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Procedía Engineering 127 (2015) 568 - 574

Procedía Engineering

www.elsevier.com/locate/procedia

International Conference on Computational Heat and Mass Transfer-2015

Effect oflnduced Magnetic Field on Natural Convection with Newtonian Heating/Cooling in Vertical Concentric Annuli

Dileep Kumar*, A. K. Singh

Department of Mathematics, Banaras Hindu University, Varanasi-221005, India

Abstract

This paper is aimed to investigate the effect of induced magnetic field and Newtonian heating/cooling on fully developed laminar natural convective flow of a viscous incompressible and electrically conducting fluid in the presence of radial magnetic field. The governing equations of the model are transformed into non-dimensional simultaneously ordinary differential equations and solved analytically. We have analyzed the influence of the Hartmann number and Biot number on the fluid velocity, induced magnetic field and induced current density by the graphs while the values of skin-friction and mass flux are given in the tabular form. We observed that the values of the velocity, induced magnetic field and induced current density have decreasing tendency with increasing the values of Hartmann number. Also, the results show that on increasing the value of Biot number leads to increase the velocity, induced magnetic field, induced current density and skin-friction in case of the Newtonian heating and decrease the velocity, induced magnetic field, induced current density and skin-friction in case of the Newtonian cooling. ©2015 The Authors. Published byElsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICCHMT - 2015

Keywords: Natural convection; induced magnetic field; Mass transfer; Newtonian heating/cooling Magnetohydrodynamics.

1. Introduction

The studies of natural convective flow along a vertical cylinder have gained importance due to its applications in the field of technology, agriculture, and geothermal power generation. Further, the study of transport phenomenon involving the annular geometry has attracted its applications in the thermal recovery of oil, solar power collectors and design of magnetohydrodynamic power generators etc. Ramamoorthy [1] has compared the classical hydrodynamic velocity with the magnetohydrodynamic velocity between two rotating co-axial cylinders in the

* Corresponding author. Tel.: 9455222528; fax: +91-542-2368174. E-mail address: dileepyadav02april@gmail.com

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICCHMT - 2015

doi: 10. 1016/j .proeng .2015.11.346

presence of a radial magnetic field by neglecting the induced magnetic field. After that, Arora and Gupta [2] have extended the problem by considering the effect of induced magnetic field. The analytical solutions for the fully developed natural convection in open ended vertical concentric annuli with mixed kind of thermal boundary conditions under a radial magnetic field have been obtained by Singh et al. [3]. Later on, the effect of induced magnetic field on natural convection in vertical concentric annuli has been investigated by Singh and Singh [4]. Further, Kumar and Singh [5] have discussed the effect of the induced magnetic field on free convection when the concentric cylinders heated/cooled asymmetrically.

In some practical situations, we find that the heat transfer from the surface is proportional to the local surface temperature. This type of flow is known as conjugate convective flow, and the term Newtonian heating/cooling is used for proportionally condition of the heat transfer to the local surface temperature. In a pioneered work, Merkin [6] studied the effect of Newtonian heating on free convection boundary layer flow over a vertical flat plate immersed in a viscous fluid. Currently, so many researchers are taking the Newtonian heating/cooling condition in their problems due to its applications in several useful engineering devices. Hussanan et al. [7] have found analytical solution of free convective flow due to heat and mass transfer past a vertical plate with Newtonian heating. The effects of heat and mass transfer in the case of hydromagnetic free convective flow over a permeable vertical stretching sheet in the presence of the radiation have been studied by Rashidi et al. [8].

Here, the aim is to discuss the effect of induced magnetic field and Newtonian heating/cooling on natural convective flow in concentric annulus with radial magnetic field. We solve the governing linear simultaneous ordinary differential equations using the boundary condition to find the solution for the velocity, induced magnetic field and temperature field. Also, we solve the given differential equations at singular point Ha = 2.0 . Finally, we analyze the effect of various parameters by using graphs and table.

2. Mathematical Formulation

In this paper, we have considered steady laminar fully developed natural convective flow of an electrically conducting fluid in the vertical concentric annulus of infinite length. The z' -axis is taken along the axis of the coaxial cylinders in the vertical upward direction and r' denotes the radial direction measured outward from the axis of cylinder. Also, a and b are the radius of inner and outer cylinders respectively. Here, we have taken Newtonian heating/cooling at the inner cylinder. Also, the applied magnetic field, taken in the form of[aH'0/r'], is directed radially outward. The considered physical model for this problem is shown in Fig. 1. Since the cylinders are of infinite length and flow of fluid is fully developed, the variables describing the flow formation depends only on the co-ordinate r' . Finally, in this case the velocity and magnetic fields are taken by [0,0,u'(r')] and

[aH'0/r' ,0, H'., (r')] respectively. Thus, for the considered model, the basic transport equations in non-dimensional form are obtained as follows:

Fig.l. Physical Model

d2u 1 du Ha2 dh

— + -— +-— + T = 0, (1)

dr r dr r dr

d2h 1 dh 1 du

—- +--+--

dr r dr r dr

2 + - — + - — = 0, (2)

d2T 1 dT n

—T +--= 0 > (3)

dr r dr

The boundary conditions in the non-dimensional form are as follows:

U = h = 0, — = BiT, atr = 1, (4)

u = h = 0, T = 1, atr = I. (5)

The non-dimensional quantities used in the above equations are defined as:

u = u'/U, r = r'/ a, X = b/a, T = (T'-Tf')/(T,; - T'), h = (H^/^aaH^U),

Ha = U = gPa2^ -Tf')/v. (6)

2.1. Solutionfor Hartmann number (Ha) ^ 2.0

The solution of Eqs.(l) - (3), using the boundary conditions given in Eqs. (4) - (5), is obtained as:

u = AjrHa + A2r Ha + A3 + [r^lnOO + B2}V(1 + Biln(^)}, (7)

h = A4 -(AjrHa - A2r Ha)/Ha + r2[B^l-21n(r)}-2B2]/[4{l + Biln(A,)}], (8)

T = {1 + Biln(r)}/{1 + Biln(^)}. (9)

The constants Aj, A2 , A3 , A4, Bj and B2 appearing in above equations are defined in appendix.

Using Eq. (7), the skin-frictions in non-dimensional form at inner surface of outer cylinder and outer surface of inner cylinder and mass flux are obtained as:

Tj ^pj = Ha(Aj -A2) + (Bj + 2B2)/{1 + Biln(X)}, (10)

xx =-| — | = Ha(A2^-Ha-1 -A1XHa-1)- l[B1{\ + 21n(r>} + 2B2]/{1 + Biln(X)}, (11)

I dr Jr=x

Q = JJrudrde = 2%

Using the Maxwell's equation, the induced current density in the 0 — direction is given by:

J0 ^ j = A1rHa-1 + A2r-Ha-1 + [{Bjln(r) + B2}r]/{1 + Biln(X)}. (13)

2.2 Solutionfor Hartmann number (Ha) = 2.0

In this section, we have solved the governing differential equations of the model for the singular point Ha = 2.0 because the mathematical expressions Bj = {— Bi/(4 — Ha2)} and B2 = {4Bi/(4 - Ha2)2 -1/(4 — Ha2)} occurring in Eqs. (7) and (8) clearly indicate that they have the singularity at Ha = 2.0 . The expressions for the velocity and induced magnetic field at Ha = 2.0 are given as:

u = Cjr2 + C2r 2 + C3 + [r2ln(r){Djln(r) + D2 }]/{l + Bi ln( X)}, (14)

h = C4 - (Cjr2 - C2r-2)ll + r2[D5 + {D6 + D7ln(r)}ln(r)]/{1 + Bi ln(X)} . (15)

In this case, the skin-friction at the cylindrical walls, the flux of fluid and induced current density are calculated

Xj = 2(Cj - C2) + D2/{1 + Biln(X)}, (16)

xx = -2(C^ - C2l-3) - [2 (Dj + D2)X ln( I) + 2DjX{ln( X)}2 + D2X]/{1 + Bi ln( 1)}, (17)

Q = 2k [Cj(X4 - l)/A + C2ln( X) + C3(X2 -1)/2 + (D8 + D9)/A{\ + Bi ln(X)}], (18)

Je = (Cjr + C2r3) + [{Djlnir) + D2}rln(r)]/{1 + Biln(X)}. (19)

The constants B5 , B6 , Cj , C2 , C3 , C4 , Dj , D2 , D5, D6 , D7 , Dg and Dg appearing in above equations are defined in appendix.

3. Results and Discussion

In order to determine the effect of different physical parameters, such as the Hartmann number (Ha) and Biot number (Bi) on the free convective flow with induced magnetic field and Newtonian heating/cooling in vertical concentric annulus, velocity field, induced magnetic field, induced current density, and the numerical values of skin-friction and mass flux are shown in the figures and table.

Figure 2 (a) shows the variation in the velocity profiles for different values of the Biot number with Ha = 2.0 (For singular point) and Ha = 3.0 (For other points). It indicates that as the Biot number increases the value of velocity increases for the Newtonian heating and decreases for the Newtonian cooling. It is because the internal diffusive resistance of the inner cylindrical wall is very high due to Biot number and hence convective heat transfer

Aj (XHa+2 - l)/(Ha + 2) + A2(X~Ha+2 - l)/(-Ha + 2) + A3(X2 -1)/2 + (B5 + 4B6)/16{1 + Biln(X)}

to the fluid is negligible in case of Newtonian heating. In the case of Newtonian cooling, internal diffusive resistance of the inner cylindrical wall is very low due to Biot number so the heat transfer to the fluid is high. It shows that as Biot number increases the convective resistance of wall reduces in Newtonian heating while it increases in Newtonian cooling. The velocity deceases with increasing value of Hartmann number because with increasing the Hartmann number electromagnetic force (Lorentz force) increases which opposes the motion of the fluid. Fig. 2 (b) shows that for higher values of Biot number velocity profiles becomes almost same.

Fig.2. (a) Velocity profile for Bi=-0.6, -0.4, 0.1 & 4.0 with Ha=2.0 & 3.0 at A,=3.0; (b) Velocity profile for Bi=5.0, 10.0 & 120.0 with

Ha=2.0 & 2.2 at ^=3.0

The variation of induced magnetic field for different values of Biot number (Bi = — 0.6, - 0.4, 0.1 & 5.0) is shown in Fig. 3 (c) at singular point Ha = 2.0 and some other value Ha = 3.2. It is found from this figure that as the values of Biot number (Bi) increase, the induced magnetic field profile increases in case of the Newtonian heating and decreases in case of the Newtonian cooling. It is because the heat transfer to the fluid is very low in Newtonian heating while in Newtonian cooling it is very high. It is observed that the value of the induced magnetic field decreases with increase in the Hartmann numbers (Ha). As Hartmann number increases the Lorentz force increases by definition, which reduces the induced magnetic field profiles. Figure 3 (d) indicates that for higher values of Biot number, the induced magnetic field profiles become almost same. The shape of induced magnetic field, near the inner cylinder, is parabolic type in upward direction and near the outer cylinder it is downward direction because the direction of induced magnetic field has changed.

Fig.3. (c) Behavior of induced magnetic field for Bi=-0.6, -0.4, 0.1 & 5.0 with Ha=2.0 & 3.01 X=3.0; (d) Behavior of induced magnetic field for

Bi= 3.0, 7.0 & 180.0 with Ha=2.0 & 2.5 at A,=3.0.

The effects of Biot number (Bi) on the induced current density are shown in Fig. 4 (e) for singularity at Ha = 2.0 and some other value Ha = 3.0 with fixed value of ratio of outer radius to inner radius (X = 3.0) . From this figure, we find that the induced current density profiles decrease with increasing the values of Biot number (Bi) because the convective heat transfer to the fluid is very high. Also, from this figure it can be noted that the positive induced current density has maximum value in the middle of both cylinder and the negative induced

current density has maximum value on the surface of both cylinders but it is more negative on the surface of inner cylinder compare to outer cylinder. Further, when the value of the Hartmann number increases, both the positive induced current density in the middle of both cylinders and the negative induced current density on the surface of both cylinders decrease due to presence of the Lorentz force. Figure 4 (f) shows that there is no difference in behavior of induced current density profiles for large values of Biot number.

Q l : i-

-j* -C 2 ¿A

r? I /:'

-¿■■■I iii

i7 1.0

Fig.4. (e) Behavior of induced current density for Bi=-0.7, -0.5, 0.1 & 4.0 with Ha=2.0 & 3.5 at X=3.0; (f) Behavior of induced current density for

Bi= 1.0, 4.5 & 200.0 with Ha=2.0 & 2.5 at ^=3.0.

The numerical values of the skin friction at the inner surface of the outer cylinder and at the outer surface of the inner cylinder, and mass flux are given in table 1. This table clearly shows that as the value of Hartman number (Ha) increases, the value of skin-friction at outer surface of inner cylinder increases and at inner surface of outer cylinder decreases. The value of mass flux has decreasing tendency with increasing value of Hartman number (Ha) . The effect of Biot number is to decrease the value of skin-friction at both surfaces and mass flux.

Table.l. Numerical values of skin-friction and mass flux

Bi "A Ha Q -G L 11=-( £ L

1.0 0.764538 0.561338 0.450467

2.0 2.0 0.747279 0.565169 0.448552

0.1 3.0 0.720878 0.571029 0.445622

1.0 8.07768 1.26389 1.26389

3.0 2.0 7.65864 1.29864 1.29864

3.0 7.08496 1.34621 1.34621

1.0 0.653655 0.444419 0.408655

2.0 2.0 0.638959 0.448601 0.406564

1.0 3.0 0.616478 0.454999 0.403365

1.0 6.79154 0.966083 0.780885

3.0 2.0 6.44200 1.00131 0.769141

3.0 5.96334 1.04957 0.753035

4. Conclusions

In the present investigation, the effect of induced magnetic field on natural convection with Newtonian heating/cooling in vertical concentric annuli is carried out. The following conclusions have been drawn from the present analysis:

• The velocity, induced magnetic field and induced current density have increasing tendency with increase in the value of Hartmann number.

• The effect of induced magnetic field is to increase the velocity, induced magnetic field and induced current density profiles.

• The effect of Biot number is to increase the values of velocity, induced magnetic field and induced current density in case of the Newtonian heating and decrease the values of velocity, induced magnetic field and induced current density in case of the Newtonian cooling.

• The velocity, induced magnetic field and induced current density profiles are almost same for higher values of Biot number.

• The skin friction at inner cylinder increases and at outer cylinder decreases due to increase in the value of Hartmann number.

• The effect of Biot number is to decrease the skin-friction and mass flux.

Appendix A.

The constants appearing in the above equations are defined as follows:

A, = (B3 + B4)/[2(l- XHa){l + Biln(X)}], A2 = (-B3 + B4)/[2(l -rHa){l + Bi ln(X)}], A3 =-[Aj + A2 + B2/{1 + Biln(X)}], A4 = (Aj - A2)/Ha-(Bj -2B2)/[4{1 + Biln(X)}], Bj =- Bi/(4 - Ha2), B2 = 4Bi/(4 - Ha2)2 -1/(4 - Ha2), B3 = Ha{(2B2 - BjX^2 -1) + 2B1X2ln(X)}/4, B4 = B^ln*» + B2(l2 -1), B5 = B^^ln^)-1} +1], B6 = B2(X4 -1), Cj = (D3 + D4)/[2(X2 -1){1 + Biln(X)}], C2 = (D3 - D4)/[2(X"2 -1){1 + Biln(X)}], C3 = -(C, + C2), C4 =- (C2 - CO/2 - D5/{1 + Biln(X)}, Dj =- Bi/8, D2 = (Bi - 4)/\6, D3 = {DjlnCX) + D2}X2ln(X), D4 = 2D3(12 -1) + Ik2{D6 + D7ln(X)}, D5 = (D2 -Dj)/4, D6 = (Dj -D2)/2, D7 =-Dj/2, D8 = D1[^4{ln(^)}2 -{X4ln(X)}/2 + (X4 -1)/8], D9 = D2{X4ln(X) - (X4 -1)/4}.

References

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[2]. KL. Arora and R.P. Gupta, Magnetohydrodynamic flow between two rotating coaxial cylinders under radial magnetic field. The Physics of Fluid. 15, 1971, pp. 1146-1148.

[3]. S.K. Singh, B.K. Jha, and A.K. Singh, Natural convection in vertical concentric annuli under a radial magnetic field. Heat and Mass Transfer. 32, 1997, pp. 399-401.

[4]. R.K. Singh and A.K. Singh, Effect of induced magnetic field on natural convection in vertical concentric annuli. Acta Mechanica Sinica. 28, 2012, pp. 315-323.

[5]. A. Kumar and A.K. Singh, Effect of Induced Magnetic Field on Natural Convection in Vertical Concentric Annuli Heated/Cooled Asymmetrically. Journal of Applied Fluid Mechanics. 6, 2013, pp. 15-26.

[6]. J. H. Merkin, Natural-convection boundaiy-layer flow on a vertical surface with Newtonian heating. International Journal of Heat and Fluid Flow. 15, 1994, pp. 392-398.

[7]. A. Hussanan, I. Khan and S. Shafie, An exact analysis of heat and mass transfer past a vertical plate with Newtonian heating. Journal of Applied Mathematics. 2013, 2013, pp. 1-9.

[8]. M.M. Rashidi, B. Rostami, N. Freidoonimehr and S. Abbasbandy, Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects. Ain Shams Engineering Journal. 5, 2014, pp. 901912.