International Journal of Mathematical Archive-5(5), 2014, Available online through ISSN

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1 International Journal of Mathematical Archive-5(5), 014, Available online through ISSN STUDY ON LIQUID METAL MHD FLOW THROUGH A TRIANGULAR DUCT UNDER THE ACTION OF VERY STRONG TRANSVERSE MAGNETIC FIELD Dipjyoti Sarma* 1 and P.N. Deka 1,Department of Mathematics, Dibrugarh University, Dibrugarh , Assam, India. (Received on: ; Revised & Accepted on: ) ABSTRACT A numerical solution for steady MHD flow of liquid metal through a right angled isosceles triangular duct has been discussed. This duct flow is under the action of strong transverse magnetic field which act normal to one of the wall of the duct. All the duct walls are considered to be electrically insulated as well as isothermal. The numerical solution for velocity, induced magnetic field and temperature distribution has been obtained by using 9-point stencil centered finite difference method. Solution for different values of Hartmann number, Magnetic Reynold number and Prandtl number for this asymmetric configuration due to action of transverse field are presented graphically and effects on various fields have been analyzed. Keywords: MHD flow, liquid metal flow, electrically insulated wall, Triangular duct, high Hartmann number. 1. INTRODUCTION During the last few decades, much interest has been given by the investigators on the flows of liquid metals through ducts, in presence of transverse magnetic field. These studies are mostly concerned with practical application such as casting of steel, aluminum reduction etc. In designing the cooling systems as well as tritium breeder systems for experimental nuclear fusion reactors, it is necessary to have appropriate knowledge of liquid metal flow behavior under the action of very strong transverse magnetic field. Hartmann and Lazarus [1] were first to study the MHD flow problem of liquid metal through a duct under the action of transverse magnetic field. In their investigations, they considered the flow of mercury as a conducting fluid in pipes of different cross sections experimentally and influence of transverse magnetic field on such a flow was elucidated. Shercliff [] calculated the ultimate steady velocity profile and its associated pressure gradient and induced potential for the case of laminar flow through a circular pipe whose walls were conducting but without contact resistance. Gold [3]obtained solution to the problem steady on dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe under the action of transverse magnetic field. Hunt [4] made an analysis on the laminar motion of a conducting liquid under the action of strong transverse magnetic field. Hunt and Stewartson [5] made an extension of the work by Hunt [4], where the effects of the duct having conducting walls were further explored. Gupta and Singh [6] obtained an exact solution for unsteady MHD flow in a circular pipe with insulating wall under a transverse field. Holroyd and Walker [7] theoretically studied the effect of wall conductivity, non-uniform magnetic fields and variable-area ducts on the liquid metal MHD flow at High Hartmann number. Holroyd [8] experimentally studied the effect of wall conductivity, non-uniform magnetic fields and variable-area ducts on the liquid metal MHD flow for High Hartmann number. Kumamaru [9] obtained the analytical solutions for MHD pressure drop and heat transfer for liquid metal flow in an annular channel with perfectly conducting walls under the action of transverse magnetic field. Walker [10] considered the liquid metal flow in a straight circular pipe with a thin metal wall under the action of strong transverse magnetic field. Kim et al. [11] developed an advanced numerical scheme based on finite difference method to obtain full solution of liquid metal MHD flow in fusion blanket. McCarthy et al. [1] obtained a computationally efficient and fast method for the analysis of liquid metal MHD flow based on the core flow approximations. Sterl [13] used finite difference method to investigate the influence of Hartmann number M, interaction parameter N, wall conductance ratio c and changing magnetic field respectively on the flow of liquid metal in rectangular ducts under the action of strong transverse magnetic field. Kunugi et al. [14] developed a computer code KAT which can analyze liquid metal MHD flow and heat transfer in relatively complex geometries. Ying and Tillack [15] numerically simulated the heat transfer issue in elongated rectangular ducts for the case of liquid metal blankets. Buhler [16] studied liquid metal MHD flow in arbitrary thin-walled channels under the action of strong transverse variable magnetic field. Corresponding author: Dipjyoti Sarma* 1 1Department of Mathematics, Dibrugarh University, Dibrugarh , Assam, India. International Journal of Mathematical Archive- 5(5), May

2 Tezer-Sezgin [17] investigated the MHD flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct under the action of transverse magnetic field by Boundary element method. Hua and Walker [18] investigated the three dimensional liquid metal MHD flow in rectangular ducts with thin conducting walls and inclined non-uniform transverse magnetic field. Takahashi et al. [19] studied the characteristics of pressure drop and heat transfer experimentally for a lithium single-phase flow in horizontal conducting rectangular channels under the action of horizontal transverse magnetic field. Barrett [0] studied MHD flow through duct under a variety of wall conductivity conditions at high Hartmann numbers by Finite element method. Smolentsev et al. [1] developed a numerical code for the analysis of pressure drop occurred in liquid metal MHD flow in a channel using various insulation techniques. Celik [] obtained the numerical solution of velocity and induced magnetic field for the case of steady state, fully developed, incompressible flow for a conducting fluid inside a rectangular duct by Chebyshev collocation method. In most of the earlier works on liquid metal MHD duct flow, investigations were carried out for rectangular and circular geometries. With the increased demand in industry, for new geometries and for efficient compact heat exchanger, the case of liquid metal MHD flow through other noncircular compact duct flow may be required. Considering such aspect of present industry requirement, and probable fusion reaction coolant design application, the investigation of MHD duct flow through a triangular duct may become useful. It is expected that a triangular duct flow will be more efficient heat exchanger than the rectangular and the circular ducts. The pressure high drop phenomena in the case of high Hartmann number duct flow could be could be controlled more efficiently in such triangular duct. Flow through a triangular duct was considered by Eckert and Irvine [3] experimentally to investigate the transition process from laminar to turbulent flow. Singh and Lal [4] considered steady MHD flow in a triangular duct with non conducting walls in presence of transverse magnetic field perpendicular to a side by using finite difference code. In a recent investigation Daschiel et al. [5] carried out Direct Numerical simulation study triangular duct flow for investigation of transition process for triangular duct with small apex angle. From the fabrication point of view triangular duct are cost effect in design and are more compact. These computed results are encouraging for construction of efficient heat exchanger as well as tritium breeder blankets is fusion reactors besides other industrial liquid metal duct flow. In this paper numerical investigations have been made for liquid metal MHD flow and associated heat transfer phenomena for the flow through a triangular duct using finite difference method.. FORMULATION OF THE PROBLEM The problem considered here is a steady motion of liquid metal through a horizontal duct of uniform cross section of right angled isosceles triangle. The MHD flow is under the action of transverse magnetic field acting normally to the 0 vertical wall of the duct. The applied transverse field act at 45 angle to the slanted wall of the duct and this field is parallel to the horizontal base of the duct as shown in Figure 1. Fig. 1: Geometrical model of the Flow problem Further, the following assumptions are considered in this investigation. (i) All fluid properties are constant and independent of the temperature, (ii) The flow and heat transfer are fully developed and (iii)the wall of the duct are all non-conducting. 014, IJMA. All Rights Reserved 136

3 The governing equations of the flow problem are V ρ + ( V. ) V + p = J B + µ V t B = ( V B) + λ B t T J ρ Cp + ( V. ) T = k T + µφ + t σ (1) ` () (3) After introducing following non-dimensional variables T Ta * x * B * V θ =, x =, B =, V = T a B V z 0 0 with V Pa 0, p = = P µ z and the following non-dimensional parameters 1 σ V0 0, m µσ e 0, c µ cp M= Ba R = VaE = T and P r µ C = k p we get the following non-dimensionalized form of governing equation of the duct flow V V M B + + = 1 x y R x m (4) B B V + + R 0 m = x y x (5) θ θ V V M PE r c B B + + PE 0 r c = x y x y R x y m (6) Here V0, B 0 and a are characteristics velocity, characteristics magnetic field strength and characteristics length while C p and µ are the fluid electrical conductivity, specific heat at constant pressure and coefficient of viscosity respectively. The dimensionless parameters M, Rm, Ec, P r are the Hartmann number, magnetic Reynold number, Eckert number and Prandtl number respectively. The boundary conditions considered are: V = 0( from no slip condition) B = 0( from externally insulated surface) θ 0( for isothermal surface) = (7) 3. NUMERICAL METHOD Here it is observed that Eq. (4)-Eq. (6) are coupled non-linear equations which are to be solved using boundary condition given in Eq. (7). In view of complexities in seeking closed form solutions, numerical solutions are considered. These equations are expressed in finite difference equation by utilizing a 5-point stencil centered finite difference scheme with mesh size h = 1 N where N is a pre-assigned positive integer. The numerical scheme to be employed will cover the computational grid points S = {( i, j) j N 1, i N j+ 1} The resulting finite difference representation for Eq. (4)-Eq. (6) are as follows V = 0.5 V + V + V + V + c B B + c (8) ( + + ) ( + ) i, j i 1, j i 1, j i, j 1 i, j 1 1 i 1, j i 1, j ( + + ) ( + ) B = 0.5 B + B + B + B + C V V (9) i, j i 1, j i 1, j i, j 1 i, j 1 3 i 1, j i 1, j 014, IJMA. All Rights Reserved 137

4 ( ) c + + ( V + V ) ( V + V ) i, j= 0.5 i 1, j+ i 1, j+ i, j 1 + i, j i 1, j i 1, j + i, j 1 i, j 1 θ θ θ θ θ ( i+ j i j) ( i j+ i j ) + c B B + B B Where 5 1, 1,, 1, 1 hm h hrm c =, c =, c =, Rm c PE r c M PE r c 4 = c5 = 16 16Rm,, (10) The numerical boundary conditions for present duct flow are as follows: Vi,1 = V1, j = ViN, + i = 0, Bi,1 = B1, j= BiN, + i = 0 and θ = θ = θ = 0 i,1 1, j in, + i The convergence of each of the computed value of variable V, B, θ at different grid points are checked by using root 7 mean square residual R s. Convergence is considered to be achieved when R s < 10. Where R s is defined as N 1 N j+ 1 n+ 1 n (,, ) R = S S s i j i j j= j= 4 RESULTS AND DISCUSSION Flow through a triangular duct is essentially asymmetric due to asymmetric boundary effects. In MHD flow through triangular ducts, transverse magnetic field effects the thickness of the boundary layers asymmetrically. Since, in a triangular duct, the transverse magnetic field acts with inclination to both or at least one of the walls, the Hartmann layer thickness, would likely to vary for different walls. The reduction of boundary layer thickness, due to suppression of vorticity out from the wall would certainly be higher for the wall for which transverse field actsnormally to the wall than the other wall. So for triangular duct, Hartmann effect is reduced over all and hence flow rate is affected through the duct. From our computed result, it has been observed that the boundary layer thickness near slanted wall of the duct will be larger than near the vertical wall. This has adverse effect on flow rate as observed in Figure a. But for MHD flow, under the action of very high transverse field, due to pressure drop phenomena, the flow rate in core flow region has been reduced. This reverse effect on flow is less intense towards slanted wall than near the vertical wall. This is observed in Figure b, where we have plotted velocity profile for different values of high Hartmann number. In Figure b, it is observed that the pressure drop effect can be more efficiently managed in the triangular duct flow. Further we observed from Figure b that with increase value of Hartmann number, velocity is less affected in region towards the slanted wall. In a fusion reactor environment, pressure drop phenomena in tritium breeder liquid metal flow viz. PbLi flow through a duct is drastically effected by pressure drop phenomena due to influence of very high plasma confining field of the reactor. This flow acts as coolant blanket as well. So successful operations of fusion reactor depend upon maintenance of PbLi or other tritium breeder liquid metal flow is essential. Our computed result for the case of MHD flow through triangular duct shows pressure drop could be effectively managed by introduction of slanted duct walls. The 3D plotting (Figure 3 and Figure 4) of velocity profile for higher values of Hartmann number indicate that for the pressure drop effect in a MHD duct flow may be effectively controlled by introduction of slanted wall for the duct. The plotting of induced magnetic field for different values of M and R are plotted in Figure 5a and Figure 5b respectively. We have plotted the temperature profile for the case of triangular duct flow for various values of Hartmann numbers and Prandtl numbers which are included in Figure 6a and Figure 6b. In the range of Hartmann number 10 M 80 temperature is higher near vertical wall than near the slanted wall. Again in the range of Prandtl number 0.01 P r 0.05 temperature is lower near the slanted wall than near the vertical wall. We have plotted 3-D graphs for temperature profile for high Hartmann number 100 M 700 Temperature is higher near the slanted wall than near the vertical wall as shown in Figure 7 and Figure 8. This reflects that due to pressure drop phenomena for the case of very high transverse field, temperature profile is effected more near the vertical wall than near the slanted. From these observations on temperature profile it may be concluded that the presence of slanted wall in a duct may make the flow more efficient heat exchanger. This observation may be useful in many industrial applications. Triangular duct flow in high Hartmann number manages pressure drop more efficiently than the rectangular duct. As heat exchanger triangular duct are more efficient than the rectangular duct Zhang [6]. In the present case for high Hartmann number flow 014, IJMA. All Rights Reserved 138 m

5 condition temperature profile is high near the slanted wall. We can conclude that triangular duct to be efficient heat exchanger in high Hartmann number flow. Fig. : Variation of Velocity profile for different values of M Fig. 3: 3D Variation of Velocity profile for M = 100,300 Fig. - 4: 3D Variation of Velocity profile for M = 500, , IJMA. All Rights Reserved 139

6 Fig. 5: Variation of induced magnetic field for different values of M and Rm Fig. 6: Variation of Temperature profile for M and P r Fig. 7: 3D Variation of Temperature profile for M = 100, , IJMA. All Rights Reserved 140

7 Fig. 8: 3D Variation of Temperature profile for M = 500,700 The exchange process is more effect in a flow under the action transverse field near the slanted wall. Triangular duct are of efficient heat exchange transverse Rao et al. [7]. In the case of MHD duct flow these aspects important particularly in the case of tritium breeder as well as coolant blanket design in nuclear reactor. Triangular duct are compact and are convenient from the fabrication consideration. From the consideration of compactness and fabrication advantage triangular duct may be favorable for very high transverse field environment. ACKNOWLEDGEMENT This work was supported by University Grant Commission, India under grant number 39-44/010(SR). REFERENCES [1] Hartmann, J. and Lazarus, F., Experimental investigations on the flow of mercury in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 15 (1937),1-45. [] Shercliff, J.A., Steady motion of conducting fluid in pipes under transverse magnetic field, Mathematical Proceeding of the Cambridge Philosophical Society, 49(1) (1953), [3] Gold, Richard R., Magnetohydrodynamic pipe flow. Part 1, Journal of Fluid Mechanics, 13(4) (196), [4] Hunt, J.C.R., Magnetohydrodynamic flow in rectangular ducts, Journal of Fluid 1(4) (1965), [5] Hunt J.C.R. and Stewartson, K., Magnetohydrodynamic flow in rectangular ducts. II, Journal of Fluid Mechanics, 3(3) (1965), [6] Gupta, S.C. and Singh, B., Unsteady Magnetohydrodynamic Flow in a Circular Pipe under a Transverse magnetic Field, The Physics of Fluids,13() (1970), [7] Holroyd, R.J. and Walker, J.S., A theoretical study of the effects of wall conductivity, non-uniform magnetic fields and variable-area ducts on liquid-metal flows at high Hartmann number, Journal of Fluid Mechanics, 84(3) (1978), [8] Holroyd, Richard R., An experimental study of the effects of wall conductivity, nonuniform magnetic fields and variable area ducts on liquid metal flows at high Hartmann number. Part. Ducts with conducting walls, Journal of Fluid Mechanics, 96() (1980), [9] Kumamaru, H., Magnetic Pressure Drop and Heat Transfer of Liquid Metal Flow in an Annular Channel under Transverse Magnetic Field, Journal of Nuclear Science and Technology,1(5) (1984), [10] Walker, J.S., Liquid-metal flow in a thin conducting pipe near the end of a region of uniform magnetic field, Journal of Fluid Mechanics, 167 (1986), [11] Kim, C.N., Hadid, A.H. and Abdou, M.A., Development of a computational method forthe full solution of MHD flow in fusion blankets, Fusion Engineering and Design, 8 (1989), , IJMA. All Rights Reserved 141

8 [1] McCarthy, K.A.,Tillack, M.S. and Abdou, M.A., Analysis of liquid metal MHD flow using an iterative method to solve the core flow equations, Fusion Engineering and Design,8 (1989), [13] Sterl, A., Numerical simulation of liquid-metal MHD flows in rectangular ducts, Journal of Fluid Mechanics, 16(1) (1990), [14] Kunugi, T., Tillack, M.S. and Abdou, M.A., Analysis of liquid metal MHD flow and heat transfer using the KAT code, Fusion Technology,19 (1991), [15] Ying, Alice Y. and Tillack, M.S., MHD heat transfer in elongated rectangular ducts forliquid metal blankets, Fusion Technology, 19(3) (1991), [16] Buhler, L., Liquid metal flow in arbitrary thin-walled channels under a strong transverse variable magnetic field,fusion Engineering and Design,17 (1991), [17] Tezer-Sezgin, M., Boundary element method solution of MHD flow in a rectangular duct, International Journal for Numerical methods in Fluids, 18 (1994), [18] Hua, T.Q., and Walker, J.S., MHD flow in rectangular ducts with inclined non-uniform transverse magnetic field, Fusion Engineering and Design, 7 (1995), [19] Takahashi, M., Aritomi, M., Inoue, A. and Matsuzaki, M., MHD pressure drop and heat transfer of lithium singlephase flow in a rectangular channel under transverse magnetic field, Fusion Engineering and Design,4 (1998), [0] Barrett, K.E., Duct flow with a transverse magnetic field at high Hartmann numbers,international journal for Numerical methods in Engineering,50(8) (001), [1]. Smolentsev, S., Morley, N. and Abdou, M., Code development for analysis of MHD pressure drop reduction in a liquid metal blanket using insulation technique based on a fully developed flow model, Fusion Engineering and Design, 73 (005), [] Celik, I., Solution of magnetohydrodynamic flow in a rectangular duct by Chebyshev collocation method, International journal for Numerical methods in Fluids, 66(10) (011), [3] Eckert, E.R.G. and Irvine Jr.,T.F., Pressure Drop and Heat Transfer in a Duct with Triangular Cross Section, Journal of Heat Transfer, 8() (1960), [4] Singh, B. and Lal, J., Magnetohydrodynamic axial flow in a Triangular pipe under transverse magnetic field, Indian Journal of Pure & Applied Mathematics, 9() (1978), [5] Daschiel, G., Frohnapfel, B., Jovanovic, J., Numerical Investigation of flow through a triangular duct: The coexistence of laminar and turbulent flow, International Journal of Heat and Fluid flow, 41 (013), [6]. Zhang, Li-Zhi, Laminar flow and heat transfer in plate-fin triangular ducts in thermally developing entry region, International Journal of Heat and Mass Transfer, 50 (007), [7] Rao, M.V., Kumar, P.V.R. and Rao, P.S.S., Laminar flow heat transfer in concentric equilateral triangular annular channels, Indian Journal of Chemical Technology,13 (006), Source of support: University Grant Commission, India under grant number 39-44/010(SR). Conflict of interest: None Declared [Copy right 014 This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.] 014, IJMA. All Rights Reserved 14