Evaluation of the Residence Time Distribution (RTD) for Flow in Ducts with Velocity Profile of Two Independent Variables

Document Type : Research Article


Chemical Engineering Department, Faculty of Engineering, Ardakan University, Ardakan, I.R. IRAN


The correct information on RTD can help in system design and evaluation. The RTD corresponding to the velocity profile is known only for certain cases, where the velocity profile depends on one coordinate only. In this research, a general procedure for the derivation of RTD corresponding to a known velocity profile is introduced. The RTD of laminar flows in different ducts as elliptic, equilateral triangular, moon-shaped, and rectangular ducts are derived. Also, it is shown that the final RTD for laminar flow in any duct, can be estimated using relation E(θ) = K θminn which is similar to laminar flow in the pipe, with their own dimensionless minimum time, qmin, where is defined as the required time for traveling the duct with the maximum velocity in a unit of space-time. The values of K and n are calculated to meet the condition of. Besides, the values of qmin for different cross-sections are studied. The results show that the RTD of elliptic ducts is precisely similar to the pipe flow. In the case of other shape ducts, the proposed model shows a suitable estimate of the numerical values. The previously published experimental data and precise analytical solutions agree with the proposed model with an acceptable consistency, except for very little time say θmin < θ < 0.7.  


Main Subjects

[2] Rajamanickam A., Balu K., Design and Development of Mathematical Model for Static Mixer, Iran. J. Chem. Chem. Eng. (IJCCE), 35 (1): 109–116 (2016).
[3] Portillo P.M., Ierapetritou M.G., Muzzio F.J., Characterization of Continuous Convective Powder Mixing Processes, Powder Technol., 182: 368–378 (2008).
[4] Rodier E., Dodds J.A., Leclerc D., Clément G., Changes in Fluid Residence Time Distribution During Deep-Bed Filtration, Chem. Eng. J., 68: 131–138 (1997).
[5] St-Pierre J., Wong A., Diep J., Kiel D., Demonstration of a Residence Time Distribution Method for Proton Exchange Membrane Fuel Cell Evaluation, J. Power Sources, 164 (2007).
[6] Hasson D., Drak A., Komlos C., Yang Q., Semiat R., Detection of Fouling on RO Modules by Residence Time Distribution Analyses, Desalination, 204: 132–144 (2007).
[8] Zhongxi C., Guogang S., Jiao J., Zheng Y., Bing G., Mingxian S., Gas Flow Behavior and Residence Time Distribution in a Rough-Cut Cyclone, Chem. Eng. J., 106: 43–52 (2005).
[10] Sheoran M., Chandra A., Bhunia H., Bajpai P.K., Pant H.J., Industrial Scale RTD Measurement Using Gold Radiotracer, Iran. J. Chem. Chem. Eng. (IJCCE), 40 (1): 313-321 (2021).
[11] Sheoran M., Goswami S., Pant H.J., Biswal J., Sharma V.K., Chandra A., Bhunia H., Bajpai P.K., Rao S.M., Dash A., Measurement of Residence Time Distribution of Liquid Phase in an Industrial-Scale Continuous Pulp Digester Using Radiotracer Technique, Appl. Radiat. Isot., 111: 10–17 (2016).
[12] Goswami S., Pant H.J., Sheoran M., Chandra A., Sharma V.K., Bhunia H., Residence Time Distribution Measurements in an Industrial-Scale Pulp Digester Using Technetium-99m as Radiotracer, J. Radioanal. Nucl. Chem., 323:1373–1379 (2020).
[13] Sheoran M., Chandra A., Bhunia H., Bajpai P.K., Pant H.J., Residence Time Distribution Studies Using Radiotracers in Chemical Industry—A Review, Chem. Eng. Commun., 205: 739-758 (2018).
[14] Günther M., Schneider S., Wagner J., Gorges R., Henkel T., Kielpinski M., Albert J., Bierbaum R., Köhler J.M., Characterisation of Residence Time and Residence Time Distribution in Chip Reactors with Modular Arrangements by Integrated Optical Detection, Chem. Eng. J., 101: 373–378 (2004).
[15] Deng Z.Q., Jung H.S., Ghimire B., Effect of Channel Size on Solute Residence Time Distributions in Rivers, Adv. Water Resour., 33: 1118–1127 (2010).
[16] Gondrexon N., Renaudin V., Petrier C., Clement M., Boldo P., Gonthier Y., Bernis A., Experimental Study of the Hydrodynamic Behaviour of a High Frequency Ultrasonic Reactor, Ultrason. Sonochem., 5: 1–6 (1998).
[17] Guangsuo Y., Zhijie Z., Qiang Q., Zunhong Y., Experimental Studying and Stochastic Modeling of Residence Time Distribution in Jet-Entrained Gasifier, Chem. Eng. Process., 41: 595-600 (2002).
[18] Galbraith S.C., Park S., Huang Z., Liu H., Meyer R.F., Metzger M., Flamm M.H., Hurley S., Yoon S., Linking Process Variables to Residence Time Distribution in a Hybrid Flowsheet Model for Continuous Direct Compression, Chem. Eng. Res. Des., 153: 85–95 (2020).
[19] Pinto J.C., Melo P.A., Biscaia J., Characterization of the Residence Time Distribution in Loop Reactors, Chem. Eng. Sci., 56: 2703–2713 (2001).
[20] Levenspiel O., "Chemical Reaction Engineering", 3rd ed., John Wiley & Sons Inc., (1998).
[21] Emami-Meibodi M., Soleimani M., Bani-Najarian S., Toward Enhancement of Rotating Packed Bed (RPB) Reactor for CaCO3 Nanoparticle Synthesis, Int. Nano Lett., 8: 189-199  (2018).
[22] Li S.Q., Chi Y., Li R.D., Yan J.H., Cen K.F., Axial Transport and Residence Time of MSW in Rotary Kilns: Part II. Theoretical and Optimal Analyses, Powder Technol., 126: 228–240  (2002).
[23] Levenspiel O., Lai B.W., Chatlynne C.Y., Tracer Curves and the Residence Time Distribution, Chem. Eng. Sci., 25: 1611–1613  (1970).
[24] Gao Y., Muzzio F.J., Ierapetritou M.G., A Review of the Residence Time Distribution (RTD) Applications in Solid Unit Operations, Powder Technol., 228: 416–423  (2012).
[27] Simcik M., Ruzicka M.C., Mota A., Teixeira J.A., Smart RTD for Multiphase Flow Systems, Chem. Eng. Res. Des., 90: 1739–1749  (2012).
[29] Scott Fogler H., "Elements of Chemical Reaction Engineering", Prentice Hall (2006)
[30] Shah R.K., London A.L., "Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data", Academic Press, New York, NY (1978).
[31] Cantu-Perez A., Bi S., Barrass S., Wood M., Gavriilidis A., Residence Time Distribution Studies In Microstructured Plate Reactors, Appl. Therm. Eng., Pergamon, 634–639 (2011).
[32] Pudjiono P.I., Tavare N.S., Garside J., Nigam K.D. P., Residence Time Distribution from a Continuous Couette Flow Device, Chem. Eng. J., 48: 101-110 (1992).