Parametric modeling of the hydrodynamics of orifices of various configurations in perforated gas distribution grids and determination of their discharge coefficients

Perforated structural elements are widely used in the chemical and other process industries due to their low cost and ease of manufacture, fulfilling various specific functions as parts of apparatuses, machines, and equipment. One of the most common functions is the uniform distribution of gas throughout the working volume. This aspect is crucial, for example, in fluidized-bed apparatuses, because the flow pattern and the efficiency of the ongoing process depend directly on it. Therefore, the design of a perforated element at the engineering stage must be clearly justified in terms of compliance with the specified process requirements for a particular application. One of the key parameters governing the behavior of a continuous-medium flow in such systems is the discharge coefficient which characterizes the deviation of the actual volumetric flow rate from the theoretically possible one. Its accurate determination makes it possible to establish a quantitative relationship between geometry and the hydrodynamic characteristics of the flow which is especially important for engineering calculations and design optimization. Six configurations of single orifices and one type of perforated element with simple cylindrical holes arranged in an equilateral-triangular pattern were investigated. Parametric computational fluid dynamics modeling was performed in the finite-element environment COMSOL Multiphysics 6.1. Dimensionless correlations were formulated in accordance with similarity theory and the Buckingham Pi theorem. The simulation results were processed using multivariate nonlinear regression methods (Levenberg Marquardt algorithm with multi-start initialization of the parameter vector). The selection of specific correlation forms was carried out using the Bayesian and Akaike information criteria. The accuracy of the resulting models was assessed using correlation and determination coefficients as well as the mean absolute percentage error. A set of dimensionless correlations for the discharge coefficient was obtained for six single-orifice configurations and for a perforated element with cylindrical holes. It was found that within the studied ranges of dimensionless parameters, the discharge coefficient is predominantly described by exponential relationships with saturation with respect to the Reynolds number. For single orifices, correlation and determination coefficients of 0.97–0.99 were achieved with a mean absolute percentage error of 3–8 %. For the perforated plate the coefficient of determination reaches 0.99 with a mean absolute percentage error of 5.48 %. The derived discharge coefficient correlations have practical value as part of broader design methodologies for flat perforated gas-distribution devices, since they provide a quantitative link between geometry, the overall hydraulic resistance of the perforated element, and the actual linear velocity in the channels. The approach proposed in this work can also be used independently to study gas-distribution devices with both regular and more complex configurations.

1. Stuppy M.L., Roe F.C., Muller R., Hartley J.R. Types of filter bottoms. Journal American Water Works Association, 1954, vol. 46, no. 6, pp. 548–554 doi: 10.1002/j.1551-8833.1954.tb20264.x

2. Mikulenok I.O. Designs of bubble caps of the contact plates of massexchange columns (review of patents). Chemical and Petroleum Engineering, 2018, vol. 54, no. 5-6, pp. 410–417. doi: 10.1007/s10556-018-0495-y

3. Baird M.H.I, Vijayan S., Rao N.V.R., Rohatgi A. Extraction and absorption with a vibrating perforated plate. The Canadian Journal of Chemical Engineering, 1989, vol. 67, no. 5, pp. 787–800. doi: 10.1002/cjce.5450670510

4. Ismail A., Khalid A., Kiong T.Y. Analysis of mixing effectiveness of perforated plate for inline static mixer. Journal of Complex Flow, 2020, vol. 2, no. 1, pp. 16–22.

5. Dang M., Jia X., Jiang W., Yin J., Wang R. Study on the uniform flow in adsorption device with perforated plate distributor. Chemical Engineering Research and Design, 2024, vol. 208, pp. 490–499. doi: 10.1016/j.cherd.2024.07.017

6. Wang R., Guo X., Zhan M., Liu Y. CFD analysis of the intensification mechanism of bubble breakup by perforated plates in a bubble column. Chemical Engineering and Processing-Process Intensification, 2025, vol. 213, pp. 110288. doi: 10.1016/j.cep.2025.110288

7. Johansen F.C. Flow through pipe orifices at low Reynolds numbers. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 1930, vol. 126, no. 801, pp. 231–245. doi: 10.1098/rspa.1930.0004

8. Hall G.W. Analytical determination of the discharge characteristics of cylindrical-tube orifices. Journal of Mechanical Engineering Science, 1963, vol. 5, no. 1, pp. 91–97. doi: 10.1243/JMES_JOUR_1963_005_013_02

9. Jorissen A.L., Newton H.T. Discharge measurements by means of cylindrical nozzles. Journal of Fluids Engineering, 1952, vol. 74, no. 5, pp. 825–835. doi: 10.1115/1.4015933

10. Spikes R.H., Pennington G.A. Discharge coefficient of small submerged orifices. Proceedings of the Institution of Mechanical Engineers, 1959, vol. 173, no. 1, pp. 661–674. doi: 10.1243/PIME_PROC_1959_173_055_02

11. Ashikhin V.I., Geller Z.I., Skobeltsyn Yu.A. The outflow of real liquid from external cylindrical nozzles. Oil Industry, 1961, no. 9, pp. 55-59. (in Russian)

12. Lichtarowicz A., Duggins R.K., Markland E. Discharge coefficients for incompressible non-cavitating flow through long orifices. Journal of Mechanical Engineering Science, 1965, vol. 7, no.2, pp. 210–219. doi: 10.1243/jmes_jour_1965_007_029_02

13. Kolodzie P.A., Van Winkle M. Discharge coefficients through perforated plates. AIChE Journal, 1957, vol. 3, no. 3, pp. 305–312. doi: 10.1002/aic.690030304

14. Smith P.L., Van Winkle M. Discharge coefficients through perforated plates at Reynolds numbers of 400 to 3,000. AIChE Journal, 1958, vol. 4, no. 3, pp. 266–272. doi: 10.1002/aic.690040306

15. Kuzmin M.I., Kushniruk D.I., Anikin A.V., Verba S.V., Zubov D.V. Applying the digital twin concept to chemical process design, modeling and control. Software & Systems, 2024, no. 4, pp. 629–637. (in Russian). doi: 10.15827/0236-235X.148.629-637

16. Huang S., Ma T., Wang D., Lin Z Study on discharge coefficient of perforated orifices as a new kind of flowmeter. Experimental Thermal and Fluid Science, 2013, vol. 46, pp. 74–83. doi: 10.1016/j.expthermflusci.2012.11.022

17. Tomaszewska-Wach B., Ligus G. Assessment of the impact of k-ε and k-ω turbulence models on the compatibility of CFD simulations with PIV measurements for flow through a measuring orifice. Applied Sciences, 2025, vol. 15, no. 22, pp. 12204. doi: 10.3390/app152212204

18. Wilcox D.C. Turbulence Modeling for CFD. DCW Industries, 2006, 522 p.