Discrete Tracer Point Method to Evaluate Turbulent Diffusion in Circular Pipe Flow

Affiliation(s)

Department of Earth Resources Engineering, Faculty of Engineering, Kyushu University, Fukuoka, Japan.

Department of Mining Engineering, Faculty of Mining and Petroleum Engineering,Institut Teknologi Bandung, Bandung, Indonesia.

Department of Earth Resources Engineering, Faculty of Engineering, Kyushu University, Fukuoka, Japan.

Department of Mining Engineering, Faculty of Mining and Petroleum Engineering,Institut Teknologi Bandung, Bandung, Indonesia.

ABSTRACT

Diffusion of a solute in turbulent flows through a circular pipe or tunnel is an important aspect of environmental safety. In this study, the diffusion coefficient of turbulent flow in circular pipe has been simulated by the Discrete Tracer Point Method (DTPM). The DTPM is a Lagrangian numerical method by a number of imaginary point displacement which satisfy turbulent mixing by velocity fluctuations, Reynolds stress, average velocity profile and a turbulent stochastic model. Numerical simulation results of points’ distribution by DTPM have been compared with the analytical solution for turbulent plug-flow. For the case of turbulent circular pipe flow, the appropriate DTPM calculation time step has been investigated using a constantβ, which represents the ratio between average mixing lengths over diameter of circular pipe. The evaluated values of diffusion coefficient by DTPM have been found to be in good agreement with Taylor’s analytical equation for turbulent circular pipe flow by givingβ=0.04 to 0.045. Further, history matching of experimental tracer gas measurement through turbulent smooth-straight pipe flow has been presented and the results showed good agreement.

Diffusion of a solute in turbulent flows through a circular pipe or tunnel is an important aspect of environmental safety. In this study, the diffusion coefficient of turbulent flow in circular pipe has been simulated by the Discrete Tracer Point Method (DTPM). The DTPM is a Lagrangian numerical method by a number of imaginary point displacement which satisfy turbulent mixing by velocity fluctuations, Reynolds stress, average velocity profile and a turbulent stochastic model. Numerical simulation results of points’ distribution by DTPM have been compared with the analytical solution for turbulent plug-flow. For the case of turbulent circular pipe flow, the appropriate DTPM calculation time step has been investigated using a constantβ, which represents the ratio between average mixing lengths over diameter of circular pipe. The evaluated values of diffusion coefficient by DTPM have been found to be in good agreement with Taylor’s analytical equation for turbulent circular pipe flow by givingβ=0.04 to 0.045. Further, history matching of experimental tracer gas measurement through turbulent smooth-straight pipe flow has been presented and the results showed good agreement.

KEYWORDS

Discrete Tracer Point Method (DTPM); Turbulent Diffusion; Pipe; Numerical Simulation; Airflow

Discrete Tracer Point Method (DTPM); Turbulent Diffusion; Pipe; Numerical Simulation; Airflow

Cite this paper

A. Widiatmojo, K. Sasaki, N. Priagung Widodo and Y. Sugai, "Discrete Tracer Point Method to Evaluate Turbulent Diffusion in Circular Pipe Flow,"*Journal of Flow Control, Measurement & Visualization*, Vol. 1 No. 2, 2013, pp. 57-68. doi: 10.4236/jfcmv.2013.12008.

A. Widiatmojo, K. Sasaki, N. Priagung Widodo and Y. Sugai, "Discrete Tracer Point Method to Evaluate Turbulent Diffusion in Circular Pipe Flow,"

References

[1] G. I. Taylor, “Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 219, No. 1137, 1953, pp. 186-203. doi:10.1098/rspa.1953.0139

[2] G. I. Taylor, “The Dispersion of Matter in Turbulent Flow through a Pipe,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 223, No. 1155, 1954, pp. 446-468. doi:10.1098/rspa.1954.0130

[3] R. Aris, “On the Dispersion of a Solute in a Fluid Flowing through a Tube,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 235, No. 1200, 1956, pp. 67-77. doi:10.1098/rspa.1956.0065

[4] S. Goldstein, “Modern Developments in Fluid Dynamics,” Dover, New York, 1965.

[5] G. I. Taylor, “Diffusion by Continuous Movements,” Proceedings of London Mathematical Society, Vol. 20, No. 1, 1954, pp. 196-211. doi:10.1112/plms/s2-20.1.196

[6] S. B. Pope, “Consistency Conditions for Random-Walk Models of Turbulent Dispersion,” Physics of Fluids, Vol. 30, No. 8, 1987, pp. 2374-2379. doi:10.1063/1.866127

[7] D. Milojevic, “Lagrangian Stochastic-Deterministic Predictions of Particle Dispersion in Turbulence,” Particle & Particle Systems Characterization, Vol. 7, No. 1-4, 1990, pp. 181-190. doi:10.1002/ppsc.19900070132

[8] C. Kr?ger and Y. Drossinos, “A Random-Walk Simulation of Thermophoretic Particle Deposition in a Turbulent Boundary Layer,” International Journal of Multiphase Flow, Vol. 26, No. 8, 2000, pp. 1325-1350. doi:10.1016/S0301-9322(99)00092-0

[9] A. K. Luhar and R. E. Britter, “A Random Walk Model for Dispersion in Inhomogeneous Turbulence in a Convective Boundary Layer,” Atmospheric Environment, Vol. 23, No. 9, 1989, pp. 1911-1924. doi:10.1016/0004-6981(89)90516-7

[10] C. N. Sittel Jr., W. D. Threadgill and K. B. Schnelle Jr., “Longitudinal Dispersion for Turbulent Flow in Pipes,” Industrial & Engineering Chemistry Fundamentals, Vol. 7, No. 1, 1968, pp. 39-43. doi:10.1021/i160025a007

[11] D. E. Hull and J. W. Kent, “Radioactive Tracers to Mark Interfaces and Measure Inter-mixing in Pipelines,” Industrial & Engineering Chemistry, Vol. 44, No. 11, 1952, pp. 2745-2750. doi:10.1021/ie50515a066

[12] J. J. Keyes Jr., “Diffusional Film Characteristics in Turbulent Flow: Dynamic Response Metho,” AiChE Journal, Vol. 1, No. 3, 1955, pp. 305-311. doi:10.1002/aic.690010306

[13] F. Davidson, D. C. Farqurharson, J. Q. Picken and D. C. Taylor, “Gas Mixing in Long Pipelines,” Chemical Engineering Science, Vol. 4, No. 5, 1955, pp. 201-205. doi:10.1016/0009-2509(55)80006-1

[14] N. P. Widodo, K. R. Sasaki, R. S. Gautama and Risono, “Mine Ventilation Measurements with Tracer Gas Method and Evaluations of Turbulent Diffusion Coefficient,” International Journal of Mining, Reclamation and Environment, Vol. 22, No. 1, 2008, pp. 60-69. doi:10.1080/17480930701474869

[15] W. B. Krantz and D. T. Wasan, “Axial Dispersion in the Turbulent Flow of Power-Law Fluids in Straight Tubes,” Industrial & Engineering Chemistry Fundamentals, Vol. 13, No. 1, 1974, pp. 56-62. doi:10.1021/i160049a011

[16] O. Levenspiel, “Longitudinal Mixing of Fluids Flowing in Circular Pipes,” Industrial & Engineering Chemistry, Vol. 50, No. 3, 1958, pp. 343-346. doi:10.1021/ie50579a034

[17] L. J. Tichacek, C. H. Barkelew and T. Baron “Axial Mixing in Pipes,” AIChE Journal, Vol. 3, No. 4, 1957, pp. 439- 442. doi:10.1002/aic.690030404

[18] C. Y. Wen and L. T. Fan, “Models for Flow System and Chemical Reactors,” Dekker, New York, 1975.

[19] K. Sasaki, A. Widiatmojo, G. Arpa and Y. Sugai, “Air-flow Measurements and Evaluation of Effective Diffusion Coefficient in Large Scale of Mine Ventilation Network Using with Tracer Gas Method,” Journal of the Mining and Materials Processing Institute of Japan, Vol. 125, No. 12, 2009, pp. 614-620. doi:10.2473/journalofmmij.125.614

[20] J. Laufer, “The Structure of Turbulent in Fully Developed Pipe Flow,” Technical Report 1174, National Committee for Aeronautics, 1954. http://naca.central.cranfield.ac.uk/report.php?NID=5843

[21] R. A. Kenyon, “Principles of Fluid Mechanics,” Ronald Press, New York, 1960.

[22] H. Reichardt, “Complete Representation of the Turbulent Velocity Distribution in Smooth Pipes,” Journal of Applied Mathematics and Mechanics, Vol. 31. No. 7, 1951, pp. 208-219. doi:10.1002/zamm.19510310704

[23] H. Rouse, “Advanced Mechanics of Fluids,” John Wiley and Sons, Inc., New York, 1959.

[24] C. F. Colebrook, “Turbulent Flow in Pipes, with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws,” Journal of the ICE, Vol. 11, No. 4, 1939, pp. 133-156. doi:10.1680/ijoti.1939.13150

[25] D. J. Thomson, W. L. Physick and R. H. Maryon, “Treatment of Interfaces in Random Walk Dispersion Models,” Journal of Applied Meteorology, Vol. 36, No. 9, 1997, pp. 1284-1285. doi:10.1175/1520-0450(1997)036<1284:TOIIRW>2.0.CO;2

[26] P. Szymczak and A. J. C. Ladd, “Boundary Conditions for Stochastic Solutions of the Convection-Diffusion Equation,” Physical Review E, Vol. 68, No. 3, 2003, Article ID: 036704. doi:10.1103/PhysRevE.68.036704

[27] P. Szymczak and A. J. C. Ladd, “Stochastic Boundary Conditions to the Convection-Diffusion Equation including Chemical Reactions at Solid Surfaces,” Physical Review E, Vol. 69, No. 3, 2004, Article ID: 036704. doi:10.1103/PhysRevE.69.036704

[28] G. Drazer and J. Koplik, “Tracer Dispersion in Two-Dimensional Rough Fractures,” Physical Review E, Vol. 63, No. 5, 2001, Article ID: 056104. doi:10.1103/PhysRevE.63.056104

[29] P. Kurowski, I. Ippolito, J. P. Hulin, J. Koplik and E. J. Hinch, “Anomalous Dispersion in a Dipole Flow Geometry,” Physics of Fluids, Vol. 6, No. 1, 1994, pp. 108-117. doi:10.1063/1.868075

[30] J. Salles, J. F. Thovert, R. Delannay, L. Prevors, J.-L. Auriault and P. M. Adler, “Taylor Dispersion in Porous Media. Determination of the Dispersion Tensor,” Physics of Fluids A: Fluid Dynamics, Vol. 5, No. 10, 1993, pp. 2348-2376. doi:10.1063/1.858751

[31] R. S. Maier, D. M. Kroll, R. S. Bernard, S. E. Howington, J. F. Peters and H. T Davis, “Pore-Scale Simulation of Dispersion,” Physics of Fluids, Vol. 12, No. 8, 2000, pp. 2065-2079. doi:10.1063/1.870452

[32] N. P. Widodo, “Study on Tracer Gas Method for Mine Ventilation Measurement and Evaluation of Gas Diffusion Coefficient,” Ph.D. Thesis, Kyushu University, Fukuoka, 2008.

[33] G. Xu, K. D. Luxbacher, S. Ragab and S. Schafrik, “Development of a Remote Analysis Method for Underground Ventilation Systems Using Tracer Gas and CFD in a Simplified Laboratory Apparatus,” Tunneling and Underground Space Technology, Vol. 33, 2013, pp. 1-11. doi:10.1016/j.tust.2012.09.001

[34] M. H. Johnson, Z. Zhai and M. Krarti, “Performance Evaluation of Network Airflow Models for Natural Ventilation,” HVAC&R Research, Vol. 18, No. 3, 2012, pp. 349-365.

[35] R. Gao, A. Li, X. Hao, W. Lei and B. Deng, “Prediction of the Spread of Smoke in a Huge Transit Terminal Subway Station under Six Different Fire Scenarios,” Tunnelling and Underground Space Technology, Vol. 21, 2012, pp. 128-138. doi:10.1016/j.tust.2012.04.013

[1] G. I. Taylor, “Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 219, No. 1137, 1953, pp. 186-203. doi:10.1098/rspa.1953.0139

[2] G. I. Taylor, “The Dispersion of Matter in Turbulent Flow through a Pipe,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 223, No. 1155, 1954, pp. 446-468. doi:10.1098/rspa.1954.0130

[3] R. Aris, “On the Dispersion of a Solute in a Fluid Flowing through a Tube,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 235, No. 1200, 1956, pp. 67-77. doi:10.1098/rspa.1956.0065

[4] S. Goldstein, “Modern Developments in Fluid Dynamics,” Dover, New York, 1965.

[5] G. I. Taylor, “Diffusion by Continuous Movements,” Proceedings of London Mathematical Society, Vol. 20, No. 1, 1954, pp. 196-211. doi:10.1112/plms/s2-20.1.196

[6] S. B. Pope, “Consistency Conditions for Random-Walk Models of Turbulent Dispersion,” Physics of Fluids, Vol. 30, No. 8, 1987, pp. 2374-2379. doi:10.1063/1.866127

[7] D. Milojevic, “Lagrangian Stochastic-Deterministic Predictions of Particle Dispersion in Turbulence,” Particle & Particle Systems Characterization, Vol. 7, No. 1-4, 1990, pp. 181-190. doi:10.1002/ppsc.19900070132

[8] C. Kr?ger and Y. Drossinos, “A Random-Walk Simulation of Thermophoretic Particle Deposition in a Turbulent Boundary Layer,” International Journal of Multiphase Flow, Vol. 26, No. 8, 2000, pp. 1325-1350. doi:10.1016/S0301-9322(99)00092-0

[9] A. K. Luhar and R. E. Britter, “A Random Walk Model for Dispersion in Inhomogeneous Turbulence in a Convective Boundary Layer,” Atmospheric Environment, Vol. 23, No. 9, 1989, pp. 1911-1924. doi:10.1016/0004-6981(89)90516-7

[10] C. N. Sittel Jr., W. D. Threadgill and K. B. Schnelle Jr., “Longitudinal Dispersion for Turbulent Flow in Pipes,” Industrial & Engineering Chemistry Fundamentals, Vol. 7, No. 1, 1968, pp. 39-43. doi:10.1021/i160025a007

[11] D. E. Hull and J. W. Kent, “Radioactive Tracers to Mark Interfaces and Measure Inter-mixing in Pipelines,” Industrial & Engineering Chemistry, Vol. 44, No. 11, 1952, pp. 2745-2750. doi:10.1021/ie50515a066

[12] J. J. Keyes Jr., “Diffusional Film Characteristics in Turbulent Flow: Dynamic Response Metho,” AiChE Journal, Vol. 1, No. 3, 1955, pp. 305-311. doi:10.1002/aic.690010306

[13] F. Davidson, D. C. Farqurharson, J. Q. Picken and D. C. Taylor, “Gas Mixing in Long Pipelines,” Chemical Engineering Science, Vol. 4, No. 5, 1955, pp. 201-205. doi:10.1016/0009-2509(55)80006-1

[14] N. P. Widodo, K. R. Sasaki, R. S. Gautama and Risono, “Mine Ventilation Measurements with Tracer Gas Method and Evaluations of Turbulent Diffusion Coefficient,” International Journal of Mining, Reclamation and Environment, Vol. 22, No. 1, 2008, pp. 60-69. doi:10.1080/17480930701474869

[15] W. B. Krantz and D. T. Wasan, “Axial Dispersion in the Turbulent Flow of Power-Law Fluids in Straight Tubes,” Industrial & Engineering Chemistry Fundamentals, Vol. 13, No. 1, 1974, pp. 56-62. doi:10.1021/i160049a011

[16] O. Levenspiel, “Longitudinal Mixing of Fluids Flowing in Circular Pipes,” Industrial & Engineering Chemistry, Vol. 50, No. 3, 1958, pp. 343-346. doi:10.1021/ie50579a034

[17] L. J. Tichacek, C. H. Barkelew and T. Baron “Axial Mixing in Pipes,” AIChE Journal, Vol. 3, No. 4, 1957, pp. 439- 442. doi:10.1002/aic.690030404

[18] C. Y. Wen and L. T. Fan, “Models for Flow System and Chemical Reactors,” Dekker, New York, 1975.

[19] K. Sasaki, A. Widiatmojo, G. Arpa and Y. Sugai, “Air-flow Measurements and Evaluation of Effective Diffusion Coefficient in Large Scale of Mine Ventilation Network Using with Tracer Gas Method,” Journal of the Mining and Materials Processing Institute of Japan, Vol. 125, No. 12, 2009, pp. 614-620. doi:10.2473/journalofmmij.125.614

[20] J. Laufer, “The Structure of Turbulent in Fully Developed Pipe Flow,” Technical Report 1174, National Committee for Aeronautics, 1954. http://naca.central.cranfield.ac.uk/report.php?NID=5843

[21] R. A. Kenyon, “Principles of Fluid Mechanics,” Ronald Press, New York, 1960.

[22] H. Reichardt, “Complete Representation of the Turbulent Velocity Distribution in Smooth Pipes,” Journal of Applied Mathematics and Mechanics, Vol. 31. No. 7, 1951, pp. 208-219. doi:10.1002/zamm.19510310704

[23] H. Rouse, “Advanced Mechanics of Fluids,” John Wiley and Sons, Inc., New York, 1959.

[24] C. F. Colebrook, “Turbulent Flow in Pipes, with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws,” Journal of the ICE, Vol. 11, No. 4, 1939, pp. 133-156. doi:10.1680/ijoti.1939.13150

[25] D. J. Thomson, W. L. Physick and R. H. Maryon, “Treatment of Interfaces in Random Walk Dispersion Models,” Journal of Applied Meteorology, Vol. 36, No. 9, 1997, pp. 1284-1285. doi:10.1175/1520-0450(1997)036<1284:TOIIRW>2.0.CO;2

[26] P. Szymczak and A. J. C. Ladd, “Boundary Conditions for Stochastic Solutions of the Convection-Diffusion Equation,” Physical Review E, Vol. 68, No. 3, 2003, Article ID: 036704. doi:10.1103/PhysRevE.68.036704

[27] P. Szymczak and A. J. C. Ladd, “Stochastic Boundary Conditions to the Convection-Diffusion Equation including Chemical Reactions at Solid Surfaces,” Physical Review E, Vol. 69, No. 3, 2004, Article ID: 036704. doi:10.1103/PhysRevE.69.036704

[28] G. Drazer and J. Koplik, “Tracer Dispersion in Two-Dimensional Rough Fractures,” Physical Review E, Vol. 63, No. 5, 2001, Article ID: 056104. doi:10.1103/PhysRevE.63.056104

[29] P. Kurowski, I. Ippolito, J. P. Hulin, J. Koplik and E. J. Hinch, “Anomalous Dispersion in a Dipole Flow Geometry,” Physics of Fluids, Vol. 6, No. 1, 1994, pp. 108-117. doi:10.1063/1.868075

[30] J. Salles, J. F. Thovert, R. Delannay, L. Prevors, J.-L. Auriault and P. M. Adler, “Taylor Dispersion in Porous Media. Determination of the Dispersion Tensor,” Physics of Fluids A: Fluid Dynamics, Vol. 5, No. 10, 1993, pp. 2348-2376. doi:10.1063/1.858751

[31] R. S. Maier, D. M. Kroll, R. S. Bernard, S. E. Howington, J. F. Peters and H. T Davis, “Pore-Scale Simulation of Dispersion,” Physics of Fluids, Vol. 12, No. 8, 2000, pp. 2065-2079. doi:10.1063/1.870452

[32] N. P. Widodo, “Study on Tracer Gas Method for Mine Ventilation Measurement and Evaluation of Gas Diffusion Coefficient,” Ph.D. Thesis, Kyushu University, Fukuoka, 2008.

[33] G. Xu, K. D. Luxbacher, S. Ragab and S. Schafrik, “Development of a Remote Analysis Method for Underground Ventilation Systems Using Tracer Gas and CFD in a Simplified Laboratory Apparatus,” Tunneling and Underground Space Technology, Vol. 33, 2013, pp. 1-11. doi:10.1016/j.tust.2012.09.001

[34] M. H. Johnson, Z. Zhai and M. Krarti, “Performance Evaluation of Network Airflow Models for Natural Ventilation,” HVAC&R Research, Vol. 18, No. 3, 2012, pp. 349-365.

[35] R. Gao, A. Li, X. Hao, W. Lei and B. Deng, “Prediction of the Spread of Smoke in a Huge Transit Terminal Subway Station under Six Different Fire Scenarios,” Tunnelling and Underground Space Technology, Vol. 21, 2012, pp. 128-138. doi:10.1016/j.tust.2012.04.013