^{1}, Steven P. Antal

^{2}and Michael Z. Podowski

^{2,a)}

### Abstract

The results of numerous studies performed to date have shown that the performance of various hydraulic systems can be significantly improved by using curved conduit geometries instead of straight tubes. In particular, the formation of Dean vortices, which enhance the development of centrifugal instabilities, has been identified as a factor behind reducing the near-wall concentration buildup in particulate flow devices (e.g., in membrane filtration modules). Still, several issues regarding the effect of conduit curvature on local multidimensional phenomena governing fluid flow still remain open. A related issue is concerned with the impact that conduit geometry makes on the concentration distribution of a dispersed phase in two-phase flows in general, and in particulate flows (solid/liquid or solid/gas suspensions) in particular. It turns out that only very limited efforts have been made in the past to understand the fluid mechanics of such flows via advanced computer simulations. The purpose of this paper is to present the results of full three-dimensional (3D) theoretical and numerical analyses of single- and two-phase dilute particle/liquid flows in U-bend and helical curved conduits. The numerical analysis is based on computational fluid dynamics(CFD) simulations performed using a state-of-the-art multiphase flow computer code, NPHASE. The major issues discussed in the first part of the paper are concerned with the effect of curved/coiled geometry on the evolution of flow field and the associated wall shear. It has been demonstrated that the primary curvature (a common factor for both the U-bend and helix geometries) may cause a substantial asymmetry in the radial distribution of the main flow velocity. This, in turn, leads to a significant, albeit highly nonuniform, increase in the wall shear stress. Specifically, the wall shear around the outer half of tube circumference may become twice the corresponding value for a straight tube, and gradually decrease to the straight tube level when approaching the inner bend location. Another important issue is concerned with the effect of the length of the curved section and of the straight tube just upstream of the bend. Specifically, the discontinuity in curvature at the straight-to-curved transition location results in a localized change in the wall shear distribution around the tube circumference. On the other hand, if the curved tube is sufficiently long, such as in the case of a helix, the asymmetric velocity profile eventually reaches a fully developed pattern. The effect of nondimensional flow parameters, the Reynolds and Dean numbers, on the entry length along the curved helix geometry is also investigated in this paper. It is shown that the predicted developing length agrees well with the existing experimental data. The objective of the second part of the paper is to investigate the mutual interactions between the liquidflow and solid particles in particulate two-phase flows in both the U-bend and helical geometries. It is shown that particle inertia causes an increase in the wall shear. At the same time, two interesting aspects are shown of Dean vortices on particle concentration under the effect of gravity. One of them is the shift in the particle settling zone from the bottom of the horizontal (or nearly horizontal) tube toward the inner bend of the tube. The other, even more important, is a dramatic reduction in peak concentration with increasing Dean number. Both effects are important for equipment design and optimization in biotechnology and process industries.

I. INTRODUCTION

II. TWO-PHASE LIQUID-PARTICULATEFLOW MODEL FORMULATION

A. Ensemble averaged conservation equations

B. Two-field dispersed-particle/continuous-liquid flow

III. DEAN VORTEX PHENOMENON

IV. OVERVIEW OF THE NPHASE CODE

V. OVERVIEW OF MODEL TESTING AND VALIDATION

VI. RESULTS AND DISCUSSION

A. Analysis of single-phase flow patterns in U-bend and helical geometries

B. Particle/liquid two-phase flow in U-bend and helical geometries

C. Effect of secondary flow on particle settling

VII. CONCLUSIONS

### Key Topics

- Multiphase flows
- 53.0
- Rotating flows
- 33.0
- Laminar flows
- 13.0
- Flow simulations
- 12.0
- Duct flows
- 10.0

## Figures

Secondary flows (Dean vortices) in a curved and coiled geometries.

Secondary flows (Dean vortices) in a curved and coiled geometries.

Computational mesh: (a) U-bend, (b) helix.

Computational mesh: (a) U-bend, (b) helix.

Single-phase wall shear plot at the outer bend of U-bend and helix-shaped curved tubes, compared against the wall shear for a straight tube at the same mass flow rate; , , , , .

Single-phase wall shear plot at the outer bend of U-bend and helix-shaped curved tubes, compared against the wall shear for a straight tube at the same mass flow rate; , , , , .

Single-phase wall shear plot at the inner bend of U-bend and helix-shaped curved tubes, compared against the wall shear for a straight tube at the same mass flow rate; , , , , .

Single-phase wall shear plot at the inner bend of U-bend and helix-shaped curved tubes, compared against the wall shear for a straight tube at the same mass flow rate; , , , , .

Streamwise velocity profiles along the cross section of U-bend geometry at various axial locations. The major geometrical parameters are: , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Streamwise velocity profiles along the cross section of U-bend geometry at various axial locations. The major geometrical parameters are: , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Streamwise velocity profiles along the cross section of U-bend geometry at various axial locations. The major geometrical parameters are: , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Streamwise velocity profiles along the cross section of helix geometry at various axial locations. The major geometrical parameters are: , , , , . “I” denotes the inner bend, “O” denotes the outer bend.

Streamwise velocity profiles along the cross section of helix geometry at various axial locations. The major geometrical parameters are: , , , , . “I” denotes the inner bend, “O” denotes the outer bend.

The effect of curvature on single-phase flow in the U-bend and helix-shaped curved tubes: (a) wall shear plotted around the circumference at the axial section along the main flow direction, (b) transverse velocity plot at the axial section along the main flow direction of the U-bend, (c) transverse velocity plot at the axial section along the main flow direction of the helix. The major geometrical parameters are: , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

The effect of curvature on single-phase flow in the U-bend and helix-shaped curved tubes: (a) wall shear plotted around the circumference at the axial section along the main flow direction, (b) transverse velocity plot at the axial section along the main flow direction of the U-bend, (c) transverse velocity plot at the axial section along the main flow direction of the helix. The major geometrical parameters are: , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Variation of developing length with Dean number for helical geometry. Calculated values (squares) are compared against the Yao and Berger, given by Eq. (32) (solid line), and Austin and Seader given by Eq. (33) (dashed line) correlations. The major geometrical parameters are: , , , (for ).

Variation of developing length with Dean number for helical geometry. Calculated values (squares) are compared against the Yao and Berger, given by Eq. (32) (solid line), and Austin and Seader given by Eq. (33) (dashed line) correlations. The major geometrical parameters are: , , , (for ).

Variation of dimensionless average wall shear with Dean number for helical geometry. NPHASE-calculated values are compared against the Mishra and Gupta [see Eq. (36)] correlation. The major geometrical parameters are: , , .

Variation of dimensionless average wall shear with Dean number for helical geometry. NPHASE-calculated values are compared against the Mishra and Gupta [see Eq. (36)] correlation. The major geometrical parameters are: , , .

Wall shear around the U-bend circumference at axial section along the main flow direction, compared against the wall shear for a straight tube at the same mass flow rate. The major geometrical parameters are: , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Wall shear around the U-bend circumference at axial section along the main flow direction, compared against the wall shear for a straight tube at the same mass flow rate. The major geometrical parameters are: , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Transverse velocity distributions at axial section along the main flow direction of the U-bend: (a) single-phase flow, (b) two-phase flow at zero gravity, (c) two-phase flow at full gravity. The major geometrical parameters are: , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Transverse velocity distributions at axial section along the main flow direction of the U-bend: (a) single-phase flow, (b) two-phase flow at zero gravity, (c) two-phase flow at full gravity. The major geometrical parameters are: , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Circumferential wall shear distribution at the axial section along the main flow direction of the helix, compared against the wall shear for a straight tube at the same mass flow rate. The major geometrical parameters are: , , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Circumferential wall shear distribution at the axial section along the main flow direction of the helix, compared against the wall shear for a straight tube at the same mass flow rate. The major geometrical parameters are: , , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Transverse velocity distributions at the axial section along the main flow direction of the helix: (a) single-phase flow, (b) two-phase flow at zero gravity, (c) two-phase at full gravity. Major geometrical parameters: , , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Transverse velocity distributions at the axial section along the main flow direction of the helix: (a) single-phase flow, (b) two-phase flow at zero gravity, (c) two-phase at full gravity. Major geometrical parameters: , , , , , , . “I” denotes the inner bend; “O” denotes the outer bend.

Particle volume fraction distributions at the axial section along the main flow direction of the U-bend, at various Dean numbers . Major geometrical parameters are: , , . “I” denotes the inner bend; “O” denotes the outer bend.

Particle volume fraction distributions at the axial section along the main flow direction of the U-bend, at various Dean numbers . Major geometrical parameters are: , , . “I” denotes the inner bend; “O” denotes the outer bend.

Deviation in settling zone at the axial section along the main flow direction of the U-bend, at various Froude numbers . Major geometrical parameters: , , . “I” denotes the inner bend; “O” denotes the outer bend.

Deviation in settling zone at the axial section along the main flow direction of the U-bend, at various Froude numbers . Major geometrical parameters: , , . “I” denotes the inner bend; “O” denotes the outer bend.

## Tables

Geometry definitions and flow parameters used for simulations.

Geometry definitions and flow parameters used for simulations.

Physical properties of particle-liquid system.

Physical properties of particle-liquid system.

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