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Magnetohydrodynamic flows in u-shaped ducts under a uniform transverse magnetic field
Fusion Engineering and Design 121 (2017) 87–99
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Research Paper
Magnetohydrodynamic flows in u-shaped ducts under a uniform transverse magnetic field Shangjing Yang a , Chang Nyung Kim b,∗ a b
Department of Mechanical Engineering, Graduate School, Kyung Hee University, Yongin-si, Gyeonggi-do, 446-701, Republic of Korea Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin-si, Gyeonggi-do, 446-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 20 October 2016 Received in revised form 2 May 2017 Accepted 5 June 2017 Keywords: LMMHD U-shaped duct Pressure drop Conductance ratio Numerical simulation
a b s t r a c t In this study, the three-dimensional liquid metal (LM) magnetohydrodynamic (MHD) flows in a u-shaped duct under a uniform magnetic field applied perpendicular to the flow plane are numerically analyzed with the use of commercial software CFX. The u-shaped duct system is made up of an inflow channal, an outflow channal and a connecting channel located between the inflow channel and outflow channel. In the current study, the effects of the length of the connecting channel and the conductance ratio on the flow characteristics are investigated. Higher velocities are observed in the side layers of the inflow, outflow and connecting channels, forming “M-shaped” velocity profiles. In addition, the velocity recirculations are created in the inner regions of the two right-angle segments just after turning due to the inertial force therein, yielding complicated distributions of the current and electric potential, and found are the diverging and converging of the velocity component parallel to the magnetic field when fluids are turning. The results show that the pressure drop is in close relationship with the conductance ratio and the length of the connecting channel. The characteristics of the fluid velocity, current, electric potential and pressure gradient of LM MHD flows in a u-shaped duct are examined in detail. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Liquid metal (LM) magnetohydrodynamic (MHD) flow has been playing an increasingly important role because of wide applications in engineering field. There has been a great interest in the study of MHD effects, especially in the design of fusion reactor blanket whose liquid metal can serve both as breeder material and as coolant [1,2]. Moving liquid metal under an applied magnetic field gives a rise to a variety of magnetohydrodynamic phenomena, having a severe impact on velocity distribution, pressure drop and pumping power for the cooling system [3]. The LM MHD effects have a great significance on the optimal design of fusion reactor blanket. For the reasons presented above, the features of LM MHD flows should be examined in detail. So far, there have been many experimental studies investigating the effects of LM MHD flows [4–8]. An experimental work on liquid metal flows in a sudden expansion under a strong magnetic field was performed by Bühler et al. [4]. An investigation on MHD and heat transfer characteristics of liquid metal flows in fusion blankets was carried out experimentally by Kirillov et al. [5]. Magnetohy-
∗ Corresponding author. E-mail address: [email protected] (C.N. Kim). http://dx.doi.org/10.1016/j.fusengdes.2017.06.007 0920-3796/© 2017 Elsevier B.V. All rights reserved.
drodynamic flows through a right-angle bend of rectangular duct at high value of the Hartmann number were presented by Stieglitz et al. [7]. Moreover, mathematical methods [9–12] have already been adopted to analyze the characteristics of MHD flows. A study of liquid metal flows in a manifold with a uniform magnetic field was shown by Moon et al. [11]. The steady and fully developed LM MHD flows in a constant cross-section rectangular duct with a uniform, transverse magnetic field were investigated mathematically in laminar as well as in turbulent conditions by Cuevas et al. [12]. Mathematical methods and experimental studies have several weaknesses, in the sense that the mathematical methods may not analyze the features of three-dimensional LM MHD flows well, and the experimental approaches, mostly with a large-scale equipment, have some limitations including a difficulty in the accurate measurement of side layer velocity profile of the liquid, and in the experimental identification of stagnant and recirculation zone. With the popularity of computer and the improvement of computer performance, it is greatly important to adopt the numerical simulation to analyze the MHD flows characteristics. Therefore, a large quantity of numerical codes have been built by different researchers [13–17]. A code named MTC-H 1.0 based on the structured grid has been used to simulate three-dimensional MHD flows in rectangular ducts by Zhou et al. [15]. Feng et al. [16] conducted the validations of the OpenFOAM code for unstructured grid,
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Table 1 Properties of the liquid metal [25].
Liquid metal (Pb-17Li)
Table 2 Cases with different lengths of the connecting channel and conductance ratios.
Density (kg/m3 )
Kinematic viscosity (m2 /s)
Electric conductivity (S/m)
9500
0.188 × 106
7.7 × 105
and the performance of the OpenFOAM code was presented to be good. Swain et al. [17] carried out a numerical analysis of threedimensional LM MHD flows by using FLUENT code, and showed the code can predict MHD flows reasonably well in a single channel with multiple right-angle bends. Three-dimensional numerical studies for LM MHD flows with the use of commercial code ANSYS CFX have been reported [18–21]. Kim et al. [19] showed the code validation for CFX, and presented that CFX has a good performance in predicting pressure drop. Kim [20] carried out the numerical analysis of LM MHD flows in a suddenly-contracting rectangular duct with two different arrangements of the Hartmann number and with a fixed Reynolds number. Mistrangelo et al. [21] performed a numerical investigation on the three-dimensional LM MHD flows in rectangular sudden expansions with walls of finite electric conductivity. The numerical tool (CFX code) used in this study gives accurate results for high Hartmann numbers (M = 1000). Though various experimental, mathematical and numerical studies have been performed in the analysis of LM MHD flows in many different geometries and earlier studies of LM MHD flows in a u-shaped duct have been performed [22–24], the detailed characteristics of LM MHD flows in a u-shaped duct such as (a) the diverging and converging of the velocity component in the direction parallel to the magnetic field; (b) the variation of pressure gradient; (c) distorted M-shaped distribution observed in different places with different conductance ratios and different lengths of the connecting channel have rarely been analyzed before. By using CFX code, this numerical study examines the three-dimensional LM MHD flows in a u-shaped duct under a uniform magnetic field applied perpendicular to the flow plane. And, cases with different conductance ratios and different lengths of the connecting channel located between the inflow channel and outflow channel are investigated. The interdependency of the fluid velocity, current, electric potential and pressure gradient of LM MHD flows in a u-shaped duct is examined in detail to better describe the electromagnetic features of LM MHD flows. Since the minimization of the pressure drop, together with proper heat extraction by the coolant, is required in the blanket, the flow design is of particular significance. MHD flows in a ushaped duct can be important as the liquid metal flowing in this duct can reach a place very near the first wall. After full understanding on the MHD flows in a u-shaped duct, detailed analysis on the MHD flows in a u-shaped multi-channel can be performed in a near-future study. 2. Problem formulation and solution method 2.1. Geometry, magnetic field, and materials In this study, three-dimensional liquid metal magnetohydrodynamic flows in a u-shaped duct are examined. The geometry of the duct under consideration is depicted in Fig. 1. As shown, the duct consists of an inflow channel, an outflow channel and a connecting channel which is located between the inflow channel and outflow channel. The properties of the liquid metal Pb-17Li [25] are given in Table 1. Cases with different arrangements of the conductance ratio and the length of the connecting channel are presented in Table 2.
subcase
d (mm)
Electric conductivity of wall (S/m)
C
Case 1
Case 1-1 Case 1-2 Case 1-3 Case 1-4
260
5.55 × 106 3.70 × 106 1.85 × 106 0.90 × 106
0.3 0.2 0.1 0.05
Case 2
Case 2-1 Case 2-2 Case 2-3 Case 2-4
100
5.55 × 106 3.70 × 106 1.85 × 106 0.90 × 106
0.3 0.2 0.1 0.05
Case 3
Case 3-1 Case 3-2 Case 3-3 Case 3-4
5
5.55 × 106 3.70 × 106 1.85 × 106 0.90 × 106
0.3 0.2 0.1 0.05
The Hartmann number, Reynolds number and conductance ratio are defined as follows: Hartmann number M = B0 L
f /
Reynolds number Re = u0 L/
(1) (2)
ConductanceratioC = (s · t) / f · Lf ,
(3)
where B0 is the magnetic field intensity, L is the characteristic length of the flow (the half length of a side of the flow channel cross-section in the direction of the applied magnetic field), Lf is the side length of the flow channel cross-section, t is the thickness of the duct wall, f is the electric conductivity of the fluid, s is the electric conductivity of the solid, is the dynamic viscosity of the fluid, is the density of the fluid, u0 is the characteristic velocity. In the current study, a uniform magnetic field strength of 0.8027T is applied in the z-direction with the Hartamnn number is 1000. The Reynolds number is 3191, which yields stable flows in the side layers [20]. 2.2. Governing equations A steady state, constant property, incompressible and laminar flows under uniform magnetic field are adopted. The system of governing equations can be given as follows: →
Conservation of mass∇ · u = 0 →
(4)
→
2→
→
→
Equation of motion u · ∇ u = −∇ p + ∇ u + J × B →
Conservation of the charge∇ · J = 0 →
→
(6) →
Ohm’slaw in fluid J = f (−∇ + u × B ) →
Ohm’slaw in solidwall J = −s ∇ ,
(5)
(7) (8)
→→
, p, J , B and are the velocity vector, pressure, current denwhereu sity vector, magnetic field intensity vector and electric potential, respectively. The substitution of Eqs. (7) and (8) into Eq. (6) gives the electrical potential equations in fluid (see Eq. (9)) and in solid wall (see Eq. (10)), respectively, as →
→
∇2 = ∇ · ( u × B ) ∇ =0 2
(9) (10)
It indicates that the equation of mass conservation and motion can be solved together with electrical potential equation for the variables of the velocity, pressure, and electrical potential in the fluid region. In the solid region, Eq. (10) can be solved with the zero velocity.
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Fig. 1. Geometry of the duct (unit: mm).
2.3. Boundary conditions In this study, no-slip condition is applied at the interface of the fluid and solid. At the inlet a uniform velocity u = 0.01 m/s is given. At the outlet, the value of pressure is given to be zero, and there is no change in the axial velocity along the main flow direction. It is considered that the whole system, including fluid
region and solid region, is electrically insulated from the outside. 2.4. Numerical method In the iteration procedure under-relaxation is adopted for the coupled governing equations. The second-order upwind scheme
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Fig. 3. Flow situation for the validation.
2.5. Validation of MHD code For the validation of the current numerical simulation with the use of commercial code CFX, a numerical study [28] can be referred to. Also, Problem A in Ref. [29] (fully developed laminar steady MHD flow) is analyzed for the verification and validation of CFX code. Here, two cases are considered: (a) non-conducting rectangular duct, and (b) two conducting walls perpendicular to the magnetic field and two non-conducting walls parallel to the magnetic field. In both the cases the Hartmann number is 500, and the rectangular duct with uniform transverse magnetic field is shown in Fig. 3. The
1
non-dimensional volume flow rate Q˜ =
˜ yd˜z obtained in Ref. Ud˜ −1
[29] and in this study are compared in Table 3, demonstrating that the numerical results are in a good agreement (the error in each case is around 3%) and the current numerical method is valid. 3. Results and discussion
Fig. 2. Axial velocity distributions in the inflow channel with different mesh systems.
is utilized to discretize the convective terms and the central difference scheme for diffusion terms. For the solution method for discretized equation, multigrid-accelerated incomplete lower upper factorization [26] is employed. Rhie-Chow interpolation method [27] gives good results for the pressure-velocity coupling in the current problems. The current study employs structured grid systems with 2,826,432 grids, 2,657,160 grids and 2,359,844 grids for Case 1, Case 2 and Case 3, respectively, chosen after grid independence tests. Three grid systems are tried for Case 1, whose grid numbers are 1,034,280 (Mesh 1), 2,826,432 (Mesh 2) and 3,926,784 (Mesh 3), respectively. At x = 0.2 m in the mid x-y plane and in the mid x-z plane of inflow channel, the axial velocities for different mesh systems are shown in Fig. 2(a) and (b), respectively. It is obvious that Mesh 1 (one cell in Hartmann layer and fifteen cells in side layer) cannot capture the peak velocity, with a lower velocity in the core region. Now, the velocities are almost the same for Mesh 2 (two cells in Hartmann layer and twenty cells in side layer) and Mesh 3 (three cells in Hartmann layer and twenty-five cells in side layer), implying that there is no meaningful difference between Mesh 2 and Mesh 3. Therefore, Mesh 2 is selected for faster calculations. Generally speaking, finer grids are used in the fluid near the solid walls and the two right-angle segments.
As described in the above, the present study investigates the LM MHD flows in a u-shaped duct with three different lengths of the connecting channel, and each case has four subcases with different conductance ratios (shown in Table 2). The results of the analyses for different cases are to be present in detail. The comparison of the pressure drops in different cases is to be discussed. 3.1. Case 1-1 →
The general equation for the current flow, Ohm’s law J = →
→
→
→
→
(−∇ + u × B ), can be expressed by J = JEF + JEM . That is to say, →
the first term JEF = − ∇ means the current induced by the gradient of the electric potential, and it can be referred to as Electric-Field Component of the Current (EFCC in abbreviation). The second term →
→
→
JEM = ( u × B ) denotes the current induced by the fluid motion in a magnetic field, and it can be called Electro-Motive Component of the Current (EMCC in abbreviation). It should be noted that the current in the fluid region is expressed by the vector summation of EFCC and EMCC, while that in the solid region is expressed only by the EFCC. Fig. 4 presents the detailed distributions of the velocity in the mid x-y plane at z = 0.065 m. As shown in Fig. 4(a), the fluid flows in the positive x-direction in the inflow channel, then changes the direction to the positive y-direction in the first right-angle segment (the first turning), and finally changes the direction again to the negative x-direction in the second right-angle segment (the second turning). It is worth noting that when the fluid changes the flow direction, counter-clcokwise velocity recirculations caused by the inertial force therein are observed in the inner regions of the right-
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Table 3 ˜ in Ref. [29] and in this study. Non-dimensional volume flow rate Q Ha Non-conducting duct Conducting duct
500
Cw (Hartmann wall) 0 0.01
Cw (Side wall) 0 0
˜ in Ref. [29] Analytical Q −3
7.680 × 10 1.405 × 10−3
˜ in this study Numerical Q 7.920 × 10−3 1.456 × 10−3
Fig. 4. Plane velocities in the mid x-y plane at z = 0.065 m in Case 1-1.
angle segments. A crescent of high velocity can be seen in each inner region of the right-angle segment, which is shown in Fig. 4(b). The velocities in the side layers of the inflow, outflow and connecting channels are higher, forming “M-shaped” velocity profiles. Fig. 5 displays the distributions of the z-directional velocity component in two different x-y planes. Fig. 5(a) shows the zdirectional velocity component in the x-y plane at z = 0.0715 m located above the mid x-y plane (at z = 0.065 m), named “the upper plane”. Fig. 5(b) shows the z-directional velocity component in the x-y plane at z = 0.0585 m located below the mid x-y plane (at
z = 0.065 m), called “the lower plane”. It can be seen from the subfigure that the upper and lower plane are located by the same distance from the mid plane. As shown in Fig. 5(a), a crescent of the positive z-directional velocity component is seen in the region of the velocity recirculation, and a water-drop shape of the negative z-directional velocity component is observed in the inner region of the velocity recirculation, surround by the crescent of the positive z-directional velocity component. The characteristics of the z-direction component of velocity in the lower plane (see Fig. 5(b)) are opposite to those in the upper plane (see Fig. 5(a)).
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Fig. 5. Z-direction component of velocities in two different x-y planes in Case 1-1.
Fig. 6 presents the distributions of different velocity components in three planes which are close to the outer solid-fluid interface. The plane located at y = 0.006 m in the x-z plane can be named “Plane 1”, the plane positioned at x = 0.524 m in the y-z plane can be called “Plane 2” and the plane at y = 0.514 m in the x-z plane can be entitled “Plane 3”. Fig. 6(a) shows the z-directional velocity component in the three planes. Eight oval-shaped regions of the z-directional velocity component are formed before and after the turning, which are pairwise symmetric (in absolute value) with respect to the mid x-y plane. The phenomenon can be explained as follows. As shown in Fig. 6(b), before the first turning, the velocity diverges in the outer side layer of the inflow channel (in Plane 1), forming positive and negative z-directional velocity in the upper and lower part, respectively, resulting in a smaller axial velocity in the region around the corner of the first right-angle segment (see Fig. 6(c)). After the first turning, near the first right-angle segment in the outer side layer of the connecting channel (in Plane 2), the velocity converges to the central flow region until the axial velocity returns to a higher value (see Fig. 6(c)). Also, in the second right-angel segment, the velocity in this part undertakes the same flow pattern. In other words, before the second turning near the second right-angle segment, the velocity diverges in the outer side layer of the connecting channel (in Plane 2), and after the second turning, the velocity converges to the
Fig. 6. Different velocity components in x-z planes and in y-z plane in Case 1-1.
central flow region. Therefore, the axial velocity is reduced when the fluid approaches the corner of the second right-angle segment, then after the second turning the axial velocity in the outer side layer of the outflow channel increases (in Plane 3) (see Fig. 6(c)). The maximum value of the z-directional velocity component reaches 20% of the inlet velocity. However, in the inner side layers, the velocity component parallel to the magnetic field converges before the first turning, and the velocity component diverges after the first turning. Also, the velocity component converges before the second turning and the velocity component diverges after the second turning. In Fig. 6(c), the x-directional velocity components in Plane 1 and Plane 3, and the y-directional velocity component in Plane 2 are displayed. The axial velocity is higher in the center region of Plane 1, Plane 2 and Plane 3, while the lower axial velocity is observed in fluid region near the corner of the two right-angle segments.
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Fig. 8. Schematic of the plane current in the mid x-y plane at z = 0.065 m in Case 1-1.
Fig. 7. Current density and electric potential in the mid x-y plane at z = 0.065 m in Case 1-1.
(also see Fig. 8(a)). Furthermore, counter-clcokwise velocity recirculations are formed near the edge of right-angle segments after turning (see Fig. 4), where the EMCC moving outwards from the center of the recirculation is generated (in a direction perpendicular to the directions of the axial velocity and the magnetic field, see the →
The distributions of the current density and electric potential in the mid x-y plane at z = 0.065 m are depicted in Fig. 7. In Fig. 7(a), the current flows to the outer wall in the whole fluid region. More specifically, in the inflow channel the current flows in the downward direction, while in the outflow channel, the current heads in the upward direction. In the connecting channel, the current moves to the right. The current in the two right-angel segments is headed outward from the edge of the inner region in a radial direction. The current density in fluid region near the edge is rather high. It →
is related with the fact that the EMCC (denoted by JEM ) is much →
superior to the EFCC (expressed by JEF ) (see Fig. 8(a)), with higher turning velocity in fluid region near the edge (see Fig. 4(b)). Meanwhile, the lowest current density is observed near the corner of the two right-angle segments, with the fact that the density of EMCC is just a little bit higher than that of EFCC (see Fig. 8(a)) with lower turning velocity in fluid region near the corner (see Fig. 4(b)). In Fig. 7(b), the electric potential is in close relationship with the direction of the EMCC, so higher electric potential is observed near the outer solid-fluid interface of the duct, and lower electric potential is induced near the inner solid-fluid interface of the duct
→
→
equation JEM = ( u × B )), which can be seen in Fig. 8(b). Therefore, the lowest electric potential is observed in the fluid recirculation regions. Fig. 9 presents the distributions of the current density, electric potential and axial velocity in the y-z plane at x = 0.4 m (at the end of the inflow channel and outflow channel). In Fig. 9(a), the current coming from the top and bottom walls converges to the center of the solid region of the connecting channel. In Fig. 9(b), the highest electric potential is seen in the outer walls of the inflow and outflow channels. In Fig. 9(c), in the inflow channel higher axial velocity is observed in the side layers, however, the velocity in the inner side layer is larger than that in the outer side layer. In the outflow channel, observed is a region of lower axial velocity in the recirculation region near the inner wall, which is located between the inner wall and the crescent of highest axial velocity (also see Fig. 4). 3.2. The comparison of the cases Fig. 10 depicts the distributions of different velocity components in the x-z planes at y = 0.124 m, y = 0.396 m and in the y-z plane at x = 0.406 m which are close to the inner solid-fluid interface in
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Fig. 9. Features in the y-z plane at x = 0.4 m in Case 1-1.
Case 1-1. In Fig. 10(a), the behavior of the z-direction component of velocity in the inner side layer is opposite to that in the outer side layer (see Fig. 6(a)). Before the first turning, the velocity converges to the central flow region. After the first turning, the velocity diverges in the inner side layer of the connecting channel. And, in the second turning segment, the velocity undertakes the same flow pattern. Due to the velocity converging and diverging, the highest axial velocity is observed in the central region of the two edges, while the lowest velocity is seen in the region of the curved axis of recirulation, which can be described in Fig. 10(b). Fig. 11 shows the distributions of different velocity components in the x-z planes at y = 0.124 m, y = 0.236 m and in the y-z plane
at x = 0.406 m in Case 2-1. The feature of the z-directional velocity component and the axial velocity are similar to that in Case 1-1 (see Fig. 10). Fig. 12 shows the distributions of different velocity components in the x-z planes at y = 0.124 m, y = 0.141 m and in the y-z plane at x = 0.406 m in Case 3-1. In fluid region near the two edges the velocity converges until the second turning and diverges after the second turning. And more notably, in the connecting channel only the velocity converging is observed. In Case 1-1, Case 2-1 and Case 3-1, the regions of the z-directional velocity component are pairwise symmetric (in absolute value) with respect to the mid x-y plane.
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Fig. 10. Different velocity components in the x-z planes at y = 0.124 m, y = 0.396 m and in the y-z plane at x = 0.406 m in Case 1-1.
The maximum magnitudes of the z-directional velocity component in the three planes close to the inner solid-fluid interface in Case 1-1, Case 2-1 and Case 3-1 correspond to 35.89%, 38.57% and 68.82% of the inlet velocity, respectively. The value increases with a decrease of the size of the connecting channel under the condition of the fixed conductance ratio. It is notable that the maximum magnitude of the z-directional velocity component in Case 3-1 is quite larger than that in Case 1-1 and Case 2-1. Fig. 13 presents the distributions of the axial velocity (absolute value) along the mid line (at z = 0.065 m) in the mid x-y plane at different cross-sections in different cases. Fig. 13(a)–(c) show the distributions of the axial velocity along Line A, Line B and Line C
Fig. 11. Different velocity components in the x-z planes at y = 0.124 m, y = 0.236 m and in the y-z plane at x = 0.406 m in Case 2-1.
in Case 1, respectively, with the same length of the connecting channel, but with different conductance ratios. And, Fig. 13(d)–(f) show the distributions of the axial velocity along Line A, Line B and Line C in Case 1-1, Case 2-1 and Case 3-1, respectively, with the same conductance ratio, but with different lengths of the connecting channel. Generally, higher axial velocity can be observed in
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position x = 0.1 m in the outflow channel. In the connecting channel, the axial velocity in the inner side layer is higher than that in the outer side layer. More specifically, in the inner side layer of the connecting channel, the velocity in Case 3-1 is much higher than that in Case 1-1 and in Case 2-1. However, the velocity in Case 3-1 is lower in the outer side layer of the connecting channel. Fig. 14 displays the distribution of the current density and electric potential in the mid x-y plane at z = 0.065 m in Case 2-1 and Case 3-1. Here, in the inflow channel the current heads to the downward direction, while in the outflow channel the direction of the current is upward. In the connecting channel the current flows to the right. In the two right-angle segments the current moves outward in a radial direction from the edge of the inner region. The lowest current density is seen near the corner of the two right-angle segments, while the highest current density can be observed near the edge of the inner region of the turning segment. And, the current density near the edge of the inner region of the turning segment in Case 3-1 is much higher than that in Case 1-1 (see Fig. 7(a)) and in Case 2-1,
Fig. 12. Different velocity components in the x-z planes at y = 0.124 m, y = 0.141 m and in the y-z plane at x = 0.406 m in Case 3-1.
the side layers, forming “M-shaped” velocity profiles. Near the end of the inflow and outflow channels, the axial velocity in the inner side layer is obviously higher than that in the outer side layer. In the inflow channel, the asymmetrical “M-shaped” velocity profile starts to appear at the position x = 0.25 m, while that vanishes at the
Fig. 13. Axial velocity distributions along different lines at z = 0.065 m.
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Fig. 13. (Continued)
which is related with higher turning velocity in fluid region near the edge. Fig. 15 shows the distributions of the pressure along the imaginary lines (marked in the sub-figure), where the value of pressure is given to be zero at the outlet. A new coordinate is established so that the center of the imaginary line in the connecting channel is regarded as the original point (S = 0). Fig. 15(a)–(c) depict the distri-
Fig. 14. Current density and electric potential in the mid x-y plane at z = 0.065 m in Case 2-1 and Case 3-1.
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S. Yang, C.N. Kim / Fusion Engineering and Design 121 (2017) 87–99 Table 4 Pressure drop along the imaginary line in cases with different lengths of the connecting channel and conductance ratios. Subcase
Total pressure drop (Pa)
Non-dimensional pressure gradients
Case 1
Case 1-1 Case 1-2 Case 1-3 Case 1-4
1816.91 1363.60 794.70 436.24
0.375 0.286 0.167 0.089
Case 2
Case 2-1 Case 2-2 Case 2-3 Case 2-4
1574.62 1182.05 689.40 378.78
0.375 0.286 0.167 0.089
Case 3
Case 3-1 Case 3-2 Case 3-3 Case 3-4
1408.44 1057.19 616.90 339.36
0.375 0.286 0.167 0.089
tance ratio (C = 0.3) is given in Fig. 15(d) (here, P* = P–PS(0) ). In the inflow and outflow channel (except for fluid region near the turning segment), the pressure gradient are the same. Table 4 shows the pressure along the imaginary lines in cases with different lengths of the connecting channel and conductance ratios. As the conductance ratio decreases (with a fixed length of the connecting channel), the pressure drop decreases. And, in the case of a fixed conductance ratio, the pressure drop decreases with the decrease of the length of the connecting channel. The average non-dimensional pressure gradients (∇ p/(f uB2 )) in the fully developed fluid region (0. 112 m ≤ x ≤ 0. 287 m) of the inflow and outflow channels in Case 1-1 (Case 2-1 and Case 3-1), Case 1-2 (Case 2-2 and Case 3-2), Case 1-3 (Case 2-3 and Case 3-3) and Case 1-4 (Case 2-4 and Case 3-4) are 0.375, 0.286, 0.167 and 0.089, respectively.
4. Conclusion
Fig. 15. Pressure gradients along the imaginary lines in different cases.
butions of the pressure in Case 1, Case 2 and Case 3, respectively. As shown, the pressure decreases almost linearly along the main flow, except for the region of right-angle segments. The pressure gradient (absolute value) is higher for larger conductance ratio. The comparison of the pressure between Case 1-1 (d = 260 mm), Case 2-1 (d = 100 mm) and Case 3-1 (d = 5 mm) with the same conduc-
This numerical study simulates the three-dimensional liquid metal (LM) magneto-hydrodynamic (MHD) flows in a u-shaped duct under a uniform magnetic field applied perpendicular to the flow plane. Commercial software ANSYS CFX is used for the numerical simulation. Considered are cases with different lengths of the connecting channel and different conductance ratios. The features of the velocity, current, electric potential and pressure gradient are analyzed in detail. The results show that higher velocities can be found in the side layers of the inflow, outflow and connecting channels, forming “Mshaped” velocity profiles. And, the velocity recirculation caused by the inertial force is observed in each inner region of the rightangle segment after turning, forming the lowest elecreic potential therein. The axial velocities are higher in the fluid region near the edges because the velocity component in the direction parallel with the magnetic field vector are converged, which is related with higher density of current therein flowing outward, while the axial velocities are lower in the corner of the two right-angle segments since the velocity component in the direction parallel with the magnetic field vector are diverged, which is linked with lower density of current therein flowing outward. The converging and diverging of the velocity component have an effect on strengthening and weakening the peak axial velocity. As expected, the pressure gradient goes down with a decrease in the conductance ratio. For a fixed conductance ratio, the pressure gradients are the same in the fully developed fluid region. Because lower pressure drop and proper heat extraction are required in the duct flow of liquid metal, the flow design is of particular importance. In this sense, the results of the current study can be utilized
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meaningfully in the flow design of the coolant system of a fusion reactor blanket. In summary, this study discusses the electromagnetic characteristics of LM MHD flows in a u-shaped duct in cases with different lengths of the connecting channel and different conductance ratios. The interdependency of the fluid velocity, current, electric potential and pressure gradient is analyzed in detail. Acknowledgements This research was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology & Ministry of Knowledge Economy (2015M1A7A1A02050613). References [1] C. Mistrangelo, L. Bühler, Numerical investigation of liquid metal flows in a rectangular sudden expansions, Fusion Eng. Des. 82 (2007) 2176–2182. [2] L. Barleon, V. Casal, L. Lenhart, MHD flow in liquid-metal-cooled blankets, Fusion Eng. Des. 14 (1991) 401–412. [3] N.B. Morley, S. Malang, I. Kirillov, Thermofluid magnetohydrodynamic issues for liquid breeders, Fusion Sci. Technol. 47 (3) (2005) 488–501. [4] L. Bühler, S. Horanyi, E. Arbogast, Experimental investigation of liquid-metal flows through a sudden expansion at fusion-relevant Hartmann numbers, Fusion Eng. Des. 82 (2007) 2239–2245. [5] I.R. Kirillov, C.B. Reed, L. Barleon, K. Miyazaki, Present understanding of MHD and heat transfer phenomena for liquid meral blankets, Fusion Eng. Des. 27 (1995) 553–569. [6] N.B. Morley, S. Smolentsev, L. Barleon, I.R. Kirillov, M. Takahashi, Liquid magnetohydrodynamics −recent progress and future directions for fusion, Fusion Eng. Des. 51-52 (2000) 701–713. [7] R. Stieglitz, L. Barleon, L. Bühler, S. Molokov, Magnetohydrodynamic flow in a right-angle bend in a strong magnetic field, J. Fluid Mech. 326 (1996) 91–123. [8] S. Horanyi, L. Bühler, E. Arbogast, Experiments on magnetohydrodynamic flows in a sudden expansion of rectangular ducts at high hartmann number, in: Joint 15th Riga and 6th PAMIR Conference on Fundamental and Applied MHD, Rigas Jurmala, Latvia, 2005. [9] J.C.R. Hunt, Magnetohydrodynamic flow in rectangular ducts, J. Fluid Mech. 21 (4) (1965) 57–590. [10] J.C.R. Hunt, J.A. Shercliff, Magnetohydrodynamics at high Hartmann number, Ann. Rev. Fluid Mech. 3 (1971) 37–62. [11] T.J. Moon, T.Q. Hua, J.S. Walker, B.F. Picologlou, Liquid metal flow in a simple manifold with a strong transverse magnetic field, Appl. Sci. Res. 49 (1) (1992) 49–65. [12] S. Cuevas, B.E. Picologlou, J.S. Walker, G. Talmage, Liquid-metal MHD flow in rectangular ducts with thin conducting or insulating walls: laminar and turbulent solutions, Int. J. Eng. Sci. 35 (5) (1997) 485–503.
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[13] J. Mao, H. Pan, Three-dimensional numerical simulation for magnetohydrodynamic duct flows in a staggered grid system, Fusion Eng. Des. 88 (3) (2013) 145–150. [14] 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 II: on an arbitrary collocated mesh, J. Comput. Phys. 227 (1) (2007) 205–228. [15] T. Zhou, Z. Yang, M. Ni, H. Chen, Code development and validation for analyzing liquid metal MHD flow in rectangular ducts, Fusion Eng. Des. 85 (2010) 1736–1741. [16] J.C. Feng, H.L. Chen, Q.Y. He, M.Y. Ye, Further validation of liquid metal MHD code for unstructured grid based on OpenFOAM, Fusion Eng. Des. 100 (2015) 260–264. [17] P.K. Swain, P. Satyamurthy, R. Bhattacharyay, A. Patel, A. Shishko, E. Platacis, A. Ziks, S. Ivanov, A.V. Despande, 3D MHD lead–lithium liquid metal flow analysis and experiments in a Test-Section of multiple rectangular bends at moderate to high Hartmann numbers, Fusion Eng. Des. 88 (2013) 2848–2859. [18] I.D. Piazza, L. Bühler, Numerical Simulations of Buoyant Magnetohydrodynamic Flows Using the CFX Code, Forschungszentrum Karlsruhe GmbH, Karlsruhe, 1999. [19] S.H. Kim, M.H. Kim, D.W. Lee, C. Choi, Code validation and development for MHD analysis of liquid metal flow in Korean TBM, Fusion Eng. Des. 87 (2012) 951–955. [20] C.N. Kim, A liquid metal magnetohydrodynamic duct flow with sudden contraction in a direction perpendicular to a magnetic field, Comput. Fluids 108 (2015) 156–167. [21] C. Mistrangelo, L. Bühler, Numerical investigation of liquid metal flows in rectangular sudden expansions, Fusion Eng. Des. 82 (2007) 2176–2182. [22] P.K. Swain, P. Satyamurthy, R. Bhattacharyay, A. Patel, A. Shishko, E. Platacis, A. Ziks, S. Ivanov, A.V. Despande, 3D MHD lead–lithium liquid metal flow analysis and experiments in a Test-Section of multiple rectangular bends at moderate to high Hartmann numbers, Fusion Eng. Des. 88 (2013) 2848–2859. [23] Q.Y. He, J.C. Feng, H.L. Chen, Numerical analysis and optimization of 3D magnetohydrodynamicflows in rectangular U-bend, Fusion Eng. Des. 109–111 (2016) 1313–1317. [24] L. Chen, M.J. Li, M.J. Ni, N.M. Zhang, MHD effects in the U-bend of DCLL blanket, Procedia Eng. 126 (2015) 461–465. [25] S. Malang, R. Mattas, Comparison of lithium and the eutectic candidate liquid metal breeder materials lead-lithium alloy, two for self-cooled blankets, Fusion Eng. Des. 27 (1995) 399–406. [26] M. Raw, Robustness of coupled algebraic multi-grid for the Navier–Stokes equations, in: AAIA 34th Aerospace Sciences Meeting and Exhibit, Reno, USA, 1996, pp. 96–0297. [27] W.Z. Shen, J.A. Michelsen, J.N. Sorensen, Improved Rhie-Chow interpolation for unsteady flow computations, AIAA J. 39 (12) (2001) 2406–2409. [28] C.N. Kim, Magnetohydrodynamic flows entering the region of a flow channel insert in a duct, Fusion Eng. Des. 89 (1) (2014) 56–68. [29] S. Smolentsev, et al., An approach to verification and validation of MHD codes for fusion applications, Fusion Eng. Des. 100 (2015) 65–72.
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Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Research Paper
Magnetohydrodynamic flows in u-shaped ducts under a uniform transverse magnetic field Shangjing Yang a , Chang Nyung Kim b,∗ a b
Department of Mechanical Engineering, Graduate School, Kyung Hee University, Yongin-si, Gyeonggi-do, 446-701, Republic of Korea Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin-si, Gyeonggi-do, 446-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 20 October 2016 Received in revised form 2 May 2017 Accepted 5 June 2017 Keywords: LMMHD U-shaped duct Pressure drop Conductance ratio Numerical simulation
a b s t r a c t In this study, the three-dimensional liquid metal (LM) magnetohydrodynamic (MHD) flows in a u-shaped duct under a uniform magnetic field applied perpendicular to the flow plane are numerically analyzed with the use of commercial software CFX. The u-shaped duct system is made up of an inflow channal, an outflow channal and a connecting channel located between the inflow channel and outflow channel. In the current study, the effects of the length of the connecting channel and the conductance ratio on the flow characteristics are investigated. Higher velocities are observed in the side layers of the inflow, outflow and connecting channels, forming “M-shaped” velocity profiles. In addition, the velocity recirculations are created in the inner regions of the two right-angle segments just after turning due to the inertial force therein, yielding complicated distributions of the current and electric potential, and found are the diverging and converging of the velocity component parallel to the magnetic field when fluids are turning. The results show that the pressure drop is in close relationship with the conductance ratio and the length of the connecting channel. The characteristics of the fluid velocity, current, electric potential and pressure gradient of LM MHD flows in a u-shaped duct are examined in detail. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Liquid metal (LM) magnetohydrodynamic (MHD) flow has been playing an increasingly important role because of wide applications in engineering field. There has been a great interest in the study of MHD effects, especially in the design of fusion reactor blanket whose liquid metal can serve both as breeder material and as coolant [1,2]. Moving liquid metal under an applied magnetic field gives a rise to a variety of magnetohydrodynamic phenomena, having a severe impact on velocity distribution, pressure drop and pumping power for the cooling system [3]. The LM MHD effects have a great significance on the optimal design of fusion reactor blanket. For the reasons presented above, the features of LM MHD flows should be examined in detail. So far, there have been many experimental studies investigating the effects of LM MHD flows [4–8]. An experimental work on liquid metal flows in a sudden expansion under a strong magnetic field was performed by Bühler et al. [4]. An investigation on MHD and heat transfer characteristics of liquid metal flows in fusion blankets was carried out experimentally by Kirillov et al. [5]. Magnetohy-
∗ Corresponding author. E-mail address: [email protected] (C.N. Kim). http://dx.doi.org/10.1016/j.fusengdes.2017.06.007 0920-3796/© 2017 Elsevier B.V. All rights reserved.
drodynamic flows through a right-angle bend of rectangular duct at high value of the Hartmann number were presented by Stieglitz et al. [7]. Moreover, mathematical methods [9–12] have already been adopted to analyze the characteristics of MHD flows. A study of liquid metal flows in a manifold with a uniform magnetic field was shown by Moon et al. [11]. The steady and fully developed LM MHD flows in a constant cross-section rectangular duct with a uniform, transverse magnetic field were investigated mathematically in laminar as well as in turbulent conditions by Cuevas et al. [12]. Mathematical methods and experimental studies have several weaknesses, in the sense that the mathematical methods may not analyze the features of three-dimensional LM MHD flows well, and the experimental approaches, mostly with a large-scale equipment, have some limitations including a difficulty in the accurate measurement of side layer velocity profile of the liquid, and in the experimental identification of stagnant and recirculation zone. With the popularity of computer and the improvement of computer performance, it is greatly important to adopt the numerical simulation to analyze the MHD flows characteristics. Therefore, a large quantity of numerical codes have been built by different researchers [13–17]. A code named MTC-H 1.0 based on the structured grid has been used to simulate three-dimensional MHD flows in rectangular ducts by Zhou et al. [15]. Feng et al. [16] conducted the validations of the OpenFOAM code for unstructured grid,
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Table 1 Properties of the liquid metal [25].
Liquid metal (Pb-17Li)
Table 2 Cases with different lengths of the connecting channel and conductance ratios.
Density (kg/m3 )
Kinematic viscosity (m2 /s)
Electric conductivity (S/m)
9500
0.188 × 106
7.7 × 105
and the performance of the OpenFOAM code was presented to be good. Swain et al. [17] carried out a numerical analysis of threedimensional LM MHD flows by using FLUENT code, and showed the code can predict MHD flows reasonably well in a single channel with multiple right-angle bends. Three-dimensional numerical studies for LM MHD flows with the use of commercial code ANSYS CFX have been reported [18–21]. Kim et al. [19] showed the code validation for CFX, and presented that CFX has a good performance in predicting pressure drop. Kim [20] carried out the numerical analysis of LM MHD flows in a suddenly-contracting rectangular duct with two different arrangements of the Hartmann number and with a fixed Reynolds number. Mistrangelo et al. [21] performed a numerical investigation on the three-dimensional LM MHD flows in rectangular sudden expansions with walls of finite electric conductivity. The numerical tool (CFX code) used in this study gives accurate results for high Hartmann numbers (M = 1000). Though various experimental, mathematical and numerical studies have been performed in the analysis of LM MHD flows in many different geometries and earlier studies of LM MHD flows in a u-shaped duct have been performed [22–24], the detailed characteristics of LM MHD flows in a u-shaped duct such as (a) the diverging and converging of the velocity component in the direction parallel to the magnetic field; (b) the variation of pressure gradient; (c) distorted M-shaped distribution observed in different places with different conductance ratios and different lengths of the connecting channel have rarely been analyzed before. By using CFX code, this numerical study examines the three-dimensional LM MHD flows in a u-shaped duct under a uniform magnetic field applied perpendicular to the flow plane. And, cases with different conductance ratios and different lengths of the connecting channel located between the inflow channel and outflow channel are investigated. The interdependency of the fluid velocity, current, electric potential and pressure gradient of LM MHD flows in a u-shaped duct is examined in detail to better describe the electromagnetic features of LM MHD flows. Since the minimization of the pressure drop, together with proper heat extraction by the coolant, is required in the blanket, the flow design is of particular significance. MHD flows in a ushaped duct can be important as the liquid metal flowing in this duct can reach a place very near the first wall. After full understanding on the MHD flows in a u-shaped duct, detailed analysis on the MHD flows in a u-shaped multi-channel can be performed in a near-future study. 2. Problem formulation and solution method 2.1. Geometry, magnetic field, and materials In this study, three-dimensional liquid metal magnetohydrodynamic flows in a u-shaped duct are examined. The geometry of the duct under consideration is depicted in Fig. 1. As shown, the duct consists of an inflow channel, an outflow channel and a connecting channel which is located between the inflow channel and outflow channel. The properties of the liquid metal Pb-17Li [25] are given in Table 1. Cases with different arrangements of the conductance ratio and the length of the connecting channel are presented in Table 2.
subcase
d (mm)
Electric conductivity of wall (S/m)
C
Case 1
Case 1-1 Case 1-2 Case 1-3 Case 1-4
260
5.55 × 106 3.70 × 106 1.85 × 106 0.90 × 106
0.3 0.2 0.1 0.05
Case 2
Case 2-1 Case 2-2 Case 2-3 Case 2-4
100
5.55 × 106 3.70 × 106 1.85 × 106 0.90 × 106
0.3 0.2 0.1 0.05
Case 3
Case 3-1 Case 3-2 Case 3-3 Case 3-4
5
5.55 × 106 3.70 × 106 1.85 × 106 0.90 × 106
0.3 0.2 0.1 0.05
The Hartmann number, Reynolds number and conductance ratio are defined as follows: Hartmann number M = B0 L
f /
Reynolds number Re = u0 L/
(1) (2)
ConductanceratioC = (s · t) / f · Lf ,
(3)
where B0 is the magnetic field intensity, L is the characteristic length of the flow (the half length of a side of the flow channel cross-section in the direction of the applied magnetic field), Lf is the side length of the flow channel cross-section, t is the thickness of the duct wall, f is the electric conductivity of the fluid, s is the electric conductivity of the solid, is the dynamic viscosity of the fluid, is the density of the fluid, u0 is the characteristic velocity. In the current study, a uniform magnetic field strength of 0.8027T is applied in the z-direction with the Hartamnn number is 1000. The Reynolds number is 3191, which yields stable flows in the side layers [20]. 2.2. Governing equations A steady state, constant property, incompressible and laminar flows under uniform magnetic field are adopted. The system of governing equations can be given as follows: →
Conservation of mass∇ · u = 0 →
(4)
→
2→
→
→
Equation of motion u · ∇ u = −∇ p + ∇ u + J × B →
Conservation of the charge∇ · J = 0 →
→
(6) →
Ohm’slaw in fluid J = f (−∇ + u × B ) →
Ohm’slaw in solidwall J = −s ∇ ,
(5)
(7) (8)
→→
, p, J , B and are the velocity vector, pressure, current denwhereu sity vector, magnetic field intensity vector and electric potential, respectively. The substitution of Eqs. (7) and (8) into Eq. (6) gives the electrical potential equations in fluid (see Eq. (9)) and in solid wall (see Eq. (10)), respectively, as →
→
∇2 = ∇ · ( u × B ) ∇ =0 2
(9) (10)
It indicates that the equation of mass conservation and motion can be solved together with electrical potential equation for the variables of the velocity, pressure, and electrical potential in the fluid region. In the solid region, Eq. (10) can be solved with the zero velocity.
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Fig. 1. Geometry of the duct (unit: mm).
2.3. Boundary conditions In this study, no-slip condition is applied at the interface of the fluid and solid. At the inlet a uniform velocity u = 0.01 m/s is given. At the outlet, the value of pressure is given to be zero, and there is no change in the axial velocity along the main flow direction. It is considered that the whole system, including fluid
region and solid region, is electrically insulated from the outside. 2.4. Numerical method In the iteration procedure under-relaxation is adopted for the coupled governing equations. The second-order upwind scheme
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Fig. 3. Flow situation for the validation.
2.5. Validation of MHD code For the validation of the current numerical simulation with the use of commercial code CFX, a numerical study [28] can be referred to. Also, Problem A in Ref. [29] (fully developed laminar steady MHD flow) is analyzed for the verification and validation of CFX code. Here, two cases are considered: (a) non-conducting rectangular duct, and (b) two conducting walls perpendicular to the magnetic field and two non-conducting walls parallel to the magnetic field. In both the cases the Hartmann number is 500, and the rectangular duct with uniform transverse magnetic field is shown in Fig. 3. The
1
non-dimensional volume flow rate Q˜ =
˜ yd˜z obtained in Ref. Ud˜ −1
[29] and in this study are compared in Table 3, demonstrating that the numerical results are in a good agreement (the error in each case is around 3%) and the current numerical method is valid. 3. Results and discussion
Fig. 2. Axial velocity distributions in the inflow channel with different mesh systems.
is utilized to discretize the convective terms and the central difference scheme for diffusion terms. For the solution method for discretized equation, multigrid-accelerated incomplete lower upper factorization [26] is employed. Rhie-Chow interpolation method [27] gives good results for the pressure-velocity coupling in the current problems. The current study employs structured grid systems with 2,826,432 grids, 2,657,160 grids and 2,359,844 grids for Case 1, Case 2 and Case 3, respectively, chosen after grid independence tests. Three grid systems are tried for Case 1, whose grid numbers are 1,034,280 (Mesh 1), 2,826,432 (Mesh 2) and 3,926,784 (Mesh 3), respectively. At x = 0.2 m in the mid x-y plane and in the mid x-z plane of inflow channel, the axial velocities for different mesh systems are shown in Fig. 2(a) and (b), respectively. It is obvious that Mesh 1 (one cell in Hartmann layer and fifteen cells in side layer) cannot capture the peak velocity, with a lower velocity in the core region. Now, the velocities are almost the same for Mesh 2 (two cells in Hartmann layer and twenty cells in side layer) and Mesh 3 (three cells in Hartmann layer and twenty-five cells in side layer), implying that there is no meaningful difference between Mesh 2 and Mesh 3. Therefore, Mesh 2 is selected for faster calculations. Generally speaking, finer grids are used in the fluid near the solid walls and the two right-angle segments.
As described in the above, the present study investigates the LM MHD flows in a u-shaped duct with three different lengths of the connecting channel, and each case has four subcases with different conductance ratios (shown in Table 2). The results of the analyses for different cases are to be present in detail. The comparison of the pressure drops in different cases is to be discussed. 3.1. Case 1-1 →
The general equation for the current flow, Ohm’s law J = →
→
→
→
→
(−∇ + u × B ), can be expressed by J = JEF + JEM . That is to say, →
the first term JEF = − ∇ means the current induced by the gradient of the electric potential, and it can be referred to as Electric-Field Component of the Current (EFCC in abbreviation). The second term →
→
→
JEM = ( u × B ) denotes the current induced by the fluid motion in a magnetic field, and it can be called Electro-Motive Component of the Current (EMCC in abbreviation). It should be noted that the current in the fluid region is expressed by the vector summation of EFCC and EMCC, while that in the solid region is expressed only by the EFCC. Fig. 4 presents the detailed distributions of the velocity in the mid x-y plane at z = 0.065 m. As shown in Fig. 4(a), the fluid flows in the positive x-direction in the inflow channel, then changes the direction to the positive y-direction in the first right-angle segment (the first turning), and finally changes the direction again to the negative x-direction in the second right-angle segment (the second turning). It is worth noting that when the fluid changes the flow direction, counter-clcokwise velocity recirculations caused by the inertial force therein are observed in the inner regions of the right-
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Table 3 ˜ in Ref. [29] and in this study. Non-dimensional volume flow rate Q Ha Non-conducting duct Conducting duct
500
Cw (Hartmann wall) 0 0.01
Cw (Side wall) 0 0
˜ in Ref. [29] Analytical Q −3
7.680 × 10 1.405 × 10−3
˜ in this study Numerical Q 7.920 × 10−3 1.456 × 10−3
Fig. 4. Plane velocities in the mid x-y plane at z = 0.065 m in Case 1-1.
angle segments. A crescent of high velocity can be seen in each inner region of the right-angle segment, which is shown in Fig. 4(b). The velocities in the side layers of the inflow, outflow and connecting channels are higher, forming “M-shaped” velocity profiles. Fig. 5 displays the distributions of the z-directional velocity component in two different x-y planes. Fig. 5(a) shows the zdirectional velocity component in the x-y plane at z = 0.0715 m located above the mid x-y plane (at z = 0.065 m), named “the upper plane”. Fig. 5(b) shows the z-directional velocity component in the x-y plane at z = 0.0585 m located below the mid x-y plane (at
z = 0.065 m), called “the lower plane”. It can be seen from the subfigure that the upper and lower plane are located by the same distance from the mid plane. As shown in Fig. 5(a), a crescent of the positive z-directional velocity component is seen in the region of the velocity recirculation, and a water-drop shape of the negative z-directional velocity component is observed in the inner region of the velocity recirculation, surround by the crescent of the positive z-directional velocity component. The characteristics of the z-direction component of velocity in the lower plane (see Fig. 5(b)) are opposite to those in the upper plane (see Fig. 5(a)).
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Fig. 5. Z-direction component of velocities in two different x-y planes in Case 1-1.
Fig. 6 presents the distributions of different velocity components in three planes which are close to the outer solid-fluid interface. The plane located at y = 0.006 m in the x-z plane can be named “Plane 1”, the plane positioned at x = 0.524 m in the y-z plane can be called “Plane 2” and the plane at y = 0.514 m in the x-z plane can be entitled “Plane 3”. Fig. 6(a) shows the z-directional velocity component in the three planes. Eight oval-shaped regions of the z-directional velocity component are formed before and after the turning, which are pairwise symmetric (in absolute value) with respect to the mid x-y plane. The phenomenon can be explained as follows. As shown in Fig. 6(b), before the first turning, the velocity diverges in the outer side layer of the inflow channel (in Plane 1), forming positive and negative z-directional velocity in the upper and lower part, respectively, resulting in a smaller axial velocity in the region around the corner of the first right-angle segment (see Fig. 6(c)). After the first turning, near the first right-angle segment in the outer side layer of the connecting channel (in Plane 2), the velocity converges to the central flow region until the axial velocity returns to a higher value (see Fig. 6(c)). Also, in the second right-angel segment, the velocity in this part undertakes the same flow pattern. In other words, before the second turning near the second right-angle segment, the velocity diverges in the outer side layer of the connecting channel (in Plane 2), and after the second turning, the velocity converges to the
Fig. 6. Different velocity components in x-z planes and in y-z plane in Case 1-1.
central flow region. Therefore, the axial velocity is reduced when the fluid approaches the corner of the second right-angle segment, then after the second turning the axial velocity in the outer side layer of the outflow channel increases (in Plane 3) (see Fig. 6(c)). The maximum value of the z-directional velocity component reaches 20% of the inlet velocity. However, in the inner side layers, the velocity component parallel to the magnetic field converges before the first turning, and the velocity component diverges after the first turning. Also, the velocity component converges before the second turning and the velocity component diverges after the second turning. In Fig. 6(c), the x-directional velocity components in Plane 1 and Plane 3, and the y-directional velocity component in Plane 2 are displayed. The axial velocity is higher in the center region of Plane 1, Plane 2 and Plane 3, while the lower axial velocity is observed in fluid region near the corner of the two right-angle segments.
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Fig. 8. Schematic of the plane current in the mid x-y plane at z = 0.065 m in Case 1-1.
Fig. 7. Current density and electric potential in the mid x-y plane at z = 0.065 m in Case 1-1.
(also see Fig. 8(a)). Furthermore, counter-clcokwise velocity recirculations are formed near the edge of right-angle segments after turning (see Fig. 4), where the EMCC moving outwards from the center of the recirculation is generated (in a direction perpendicular to the directions of the axial velocity and the magnetic field, see the →
The distributions of the current density and electric potential in the mid x-y plane at z = 0.065 m are depicted in Fig. 7. In Fig. 7(a), the current flows to the outer wall in the whole fluid region. More specifically, in the inflow channel the current flows in the downward direction, while in the outflow channel, the current heads in the upward direction. In the connecting channel, the current moves to the right. The current in the two right-angel segments is headed outward from the edge of the inner region in a radial direction. The current density in fluid region near the edge is rather high. It →
is related with the fact that the EMCC (denoted by JEM ) is much →
superior to the EFCC (expressed by JEF ) (see Fig. 8(a)), with higher turning velocity in fluid region near the edge (see Fig. 4(b)). Meanwhile, the lowest current density is observed near the corner of the two right-angle segments, with the fact that the density of EMCC is just a little bit higher than that of EFCC (see Fig. 8(a)) with lower turning velocity in fluid region near the corner (see Fig. 4(b)). In Fig. 7(b), the electric potential is in close relationship with the direction of the EMCC, so higher electric potential is observed near the outer solid-fluid interface of the duct, and lower electric potential is induced near the inner solid-fluid interface of the duct
→
→
equation JEM = ( u × B )), which can be seen in Fig. 8(b). Therefore, the lowest electric potential is observed in the fluid recirculation regions. Fig. 9 presents the distributions of the current density, electric potential and axial velocity in the y-z plane at x = 0.4 m (at the end of the inflow channel and outflow channel). In Fig. 9(a), the current coming from the top and bottom walls converges to the center of the solid region of the connecting channel. In Fig. 9(b), the highest electric potential is seen in the outer walls of the inflow and outflow channels. In Fig. 9(c), in the inflow channel higher axial velocity is observed in the side layers, however, the velocity in the inner side layer is larger than that in the outer side layer. In the outflow channel, observed is a region of lower axial velocity in the recirculation region near the inner wall, which is located between the inner wall and the crescent of highest axial velocity (also see Fig. 4). 3.2. The comparison of the cases Fig. 10 depicts the distributions of different velocity components in the x-z planes at y = 0.124 m, y = 0.396 m and in the y-z plane at x = 0.406 m which are close to the inner solid-fluid interface in
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Fig. 9. Features in the y-z plane at x = 0.4 m in Case 1-1.
Case 1-1. In Fig. 10(a), the behavior of the z-direction component of velocity in the inner side layer is opposite to that in the outer side layer (see Fig. 6(a)). Before the first turning, the velocity converges to the central flow region. After the first turning, the velocity diverges in the inner side layer of the connecting channel. And, in the second turning segment, the velocity undertakes the same flow pattern. Due to the velocity converging and diverging, the highest axial velocity is observed in the central region of the two edges, while the lowest velocity is seen in the region of the curved axis of recirulation, which can be described in Fig. 10(b). Fig. 11 shows the distributions of different velocity components in the x-z planes at y = 0.124 m, y = 0.236 m and in the y-z plane
at x = 0.406 m in Case 2-1. The feature of the z-directional velocity component and the axial velocity are similar to that in Case 1-1 (see Fig. 10). Fig. 12 shows the distributions of different velocity components in the x-z planes at y = 0.124 m, y = 0.141 m and in the y-z plane at x = 0.406 m in Case 3-1. In fluid region near the two edges the velocity converges until the second turning and diverges after the second turning. And more notably, in the connecting channel only the velocity converging is observed. In Case 1-1, Case 2-1 and Case 3-1, the regions of the z-directional velocity component are pairwise symmetric (in absolute value) with respect to the mid x-y plane.
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Fig. 10. Different velocity components in the x-z planes at y = 0.124 m, y = 0.396 m and in the y-z plane at x = 0.406 m in Case 1-1.
The maximum magnitudes of the z-directional velocity component in the three planes close to the inner solid-fluid interface in Case 1-1, Case 2-1 and Case 3-1 correspond to 35.89%, 38.57% and 68.82% of the inlet velocity, respectively. The value increases with a decrease of the size of the connecting channel under the condition of the fixed conductance ratio. It is notable that the maximum magnitude of the z-directional velocity component in Case 3-1 is quite larger than that in Case 1-1 and Case 2-1. Fig. 13 presents the distributions of the axial velocity (absolute value) along the mid line (at z = 0.065 m) in the mid x-y plane at different cross-sections in different cases. Fig. 13(a)–(c) show the distributions of the axial velocity along Line A, Line B and Line C
Fig. 11. Different velocity components in the x-z planes at y = 0.124 m, y = 0.236 m and in the y-z plane at x = 0.406 m in Case 2-1.
in Case 1, respectively, with the same length of the connecting channel, but with different conductance ratios. And, Fig. 13(d)–(f) show the distributions of the axial velocity along Line A, Line B and Line C in Case 1-1, Case 2-1 and Case 3-1, respectively, with the same conductance ratio, but with different lengths of the connecting channel. Generally, higher axial velocity can be observed in
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position x = 0.1 m in the outflow channel. In the connecting channel, the axial velocity in the inner side layer is higher than that in the outer side layer. More specifically, in the inner side layer of the connecting channel, the velocity in Case 3-1 is much higher than that in Case 1-1 and in Case 2-1. However, the velocity in Case 3-1 is lower in the outer side layer of the connecting channel. Fig. 14 displays the distribution of the current density and electric potential in the mid x-y plane at z = 0.065 m in Case 2-1 and Case 3-1. Here, in the inflow channel the current heads to the downward direction, while in the outflow channel the direction of the current is upward. In the connecting channel the current flows to the right. In the two right-angle segments the current moves outward in a radial direction from the edge of the inner region. The lowest current density is seen near the corner of the two right-angle segments, while the highest current density can be observed near the edge of the inner region of the turning segment. And, the current density near the edge of the inner region of the turning segment in Case 3-1 is much higher than that in Case 1-1 (see Fig. 7(a)) and in Case 2-1,
Fig. 12. Different velocity components in the x-z planes at y = 0.124 m, y = 0.141 m and in the y-z plane at x = 0.406 m in Case 3-1.
the side layers, forming “M-shaped” velocity profiles. Near the end of the inflow and outflow channels, the axial velocity in the inner side layer is obviously higher than that in the outer side layer. In the inflow channel, the asymmetrical “M-shaped” velocity profile starts to appear at the position x = 0.25 m, while that vanishes at the
Fig. 13. Axial velocity distributions along different lines at z = 0.065 m.
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Fig. 13. (Continued)
which is related with higher turning velocity in fluid region near the edge. Fig. 15 shows the distributions of the pressure along the imaginary lines (marked in the sub-figure), where the value of pressure is given to be zero at the outlet. A new coordinate is established so that the center of the imaginary line in the connecting channel is regarded as the original point (S = 0). Fig. 15(a)–(c) depict the distri-
Fig. 14. Current density and electric potential in the mid x-y plane at z = 0.065 m in Case 2-1 and Case 3-1.
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S. Yang, C.N. Kim / Fusion Engineering and Design 121 (2017) 87–99 Table 4 Pressure drop along the imaginary line in cases with different lengths of the connecting channel and conductance ratios. Subcase
Total pressure drop (Pa)
Non-dimensional pressure gradients
Case 1
Case 1-1 Case 1-2 Case 1-3 Case 1-4
1816.91 1363.60 794.70 436.24
0.375 0.286 0.167 0.089
Case 2
Case 2-1 Case 2-2 Case 2-3 Case 2-4
1574.62 1182.05 689.40 378.78
0.375 0.286 0.167 0.089
Case 3
Case 3-1 Case 3-2 Case 3-3 Case 3-4
1408.44 1057.19 616.90 339.36
0.375 0.286 0.167 0.089
tance ratio (C = 0.3) is given in Fig. 15(d) (here, P* = P–PS(0) ). In the inflow and outflow channel (except for fluid region near the turning segment), the pressure gradient are the same. Table 4 shows the pressure along the imaginary lines in cases with different lengths of the connecting channel and conductance ratios. As the conductance ratio decreases (with a fixed length of the connecting channel), the pressure drop decreases. And, in the case of a fixed conductance ratio, the pressure drop decreases with the decrease of the length of the connecting channel. The average non-dimensional pressure gradients (∇ p/(f uB2 )) in the fully developed fluid region (0. 112 m ≤ x ≤ 0. 287 m) of the inflow and outflow channels in Case 1-1 (Case 2-1 and Case 3-1), Case 1-2 (Case 2-2 and Case 3-2), Case 1-3 (Case 2-3 and Case 3-3) and Case 1-4 (Case 2-4 and Case 3-4) are 0.375, 0.286, 0.167 and 0.089, respectively.
4. Conclusion
Fig. 15. Pressure gradients along the imaginary lines in different cases.
butions of the pressure in Case 1, Case 2 and Case 3, respectively. As shown, the pressure decreases almost linearly along the main flow, except for the region of right-angle segments. The pressure gradient (absolute value) is higher for larger conductance ratio. The comparison of the pressure between Case 1-1 (d = 260 mm), Case 2-1 (d = 100 mm) and Case 3-1 (d = 5 mm) with the same conduc-
This numerical study simulates the three-dimensional liquid metal (LM) magneto-hydrodynamic (MHD) flows in a u-shaped duct under a uniform magnetic field applied perpendicular to the flow plane. Commercial software ANSYS CFX is used for the numerical simulation. Considered are cases with different lengths of the connecting channel and different conductance ratios. The features of the velocity, current, electric potential and pressure gradient are analyzed in detail. The results show that higher velocities can be found in the side layers of the inflow, outflow and connecting channels, forming “Mshaped” velocity profiles. And, the velocity recirculation caused by the inertial force is observed in each inner region of the rightangle segment after turning, forming the lowest elecreic potential therein. The axial velocities are higher in the fluid region near the edges because the velocity component in the direction parallel with the magnetic field vector are converged, which is related with higher density of current therein flowing outward, while the axial velocities are lower in the corner of the two right-angle segments since the velocity component in the direction parallel with the magnetic field vector are diverged, which is linked with lower density of current therein flowing outward. The converging and diverging of the velocity component have an effect on strengthening and weakening the peak axial velocity. As expected, the pressure gradient goes down with a decrease in the conductance ratio. For a fixed conductance ratio, the pressure gradients are the same in the fully developed fluid region. Because lower pressure drop and proper heat extraction are required in the duct flow of liquid metal, the flow design is of particular importance. In this sense, the results of the current study can be utilized
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meaningfully in the flow design of the coolant system of a fusion reactor blanket. In summary, this study discusses the electromagnetic characteristics of LM MHD flows in a u-shaped duct in cases with different lengths of the connecting channel and different conductance ratios. The interdependency of the fluid velocity, current, electric potential and pressure gradient is analyzed in detail. Acknowledgements This research was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology & Ministry of Knowledge Economy (2015M1A7A1A02050613). References [1] C. Mistrangelo, L. Bühler, Numerical investigation of liquid metal flows in a rectangular sudden expansions, Fusion Eng. Des. 82 (2007) 2176–2182. [2] L. Barleon, V. Casal, L. Lenhart, MHD flow in liquid-metal-cooled blankets, Fusion Eng. Des. 14 (1991) 401–412. [3] N.B. Morley, S. Malang, I. Kirillov, Thermofluid magnetohydrodynamic issues for liquid breeders, Fusion Sci. Technol. 47 (3) (2005) 488–501. [4] L. Bühler, S. Horanyi, E. Arbogast, Experimental investigation of liquid-metal flows through a sudden expansion at fusion-relevant Hartmann numbers, Fusion Eng. Des. 82 (2007) 2239–2245. [5] I.R. Kirillov, C.B. Reed, L. Barleon, K. Miyazaki, Present understanding of MHD and heat transfer phenomena for liquid meral blankets, Fusion Eng. Des. 27 (1995) 553–569. [6] N.B. Morley, S. Smolentsev, L. Barleon, I.R. Kirillov, M. Takahashi, Liquid magnetohydrodynamics −recent progress and future directions for fusion, Fusion Eng. Des. 51-52 (2000) 701–713. [7] R. Stieglitz, L. Barleon, L. Bühler, S. Molokov, Magnetohydrodynamic flow in a right-angle bend in a strong magnetic field, J. Fluid Mech. 326 (1996) 91–123. [8] S. Horanyi, L. Bühler, E. Arbogast, Experiments on magnetohydrodynamic flows in a sudden expansion of rectangular ducts at high hartmann number, in: Joint 15th Riga and 6th PAMIR Conference on Fundamental and Applied MHD, Rigas Jurmala, Latvia, 2005. [9] J.C.R. Hunt, Magnetohydrodynamic flow in rectangular ducts, J. Fluid Mech. 21 (4) (1965) 57–590. [10] J.C.R. Hunt, J.A. Shercliff, Magnetohydrodynamics at high Hartmann number, Ann. Rev. Fluid Mech. 3 (1971) 37–62. [11] T.J. Moon, T.Q. Hua, J.S. Walker, B.F. Picologlou, Liquid metal flow in a simple manifold with a strong transverse magnetic field, Appl. Sci. Res. 49 (1) (1992) 49–65. [12] S. Cuevas, B.E. Picologlou, J.S. Walker, G. Talmage, Liquid-metal MHD flow in rectangular ducts with thin conducting or insulating walls: laminar and turbulent solutions, Int. J. Eng. Sci. 35 (5) (1997) 485–503.
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