In this paper, the optimal condition of mixed magnetohydrodynamic convection in a ventilated square cavity filled with a Cu-water nanofluid with diffe...

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THE EUROPEAN PHYSICAL JOURNAL PLUS

Regular Article

Mixed magnetohydrodynamic convection in a Cu-waternanoﬂuid–ﬁlled ventilated square cavity using the Taguchi method: A numerical investigation and optimization 2,a ¨ Kamel Milani Shirvan1 , Hakan F. Oztop , and Khaled Al-Salem3 1 2 3

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia Received: 13 January 2017 / Revised: 27 February 2017 c Societ` Published online: 5 May 2017 – a Italiana di Fisica / Springer-Verlag 2017 Abstract. In this paper, the optimal condition of mixed magnetohydrodynamic convection in a ventilated square cavity ﬁlled with a Cu-water nanoﬂuid with diﬀerent positions of inlet and outlet port is analyzed. To serve the purpose, the L16 (43 ) orthogonal Taguchi array is used by means of the Taguchi method. Discretization of the governing equations is obtained through the ﬁnite volume method and then solved with the SIMPLE algorithm. A very eﬀective mode, namely the Taguchi method, is used for the L16 (43 ) orthogonal Taguchi array. The eﬀects of Richardson number (0.01–10), Hartmann number (0–50), and inlet and outlet positions (0–0.9 H) at Φ = 1% are investigated. The entire analysis is performed for ﬁxed Grashof number 104 . It is found that the mean Nusselt number decreases by increasing the Richardson number, Hartmann number and the position of the inlet port, whereas the position of the outlet port has an increment eﬀect on the mean Nusselt number. The optimal distance of outlet port position from the upper wall of the cavity is 0.9 H at Richardson number 0.01, while the Hartmann number of optimal design for heat transfer is noted as 30. Comparison with existing studies is made as a limiting case of the considered problem.

Nomenclature B0 Cp g

Magnitude of magnetic ﬁeld Speciﬁc heat (J/kg K) Gravitational acceleration (m/s2 )

Gr h H Ha k Nu p P Pr q0 Ri Re T u, v

kbf Grashof number, ϑbf 2 Heat transfer coeﬃcient (W/m2 k) Height of the cavity (m) Hartmann number, B0 W μσff Thermal conductivity (W/mK) Nusselt number Pressure (N m−2 ) Dimensionless pressure Prandtl number Heat ﬂux (W/m2 ) Richardson number Reynolds number, Temperature (K) Velocity components along the xand y-axis, respectively (m/s) Dimensionless of velocity component

u∗ , v ∗ a

gβbf (

q0 W

W x, y x∗ , y ∗

Width of cavity (m) x- and y-axis coordinates, respectively Dimensionless Cartesian coordinates

)W 3

e-mail: [email protected] (corresponding author)

Greek symbols α Thermal diﬀusivity, k/(ρcp ) (m2 /s) β Thermal expansion coeﬃcient, (1/K) φ Nanoparticle volume fraction (%) θ Dimensionless temperature σ Electrical conductivity (Ω −1 /m) μ Dynamic viscosity (Pa s) υ Kinematics viscosity (m2 /s) ρ Density (kg/m3 ) Subscripts bf p nf s

Base ﬂuid Particle Nanoﬂuid Surface

Page 2 of 11

Eur. Phys. J. Plus (2017) 132: 204

1 Introduction Heat convection in cavities has numerous engineering uses including cooling of electronic components, heat exchangers, heat transfer in solar ponds, chemical and food industries, nuclear reactors, lubrication systems, solar collectors, oil and gas industries, and many others. Studies have been conducted on the optimization of closed cavities in refs. [1–4]. Usually ventilated cavities are used for reducing drag and increasing the ﬂow stability. Therefore considering these advantages, the use of ventilated cavities has been studied since the middle of the last century for mainly military, but also for other commercial, implementation [5]. Recently, an increased number of researches focused on augmentation of mixed convection heat transfer in diﬀerent shape of cavities. Cho et al. [6] investigated, numerically, mixed convection of nanoﬂuids in wavy walled lid-driven cavity. They used diﬀerent types of nanoﬂuids in their study. They concluded that for all given Richardson numbers, the increase in the volume fraction of nanoparticles causes an enhancement of the mean Nusselt number. Chamkha and Abu-Nada [7] investigated mixed convection ﬂow in single- and double-lid–driven ﬂows in square cavities ﬁlled with a water-Al2 O3 nanoﬂuid. They deduced that the heat transfer increases signiﬁcantly as the nanoparticle volume fractions increase. Sourtiji et al. [8] carried out a numerical study on the mixed convection heat transfer in a ventilated cavity ﬁlled with nanoﬂuid. They considered diﬀerent positions of the outlet port. They concluded that the improvement in the mean Nusselt number caused by enhancing the Reynolds number, the Richardson number and the volume fraction of the Al2 O3 -water nanoﬂuid. Mahmoudi et al. [9] numerically investigated a study about inﬂuence of inlet and outlet port position on the mixed convective cooling within the ventilated square cavity ﬁlled with Cu-water nanoﬂuid. They found that the addition of nanoparticles, when the Reynolds and Richardson numbers are at their highest values, causes a signiﬁcant enhancement in the heat transfer. Other investigations on nanoﬂuid in cavities can be found in [10–19] and several references therein. In recent years, the eﬀect of convection heat transfer in nanoﬂuids under the inﬂuence of magnetic ﬁeld is widely investigated. Magnetic ﬁeld is an important factor in many industrial ﬂows. Magnetohydrodynamics (MHD) is the study of the interactions between a magnetic ﬁeld and a moving conductor ﬂuid. MHD eﬀects often occur in many industrial processes, like nuclear reactors, geothermal reservoirs, thermal insulations and reﬁnery reservoirs. Many investigators have studied the MHD eﬀects on natural convection and a few focused on the mixed convection heat transfer in ventilated cavities in the presence of magnetic ﬁeld. Mahmoudi et al. [20] considered the MHD natural convection and entropy generation in a trapezoidal enclosure by using a Cu-water nanoﬂuid in the presence of a constant magnetic ﬁeld. They found that the entropy generation decreases as the external magnetic ﬁeld increases. Heidary et al. [21] numerically analyzed the heat transfer and ﬂuid ﬂow in a straight channel ﬁlled with a nanoﬂuid under the inﬂuence of a magnetic ﬁeld. They found that the heat transfer in channels increases by 75% by using nanoﬂuids in the presence of a magnetic ﬁeld. Rahman et al. [22] made a ﬁnite element analysis to study MHD mixed convection in an open channel with a square cavity while its left side was partially or fully heated. Their results showed that at higher values of the Hartmann number the length of heater has an insigniﬁcant eﬀect on the ﬂow ﬁeld. Rahman et al. [23] numerically investigated a study on the magnetohydrodynamic mixed convection in a horizontal channel with a bottom-heated open cavity. They found that the ﬂow and temperature ﬁeld within the cavity are aﬀected by the mixed convection parameters signiﬁcantly. Some relevant studies on convection are listed in [24–27]. In the absence of a magnetic ﬁeld, thermally induced motion orients the dipole particles and the nanoﬂuid is not magnetized. When the nanoﬂuid is exposed to a uniform or non-uniform magnetic ﬁeld, the nanoﬂuid is magnetized [28]. The magnetic ﬁeld, also, changes the velocity and temperature proﬁles of the nanoﬂuid, hence, decreases or increases the heat transfer [29–31], There is also the eﬀect of magnetic ﬁeld on the thermo-physical properties of the ﬂuid. Such changes cause enhancement or reduction in the thermal boundary layer thickness. As a result, decreases or improves the temperature gradients, leading to an improvement or decrease of the Nusselt number. Mixed convection heat transfer has attracted the attention of many researchers. Optimization in mixed heat transfer in square cavities, however, is not widely investigated. Designing experiments to obtain optimal conditions for heat transfer in cavities can be found in [32]. Since 1980, the Taguchi method has been widely used in the mechanical industry as a powerful optimization tool [33]. In the Taguchi method, experiments are analyzed in order to achieve goals, determine the optimal conditions, evaluate the eﬀect of each of the factors aﬀecting the response, and to estimate the response under optimal conditions. Results are evaluated through the signal-to-noise ratio (S/N) method. This ratio is in fact the ratio of ﬁxed operational factors to the uncontrollable disturbance factors. The Taguchi method employs the analysis of variance (ANOVA) method to analyze the results. In other words, in this method, ANOVA can also be used to obtain the relative importance of factors to each other. The results of the analysis are usually tabulated. The tables contain useful information, such as degree of freedom of each factor and the error caused by the factor, sum of squares, variance, and the percentage eﬀect of factors on the response [34]. Literature review showed that no study has so far been conducted on the optimization of square cavities by considering the four parameters of the distance from the inlet port to the bottom wall, the distance from the outlet port to the upper wall of the cavity, Richardson number, and Hartmann number. Studies have mainly addressed the

Eur. Phys. J. Plus (2017) 132: 204

Page 3 of 11

Fig. 1. Physical model of considered ventilated square cavity.

eﬀects of parameters inﬂuencing heat transfer without optimization, and no study has been conducted to simultaneously optimize the parameters. The novelty of the present research lies in the numerical investigations performed to optimize the mixed convection heat transfer rates in square cavities by simultaneously considering four parameters eﬀective in heat transfer. This paper used the Taguchi method to study four parameters inﬂuencing the heat transfer rate. Finally, the optimal geometry was achieved to maximize the rate of heat transfer. The eﬀects of the distance from inlet port to the lower wall of the cavity and the distance from outlet port to the upper wall of the cavity in diﬀerent Richardson numbers (0.01–10) and Hartmann numbers (0–50) in a constant Grashof number (104 ) on the mixed convection heat transfer on the hot wall in a square cavity was studied.

2 Governing equations and problem description Figure 1 shows the physical model under consideration. H and W are the height and width of the cavity, respectively. The distance from bottom and top walls are H and H , respectively. All walls of the cavity are insulated except for the bottom wall which is maintained at a constant heat ﬂux. The ﬂuid entering the cavity has temperature (Ti ) and velocity (Ui ). The inlet and outlet ports have variable positions and located on the right and the left walls of cavity, respectively. The diameters of the inlet and outlet ports (D and D , respectively) are 0.1 H. The magnetic ﬁeld is uniform with constant magnitude, B0 , and directed horizontally. A Cu-water nanoﬂuid is considered as the working ﬂuid in the present investigation with volume fraction of nanoparticles of 1% (Φ = 1%). The conservation equations for mass, momentum and energy are considered as governing equations. In the energy equation, the thermal radiation, viscous dissipation, induced electric current and Joule heating is ignored. The dimensional transport equations are written as follows [35,36]: ∂u ∂v + ∂x ∂y ∂u ∂u +v u ∂x ∂y ∂v ∂v u +v ∂x ∂y ∂T ∂T +v u ∂x ∂y

=0 2 1 ∂p ∂ u ∂2u + vnf + ρnf ∂x ∂x2 ∂y 2 2 1 ∂p ∂ v g ∂2v =− + vnf − + (T − Tin ) × [φρp βp + (1 − φ)ρbf βbf ] − σnf B02 v 2 2 ρnf ∂y ∂x ∂y ρnf 2 ∂ T ∂2T = αnf , + ∂x2 ∂y 2 =−

(1) (2) (3) (4)

, is the eﬀective thermal diﬀusivity of the nanoﬂuid. Here, subscripts nf, bf and p are nanoﬂuid, where αnf = (ρCknf p )nf base ﬂuid and nanoparticles characteristics, respectively. Also φ is the volume fraction of Cu nanoparticles in the base ﬂuid.

Page 4 of 11

Eur. Phys. J. Plus (2017) 132: 204 Table 1. Thermo-physical properties of water and nanoparticles [39]. Property Speciﬁc heat, Cp (J/kgK) Thermal conductivity, k (W/mK) Thermal expansion coeﬃcient, β (1/K)

Water

Cu nanoparticles

4179

385

0.613

400

2.1 × 10−4

1.67 × 10−5

Density, ρ (kg/m3 )

997.1

8933

Electrical conductivity, σ (Ω · m)−1

0.05

5.96 × 107

The thermo-physical properties of the nanoﬂuid for solving the governing equations (1) to (4) was obtained from the following eqs. (5)–(10) [37,38]. Equations (7) and (8) are related to the eﬀective viscosity of the nanoﬂuid (Brinkman equation) and the eﬀective thermal conductivity of the nanoﬂuid (Maxwell-Garnett model), respectively, ρnf = (1 − φ)ρbf + φρp (ρCp )nf = (1 − φ)(ρCp )bf + φ(ρCp )p μbf μnf = (1 − φ)2.5 (kp + 2kbf ) − 2φ(kbf − kp ) knf = kbf (kp + 2kbf ) + φ(kbf − kp ) knf αnf = (ρCp )nf σnf = (1 − φ)σbf + φσp .

(5) (6) (7) (8) (9) (10)

Thermo-physical properties of water (as the base ﬂuid) and properties of the Cu nanoparticles are shown in table 1. Equations (1)–(4) can be non-dimensionalized by deﬁning the following non-dimensional variables: x∗ =

x , W

y∗ =

y , W

u∗ =

u , Uin

v∗ =

v , Uin

p∗ =

p , ρbf Uin 2

θ=

T − Tin q0 W kbf

.

(11)

By use of the dimensionless variables above, the dimensionless forms of the governing equations are ∂v ∗ ∂u∗ + =0 ∂x∗ ∂y ∗ ∗ ∗ ρbf ∂p∗ ϑnf 1 ∂ 2 u∗ ∂ 2 u∗ ∗ ∂u ∗ ∂u +v =− + + ∗2 u ∂x∗ ∂y ∗ ρnf ∂x∗ ϑbf Re ∂x∗ 2 ∂y 2 ∗ ∗ ∗ ∗ 2 ∗ ∂ (1 − φ)(ρβ)bf + φ(ρβ)p ρ ϑ ∂ σnf Ha2 ∗ ∂v ∂v ∂p 1 v v bf nf v + + + Ri · θ − u∗ ∗ + v ∗ ∗ = − ∂x ∂y ρnf ∂y ∗ ϑbf Re ∂x∗ 2 ρnf βbf σf Re ∂y ∗ 2 2 ∂ θ αnf ∂2θ ∂θ ∂θ 1 . + u∗ ∗ + v ∗ ∗ = ∂x ∂y αbf Re · Pr ∂x∗ 2 ∂y ∗ 2

(12) (13) (14) (15)

The Prandtl number (Pr), Re number, Ri number, Grashof number and Hartmann number are written as ϑbf αbf ρbf Uin W Re = μbf gβbf q0kbfW W Gr = Ri = 2 U Re2 in gβbf q0kbfW W 3 Gr = ϑ 2 bf σf Ha = B0 W . μf Pr =

(16) (17)

(18)

(19) (20)

The speciﬁed boundary conditions in form of dimensionless parameters for the present problem are the following.

Eur. Phys. J. Plus (2017) 132: 204

Page 5 of 11

Table 2. Results of grid independence examination for three conﬁguration of cavity.

Inlet port: Outlet port:

Number of grids in X-Y

Nu number

41 × 41

5.578907

61 × 61

5.535904

81 × 81

5.530604

101 × 101

5.528104

121 × 121

5.528001

u∗ = 1, ∂u∗ = 0, ∂X

v ∗ = 0, ∂v ∗ = 0, ∂X

Left solid wall:

θ = 0.

(21)

∂θ = 0. ∂x∗

(22)

u∗ = 0,

v ∗ = 0,

∂θ = 0. ∂x∗

(23)

u∗ = 0,

v ∗ = 0,

∂θ = 0. ∂x∗

(24)

u∗ = 0,

v ∗ = 0,

∂θ = 0. ∂y ∗

(25)

∂θ kbf = . ∗ ∂y knf

(26)

Right solid wall:

Top solid walls:

Bottom solid wall: u∗ = 0,

v ∗ = 0,

The convection heat transfer enhancement between the bottom hot wall and the nanoﬂuid can be determined by the Nusselt number (Nu). The local Nu number on the heated wall can be obtained as follows: Nu =

hW . kbf

Here, h is the convection heat transfer coeﬃcient and it equals the Nusselt number can be calculated as follows: 1 Nu = . θs

(27) q0 Ts −TL .

By substituting θ from eq. (11) into eq. (27), (28)

The mean Nusselt number (Num ) is achieved by the integration of Nu along the heated surface as Num =

1

Nu dX.

(29)

0

3 Numerical survey and validation investigation The governing equations are discretized and solved using the ﬁnite volume method. The velocity and pressure in the momentum equation is coupled by using the SIMPLE algorithm [40]. For discretizing the convective terms in these equations we used the second order upwind scheme. In this paper, the convergence criteria is considered when the residuals between iterations are less than 10−6 for every equation and every discrete control volume. A square cavity was considered as the solution domain. The mean Nusselt numbers from the diﬀerent uniform grids for a grid independent test are shown as can be seen in table 2 within the present cavities. The grid study test was performed at Ri = 10, Gr = 104 and Ha = 30 for deferent Reynolds numbers. From table 2, it is seen that a 101 × 101 uniform grid for square cavity is ﬁne enough to achieve accurate results. Consequently, this grid is selected for the computations.

Page 6 of 11

Eur. Phys. J. Plus (2017) 132: 204 Table 3. Comparison of the present study results for the mean Nusselt number [41]. Ri

The mean Nu number of the present study

The mean Nu number of ref. [41]

0.1

5.92

6.00

1

4.39

4.50

10

3.12

3.20

Table 4. Studied parameters and their values in the Taguchi analysis. Eﬀective parameters

Levels 1

2

3

4

1

Ri

0.01

0.1

1

10

2

Ha

0

10

30

50

0

0.3 H

0.6 H

0.9 H

0

0.3 H

0.6 H

0.9 H

3

H

4

H

The results from the present model are compared with those from ref. [41] for validation. In ref. [41], the authors have investigated the mixed convection problem in a lid-driven nanoﬂuid-ﬁlled square cavity. This cavity is exposed to a magnetic ﬁeld. The working ﬂuid in that study is a Cu-water nanoﬂuid and the vertical walls of cavity are kept at constant temperatures, the horizontal walls of the cavity are adiabatic and the top wall of the cavity is moving from left to right. The results of the comparisons are shown in table 3 for Ha = 30. A very good agreement is found between the result of the present model and those of ref. [41], as can be seen in table 3.

4 Design using the Taguchi method As discussed in the introduction part, the mixed convection heat transfer was analyzed using the Taguchi method while considering four eﬀective parameters. The purpose of this study was to ﬁnd the optimal geometry by taking into account the eﬀect of four parameters and evaluate the eﬀect of the parameters studied in this paper on the heat transfer rate inside the cavity. The used dimensionless parameters in the simulation are: – – – – – –

Volume fraction of nanoparticles (Φ): Φ is considered 1%. Richardson number (Ri): Ri varies from 10−2 to 101 . Grashof number (Gr): Gr is considered 104 . Hartmann Number: Hartmann number varies from 0 to 50. The distance from inlet port to the lower wall of the cavity (H ): H varies from 0 to 0.9 H. The distance from outlet port to the upper wall of the cavity (H ): H varies from 0 to 0.9 H.

To determine the optimum heat transfer conditions, four factors, i.e. H , H , Richardson number, and Hartmann number, are evaluated at four diﬀerent levels. Table 4 shows the values of each of the four factors studied on the four levels of 1, 2, 3 and 4. Sixteen samples were designed using the Taguchi method (L16 orthogonal array). Samples are shown in table 5. As already mentioned, in the Taguchi method in the statistical analysis of results, a transformed response function is used as the ratio of eﬀect (S) to the eﬀect caused by the error (N ). A major advantage of using such response in statistical analysis is that the magnitude of the eﬀects of each assumed parameter can be compared with eﬀects of the factors causing error and disturbance [42]. When deﬁning the signal-to-noise ratio, it can be seen the computed amounts will be diﬀerent depending on the optimization objectives. Since we want to achieve enhancement in the heat transfer rate, the ratio was calculated according to eq. (30), 1 S = −10 × log n . (30) N yn 2 In eq. (30), yn is the measured response for each sample in each experiment, and n is the number of sample iterations. The signal-to-noise ratio responses in the Taguchi analysis for each sample are shown in table 6.

Eur. Phys. J. Plus (2017) 132: 204

Page 7 of 11 Table 5. The samples designed.

Sample number

Factors Ri

Ha

Result

H

H

Nu

1

0.01

0

0

0

88.44372

2

0.01

10

0.3 H

0.3 H

15.25008

3

0.01

30

0.6 H

0.6 H

17.72146

4

0.01

50

0.9 H

0.9 H

33.50982

5

0.1

0

0.3 H

0.6 H

9.066127

6

0.1

10

0

0.9 H

66.72104

7

0.1

30

0.9 H

0

14.52213

8

0.1

50

0.6 H

0.3 H

6.300927

9

1

0

0.6 H

0.9 H

8.26227

10

1

10

0.9 H

0.6 H

4.6336

11

1

30

0

0.3 H

27.36768

12

1

50

0.3 H

0

3.447516

13

10

0

0.9 H

0.3 H

3.190126

14

10

10

0.6 H

0

3.57601

15

10

30

0.3 H

0.9 H

5.528104

16

10

50

0

0.6 H

15.9096

Table 6. Table of signal-to-noise ratio for each sample. Sample number

Ratio of signal-to-noise related to the Nusselt number (S/N )

1

38.933

2

23.665

3

24.969

4

30.503

5

19.148

6

36.485

7

23.24

8

15.986

9

18.341

10

13.317

11

28.744

12

10.748

13

10.075

14

11.067

15

14.851

16

24.032

5 Analysis of results In this section the response of each sample was transformed to an S/N rate using eq. (30). Table 7 and ﬁg. 2 show the inﬂuence of the considered variables on the transformed system response and also they show how the transformed system response is aﬀected by the eﬀective parameters. Since the Taguchi analysis was aimed at maximizing the heat transfer rate, the highest signal-to-noise ratio was considered in the analysis. Figure 2 shows the amount of the obtained signal-to-noise ratio from the Taguchi analysis considering the maximum goal parameter (heat transfer rate). Higher average amounts of signal-to-noise ratio of level of a factor means that the

Page 8 of 11

Eur. Phys. J. Plus (2017) 132: 204 Table 7. Variation of signal-to-noise ratio. Eﬀective parameter

The magnitude of signal-to-noise ratio (S/N ) Level 1

Level 2

Level 3

Level 4 15.006

Ri

29.517

23.715

17.788

Ha

21.624

21.133

22.951

20.317

H

32.048

17.103

17.591

19.284

H

20.997

19.618

20.367

25.045

Fig. 2. The mean amount of signal-to-noise ratio for various levels (Nu). Table 8. The importance of eﬀective parameter in terms of diﬀerences in the magnitudes of signal-to-noise ratios (Nu). Eﬀective

Diﬀerence between maximum

The importance rank

parameters

and minimum magnitudes of S/N

of each parameter

Ri

14.511

2

Ha

2.634

4

H

14.945

1

H

5.427

3

Table 9. The designed sample for obtaining the optimal geometry. Number of design

Factors Ri

Result

Ha

H

H

Nu

1

0.01

0

0

0.9 H

179.4319

2

0.01

10

0

0.9 H

179.3987

3

0.01

30

0

0.9 H

179.5547

4

0.01

50

0

0.9 H

179.2072

level has a higher eﬀect. In order words, the optimal value of each parameter is the corresponding value of the level with the highest average signal-to-noise ratio. In table 8, the importance and ranking of each eﬀective parameter is listed. As can be seen in table 8, if the goal is to optimize goal parameter, heat transfer rate in this study, distance from the inlet port to the bottom wall (H ), Richardson number (Ri), the distance from the outlet port to the upper wall (H ) and the Hartmann number (Ha) have the highest impact, respectively. As can be seen in table 8, the maximum signal-to-noise ratio is not conclusive for Hartmann number. On the other hand, as can be seen in this table, the maximum value and the next value diﬀer very little; therefore, another sample was selected. The new selected sample and the concluding results from this sample are presented in table 9.

Eur. Phys. J. Plus (2017) 132: 204

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Table 10. Results of analysis of variance (ANOVA) for heat transfer. Eﬀective parameters Ri Ha H H

DOF (f ) 3 3 3 3

Sum of Squares (S) 500.525 14.614 603.183 70.587

Variance (V ) 166.841 4.871 201.061 23.529

Percent P (%) 43.113 0.278 51.667 4.942

As indicated in table 9, in this design, the level of the Hartmann number parameter changes in each sample and the values of the other parameters are selected based on the highest magnitude between levels of parameters, then the results were obtained. Results show that the level obtained by the Taguchi method (Ha = 30) is the optimal level. In other words, as is shown in table 7 and ﬁg. 2 (for heat transfer as a goal parameter): – For the Hartmann number, the optimal level is seen in level 3; the other optimal levels are levels 1, 2 and 4. – For the Richardson number, the optimal level is seen in level 1; the other optimal levels are levels 2, 3 and 4. – For the distance from the inlet port to the bottom wall (H ), the optimal level is seen in level 1; the other optimal levels are levels 4, 3 and 2. – For the distance from the outlet port to the upper wall (H ), the optimal level is seen in level 4; the other optimal levels are levels 1, 3 and 2. 5.1 ANOVA results The results are analyzed via the ANOVA table in this section. The purpose of the analysis was to ﬁnd the ratio of variance of each parameter to the total variance. The ANOVA analysis of the results is shown in table 10. Since four levels were considered for the analysis of parameters, their degree of freedom was equal to three. Error variance was calculated by dividing the sum of squares of error by the degrees of freedom. Table 10 shows the percentage eﬀect of each parameter on heat transfer. As can be seen, all factors had somehow aﬀected the response, while the parameter of the distance from the inlet port to the bottom wall (H ) had the highest impact on the heat transfer.

6 The eﬀect of the parameters aﬀecting the results This section examines the eﬀect of each selected factor, as eﬀective parameters in heat transfer, with respect to the signal-to-noise ratios in ﬁg. 2. For this purpose, the impact of parameters on Hartmann number, Richardson number, the distance from the inlet port to the bottom wall (H ) and the distance from the outlet port to the upper wall of the cavity (H ) are examined. As can be seen in ﬁg. 2, enhancing the levels of Ri number, Hartmann number, distance from the inlet port to the bottom wall (H ) and distance from the outlet port to the upper wall (H ) reduces the signal-to-noise ratio. This means that the higher the values of the eﬀective parameters, the lower the heat transfer rate. Figure 3 shows the streamline and isothermal line contours in the studied geometry with a hot wall subjected to a constant ﬂux. Contours are drawn for diﬀerent Ri numbers in the ranges of 0.01, 0.1, 1 and 10 at Grashof number of 104 with solid volume fraction of 1%. The ﬁgure shows that as the Ri number increases, the ﬂow intensity decreases. This reduction in the ﬂow velocity causes the convection heat transfer rate to decrease and that, in turn, reduces the Nu number. Therefore the natural convection mechanism becomes dominant. At high Richardson numbers, the buoyancy-driven motions are dominant. Therefore, the streamlines have a symmetrical distribution (see ﬁg. 3(c), (d)). Also, this eﬀect can be seen within the distribution of isotherm contours. By decreasing the Richardson number, the thermally induced buoyancy eﬀect is less powerful, and therefore the streamlines are twisted (see ﬁg. 3(a)). Also, both the natural and forced convection mechanisms become present. The velocity and temperature ﬁelds are more aﬀected by forced convection at lower Richardson numbers. This is evident in the isotherm contours in ﬁg. 3(a). The results show that the mean Nu number decreases with increasing the Ha number. The reason for this is that as the Ha number increases, the isotherms, as can be seen in ﬁg. 4, are compressed together and close to the hot wall. As a result, the temperature on the hot wall increases and, according to eq. (28), less heat transfer is obtained between the ﬂuid and the hot wall.

7 Conclusion In this paper, the mixed magnetohydrodynamic convection heat transfer characteristic in a nanoﬂuid ﬁlled square cavity with constant heat ﬂux boundary condition was investigated. This study was conducted using the Taguchi method and the ﬁnite-volume method and it was aimed at obtaining the optimal geometry in terms of heat transfer and presenting and analyzing the results using the signal-to-noise ratio (S/N ) values.

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Fig. 3. The eﬀect of the Richardson number on ﬂow streamlines (left) and isotherms (right). (a) Ri = 10−2 ; (b) Ri = 10−1 ; (c) Ri = 1; (d) Ri = 10; Φ = 0.01% and Gr = 104 .

Fig. 4. The eﬀect of the Hartmann number on isotherms contours: (a) Ha = 0; (b) Ha = 10; (c) straight line (Ha = 0) and dashed line (Ha = 10) at Ri = 102 ; Φ = 0.01% and Gr = 104 .

This paper used ANOVA for analysis through the Taguchi method. In other words, the ANOVA method was used to obtain the relative importance of each factor. The results of this analysis were presented in tables. The tables include useful information, such as the degree of freedom of each factor, the sum of squared error, variance and percentage eﬀect of each factor on the response. In the studied geometry, all walls were insulated, except for the bottom wall, which was under constant heat ﬂux. The numerical results are deduced as follows: – For the Hartmann number, the optimal level is seen in level 3; the other optimal levels are levels 1, 2 and 4. – For the Richardson number, the optimal level is seen in level 1; the other optimal levels are levels 2, 3 and 4. – For the distance from the inlet port to the bottom wall (H ), the optimal level is seen in level 1; the other optimal levels are levels 4, 3 and 2. – For the distance from the outlet port to the upper wall (H ), the optimal level is seen in level 4; the other optimal levels are levels 1, 3 and 2.

Eur. Phys. J. Plus (2017) 132: 204

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Analysis of the obtained results by the Taguchi method concludes that the cavity with the distance from the inlet port to the bottom wall (H ) 0 and the distance from the outlet port to the upper wall (H ) 0.9 H at Richardson number 0.01 and Hartmann number 30 is the optimal design for heat transfer in the current conﬁguration. Increase in the levels of Richardson number, Hartmann number and the distance from the inlet port to the bottom wall (H ) reduces the signal-to-noise ratio. This means that higher values of the eﬀective parameters reduce the heat transfer rate. The signal-to-noise ratio increases with increasing the distance from the outlet port to the upper wall of the cavity (H ). This means that higher values of eﬀective parameters increase the heat transfer rate. Reduction in the Richardson number causes the forced and natural convections to be mixed, and isothermal lines start to rotate. At lower Ri numbers, velocity and temperature ﬁelds are more aﬀected by forced convection, creating a vortex structure. As the Richardson number gets smaller, the forced convection mechanism becomes dominant. As a result, the Nusselt number increases. The second and third authors extend their appreciation to the International Scientiﬁc Partnership Program ISPP at King Saud University for funding this research work through ISPP# 0030.

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