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MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater
International Journal of Heat and Mass Transfer 103 (2016) 955–964
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MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater M.A. Sheremet a,b, H.F. Oztop c,⇑, I. Pop d, K. Al-Salem e a
Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey d Department of Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania e Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 12 June 2016 Received in revised form 12 July 2016 Accepted 4 August 2016 Available online 16 August 2016 Keywords: Open wavy cavity Natural convection Magnetic field Porous medium Nanofluid Corner heater Numerical results
a b s t r a c t A numerical analysis of MHD natural convection in a wavy open porous tall cavity filled with a Cu–water nanofluid in the presence of an isothermal corner heater has been carried out. The cavity is cooled from the left wavy wall and heated from the right bottom corner while the bottom wall is adiabatic. Uniform magnetic field affects the heat transfer and fluid flow with an inclination angle to the axis x. Mathematical model formulated using the single-phase nanofluid approach in dimensionless variables stream function, vorticity and temperature has been solved by finite difference method of the second order accuracy in a wide range of governing parameters: Rayleigh number (Ra = 100–1000), Hartmann number (Ha = 0–100), inclination angle of the magnetic field (c = 0–p) and solid volume fraction parameter of nanoparticles (u = 0.0–0.05). Main efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms and average Nusselt numbers. It has been found heat transfer enhancement with Rayleigh number and heat transfer reduction with Hartmann number, while magnetic field inclination angle leads to non-monotonic changes of the heat transfer rate. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection in different technologies finds an important place for engineering analysis. It has wide applications in engineering such as solar applications, building applications and electronic industry. The problem can be find for industrial boilers or ovens with porous materials. Also, boundaries of open or closed geometries can be non-linear. Also, the working fluid can be pure or nanofluid filled under magnetic field. In this context, a wide review has been performed by Mahdi et al. [1]. They exhibited studies on convection heat transfer and fluid flow in porous media with nanofluid. Bilgen and Oztop [2] analyzed the natural convection in partially open inclined square cavities by using finite volume technique. They observed that inclination angle can be chosen as control parameter for flow rate and heat transfer enhancement. Arbin et al. [3] solved the double-diffusive convection problem in an open cavity and they applied heatline approach to see the heat ⇑ Corresponding author. E-mail address: [email protected] (H.F. Oztop). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.006 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
transport way. Sheremet et al. [4] studied double-diffusive mixed convection flow in a porous open cavity filled with a nanofluid using two-phase nanofluid model. They revealed that average Nusselt number is an increasing function of the Rayleigh and Reynolds numbers, and a decreasing function of the usual Lewis number, while the average Sherwood number is an increasing function of the Reynolds and usual Lewis. Mehrez et al. [5] presented a numerical study of entropy generation and mixed convection heat transfer of Cu–water nanofluid flow in an inclined open cavity by solving governing equations via finite volume method. Their results showed that the inclination angle affects the flow field, the temperature distribution, the heat transfer mode, the heat transfer and the entropy generation rates, and the magnitude of irreversibilities in the entropy generation. Numerical simulation of natural convection in partially C-shape open ended enclosure filled with nanofluid has been performed by Bakier [6]. He found that existence of nanoparticles increases the rate of heat and mass transfer through the opening boundaries for low Rayleigh number. Sheremet et al. [7] investigated unsteady natural convection in a differentially heated wavy-walled open cavity filled with a nanofluid. The obtained results showed that an increase in the undula-
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tions number leads to a decrease in the average Nusselt number at wavy wall due to the significant heating of the wave troughs. Effect of magnetic field on convection heat transfer is discussed by Heidary et al. [8]. They observed that the heat transfer in channels can enhance up to 75% due to the presence of nanoparticles and magnetic field in channels. Kefayati [9] solved the effect of a magnetic field on natural convection in an open enclosure which subjugated to water/alumina nanofluid using Lattice Boltzmann method (LBM). Obtained results showed that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and volume fractions and the magnetic field augments the effect of nanoparticles at Rayleigh number of Ra = 106 regularly. Mejri and Mahmoudi [10] examined the natural convection in an open cavity filled with a water–Al2O3 nanofluid and subjected to a magnetic field with a sinusoidal thermal boundary condition by using LBM. It was found that the heat transfer rate decreases with an increase in Hartmann number and increases with the rise of Rayleigh number. Bondareva et al. [11] analyzed unsteady natural convection of a water based nanofluid in a trapezoidal cavity under the influence of a uniform inclined magnetic field using the two-phase nanofluid model. It was ascertained that low Lewis and high Hartmann numbers reflect essential nonhomogeneous distribution of the nanoparticles inside the cavity. Therefore the considered range of Le and Ha characterizes a competency of utilizing of the non-homogeneous model. Mahmoudi et al. [12] worked on natural convection in nanofluid filled cavity under magnetic field including heat generation/absorption boundary conditions. Wavy walled cavities can be seen in different engineering applications such as heat exchanger, solar energy applications or buildings [13]. In this context, Cho et al. [14] presented an application on natural convection and entropy generation in a nanofluid filled cavities. Billah et al. [15] made a numerical simulation on buoyancy-driven heat and fluid flow in a nanofluid filled triangular enclosure as an application of curvilinear boundaries. They used Galerkin finite element method to solve governing equations and found that heat transfer was increased by 28% as volume fraction d increases from 0% to 20% at Gr = 105. Other applications on wavy-walled enclosure filled with nanofluid can be found for wavy-walled porous cavity with a nanofluid presented by Sheremet et al. [16] and combined convection flow in triangular wavy chamber by Nasrin et al. [17]. Partial heater can be seen especially in electronic cooling and building heating applications [18,19]. These applications are reviewed by Oztop et al. [20]. The corner heater is a specific application to see the effects of heater dimensions on natural convection. There are many works on this application such as natural convection in inclined enclosure with a corner heater by Varol et al. [21], natural convection coupled with radiation in an inclined porous cavity by Ahmed et al. [22]. Other related paper can be found in Refs. [23–25]. The main aim of this paper is to investigate the natural convection heat transfer and fluid flow in an open nanofluid filled porous wavy cavity in the presence of corner heater and magnetic field. In the study, Cu–water nanofluid is chosen because its cost is very low and copper is common material and it can be prepared easily.
2. Basic equations The natural convective heat transfer in a porous medium saturated with an electrically, conducting Cu–water nanofluid located in a partially open cavity with left wavy, right and bottom flat solid walls is analyzed. The considered domain of interest is presented in Fig. 1. The analyzed cavity includes isothermal left wavy wall of cold temperature and a heater of constant temperature in the right
Fig. 1. Physical model and coordinate system.
bottom corner. An inclined uniform magnetic field affects natural convection inside a porous cavity. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. It is worth noting that the left wavy wall and right flat wall of the open cavity =HÞ are defined by the relations such as: x1 ¼ L L½a þ b cosð2pjy is the left wavy wall; x2 ¼ L is the right flat wall; ¼ =HÞ is the distance between the D x2 x1 ¼ L½a þ b cosð2pjy right flat and left wavy walls. It is assumed that the nanofluid temperature is equal to the solid matrix temperature everywhere in the homogeneous and isotropic porous medium, and the local thermal equilibrium model is used. In the present study, the Darcy–Boussinesq model has been adopted in the governing equations of the problem. Taking into account these assumptions the governing equations can be written in dimensional Cartesian coordinates
¼0 rV 0 ¼ rp
ð1Þ
lnf V ðqbÞnf ðT T 0 Þg þ I B K
2 2 rTÞ ¼ amnf @ T þ @ T ðV 2 @ x2 @ y
ð2Þ
! ð3Þ
rI¼0
ð4Þ
I ¼ rnf ðr1 þ V BÞ
ð5Þ
is the dimensional velocity vector, p is the pressure, T is the where V fluid temperature, g is the gravity vector, B is the external magnetic field vector, I is the electric current vector, f is the electric potential, lnf is the dynamic viscosity of nanofluid, amnf is the effective nanofluid thermal diffusivity saturated in porous medium, b is the coefficient of thermal expansion, rnf is the electrical conductivity of nanofluid and r1 is the associated electric field. As discussed by Revnic et al. [26], Eqs. (4) and (5) reduce to r2 1 ¼ 0. Eqs. (1)–(5) for the problem under consideration can be written, after the pressure p is eliminated by cross-differentiation, in Carte as sian coordinates x and y
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@ v @u þ ¼0 @ x @ y
lnf @ u @ v K
u
@y
@ x
ð6Þ
@T @ v þ rnf B20 ¼ gðqbÞnf cos2 ðcÞ @ x @ x @ v @ u @u 2 sinðcÞ cosðcÞ sin ðcÞ þ @ x @y @y
@T @T @2T @2T þ v ¼ amnf þ 2 @ x @y @ x2 @ y
(
ð7Þ
ð8Þ
and v are the velocity components along x and y directions, where u respectively, B0 is the magnitude of B. The physical properties of the base fluid, nanoparticles and solid matrix of porous medium used in the present study have been described in detail previously (see Sheremet et al. [27]). It should be noted that for the electrical conductivity of nanofluid we used the following expression [28,29]:
rf
ð9Þ
rf
y xÞ and the ¼ @ w=@ ; v ¼ @ w=@ Introducing the stream function ðu following dimensionless variables
x ¼ x=L;
=L; y¼y
amnf ; w ¼ w=
h ¼ ðT T c Þ=ðT h T c Þ
pjy=AÞ 1 n ¼ xx ¼ x1þaþbcosð2 ; D aþbcosð2pjy=AÞ
ð10Þ
As a result the governing Eqs. (11) and (12) can be written as follows:
" 2 # 2 2 @n @n @ w @n @ 2 w @ 2 w @ 2 n @w þ þ2 þ þ 2 @x @y @y @n@ g @ g2 @y2 @n @n " ( 2 @n @ 2 w @n @n @ 2 w @n @ 2 w 2 ¼ Ha H1 ðuÞ cos ð c Þ þ 2 þ 2 @x @n @x @y @n2 @x @n@ g ) @ 2 n @w sinðcÞ cosðcÞ þ @x@y @n ( ) # 2 @n @ 2 w @n @ 2 w @ 2 w @ 2 n @w 2 sin þ þ 2 þ þ ð c Þ @y @n2 @y @n@ g @ g2 @y2 @n Ra H2 ðuÞ
" @2w @2w @2w @2w sinðcÞ cosðcÞ þ ¼ Ha H1 ðuÞ cos2 ðcÞ þ 2 @x2 @y2 @x2 @x@y # @2w @h 2 ð11Þ þ 2 sin ðcÞ Ra H2 ðuÞ @y @x
ð12Þ
Taking into account the considered dimensionless variables (10) the left wavy and the right flat walls of the cavity are described by the following relations: x1 ¼ 1 a b cosð2pjy=AÞ is the left wavy wall; x2 ¼ 1 is the right flat wall; D ¼ x2 x1 ¼ a þ b cosð2pjy=AÞ is the distance between vertical walls. The corresponding boundary conditions for these equations are
h ¼ 0 at x ¼ x1
w ¼ 0;
h ¼ 1:0 at x ¼ x2
w ¼ 0; w ¼ 0;
@h=@x ¼ 0:0 at x ¼ x2 over the heat source h ¼ 1:0 at y ¼ 0 along the heat source
w ¼ 0;
@h=@y ¼ 0 at y ¼ 0 before the heat source
along the heat source ð13Þ
@h=@y ¼ 0 at y ¼ A
Here Ra ¼ gKðqbÞf ðT h T c ÞL=ðam lf Þ is the Rayleigh number for the porous medium, am ¼ km =ðqC p Þf is the thermal diffusivity of the
clear fluid saturated porous medium, Ha ¼ rf KB20 =lf is the Hartmann number for the porous medium, A ¼ H=L is the aspect ratio of the semi-open cavity, and the functions H1 ðuÞ and H2 ðuÞ are given by
3 r 3 rp 1 u f 5ð1 uÞ2:5 H1 ðuÞ ¼ 41 þ rp rp þ 2 1 u r r 2
f
H2 ðuÞ ¼
ð14Þ
f
½1 u þ uðqbÞp =ðqbÞf ½1 u þ uðqC p Þp =ðqC p Þf 3euk ðk k Þ
1 km ½kp þ2kf þfuðkpkp Þ f
ð17Þ "
þ
2 2 # 2 @n @n @ h @n @ 2 h þ þ 2 @x @y @y @n@ g @n2
@ 2 h @ 2 n @h þ @ g2 @y2 @n
ð18Þ
with corresponding boundary conditions
w ¼ 0; w ¼ 0;
h ¼ 0 on n ¼ 0 h ¼ 1:0 on n ¼ 1 along the heat source
w ¼ 0;
@h=@n ¼ 0:0 on n ¼ 1 over the heat source
w ¼ 0;
h ¼ 1:0 on g ¼ 0 along the heat source
w ¼ 0;
@h=@ g ¼ 0 on g ¼ 0 before the heat source
@w=@ g ¼ 0;
ð19Þ
@h=@ g ¼ 0 on g ¼ A
For the definition of the local Nusselt numbers Nux and Nuy along the heat source surface we used the following expressions
Nux ¼
kmnf km
@n @h ; @x @n n¼1
Nuy ¼
kmnf @h km @ g g¼0
ð20Þ
Therefore the average Nusselt number Nu can be defined in the following way
w ¼ 0;
@w=@y ¼ 0;
@n @h @x @n
@n @w @h @n @w @h ¼ @x @ g @n @x @n @ g
the governing equations in dimensionless form can be written as follows:
@w @h @w @h @ 2 h @ 2 h ¼ þ @y @x @x @y @x2 @y2
ð16Þ
g¼y
!
r 3 rp 1 u rnf f ¼1þ rp r rf þ2 p1 u
Further on we introduce the following algebraic transformation with new independent variables n and g:
ð1 uÞ2:5
f
ð15Þ
Nu ¼
L hhs
Z 0
hhs =L
jNux j dg þ
L lhs
Z
1 1lhs =L
jNuy j dn
ð21Þ
The partial differential Eqs. (17) and (18) with corresponding boundary conditions (19) have been solved using an in–house computational fluid dynamics code (see Sheremet et al. [4,7,16,24]; Bondareva et al. [11]). We have validated the developed computational code against the works of Sarris et al. [30], Pirmohammadi and Ghassemi [31], and Al-Najem et al. [32] for steady-state MHD natural convection in a square cavity for Rayleigh numbers 7103 and 7105 and for Hartmann numbers in the range 0–100. Figs. 2 and 3 show a good agreement between the obtained streamlines and isotherms for different Rayleigh and Hartmann numbers and the numerical results of Sarris et al. [30], Pirmohammadi and Ghassemi [31]. The comparison between the results of the present model and those in [30,32] are shown in Table 1. The maximum horizontal velocity in the midsection (Uc,max) and the average Nusselt number are practically the same for ever value of Ha for the case of Ra = 7103.
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Fig. 2. Comparison of streamlines W and isotherms H at Ra = 7103, Ha = 25: numerical data of Sarris et al. [30] – a, numerical data of Pirmohammadi and Ghassemi [31] – b, present results – c.
Fig. 3. Comparison of streamlines W and isotherms H at Ra = 7105, Ha = 100: numerical data of Sarris et al. [30] – a, numerical data of Pirmohammadi and Ghassemi [31] – b, present results – c.
Table 2 compares the accuracy of the average Nusselt number for different values of the Rayleigh number with some numerical solutions reported by different authors [33–35] for the case of pure natural convection in a differentially heated porous cavity.
We have conducted also the grid independent test, analyzing the steady-state free convection in an open porous cavity filled with a Cu–water nanofluid at Ra = 100, Ha = 10, c = 0, u = 0.01, e = 0.5, j = 3, a = 0.9, A = 3 where the solid matrix of the porous
M.A. Sheremet et al. / International Journal of Heat and Mass Transfer 103 (2016) 955–964 Table 1 Comparison of present calculations with those of Sarris et al. [30] and Al-Najem et al. [32]. Authors
Sarris et al. [30] Al-Najem et al. [32] Present results
Ha = 10
Ha = 50
Uc,max
Nu
Uc,max
Nu
0.138 0.135 0.138
1.738 1.75 1.683
0.025 0.026 0.026
1.022 1.05 1.019
Table 2 Variations of the average Nusselt number for a square cavity filled with a regular porous medium in comparison with results from the open literature. Authors
Baytas and Pop [33] Manole and Lage [34] Bejan [35] Present results
Ra 10
100
1000
10,000
1.079 – – 1.079
3.16 3.118 4.20 3.115
14.06 13.637 15.80 13.667
48.33 48.117 50.80 48.823
medium is the aluminum foam. Three cases of the uniform grid are tested: a grid of 50 150 points, a grid of 100 300 points, and a much finer grid of 200 600 points. Figs. 4 and 5 show the effect of the mesh parameters on the stream function and temperature profiles along middle cross-section x = 0.5. Taking into account the conducted verifications the uniform grid of 100 300 points has been selected for the further investigation.
3. Results and discussion Numerical analysis has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 100, 500, 1000), Hartmann number (Ha = 0–100), inclination angle of the magnetic field (0 6 c < p), solid volume fraction parameter of nanoparticles (u = 0.0–0.05), e = 0.5, j = 3, a = 0.9, A = 3, lhs =L ¼ 0:5, hhs =L ¼ 0:5. Particular efforts have been focused on the effects of the Rayleigh and Hartmann numbers, inclination angle of the magnetic field and nanoparticles volume fraction on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms and average Nusselt number for different values of govern-
959
ing parameters mentioned above are illustrated in Figs. 6–11 and Table 3. Fig. 6 demonstrates streamlines and isotherms for Ha = 10, u = 0.02, c = 0 and different values of the Rayleigh number. It should be noted that Rayleigh number characterizes an effect of the buoyancy force and an increase in this parameter leads to an intensification and chaotization of fluid flow. In the case of low value of Ra (Ra = 100) one can find a formation of a weak convective cell inside the cavity close to the bottom wall where a heater is located. This circulation does not allow to refresh the medium inside the cavity taking into account the presence of the open top boundary. Moreover a weak circulation can be explained by the presence of the solid matrix and low temperature difference. The latter reflects a dominating of the heat conduction inside the cavity. An increase in the Rayleigh number (Fig. 6b) leads to an essential intensification of convective flow and formation of several recirculations due to the considered aspect ratio and heater sizes. In this case some fluid portions can leave the cavity. At the same time one can find more intensive heating of the cavity with ascending flow near the right flat wall and descending flow close to the left wavy wall. Further increase in the buoyancy force magnitude leads to an intensification of convective flow and heat transfer (Fig. 6c). It is worth noting that the convective cores shift insignificantly along the vertical axis with contraction of the bottom recirculation. The effect of the Hartmann number is presented in Fig. 7 for Ra = 500, u = 0.02, c = 0. It should be noted that this parameter characterizes an influence of the magnetic field on the flow and hear transfer of electrically conducting fluid. It is well known that the magnetic field aspires to suppress the convective flow and heat transfer [36]. In the case of absence of the magnetic field one can find a formation of intensive descending flow near the left wavy wall and ascending flow close to the right flat wall with small cores in the upper part of the cavity with thin thermal boundary layers near the heater. At the same time it is interesting to note an essential influence of the descending cold fluid on the thermal plume from the heater in the bottom part of the cavity. An appearance of the horizontal magnetic field of intensity Ha = 10 (Fig. 7b) leads to an essential attenuation of the convective flow taking into account the values of jwjmax with an increase in the thermal boundary layers thickness near the heater surface. Also it is possible to find less intensive cooling of the bottom part from the left isothermal wall. Further increase in the magnetic field intensity leads to an essential attenuation of the convective flow with a formation
Fig. 4. Profiles of stream function along the middle cross-section x = 0.5 for different mesh parameters.
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Fig. 5. Profiles of temperature along the middle cross-section x = 0.5 for different mesh parameters.
Fig. 6. Streamlines W and isotherms h for Ha = 10, u = 0.02, c = 0: Ra = 100 – a, Ra = 500 – b, Ra = 1000 – c.
of a single-core circulation close to the bottom wall with dominating of the heat conduction. Fig. 8 presents variation of the average Nusselt number along the heat source surface (see Eq. (21)) with Hartmann and Rayleigh numbers. As it has been mentioned above an increase in the buoyancy force magnitude leads to heat transfer enhancement while an increase in the magnetic field intensity characterizes the heat transfer reduction. It should be noted that for high values of the Hartmann number there is no differences between the average Nusselt numbers for Ra = 100 and Ra = 500. An essential effect of the magnetic field inclination angle on streamlines and isotherms is presented in Fig. 9 for Ra = 500, Ha = 10, u = 0.02. It should be noted that the direction of the magnetic field also characterizes an interaction of the magnetic and gravity fields that can lead to not so essential attenuation of the convective flow and heat transfer. An increase in the inclination angle c from 0 to p/2 leads to intensification of convective flow and combination of recirculations inside the cavity. While for c = p/2 one can find the fluid flow and heat transfer that are similar to the case without magnetic field (Fig. 7a) but not so intensive. This case is characterized by opposite gravity and magnetic forces. Further inclination of magnetic field reflects reduction of heat transfer and convective flow rates with an appearance of several recirculations inside the cavity. Fig. 10 demonstrates the effects of Hartmann number and magnetic field inclination angle on the average Nusselt number. As it has been mentioned above in the case of vertical magnetic field one can find an essential intensification of convective heat transfer that can be used in different technologies where magnetic fields are used for control of heat transfer and fluid flow [37]. An influence of the nanoparticles volume fraction on distributions of stream function and temperature is presented in Fig. 11 for horizontal magnetic field with Ha = 10 and Ra = 500. It is interesting to note that an increase in u leads to a weak intensification of convective flow in the central and bottom parts of the cavity with more intensive heating of the upper part taking into account the position of isotherm h = 0.4. At the same time one can find in the upper part an intensification of recirculation flow with nanoparticle volume fraction. Observed changes in streamlines and isotherms can be explained by an increase in the thermal conductivity of nanofluid that leads to a growth of the temperature gradient in the bottom part. Moreover, a rise of nanoparticles volume fraction leads to an increase in thermal conductivity and dynamic viscosity of nanofluid. It is well known that an increase
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Fig. 7. Streamlines W and isotherms h for Ra = 500, u = 0.02, c = 0: Ha = 0 – a, Ha = 10 – b, Ha = 50 – c, Ha = 100 – d.
in dynamic viscosity of the fluid has to lead to suppression of the convective flow while we have the opposite effect. This behavior is defined by more essential effect of the thermal field due to an increase in the thermal conductivity of the nanofluid. It should be noted also that in the case of magnetic field effect an increase in the nanoparticle volume fraction leads to a weak heat transfer enhancement up to 1.23% for Ha = 50 while for Ha = 0 one can find the heat transfer reduction up to 8.4% (see Table 3). Effect of the undulation number on fluid flow and heat transfer has been analyzed earlier in the case of pure natural convection without magnetic field and corner heater (see papers by Sheremet et al. [7,38] and book by Shenoy et al. [39]). 4. Conclusions
Fig. 8. Variations of the average Nusselt number along the heat source surface with Rayleigh and Hartmann numbers for u = 0.02, c = 0.
MHD natural convection of Cu–water nanofluid in a wavy porous open tall cavity with a corner heater of constant temperature was solved numerically on the basis of finite difference method of second order accuracy. Distributions of streamlines, isotherms and average Nusselt number along the heat source surface in dependence on the Rayleigh and Hartmann numbers, magnetic
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Fig. 9. Streamlines W and isotherms h for Ra = 500, Ha = 10, u = 0.02: c = 0 – a, c = p/4 – b, c = p/2 – c, c = 3p/4 – d.
field inclination angle and nanoparticle volume fraction have been obtained. The main findings can be listed as:
Fig. 10. Variations of the average Nusselt number along the heat source surface with Hartmann number and magnetic field inclination angle for Ra = 500, u = 0.02.
(1) The average Nusselt number is an increasing function of the Rayleigh number and a decreasing function of the Hartmann number. The effects of the magnetic field inclination angle and nanoparticles volume fraction on the heat transfer rate are non-monotonic. (2) An increase in the Hartmann number leads to a reduction of convective flow rate and a formation of a circulation close to the bottom wall. There is no differences between the average Nusselt numbers for Ra = 100 and Ra = 500 for high values of the Hartmann number. (3) An increase in the inclination angle c from 0 to p/2 leads to an intensification of convective flow and combination of recirculations inside the cavity due to an increase in the part of the Lorentz force that counteracts to the buoyancy force. (4) An increase in u leads to a weak intensification of convective flow in the central and bottom parts of the cavity with more intensive heating of the upper part taking into account the position of isotherm h = 0.4. In the case of magnetic field
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References
Fig. 11. Streamlines W and isotherms h for Ra = 500, Ha = 10, c = 0: u = 0.0 – a, u = 0.02 – b, u = 0.05 – c.
Table 3 Variations of the average Nusselt number along the heat source surface with Hartmann number and nanoparticles volume fraction for Ra = 500, c = 0.
u = 0.0 u = 0.02 u = 0.05
Ha = 0
Ha = 10
Ha = 50
Ha = 100
24.648 23.81 22.574
6.789 6.817 6.852
2.933 2.947 2.969
2.627 2.632 2.641
effect an increase in the nanoparticle volume fraction leads to a weak heat transfer enhancement while for Ha = 0 one can find the heat transfer reduction.
Acknowledgement This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K.
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Pop, Magnetic field effect on the unsteady natural convection in a right-angle trapezoidal cavity filled with a nanofluid, Int. J. Numer. Methods Heat Fluid Flow 25 (2015) 1924–1946. [12] A. Mahmoudi, I. Mejri, M.A. Abbassi, A. Omri, Analysis of MHD natural convection in a nanofluid-filled open cavity with non-uniform boundary condition in the presence of uniform heat generation/absorption, Powder Technol. 269 (2015) 275–289. [13] H. Heidary, M.J. Kermani, Effect of nanoparticles on forced convection in sinusoidal-wall channel, Int. Commun. Heat Mass Transfer 37 (2010) 1520– 1527. [14] C.C. Cho, C.L. Chen, C.K. Chen, Natural convection heat transfer and entropy generation in wavy-wall enclosure containing water-based nanofluid, Int. J. Heat Mass Transfer 61 (2013) 749–758. [15] M.M. Billah, M.M. Rahman, Uddin M. Sharif, M.N. Islam, Numerical simulation on buoyancy-driven heat transfer enhancement of nanofluids in an inclined triangular enclosure, Procedia Eng. 90 (2014) 517–523. [16] M.A. Sheremet, I. Pop, N. Bachok, Effect of thermal dispersion on transient natural convection in a wavy-walled porous cavity filled with a nanofluid: Tiwari and Das’ nanofluid model, Int. J. Heat Mass Transfer 92 (2016) 1053– 1060. [17] R. Nasrin, M.A. Alim, A.J. Chamkha, Combined convection flow in triangular wavy chamber filled with water–CuO nanofluid: effect of viscosity models, Int. Commun. Heat Mass Transfer 39 (2012) 1226–1236. [18] S.G. Martyushev, M.A. Sheremet, Conjugate natural convection combined with surface thermal radiation in an air filled cavity with internal heat source, Int. J. Therm. Sci. 76 (2014) 51–67. [19] S.G. Martyushev, M.A. Sheremet, Conjugate natural convection combined with surface thermal radiation in a three-dimensional enclosure with a heat source, Int. J. Heat Mass Transfer 73 (2014) 340–353. [20] H.F. Oztop, P. Estelle, W.M. Yan, K. Al-Salem, J. Orfi, O. Mahian, A brief review of natural convection in enclosures under localized heating with and without nanofluids, Int. Commun. Heat Mass Transfer 60 (2015) 37–44. [21] Y. Varol, H.F. Oztop, A. Koca, F. Ozgen, Natural convection and fluid flow in inclined enclosure with a corner heater, Appl. Therm. Eng. 29 (2009) 340–350. [22] S.E. Ahmed, H.F. Oztop, K. Al-Salem, Natural convection coupled with radiation heat transfer in an inclined porous cavity with corner heater, Comput. Fluids 102 (2014) 74–84. [23] F. Selimefendigil, H.F. Öztop, Conjugate natural convection in a cavity with a conductive partition and filled with different nanofluids on different sides of the partition, J. Mol. Liq. 216 (2016) 67–77. [24] F. Selimefendigil, H.F. Öztop, Mixed convection in a two-sided elastic walled and SiO2 nanofluid filled cavity with internal heat generation: effects of inner rotating cylinder and nanoparticle’s shape, J. Mol. Liq. 212 (2015) 509–516. [25] F. Garoosi, F. Hoseininejad, Numerical study of natural and mixed convection heat transfer between differentially heated cylinders in an adiabatic enclosure filled with nanofluid, J. Mol. Liq. 215 (2016) 1–17. [26] C. Revnic, T. Grosan, I. Pop, D.B. Ingham, Magnetic field effect on the unsteady free convection flow in a square cavity filled with a porous medium with a constant heat generation, Int. J. Heat Mass Transfer 54 (2011) 1734–1742. [27] M.A. Sheremet, T. Grosan, I. Pop, Free convection in a square cavity filled with a porous medium saturated by nanofluid using Tiwari and Das’ nanofluid model, Transp. Porous Media 106 (2015) 595–610. [28] GH.R. Kefayati, Simulation of heat transfer and entropy generation of MHD natural convection of non-Newtonian nanofluid in an enclosure, Int. J. Heat Mass Transfer 92 (2016) 1066–1089.
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[29] I. Fersadou, H. Kahalerras, M. El Ganaoui, MHD mixed convection and entropy generation of a nanofluid in a vertical porous channel, Comput. Fluids 121 (2015) 164–179. [30] I.E. Sarris, G.K. Zikos, A.P. Grecos, N.S. Vlachos, On the limits of validity of the low magnetic Reynolds number approximation in MHD natural-convection heat transfer, Numer. Heat Transfer Part B 50 (2006) 157–180. [31] M. Pirmohammadi, M. Ghassemi, Effect of magnetic field on convection heat transfer inside a tilted square enclosure, Int. Commun. Heat Mass Transfer 36 (2009) 776–780. [32] N.M. Al-Najem, K.M. Khanafer, M.M. El-Rafaee, Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field, Int. J. Numer. Methods Heat Fluid Flow 8 (1998) 651–672. [33] A.C. Baytas, I. Pop, Natural convection in a trapezoidal enclosure filled with a porous medium, Int. J. Eng. Sci. 39 (2001) 125–134.
[34] D.M. Manole, J.L. Lage, Numerical benchmark results for natural convection in a porous medium cavity, in: ASME Heat Mass Transfer Porous Med. Conf., 1992, pp. 55–60. [35] A. Bejan, On the boundary layer regime in a vertical enclosure filled with a porous medium, Lett. Heat Mass Transfer 6 (1979) 93–102. [36] N.S. Bondareva, M.A. Sheremet, Influence of uniform magnetic field on laminar regimes of natural convection in an enclosure, Thermophy. Aeromech. 22 (2015) 203–216. [37] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001. [38] M.A. Sheremet, I. Pop, A. Shenoy, Natural convection in a wavy open porous cavity filled with a nanofluid: Tiwari and Das’ nanofluid model, Eur. Phys. J. Plus 131 (2016) (12 pages) 62. [39] A. Shenoy, M. Sheremet, I. Pop, Convective Flow and Heat Transfer from Wavy Surfaces: Viscous Fluids, Porous Media and Nanofluids, CRC Press, Taylor & Francis Group, Boca Raton, New York, 2016.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater M.A. Sheremet a,b, H.F. Oztop c,⇑, I. Pop d, K. Al-Salem e a
Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey d Department of Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania e Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 12 June 2016 Received in revised form 12 July 2016 Accepted 4 August 2016 Available online 16 August 2016 Keywords: Open wavy cavity Natural convection Magnetic field Porous medium Nanofluid Corner heater Numerical results
a b s t r a c t A numerical analysis of MHD natural convection in a wavy open porous tall cavity filled with a Cu–water nanofluid in the presence of an isothermal corner heater has been carried out. The cavity is cooled from the left wavy wall and heated from the right bottom corner while the bottom wall is adiabatic. Uniform magnetic field affects the heat transfer and fluid flow with an inclination angle to the axis x. Mathematical model formulated using the single-phase nanofluid approach in dimensionless variables stream function, vorticity and temperature has been solved by finite difference method of the second order accuracy in a wide range of governing parameters: Rayleigh number (Ra = 100–1000), Hartmann number (Ha = 0–100), inclination angle of the magnetic field (c = 0–p) and solid volume fraction parameter of nanoparticles (u = 0.0–0.05). Main efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms and average Nusselt numbers. It has been found heat transfer enhancement with Rayleigh number and heat transfer reduction with Hartmann number, while magnetic field inclination angle leads to non-monotonic changes of the heat transfer rate. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection in different technologies finds an important place for engineering analysis. It has wide applications in engineering such as solar applications, building applications and electronic industry. The problem can be find for industrial boilers or ovens with porous materials. Also, boundaries of open or closed geometries can be non-linear. Also, the working fluid can be pure or nanofluid filled under magnetic field. In this context, a wide review has been performed by Mahdi et al. [1]. They exhibited studies on convection heat transfer and fluid flow in porous media with nanofluid. Bilgen and Oztop [2] analyzed the natural convection in partially open inclined square cavities by using finite volume technique. They observed that inclination angle can be chosen as control parameter for flow rate and heat transfer enhancement. Arbin et al. [3] solved the double-diffusive convection problem in an open cavity and they applied heatline approach to see the heat ⇑ Corresponding author. E-mail address: [email protected] (H.F. Oztop). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.006 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
transport way. Sheremet et al. [4] studied double-diffusive mixed convection flow in a porous open cavity filled with a nanofluid using two-phase nanofluid model. They revealed that average Nusselt number is an increasing function of the Rayleigh and Reynolds numbers, and a decreasing function of the usual Lewis number, while the average Sherwood number is an increasing function of the Reynolds and usual Lewis. Mehrez et al. [5] presented a numerical study of entropy generation and mixed convection heat transfer of Cu–water nanofluid flow in an inclined open cavity by solving governing equations via finite volume method. Their results showed that the inclination angle affects the flow field, the temperature distribution, the heat transfer mode, the heat transfer and the entropy generation rates, and the magnitude of irreversibilities in the entropy generation. Numerical simulation of natural convection in partially C-shape open ended enclosure filled with nanofluid has been performed by Bakier [6]. He found that existence of nanoparticles increases the rate of heat and mass transfer through the opening boundaries for low Rayleigh number. Sheremet et al. [7] investigated unsteady natural convection in a differentially heated wavy-walled open cavity filled with a nanofluid. The obtained results showed that an increase in the undula-
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tions number leads to a decrease in the average Nusselt number at wavy wall due to the significant heating of the wave troughs. Effect of magnetic field on convection heat transfer is discussed by Heidary et al. [8]. They observed that the heat transfer in channels can enhance up to 75% due to the presence of nanoparticles and magnetic field in channels. Kefayati [9] solved the effect of a magnetic field on natural convection in an open enclosure which subjugated to water/alumina nanofluid using Lattice Boltzmann method (LBM). Obtained results showed that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and volume fractions and the magnetic field augments the effect of nanoparticles at Rayleigh number of Ra = 106 regularly. Mejri and Mahmoudi [10] examined the natural convection in an open cavity filled with a water–Al2O3 nanofluid and subjected to a magnetic field with a sinusoidal thermal boundary condition by using LBM. It was found that the heat transfer rate decreases with an increase in Hartmann number and increases with the rise of Rayleigh number. Bondareva et al. [11] analyzed unsteady natural convection of a water based nanofluid in a trapezoidal cavity under the influence of a uniform inclined magnetic field using the two-phase nanofluid model. It was ascertained that low Lewis and high Hartmann numbers reflect essential nonhomogeneous distribution of the nanoparticles inside the cavity. Therefore the considered range of Le and Ha characterizes a competency of utilizing of the non-homogeneous model. Mahmoudi et al. [12] worked on natural convection in nanofluid filled cavity under magnetic field including heat generation/absorption boundary conditions. Wavy walled cavities can be seen in different engineering applications such as heat exchanger, solar energy applications or buildings [13]. In this context, Cho et al. [14] presented an application on natural convection and entropy generation in a nanofluid filled cavities. Billah et al. [15] made a numerical simulation on buoyancy-driven heat and fluid flow in a nanofluid filled triangular enclosure as an application of curvilinear boundaries. They used Galerkin finite element method to solve governing equations and found that heat transfer was increased by 28% as volume fraction d increases from 0% to 20% at Gr = 105. Other applications on wavy-walled enclosure filled with nanofluid can be found for wavy-walled porous cavity with a nanofluid presented by Sheremet et al. [16] and combined convection flow in triangular wavy chamber by Nasrin et al. [17]. Partial heater can be seen especially in electronic cooling and building heating applications [18,19]. These applications are reviewed by Oztop et al. [20]. The corner heater is a specific application to see the effects of heater dimensions on natural convection. There are many works on this application such as natural convection in inclined enclosure with a corner heater by Varol et al. [21], natural convection coupled with radiation in an inclined porous cavity by Ahmed et al. [22]. Other related paper can be found in Refs. [23–25]. The main aim of this paper is to investigate the natural convection heat transfer and fluid flow in an open nanofluid filled porous wavy cavity in the presence of corner heater and magnetic field. In the study, Cu–water nanofluid is chosen because its cost is very low and copper is common material and it can be prepared easily.
2. Basic equations The natural convective heat transfer in a porous medium saturated with an electrically, conducting Cu–water nanofluid located in a partially open cavity with left wavy, right and bottom flat solid walls is analyzed. The considered domain of interest is presented in Fig. 1. The analyzed cavity includes isothermal left wavy wall of cold temperature and a heater of constant temperature in the right
Fig. 1. Physical model and coordinate system.
bottom corner. An inclined uniform magnetic field affects natural convection inside a porous cavity. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. It is worth noting that the left wavy wall and right flat wall of the open cavity =HÞ are defined by the relations such as: x1 ¼ L L½a þ b cosð2pjy is the left wavy wall; x2 ¼ L is the right flat wall; ¼ =HÞ is the distance between the D x2 x1 ¼ L½a þ b cosð2pjy right flat and left wavy walls. It is assumed that the nanofluid temperature is equal to the solid matrix temperature everywhere in the homogeneous and isotropic porous medium, and the local thermal equilibrium model is used. In the present study, the Darcy–Boussinesq model has been adopted in the governing equations of the problem. Taking into account these assumptions the governing equations can be written in dimensional Cartesian coordinates
¼0 rV 0 ¼ rp
ð1Þ
lnf V ðqbÞnf ðT T 0 Þg þ I B K
2 2 rTÞ ¼ amnf @ T þ @ T ðV 2 @ x2 @ y
ð2Þ
! ð3Þ
rI¼0
ð4Þ
I ¼ rnf ðr1 þ V BÞ
ð5Þ
is the dimensional velocity vector, p is the pressure, T is the where V fluid temperature, g is the gravity vector, B is the external magnetic field vector, I is the electric current vector, f is the electric potential, lnf is the dynamic viscosity of nanofluid, amnf is the effective nanofluid thermal diffusivity saturated in porous medium, b is the coefficient of thermal expansion, rnf is the electrical conductivity of nanofluid and r1 is the associated electric field. As discussed by Revnic et al. [26], Eqs. (4) and (5) reduce to r2 1 ¼ 0. Eqs. (1)–(5) for the problem under consideration can be written, after the pressure p is eliminated by cross-differentiation, in Carte as sian coordinates x and y
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@ v @u þ ¼0 @ x @ y
lnf @ u @ v K
u
@y
@ x
ð6Þ
@T @ v þ rnf B20 ¼ gðqbÞnf cos2 ðcÞ @ x @ x @ v @ u @u 2 sinðcÞ cosðcÞ sin ðcÞ þ @ x @y @y
@T @T @2T @2T þ v ¼ amnf þ 2 @ x @y @ x2 @ y
(
ð7Þ
ð8Þ
and v are the velocity components along x and y directions, where u respectively, B0 is the magnitude of B. The physical properties of the base fluid, nanoparticles and solid matrix of porous medium used in the present study have been described in detail previously (see Sheremet et al. [27]). It should be noted that for the electrical conductivity of nanofluid we used the following expression [28,29]:
rf
ð9Þ
rf
y xÞ and the ¼ @ w=@ ; v ¼ @ w=@ Introducing the stream function ðu following dimensionless variables
x ¼ x=L;
=L; y¼y
amnf ; w ¼ w=
h ¼ ðT T c Þ=ðT h T c Þ
pjy=AÞ 1 n ¼ xx ¼ x1þaþbcosð2 ; D aþbcosð2pjy=AÞ
ð10Þ
As a result the governing Eqs. (11) and (12) can be written as follows:
" 2 # 2 2 @n @n @ w @n @ 2 w @ 2 w @ 2 n @w þ þ2 þ þ 2 @x @y @y @n@ g @ g2 @y2 @n @n " ( 2 @n @ 2 w @n @n @ 2 w @n @ 2 w 2 ¼ Ha H1 ðuÞ cos ð c Þ þ 2 þ 2 @x @n @x @y @n2 @x @n@ g ) @ 2 n @w sinðcÞ cosðcÞ þ @x@y @n ( ) # 2 @n @ 2 w @n @ 2 w @ 2 w @ 2 n @w 2 sin þ þ 2 þ þ ð c Þ @y @n2 @y @n@ g @ g2 @y2 @n Ra H2 ðuÞ
" @2w @2w @2w @2w sinðcÞ cosðcÞ þ ¼ Ha H1 ðuÞ cos2 ðcÞ þ 2 @x2 @y2 @x2 @x@y # @2w @h 2 ð11Þ þ 2 sin ðcÞ Ra H2 ðuÞ @y @x
ð12Þ
Taking into account the considered dimensionless variables (10) the left wavy and the right flat walls of the cavity are described by the following relations: x1 ¼ 1 a b cosð2pjy=AÞ is the left wavy wall; x2 ¼ 1 is the right flat wall; D ¼ x2 x1 ¼ a þ b cosð2pjy=AÞ is the distance between vertical walls. The corresponding boundary conditions for these equations are
h ¼ 0 at x ¼ x1
w ¼ 0;
h ¼ 1:0 at x ¼ x2
w ¼ 0; w ¼ 0;
@h=@x ¼ 0:0 at x ¼ x2 over the heat source h ¼ 1:0 at y ¼ 0 along the heat source
w ¼ 0;
@h=@y ¼ 0 at y ¼ 0 before the heat source
along the heat source ð13Þ
@h=@y ¼ 0 at y ¼ A
Here Ra ¼ gKðqbÞf ðT h T c ÞL=ðam lf Þ is the Rayleigh number for the porous medium, am ¼ km =ðqC p Þf is the thermal diffusivity of the
clear fluid saturated porous medium, Ha ¼ rf KB20 =lf is the Hartmann number for the porous medium, A ¼ H=L is the aspect ratio of the semi-open cavity, and the functions H1 ðuÞ and H2 ðuÞ are given by
3 r 3 rp 1 u f 5ð1 uÞ2:5 H1 ðuÞ ¼ 41 þ rp rp þ 2 1 u r r 2
f
H2 ðuÞ ¼
ð14Þ
f
½1 u þ uðqbÞp =ðqbÞf ½1 u þ uðqC p Þp =ðqC p Þf 3euk ðk k Þ
1 km ½kp þ2kf þfuðkpkp Þ f
ð17Þ "
þ
2 2 # 2 @n @n @ h @n @ 2 h þ þ 2 @x @y @y @n@ g @n2
@ 2 h @ 2 n @h þ @ g2 @y2 @n
ð18Þ
with corresponding boundary conditions
w ¼ 0; w ¼ 0;
h ¼ 0 on n ¼ 0 h ¼ 1:0 on n ¼ 1 along the heat source
w ¼ 0;
@h=@n ¼ 0:0 on n ¼ 1 over the heat source
w ¼ 0;
h ¼ 1:0 on g ¼ 0 along the heat source
w ¼ 0;
@h=@ g ¼ 0 on g ¼ 0 before the heat source
@w=@ g ¼ 0;
ð19Þ
@h=@ g ¼ 0 on g ¼ A
For the definition of the local Nusselt numbers Nux and Nuy along the heat source surface we used the following expressions
Nux ¼
kmnf km
@n @h ; @x @n n¼1
Nuy ¼
kmnf @h km @ g g¼0
ð20Þ
Therefore the average Nusselt number Nu can be defined in the following way
w ¼ 0;
@w=@y ¼ 0;
@n @h @x @n
@n @w @h @n @w @h ¼ @x @ g @n @x @n @ g
the governing equations in dimensionless form can be written as follows:
@w @h @w @h @ 2 h @ 2 h ¼ þ @y @x @x @y @x2 @y2
ð16Þ
g¼y
!
r 3 rp 1 u rnf f ¼1þ rp r rf þ2 p1 u
Further on we introduce the following algebraic transformation with new independent variables n and g:
ð1 uÞ2:5
f
ð15Þ
Nu ¼
L hhs
Z 0
hhs =L
jNux j dg þ
L lhs
Z
1 1lhs =L
jNuy j dn
ð21Þ
The partial differential Eqs. (17) and (18) with corresponding boundary conditions (19) have been solved using an in–house computational fluid dynamics code (see Sheremet et al. [4,7,16,24]; Bondareva et al. [11]). We have validated the developed computational code against the works of Sarris et al. [30], Pirmohammadi and Ghassemi [31], and Al-Najem et al. [32] for steady-state MHD natural convection in a square cavity for Rayleigh numbers 7103 and 7105 and for Hartmann numbers in the range 0–100. Figs. 2 and 3 show a good agreement between the obtained streamlines and isotherms for different Rayleigh and Hartmann numbers and the numerical results of Sarris et al. [30], Pirmohammadi and Ghassemi [31]. The comparison between the results of the present model and those in [30,32] are shown in Table 1. The maximum horizontal velocity in the midsection (Uc,max) and the average Nusselt number are practically the same for ever value of Ha for the case of Ra = 7103.
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Fig. 2. Comparison of streamlines W and isotherms H at Ra = 7103, Ha = 25: numerical data of Sarris et al. [30] – a, numerical data of Pirmohammadi and Ghassemi [31] – b, present results – c.
Fig. 3. Comparison of streamlines W and isotherms H at Ra = 7105, Ha = 100: numerical data of Sarris et al. [30] – a, numerical data of Pirmohammadi and Ghassemi [31] – b, present results – c.
Table 2 compares the accuracy of the average Nusselt number for different values of the Rayleigh number with some numerical solutions reported by different authors [33–35] for the case of pure natural convection in a differentially heated porous cavity.
We have conducted also the grid independent test, analyzing the steady-state free convection in an open porous cavity filled with a Cu–water nanofluid at Ra = 100, Ha = 10, c = 0, u = 0.01, e = 0.5, j = 3, a = 0.9, A = 3 where the solid matrix of the porous
M.A. Sheremet et al. / International Journal of Heat and Mass Transfer 103 (2016) 955–964 Table 1 Comparison of present calculations with those of Sarris et al. [30] and Al-Najem et al. [32]. Authors
Sarris et al. [30] Al-Najem et al. [32] Present results
Ha = 10
Ha = 50
Uc,max
Nu
Uc,max
Nu
0.138 0.135 0.138
1.738 1.75 1.683
0.025 0.026 0.026
1.022 1.05 1.019
Table 2 Variations of the average Nusselt number for a square cavity filled with a regular porous medium in comparison with results from the open literature. Authors
Baytas and Pop [33] Manole and Lage [34] Bejan [35] Present results
Ra 10
100
1000
10,000
1.079 – – 1.079
3.16 3.118 4.20 3.115
14.06 13.637 15.80 13.667
48.33 48.117 50.80 48.823
medium is the aluminum foam. Three cases of the uniform grid are tested: a grid of 50 150 points, a grid of 100 300 points, and a much finer grid of 200 600 points. Figs. 4 and 5 show the effect of the mesh parameters on the stream function and temperature profiles along middle cross-section x = 0.5. Taking into account the conducted verifications the uniform grid of 100 300 points has been selected for the further investigation.
3. Results and discussion Numerical analysis has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 100, 500, 1000), Hartmann number (Ha = 0–100), inclination angle of the magnetic field (0 6 c < p), solid volume fraction parameter of nanoparticles (u = 0.0–0.05), e = 0.5, j = 3, a = 0.9, A = 3, lhs =L ¼ 0:5, hhs =L ¼ 0:5. Particular efforts have been focused on the effects of the Rayleigh and Hartmann numbers, inclination angle of the magnetic field and nanoparticles volume fraction on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms and average Nusselt number for different values of govern-
959
ing parameters mentioned above are illustrated in Figs. 6–11 and Table 3. Fig. 6 demonstrates streamlines and isotherms for Ha = 10, u = 0.02, c = 0 and different values of the Rayleigh number. It should be noted that Rayleigh number characterizes an effect of the buoyancy force and an increase in this parameter leads to an intensification and chaotization of fluid flow. In the case of low value of Ra (Ra = 100) one can find a formation of a weak convective cell inside the cavity close to the bottom wall where a heater is located. This circulation does not allow to refresh the medium inside the cavity taking into account the presence of the open top boundary. Moreover a weak circulation can be explained by the presence of the solid matrix and low temperature difference. The latter reflects a dominating of the heat conduction inside the cavity. An increase in the Rayleigh number (Fig. 6b) leads to an essential intensification of convective flow and formation of several recirculations due to the considered aspect ratio and heater sizes. In this case some fluid portions can leave the cavity. At the same time one can find more intensive heating of the cavity with ascending flow near the right flat wall and descending flow close to the left wavy wall. Further increase in the buoyancy force magnitude leads to an intensification of convective flow and heat transfer (Fig. 6c). It is worth noting that the convective cores shift insignificantly along the vertical axis with contraction of the bottom recirculation. The effect of the Hartmann number is presented in Fig. 7 for Ra = 500, u = 0.02, c = 0. It should be noted that this parameter characterizes an influence of the magnetic field on the flow and hear transfer of electrically conducting fluid. It is well known that the magnetic field aspires to suppress the convective flow and heat transfer [36]. In the case of absence of the magnetic field one can find a formation of intensive descending flow near the left wavy wall and ascending flow close to the right flat wall with small cores in the upper part of the cavity with thin thermal boundary layers near the heater. At the same time it is interesting to note an essential influence of the descending cold fluid on the thermal plume from the heater in the bottom part of the cavity. An appearance of the horizontal magnetic field of intensity Ha = 10 (Fig. 7b) leads to an essential attenuation of the convective flow taking into account the values of jwjmax with an increase in the thermal boundary layers thickness near the heater surface. Also it is possible to find less intensive cooling of the bottom part from the left isothermal wall. Further increase in the magnetic field intensity leads to an essential attenuation of the convective flow with a formation
Fig. 4. Profiles of stream function along the middle cross-section x = 0.5 for different mesh parameters.
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Fig. 5. Profiles of temperature along the middle cross-section x = 0.5 for different mesh parameters.
Fig. 6. Streamlines W and isotherms h for Ha = 10, u = 0.02, c = 0: Ra = 100 – a, Ra = 500 – b, Ra = 1000 – c.
of a single-core circulation close to the bottom wall with dominating of the heat conduction. Fig. 8 presents variation of the average Nusselt number along the heat source surface (see Eq. (21)) with Hartmann and Rayleigh numbers. As it has been mentioned above an increase in the buoyancy force magnitude leads to heat transfer enhancement while an increase in the magnetic field intensity characterizes the heat transfer reduction. It should be noted that for high values of the Hartmann number there is no differences between the average Nusselt numbers for Ra = 100 and Ra = 500. An essential effect of the magnetic field inclination angle on streamlines and isotherms is presented in Fig. 9 for Ra = 500, Ha = 10, u = 0.02. It should be noted that the direction of the magnetic field also characterizes an interaction of the magnetic and gravity fields that can lead to not so essential attenuation of the convective flow and heat transfer. An increase in the inclination angle c from 0 to p/2 leads to intensification of convective flow and combination of recirculations inside the cavity. While for c = p/2 one can find the fluid flow and heat transfer that are similar to the case without magnetic field (Fig. 7a) but not so intensive. This case is characterized by opposite gravity and magnetic forces. Further inclination of magnetic field reflects reduction of heat transfer and convective flow rates with an appearance of several recirculations inside the cavity. Fig. 10 demonstrates the effects of Hartmann number and magnetic field inclination angle on the average Nusselt number. As it has been mentioned above in the case of vertical magnetic field one can find an essential intensification of convective heat transfer that can be used in different technologies where magnetic fields are used for control of heat transfer and fluid flow [37]. An influence of the nanoparticles volume fraction on distributions of stream function and temperature is presented in Fig. 11 for horizontal magnetic field with Ha = 10 and Ra = 500. It is interesting to note that an increase in u leads to a weak intensification of convective flow in the central and bottom parts of the cavity with more intensive heating of the upper part taking into account the position of isotherm h = 0.4. At the same time one can find in the upper part an intensification of recirculation flow with nanoparticle volume fraction. Observed changes in streamlines and isotherms can be explained by an increase in the thermal conductivity of nanofluid that leads to a growth of the temperature gradient in the bottom part. Moreover, a rise of nanoparticles volume fraction leads to an increase in thermal conductivity and dynamic viscosity of nanofluid. It is well known that an increase
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Fig. 7. Streamlines W and isotherms h for Ra = 500, u = 0.02, c = 0: Ha = 0 – a, Ha = 10 – b, Ha = 50 – c, Ha = 100 – d.
in dynamic viscosity of the fluid has to lead to suppression of the convective flow while we have the opposite effect. This behavior is defined by more essential effect of the thermal field due to an increase in the thermal conductivity of the nanofluid. It should be noted also that in the case of magnetic field effect an increase in the nanoparticle volume fraction leads to a weak heat transfer enhancement up to 1.23% for Ha = 50 while for Ha = 0 one can find the heat transfer reduction up to 8.4% (see Table 3). Effect of the undulation number on fluid flow and heat transfer has been analyzed earlier in the case of pure natural convection without magnetic field and corner heater (see papers by Sheremet et al. [7,38] and book by Shenoy et al. [39]). 4. Conclusions
Fig. 8. Variations of the average Nusselt number along the heat source surface with Rayleigh and Hartmann numbers for u = 0.02, c = 0.
MHD natural convection of Cu–water nanofluid in a wavy porous open tall cavity with a corner heater of constant temperature was solved numerically on the basis of finite difference method of second order accuracy. Distributions of streamlines, isotherms and average Nusselt number along the heat source surface in dependence on the Rayleigh and Hartmann numbers, magnetic
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Fig. 9. Streamlines W and isotherms h for Ra = 500, Ha = 10, u = 0.02: c = 0 – a, c = p/4 – b, c = p/2 – c, c = 3p/4 – d.
field inclination angle and nanoparticle volume fraction have been obtained. The main findings can be listed as:
Fig. 10. Variations of the average Nusselt number along the heat source surface with Hartmann number and magnetic field inclination angle for Ra = 500, u = 0.02.
(1) The average Nusselt number is an increasing function of the Rayleigh number and a decreasing function of the Hartmann number. The effects of the magnetic field inclination angle and nanoparticles volume fraction on the heat transfer rate are non-monotonic. (2) An increase in the Hartmann number leads to a reduction of convective flow rate and a formation of a circulation close to the bottom wall. There is no differences between the average Nusselt numbers for Ra = 100 and Ra = 500 for high values of the Hartmann number. (3) An increase in the inclination angle c from 0 to p/2 leads to an intensification of convective flow and combination of recirculations inside the cavity due to an increase in the part of the Lorentz force that counteracts to the buoyancy force. (4) An increase in u leads to a weak intensification of convective flow in the central and bottom parts of the cavity with more intensive heating of the upper part taking into account the position of isotherm h = 0.4. In the case of magnetic field
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References
Fig. 11. Streamlines W and isotherms h for Ra = 500, Ha = 10, c = 0: u = 0.0 – a, u = 0.02 – b, u = 0.05 – c.
Table 3 Variations of the average Nusselt number along the heat source surface with Hartmann number and nanoparticles volume fraction for Ra = 500, c = 0.
u = 0.0 u = 0.02 u = 0.05
Ha = 0
Ha = 10
Ha = 50
Ha = 100
24.648 23.81 22.574
6.789 6.817 6.852
2.933 2.947 2.969
2.627 2.632 2.641
effect an increase in the nanoparticle volume fraction leads to a weak heat transfer enhancement while for Ha = 0 one can find the heat transfer reduction.
Acknowledgement This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K.
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