MHD natural convection flow in a vertical parallel ...

Ain Shams Engineering Journal (2014) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

MECHANICAL ENGINEERING

MHD natural convection flow in a vertical parallel plate microchannel Basant K. Jha, Babatunde Aina *, A.T. Ajiya Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria Received 29 June 2014; revised 15 September 2014; accepted 26 September 2014

KEYWORDS MHD; Natural convection; Micro-channel; Velocity slip; Temperature jump

Abstract The fully developed steady natural convection flow of conducting fluid in a vertical parallel plate micro-channel in the presence of transverse magnetic field is considered. The velocity slip and temperature jump at the plates are taken in to consideration. The effect of various flow parameters entering into the problem such as Hartmann number, rarefaction parameter, fluid-wall interaction parameter and wall-ambient temperature difference ratio are discussed with the aid of line graphs. During the course of computation, it is found that increase in the effects of rarefaction and fluid wall interaction leads to increase in volume flow rate while the volume flow rate decreases with increase in Hartmann number. Ó 2014 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction Micro-electro-mechanical systems (MEMS) and nano-electromechanical systems (NEMS) have received great interest in recent years. These highly integrated systems are now widely applied to various applications. As a basic element in MEMS/NEMS devices, micro-channel is often found to be used for integrated cooling or heating in micro-reactor devices. Current applications for such devices include micro-channel heatsink, microjet impingement cooling and micro heat pipe.

* Corresponding author. E-mail addresses: [email protected] (B.K. Jha), [email protected] (B. Aina), [email protected] (A.T. Ajiya). Peer review under responsibility of Ain Shams University.

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Most of these proposed designs involve internal microchannel flows. Therefore, a thorough understanding of the flow behaviors in micro-channel is becoming increasingly important for accurate prediction of performance during the design process. Although there are a remarkably growing number of realized scientific and engineering applications found for MEMS devices, the understanding on the fluid dynamics and heat transfer processes in such MEMS devices is still far from being thorough. On the other hand, the performance of MEMS often defies predictions made using scaling laws developed for large systems. On the other hand, heat can be easily built up in a densely packed MEMS protective housing, which may cause undesirable or even destructive deformation. Therefore, there is a pressing need of reliable computational capabilities for accurate predictions of these devices. The key quantity in micro-channel analysis is Knudsen number (Kn), defined as the ratio of molecular mean free path to the characteristic length ðKn ¼ k=bÞ. Knudsen number is very small for continuum flows. However, for microscale gas flows where the gas mean free path become comparable with

http://dx.doi.org/10.1016/j.asej.2014.09.012 2090-4479 Ó 2014 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Please cite this article in press as: Jha BK et al., MHD natural convection flow in a vertical parallel plate microchannel, Ain Shams Eng J (2014), http://dx.doi.org/ 10.1016/j.asej.2014.09.012

2

B.K. Jha et al.

Nomenclature b B0 Cp,Cv ft,fv g In Kn m Q M Nu Pr T T0 u U

channel width constant magnetic flux density specific heats at constant pressure and constant volume, respectively thermal and tangential momentum accommodation coefficients, respectively gravitational acceleration fluid–wall interaction parameter, bt/bv Knudsen number, k=b volume flow rate dimensionless volume flow rate Hartmann number dimensionless heat transfer rate (Nusselt number) Prandtl number temperature of fluid reference temperature velocity components in x direction dimensionless velocity

the characteristic dimension of the duct, the Knudsen number may be greater than 0.001. Micro-channel with characteristic lengths on the order of 100 lm would produce flows inside the slip regime for gas with a typical mean free path of approximately 100 nm at standard condition. A classification of different flow regimes based on Kn is given in Schaaf and Chambre [1]. A series of investigations have been conducted recently in the field of micro geometry flow. However, to cite a few works in this direction, Chen and Weng [2] analytically studied the fully developed natural convection in open-ended vertical parallel plate micro-channel with asymmetric wall temperature distribution in which the effect of rarefaction and fluid wall interaction was shown to increase the volume flow rate and decrease the heat transfer. This result is further extended by taking into account suction/injection on the micro-channel walls by Jha et al. [3]. They concluded that skin-friction as well as rate of heat transfer is strongly dependent on suction/injection parameter. In another article, Larrode et al. [4] and Yu and Ameel [5] considered the temperature jump condition and found that the effect of fluid–wall interaction is very important. The effect of the hyperbolic heat conduction model on the transient hydrodynamics and thermal behaviors of fluid flow in an open-ended vertical parallel plate microchannel was analytically studied by Khadrawi et al. [6]. Biswal et al. [7] numerically investigated the flow and heat transfer characteristics in the developing region of an isothermal planar microchannel by using the semi-implicit method for pressure linked equations. Recently, Chen and Weng [8] modeled the developing convection in an asymmetrically heated planar microchannel based on the second-order Beskok–Karniadakis slip/jump boundary conditions by using a marching implicit procedure. On the other hand, the flow of an electrically conducting fluid through a channel or a circular pipe in the presence of a transverse magnetic field is encountered in a variety of applications such as magnetohydrodynamic (MHD) generators,

Greek letters a thermal diffusivity b thermal expansion coefficient bt,bm dimensionless variables cs ratio of specific heats (Cp/Cv) l dynamic viscosity h dimensionless temperature n wall-ambient temperature difference ratio q density m fluid kinematic viscosity r electrical conductivity of the fluid Subscripts 1 hotter wall values 2 cooler wall values

pumps, accelerators, and flow meters. Besides these applications, when the fluid is electrically conducting, the free convection flow is appreciably influenced by an imposed magnetic field. In the pasts, several MHD free convection solutions were obtained by Cramer and Pai [9], Chawla [10], Soundalgekar and Takhar [11], and so on. Sheikholeslami et al. [12] investigated the magnetic field effect on nanofluid flow and heat transfer in a semi-annulus enclosure via control volume based finite element method. Sheikholeslami and Gorji-Bandpy [13] presented the numerical solution for free convection of ferrofluid in a cavity heated from below in the presence of external magnetic field, while the magnetohydrodynamic natural convection of nanofluid in a concentric annulus between a cold outer square cylinder and a heated inner circular is investigated by Sheikholeslami et al. [14]. Some recent works related to the present investigation are found in literature [15–21], In [15], Sheikholeslami et al. conduct a numerical investigation of MHD natural convection heat transfer in an L-shape inclined enclosure filled with nanofluid. Sheikholeslami et al. [16] studied magnetohydrodynamic effect on natural convection heat transfer of Cu–water nanofluid in an enclosure with hot elliptic cylinder. Also, Sheikholeslami et al. [17] numerically examine the natural convection of nanofluids in a concentric annulus between a cold outer square cylinder and a heated inner circular. Flow and heat transfer of a nanofluid over a stretching cylinder in the presence of magnetic field has been investigated by Ahorynejad et al. [18]. Recently, Sheikholeslami et al. [19] presented the numerical solution for natural convection of nanofluids in a cold outer circular enclosure containing a hot inner sinusoidal cylinder. Sheikholeslami et al. [20] carried out a numerical investigation on natural convection nanofluid flow in a half annulus enclosure with one wall under constant heat flux in the presence of magnetic field. In another work, control based finite element method (CVFEM) is applied to investigate flow and heat transfer of Cu–water nanofluid in the presence of magnetic by Sheikholeslami et al. [21], In all

Please cite this article in press as: Jha BK et al., MHD natural convection flow in a vertical parallel plate microchannel, Ain Shams Eng J (2014), http://dx.doi.org/ 10.1016/j.asej.2014.09.012

MHD natural convection flow in a vertical parallel plate microchannel the works mentioned above, there was none that considered the fully developed steady natural convection flow of conducting fluid in a vertical parallel plate micro-channel in the presence of transverse magnetic field, hence the motivation of this work. Therefore, the main goal of the present work is to investigate the influence of externally applied transverse magnetic field on steady natural convection flow of conducting fluid in a vertical micro-channel. The effects of rarefaction parameter, fluid–wall interaction parameter, wall-ambient temperature difference ratio and Hartmann number are discussed with the aid of line graphs. The present work extends the work of Chen and Weng [2] by considering the influence of externally applied transverse magnetic field on steady natural convection flow of conducting fluid in a vertical micro-channel. 2. Mathematical analysis Consider the fully developed steady natural convection flow of viscous, incompressible, electrically conducting fluid in a micro-channel formed by two vertical plates under the effect of transverse magnetic field. The x-axis is parallel to the gravitational acceleration g but in the opposite direction while the y-axis is orthogonal to the vertical parallel plates. A magnetic field of uniform strength (0, B0, 0) is assumed to be applied in the direction perpendicular to the direction of flow. It is assumed that the magnetic Reynolds number is very small, which corresponds to negligibly induced magnetic field compared to the externally applied one. The plates are heated asymmetrically with one plate maintained at a temperature T1 while the other plate at a temperature T1 where T1 > T2. Due to this temperature gradient between the plates, natural convection flow occurs in the channel. The geometry of the system under consideration in this present study is shown schematically in Fig. 1. Following Chen and Weng [2], by taking into account the conducting fluid and transverse magnetic field, the governing equations for the transport processes in dimensionless form in the presence of velocity slip and temperature jump under Boussinesq’s approximation are obtained as follows: d2 U M2 þ h ¼ 0 dY2

ð1Þ

3

d2 h ¼0 dY2

ð2Þ

The dimensionless quantities used in the above equations are: T T0 ; T1 T0

y Y¼ ; b

h¼

m Pr ¼ ; a

where U0 ¼

U¼

u ; U0

M2 ¼

rB20 b2 qm

qgbðT1 T0 Þb2 l

The physical quantities used in the above equations are defined in the nomenclature. The boundary conditions which describe velocity slip and temperature jump conditions at the fluid–wall interface are [22–24] dU dU Uð0Þ ¼ bm Kn ; Uð1Þ ¼ bm Kn ; dY Y¼0 dY Y¼1 ð4Þ dh dh hð0Þ ¼ n þ bm KnIn ; hð1Þ ¼ 1 bm KnIn ; dY Y¼0 dY Y¼1 where: 2 fv 2 ft 2cs 1 ; bt ¼ ; fv ft cs þ 1 Pr b T2 To ln ¼ t ; n ¼ : bv T1 To bv ¼

k Kn ¼ ; b

Here cs is the ratio of specific heats, Pr is the Prandtl number, fv and ft are the tangential momentum and thermal accommodation coefficients, respectively, and range from near 0 to 1, k is the molecular mean free path, Kn is the Knudsen number, ln is the fluid–wall interaction parameter, and n is the wall-ambient temperature difference ratio. Referring to the values of fv and ft given in Eckert and Drake [22] and Goniak and Duffa [24], the value of bv is near unity, and the value of bt ranges from near 1 to more than 100 for actual wall surface conditions and is near 1.667 for many engineering applications, corresponding to fv = 1; ft = 1; cs = 1.4 and Pr = 0.71 (bv = 1; bt = 1.667). Eqs. (1) and (2), subject to the boundary conditions, (4) have the following exact solutions: hðYÞ ¼ A0 þ A1 ðYÞ

ð5Þ

B0 UðYÞ ¼ A2 expðMYÞ þ A3 expðMYÞ þ

g

A0 ¼ n þ

x y

Figure 1

bm Knlnð1 nÞ ; 1 þ 2bv Knln

A2 ¼ F5

T0

A0 A1 Y þ M2 M2

ð6Þ

where:

b

T2

ð3Þ

T1

Flow configuration and coordinate system.

A1 ¼

bv KnMF7 F7 ; F4 F6 F4 F6

1n ; 1 þ 2bv Knln

and A3 ¼

F7 F6

Two important parameters for buoyancy – induced microflow and microheat transfer are the volume flow rate m and heat transfer rate q, respectively. The dimensionless volume flow rate is:


4

B.K. Jha et al.

Q¼

m ¼ bU0

Z

1

ð7Þ

UdY 0

Substituting Eq. (6) into Eq. (7), one obtain Q¼

A2 expðMÞ A3 expðMÞ A0 A1 A2 A3 þ 2þ þ SPrF3 M M M M M

ð8Þ

And rate of heat transfer which is expressed as the Nusselt number are: qb dh dh Nu ¼ ¼ ¼ ¼ A1 ð9Þ ðT1 T0 Þk dY Y¼0 dY Y¼1 Using expression (6), we obtain the skin – friction (s) on the micro-channel plates as follows: dU s0 ¼ ¼ MA2 MA3 þ F3 ð10Þ dYY¼0 s1 ¼

dU ¼ MA2 expðMÞ MA3 expðMÞ þ F3 dYY¼1

ð11Þ

All the constants are declared in Appendix. 3. Results and discussion To see the physical impact of the Hartmann number, rarefaction parameter, and fluid–wall interaction parameter on the flow, we have plotted the line graphs for velocity, volume flow rate and skin-friction. The present parametric study has been performed over reasonable ranges of 0 6 bvKn 6 0.1, and 0 6 ln 6 10. The product of bvKn represents a measure of the departure from the continuum regime, while ln represents a property of the fluid–wall interaction. The selected reference values of bvKn, and ln for the analysis are 0.05 and 1.667 as presented by Chen and Weng [2]. In addition, the values of Hartmann number (M) are taken over the range of 1 6 M 6 2 with a reference value of M = 2.

The expression for the temperature in Eq. (5), the effects of the rarefaction parameter (bvKn), and fluid–wall interaction parameter (ln) on the temperature profile and rate of heat transfer which are expressed as the Nusselt number are exactly the same as those given by Chen and Weng [2]. Fig. 2 shows the velocity variation for different values of rarefaction parameter (bvKn). It is observed that as bvKn increases, the velocity slip at the wall increases which reduces the retarding effect of the wall. This yields an observable increase in the gas velocity near the wall. Also, as bvKn increases, the temperature jump increases and reduces the amount of heat transfer from the wall to the fluid. This reduction in heat transfer reduces the buoyancy effect, which derives the flow and hence reduces the gas velocity far from the wall. The reduction in velocity due to the reduction in heat transfer is offset by the increase in velocity due to the reduction in the frictional retarding forces near the wall. In addition, as the wall-ambient temperature difference ratio (n) increases, the effect of rarefaction parameter (bvKn) on the microchannel slip velocity becomes significant. Fig. 3 shows the velocity variation for different values of fluid–wall interaction parameter (ln). It clearly reveals that velocity slip increases on the microchannel surfaces with increase of fluid wall interaction parameter while the impact of fluid–wall interaction parameter on the microchannel slip velocity become significant with the decrease of the wall-ambient temperature difference ratio (n). Fig. 4 shows the velocity variation for different values of Hartmann number (M). It is found from Fig. 4 that the effect of the Hartmann number is to decrease the fluid velocity. This is the classical Hartmann effect. Furthermore, the effect of Hartmann number on the microchannel slip velocity becomes significant with the increase of the wall-ambient temperature difference ratio (n). The variation of volume flow rate is displayed in Fig. 5 with respect to fluid–wall interaction parameter (ln) and the depar-

0.12

0.1

ξ=1 ξ=0 ξ=-1

0.1

ξ=1 ξ=0 ξ=-1

0.08

0.08

Velocity (U)

Velocity (U)

0.06 0.06

0.04

0.04

0.02 0.02

βv Kn=0.0,0.05,0.1

0

-0.02

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.9

1

-0.02

In=0,5,10 0

Y

Figure 2 M = 2.0.

Velocity profile as a function bvKn for ln = 1.667,

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

Figure 3 M = 2.0.

Velocity profile as a function ln for bvKn = 0.05,


MHD natural convection flow in a vertical parallel plate microchannel

5 1

0.14

ξ=1 ξ=0 ξ=-1

0.12

ξ=1 ξ=0 ξ=-1

0.9

M=1.0,1.5,2.0

0.8

M=1.0,1.5,2.0 0.1

Volume flow rate

0.7

Velocity (U)

0.08

0.06

0.04

0.6 0.5 0.4 0.3

0.02 0.2 0

-0.02

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.01

0.02

0.03

0.04

Y

0.05

0.06

0.07

0.08

Figure 4 Velocity profile as a function M for bvKn = 0.05, ln = 1.667.

Figure 6

0.1

Volume flow rate versus bvKn different values of M.

0.5

ξ=1 ξ=0 ξ=-1

0.3

ξ=1 ξ=0 ξ=-1

In=0,5,10

0.25

0.09

βv Kn

0.4

0

skin friction (τ )

Volume flow rate

0.3

0.2

0.15

0.2

0.1

0

0.1

-0.1 In=0,5,10

0.05

-0.2 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0

Volume flow rate versus bvKn different values of ln.

ture from the continuum regime (bvKn) for different values of wall-ambient temperature difference ratio (n). It is obvious from Fig. 5, increasing bvKn yields an increase in volume flow rate. It is interesting to note that as the value of wall–ambient temperature difference ratio (n) increases, there is decrease in the volume flow. Fig. 6 presents variation of volume flow rate with respect to Hartmann number (M) and the departure from the continuum regime (bvKn) for different values of wall-ambient temperature difference ratio (n). From this figure, we concluded that as Hartmann number (M) increases, and the volume flow rate decreases. This is due to the fact that,

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

βv Kn

βv Kn

Figure 5

0.01

Figure 7 (Y = 0).

Variation of skin friction (s0) for different values of ln

as the Hartmann number increases, the velocity decreases. It is also interesting to note that the role of Hartmann number is insignificant for the case of symmetric heating (n = 1). The variation of skin-friction is revealed in Figs. 7 and 8 with respect to fluid–wall interaction parameter (ln) and the departure from the continuum regime (bvKn) for different values of wall-ambient temperature difference ratio (n) at the microchannel plates Y = 0 and Y = 1, respectively. It is observed from these figures that the skin-friction increases with the increase of the values of fluid–wall interaction parameter


6

B.K. Jha et al. 0

-0.05

ξ=1 ξ=0 ξ=-1

-0.05

-0.1

ξ=1 ξ=0 ξ=-1

-0.15

-0.1

1

skin friction ( τ )

1

skin friction ( τ )

-0.2 -0.15

-0.2

-0.25

-0.25 -0.3 -0.35

-0.3

-0.4

-0.35

-0.45

M=1.0,1.5,2.0

In=0,5,10 -0.4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-0.5

0.1

0

0.01

βv Kn

0.5

0.4

ξ=1 ξ=0 ξ=-1

0

skin friction (τ )

0.3

0.2

0.1

0

-0.1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

βv Kn

Figure 9 (Y = 0).

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 10 Variation of skin friction (s1) for different values of M (Y = 1) (Y = 1).

4. Conclusions

M=1.0,1.5,2.0

0

0.03

βv Kn

Figure 8 Variation of skin friction (s1) for different values of ln (Y = 1).

-0.2

0.02

Variation of skin friction (s0) for different values of M

(ln). Further, it is found that the impact of fluid–wall interaction parameter (ln) is insignificant for symmetric heating of microchannel plates (n = 1). Figs. 9 and 10 show the combined effects of Hartmann number (M) and the departure from the continuum regime (bvKn) for different values of wall-ambient temperature difference ratio (n) at the microchannel plates Y = 0 and Y = 1, respectively. It is evident from figures that the skin-friction decreases with the increase of the values of Hartmann number (M). In addition, the magnitude of skin-friction is higher in case of symmetric heating (n = 1) in comparison with asymmetric heating of microchannel plates (n = 0).

A theoretical study on steady magnetohydrodynamic fully developed natural convection flow in a micro-channel has been presented in this work. Closed-form expressions for velocity, temperature, volume flow rate, skin-friction and rate of heat transfer which is expressed as a Nusselt number are obtained by solving the present mathematical model. The effect of rarefaction parameter, fluid–wall interaction parameter, and Hartmann number is discussed with the aid of line graphs. Results show that increase in the effects of rarefaction and fluid wall interaction leads to increase of the volume flow rate while it decreases with increase of Hartmann number (M). It is also interesting to note that the role of Hartmann number on volume flow rate is insignificant for the case of symmetric heating (n = 1). In addition, it is observed that the skin-friction decreases with the increase of the values of Hartmann number (M) at Y = 0 while the reverse phenomenon occurs at the Y = 1. Furthermore, the magnitude of skin-friction is higher in case of symmetric heating (n = 1) in comparison with asymmetric heating of microchannel plates (n = 0). This study exactly agrees with the finding of Chen and Weng [2] in the absence of magnetic field.

Appendix A A0 A1 A1 þ ; F3 ¼ 2 ; F4 ¼ 1 bv KnM; M2 M2 M b KnF3 F1 2b KnM expðMÞ F5 ¼ v ; F6 ¼ exp ðMÞ v F4 F4 2 2 expðMÞ ðbv KnÞ M expðMÞ bv KnM expðMÞ; F4 F4 F7 ¼ bv KnMF5 exp ðMÞ bv KnF3 F5 exp ðMÞ F2 F1 ¼

A0 ; M2

F2 ¼


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between a circular and a sinusoidal cylinder in the presence of magnetic field. Int Commun Heat Mass Transfer 2012;39:1435–43. Sheikholeslamia M, Gorji-Bandpy M, Ganji DD, Soleiman Soheil. Effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with Cu–water nanofluid using CVFEM. Adv Powder Technol 2013;24:980–91. Sheikholeslamia M, Gorji-Bandpy M, Ganji DD, Soleiman Soheil. Natural convection heat transfer in a cavity with sinusoidal wall filled with CuO–water nanofluid in presence of magnetic field. J Taiwan Inst Chem Eng 2014;45:40–9. Eckert ERG, Drake Jr RM. Analysis of heat and mass transfer. New York: Mcgraw-Hill; 1972. Rohsenow WM, Hartnett JP. Handbook of heat transfer. New York: McGraw-Hill; 1973, Chap. 9. Goniak R, Duffa G. Corrective term in wall slip equations for Knudsen layer. J Thermophys Heat Transfer 1995;9:383–4.

Basant Kumar Jha was born in Darbhanga, Bihar, India in 1964. He is a professor in Applied Mathematics at Ahmadu Bello University, Zaria, Nigeria. He was awarded PhD degree in Mathematics from Banaras Hindu University, Varanasi, India in 1992. Dr. Jha has published more than 140 papers in reputed National/International journals. His research interest includes flow through porous media, magneto hydrodynamics, computational fluid dynamics, and heat and mass transfer.

Babatunde Aina completed his master degree from Ahmadu Bello University, Zaria, Nigeria. He is pursuing Ph.D. in the field of Applied Mathematics at Ahmadu Bello University, Zaria, Nigeria. He is currently working in kogi State Polytechnic, Lokoja, Nigeria. He has a teaching experience of 2 year and a research experience of 3 years.

A.T. Ajiya He is pursuing his master degree at Ahmadu Bello University, Zaria, Nigeria. He is currently working in SBRS, Ahmadu Bello University, Zaria, Nigeria. He has a teaching experience of 10 year and a research experience of 3 years.
