Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes
Abstract
:1. Introduction
2. Materials and Methods
2.1. Proposed Mathematical Model Based on Dimensional Parameters, Velocity, and Roughness: The Churchill B(V,ε) Function
Statistical Methodology
2.2. Proposed Mathematical Model Based on Dimensionless Parameters, Reynolds Number and Relative Roughness: The Churchill B(Re) Function
2.2.1. GRG Nonlinear Optimization
- Formulation of the Optimization Problem
- b.
- Concept of Reduced Gradients
- c.
- Feasible Direction
Jh(xk) d ≤ 0
- d.
- Lagrangian Function
- e.
- Karush–Kuhn–Tucker (KKT) ConditionsFor a point to be optimal, it must satisfy the KKT conditions:
- ○
- Stationarity Condition, Equation (18):
- ○
- Primal Feasibility Condition:
- ○
- Dual Feasibility Condition:
- ○
- Complementarity Condition:
- f.
- GRG Method Algorithm
- Initialization: Choose an initial feasible point x0. Set initial values for the Lagrange multipliers.
- Search Direction: Compute the search direction dk using the reduced gradient.
- Update: Update the current point: , where is the step size determined by a line search procedure.
- Convergence Check: Check if the KKT conditions are satisfied within a predefined tolerance.
- Iteration: If not converged, update the Lagrange multipliers and repeat from Step 2.
- g.
- Implementation Details
2.2.2. Optimization of Churchill B(Re)’s Friction Factor Equation Using Generalized Reduced Gradient Method
2.2.3. Proposed Function for the Churchill B(Re) Function
2.3. Other Equations Used for Comparative Analysis
2.4. Test Cases for the Validation of the Churchill Functions
2.4.1. Input Parameters and Calculation Method
2.4.2. Development of Churchill Functions through Test Case Generation
2.4.3. Test Case Generation for Validation of Churchill Functions
3. Results
3.1. Precisions and Errors of the Selected Methods
3.1.1. Relative Errors
3.1.2. Absolute Errors
4. Discussion and Comparative Results
4.1. Normal Probability Plot
4.2. Histogram of the Residuals
4.3. Error Comparison
4.3.1. Absolute Error Comparison for Churchill B(V,ε) Function
4.3.2. Relative Error Comparison for Churchill B(Re) Function
4.4. Statistical Significance
4.5. Analysis of Precision of the Churchill B(V,ε) Function in Comparative Cases
Comparative Analysis
5. Conclusions
5.1. Limitations of the Study
5.1.1. Sample Size and Flow Conditions
5.1.2. Fluid Properties
5.1.3. Comparison Scope
5.2. Future Work
5.2.1. Expand the Dataset and Flow Conditions
5.2.2. Explore Fluid Property Effects
5.2.3. Evaluating the Practicality and Impact of Incorporating the New Churchill B(V,ε) and Churchill B(Re) Expression in WDS Simulation Software (in the Most Current Version Available at the Time of the Future Evaluation)
5.2.4. Broaden the Comparison Scope
5.2.5. Experimentally Validate and Explore Advanced Techniques
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Comparative Analysis of the Precision of Friction Factor Estimation Methods: Evaluation of Absolute Error Grouped by Reynolds Number
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ε (mm) | A | C | F | G | H | J | K | M |
---|---|---|---|---|---|---|---|---|
0.001500 | 123.7852122 | 212.2573135 | 1.0055491 | 1.6833766 | 0.4692636 | 1.9972844 | 0.0997693 | 0.5860833 |
0.004125 | 123.7851978 | 212.2578596 | 1.0072233 | 1.6866568 | 0.4693298 | 1.9951957 | 0.0988583 | 0.5927161 |
0.008250 | 123.7610000 | 212.3220000 | 1.0073013 | 1.6916093 | 0.4700108 | 1.9885561 | 0.0975234 | 0.5808196 |
0.012375 | 113.0025564 | 212.8032189 | 1.0056746 | 1.6999204 | 0.4711697 | 1.9602235 | 0.0994546 | 0.4840345 |
0.015000 | 111.3241942 | 212.9999935 | 1.0059032 | 1.7143822 | 0.4679270 | 1.9670447 | 0.0967506 | 0.4961413 |
0.020000 | 110.3689523 | 213.2338135 | 1.0053641 | 1.7355369 | 0.4655973 | 1.9543190 | 0.0948980 | 0.4678421 |
0.041250 | 97.7282841 | 213.4999993 | 1.0036670 | 1.7961942 | 0.4565267 | 1.9213918 | 0.0928863 | 0.3869753 |
0.082500 | 93.3690383 | 214.0246813 | 1.0021490 | 1.8760483 | 0.4455154 | 1.8611599 | 0.0929679 | 0.2993959 |
0.123750 | 82.0499806 | 215.6486086 | 1.0013210 | 1.9256581 | 0.4330146 | 1.8263892 | 0.1023031 | 0.2445000 |
0.150000 | 81.7039696 | 215.6175947 | 1.0010905 | 1.9446166 | 0.4291668 | 1.8078201 | 0.1053104 | 0.2229616 |
0.225000 | 54.1653832 | 216.6564593 | 1.0006130 | 2.0735259 | 0.4004915 | 1.8259778 | 0.1050203 | 0.2008119 |
0.300000 | 36.9417419 | 217.8925943 | 1.0004194 | 2.0199180 | 0.4036004 | 1.7961138 | 0.1183420 | 0.1633125 |
0.400000 | 14.6831034 | 218.4900000 | 1.0002137 | 2.1366956 | 0.3792815 | 1.8614158 | 0.1124163 | 0.2019214 |
0.500000 | 14.6831034 | 219.5796607 | 1.0000903 | 2.0978096 | 0.3792815 | 1.8614158 | 0.1208657 | 0.1939013 |
Variable | Symbol | SI Units |
---|---|---|
Absolute Roughness | ε | m |
Internal Diameter | D | m |
Velocity | V | m/s |
Kinematic Viscosity | ν | m²/s |
Pipe Roughness ε (mm) | % Average Relative Error Churchill B(Re) | % Average Relative Error Churchill B(V,ε) |
---|---|---|
0.001500 | 0.019808 | 0.437443 |
0.004125 | 0.018866 | 0.531191 |
0.008250 | 0.021176 | 0.671271 |
0.012375 | 0.023488 | 0.769550 |
0.015000 | 0.023097 | 0.817256 |
0.020000 | 0.025254 | 0.885935 |
0.041250 | 0.032443 | 1.013247 |
0.082500 | 0.036022 | 1.033179 |
0.123750 | 0.034340 | 0.996127 |
0.150000 | 0.032512 | 0.967613 |
0.225000 | 0.027243 | 0.889976 |
0.300000 | 0.023956 | 0.824762 |
0.400000 | 0.019154 | 0.754921 |
0.500000 | 0.016149 | 0.699515 |
Model | Advantages | Disadvantages |
---|---|---|
Churchill B(Re) Function |
|
|
Churchill B(V,ε) Function |
|
|
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Benavides-Muñoz, H.M. Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes. Water 2024, 16, 2328. https://doi.org/10.3390/w16162328
Benavides-Muñoz HM. Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes. Water. 2024; 16(16):2328. https://doi.org/10.3390/w16162328
Chicago/Turabian StyleBenavides-Muñoz, Holger Manuel. 2024. "Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes" Water 16, no. 16: 2328. https://doi.org/10.3390/w16162328
APA StyleBenavides-Muñoz, H. M. (2024). Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes. Water, 16(16), 2328. https://doi.org/10.3390/w16162328