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Influence of flow channel insert with pressure equalization opening on MHD flows in a rectangular duct
Fusion Engineering and Design 88 (2013) 271–275
Contents lists available at SciVerse ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Influence of flow channel insert with pressure equalization opening on MHD flows in a rectangular duct Shi-Jing Xu, Nian-Mei Zhang, Ming-Jiu Ni ∗ School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
h i g h l i g h t s We conduct 3D simulation of MHD flows in flow channel insert (FCI) with holes or slots. Pressure drop and velocity distribution are shown to illustrate the holes and plots effects on MHD flows of FCI. Pressure fluctuation for MHD flows in FCI with holes may introduce MHD instability.
a r t i c l e
i n f o
Article history: Received 16 September 2011 Received in revised form 26 January 2013 Accepted 6 February 2013 Available online 12 April 2013 Keywords: Flow channel insert Pressure equalization slot Pressure equalization hole Liquid metal blanket
a b s t r a c t Direct simulation of 3D MHD flows in a duct with flow channel insert (FCI) relevant to R&D of fusion blanket has been conducted based on an electrical potential formula by using a consistent and conservative scheme. Comparison study of the pressure and velocity distributions of liquid metal in a poloidal duct with FCI, which has pressure equalization slot (PES) and pressure equalization holes (PEHs) with the same total area at the corresponding walls, is conducted. Both the PES and the PEHs have two kinds of locations, either in a Hartmann wall or in a side wall. 3D pressure and velocity distributions of the different cases have been given. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The dual coolant lead lithium (DCLL) blanket concept proposes the use of a flow channel insert (FCI) [1] made of SiCf /SiC composite in the breeder region flow channels in order to: (1) thermally insulate the self-cooled breeder region from the helium cooled ferritic steel walls; (2) electrically insulate the PbLi flow from current closure paths in the ferritic steel walls; (3) provide a nearly stagnant PbLi layer near the ferritic steel wall that has increased corrosion temperature limits [2]. It has been verified [3] that the liquid metal fluid in rectangular duct relevant to fusion blanket using FCI made of conductive or insulated walls with/without slot is a fully 3D flow, which can be divided into two stages of developing flows with changing pressure gradient and with constant pressure gradient in the flow direction. The first stage of developing length is around 4–5 times of the half channel width. Insulated FCI in the blanket can effectively reduce MHD pressure drop, while highly conductive FCI cannot effectively reduce the MHD pressure drop [3,4].
∗ Corresponding author. Tel.: +86 01088256072. E-mail addresses: [email protected], [email protected] (M.-J. Ni). 0920-3796/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fusengdes.2013.02.015
Velocity profiles of MHD flow in the front poloidal channel of the DCLL blanket were studied by Smolentsev et al. [5] based on a 2D model for a fully developed flow. It’s verified that MHD pressure drop reduction at = 500 (m)−1 is a factor of 10 and at = 5 (m)−1 is a factor of 200–400, compared to the case without insulation. Sutevski et al. [6] obtained a 3D result using an unstructured, parallel MHD solver HIMAG [7]. These results confirm a substantial reduction in MHD pressure drop by the FCI. Bühler [8] and Bühler and Norajitra [9] analyze the pressure drop in HCLL blanket and DCLL blanket, in which it is indicated that the major MHD pressure drop arises in the poloidal manifold connecting breeder units. An experimental GaInSn flows in a toroidally oriented manifold is numerically conducted by Morley et al. [10], which illustrated 3D effects of MHD flows. Smolentsev et al. studied the MHD and thermal issues for the SiCf /SiC FCI. Their mathematic model includes the two dimensional momentum and induction equations for a fully developed flow and the three dimensional energy equations [11]. The qualification of MHD effects in the DRM DEMO blanket has been assessed in Ref. [12]. The flow in a toroidal plane of the DRM blanket is studied by solving the 2D flow model from Sommeria and Moreau [14] coupled sequentially with temperature equation. Mistrangelo [13] has analyzed the influence of various parameters such as electrical conductivity of the SiC insert
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S.-J. Xu et al. / Fusion Engineering and Design 88 (2013) 271–275
2. Solver development for MHD flows on a collocated mesh The flow of electrically conducting fluid under the influence of an external magnetic field is governed by the following NavierStokes equation and continuity equation, which represent the conservation of momentum and mass:
Fig. 1. Cross-section of a poloidal duct with silicon carbide (SiC) flow channel insert (FCI).
and the orientation of the magnetic fields based on the finite volume method. In this paper, a direct simulation of MHD flow in a duct relevant to liquid metal fusion blanket is conducted to study the distribution of pressure and velocity influenced by different FCI structures. Considering laminar flows of liquid metal through a rectangular channel in a transverse strong uniform magnetic field, the sketch of a cross-section of the blanket channel with width 2a = 68 mm and 2b = 60 mm, length l = 1000 mm is shown in Fig. 1. Liquid metal flows along x-direction. The magnetic field strength is B0 = 1.852T and the inlet velocity of liquid metal is u0 = 0.0675 m/s. The liquid metal is GaInSn with density = 6363 kg/m3 , electrical conductivity = 3.3074 × 106 (m)−1 , viscosity = 2.545 × 10−3 kg/(ms). The thickness of the reduced activation ferritic wall is 2 mm with conductivity of w = 1.23 × 106 (m)−1 . The velocity and pressure distributions of FCI with slot or slots in Hartmann wall(s) and side wall(s) have been conducted by Xu and Ni [4]. The numerical results show that a slot in a Hartmann wall can effectively balance the pressure difference between the two sides of a flow channel insert and can greatly reduce the MHD pressure drop. The typical arrangement of the FCI in a poloidal duct is shown in Fig. 2. The FCI is loosely fitted into the poloidal channels. Close to the duct walls, there are narrow gaps filled with liquid metal. These gaps can avoid strong mechanical interaction between the flow channel insert and the wall. To minimize the primary stresses on the FCI, pressure equilibrium holes (PEHs) or pressure equilibrium slot (PES) is opened in the wall of FCI, connecting gap and bulk flow to equalize the pressure in these two fluid domains. Both the PEHs and the PES have the same total area in the paper.
1 ∂u + u · ∇ u = (−∇ p + ∇ · (∇ u) + (J × B)) ∂t
(1)
∇ ·u = 0
(2)
where u, p are the non-dimensional velocity vector and kinetic pressure, respectively. J represents the current density, and B is the applied magnetic field. Re = u0 a/ is the Reynolds number; Ha = aB0 / is the Hartmann number; N = Ha2 /Re is the interaction parameter; ϕ is the electrical potential. Also we define the magnetic Reynolds number here as Rem = au0 . Here is the permeability of the fluid and the walls. With the inlet average velocity as a characteristic velocity, the Hartmann number is 2270 and Reynolds number is 5738 in our simulation. At a low magnetic Reynolds number, the flow of electrically conducting fluid under an external magnetic field with negligible induced field, the current density can be calculated through Ohm’s law [15,16]: J = (−∇ ϕ + u × B)
(3)
The charge is conservative, such that:
∇ ·J = 0
(4)
From Eqs. (3) and (4), we can get the electrical potential Poisson equation as:
∇ · ( ∇ ϕ) = ∇ · (u × B)
(5)
The non-conservative form of the Lorentz force ( J × B) has been reformulated into a divergence form [7] with distance vector r introduced as: J × B = −∇ · (J(B × r)) + ∇ · (JB) × r
(6)
The Lorentz force in Eq. (1) can globally conservative the momentum when the applied magnetic field is constant. The divergence form of the Lorentz force of Eq. (6) has been employed for development of a consistent and conservative scheme on an arbitrary collocated mesh [7], on a rectangular collocated mesh [17] and on a staggered grid system [18], which can conserve the charge and the momentum and can be employed for accurate simulation of MHD flows with high Ha. A uniform velocity is given as an inlet boundary condition, and the pressure at outlet is specified as zero. On the solid walls, nonslip boundary conditions are applied for velocity with Neumann conditions set up for pressure on the inner walls. All of the walls are meshed with the electrical potential Poisson equation and Ohm’s equation solved and Neumann condition specified for electrical potential at the out walls. 3. Results and analysis
Fig. 2. The sketch of the blanket channel with FCI.
Fig. 3 displays the non-dimensional pressure distribution corresponding to the central position in a cross section along the flow direction, from which we can see the non-dimensional pressure drop in FCI with holes in a side wall is the biggest. This is because the appearance of the holes locally causes disturbance of the flow as well as the discontinuity of the conductivity. The MHD pressure drop using FCI with PES is less than the drop with PEHs as illustrated in Fig. 3(b). As illustrated in Fig. 4, the pressure distributions along the flow direction at the center and in the slot of the same cross-section
S.-J. Xu et al. / Fusion Engineering and Design 88 (2013) 271–275
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Fig. 3. Non-dimensional pressure distribution along the flow direction (at the center of FCI channel).
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are very different. The x-component of pressure gradient is nonuniformly distributed in a cross-section. There are two flow regions from the pressure distributions shown in Fig. 4(c) and (d) for FCI with PES. The pressure gradient is a function of (x, y, z) near the inlet, which gradually changes at the flow direction. Downstream around 4–5 times the width of the flow channel, the x-component of pressure gradient is just a function of (y, z), which keeps constant along the flow direction. However, the 2D fully developed assumption is not the real case of MHD flows in FCI. We can see from Fig. 4 that the flows of liquid metal in a duct with FCI are typical 3D flows with the x-component of pressure gradient nonuniformly distributed in a cross-section as firstly illustrated in Ref. [3]. To understand the MHD effect on the pressure drop and velocity distribution of liquid metal fusion blanket with FCI, one must conduct fully three-dimensional analysis. It is interesting to notice that there is a violent pressure fluctuation at the top gap of the duct with PEHs in a Hartmann wall in Fig. 4(b). This violent fluctuation of pressure may cause deformation and relative motion of FCI. Pressure contours at y–z cross-section are shown in Fig. 5, respectively. As shown in Fig. 5(a), PEH in a side wall can balance the pressure well between two sides of FCI, and only a small
0.4
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(d) PES in Hartmann wall Fig. 4. Non-dimensional pressure distribution along the flow direction (at the center of FCI).
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Velocity contours at y–z cross-section of FCIs in a duct near the exit with PES or PEHs opened in a Hartmann wall or a side wall are shown in Fig. 6, respectively. With PES in a side wall, the strong component of current density Jz formed in PES introduce a strong Lorentz force and reversed velocities in the PES. As seen in Fig. 6(c), velocity near inner wall of FCI is larger than that in the core flow; velocity along the side wall without PES is slightly higher than that along the side wall with PES, which is caused by imperfect insulation. With PES in a Hartmann wall, there is no reverse velocity in PES, while a weak velocity jump forms around PES as seen in Fig. 6(d). With holes in a side wall or a Hartmann wall, neither reversed flow nor increased velocity along the side wall occurs. 4. Conclusions
Fig. 5. Pressure contours at the cross section perpendicular to flow direction (X = 500).
pressure jump occurs, while the average pressure and the MHD pressure drop increase. Big pressure jump between two sides of FCI is shown in Fig. 5(b) for PEHs in a Hartmann wall, Fig. 5(c) for PES in a side wall and for PES in a Hartmann wall, which may be one of the reasons to cause instability. With PES in a side wall, pressure decreases linearly outside FCI along z direction and increases linearly inside FCI along z direction. PEH and PES opened in a side wall can effectively balance the pressure difference between inside and outside of FCI, with PEH in a side wall more effectively. With PES in a Hartmann wall, pressure has a minimum value at PES, and pressure has large difference between inside and outside of FCI. As shown in Fig. 5(d), although pressure between two sides of FCI still has big difference, MHD pressure drop is greatly reduced comparing with Fig. 5(c).
Direct simulation of 3D MHD flows of liquid metal in a duct with flow channel insert, which is relevant to R&D of liquid metal fusion blanket, is conducted by using a consistent and conservative scheme for calculation of the Lorentz force. The numerical results show that: (1) The flows of liquid metal in a duct with FCI are typically 3D flows with the x-component of pressure gradient nonuniformly distributed in a cross-section; (2) PEHs and PES opened in a side wall can effectively balance the pressure difference between inside and outside of FCI, with PEHs in a side wall more effectively; (3) PES in a Hartmann wall can greatly reduce the MHD pressure drop comparing with PES opened in a side wall, but the pressure between inside and outside of FCI near the side wall is quite different; (4) Strong reverse flow appears in PES in a side wall, which may induce unsteady flows of MHD at a strong applied magnetic field; (5) With PES in a Hartmann wall, there is no reverse velocity in PES, while a weak velocity jump formed around PES; (6) Pressure fluctuation occurs inside FCI with PEHs in a Hartmann wall rather than in a side wall. Making the average pressure increase, PEH in side wall can balance the pressure at two sides of FCI wall, and neither pressure jump nor reversed flow occurs. And PEH in side wall is regarded as a better choice than any other cases in the design of the DCLL blanket. Although GaInSn considered as the liquid metal of flows in a duct, the MHD effects on pressure drop and velocity profile are analyzed based on dimensionless parameters such as Hartmann number and Reynolds number in this paper. It is believed that the results are useful in R&D of liquid metal fusion blanket. Based on the preliminary analysis, we are now conducting fully 3D numerical study of thermal-fluid-solid of FCI with PbLi as liquid metal and reaction heat included in a fusion blanket duct. Acknowledgements The supports from the NSFC under Grants 50936006 and 11125212 and from National Magnetic Confinement Fusion Science Program in China with Grant 2009GB10401 are acknowledged. References
Fig. 6. Velocity contours at the cross section perpendicular to flow direction (X = 500).
[1] S. Malang, E. Bojarsky, L. Bühler, H. Deckers, U. Fisher, P. Norajira, H. Reiser, Dual coolant liquid metal breeder blanker, in: Proceedings of the 17th Symposium on Fusion Technology, Rome, Italy, September 14–18, 1992, 1992, pp. 1424–1428. [2] M.A. Abdou, N.B. Morley, A.Y. Ying, S. Smolentsev, P. Calderoni, Overview of fusion blanket R&D in the US over the last decade, Nuclear Engineering and Technology 37 (2005) 401–422. [3] M.-J. Ni, S.-J. Xu, Z.-H. WANG, N.-M. Zhang, Direct simulation of three dimensional MHD flows in liquid metal blanket with flow channel insert, Fusion Science and Technology 60 (2011) 292–297. [4] S.-J. Xu, M.-J. Ni, Direct simulation of MHD flows in dual-coolant liquid metal fusion blanket using a consistent and conservative scheme, Theoretical and Applied Mechanics Letters 1 (2011) 012006.
S.-J. Xu et al. / Fusion Engineering and Design 88 (2013) 271–275 [5] S. Smolentsev, M.A. Abdou, N.B. Morley, M. Sawan, S. Malang, C. Wong, Numerical analysis of MHD flow and heat transfer in a poloidal channel of the DCLL blanket with a SiC flow channel insert, Fus. Eng. Des. 81 (2006) 549–553. [6] D.J. Sutevski, S. Smolentsev, N.B. Morley, M.A. Abdou, 3D numerical study of MHD flow in a rectangular duct with a flow channel insert, Fusion Science and Technology 60 (2011) 513–517. [7] M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic reynolds number. Part II: on an arbitrary collocated mesh, Journal of Comparative Physiology 227 (2007) 205–228. [8] L. Bühler, Magnetohydrodynamics pressure-driven flows in the HCLL blanket, Fusion Engineering and Design 75–79 (2005) 923–926. [9] L. Bühler, P. Norajitra, Magnetohydrodynamic flow in the dual coolant blanket, FZKA 6802, 2003. [10] N. Morley, M.-J. Ni, R. Munipalli, P. Huang, M. Abdou, MHD simulations of liquid metal flow through a toroidally oriented manifold, Fusion Engineering and Design 83 (2008) 1335–C1339. [11] S. Smolentsev, N.B. Morley, M.A. Abdou, Magnetohydrodynamic and thermal issues of the SiCf /SiC flow channel insert, Fusion Science and Technology 50 (2006) 107–119.
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[12] E. Mas de les Valls, E. Batet, V. de Medina, J. Fradera, L.A. Sedano, Qualification of MHD effects in dual-coolant DEMO blanket and approaches to their modeling, Fusion Engineering and Design 86 (2011) 2326–2329. [13] C. Mistrangelo, MHD and corrosion analysis of dual coolant PbLi blanket modules for ARIES-CS, FZKA 7311, 2007. [14] J. Sommeria, R. Moreau, Why, how and when MHD turbulence becomes twodimensional, Journal of Fluid Mechanics 118 (1982) 507. [15] R.J. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, P.O.Box 17, 3300 AA Dordrecht, The Netherlands, 1990. [16] U. Müller, L. Bühler, Magnetofluiddynamics in Channels and Containers, Springer-Verlag, Berlin, Heidelberg, Germany, 2001. [17] M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic reynolds number. Part I: on a rectangular collocated grid system, Journal of Comparative Physiology 227 (2007) 174. [18] M.-J. Ni, J.-F. Li, A consistent and conservative scheme for incompressible MHD flows at a low magnetic reynolds number. Part III: on a staggered mesh, Journal of Comparative Physiology 231 (2011) 281.
Contents lists available at SciVerse ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Influence of flow channel insert with pressure equalization opening on MHD flows in a rectangular duct Shi-Jing Xu, Nian-Mei Zhang, Ming-Jiu Ni ∗ School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
h i g h l i g h t s We conduct 3D simulation of MHD flows in flow channel insert (FCI) with holes or slots. Pressure drop and velocity distribution are shown to illustrate the holes and plots effects on MHD flows of FCI. Pressure fluctuation for MHD flows in FCI with holes may introduce MHD instability.
a r t i c l e
i n f o
Article history: Received 16 September 2011 Received in revised form 26 January 2013 Accepted 6 February 2013 Available online 12 April 2013 Keywords: Flow channel insert Pressure equalization slot Pressure equalization hole Liquid metal blanket
a b s t r a c t Direct simulation of 3D MHD flows in a duct with flow channel insert (FCI) relevant to R&D of fusion blanket has been conducted based on an electrical potential formula by using a consistent and conservative scheme. Comparison study of the pressure and velocity distributions of liquid metal in a poloidal duct with FCI, which has pressure equalization slot (PES) and pressure equalization holes (PEHs) with the same total area at the corresponding walls, is conducted. Both the PES and the PEHs have two kinds of locations, either in a Hartmann wall or in a side wall. 3D pressure and velocity distributions of the different cases have been given. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The dual coolant lead lithium (DCLL) blanket concept proposes the use of a flow channel insert (FCI) [1] made of SiCf /SiC composite in the breeder region flow channels in order to: (1) thermally insulate the self-cooled breeder region from the helium cooled ferritic steel walls; (2) electrically insulate the PbLi flow from current closure paths in the ferritic steel walls; (3) provide a nearly stagnant PbLi layer near the ferritic steel wall that has increased corrosion temperature limits [2]. It has been verified [3] that the liquid metal fluid in rectangular duct relevant to fusion blanket using FCI made of conductive or insulated walls with/without slot is a fully 3D flow, which can be divided into two stages of developing flows with changing pressure gradient and with constant pressure gradient in the flow direction. The first stage of developing length is around 4–5 times of the half channel width. Insulated FCI in the blanket can effectively reduce MHD pressure drop, while highly conductive FCI cannot effectively reduce the MHD pressure drop [3,4].
∗ Corresponding author. Tel.: +86 01088256072. E-mail addresses: [email protected], [email protected] (M.-J. Ni). 0920-3796/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fusengdes.2013.02.015
Velocity profiles of MHD flow in the front poloidal channel of the DCLL blanket were studied by Smolentsev et al. [5] based on a 2D model for a fully developed flow. It’s verified that MHD pressure drop reduction at = 500 (m)−1 is a factor of 10 and at = 5 (m)−1 is a factor of 200–400, compared to the case without insulation. Sutevski et al. [6] obtained a 3D result using an unstructured, parallel MHD solver HIMAG [7]. These results confirm a substantial reduction in MHD pressure drop by the FCI. Bühler [8] and Bühler and Norajitra [9] analyze the pressure drop in HCLL blanket and DCLL blanket, in which it is indicated that the major MHD pressure drop arises in the poloidal manifold connecting breeder units. An experimental GaInSn flows in a toroidally oriented manifold is numerically conducted by Morley et al. [10], which illustrated 3D effects of MHD flows. Smolentsev et al. studied the MHD and thermal issues for the SiCf /SiC FCI. Their mathematic model includes the two dimensional momentum and induction equations for a fully developed flow and the three dimensional energy equations [11]. The qualification of MHD effects in the DRM DEMO blanket has been assessed in Ref. [12]. The flow in a toroidal plane of the DRM blanket is studied by solving the 2D flow model from Sommeria and Moreau [14] coupled sequentially with temperature equation. Mistrangelo [13] has analyzed the influence of various parameters such as electrical conductivity of the SiC insert
272
S.-J. Xu et al. / Fusion Engineering and Design 88 (2013) 271–275
2. Solver development for MHD flows on a collocated mesh The flow of electrically conducting fluid under the influence of an external magnetic field is governed by the following NavierStokes equation and continuity equation, which represent the conservation of momentum and mass:
Fig. 1. Cross-section of a poloidal duct with silicon carbide (SiC) flow channel insert (FCI).
and the orientation of the magnetic fields based on the finite volume method. In this paper, a direct simulation of MHD flow in a duct relevant to liquid metal fusion blanket is conducted to study the distribution of pressure and velocity influenced by different FCI structures. Considering laminar flows of liquid metal through a rectangular channel in a transverse strong uniform magnetic field, the sketch of a cross-section of the blanket channel with width 2a = 68 mm and 2b = 60 mm, length l = 1000 mm is shown in Fig. 1. Liquid metal flows along x-direction. The magnetic field strength is B0 = 1.852T and the inlet velocity of liquid metal is u0 = 0.0675 m/s. The liquid metal is GaInSn with density = 6363 kg/m3 , electrical conductivity = 3.3074 × 106 (m)−1 , viscosity = 2.545 × 10−3 kg/(ms). The thickness of the reduced activation ferritic wall is 2 mm with conductivity of w = 1.23 × 106 (m)−1 . The velocity and pressure distributions of FCI with slot or slots in Hartmann wall(s) and side wall(s) have been conducted by Xu and Ni [4]. The numerical results show that a slot in a Hartmann wall can effectively balance the pressure difference between the two sides of a flow channel insert and can greatly reduce the MHD pressure drop. The typical arrangement of the FCI in a poloidal duct is shown in Fig. 2. The FCI is loosely fitted into the poloidal channels. Close to the duct walls, there are narrow gaps filled with liquid metal. These gaps can avoid strong mechanical interaction between the flow channel insert and the wall. To minimize the primary stresses on the FCI, pressure equilibrium holes (PEHs) or pressure equilibrium slot (PES) is opened in the wall of FCI, connecting gap and bulk flow to equalize the pressure in these two fluid domains. Both the PEHs and the PES have the same total area in the paper.
1 ∂u + u · ∇ u = (−∇ p + ∇ · (∇ u) + (J × B)) ∂t
(1)
∇ ·u = 0
(2)
where u, p are the non-dimensional velocity vector and kinetic pressure, respectively. J represents the current density, and B is the applied magnetic field. Re = u0 a/ is the Reynolds number; Ha = aB0 / is the Hartmann number; N = Ha2 /Re is the interaction parameter; ϕ is the electrical potential. Also we define the magnetic Reynolds number here as Rem = au0 . Here is the permeability of the fluid and the walls. With the inlet average velocity as a characteristic velocity, the Hartmann number is 2270 and Reynolds number is 5738 in our simulation. At a low magnetic Reynolds number, the flow of electrically conducting fluid under an external magnetic field with negligible induced field, the current density can be calculated through Ohm’s law [15,16]: J = (−∇ ϕ + u × B)
(3)
The charge is conservative, such that:
∇ ·J = 0
(4)
From Eqs. (3) and (4), we can get the electrical potential Poisson equation as:
∇ · ( ∇ ϕ) = ∇ · (u × B)
(5)
The non-conservative form of the Lorentz force ( J × B) has been reformulated into a divergence form [7] with distance vector r introduced as: J × B = −∇ · (J(B × r)) + ∇ · (JB) × r
(6)
The Lorentz force in Eq. (1) can globally conservative the momentum when the applied magnetic field is constant. The divergence form of the Lorentz force of Eq. (6) has been employed for development of a consistent and conservative scheme on an arbitrary collocated mesh [7], on a rectangular collocated mesh [17] and on a staggered grid system [18], which can conserve the charge and the momentum and can be employed for accurate simulation of MHD flows with high Ha. A uniform velocity is given as an inlet boundary condition, and the pressure at outlet is specified as zero. On the solid walls, nonslip boundary conditions are applied for velocity with Neumann conditions set up for pressure on the inner walls. All of the walls are meshed with the electrical potential Poisson equation and Ohm’s equation solved and Neumann condition specified for electrical potential at the out walls. 3. Results and analysis
Fig. 2. The sketch of the blanket channel with FCI.
Fig. 3 displays the non-dimensional pressure distribution corresponding to the central position in a cross section along the flow direction, from which we can see the non-dimensional pressure drop in FCI with holes in a side wall is the biggest. This is because the appearance of the holes locally causes disturbance of the flow as well as the discontinuity of the conductivity. The MHD pressure drop using FCI with PES is less than the drop with PEHs as illustrated in Fig. 3(b). As illustrated in Fig. 4, the pressure distributions along the flow direction at the center and in the slot of the same cross-section
S.-J. Xu et al. / Fusion Engineering and Design 88 (2013) 271–275
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Fig. 3. Non-dimensional pressure distribution along the flow direction (at the center of FCI channel).
0 0
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(c) PES in side wall cent re cent re cent re cent re cent re
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of of of of of
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bulk flow top ga p left gap right gap bottom ga p
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P=p/σuaB
are very different. The x-component of pressure gradient is nonuniformly distributed in a cross-section. There are two flow regions from the pressure distributions shown in Fig. 4(c) and (d) for FCI with PES. The pressure gradient is a function of (x, y, z) near the inlet, which gradually changes at the flow direction. Downstream around 4–5 times the width of the flow channel, the x-component of pressure gradient is just a function of (y, z), which keeps constant along the flow direction. However, the 2D fully developed assumption is not the real case of MHD flows in FCI. We can see from Fig. 4 that the flows of liquid metal in a duct with FCI are typical 3D flows with the x-component of pressure gradient nonuniformly distributed in a cross-section as firstly illustrated in Ref. [3]. To understand the MHD effect on the pressure drop and velocity distribution of liquid metal fusion blanket with FCI, one must conduct fully three-dimensional analysis. It is interesting to notice that there is a violent pressure fluctuation at the top gap of the duct with PEHs in a Hartmann wall in Fig. 4(b). This violent fluctuation of pressure may cause deformation and relative motion of FCI. Pressure contours at y–z cross-section are shown in Fig. 5, respectively. As shown in Fig. 5(a), PEH in a side wall can balance the pressure well between two sides of FCI, and only a small
0.4
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(d) PES in Hartmann wall Fig. 4. Non-dimensional pressure distribution along the flow direction (at the center of FCI).
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Velocity contours at y–z cross-section of FCIs in a duct near the exit with PES or PEHs opened in a Hartmann wall or a side wall are shown in Fig. 6, respectively. With PES in a side wall, the strong component of current density Jz formed in PES introduce a strong Lorentz force and reversed velocities in the PES. As seen in Fig. 6(c), velocity near inner wall of FCI is larger than that in the core flow; velocity along the side wall without PES is slightly higher than that along the side wall with PES, which is caused by imperfect insulation. With PES in a Hartmann wall, there is no reverse velocity in PES, while a weak velocity jump forms around PES as seen in Fig. 6(d). With holes in a side wall or a Hartmann wall, neither reversed flow nor increased velocity along the side wall occurs. 4. Conclusions
Fig. 5. Pressure contours at the cross section perpendicular to flow direction (X = 500).
pressure jump occurs, while the average pressure and the MHD pressure drop increase. Big pressure jump between two sides of FCI is shown in Fig. 5(b) for PEHs in a Hartmann wall, Fig. 5(c) for PES in a side wall and for PES in a Hartmann wall, which may be one of the reasons to cause instability. With PES in a side wall, pressure decreases linearly outside FCI along z direction and increases linearly inside FCI along z direction. PEH and PES opened in a side wall can effectively balance the pressure difference between inside and outside of FCI, with PEH in a side wall more effectively. With PES in a Hartmann wall, pressure has a minimum value at PES, and pressure has large difference between inside and outside of FCI. As shown in Fig. 5(d), although pressure between two sides of FCI still has big difference, MHD pressure drop is greatly reduced comparing with Fig. 5(c).
Direct simulation of 3D MHD flows of liquid metal in a duct with flow channel insert, which is relevant to R&D of liquid metal fusion blanket, is conducted by using a consistent and conservative scheme for calculation of the Lorentz force. The numerical results show that: (1) The flows of liquid metal in a duct with FCI are typically 3D flows with the x-component of pressure gradient nonuniformly distributed in a cross-section; (2) PEHs and PES opened in a side wall can effectively balance the pressure difference between inside and outside of FCI, with PEHs in a side wall more effectively; (3) PES in a Hartmann wall can greatly reduce the MHD pressure drop comparing with PES opened in a side wall, but the pressure between inside and outside of FCI near the side wall is quite different; (4) Strong reverse flow appears in PES in a side wall, which may induce unsteady flows of MHD at a strong applied magnetic field; (5) With PES in a Hartmann wall, there is no reverse velocity in PES, while a weak velocity jump formed around PES; (6) Pressure fluctuation occurs inside FCI with PEHs in a Hartmann wall rather than in a side wall. Making the average pressure increase, PEH in side wall can balance the pressure at two sides of FCI wall, and neither pressure jump nor reversed flow occurs. And PEH in side wall is regarded as a better choice than any other cases in the design of the DCLL blanket. Although GaInSn considered as the liquid metal of flows in a duct, the MHD effects on pressure drop and velocity profile are analyzed based on dimensionless parameters such as Hartmann number and Reynolds number in this paper. It is believed that the results are useful in R&D of liquid metal fusion blanket. Based on the preliminary analysis, we are now conducting fully 3D numerical study of thermal-fluid-solid of FCI with PbLi as liquid metal and reaction heat included in a fusion blanket duct. Acknowledgements The supports from the NSFC under Grants 50936006 and 11125212 and from National Magnetic Confinement Fusion Science Program in China with Grant 2009GB10401 are acknowledged. References
Fig. 6. Velocity contours at the cross section perpendicular to flow direction (X = 500).
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