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Natural convection heat transfer from a vertical single cylinder with eccentric grid spacer in liquid sodium

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Abstract

Natural convection heat transfer from a vertical single-cylinder with eccentric grid spacer in liquid sodium was numerically analyzed on the following seven effects, the influences of grid spacer, surface heat flux, the gap between grid spacer and heated cylinder surface, grid spacer thickness, height, eccentricity, and material on heat transfer. The unsteady laminar three-dimensional basic equations for natural convection heat transfer caused by a step heat flux were numerically solved until the numerical solution reaches a steady state. The PHOENICS code was used for the calculation considering the temperature dependence of thermo-physical properties concerned. The heated cylinder for diameter (D = 7.6 mm), heated length (L = 200 mm) and L/d (= 26.32), and the round cell type grid spacers for gaps (0.05 to 0.5 mm), inner diameters (di = 7.7 to 8.6 mm), thickness (w = 0.2, 0.3 and 0.5 mm), outer diameters (do = 8.1 to 9.2 mm), heights (h = 10, 30 and 50 mm), eccentricity (B = 0, 0.3333, 0.6667 and 0.8333: s = 0, 0.1, 0.2 and 0.25 mm) and materials (SUS304, SUS316, Hastelloy X and Adiabatic) were used in this work. The surface heat fluxes for each cylinder with eccentric grid spacer were equally given for a modified Rayleigh number, (Raf,L)g-s, ranging from 3.057 × 104 to 3.171 × 107 (q = 1 × 104 to 7 × 106 W/m2) in liquid temperature (TL = 673.15 K). The values of gap were given 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5 mm at w = 0.2 mm, h = 30 mm, B = 0 and SUS316, those of thickness were given 0.2, 0.3 and 0.5 mm at a gap of 0.3 mm, h = 30 mm, B = 0 and SUS316, those of height were given 20, 30 and 50 mm at a gap of 0.3 mm, w = 0.2 mm, B = 0 and SUS316, those of eccentricity were given 0, 0.3333, 0.6667 and 0.8333 at a gap of 0.3 mm, w = 0.2 mm, h = 30 mm and SUS316, and those of material were given SUS304, SUS316, Hastelloy X and Adiabatic at a gap of 0.3 mm, w = 0.2 mm, h = 30 mm and B = 0 on each cylinder with eccentric grid spacer. The spatial distributions of local and average Nusselt numbers, (Nuθ,z)g-s and (Nuav)g-s, on vertical single-cylinder with eccentric grid spacer, were clarified. The values of average Nusselt number, (Nuav)g-s, for a vertical single-cylinder with eccentric grid spacer were calculated to examine the effects of grid spacer, (Raf,L)g-s, gap, thickness, height, eccentricity, and material on heat transfer. The correlation for (Nuav)g-s for a vertical single-cylinder with eccentric grid spacer above mentioned, including the effects of grid spacer, (Raf,L)g-s, gap, thickness, height, eccentricity, and the material was developed. The correlation can describe the theoretical values of (Nuav)g-s for a vertical single-cylinder with eccentric grid spacer within -1.897 to 1.432% differences.

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Abbreviations

B :

 = 2 s/(E-1)/D, Eccentricity

C :

Parameter in Eqs. (14), (19) and (33)

C 1 :

Constant in Eqs. (30), (31), (33) and (34)

c:

Specific heat, J/kg K

c p :

Specific heat at constant pressure, J/kg K

D :

Cylinder diameter and rod diameter, m

d :

Helical wire diameter, m

d i :

Inner diameter of grid spacer, m

d o :

Outer diameter of grid spacer, m

E :

 = di/D, Expansion rate

Gr L * :

 = g βl q L4/λl νl2, Grashof number for constant heat flux without eccentric grid spacer

( Gr L * ) g-s :

 = g βl q L4/λl νl2, Grashof number for constant heat flux with eccentric grid spacer

Gr z * :

 = g βl q z4/λl νl2, Local Grashof number for constant heat flux without eccentric grid spacer

( Gr z * ) g-s :

 = g βl q z4/λl νl2, Local Grashof number for constant heat flux with eccentric grid spacer

g :

Acceleration of gravity, m/s2

H :

Height, liquid head, m

h :

=(ρcλ)0.5, thermal activity factor, J/m2Ks0.5

h av :

Average heat transfer coefficient for a vertical single-cylinder without eccentric grid spacer, W/(m2K)

( h av ) g-s :

Average heat transfer coefficient for a vertical single-cylinder with eccentric grid spacer, W/(m2K)

h z :

Local heat transfer coefficient for a vertical single-cylinder without helical wire spacer, W/(m2K)

( h z ) g-s :

Local heat transfer coefficient for a vertical single-cylinder with eccentric grid spacer, W/(m2K)

L :

Heated length, m

N r :

Grid point for the r-component

N z :

Grid point for the z-component

N θ :

Grid point for the θ-component

n :

Exponent in Eqs. (30), (31), (33) and (35)

Nu :

Nusselt number

Nu av :

Average Nusselt number for a vertical single-cylinder without eccentric grid spacer

( Nu av ) g-s :

Average Nusselt number for a vertical single-cylinder with eccentric grid spacer

Nu z :

= q z/λl (TS-TL), Local Nusselt number for a vertical cylinder without eccentric grid spacer

Nu θ , z :

= q z/λl (TS-TL), Local Nusselt number for a vertical cylinder without eccentric grid spacer at a peripheral angle of the heated cylinder, θ

( Nu θ , z ) g-s :

= q z/λl (TS-TL), Local Nusselt number for a vertical cylinder at a peripheral angle of the heated cylinder, θ

Pr :

= cpl μl/λl, Prandtl number

q :

Heat flux, W/m2

Ra L * :

 = GrL*Pr, Rayleigh number for constant heat flux without eccentric grid spacer

( Ra L * ) g-s :

 = GrL*Pr, Rayleigh number for constant heat flux with eccentric grid spacer

Ra z * :

= Grz*Pr, Local Rayleigh number for constant heat flux without eccentric grid spacer

( Ra z * ) g-s :

= Grz*Pr, Local Rayleigh number for constant heat flux with eccentric grid spacer

R a f ,z :

= Grz*Pr2/(4 + 9Pr1/2 + 10Pr), Modified local Rayleigh number without eccentric grid spacer

{( R a f ) θ ,z } g-s :

= Grz*Pr2/(4 + 9Pr1/2 + 10Pr), Modified local Rayleigh number with eccentric grid spacer

R a f,L :

 = GrL*Pr2/(4 + 9Pr1/2 + 10Pr), Modified Rayleigh number without eccentric grid spacer

( R a f,L ) g-s :

 = GrL*Pr2/(4 + 9Pr1/2 + 10Pr), Modified Rayleigh number with eccentric grid spacer

r :

Radius of a cylinder, m

Δr :

First control volume width for r-component, m

s :

Distance between central axes, m

T :

Temperature, K and °C

TEM :

Analyzed liquid temperature of the first control volume, K

T f :

Liquid temperature, K

T L :

Bulk liquid temperature, K

T s :

Heater surface temperature, K

( T s ) av :

Average surface temperature without eccentric grid spacer, K

{( T s ) av } g-s :

Average surface temperature with eccentric grid spacer, K

( T s ) z :

Local surface temperature without eccentric grid spacer, K

{( T s ) θ ,z } g-s :

Local surface temperature with eccentric grid spacer, K

T ∞ :

Temperature of the fluid outside the thermal boundary layer, K

t :

Time, s

Δ t :

Time step, s

U ∞ :

Fictitious velocity, m/s

(v θ ) g -s :

Velocity component in θ-direction, m/s

(v r ) g -s :

Velocity component in r-direction, m/s

(v z ) g -s :

Velocity component in z-direction, m/s

w:

Thickness of grid spacer, m

z :

Cylinder height and the vertical distance from the leading edge of the heated section, m

β :

Volumetric expansion coefficient, K−1

δ :

Hydrodynamic boundary layer thickness, m

δ CSL :

Thickness of conductive sublayer, m

δ t :

Thermal boundary layer thickness, m

θ :

Peripheral angle from inside on the r-θ plane, degree

λ :

Thermal conductivity, W/(m K)

μ :

Viscosity, Ns/m2

ν :

Kinematic viscosity, m2/s

ρ :

Density, kg/m3

l :

Liquid

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Acknowledgements

This work was supported by the “Joint Usage/Research Program on Zero-Emission Energy Research, Institute of Advanced Energy, Kyoto University, Japan, ZE30B-03, 2018 and ZE31B-05, 2019.”

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  1. Graduate School of Maritime Sciences, Kobe University, 5-1-1, Fukae-minami, Kobe, Hyogo, 658-0022, Japan

    Koichi Hata & Makoto Shibahara

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  1. Koichi Hata
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Correspondence to Koichi Hata.

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Hata, K., Shibahara, M. Natural convection heat transfer from a vertical single cylinder with eccentric grid spacer in liquid sodium. Heat Mass Transfer 58, 1607–1636 (2022). https://doi.org/10.1007/s00231-022-03199-x

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