This study presents the application of statistical thermodynamics in modeling orifice discharge function. Micro-based variables such as the velocity of fluid flow at the midpoint of the reservoir elevated to the mid-height of the orifice and the head loss due to sudden expansion that was neglected in the classical discharge model were considered in the modeling operation. Bernoulli’s equation was used to determine the velocity model at the exit point of the orifice. Some classical thermodynamic models were used to compute certain parameters like: the flow pressure of fluid (water) (Pf), pressure drop (Pd), exit temperature, polytropic work (W), heat absorption/rejection (Q), and flow energy (E). The grand canonical ensemble which applies to open systems was used to establish a relationship between the flow discharge and energy variables according to the micro-based behavior of fluid flow through the orifice that generates the observable macro-flow behavior of a fluid. The discharge values obtained using the experimental (Qe), classical (Q(c,)) and statistically derived orifice discharge (Qs) models were compared and accessed statistically using the indices of: mean bias error (MBE), mean percentage error (MPE), root mean square error (RMSE), Nash-Sutcliffe equation (NSE), and coefficient of correlation (R). The results of the study showed that the new model that was derived using a statistical thermodynamics approach outperformed the classical orifice model as its MBE value of 5.032E-05, MPE of 5.62, RMSE of 5.24E-05, NSE of 0.873 and R of 0.999 were all better than that of the classical model having an MBE value of -1.0042E-04, MPE of -11.30 (underestimation), RMSE of 5.814E-05, and R of 0.999. In addition, the values of Pf, T2, W, Q, and E increased as the flow head increased. The polytropic work, W was negative, indicating that the system did some work. The positive values of Q and E showed that the system absorbed energy from its environment during the fluid flow operation. Furthermore, all the orifice discharge functions (Qs, Q(c) and Qe) had a direct linear relationship with flow energy, E which therefore satisfied the grand canonical ensemble model for the open system’s thermodynamic micro-variable description. Hence, the new orifice discharge model is recommended for the industry's volumetric flow rate measurement of Newtonian fluids.
| Published in | American Journal of Mechanical and Industrial Engineering (Volume 10, Issue 1) |
| DOI | 10.11648/j.ajmie.20251001.11 |
| Page(s) | 1-11 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Classical Thermodynamics, Discharge, Grand Canonical Ensemble, Orifice, Statistical Thermodynamics
| [1] | R. K. Rajput, (2011). A textbook of fluid mechanics and hydraulic machines, 4th edition, New Delhi, S. Chand & Company Ltd. Pp. 297-306. |
| [2] | L. T. Chang, and H. L. John, (2017). Statistical thermodynamics, 4th edition, NewYork, Hemisphere Publishing Corporation, p. 3. |
| [3] | A. Gans, P. Aussillous, B. Dalloz, and M. Nicolas, “The effect of cohesion on the discharge of a granular material through the orifice of a silo,” EPJ Web of Conferences, vol. 249, p. 08014, 2021, |
| [4] | A. Hotupan and A. Hadarean, “Experimental Determination of the Discharge Coefficient Through Circular Orifice in PVC Pipes,” Journal of Applied Engineering Sciences, vol. 10, no. 2, pp. 133–138, Nov. 2020, |
| [5] | M. H. Assran, B. shenouda, and H. I. Mohamed, “Effect of orifice-meter shape on discharge coefficient and head loss through it,” Research Square Platform LLC, Jun. 2022. Accessed: Jan. 06, 2025. [Online]. Available: |
| [6] | R. Cao, Y. Liu, and C. Yan, “A criterion for flow mechanisms through vertical sharp-edged orifice and model for the orifice discharge coefficient,” Petroleum Science, vol. 8, no. 1, pp. 108–113, Feb. 2011, |
| [7] | C. D. Jan and Q. T. Nguyen, “Discharge Coefficient for a Water Flow through a Bottom Orifice of a Conical Hopper,” Journal of Irrigation and Drainage Engineering, vol. 136, no. 8, pp. 567–572, Aug. 2010, |
| [8] | C. H. Achebe, O. C. Okafor, and E. N. Obika, “Design and implementation of a crossflow turbine for Pico hydropower electricity generation,” Heliyon, vol. 6, no. 7, p. e04523, Jul. 2020, |
| [9] | O. C. Okafor and E. I. Echezona, “Mathematical Modelling of the Thermodynamical Functional Parameters of a Typical Hand-Driven Reciprocating Compressor,” The International Journal of Science & Technoledge, vol. 7, no. 9, Sep. 2019, |
| [10] | V. O. Onyeka, C. C. Nwobi-Okoye, O. C. Okafor, K. E. Madu, and O. M. Mbah (2021). Estimation of global solar radiation using empirical models. Journal of Engineering Sciences, Vol. 8(2), pp. 11-19, |
| [11] | O. M. Mbah, C. I. Madueke, R. Umunakwe, and C. O. Okafor, “Machine Learning Approach for Solar Irradiance Estimation on Tilted Surfaces in Comparison with Sky Models Prediction,” Journal of Engineering Sciences, vol. 9, no. 2, pp. G1–G6, 2022, |
| [12] | H. O. Nnabuenyi, L. N. Okoli, F. C. Nwosu, and G. Ibe (2017). Estimation of global solar radiation using sunshine and temperature-based models for Oko town in Anambra State, Nigeria. American Journal of Renewable and Sustainable Energy, 3(2): 8-14. |
| [13] | O. N. Akpenyi-Aboh, and O. C. Okafor (2024). Modeling the Solar Intensity of Asaba Town in Nigeria Using Response Surface Methodology and Machine Learning Techniques. American Journal of Mechanical and Industrial Engineering, 9(4), 63-74. |
| [14] | E. Schrodinger, E. (1946). Statistical Thermodynamics. Dover Publications, Inc. |
| [15] | Chukwuneke, J. L., Achebe, C. H., Okolie, P. C., Ajike, C. O. (2012). An application of statistical thermodynamics to an open flow system. International Journal of Scientific and Engineering Research, 13(5): 1-6. |
APA Style
Okafor, O. C., Achebe, C. H., Mbaka, U. O. (2025). Application of Statistical Thermodynamics to Modelling Fluid Discharge Through a Circular Orifice System. American Journal of Mechanical and Industrial Engineering, 10(1), 1-11. https://doi.org/10.11648/j.ajmie.20251001.11
ACS Style
Okafor, O. C.; Achebe, C. H.; Mbaka, U. O. Application of Statistical Thermodynamics to Modelling Fluid Discharge Through a Circular Orifice System. Am. J. Mech. Ind. Eng. 2025, 10(1), 1-11. doi: 10.11648/j.ajmie.20251001.11
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Download@article{10.11648/j.ajmie.20251001.11,
author = {Obiora Clement Okafor and Chinonso Hubert Achebe and Uchechukwu Ogbu Mbaka},
title = {Application of Statistical Thermodynamics to Modelling Fluid Discharge Through a Circular Orifice System},
journal = {American Journal of Mechanical and Industrial Engineering},
volume = {10},
number = {1},
pages = {1-11},
doi = {10.11648/j.ajmie.20251001.11},
url = {https://doi.org/10.11648/j.ajmie.20251001.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmie.20251001.11},
abstract = {This study presents the application of statistical thermodynamics in modeling orifice discharge function. Micro-based variables such as the velocity of fluid flow at the midpoint of the reservoir elevated to the mid-height of the orifice and the head loss due to sudden expansion that was neglected in the classical discharge model were considered in the modeling operation. Bernoulli’s equation was used to determine the velocity model at the exit point of the orifice. Some classical thermodynamic models were used to compute certain parameters like: the flow pressure of fluid (water) (Pf), pressure drop (Pd), exit temperature, polytropic work (W), heat absorption/rejection (Q), and flow energy (E). The grand canonical ensemble which applies to open systems was used to establish a relationship between the flow discharge and energy variables according to the micro-based behavior of fluid flow through the orifice that generates the observable macro-flow behavior of a fluid. The discharge values obtained using the experimental (Qe), classical (Q(c,)) and statistically derived orifice discharge (Qs) models were compared and accessed statistically using the indices of: mean bias error (MBE), mean percentage error (MPE), root mean square error (RMSE), Nash-Sutcliffe equation (NSE), and coefficient of correlation (R). The results of the study showed that the new model that was derived using a statistical thermodynamics approach outperformed the classical orifice model as its MBE value of 5.032E-05, MPE of 5.62, RMSE of 5.24E-05, NSE of 0.873 and R of 0.999 were all better than that of the classical model having an MBE value of -1.0042E-04, MPE of -11.30 (underestimation), RMSE of 5.814E-05, and R of 0.999. In addition, the values of Pf, T2, W, Q, and E increased as the flow head increased. The polytropic work, W was negative, indicating that the system did some work. The positive values of Q and E showed that the system absorbed energy from its environment during the fluid flow operation. Furthermore, all the orifice discharge functions (Qs, Q(c) and Qe) had a direct linear relationship with flow energy, E which therefore satisfied the grand canonical ensemble model for the open system’s thermodynamic micro-variable description. Hence, the new orifice discharge model is recommended for the industry's volumetric flow rate measurement of Newtonian fluids.},
year = {2025}
}
TY - JOUR T1 - Application of Statistical Thermodynamics to Modelling Fluid Discharge Through a Circular Orifice System AU - Obiora Clement Okafor AU - Chinonso Hubert Achebe AU - Uchechukwu Ogbu Mbaka Y1 - 2025/01/24 PY - 2025 N1 - https://doi.org/10.11648/j.ajmie.20251001.11 DO - 10.11648/j.ajmie.20251001.11 T2 - American Journal of Mechanical and Industrial Engineering JF - American Journal of Mechanical and Industrial Engineering JO - American Journal of Mechanical and Industrial Engineering SP - 1 EP - 11 PB - Science Publishing Group SN - 2575-6060 UR - https://doi.org/10.11648/j.ajmie.20251001.11 AB - This study presents the application of statistical thermodynamics in modeling orifice discharge function. Micro-based variables such as the velocity of fluid flow at the midpoint of the reservoir elevated to the mid-height of the orifice and the head loss due to sudden expansion that was neglected in the classical discharge model were considered in the modeling operation. Bernoulli’s equation was used to determine the velocity model at the exit point of the orifice. Some classical thermodynamic models were used to compute certain parameters like: the flow pressure of fluid (water) (Pf), pressure drop (Pd), exit temperature, polytropic work (W), heat absorption/rejection (Q), and flow energy (E). The grand canonical ensemble which applies to open systems was used to establish a relationship between the flow discharge and energy variables according to the micro-based behavior of fluid flow through the orifice that generates the observable macro-flow behavior of a fluid. The discharge values obtained using the experimental (Qe), classical (Q(c,)) and statistically derived orifice discharge (Qs) models were compared and accessed statistically using the indices of: mean bias error (MBE), mean percentage error (MPE), root mean square error (RMSE), Nash-Sutcliffe equation (NSE), and coefficient of correlation (R). The results of the study showed that the new model that was derived using a statistical thermodynamics approach outperformed the classical orifice model as its MBE value of 5.032E-05, MPE of 5.62, RMSE of 5.24E-05, NSE of 0.873 and R of 0.999 were all better than that of the classical model having an MBE value of -1.0042E-04, MPE of -11.30 (underestimation), RMSE of 5.814E-05, and R of 0.999. In addition, the values of Pf, T2, W, Q, and E increased as the flow head increased. The polytropic work, W was negative, indicating that the system did some work. The positive values of Q and E showed that the system absorbed energy from its environment during the fluid flow operation. Furthermore, all the orifice discharge functions (Qs, Q(c) and Qe) had a direct linear relationship with flow energy, E which therefore satisfied the grand canonical ensemble model for the open system’s thermodynamic micro-variable description. Hence, the new orifice discharge model is recommended for the industry's volumetric flow rate measurement of Newtonian fluids. VL - 10 IS - 1 ER -
Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria
Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria
Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria
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