Numerical Investigation on the Characteristics of Vapor-liquid Flow in

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Numerical Investigation on the Characteristics of Vapor-liquid Flow in the Heater of Self-evaporatingdriving Liquid Metal Magneto-hydro-dynamic System PENG LU, Xingwen Zheng, and Hulin Huang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02550 • Publication Date (Web): 10 Aug 2017 Downloaded from http://pubs.acs.org on August 13, 2017

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Industrial & Engineering Chemistry Research

Numerical Investigation on the Characteristics of Vapor-liquid Flow in the Heater of Self-evaporating-driving Liquid Metal Magneto-hydro-dynamic System Peng Lua*, Xingwen Zhenga, and Hulin Huangb Jiangsu Province Key Laboratory of Aerospace Power System, Key Laboratory of Thermal Environment and Structure of Ministry of Industry and Information, College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China b College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing , Jiangsu 210016, China a

ABSTRACT: Liquid metal magneto-hydro-dynamic (LMMHD) system is a promising power generation converter for a variety of heat sources. This paper presents a novel concept of self-evaporating-driving LMMHD system, in which the liquid metal is driven by the vapor from its own evaporation. It has advantages of safety, higher electrical conductivity, a simplified structure without mixer and separator, and therefore a more reliable and higher efficiency. Firstly, the basic principle of this system was introduced, in comparison with that of a conventional LMMHD system. Secondly, simulations were conducted to study the physical process of the evaporation of liquid metal in the heater and the characteristics of vapor-liquid flow in the following tube. Finally, three main impacting factors, namely inlet velocity, inlet temperature of liquid sodium, and temperature of pipe wall were discussed respectively. The findings verify the feasibility of the proposed self-evaporating-driving LMMHD system, and present the outlook for future research.

1. INTRODUCTION Nowadays, the rising price of energy, scarcity of fossil fuel resources and environmental pollution have led to the worldwide development of high-efficiency energy generation technologies.1 MHD (MagnetoHydroDynamic) power generation systems are built based on the extension of Faraday's law of induction to conductive fluid. Based on the types of conductive fluids, they can be divided into high-temperature plasma systems and liquid metal magneto-hydro-dynamic (LMMHD) systems. A high-temperature plasma power generation system requires a very high temperature to ionize the gas before power generating, which has a high efficiency without mechanical components. The main restriction is that in order to obtain a higher electrical conductivity, a high temperature is required.2 For example, the experimental ionization temperature for xenon plasma power generation is 5000~11500 K,3 and for argon plasma is 9000 K.4 Although ionization seeds can be added to lower this temperature, it still exceeds 2000~2500 K.5 Otherwise, the electrical conductivity of the gas would be too low to carry out the MHD power generating process. Additionally, high-temperature plasma power generation systems are stilling facing some technical problems to resolve, such as low conductivity and slagging. 6 By comparison, an LMMHD power generation system does not suffer from the above problems, which is based on an extension of Faraday's law of induction to liquid metal and has a few advantages. The thermal efficiency of an LMMHD power generation system is very close to the Carnot cycle efficiency owing to sustained heating on the expanding gas (known as the thermodynamic fluid) by the liquid metal (known as the power generation fluid). The liquid metal acts as an infinite heat source due to its high thermal capacity, while the

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expanding gas pushes the liquid metal to flow through the MHD channel to produce electricity. The gas expansion can be regarded as isothermal, which contributes to a high-efficiency conversion.7 Besides, an LMMHD system has few mechanical moving parts, leading to a quiet operation and a lower manufacturing and maintenance cost . Last but not the least, LMMHD systems can work under a wide temperature range from 300 K to 3000 K, which means that they can utilize a variety of heat sources, for example, solar, geothermal, fossil fuels, nuclear, etc.8

Pump Low-boiling point working medium loop Mixer

Condenser Cooling medium

N S

MHD generator Nozzle

Separator

Liquid metal loop Heat resource N

Heater S

MHD pump

Figure 1. Schematic of a general LMMHD system.

The schematic of a typical LMMHD system is shown in Figure 1. The colors indicate the temperatures at different locations, approximately. Red means high temperature, whereas blue means low temperature. This system has a liquid metal loop and a low-boiling point working medium loop. The liquid metal is heated in the heater. The low-boiling point working medium in liquid form enters the mixer at an appropriate pressure, where it contacts and mixes with the high-temperature liquid metal and evaporates. Then a vapor-liquid two phase flow is created inside the mixer and the follow-up pipeline. The vapor expands continuously and pushes the vapor-liquid flow through the MHD generator until it is separated from the liquid metal in the separator. Afterwards, the separated vapor enters a condenser, forming the low-boiling point working medium loop; and the liquid metal alone goes into the next liquid metal loop. LMMHD systems are gaining increasing attention in various institutions. Topics include flow and heat transfer,9 magnetic effects,10 working fluids effects,11 power generating characteristics,12 and bubble behaviours,13,

14

etc. However, the researches mentioned above are restricted to the

LMMHD system in which the flow is two phases of the low-boiling pointworking fluid and the liquid metal. The main downside is that the two-phase flow needs to be separated in a separator after power generation, which increases the complexity and the system cost. In addition, it should be pointed out that the low-boiling point working medium is usually flammable, combustible and toxic. In the present paper, as depicted in Figure 2, we propose a single loop LMMHD system, in which the liquid metal is driven by the vapor from its own evaporation, when the temperature is high enough to evaporate the liquid metal, as depicted in Figure 2. In our LMMHD system, the electrical conductivity of the fluid is dramatically increased. As the


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electric current density of MHD power generation is directly proportional to the flow velocity,15 the characteristics related to flow velocity distribution, along with the impacting factors, are major concerns. This paper is organized as follows: basic principles of the self-evaporating-driving LMMHD system are presented first, followed by the characteristics of the self-evaporating-driving two-phase flow, including the flow velocity, volume fraction distribution and the flow stability, etc.

2. BASIC PRINCIPLE OF SELF-EVAPORATION-DRIVING LMMHD SYSTEM Heat resource N

Heater

S

MHD generator

N

S

MHD pump

Condenser Cooling medium

Figure 2. Schematic of self-evaporating-driving LMMHD system.

The schematic of the self-evaporating-driving LMMHD system is depicted in Figure 2. It includes a heater, an MHD generator, a condenser, an MHD pump, etc. Compared to the conventional LMMHD system shown in Figure 1, two main components —the mixer and the separator are eliminated, and this system is single-loop, compact and low-cost. Thermal energy is added directly to the liquid metal, leading to a temperature higher than its boiling point in the heater, where bubbles are formed within the liquid metal. The expanding bubbles then push the liquid metal to flow through the LMMHD generator channel (where electrical power is obtained). After that, the vapor-liquid flow enters a condenser to be cooled to single-phase liquid metal and pumped back to the heater, thus completing the loop. Like the conventional LMMHD system, the vapor expands isothermally. Besides, due to the absence of mixer and separator, both the financial investment on the whole system and the energy loss associated with flow resistance can be reduced significantly. Furthermore, nonuse of the low-boiling point working medium, which is flammable, combustible and toxic, will improve the safety of the system. All in all, the thermodynamic cycle of the self-evaporating-driving LMMHD system is efficient and environmentally friendly. The present paper will simulate the physical process of the evaporation of liquid metal in the heater (nozzle) and the accelerated vapor-liquid flow in the following tube. Although the data-driven modelling has become increasingly popular within process system engineering due to its simplicity and efficacy, it requires clean data to build a reliable model.16 In this work, based on mass/energy balance modelling, we numerically investigated the characteristics of vapor-liquid two phase flow in the heater, in order to understand the impacts of the key inlet



conditions. Here the liquid sodium is selected as the working medium, and its physical properties are listed in Table 1. Table 1. Physical Properties of Liquid Sodium and Its Vapor. Physical properties liquid sodium Molar mass (g·mol-1) 22.9898 Density (kg·m-3) Boiling point (K) Latent heat of vaporization Heat capacity

(kJ·kg-1)

(J·kg-1·K-1)

sodium vapor 22.9898

927

1.686

1156

—

3521.5

—

1281.96

1300

Heat conductivity (W·m-1·K-1)

50

59.2

Viscosity (kg·m-1·s-1)

0.15×10-3

3.26×10-6

3. MODELING 3.1. Model Setup. A de Laval nozzle is employed here to better accelerate the fluid. To prevent the backflow phenomenon that may occur at the exit of the nozzle, a follow-up extended adiabatic tube is connected to stabilize the vapor-liquid two-phase flow. The structural model built by 3D Unigraphics (UG) software is plotted in Figure 3. 0.4m

0.2m

0.01m

Y

0.04m

Inlet Z

X

High temperature nozzle

Adiabatic tube

Outlet

0.2m

Figure 3. Structural model built by UG.

3.2. Mesh Generation. The above model was imported into Gambit software. Hexahedral mesh grids were structured with the Cooper method, and the mesh quantity was set to be 1.1×105, 2.6×105, 5.3×105 and 8.5×105 respectively. The pressure distribution and velocity distribution of liquid sodium along the central axis were obtained under different mesh quantities, as shown in Figure 4. (a) 6

×105 5 1.1×10 5 2.6×10 5 5.3×10 5 8.5×10

4

Pressure/Pa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2 0 -2 -4 -6 0.0

0.1

0.2

0.3

0.4

0.5

X/m


0.6

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(b) 5 1.1×10 5 2.6×10 5 5.3×10 5 8.5×10

50 40

Velocity/(m·s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


30 20 10 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

X/m Figure 4. The grid independence test and verification: (a) Pressure distribution; (b) Velocity distribution.

3.3. Simulation Process. Firstly, the above mesh was introduced into FLUENT software. Separation implicit solver and SIMPLE scheme were applied to resolve pressure-velocity coupling equation. Combined with the k-ε two-equation turbulent model, standard wall functions were employed to the flow in near-wall region. The second-order upwind difference scheme was also applied to improve the computational accuracy by discretizing the physical parameters of volume control interface, with the convergence precision 10 -6 of the residual monitor for energy equation. Secondly, conversion of mass source and energy source were compiled by User Defined Function (UDF) and coupled to flow functions. Thirdly, a mixture model was selected with inter-phase velocity slip taken into account. Finally, governing equations can be described as follows. 1) Continuity Equation The continuity equation for the mixture is

  m     m v m  0 t



where

vm

vm 

(1)

is the mass-averaged velocity:

n

 k 1



k

k v k

(2)

m

and ρm is the mixture density: n

m    k k

(3)

k 1

where αk is the volume fraction of phase k. 2) Momentum Equation The momentum equation for the mixture can be obtained by summing the individual momentum equations for all phases.17 It can be expressed as







T   n  m vm   m vm vm  p    m v m  v m   m g  F     k k vdr,k vdr,k  ;   t  k 1 



 



(4)

where n is the number of phases, F is a body force, and µm is the viscosity of the mixture: n

m    k k

(5)

k 1

v dr ,k

is the drift velocity for secondary phase k:

vdr,k  v k  v m

(6)

3) Energy Equation n  n  k v k  k Ek  p       keff T   SE ;   E       k k k    t k 1 k 1

n

keff is the effective conductivity

 k 1

k

(7)

(kk  kt ) , where kt is the turbulent thermal conductivity,

and is defined according to the turbulence model being used. The first term on the right-hand side of Equation (7) represents energy transfer due to conduction. SE includes any other volumetric heat sources which do not exist in the present paper. The model (including turbulence model and multiphase model etc.) in the paper was used to simulate the gas-liquid flow process through a round tube with a smooth linear contraction in literature.18 The results are presented in Figure 5, which shows the pressure distribution along the axial direction at different Reynolds number. It can be clearly seen that both the variation trend and the values are in excellent agreement with those in literature18, with an error between nearly 0 and 10%. According to Bernoulli equation, the significant drop of pressure at around z=0.25 m is caused by fluid acceleration near the converging section. Num.(ReL=136000, β=0)

140 Pressure, P (mbar)

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Num.(ReL=79300, β=0) Num.(ReL=95100, β=0.11)

120 100 80 60 40 20 0 0.0

0.1

0.2

0.3

0.4

Axial position, z(m)


0.5

0.6

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Figure 5. The numerical simulation results by the model and method in the present work.

3.4. Boundary Conditions Setting. The inlet boundary condition of the computational domain was set as velocity-inlet, while the outlet condition was pressure-outlet with standard atmospheric pressure. The boundary conditions of pipe wall were set as impermeable and non-slip, with constant wall temperature, adiabatic outside surface, and wall roughness of 5×10 -5 m. We assume that: 1) The physical properties of liquid metal (liquid sodium) and sodium vapor are constant; 2) The radiation heat transfer between liquid and gas is ignored; 3) There is no heat exchange between the system and the outside.

4. RESULTS AND DISCUSSION 4.1. Overall Impacts of Boundary Conditions on the Vapor-liquid Flow Process. The results of numerical simulation show that generally the boundary conditions of inlet velocity and initial temperature of liquid sodium, and pipe wall temperature have minimal impacts on the distributions of velocity, temperature and volume fraction of vapor-liquid flow, except for the changes in magnitude. For instance, Figure 6 exhibits the distributions of velocity, temperature and volume fraction of liquid sodium along the longitudinal section under two different inlet velocities of 1.8 m·s-1 and 3.0 m·s-1, with initial temperature of liquid sodium at 1155 K and pipe wall temperature at 1773 K.

(a)

(b)

(c)

Figure 6. Distributions of three key parameters under inlet velocities of 1.8 m·s-1 and 3.0 m·s-1 : (a) Velocity(m·s-1); (b) Temperature(K); (c) Volume fraction of liquid sodium.

Figure 6 shows that although the magnitudes may differ, the vapor-liquid mixture flows faster as the inlet-velocity of liquid sodium increases. Due to the higher velocity of vapor-liquid mixture flow, the duration time of heat transfer form high -temperature wall



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pipe to the fluid is shorter, which leads to the higher volume fraction distribution of liquid sodium, shown in Figure 6c. It is noticeable from Figure 6c that the evaporation from liquid sodium to sodium vapor occurs in the near-wall region, and the vapor accounts for much more proportion at the back part of the pipe line. As an example, the liquid sodium flow and heat transfer process throughout the selected region under the conditions of inlet velocity at 2.4 m·s -1, inlet temperature at 1155 K and wall temperature at 1773 K was discussed as follows. Figure 7b and Figure 8b show that the liquid sodium near the wall evaporates first due to heat from high-temperature pipe walls. As a result of vapor expansion and cross section narrowing, the fluid velocity increase s rapidly, especially for the fluid adjacent to the wall (Figure 7a and Figure 8a). In the follow-up diverging pipeline, although the fluid velocity drops due to the increasing cross-section area, the outlet fluid velocity remains higher than the inlet because of the continuous heating from the hot pipe wall. Most noticeably, the variation of flow regime and stability can be inferred from Figure 7c and Figure 8c. With the consistent evaporation of liquid sodium by heat, the volumetric fraction of sodium vapor keeps rising, particularly in the area adjacent to the wall, which means that the main liquid sodium flows in the central pipe line and is isolated from the wall by the annular sodium vapor. This annular flow regime may negatively affect the subsequent power generation in magnetic channel, according to Q. Wu.7 The extended adiabatic tube, as shown Figure 7c, plays an important role in the flow stabilization, in which the annular flow transforms into a much more uniform flow pattern.

(a)

(b)

(c)

0

0.1

0.2

0.3

0.5

0.4


Adiabatic tube

Figure 7. Fluid (a)velocity, (b)temperature and (c)VF at different X position.


0.6

X/m

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(a) X = 0m X = 0.1m X = 0.2m X = 0.3m X = 0.4m

60

v/(m·s-1)

50 40 30 20 10 0 -0.02

-0.01

0.00

0.01

0.02

Z/m (b) 1755

X = 0m X = 0.1m X = 0.2m X = 0.3m X = 0.4m

1655

T/K

1555 1455 1355 1255 1155 -0.02

-0.01

0.00

0.01

0.02

Z/m

(c) 100 X = 0m X = 0.1m X = 0.2m X = 0.3m X = 0.4m

80

VF/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60 40 20 0 -0.02

-0.01

0.00

0.01

0.02

Z/m Figure 8. Fluid (a)velocity, (b)temperature and (c)VF distribution along Y=0 at different X position.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

(b)

Figure 9. The typical flow field along the longitudinal section of pipeline: (a) With gravity; (b) Without gravity.

Further, a typical flow field along the longitudinal section of pipeline was obtained (Figure 9a). It is noted that two different sized vortexes can be observed very close to the pipe wall of nozzle divergence cone. The bigger anticlockwise one is shown on the top and the smaller clockwise one is given on the bottom. The vortexes may be caused by pressure changes. In the diverging section of the pipeline, the fluid velocity will be reduced due to the increasing cross-sectional area, based on continuity equation. As a result, the corresponding local pressure will rise according to the Bernoulli equation. Particularly, at the upper near-wall region of nozzle divergence cone, the pipe wall resistance can further decrease the fluid flow velocity to a minimal value, where the local pressure will be much higher. This high pressure will then drive the fluid off the wall and backward to the main stream, which forms an anticlockwise vortex. Similarly, the vortex direction will be cl ock wise at the bottom near-wall region. Additionally, it is noted that the lower vortex is smaller than the upper one . To explain this, Figure 9b depicts the flow field along the longitudinal section of pipeline with zero gravity, from which we can see that the lower vortex under gravity effect is smaller but more congested.

4.2. Impacts of Inlet Velocity of Liquid Sodium on the Outlet


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Characteristics. (a)

14

1700 vout

13

Tout

1600

11 10

1500

Tout/K

vout/(ms-1)

12

9 8

1400

7 6

1300 0.6

1.2

1.8

2.4

3.0

vin/(ms-1)

(b)

25

2.5 VFout

20

2.0

15

1.5

10

1.0

5

Er/%

Er

VFout/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5 0.6

1.2

1.8

2.4

3.0

vin/(ms-1) Figure 10. Impacts of inlet velocity of liquid sodium on the outlet characteristics: (a) Velocity and temperature; (b) Volume fraction of liquid sodium and evaporation rate.

How the inlet velocity of liquid sodium affects the outlet parameters of velocity, temperature, volume fraction of liquid sodium and evaporation rate are depicted in Figure 10. As the inlet velocity of liquid sodium increases from 0.6 m∙s-1 to 3.0 m∙s-1, the outlet velocity rises from 6.9 m∙s-1 to 12.9 m∙s-1, and the volume fraction of liquid from 8.3% to 23.1%. By contrast, the outlet temperature drops from 1588.4 K to 1384.5 K, and evaporation rate from 1.92% to 0.60%. The reason is understandable as the duration time of heat transfer from high-temperature pipe wall to the fluid shortens. Although the relatively high outlet velocity and volume fraction of liquid sodium will lead to better power generation in the follow-up magnetic channel, it should be pointed out that higher inlet velocity of liquid sodium need to consume more MHD pump power, which in turn will increase the running cost of the whole system, and will reduce the overall efficiency.

4.3. Impacts of Inlet Temperature of Liquid Sodium on the Outlet Characteristics.



(a)

1500 vout

12

Tout

1200

6 4

1100 755

(b)

Tout/K

1300

8

855

955 Tin/K

1055

1155

60

0.8 VFout Er

50

0.6

40 0.4

Er/%

vout/(m·s-1)

1400 10

VFout/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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30 0.2

20 10

0.0 755

855

955 Tin/K

1055

1155

Figure 11. Impacts of inlet temperature of liquid sodium on the outlet characteristics: (a) Velocity and temperature; (b) Volume fraction of liquid sodium and evaporation rate.

With the inlet temperature of liquid sodium growing, the variations in the outlet parameters of velocity, temperature, volume fraction of liquid sodium, and evaporation rate are illustrated in Figure 11. It can be seen from the figure that, with the inlet temperature of liquid sodium increasing, three outlet parameters increase, i.e., the outlet velocity, outlet temperature and evaporation rate; while one parameter volume fraction of liquid sodium declines. It is understandable, because liquid sodium with a higher inlet temperature means that it is more easily transformed into sodium vapor. As a result, the evaporation rate increases from 0.16% to 0.72%, which speeds up the outlet velocity dramatically from 4.4 m∙s-1 to 11.9 m∙s-1. However, the adverse effect of this process is that volume fraction of liquid sodium plummets from 53.6% to 19.9%, which may result in a large proportion of continuous sodium vapor film between the liquid sodium and pipe wall. In other words, the undesirable flow regime of annular flow may appear, causing a poor contact between the liquid sodium and the electrodes on the wall of the follow-up magnetic power generation channel. The poor contact may hinder or even break the electric current loop, and therefore reduces power generation efficiency.


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(a)

(b)

(c)

Figure 12. VFout on the circular cross-section under three different Tin: (a) Tin=755 K; (b) Tin=955 K; (c) Tin=1155 K.

More specifically, Figure 12 provides the volume fractions of liquid sodium on the circular

cross-section under three different inlet temperature of liquid sodium. Clearly, the annular flow of sodium vapor is observed when the inlet temperature of liquid sodium reaches 1155 K, with the volume fraction of liquid sodium only at 20% approximately. As the inlet temperature of liquid sodium decreases to 755 K, the volume fraction of liquid sodium increases to about 54%. Wallis 19 pointed out that the volume fraction of liquid sodium should be kept above 20% to avoid the annular flow.

4.4. Impacts of Temperature of Pipe Wall on the Outlet Characteristics. (a)

13

1500 vout

12

Tout

11

1400 Tout/K

vout/(ms-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 9

1300

8 7

1200 1373

1473

1573 Tw/K

1673


1773


(b)

40

0.8 VFout

0.7

Er

30

0.6 0.5

20

Er/%

VFout/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4 0.3

10

0.2 1373

1473

1573 Tw/K

1673

1773

Figure 13. Impacts of temperature of pipe wall on the outlet characteristics: (a) Velocity and temperature; (b) Volume fraction of liquid sodium and evaporation rate.

The variation tendency of the outlet parameters with the increasing temperature of pipe wall is plotted in Figure 13. With the temperature of pipe wall increasing from 1373 K to 1773 K, the outlet velocity rises from 7.7 m∙s-1 to 11.9 m∙s -1, the outlet temperature from 1248.5 K to 1407.5 K, and evaporation rate from 0.40% to 0.72%. Conversely, the volume fraction of liquid sodium reduces from 31.0% to 19.9%. Table 2. The Heat-transfer Coefficients under Different Temperature of Pipe Wall. Tw/K Heat-transfer coefficient W/(m2•K)

1373

1473

1573

1673

1773

12258

16014

19122

21814

24179

The reason for the above results can be further interpreted from the perspective of heat-transfer coefficient. The heat-transfer coefficients under different temperature of pipe wall are presented in Table 2. It indicates that with the temperature of pipe wall increasing from 1373 K to 1773 K, the heat-transfer coefficient almost doubles, which means more evaporation of liquid sodium, thus more sodium vapor produced acting as driving gas, and a higher outlet velocity as a result. Yet, it should be noted that the volume fraction of liquid sodium may drop to below 20% due to the excessively high temperature of pipe wall. Therefore, the temperature of heater in Figure 2 should be heated up to a reasonable range in a controlled way.

5. CONCLUSIONS A novel concept of self-evaporating-driving LMMHD system is proposed in the present work. In this system, the power generation working medium liquid metal is driven by the vapor from its own evaporation. In this way, the electrical conductivity of the fluid can be dramatically increased. More importantly, nonuse of the low-boiling point working medium, mixer and separator makes the system less expensive, safer, more reliable and efficient. A conceptional scheme and basic working principles of self-evaporating-driving LMMHD system was introduced in the first place. In addition, by simulation investigation, more discussions were provided on the physical process of the evaporation of liquid metal in the heater (nozzle) and characteristics of the accelerated vapour-liquid flow until the inlet of magnetic power generation channel. The objective of such an investigation is to verify the feasibility of the self-evaporating-driving LMMHD system.


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The present results indicate that the boundary conditions have minimal impacts on the distributions of velocity, temperature and volume fraction of vapor-liquid flow, with only differences in value. Improving the inlet velocity of liquid sodium can lead to a higher outlet velocity and a higher volume fraction of liquid, which is beneficial to the follow-up power generation process. However, more MHD pump power will be needed. Moreover, raising the inlet temperature of liquid sodium and temperature of pipe wall will, on the one hand speed up the outlet velocity; on the other hand, it will lower the volume fraction of liquid sodium. To avoid the undesirable annular flow regime, the inlet temperature of liquid sodium and temperature of pipe wall should be controlled in a reasonable scope. Further work should be focused on engineering design, economic studies and microscopic bubble behaviors as well.

 AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

ORCID Peng Lu: 0000-0002-0938-1351

Notes The authors declare no competing financial interest.

 NOMENCLATURE B = strength of the magnetic field, T Er = evaporation rate, % F = body force, N

g = gravitational acceleration, m·s-2

J = current density, A·m-2 keff = effective heat transfer conductivity, W·m-1·K-1 kk = heat transfer conductivity of phase k, W·m-1·K-1 kt = turbulent thermal conductivity, W·m-1·K-1 n = number of phases p = pressure, Pa ReL = Reynolds number of liquids SE = volumetric heat sources, W·m-3 T = temperature, K Tin = inlet temperature of liquid metal, K Tout = outlet temperature of liquid metal, K Tw = temperature of pipe wall, K v = velocity, m·s-1

v dr,k = drift velocity of phase k, m·s-1 vin = inlet velocity of liquid metal, m·s-1

v k = mass-average velocity of phase k, m·s-1 v m = mass-average velocity of mixture, m·s-1

vout = outlet velocity of liquid metal, m·s-1 VF = volume fraction of liquid metal, % VFout = outlet volume fraction of liquid metal, % X = axis X, m



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Y = axis Y, m Z = axis Z, m Greek Symbols αk = volume fraction of phase k, % β = void fraction σ = electrical conductivity, S·m-1 ρk = density of phase k, kg·m-3 ρm = density of mixture, kg·m-3 μk = viscosity of phase k, kg·m-1·s-1 μm = viscosity of mixture, kg·m-1·s-1

 ACKNOWLEDGMENTS This work is supported by "National Natural Science Foundation of China'' (NO. 11675077; 51506087); “Fundamental Research Funds” (NO. JCKY2013203B003); "China Postdoctoral Science Foundation" (NO. 2015M571747); “the Fundamental Research Funds for the Central Universities” (No. NJ20160041; NS2015017), and “Foundation of Graduate Innovation Center in NUAA” (NO. kfjj20160210).

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(10) Miao, X.; Lucas, D.; Ren, Z.; Eckert, S.; Gerbeth, G. Numerical modeling of bubble-driven liquid metal flows with external static magnetic field. Int. J. Multiphase Flow. 2013, 48, 32-45. (11) Hu, L.; Kobayashi, H.; Okuno, Y. Performance of a liquid metal MHD power generation system for various external forces. In 12th International Energy Conversity Engineering Conference, 2014; pp 1-8. (12) Kim, C. N. Numerical examination of liquid metal magnetohydrodynamic flow in multiple channels in the plane perpendicular to the magnetic field. J. Mech. Sci. Technol. 2014, 28, 4959-4968. (13) Zhang, J.; Ni, M.-J.; Moreau, R. Rising motion of a single bubble through a liquid metal in the presence of a horizontal magnetic field. Phys. Fluids. 2016, 28, 032101-1-032101-17. (14) Pavlovs, S.; Jakovics, A.; Baake, E.; Sushkovs, V. Gas bubbles and liquid metal flow influenced by uniform external magnetic field. Int. J. Appl. Electromagn. Mech. 2017, 53, 31-41. (15) Hollenstein, L.; Jain, R. K.; Urban, F. R. Cosmological Ohm's law and dynamics of non-minimal electromagnetism. J. Cosmol. Astropart. Phys. 2013, (1), 013-1-013-23. (16) Xu, S.; Lu, B.; Baldea, M.; Edgar, T. F.; Wojsznis, W.; Blevins, T.; Nixon, M. Data cleaning in the process industries. Rev. Chem. Eng. 2015, 31, 453-490. (17) Wang, P.; Sun, H.; Wong, P. Y.; Fukuda, H.; Ando, T. Modeling of droplet-based processing for the production of high-performance particulate materials using the level set method. Numer. Heat Transfer, Part A. 2012, 61, 401-416. (18) Ahmadpour, A.; Noori Rahim Abadi, S. M. A.; Kouhikamali, R. Numerical simulation of two-phase gas–liquid flow through gradual expansions/contractions. Int. J. Multiphase Flow. 2016, 79, 31-49. (19) Wallis, G. B. One-dimensional two-phase flow. McGraw-Hill: New York, 1969.



TOC (For Table of Contents Only)

v (m·s-1) at different X position 0

0.3

X

0.4

0.5

0.6

X/m

Adiabatic tube

0.01m


Inlet Z

0.2

0.1

Y

0.04m

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Outlet

Numerical Investigation on the Characteristics of Vapor-liquid Flow in

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