Modeling friction factor in pipeline flow using a GMDH-type neural network

Figures & data

Table 1. Literature review summary

References	Description	Method	Data-set	Source for data-set	Range—ϵ/D	Range—Re
Shayya et al. (2005)	Authors developed ANN to find Fanning friction factor for Herschel–Bulkley fluids in closed pipes. Input parameters were Re and He.	Feed-forward ANN	1,991 (training), 29,507 (validation)	Regula-Falsi iterative method to solve an implicit functions based on mechanical energy balance, laminar flow function, pressure gradient and friction factor. Solutions apply for laminar, transition and turbulent flows.	n/a	2–1E8
Sablani et al. (2003)	Authors develop an ANN to find Fanning friction factor for flow of Bingham plastic fluids in closed pipes. Input parameters were Re and He.	Feed-forward ANN	1,177 (training), 32,141 (validation)	Regula-Falsi iterative method was used to solve for f given Re and He in laminar and turbulent conditions.	n/a	2–1E8
Sablani and Shayya (2003)	Authors developed an ANN to find the Fanning friction factor for power law (shear thinning) fluids in closed pipes.	Feed-forward ANN	7,420 (training), 72,400 (validation)	Regula-Falsi iterative method was used to solve an equation the authors developed based on the von Karman equation, which is a function of Re and n, where n is the flow behavior index.	n/a	2E3–1E8
Shayya and Sablani (1998)	Authors developed an ANN to find the Darcy friction factor for flow of Newtonian fluids in closed pipes.	Feed-forward ANN	1,720 (training), 82,100 (validation)	Regula-Falsi iterative method was used to solve the Colebrook–White equation for specified ranges of relative roughness and Reynolds number.	1E-6 to 5E-2	2E3–1E8
Tufail and Ormsbee (2006)	Authors develop a FFSGA based on genetic algorithms and least squares optimization to find Darcy friction factor. The method combines numerical coefficients, functions of decision variables, and mathematical operators.	Fixed functional set genetic algorithm (FFSGA)	100	Authors used the Swamee-Jain equation (explicit version of Colebrook–White equation, limited to full flow circular pipe) to generate data-sets, with specified ranges of relative roughness and Re.	0.001–0.01	1E5–1E6
Mittal and Zhang (2007)	Authors created 4 ANNs to find friction factor for Newtonian, Bingham plastic, Power law, and Herschel–Bulkely fluids. Inputs were mean velocity, viscosity, density, and pipe diameter.	Back-propogation ANN	45,747 (training), 6,862 (validation)	Authors used an iterative solution to yield friction factor from the von Karman correlation.	n/a	n/a
Bilgil and Altun (2008)	Authors develop an ANN to predict friction factor in open channel flow for a Newtonian fluid in comparison with prediction of the Manning equation, given experimental data.	Feed-forward ANN	n/a	Authors created an experimental setup with an adjustable-slope open channel and varied the W/h ratio of the channel and water discharge. They compared their experimental results to their ANN as well as the Manning equation.	n/a	8,291–217,791
Özger and Yildirim (2009)	Authors used a set of 9 “fuzzy” rules (IF THEN statements) relating Re, relative roughness and f, whereby the rules are structured similar to an ANN to predict f.	Adaptive neuro-fuzzy inference system (ANFIS)	1,042 (training), 700 (validation)	Authors used the graph reading software Techdig 2.0 to sensitively read data from the Moody diagram.	1E-6–5E-2	4E3–1E8
Yazdi and Bardi (2011)	Authors developed an ANN to predict Darcy friction factor, given Re and relative roughness as inputs. They also compared predictions of f depending on the training algorithm used.	Feed-forward ANN	1,600 (training), 400 (validation)	Authors generated a data-set from the Haaland equation, which is explicit in f, given specified ranges of relative roughness and Re.	n/a	3E3–1E8
Fadare and Ofidhe (2009)	Authors developed an ANN to predict Darcy friction factor, given Re and relative roughness as inputs. They also compared predictions of f depending on the training algorithm used.	Feed-forward ANN	1,920 (training), 640 (validation)	Authors generated a data-set from the Haaland equation, which is explicit in f, given specified ranges of relative roughness and Re.	5E-6–7E-2	2.5E3–1E8

Acronyms: MRE = Mean relative error; MSE = Mean square error; MAPE = Mean absolute percentage error; SSE = Sum of square error; R = Correlation coefficient; R² = Coefficient of determination; STD = Standard deviation.

MAE, STD (of MAE), MRE, STD (of MRE), R².

MAE, STD (of MAE), MRE, STD (of MRE), R².

MAE, STD (of MAE), MRE, STD (of MRE), R².

MAE, STD (of MAE), MRE, STD (of MRE), R².

MSE, Max absolute error.

Avg. Absolute error, R.

R².

MSE, R, Max % Error, Time, Iterations.

MSE, MAPE (%), SSE, R.

Figure 1. Example GMDH ANN.

Figure 2. Distribution of samples over the range of Reynolds number and relative roughness.

Figure 3. Optimal structure of GS-GMDH network for modeling of friction factor.

Figure 4. (a) Correlation plot of the model versus the Colebrook–White equation for the entire dataset (b) A simplified reproduction of the classical Moody diagram using the ANN model (solid and dashed lines) with Colebrook–White data points (X) overlain.

Table 2. Comparison of error and network complexity between this paper and other investigations

References	Structure	No. of neurons	R^2	MRE	MAE
This study	2-1-1-1	5	0.9954	0.0344	0.0011
Shayya et al. (2005)	2-12-12-12-1	39	0.9899	0.0173	0.00013
Sablani et al. (2003)	2-8-8-1	19	0.9870	0.0201	0.000079
Sablani et al. (2003)	2-4-1	7	0.972	0.0316	0.00105
Sablani and Shayya (2003)	2-12-12-1	27	0.9999	0.0019	2.0E-7
Shayya and Sablani (1998)	2-14-14-14-1	45	0.9996	0.0122	3.0E-6
Bilgil and Altun (2008)	6-7-15-1	29	0.9926	–	–
Özger and Yildirim (2009)	2-6-9-9-1-1	28	–	0.0068	–

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Mittal, G. S., & Zhang, J. (2007). Friction factor prediction for Newtonian and non-Newtonian fluids in pipe flows using neural networks. International Journal of Food Engineering, 3(1), 1–18.

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Özger, M., & Yildirim, G. (2009). Determining turbulent flow friction coefficient using adaptive neuro-fuzzy computing techniques. Advances in Engineering Software, 40, 281–287.

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Yazdi, M., & Bardi, A. (2011). Estimation of friction factor in pipe flow using artificial neural networks. Canadian Journal on Automation, Control and Intelligent Systems, 2, 52–56.

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Fadare, D. A., & Ofidhe, U. I. (2009). Artificial neural network model for prediction of friction factor in pipe flow. Journal of Applied Sciences, 5, 662–670.

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