 0.3% at Ra= 10⁴. Considering the shape, wavelength changes and no effects of volume concentrations on the velocity profiles are obtained, thus the velocity of nanofluid is zero near the upper and lower walls within the enclosure and the velocity is maximum at Y &0.2 and Y&0.8. Figure7 shows U profiles on the vertical center line of the enclosure for u= 0.1% and u= 0.3% at Ra= 10⁵ for various wavelengths. As shown in Fig.6, volume concentration has no more effect on U profiles. Also same results are obtained from Fig. 7. But it is observed that with the increase in wavelength higher velocity is obtained due to increased radiation effect on the buoyancy forces. Figure 8shows the linear profiles of h on the vertical center line of the enclosure for u= 0.1% and u= 0.3% at Ra= 10⁴ and different wavelengths. It is concluded that changes in volume concentration and wavelength are not affected.

Y

0.0 0.2 0.4 0.6 0.8 1.0

Ra = 10⁵, = 0.1%, = 200 nm Ra = 10⁵, = 0.1%, = 300 nm Ra = 10⁵, = 0.1%, = 600 nm

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%, = 200 nm Ra = 10⁵, = 0.3%, = 300 nm Ra = 10⁵, = 0.3%, = 600 nm

Fig. 9 Dimensionless temperature profiles at vertical central line at Ra= 10⁵

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 20 40 60 80 100 120 140 160 180

Rad. (Ra = 10⁴, = 0.1%, = 200 nm) Rad.+ Con. (Ra = 10⁴, = 0.1%, = 200 nm)

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 50 100 150 200 250

Rad. (Ra = 10⁴, = 0.1%, = 600 nm) Rad.+Con. (Ra = 10⁴, = 0.1%, = 600 nm)

Heat flux/wm–2 Heat flux/wm–2

Fig. 10 Radiation heat flux and radiation together convection heat flux on hot wall atRa= 10⁴

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Similarly (for Ra= 10⁵), the linear profiles of h on the vertical center line of the enclosure which is presented in Fig.9 are the same as the previous results.

In Fig.10, the local heat flux of the radiation and the local heat flux of the total radiation and natural convection on the hot wall of the enclosure are presented at a constant volume concentration of nanoparticles and Ra= 10⁴ for k= 200 nm and k = 600 nm. As it can be seen, with the increase in the wavelength of the radiation, the value of convective heat flux and radiation will increase. Moreover, For X\0.75 the heat flux is maximum and then decli- nes insignificantly. The same trend is also observed for the heat flux in Ra= 10⁵ which is presented in Fig.11. However, by comparison between the results presented in Fig.10 and 11, it can be seen that with increasing Ra number the heat flux of convection and radiation also increases due to increased floating force, but it is noteworthy to express that the heat radiation flux in the constant wavelength increases significantly.

To investigate the effect of the wavelength on the heat flux, the radiation heat flux on the hot wall is calculated in Fig.12atRa= 10⁴andRa= 10⁵. As it can be seen, with the increase in the wavelength of the heat transfer, the radiation increases which is due to the declination in the absorption and scattering coefficient. Also, this effect is more pronounced with the increase in Ra number. In order to investigate the effects of the nanoparticles volume concentration on the heat flux, the local heat flux of radiation and the total natural convective heat flux on the hot wall of the enclosure are presented in Fig.13for different kanduatRa= 10⁴andRa= 10⁵. It can be seen that with increasing the wavelength and Ra number, the heat flux of the total convection and radiation increases significantly.

On the other hand, in the case of k= 300 nm, the convective and radiative heat transfer decrease with increasing nanoparticles volume concentration. According to Table2, with the increasing nanoparticles volume concentration, an X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 50 100 150 200 250

300 Rad. (Ra = 10⁵, = 0.1%, = 200 nm)

Rad.+Con. (Ra = 10⁵, = 0.1%, = 200 nm)

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 100 200 300

Rad. (Ra = 10⁵, = 0.1%, = 600 nm) Rad.+Con. (Ra = 10⁵, = 0.1%, = 600 nm)

Heat flux/wm–2Heat flux/wm–2

Fig. 11 Radiation heat flux and radiation together convection heat flux on hot wall atRa= 10⁵

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 20 40

60 Rad. (Ra = 10⁴, = 0.1%, = 200 nm)

Rad. (Ra = 10⁴, = 0.1%, = 300 nm) Rad. (Ra = 10⁴, = 0.1%, = 600 nm)

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 20 40 60 80 100

Rad. (Ra = 10⁵, = 0.1%, = 200 nm) Rad. (Ra = 10⁵, = 0.1%, = 300 nm) Rad. (Ra = 10⁵, = 0.1%, = 600 nm)

Fig. 12 Heat flux of radiation on the hot wall at Ra= 10⁴ and Re= 10⁵

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increase in scattering coefficient is resulted that reduces the heat transfer of radiation and increases the natural convection. Fork = 600 nm, the scattering coefficient does not change, and therefore the heat flux would not change significantly.

To investigate the effect of the nanoparticles volume concentration on the heat flux of radiation, it is given foru andkatRa= 10⁴andRa= 10⁵in Fig.14on the hot wall.

According to the results and as previously mentioned for other condition with the increase in volume concentration

of nanoparticles, the heat transfer of radiation decreases at k = 300 nm and it does not change at k = 600 nm. In Fig.15, the radiation heat flux and the total radiation and convection heat flux on the hot wall of the enclosure are presented for different values of Ra,uandk. As it can be seen, with the increase in volume concentration at k = 300 nm, the maximum effect is resulted on heat transfer also atk = 200 nm less effect is obtained and also can be neglected atk= 600 nm.

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 100 200 300 400

Rad.+ Con. (Ra = 10⁴, = 0.1%, = 300 nm) Rad.+ Con. (Ra = 10⁵, = 0.1%, = 300 nm) Rad.+ Con. (Ra = 10⁴, = 0.3%, = 300 nm) Rad.+ Con. (Ra = 10⁵, = 0.3%, = 300 nm)

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 100 200 300

400 Rad.+ Con. (Ra = 10⁴, = 0.1%, = 600 nm)

Rad.+ Con. (Ra = 10⁵, = 0.1%, = 600 nm) Rad.+ Con. (Ra = 10⁴, = 0.3%, = 600 nm) Rad.+ Con. (Ra = 10⁵, = 0.3%, = 600 nm)

Fig. 13 Total radiation and natural convection heat flux on hot wall at Ra= 10⁴andRa= 10⁵

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 20 40 60

Rad. (Ra = 10⁴, = 0.1%, = 300 nm) Rad. (Ra = 10⁵, = 0.1%, = 300 nm) Rad. (Ra = 10⁴, = 0.3%, = 300 nm) Rad. (Ra = 10⁵, = 0.3%, = 300 nm)

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Heat flux/wm–2 Heat flux/wm–2

0 20 40 60 80

Rad. (Ra = 10⁴, = 0.1%, = 600 nm) Rad. (Ra = 10⁵, = 0.1%, = 600 nm) Rad. (Ra = 10⁴, = 0.3%, = 600 nm) Rad. (Ra = 10⁵, = 0.3%, = 600 nm)

Fig. 14 Heat flux of radiation on hot wall atRa= 10⁴andRa= 10⁵

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Conclusions

Natural convection with radiation heat transfers was studied numerically, and the following results were obtained:

1. More accurate results were achieved in comparison with the previous researches used Roseland approximation, because of applying discrete ordinates method and considering the adsorption and scattering coefficients of nanoparticles.

2. The nanoparticles volume concentration did not have a significant effect on U profiles, but it increased with the wavelength and buoyancy forces. Concentration and wavelength had not significant impact on temperature profiles. Moreover, the heat flux of the radiation with convection mechanisms increased at more wavelengths.

3. Because of the decrease in absorption and scattering coefficients at higher wavelengths, the scattering radiation of the medium reduced and this process would increase the heat flux of the radiation on the walls. Increasing the Ra number made this phenomenon with more intensity.

4. With increasing nanoparticles volume concentration, the scattering coefficient increased; hence, the heat transfer of radiation declined. On the other hand, it enhanced the natural convective heat transfer. By coupling these two conclusions, it was observed that more volume concentration corresponded to less total heat transfer which implied the main outcome of the present work and made it different from the present works used nanofluid.

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