Effects of magnetohydrodynamic mixed convection on fluid flow and structural stresses in the DCLL blanket

International Journal of Heat and Mass Transfer 135 (2019) 847–859

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effects of magnetohydrodynamic mixed convection on fluid flow and structural stresses in the DCLL blanket Zhi-Hong Liu, Long Chen, Ming-Jiu Ni, Nian-Mei Zhang ⇑ School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e

i n f o

Article history: Received 4 August 2018 Received in revised form 3 January 2019 Accepted 9 February 2019

2018 MSC: 00-01 99-00 Keywords: Mixed convection MHD Volumetric heat source DCLL blanket

a b s t r a c t In this study, numerical simulations are conducted to investigate magnetohydrodynamic (MHD) mixed convection for buoyancy-assisted flows under strong magnetic field and large volumetric heat sources in the Dual-Coolant Lead-Lithium (DCLL) blanket. A magnetic-convection code based on the finite volume method is developed and validated using benchmark solutions. A consistent and conservative scheme is applied to deal with the electric current conservation issues. The PISO algorithm on unstructured collocated meshes is employed to solve the N-S equations considering the Lorentz force effect. Deformations and stresses of flow channel insert (FCI) are analyzed using the finite element method (FEM). Cases with high Hartmann number of 9600–19,200, high Reynold number of 31,000 and high Grashof number of 3:5
1011 are used in the numerical simulations. The buoyancy effects as well as electric conductivity of the FCI on poloidal flows in rectangular channel with a SiC FCI are analyzed, considering nonuniform exponential volumetric heat source and toroidal magnetic field. The deformation field and stress field of the FCI are calculated under MHD mixed convection effects. Results demonstrate that a reverse flow occurs near the cold wall in the bulk region, which is a special phenomenon resulted from buoyancy. Compared to MHD forced convection, buoyancy delivers enhanced temperature uniformity, a drastically changed velocity distribution, and a slightly elevated pressure drop. At the same time, the pressure drop between inlet and outlet has a linear relation with eB=5 . Mixed convection temperatures is insensitive to FCI electrical conductivity, and only velocity near the cold wall appears sensitive. In the FCI, both magnetic field and electrical conductivity positively correlate with thermal stresses. Simulations also suggest the buoyancy effect reduces temperature difference across FCI and thermal stress. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Blanket is one of the key components of energy conversion in Tokamak fusion reactor. Dual-Coolant Lead-Lithium (DCLL) blanket is a helium-cooled ferritic structure with self-cooled PbLi breeder zone that uses SiCf/SiC flow channel insert as MHD and thermal insulator, and is one of the most competitive candidate blankets [1]. This blanket is impacted under an environment of strong magnetic field and strong internal heat source. Therefore, the mechanical behavior of DCLL blanket is a multi-physical field problem composed of magnetic, thermal, flow and structure field. Without considering buoyancy in DCLL blanket, numerous forced or natural convection studies have been conducted in past decades. Smolentsev et al. [2] utilized two-dimensional model for a fully developed flow, and their simulation suggests that the FCI ⇑ Corresponding author. E-mail address: [email protected] (N.-M. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2019.02.019 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

decreases MHD pressure drop and improves heat transport. Sutevski et al. [3] used an unstructured solver to acquire 3D results which corresponded with experimental data. Their results indicate that axial currents are the main reason why 2D results have discrepancies among experimental results. Tagawa et al. [4] proposed analytical solutions for long vertical enclosure under perpendicular and parallel magnetic field. Authié et al. [5] found that heat transfer could be enhanced when fluid transited from 3D to Q2D flow. Chen et al. [6] have studied depth than Authié [5]. Their results suggested the physical mechanism of transition from 3D to Q2D and proposed a more suitable relations among Nu, Ha and Gr number. Some analytical solutions of MHD natural convection have been proposed, such as a shallow cavity with heat source [7], a vertical channel [8,9] and a horizontal cylinder [10]. Smolentsev et al. [11] concluded some MHD numerical benchmarks for fusion applications, including MHD laminar, buoyancy convection, Q2D and 3D turbulence cases. Bondareva and Sheremet [12] studied MHD natural convection in an enclosure under the effect of inclined mag-

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netic field and enclosure relative length. In a review [13], Abdou et al. described key recent developments and physical phenomena in the DCLL blanket, they concluded that great progress has been made in the study of the forced convection, although research into mixed convection under the blanket conditions is still in its infancy. Smolentsev et al. [14] pointed out that due to the big neutron heat source, Grashof number can reach to 2:0
1012 , mixed convection plays a key role on MHD heat, mass, and momentum transfer in the DCLL blanket conditions. However, compared to MHD forced convective flows, mixed convective flows affected by nonuniform internal heat source have been ignored for a long time. The situation has changed in recent years. Vetcha et al. [15] proposed quasi-two-dimensional (Q2D) approximate solution for fully-established steady upward mixed convection associated with exponential volumetric internal heating and transverse magnetic field. Hudoba et al. [16] proposed an analytical solution of velocity in a vertical channel under uniform internal heat source and transverse magnetic field. Valls et al. [17] studied mixed convection flow behaviors of a U-bend under neutron heat source and magnetic field. Zhang and Zikanov [18] analyzed a horizontal duct aligned with magnetic fields while the duct was heated by a neutron volumetric heat source and cooled by constant temperature wall. Their results suggested that mixed convection effects is benefit for blanket’s operation, because turbulence and mixed convection result in more uniform heat transfer towards the cold wall. Zikanov and Listratov [19] simulated a downward vertical pipe associated with a half of heating pipe’s wall and a horizontal magnetic field, their results showed an anomalous high-amplitude temperature fluctuations in mixed convection which also observed in experiments [20,21]. Umavathi and Sheremet studied the effect of heat source/sink [22] and first order chemical reaction [23] to MHD heat transfer in a vertical channel. As to solid structure analysis of DCLL blanket, Vitkovsky et al. [24] developed a linear structural analysis technique using 3D finite element model for blanket, consisting of thin shell and beam elements. Results confirmed that the structure is safe in normal working condition. Sharafat et al. [25] analyzed mechanical behavior of FCI structures, with sequential coupling method. In their study, the magnetic fluid mechanics analysis was simulated firstly, then extracted the surface temperature distributions for the structure analysis. Ying et al. [26] proposed an Integrated multi-physics Simulation Predictive Capability (ISPC). Through structure

simulation, they pointed out that the maximum structural stress in blankets exceeded the material limit, and this unsafe situation illustrated that an optimal design of the structure was necessary. Li et al. [27] studied coupled thermal, hydrodynamic, elastic issues and structural safety of FCI in DCLL blanket by applying FEM and FVM method. The aim of the present work is to study the fully developed flow state of buoyancy-aided up flow and mechanical behaviors of structure in magneto-thermo-fluid-structure multi-physics coupled field in DCLL blanket. The effects of magnetic field, FCI’s electrical and thermal conductivity on mixed convection are investigated. The results show that the buoyancy force enormously influenced the flow of metal fluid and heat transfer, even the thermal stresses of FCI. And the coupling effect of magnetic field magnitude and electrical conductivity of FCI has impact on the thermal deformation of FCI.

2. Physical model This study focuses on buoyancy-assisted and pressure-driven MHD mixed convection in DCLL blanket. The physical model studied at present is a straight channel with FCI inside it, as shown in Fig. 1, the structure parameters are referred to [28]. Effort is made to keep simulation parameters, such as Reynold, Hartmann and Grashof number, close to the actual DCLL blanket conditions. Liquid metal PbLi flows along Z-axis, the inlet velocity and temperature are 0.06 m/s, 733 K. The X-axis is defined as the radial direction, Y-axis as toroidal direction and Z-axis as poloidal direction. The magnetic field is parallel to the toroidal direction. As shown in Fig. 1, from the inside to outside, regions are defined as bulk region, flow channel insert (FCI), gap region, Fe wall, respectively. Besides, the top gap and the bottom gap are Hartmann gaps which are perpendicular to the magnetic field. The left and right gaps are called as side gaps. The Fe wall is a thin-walled channel with radial length 0.224 m, toroidal width 0.324 m and poloidal height 2 m. Thickness of Fe wall and FCI, gap width are 5 mm, 5 mm and 8 mm, respectively. The outside of Fe wall is helium gas, whose temperature is set at 673 K, and the convective heat
 transfer coefficient of helium is 4000 W= m2  K . According to the Hartmann–Reynolds number diagram in [11], we predict flow regime in this paper stays in a laminar state; although buoyancy has strong influence on the velocity profile, fluid flows in these

Fig. 1. Geometry of the rectangular channel flows with FCI.

Z.-H. Liu et al. / International Journal of Heat and Mass Transfer 135 (2019) 847–859

   @T ! q0 C p ¼ r  ðjDT Þ þ q000 þr uT @t

849

ð6Þ

Hudoba and Molokov [16] concluded that in a channel of width d ¼ 5 cm, the ratio of the Joule heating to heating source is of order 107  108  Ha2 =Gr. Compared with the heat source, Joule heat! ing ( j 2 =r) is much smaller, it can be neglected. Electric currents, which generated in the flowing PbLi fluid under toroidal magnetic ! field B , should satisfy Ohms law (Eq. (7)). By using the electric charge conservation (Eq. (8)), we can obtain electric Poisson equation (Eq. (9)).

  ! ! ! j ¼ r ru þ u
B

ð7Þ

!

r j ¼0

Fig. 2. Internal volumetric heat source of fluid region profile.

ð8Þ 

!

r  ðrruÞ ¼ r  r! u
B

ð9Þ

simulations remain laminar. The material property is referred in [29–32]. The PbLi fluid in DCLL blanket is heated by the neutronic thermal source which comes from the neutron reaction between neutrons and PbLi fluid. An approximate expression of internal heating q000 ðxÞ was proposed by Smolentsev and Morley Abdou [28], and shown in Eq. (1) and Fig. 2. Here, a is half width of the bulk region, and q0 is the maximum volumetric heat source value,
 q0 ¼ 3
107 J= m3  s is adopted in this simulation; the heat source is exponential distribution along the radial direction.

! ! where B denotes the imposed magnetic field, j the induced cur! rent density, u the electric potential and g the acceleration of gravity. Here, q; m; b; C p , and k are fluid density, kinematic viscosity, volumetric thermal expansion coefficient, specific heat capacity, and thermal conductivity correspondingly. T ref ðzÞ is the reference temperature, it is the mean temperature of a cross section perpendicular to the flow direction and calculated as below:

 x þ a q000 ðxÞ ¼ q0  exp  a

T ref ðzÞ ¼

ð1Þ

Non-dimensional temperature h is defined as

T  T0 T  T0 h¼ ¼ ; T 0  T He DT

ð2Þ

where T 0 is the inlet temperature, T He is the temperature of helium @h and the wall gas. The local Nusselt number is defined as Nu ¼ @n Nusselt number is defined as

R NudA : Nuwall ¼ R dA

ð3Þ

In simple geometry whose wall temperature is fixed at a constant value, such as in an enclosure [5], single channel [33], the range of non-dimensional temperature h is usually defined to be 0–1. However, the range of h in DCLL blanket cannot be defined to 0–1 due to the complex geometry and the conjugate heat transfer boundary condition of FCI. In this work, we adopted two constant temperatures, the fluid inlet temperature and the helium gas temperature, to scale the non-dimensional temperature. 3. Numerical solution 3.1. Basic equations 3.1.1. Fluid region The Boussinesq approximation is empolyed for the buoyancy calculation. Continuity equation, incompressible Navier-Stokes equation considering buoyancy and Lorentz force and energy equation are shown below:

r! u ¼0

ð4Þ

!    1 ! ! @ u ! 1 ! ! þ u  r u ¼  rP þ r m D u þ j
B @t q0 q0 
 þ g 1  b T  T ref

ð5Þ

1 4ab

Z

a

a

Z

b

b

T ðx; y; zÞdz:

ð10Þ

! j is electric current, which is determined by Ohm’s law (Eq. (7)), u is electric potential. The non-dimensional parameters are: the Reynold number Re ¼ UL=m, the Hartmann number pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Ha ¼ BL r=qm ¼ Bb r=qm , the Prandtl number Pr ¼ mqC p =j, andthe Grashof number Gr ¼ gbDTL3 =m2 ¼ gbDTa3 =m2 . The DT in Gr
a2 =j, and it is measured by the average number is defined as DT ¼ q R a 000 1
(q
¼ 4ab q ðx; yÞdxdy). Hartmann numvolumetric heat source q a ber scale the thickness of Hartmann layer and side layer. Hartmann layer of thickness  L=Ha is located at the walls perpendicular to pffiffiffiffiffiffi magnetic field, while side layer of thickness  L= Ha is located at the walls parallel to magnetic field. DCLL blanket DEMO is a Fusion Energy Demonstration Reactor, commonly called DEMO. In DCLL DEMO, the Hartmann, Reynold and Grashof number can reach to 12,000, 60,000 and 2:0
1012 , respectively [14]. Three-dimensional buoyancy assisted mixed convection cases in the DCLL blanket are studied in this work for various magnetic field ranging from 3 T to 6 T, corresponding to Hartmann number ranging from 9600 to 19,200; The inlet Reynold number is 31,000, and Grashof number is 3:5
1011 . The discretization is a second-order accuracy in time and space. For the time term, a Crank–Nicholson scheme is used. Centraldifference formula is used for the discretization of the diffusive terms in the Navier-Stokes equation and energy equation. The finite volume method is employed to solve N–S equation (Eq. (5)) of flow field, energy equation (Eq. (6)) and electric potential equation (Eq. (9)). A PISO algorithm with an unstructured collocated mesh is used to solve pressure Poisson equation. The Lorentz force is calculated using the consistent and conservative scheme [34,35] to guarantee the satisfaction of the current conservation law and momentum conservation law. Before investigating the influences of the buoyancy on flows in the DCLL blanket, a mesh convergence investigation is carried out firstly, with parameters of B = 4 T, r ¼ 20 S=m;

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Ha ¼ 12; 800; Re ¼ 31; 000 and Gr ¼ 3:5
1011 . In order to describe the flow and heat transfer more precisely in Hartmann layer and side layer, at least 4 layers and 6 layers meshes are discretized in Hartmann layer and side layer, respectively. Mesh sensitivity study is conducted using three kinds of mesh (as shown in Table 1). The results of the Nu with different spatial discretization are obtained. The table shows that the relative error between medium and fine mesh is 0:54%, indicating the grid independence of the numerical results. Medium mesh is adopted in the following DCLL blanket simulations. 3.1.2. Solid region The steady fluid-structure interaction is investigated based on the consideration of the influence of fluid fields on the FCI mechanical behaviors. In this study, the influence of solid deformation to fluid fields is neglected, which would be investigated in a following-up study. The main objective is to investigate the influence of buoyancy on the FCI deformations and stresses of the FCI structure. It is supposed that the FCI has small deformations, under fluid field and thermal field. The relation of strain and displacement satisfies following formula:

eij

 1
¼ di;j þ dj;i þ aDTdij ; 2

ð11Þ

where eij ; di ; a; DT and dij denote strain tensor, displacement vector, thermal expansion coefficient, temperature variation of solid region and Kronecker delta, respectively. The constitutive equation for FCI is described below:

rij ¼ 2Geij þ kHdij :

ð12Þ

Here, rij ; G; k and H represent stress tensor, shear modulus, Lame constant and volumetric strain, respectively. And the Lame constant is defined as k ¼ El=½ð1 þ lÞð1  lÞ, where l is Poisson’s ratio. The equilibrium equation of structure satisfies

rij;j þ F i ¼ 0;

ð13Þ

where F i is the ith component of volumetric force. To investigate the thermal deformation of the FCI in the multiphysics coupled fields, static fluid-structure interaction is analyzed with the help of sequential coupling method. Firstly, the FVM is employed to solve fluid field and the conjugate heat transfer. Then, interface temperature between the fluid region and the FCI structure is extracted as boundary data for structure analysis. Finally, FEM is applied to analyze deformation field and stress field in FCI. The working process is concluded in Fig. 3.

Fig. 3. The working process of static fluid-structure interaction.

is employed for outlet velocity. It is an opening boundary condition which prevent outlet flows from generating significant distortion. On the interface between fluid and solid regions, for the continuity of the temperature, electric potential and current, the boundary conditions are given below:

T fluid ¼ T solid ;

ufluid ¼ usolid ; ! ! ! ! j fluid  n fluid ¼ j solid  n solid :

ð15Þ

The coolant helium surrounding the blanket is used to cool the blanket. The outside wall of blanket meet the needs of electric insulation (Eq. (16)) and the robin boundary condition (Eq. (17)):

@u ¼0 @n

j

@T þ hðT  T He Þ ¼ 0 @n

ð16Þ ð17Þ

3.2. Code validation 3.2.1. Reverse flow A vertical channel fulled with fluid shown in Fig. 4(a), is heated to keep constant temperatures, left wall having a lower temperature than right wall. The parameter r T ¼ ðT 1  T 0 Þ=ðT 2  T 0 Þ is used to measure wall temperature difference ratio. T 0 ; T 1 and T 2 represent fluid, cold wall and hot wall temperature, respectively, with T 1 < T 0 < T 2 . The case adopts the parameter r T ¼ 0:5 and nondimensional number Gr=Re ¼ 250. The results show that a reverse flow took place at cold wall while velocity accelerated near the hot wall. An excellent agreement of fully developed flow velocity at outlet between simulation and analytical solution [37] can be seen in Fig. 4(b). The maximum relative error between simulation and analytical solution for the flow in core region is about 0.15%. 3.2.2. Mixed convection with MHD and heat source Comparisons between the present simulation and the Q2D approximate solution proposed by Vetcha [15] are shown in

3.1.3. The initial conditions and boundary conditions Uniform initial fields are applied for velocity, electric potential, temperature and pressure fields. At the inlet, uniform constant field are applied for velocity and temperature of metal fluid, that is, U ¼ 0:06 m=s and T 0 ¼ 733 K. Neumann boundary conditions @p are adopted for electric potential (@@nu ¼ 0) and pressure (@n ¼ 0). At the outlet, electric potential and pressure are fixed at zero, Neu¼ 0). mann boundary conditions are adopted for temperature (@T @n Convective boundary condition [36]:

@u @u þ u0 ¼0 @t @n

ð14Þ

Table 1 The Computational Results under different spacial discretization. Mesh

Grid

Nu

Relative error

Coarse Medium Fine

105
136
80 135
177
100 178
229
120

46:96 46:45 46:20

1:64% 0:54% –

Fig. 4. r T ¼ 0:5; Gr=Re ¼ 250 (a) Schematic diagram of the vertical plane channel, velocity contour and streamline; (b) comparison of present values and analytical solution.

Z.-H. Liu et al. / International Journal of Heat and Mass Transfer 135 (2019) 847–859

Fig. 5. Liquid metal fluid PbLi is used in the DCLL blanket. Grashof number is measured by the internal heat source. At present simulation, Hartmann number and Reynold number are fixed to be Ha ¼ 1000 and Re ¼ 1000, and four different Grashof number cases (Gr ¼ 1
107 ; Gr ¼ 1
108 ; Gr ¼ 1
109 ; Gr ¼ 1
1010 ) are applied in simulation. Four simulation results have good agreement with the approximate solution (Fig. 5). The maximum relative errors for the four cases in core region are 7.1%, 0.1%, 0.1% and 3.4%, respectively. As the approximate solution cannot accurately predict velocity in boundary layer, and the maximum relative error locate near boundary layer, the error between present simulation and approximate solution is acceptable.

851

4.1. The effects of magnetic field on MHD mixed convective in blanket In this research, the finite volume method is employed to solve Navier-Stokes equation concerning the Lorentz force and buoyancy and electric potential equation. The Boussinesq approximation is applied for the buoyancy calculation in simulation, which is typically used in blanket studies [15,18,19]. It is well-known that the Boussinesq approximation is conditional, the range of validity is usually limited by [39]

bDT max 6 0:1;

ð18Þ

4. Results As described by the momentum equation Eq. (5), the Lorentz force would affect the velocity field of fluid. Due to the small magnetic Reynolds number, the magnitude of external magnetic field and electrical conductivity of structure play important roles in the velocity profile of metal fluid. In the cases of forced convection [13,38], the coupling effects of magnetic fields and electrical conductivities of FCI impact the flow velocity and temperature fields. In this study, for the mixed convective cases, the effects of magnitudes of magnetic fields and electrical conductivities of the FCI on metal fluid flow and heat transfer and mechanical behavior of FCI are investigated through simulations on the basis of finite volume and finite element methods. Meanwhile, the buoyancy effects are analyzed. The cold wall is used to represent the side wall near small heat source, and hot wall to represent the side wall near big heat source in the following analysis.

Fig. 6. (a) Pressure and velocity streamline of center slice, B ¼ 4 T; vortex of Q ¼ 0:3; B ¼ 3—6 T; r ¼ 20 S=m.

Fig. 5. Comparisons between the present simulation (solid line) and approximate solution (dashed line).

r ¼ 20 S=m; (b)

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Fig. 7. (a) Velocity profile of bulk region; (b) temperature profile of bulk region.

Table 2 Wall Nusselt number of mixed convection and forced convection. Magnetic field

Numixed

Nuforced

3T 4T 5T 6T

47.05 46.45 45.62 45.04

45.86 44.39 43.54 42.99


 where DT max ¼ T  T ref ðzÞ max is the maximum temperature difference in a cross section perpendicular to the flow direction. With the limit, as the thermal expansion coefficient b of fluid PbLi is enough small, the maximum temperature difference of PbLi fluid can be almost 500 K. According to the research about the forced convection [38], the maximum temperature difference of fluid was within the limit. So, it is rational to employ Boussinesq approximation in simulation of MHD mixed convection. 4.1.1. MHD flow and heat transfer To study the effects of magnetic field effects on the flow, we apply four magnitudes of magnetic fields B ¼ 3 T, 4 T, 5 T and 6 T with fixed electric conductivity of FCI r ¼ 20 S=m. The velocity profile of different magnetic field is shown in Figs. 6(a) and 7(a). The velocity streamlines display that the reverse flow occurs under all the four magnetic fields and the reverse flow velocity vanishes at about 1/3 channel length from the outlet (Fig. 6(a)).

Fig. 8. B = 4 T,

Based on Q-Criterion, the vortex of Q ¼ 0:3 is extracted, and shown in Fig. 6(b), it is suppressed and move towards the outlet with the enhanced magnetic field. According to the previous studies about forced convection in DCLL blanket [2,3], the reverse flow has never been observed in bulk region. It is buoyancy that leads to the reverse flow near cold wall in the DCLL blanket. The smaller the reverse velocity is, the easier it is for fluid to transport heat to outlet, which leads to a lower temperature near the cold wall as shown in Fig. 7(b). Under different magnetic fields, the maximum temperature at outlet always lies near the hot wall (Fig. 7(b)), which change quite slightly. Besides, the upward velocity near centerline is inhibited and heat transport to outlet is more difficult. Therefore, the temperature near centerline increases. Near the hot wall, the velocity and temperature remain almost unchanging. These results show that under exponential heat source and buoyancy effects, the strength of magnetic field has a strong influence on temperature and velocity in the vicinity of the cold wall, although there is no significant effect in the vicinity of the hot wall.
 Besides, in our result (Fig. 7(b)), DT max ¼ T  T ref ðzÞ max of PbLi at outlet is 36–45 K. In the forced convection, high temperature difference exists ( 102 K). However, the temperature difference in mixed convection is quite small. The result further indicates that Boussinesq approximation is reasonable in the DCLL blanket simulations. Nusselt numbers represent the ratio of convective to conductive heat transfer. The wall Nusselt numbers are shown in Table 2.

r ¼ 20 S=m, (a) velocity profile of bulk and gap region at outlet; (b) temperature profile of all region at outlet.

Z.-H. Liu et al. / International Journal of Heat and Mass Transfer 135 (2019) 847–859

Fig. 9. Comparison between mixed convection and forced convection, B = 4 T,

853

r ¼ 20 S=m (a) velocity; (b) temperature.

Fig. 10. Outlet electric potential and current streamline distributions, (a) mixed convection; (b) forced convection.

Numixed is defined as the wall Nusselt number of bulk region to FCI of mixed convection, while Nuforced is defined in terms of the forced convection. As shown in numerical results, with increasing magnetic field strength, wall Nusselt numbers of both mixed and forced convection are decreased, and the heat transfer is suppressed. The goal of DCLL blanket is to provide higher energy conversion and to keep structure safety. Numixed is always larger than Nuforced , this reflected buoyancy mixed the fluid flow and increased the heat loss through the wall. Fig. 8 displays the contours of velocity profile and temperature distribution in detail with fixed B = 4 T and r ¼ 20 S=m. The outlet velocity profile of bulk region and gap region are shown in Fig. 8(a). It can be seen that the velocity in bulk region is a quasi-2D distribution, while 3D distribution existed within gap region. In Hartmann gap, current component perpendicular to magnetic field is much higher than side gap (Fig. 10(a)). Therefore, the suppression in Hartmann gap is much higher than the suppression in the side

Fig. 11. A relation between pressure drop and magnetic field of mixed convection, r ¼ 20 S=m.

gap, these led to the occurrence of a jet flow in side gap. In Hartmann gap, Lorentz force is sufficiently strong, fluid proceeded to an almost stagnant state.

Table 3 R of bulk region. Magnetic field

ð DpÞmixed ½Pa

ð DpÞforced ½Pa

R

3T 4T 5T 6T

773 1174 1701 2358

626 1062 1608 2263

1.23 1.11 1.06 1.04

4.1.2. The effects of buoyancy In order to figure out the influence of buoyancy on DCLL blanket, velocity and temperature between mixed and forced convection are compared (Fig. 9). Results show that near the hot wall and the centerline, the velocity increases slightly. Near the cold

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Fig. 12. Pressure distribution along stream-wise center line, (a) bulk region; (b) gap region. Linear decreased pressure is at side gap, and pressure which have a rapid drop at inlet is at Hartmann gap.

Fig. 13. Convection structure with different Gr number under B ¼ 4 T (a) forced convection, Gr ¼ 0; (b) Gr ¼ 3:5
1010 ; (c) Gr ¼ 3:5
1011 .

gests that buoyancy has a substantial influence on fluid flows in DCLL blanket. Therefore, in order to investigate actual flow states under working conditions, buoyancy cannot be ignored. Electric current streamlines of mixed convection and forced convection are compared in Fig. 10. Obviously, the current streamline distributions are symmetric about y-axis in forced convection. However, buoyancy causes the asymmetricity in the current streamlines of mixed convection and the reverse flow lead to the bending of current near cold wall. In the DCLL blanket, the pressure drop is a key parameter used to evaluate the efficiency of blanket. When metal fluid flows in conducting channel, the Lorentz force impedes the movement of liquid metal, resulting in a much higher pressure drop than a non-conducting fluid flow. A pressure drop reduction factor R ¼ ðdp=dxÞ0 =ðdp=dxÞ is used in [28,40], where ðdp=dxÞ0 and ðdp=dxÞ represent pressure gradient without FCI and with FCI, respectively. The R can reach about 120, when the FCI with electrical conductivity is r ¼ 20 S=m and FCI width is 5 mm. Their findings suggest that FCI plays a key role in the reduction of pressure drop. In this paper, R ¼ ð DpÞmixed =ð DpÞforced is used to compare the pressure drop in bulk region between mixed convection and forced convection. The R is shown in Table 3, the pressure drop of mixed convection is slightly larger than forced convection, and R decreases with increasing magnetic field. ð DpÞmixed is larger than ð DpÞforced , because buoyancy influences the distribution of velocity, the velocity causes the increase of Lorentz force, and in turn, the increased Lorentz force caused the increase of pressure drop. Additionally, we find a relation between pressure drop and magnetic field of mixed convection in bulk region (Fig. 11). The pressure drop has a linear relation with eB=5 :

ðDpÞmixed ¼ 1072:79eB=5  1207

Fig. 14. Velocity profile with thermal insulation FCI, B = 4 T,

r ¼ 20 S=m.

wall, the mixed convection velocity magnitude increases significantly compared to the forced convective velocity magnitude, and the reverse flow peak velocity is approximately 4 times as high as the peak velocity of forced convection. Besides, it can be seen that the temperatures of the mixed convection are almost uniform, while temperature in forced convection severely change. This sug-

ð19Þ

As shown in Figs. 6(a) and 12, the pressure drop in bulk region linearly decreases along the stream-wise, except in the vicinity of vortexes caused by the reverse flow. Non-monotonical pressure fluctuation is caused by the reverse flow, and as the vortex is suppressed by the magnetic field (see in Fig. 6(b)), pressure fluctuations decreased. Even in the gap region, the pressure drop in the side gap is different from that in the Hartmann gap. In the Hartmann gap, the velocity decreases rapidly at the inlet. In contrast, the velocity in the side gap has a monotonically decrease at the inlet. As has been mentioned, near the inlet of Hartmann gap, current component which is perpendicular to magnetic field is much higher than that in side gap, and therefore, cause the increase of both Lorentz force and pressure. A comparison of convection structure between different Grashof numbers is shown in Fig. 13. The dynamic process influenced

Z.-H. Liu et al. / International Journal of Heat and Mass Transfer 135 (2019) 847–859

855

Fig. 15. (a) Velocity profile and (b) temperature profile with different FCI electrical conductivity, B = 4 T.

stronger reverse flows, bigger reverse flow region and negative peak velocity. The temperature contours illustrate that buoyancy can lead to more uniform temperature distribution. 4.2. The influences of FCI’s electrical conductivity on MHD flow and heat transfer

Fig. 16. Temperature distribution of Fe wall, ‘‘left” wall is the wall near big heat source, T max ¼ 733 K, B = 4 T, r ¼ 20 S=m.

Fig. 17. Maximum temperature difference across the FCI of forced convection and mixed convection, r ¼ 20 S=m.

by buoyancy effects is similar to Fig. 13(a)–(c). If forced convection is only considered, the velocity field is symmetrical about the poloidal central axis of the channel. The exponential heat source cause higher temperature near the left wall of the channel. With the Gr number increasing, the fluid near the left wall is accelerated by buoyancy, meanwhile the fluid near the right wall is decelerated. Along with the fluid flows, the velocity of the fluid near the right wall firstly decreases to zero and then becomes negative. The reverse flow occurs. The buoyancy effects can result in

In order to reduce the MHD pressure drop and interface temperatures between metal fluid and Fe wall, researchers [28,40,41] have suggested introducing flow channel inserts (FCI) in flow channel. Usually, FCI are made of the silicon carbide composite material (SiCf/SiC) for thermal and electrical insulation; however, due to the manufacturing process, FCI still have low electrical and thermal conductivities. Electrical conductivity of solid wall affects the currents in both solid and fluid regions, then the currents changes the velocity field. In the present study, we performed calculations with the electrical conductivities of FCI varying from 1 S/m to 100 S/m at a magnetic field fixed at 4 T. Mixed convection of MHD with thermal insulating FCI is simulated with the help of magnetic-convection code. Results show that no reverse flow took place in the DCLL blanket (Fig. 14), because heat could not be transferred across side walls or Hartmann walls of thermal insulating FCI. These situations are the same as the validation cases in Fig. 5. The occurrence of reverse flow depends on both buoyancy effects and thermal conductivity of the FCI. Fig. 15 shows the variation of the velocities and temperatures as the magnetic field fixed at 4T and electric conductivities changes from 1  100 S=m. With regard to the effect of FCI’s conductivity on the Nusselt number, increasing the conductivity would decrease the Nusselt number, and Nusselt numbers at different conductivities are Nur¼1 ¼ 47:93, Nur¼5 ¼ 47:68, Nur¼20 ¼ 46:45, Nur¼100 ¼ 43:21, respectively. These results show that a higher electric conductivity of FCI lead to a lower heat loss. When the electric conductivities increased, the reverse flows and the other two peak velocities (shown in Fig. 15(a)) are weakened. Smolentsev [42] studied changes of velocity and temperature with electric conductivity of FCI in forced convection, and reported that the velocities close to both side walls were very sensitive to conductivities. However, in the case of mixed convection, the situation is somewhat different. The reverse flow near cold wall is sensitive to FCI conductivity changes, while peak velocity near hot wall is not. Paper [42] described the relation between temperature and electric conductivity of FCI in forced convection. Their results suggested that temperature in forced convection was sensitive to electric conductivity of FCI. However, temperature in mixed convection is not sensitive to electric conductivity of FCI, as shown in Fig. 15 (b). In bulk region, temperature near center increased slightly, and has a slight decrease near the hot wall and the cold wall with

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Fig. 18. (a) Displacements component of FCI with different magnetic fields in radial direction; (b) in toroidal direction; (c) displacement in three direction; (d)3D view of displacement, B = 4 T; (e)3D view of Von-Mises stress, B = 4 T.

Fig. 19. Von-Mises stress and three principal stresses of FCI in the mixed and forced convection, B = 4 T, r ¼ 20 S=m, (a) along centerline on Hartmann wall, mixed convection; (b) along centerline on side wall, mixed convection; (c) along centerline on Hartmann wall, forced convection; (d) along centerline on side wall, forced convection.

Z.-H. Liu et al. / International Journal of Heat and Mass Transfer 135 (2019) 847–859

the increases of electric conductivity. The lower temperatures at both hot wall and cold wall surfaces lead to a decrease in temperature difference across the FCI. Conclusions can be drawn that velocity near hot wall and temperature are insensitive to FCI conductivity changes, despite the reverse flow (near cold wall). 4.3. Analysis on mechanical behaviors of Flow Channel Inserts The solid region temperature should be kept within a safe range to ensure that thermal stress and the corrosion do not exceed acceptable thresholds. According to the allowable corrosion limit, Fe wall temperature is usually limited to below  753 K. When B ¼ 4 T, the Fe wall temperature is shown Fig. 16. The temperature along stream-wise decreases first and then increases near outlet. The maximum temperature is located

at inlet, and T max ¼ 733 K. In other cases, with different magnetic fields and electrical conductivity of FCI, T max is also equal to 733 K. Under the present simulation parameters and buoyancy effects, the Fe wall is safe. According to the thermal deformation and stress limits of FCI, temperature difference across the FCI is usually limited to 200 K [28]. A comparison of the maximum temperature difference across the FCI between forced convection and mixed convection is shown in Fig. 17. In forced convection, DT max has far exceed temperature limit. Compared to forced convection, DT max in mixed convection is substantially reduced. Due to buoyancy effects, fluid temperature is more uniform, and internal wall temperature of FCI is lower than forced convection. The comparison of thermal stresses in forced convection and mixed convection cases would be analyzed later.

Fig. 20. Von-Mises stress and three principal stresses of FCI, (a) along centerline on Hartmann wall, r ¼ 20 S=m, B = 3–6 T; (b) along centerline on side wall, B = 3–6 T; (c) along centerline on Hartmann wall, r ¼ 1  100 S=m, B = 4 T; (d) along centerline on side wall, r ¼ 1—100 S=m, B = 4 T.

Fig. 21. Von-Mises stress of FCI, (a) under different magnetic field,

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r ¼ 20 S=m; (b) with different electrical conductivity, B = 4 T.

r ¼ 20 S=m,

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Table 4 Some important data with different magnetic field,

r ¼ 20 S=m.

B

DP [Pa]

DT max [K]

U max [m/s]

T max [K]

rmax [MPa]

Dmax;1 [mm]

Dmax;2 [mm]

3T 4T 5T 6T

772 1174 1700 2358

208.4 205.6 204.3 203.2

0.498 0.378 0.286 0.224

960.19 958.86 959.73 962.06

168.20 170.24 174.81 179.54

0.135 0.131 0.124 0.106

0.219 0.264 0.313 0.360

When deformations are small and stresses low, SiC can be considered isotropic materials with linear elasticity [43]. Under structural analysis, the following boundary conditions based on the installation of FCI in the actual operating conditions are adopted. The poloidal displacement is constrained at the inlet end, while the toroidal and radial displacements are constrained at the outlet end. The FCI displacements of the left side wall centerline and the Hartmann bottom wall are shown in Fig. 18. Numerical results suggest that the stronger magnetic field is, the larger radial displacements and the smaller toroidal displacements would be. The 3D displacements view in Fig. 18(d) clearly shows that side walls of FCI concave inward at radial direction and Hartmann walls expand outward at toroidal direction. Experimental results [44] illustrate that the tensile and compression strength of SiC are quite different, thus, we discuss not only the tensile stress but also the compressive stress. In Fig. 20, FCI stresses data is presented. Three principal stresses and VonMises stresses along the centerline of left side wall and bottom Hartmann wall are analyzed. The maximum Von-Mises stress and three principal stresses between forced convection and mixed convection at B = 4 T (Fig. 19) are compared. The results suggest that all stresses are greatly reduced due to buoyancy effects. At the centerline of the bottom Hartmann wall, the buoyancy effect reduces the maximum Von-Mises stress by 168.6 MPa; at the centerline of the left side wall, the buoyancy reduces the maximum Von-Mises stress by 40.9 MPa. The reduction of the maximum Von-Mises stress corresponds to the decrease in temperature difference across the FCI. It can be concluded that buoyancy effects make the FCI safer. The curves in Figs. 19(a), (b) and 20(a)–(d) demonstrate that the maximum Von-Mises stress in bottom Hartmann wall appeared near the hot side wall, because of the exponential distribution of heat source along radial direction. The first principal stress is small tensile stress, while the second and third principal stresses are compression stresses which are all about 140 MPa. With the magnetic field increasing, Von-Mises stress increased in Hartmann wall and decreased slightly in side wall. Fig. 20(c) and (d) show that, with the increasing of electrical conductivity of the FCI, the changes of Von-Mises stresses in both the Hartmann wall and the side wall are similar to the changes of increasing magnetic field. The maximum Von-Mises stresses of FCI under different magnetic fields and with different electrical conductivities are shown in Fig. 21. The maximum stress in Hartmann wall is greatly affected by magnetic field and FCI’s electrical conductivity, while the maximum stress in side wall is slightly affected by magnetic field and FCI’s electrical conductivity. Nonlinear effects of magnetic field and FCI’s electrical conductivity can be clearly observed in Hartmann wall. FCI with a larger electrical conductivity under a strong magnetic field would have maximum thermal stress, which may lead to the dangerous state of the structure. 5. Conclusions Three-dimensional simulations of DCLL blanket MHD mixed convection are conducted in this paper. The effects of magnetic field strength and FCI’s electric conductance ratio on the velocity, temperature and Nusselt Number distributions in DCLL blanket

are investigated. Several important results have been summarized in Table 4, where DP is pressure drop between inlet and outlet, DT max is the maximum temperature difference across FCI, U max is the maximum reverse velocity [m/s], T max is the maximum temperature at outlet, rmax is the maximum Von-Mises stress of FCI Hartmann wall, Dmax;1 is the maximum displacement of FCI Hartmann wall, and Dmax;2 is the maximum displacement of FCI side wall. The most significant results can be concluded as follows: (1) Due to exponential neutron heat source and conductive FCI, a reverse flow occurs near cold wall in the bulk region and is suppressed by increasing magnetic field. The reverse flow depends on both buoyancy and the thermal and electrical conductivity of the FCI. (2) Comparing with MHD forced convection, buoyancy effects bring a more uniform temperature field, a drastically changed velocity distribution and a slightly elevated pressure drop. The pressure drop has a linear relation with eB=5 . (3) The temperature and velocity near side walls in forced convection are affected substantially by the electrical conductivity of FCI. In contrast, temperature in mixed convection is not sensitive to the electrical conductivity of FCI, and only velocity near cold wall is sensitive to FCI conductivity. (4) With an increasing magnetic field or the electrical conductivity of FCI, the thermal stresses of FCI increased, FCI with a large electrical conductivity or in strong magnetic field may reach dangerous state. (5) Buoyancy effects reduce the temperature difference across the FCI, and thus reduce the thermal stress of FCI. Buoyancy effects make the FCI safer. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgements The authors gratefully acknowledge the support from the NSFC (Natural Science Foundation of China) under Grants #51776194 and #51376175, National Key Research and Development Program of China (No. 2017YFE0301300). References [1] M. Abdou, D. Sze, C. Wong, M. Sawan, A. Ying, N. Morley, S. Malang, US plans and strategy for ITER blanket testing, Fusion Sci. Technol. 47 (3) (2005) 475– 487. [2] S. Smolentsev, N. Morley, M. Abdou, Code development for analysis of MHD pressure drop reduction in a liquid metal blanket using insulation technique based on a fully developed flow model, Fusion Eng. Des. 73 (1) (2005) 83–93. [3] D. Sutevski, S. Smolentsev, N. Morley, M. Abdou, 3d numerical study of MHD flow in a rectangular duct with a flow channel insert, Fusion Sci. Technol. 60 (2) (2011) 513–517. [4] T. Tagawa, G. Authié, R. Moreau, Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 1. Fully-established flow, Eur. J. Mech. B/Fluids 21 (4) (2002) 383–398. [5] G. Authié, T. Tagawa, R. Moreau, Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures, Eur. J. Mech. B/Fluids 22 (3) (2003) 203–220.

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