Pressure drop force due to a non-closely fitting sphere settling along the central line in long rectangular tubes

doi:10.1016/j.cjche.2025.02.020

Abstract

Abstract: The ratio of the pressure drop force to the drag force, C_P, is concerned for a non-closely fitting spherical particle settling along the central line in long rectangular tubes with different A_r (A_r is W/H, where W, H is length of the longer and shorter side of the rectangle respectively). Under Stokes flow conditions, C_P0 for an infinitely small sphere in long rectangular tubes and C_P for a sphere in a long channel between two parallel layered barriers are both calculated. Then C_P of a sphere settling in long rectangular tubes are conducted with the direct-forcing fictitious domain (DF/FD) method. At large Reynolds number, the sphere settles unstably with a fluctuating velocity and C_P. The fluctuation of C_P is much stronger than that of velocity and both fluctuations are stronger for less confined sphere. The influences of the particle Reynolds number (Re_p) on C_P is similar to the existing experimental results in long circular tubes. At low Re_p, C_P is a determined value and is calculated. For a given d/H (d sphere diameter), C_P gets its maximum value at one A_r in the range of [1, 1.5]. For a given A_r, C_P is a quadratic function of d/H similar to that in a circular tube, and parameters of the quadratic function are got by curve fitting from numerical data. The constant term coefficients got have almost no difference with C_P0 and are furtherly replaced by the latter to get new quadratic coefficients C_P1. Lastly, an algebraic correlation of C_P1 to A_r is developed. The predictions of C_P are good with a maximum relative error about 1.5% for a sphere with d/H not greater than 0.7, compared to numerical results.

Key words: Pressure drop, Numerical simulation, Stokes flow, Non-closely fitting, Sedimentation, Particle

摘要： The ratio of the pressure drop force to the drag force, C_P, is concerned for a non-closely fitting spherical particle settling along the central line in long rectangular tubes with different A_r (A_r is W/H, where W, H is length of the longer and shorter side of the rectangle respectively). Under Stokes flow conditions, C_P0 for an infinitely small sphere in long rectangular tubes and C_P for a sphere in a long channel between two parallel layered barriers are both calculated. Then C_P of a sphere settling in long rectangular tubes are conducted with the direct-forcing fictitious domain (DF/FD) method. At large Reynolds number, the sphere settles unstably with a fluctuating velocity and C_P. The fluctuation of C_P is much stronger than that of velocity and both fluctuations are stronger for less confined sphere. The influences of the particle Reynolds number (Re_p) on C_P is similar to the existing experimental results in long circular tubes. At low Re_p, C_P is a determined value and is calculated. For a given d/H (d sphere diameter), C_P gets its maximum value at one A_r in the range of [1, 1.5]. For a given A_r, C_P is a quadratic function of d/H similar to that in a circular tube, and parameters of the quadratic function are got by curve fitting from numerical data. The constant term coefficients got have almost no difference with C_P0 and are furtherly replaced by the latter to get new quadratic coefficients C_P1. Lastly, an algebraic correlation of C_P1 to A_r is developed. The predictions of C_P are good with a maximum relative error about 1.5% for a sphere with d/H not greater than 0.7, compared to numerical results.

关键词: Pressure drop, Numerical simulation, Stokes flow, Non-closely fitting, Sedimentation, Particle

Yelong Wang, Zhaosheng Yu, Jianzhong Lin. Pressure drop force due to a non-closely fitting sphere settling along the central line in long rectangular tubes[J]. Chinese Journal of Chemical Engineering, 2025, 82(6): 165-175.

Yelong Wang, Zhaosheng Yu, Jianzhong Lin. Pressure drop force due to a non-closely fitting sphere settling along the central line in long rectangular tubes[J]. 中国化学工程学报, 2025, 82(6): 165-175.

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