Abstract
Numerical method is used to investigate fully developed laminar flow in helically coiled circular tube in this paper. The non-dimensional parameter (secondary flow Reynolds number Se) based on absolute vorticity flux along the mainstream is used to indicate the intensity of secondary flow caused by the centrifugal effect in helically coiled circular tube. The relationship between the intensity of secondary flow and the intensity of laminar convective heat transfer is studied. The effects of curvature and torsion on the enhancement of heat transfer are also considered. The results reveal that the absolute vorticity flux along the mainstream can be used to indicate the local or averaged intensity of secondary flow; the non-dimensional parameter of the absolute vortex along the main flow determines the convective heat transfer and friction factor. The relationships of Nusselt number and friction factor with the Se are obtained. The effect of curvature on Nusselt number is obvious, but the effect of torsion on Nusselt number is less obvious.
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Abbreviations
- A :
-
coefficient of the discretization equation, depending on equation
- A :
-
area of cross section/m2
- A, A′, A0 :
-
circle center mark
- a :
-
coefficient of the discretization equation, depending on equation
- a :
-
radius of helically coiled tube/m
- b :
-
source term of discretization equation, depending on equation
- c p :
-
heat capacity/kJ·(kg·K)−1
- D :
-
hydraulic diameter of the tube /m
- Dn :
-
Dean number
- d :
-
tube diameter/m
- \({{\boldsymbol{e}}_{{x^\prime}}},{{\boldsymbol{e}}_{{y^\prime}}},{{\boldsymbol{e}}_{{z^\prime}}}\) :
-
unit vector along the coordinate x′, y′, z’
- f :
-
Fanning friction factor
- H :
-
tube pitch/m
- h :
-
convective heat transfer coefficient/W·(m2·K)−1
- i, j, k :
-
unit vector along the coordinate x′, y′, z′/m
- J :
-
Jacobi factor
- J n :
-
vorticity flux along the normal direction of cross section/s
- \(J_{{\rm{ABS}}}^{\rm{n}}\) :
-
absolute vorticity flux along the normal direction of cross section/s−1
- \(J_{{\rm{ABS}},{\rm{V}}}^{\rm{n}}\) :
-
volumetrically averaged absolute vorticity flux/s−1
- K :
-
tube pitch per unit rad/m
- L :
-
Axial length/m
- L, M, N :
-
grid number
- n :
-
unit vector, normal direction of the cross section or wall surface
- Nu :
-
Nusselt number
- p :
-
static pressure/Pa
- Pr :
-
Prandtl number
- q :
-
heat flux/W·m−2
- Q :
-
crossly averaged velocity component along the main flow direction/m·s−1
- R :
-
radius of helically coiled tube center line/m
- Re :
-
Reynolds number
- Re c :
-
Critical Reynolds number
- r :
-
radial coordinate/m
- S :
-
source term /depending on equation
- S ad :
-
additional source term /depending on equation
- Se :
-
Reynolds number of secondary flow
- T :
-
temperature/K
- U, V, W :
-
contravariant velocity/m·s−1
- u′, v′, w′ :
-
velocity components along the coordinates x′, y′, z′/m·s−1
- u, v, w :
-
velocity components along the coordinates x, y, z/m·s−1
- v :
-
velocity vector/m·s−1
- x, y, z :
-
coordinates axes/m
- x′, y′, z′:
-
rotation axis/m
- α, β, γ :
-
coefficient of the discretization equation
- Γ :
-
velocity loop/m2·s−1
- Γ φ :
-
generalized diffusion coefficient, depending on equation
- ζ, η, ξ :
-
body-fitted coordinate axis/m
- Θ :
-
Dimensionless temperature
- κ :
-
dimensionless curvature
- µ :
-
dynamic viscosity/Pa·s
- ρ :
-
fluid density/kg·m−3
- τ :
-
dimensionless torsion
- φ :
-
general variable, cross section angle, depending on equation
- ω :
-
vorticity component/s−1
- ω :
-
vorticity vector/s−1
- ABS:
-
absolute value
- B:
-
lower node
- bulk:
-
cross averaged value
- E:
-
east node
- h:
-
helically coiled tube
- in:
-
inlet
- local:
-
local value
- m:
-
averaged value
- N:
-
north node
- out:
-
outlet
- P:
-
major node
- p:
-
circumferential local value
- S:
-
south node
- s:
-
straight pipe
- T:
-
upper node
- W:
-
west node
- w:
-
wall
- ′:
-
corrected value
- n:
-
normal direction of cross section
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Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 51776093, No. 52066009), Transformation of S&T achievements in Universities of Gansu Province of China (No. 2019C-06), Major Special Projects of Gansu Province of China (21ZD4GA027), Young Scientists Fund of Lanzhou Jiaotong University (2020038).
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Zhang, J., Zhao, C. & Wang, L. Relationship between the Intensity of Secondary Flow and Convection Heat Transfer in a Helically Coiled Circular Tube with Uniform Wall Temperature. J. Therm. Sci. 32, 1007–1022 (2023). https://doi.org/10.1007/s11630-023-1794-y
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DOI: https://doi.org/10.1007/s11630-023-1794-y