Numerical study of natural convection in a liquefied ... - Springer Link

Journal of Mechanical Science and Technology 26 (10) (2012) 3133~3140 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0820-x

Numerical study of natural convection in a liquefied natural gas tank† Sangeun Roh and Gihun Son* School of Mechanic Engineering, Sogang University, Seoul, 121-742, Korea (Manuscript Received December 15, 2011; Revised May 12, 2012; Accepted June 17, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Natural convection in a liquefied natural gas storage tank, which is caused by the heat absorption from the surrounding environment, is studied numerically by solving the conservation equations for mass, momentum, and energy with a commercial computational fluid dynamics code, FLUENT. The present numerical results show that the interfacial heat transfer rate, which is directly related to the boil-offgas generation rate, strongly depends on the liquid-solid contact area. The contribution of the heat transfer rate from the vapor region is negligible compared with that from the liquid region. The effects of the external convection coefficient, tank size, and tank shape on the flow and temperature fields and on the interfacial heat flux are quantified. Keywords: Boil-off gas; Cryogenic storage tank; Liquefied natural gas; Natural convection ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Natural gas has recently attracted increasing attention as a clean fuel source. This resource is typically stored and transported in the liquid form because liquefied natural gas (LNG) has approximately one-six hundredth of its volume when in the gaseous state. One of the main issues in the treatment of LNG storage tanks, which should be kept at very low temperatures, is the vaporization of the liquid or boil-off gas (BOG) caused by heat absorption from the surrounding environment. A fundamental understanding of BOG generation is essential for the reduction of financial loss and for the prevention of possible safety risks, such as fractures in tanks. BOG generation is directly related to the natural convection heat transfer in a cryogenic storage tank. Despite the extensive studies on natural convection, few reports have been made on cryogenic storage tanks. Boukeffa et al. [1] and Khemis et al. [2] conducted an experimental analysis of heat transfer in a liquid nitrogen cryostat. The estimation of heat transfer can be used to predict the evaporation of stored cryogenic fluids and to minimize heat loss from a cryostat. Kanazawa et al. [3] experimentally investigated the fluid flow and heat transfer in an LNG tank under high-Rayleighnumber conditions (7.5 × 1010 < Ra < 1.5 × 1013). Using the Schlieren technique, they visualized the convective flow pattern in the tank and observed that the flow circulation pattern *

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depends on the heat transfer conditions at the bottom and side walls. Belmedani et al. [4] conducted an experimental and theoretical study on the natural convection in a liquid nitrogen tank. They developed a theoretical model for the wall and interfacial thermal boundary layers using the integral approach and their experimental data. The model provided the velocity and temperature profiles at various heat fluxes, with the boundary thickness as a function of evaporation rate. Prasanth–Kumar et al. [5] performed a numerical simulation of natural convection and evaporation in a liquid hydrogen tank using a homogeneous two-phase model. The computation shows the effect of interfacial evaporation on the thermal stratification in the tank. Khelifi–Touhami et al. [6] presented a more detailed computational work on the laminar natural convection flow in an LNG storage tank under low Rayleigh number conditions (103 < Ra < 105). The liquid-vapor interface was treated as one of the computational boundaries, and the evaporative heat flux was obtained from the Hashemi-Wesson’s law. The numerical results show that the circulation flow becomes stronger as the Rayleigh number increases and that the evaporative heat flux is not uniform along the liquid-vapor interface. In this study, we investigate the natural convection in an LNG storage tank and the associated heat flux at the liquidvapor interface by determining the BOG generation rate. The conservation equations for mass, momentum, and energy are solved using a commercial computational fluid dynamics (CFD) code, FLUENT. The effects of the external convection coefficient, the tank size, and the tank shape on the flow and

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S. Roh and G. Son / Journal of Mechanical Science and Technology 26 (10) (2012) 3133~3140

Gk = µt ∇u + (∇u )T  : (∇u)

C1ε = 1.44 C2ε = 1.92 Cµ = 0.09

σ t = 0.85 σ k = 1.0 σ ε = 1.3 . 2.2 Boundary condition

Fig. 1. Configuration of the storage tank used in the present computation.

temperature fields, and on the interfacial heat flux are quantified.

u = 0,

∂v ∂T = =0. ∂x ∂x

At the side wall ( x = 0.5Wt ),

2. Numerical analysis Fig. 1 shows the configuration of the LNG storage tank used in this study. The flow is treated as two-dimensional. All of the fluid properties, except for density, are considered constant. The density variation attributed to temperature is described by the Boussinesq approximation for the evaluation of buoyancy force. In the computation for the natural convection in a storage tank, the liquid-vapor interface is assumed to remain stationary. This assumption may not be so restrictive because the interface velocity attributed to the liquid-vapor phase change is relatively small in comparison with the mean liquid and vapor velocities. 2.1 Governing equation Based on the Boussinesq approximation for the buoyancy effect and the k -ε turbulence modeling for high Rayleigh number conditions, the equations governing the conservation of mass, momentum, energy, turbulent kinetic energy (k), and turbulent dissipation rate (ε) for the liquid and vapor phases are written as ∇ ⋅u = 0

(1)

∇ ⋅ ( ρ uu) = −∇p − ρβ g (T − Tref ) + ∇ ⋅ ( µ + µt ) ∇u + (∇u)T 

(2)  c p µt ∇ ⋅ ( ρ c puT ) = ∇ ⋅  λ + σt 

   ∇T   

 µ ∇ ⋅ ( ρ uk ) = ∇ ⋅  µ + t σ k 

  µt  ∇k  + Gk + β g ⋅ ∇T − ρε σ t  

 µ ∇ ⋅ ( ρ uε ) = ∇ ⋅  µ + t σ ε 

  ε ε2  ∇ε  + Gk + C1ε ρ Gk − C2ε ρ k k  

u = v = 0, λ

∂T = ha (Ta − T ) . ∂x

At the bottom wall ( y = 0 ), u = v = 0, − λ

∂T = ha (Ta − T ). ∂y

At the liquid-vapor interface ( y = H l ), ∂u = 0, v = 0, T = Tsat . ∂y

Here, Ta is the ambient temperature, and ha is an external heat transfer coefficient that includes the ambient convection and the thermal resistances of the tank wall and its insulation. For the LNG (or methane) used in the present computation, the saturation temperature at 1 atm is 113.36 K. All of the fluid properties, including the density, viscosity, specific heat, thermal conductivity, and the thermal expansion coefficient, are evaluated at the reference temperature. For the vapor region, the boundary conditions are as follows (refer to Fig. 1): At the symmetry plane ( x = 0 ),

(3) u = 0,

∂v ∂T = =0. ∂x ∂x

(4) At the side wall ( x = 0.5Wt ), u = v = 0, λ

(5)

∂T = ha (Ta − T ) . ∂x

At the liquid-vapor interface ( y = H l ),

where

µt = cµ ρ

Computations are separately performed for the liquid and vapor regions. For the liquid region, the boundary conditions are as follows (refer to Fig. 1): At the symmetry plane ( x = 0 ),

k2

ε

∂u = 0, v = ve , T = Tsat . ∂y


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At the vent ( 0 ≤ x < 0.5Wv and y = H t ), ∂u ∂v ∂T = = =0. ∂y ∂y ∂y

At the top wall ( 0.5Wv ≤ x < 0.5Wt and y = H t ), u = v = 0, − λ

∂T = ha (T − Ta ) . ∂y

Here, ve is the vapor velocity caused by the evaporation at the liquid-vapor interface, which is evaluated from the energy balance at the interface, ve = qliq /ρghfg.

(a)

2.3 Numerical scheme The commercial CFD code FLUENT 6.3.2 is employed to solve the conservation equations. The Semi-Implicit Method for Pressure Linked EquationConsistent pressure-velocity coupling algorithm, which has an improved behavior over the Semi-Implicit Method for Pressure Linked Equation algorithm, was chosen for the current calculations to obtain the converged solutions more quickly. For the pressure interpolation required to solve the momentum equation, the Pressure Staggering Option scheme is used for its effectiveness in solving high-Rayleigh-number natural convection problems. A second-order upwind scheme is chosen for the convection terms in the conservation equations. In the turbulent k-ε model, the logarithmic wall function is used at the side and bottom walls. For the present computations, the results from the standard wall function and the enhanced wall models show no remarkable differences. For the sufficient convergence of the iterative solutions, 10-7 was used for the energy equation, and 10-5 was used for the other conservation equations.

(b) Fig. 2. Effect of mesh size (∆) on (a) the velocity field and on (b) the temperature field.

3. Results and discussion The computations are performed for the steady natural convection in a liquefied methane gas tank. The following values were chosen as the base case: Wt = 1 m, Hl = 1 m, Ht = 2 m, ha = 1 W/m2K, Ta = 300 K, and Τsat = 113.36 K. 3.1 Liquid-side natural convection The present computations are performed by using two grid systems with 8450 and 11476 cells, respectively, to determine an appropriate mesh size for the computation of natural convection. Fine meshes are used in the thin boundary layers near the side and bottom walls, as well as in the liquid-vapor interface, whereas coarse meshes for the center of tank to save computational time. The minimum mesh sizes (∆) for the fine and coarse meshes are 0.1 and 0.2 mm, respectively. The results are plotted in Figs. 2 and 3. The velocity and temperature

Fig. 3. Effect of mesh size (∆) on the interfacial heat flux distribution.

fields, as well as the interfacial heat fluxes, for the two grid systems show no significant differences. The use of a grid system with a minimum mesh size of 0.2 mm provides a good compromise between the simulation effort and the numerical accuracy. The heated fluid is observed to move upward along the bottom and side walls because of the buoyancy force that pulls the lighter fluid up, whereas the fluid is cooled near the liquid-vapor interface and moves downward along the symmetry plane. This process induces a counter clockwise circulation occupying the entire

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(a) Fig. 4. Pressure variation along the interface.

liquid region of the tank. The liquid is superheated to 115.39 K, which corresponds to a superheat of 2.04 K. The heat transferred to the interface from the bulk liquid contributes to BOG generation. Fig. 3 shows that the interfacial heat flux has a slow variation along the interface, except near the wall and the symmetry plane, where the upward and downward flows are evident. The heat transfer rate integrated over the interface area (interface line in two dimensions) is 278.4 W/m. In this study, we assume that the liquid-vapor interface temperature is constant (Tsat). In reality, the saturation temperature is a function of the saturation pressure. To verify the constant interface temperature assumption, the pressure distribution is plotted along the liquid-vapor interface in Fig. 4. The pressure is observed to vary within 0.40 Pa, and the corresponding interface temperature variation is less than 10-4 K.

(b) Fig. 5. Vapor-side velocity and temperature fields with different vapor inlet velocities: (a) ve = 0 m/s; (b) ve = 6 × 10-4 m/s.

3.2 Vapor-side natural convection The calculation of vapor-side natural convection requires the vapor velocity ve caused by the evaporation at the liquidvapor interface. The vapor velocity can be evaluated from the energy balance at the interface as ve = qliq /ρghfg ≈ 6.0 × 10-4, assuming that the vapor-side contribution is small. Figs. 5 and 6 present the results for two cases: ve = 0.0 m/s and ve = 6 × 104 m/s. For the case with no evaporation (ve = 0), the temperature field is stratified, as observed in the positive (or stabilizing) temperature distribution in the vertical direction. The associated vapor velocity is nearly stationary, and its maximum velocity is 0.01 m/s. The total heat transfer rate integrated over the interface area is 14.2 W/m, which is 5% of that from the liquid region. This result implies that the vapor-side contribution to the interfacial heat transfer and thus, the BOG generation, is negligible compared with the liquid-side contribution. For the case of ve = 6.0 × 10-4 m/s, where the evaporation effect is included, the velocity and temperature fields are changed significantly, as depicted in Fig. 5(b). The isotherms are pushed upwards by the vertical vapor velocity. The vapor temperature at the vent is 0.21 K higher than the saturation temperature at the interface. The heat rate absorbed from the ambient convection for the sensible heat is 240 W/m. The

Fig. 6. Interfacial heat flux distribution in the vapor region.

interfacial heat transfer rate is reduced to 0.018 W/m. This result confirms that the vapor-side contribution to BOG generation is negligible. Thus, the vapor-side natural convection is not included in the analysis of other cases. 3.3 Effect of external heat transfer coefficient The external heat transfer coefficient ha depends on the ambient convection and on the thermal resistances of the tank wall and the insulation. To investigate the effect of ha on the liquid-side natural convection in a tank, two cases are tested with ha = 0.2 W/m2K and ha = 5 W/m2K. The results are plot-

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Table 1. Comparison of the actual and ideal interfacial heat transfer rates for various external heat transfer coefficients. q' [W/m]

q'max [W/m]

q'/q'max

55.9

56.1

0.996

ha = 1.0 [W/m K]

278.4

280.5

0.992

ha = 5.0 [W/m2 K]

1375

1403

0.980

ha = 0.2 [W/m2 K] 2

(a)

(a)

(b) Fig. 7. Effect of external heat transfer coefficient on the velocity and temperature fields: (a) ha = 0.2 W/m2K; (b) ha = 5.0 W/m2K.

(b) Fig. 9. Effect of tank size on the velocity and temperature field: (a) Slength = 2; (b) Slength = 4.

q 'max = ha Lsl (Ta − Tsat )

Fig. 8. Effect of external heat transfer coefficient on the interfacial heat flux distribution.

ted in Figs. 7 and 8. As the external heat transfer coefficient increases, the flow velocity increases, although its direction is invariant. As the ha increases from 0.2 W/m2K to 1 W/m2K and 5 W/m2K, the maximum liquid superheat increases from 0.63 K to 2.04 K and 6.51 K, respectively. The respective total interfacial heat rate also increases from 55.9 W/m to 278.4 W/m and 1374.7 W/m. The interfacial heat rate, as well as the BOG generation rate, varies with ha. When the superheating of the LNG is negligible, the ideal interfacial heat transfer rate per unit of the tank length (z-directional length), designated as q'max, is given from the energy balance as:

where Lsl is the liquid-solid contact area per unit of tank length. The actual interfacial heat transfer rates and the ideal interfacial heat transfer rates are compared in Table 1. The actual heat transfer rates are shown to deviate from the ideal values as ha increases. 3.4 Effect of tank size To study the effect of tank size on the interfacial heat rate, computations are performed for the tank geometry enlarged by the same ratio, Slength, in both the x and y directions. Fig. 9 shows the flow and temperature fields in the tanks scaled up by factors of two (Slength = 2) and four (Slength = 4) while other conditions are kept constant. The flow and temperature patterns are similar to those shown in Fig. 2 for the base case. The maximum liquid superheats for Slength values of 2 and 4 are 1.96 and 2.07 K, respectively. The associated interfacial

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Fig. 10. Effect of tank size on the interfacial heat flux distribution.

Fig. 12. Effect of the aspect ratio of the tank on the interfacial heat flux distribution.

(a) (a)

(b) Fig. 11. Velocity and temperature fields with a large aspect ratio of Wt/Hl = 4. (b)

heat flux is plotted in Fig. 10. The heat flux is observed to depend weakly on the length scale. The total heat rates integrated over the interface area for Slength values of 1, 2, and 4, are 278.4, 556.8, and 1,112 W/m, respectively. The interfacial heat transfer rate is linearly proportional to the Slength. 3.5 Effect of tank aspect ratio Figs. 11 and 12 show the effect of the tank aspect ratio on the natural convection in the tank when the liquid volume and the liquid-solid contact area are both kept constant. When the aspect ratio Wt/Hl is increased to 4, the maximum superheat is slightly decreased to 1.78 K. The interfacial heat flux decreases with Wt/Hl, but the total heat rate integrated over the interface area is identical within a 0.13% deviation. This result indicates that the interfacial heat rate, as well as the BOG generation rate, is independent of the interfacial area as long as the liquid-solid contact area is unchanged.

Fig. 13. Schematic of the storage tank with circular cross-sections: (a) Hl = Rt; (b) Hl = 1.5Rt .

3.6 Effect of tank shape In this work, efforts are made to reduce the interfacial heat transfer rate coupled with BOG generation by using other tank geometries. Two tank geometries with circular cross-sections were tested, wherein the total liquid volume was kept constant, as depicted in Fig. 13. The results are plotted in Figs. 14 and 15. The flow velocity is larger for Hl = 1.5Rt than for Hl = Rt. The maximum superheats for Hl = Rt and Hl = 1.5Rt are 1.40 and 1.42 K, respectively. The total heat transfer rates integrated over the interface area for Hl = Rt and Hl = 1.5Rt are 233.3 and 245.6 W/m, respectively. This condition indicates that the use of a tank with a semicircular cross-section (Hl = Rt) can reduce the interfacial heat transfer rate or the BOG generation rate by 16% in comparison with a rectangular tank.


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Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) Grant No. 20090083510 funded by the Korean government (MEST) through the Multi-phenomena CFD Engineering Research Center.

Nomenclature-----------------------------------------------------------------------(a)

(b) Fig. 14. Velocity and temperature fields in the storage tank with circular cross-sections: (a) Hl = Rt; (b) Hl = 1.5Rt .

cp g Hl Ht ha hfg k Lsl Slength p Rt T Ta Tref Tsat u ve Wt β ε λ µ µt ρ σ

: Specific heat : Gravity : Height of liquefied gas : Height of tank : Convective heat transfer coefficient : Latent heat of evaporation : Turbulent kinetic energy : Liquid-solid contact area per unit tank length : Geometry ratio : Pressure : Tank Radius : Temperature : Air temperature : Reference temperature : Saturation temperature : Velocity vector (u, v) : Vapor velocity : Width of tank : Thermal expansion coefficient : Turbulent dissipate rate : Thermal conductivity : Viscous : Turbulent viscous : Density : Surface tension

References Fig. 15. Interfacial heat flux distribution for circular cross sectional tank heat flux.

4. Conclusion The natural convection in a cryogenic tank was numerically studied by solving the conservation equations for the mass, momentum energy, turbulent kinetic energy, and the turbulent dissipation rate. The computations demonstrated that the vapor-side contribution to the interfacial heat transfer rate, which is directly related to the BOG generation rate, is negligible compared with the liquid-side contribution. The interfacial heat rate is linearly proportional to the external heat transfer coefficient. The interfacial heat transfer and the BOG generation rate strongly depend on the liquid-solid contact area, but are independent of the liquid-vapor interface area. A tank geometry with a semicircular cross-section is found to reduce the interfacial heat transfer rate and the BOG generation rate effectively.

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Blay, Laminar natural convection flow in a cylindrical cavity application to the storage of LNG, J. of Petroleum science and Engineering 71(2001) 126-132.

Sangeun Roh received a B.S. degree in Mechanical Engineering from Sogang University in 2011. He is a graduate student of Mechanical Engineering at Sogang University in Seoul, Korea. Roh’s research interests are in the areas of multiphase flows and heat transfer.

Gihun Son received his B.S. and M.S. degrees in Mechanical Engineering from Seoul National University in 1986 and 1988, respectively. He received his Ph.D in Mechanical Engineering from the University of California-Los Angeles in 1996. Dr. Son is currently a professor of Mechanical Engineering at Sogang University in Seoul, Korea. His research interests are in the areas of multiphase dynamics, heat transfer, and power system simulation.