# Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Calculation Algorithm Concept

_{i}) reaching the input layer were multiplied by the values of weights w

_{ij}. The obtained sums were transformed using the linear or nonlinear activation function (f) and then were transmitted to the neuron(s) of the output layer. The disadvantage of the neural network method was that the initial values of the weights affected the learning process, which may create problems with finding the global minimum of a function [74].

#### 2.2. Differential Equation of Stormwater Volume Balance

_{or}. For the assumptions mentioned above, the variability in the amount of stormwater flowing into the tanks during the growth phase (t

_{p}) and decline phase (t

_{op}) can be described with the following dependency (see also the graphical presentation in Figure 3):

^{3}s

^{−1}), I

_{m}is the peak flow to the tank (m

^{3}s

^{−1}), t

_{p}is the peak flow time (min), t

_{op}is the time of water lowering, and t is the time (min).

_{z}is the geometric parameter of the tank described as L

_{ret}·D

_{ret}

^{0.60}(where L

_{ret}is the length of the retention chamber (m), and D

_{ret}is the diameter of the retention chamber (m)), B

_{or}is the parameter describing the outlet orifice (expressed as μ·π·4

^{−1}·D

_{out}

^{2}·(2g)

^{0.50}), and V is the tank volume variable in time t (m

^{3}).

**Figure 2.**Scheme of a pipe tank. Where: D

_{ret}is the diameter of the retention chamber, L

_{ret}is the length of the retention chamber (m), D

_{or}is the diameter of the outlet orifice from the pipe chamber (m), D

_{in}is the diameter of the inlet channel (m), D

_{out}is the diameter of the outlet chamber (m), h is the filling level of the retention chamber (m), i is the longitudinal slope of the pipe chamber (-), U

_{av}is the average flow velocity of the stormwater in the retention chamber (ms

^{−1}), I

_{in}is the stormwater discharge flowing into the tank (m

^{3}s

^{−1}), I

_{out}is the stormwater discharge flowing out of the retention chamber (m

^{3}s

^{−1}), 1 is the inlet channel, 2 is the inlet chamber, 3 is the retention chamber, 4 is the safety overflow, and 5 is the outlet chamber.

#### 2.3. Dimensionless Differential Equation of Volume Balance

- -
- Dimensionless time from the start (t*):

- -
- Dimensionless flow rate (I*):

- -
- Dimensionless tank volume (V*):

_{c}is the total inflow hydrograph volume (m

^{3}), V

_{c}= I

_{max}(t

_{p}+ t

_{op})/2.

_{ret}. When the retention chamber is completely filled, the maximal flow rate through the outlet orifice equals:

_{c}= 900 ÷ 45,000 m

^{3}, a peak flow of (I

_{m}= 1.5 ÷ 7.5 m

^{3}s

^{−1}), and peak flow time t

_{p}= 10 ÷ 150 min. For the assumed hydrographs, the obtained values of the ω parameter ranged from 0.15 to 1.00. This assumption taken for the calculation actually meant that the peak time t

_{p}was assumed to be maximal at one-half of the end hydrograph time. That assumption was related to the fact that, to determine the nomogram with the widest possible range of parameters enabling all possible cases were taken into account, even the unlikely ones, to conditions of maximal tank capacity. The length of the retention chamber was calculated assuming β = 0.1–0.9 and the diameter of the outlet orifice D

_{or}= 0.25 ÷ 1.25 m. The differential equation V

_{(h)}= f

_{(t)}was solved using the explicit Runge–Kutta method, in relation to retention chamber length L

_{ret}[79].

#### 2.4. Local Sensitivity Analysis

_{i}which is the product of the standard deviation σ

_{Xi}and the derivative of partial primary function ∂y/∂x

_{i}. The number of sensitivity functions that can be created for one primary function is equal to the number of variables (and parameters) in the primary function.

_{tp}

_{,}S

_{Im}, S

_{λ}, S

_{η}

_{,}where

_{λ}and

_{η}are sensitivity coefficients) were highly sensitive, a local sensitivity analysis was employed to assess the influence of the influent hydrograph parameters (t

_{p}, I

_{m}, λ = t

_{op}·t

_{p}

^{−1}) on the length of the retention chamber. The standard deviation (σ

_{Xi}) of the aforementioned variables in the presented methodology was determined using the Monte Carlo method, by sampling 1000 values from continuous uniform distributions, where the independent variables changed within appropriately assumed ranges. In this case, the sensitivity function described with formula (14) illustrates the change to the retention chamber length in relation to the unit change of parameter x

_{i}. On the basis of the above-mentioned dependencies (see Equation (10)), the relationship describing the length of the chamber was determined and, then, the successive partial derivatives were determined. The following sensitivity functions (S) were developed, enabling us to evaluate the effect of the influent hydrograph parameters and η on the length of the retention chamber:

_{Im}= f(I

_{m}, λ), S

_{tp}= f(t

_{p}, λ), S

_{λ}= f(t

_{p}, λ), and S

_{η}= f(t

_{p}, η).

#### 2.5. Influence of Hydrograph Parameters and Outlet Devices on Calculation Errors

_{ret[mod]}is the tank volume obtained basing on numerical simulations (m

^{3}), V

_{ret[nom]}is the tank volume determined based on the proposed nomograms (m

^{3}), L

_{ret[mod]}is the tank length calculated using the simulations (m), L

_{ret[nom]}is the tank length determined based on the proposed nomogram (see Figure 4) (m), F

_{ret}is the transverse cross-section area of the retention chamber (m

^{2}), and ∆L

_{ret}is the absolute error of the predicted tank length.

_{mod}is the the value of the parameter determined on the basis of a simulation using a hydrodynamic model, i.e., the results from SWMM were taken, and the η was directly substituted and determined here based on data from the hydrograph, the diameter of the retention pipe chamber and the diameter of the outlet orifice from the pipe chamber. The range of parameter changed from 0.0328 to 0.7032.

#### 2.6. Modeling of the Errors in the Predicted Tank Length Using Data Mining Methods

## 3. Results and Discussion

#### 3.1. Dependence η = f(ω, β)

^{2}) in Equation (22) ranged from 0.989 to 0.996 for β = 0.1 ÷ 0.9. The determined curves showed that the values of η increased along with β. The η = f(ω, β = const) curves determined in the form of the nomogram indicated that the greater the value of peak flow reduction and influent hydrograph asymmetry coefficients, the higher the increase in the dispersion of the η parameter values for the analyzed wave asymmetry coefficient (ω = 0.2 ÷ 1.0), and thus the higher the prediction error of the tank volume.

**Figure 4.**Influence of the η parameter and wave asymmetry coefficients (ω) on the peak flow reduction coefficient (β). Points correspond to the values acquired from simulations, whereas curves are related to the values obtained by means of Formula (21).

_{Im}= f(I

_{m}, λ) in the analyzed range. It should be noted that in the range I

_{m}= 0.0 ÷ 3.0 m

^{3}s

^{−1}, the S

_{Im}values for t

_{p}= 500 s and t

_{p}= 1500 s changed in the range from −0.75 × 10

^{6}to −0.25 × 10

^{6}, and from −2.5 × 10

^{6}to −0.30 × 10

^{6}, respectively. In turn, for I

_{m}> 3.0 m

^{3}s

^{−1}the changes to S

_{Im}in the above-mentioned ranges did not exceed 10%. The performed calculations (Figure 5b) indicate that the increase in time from t

_{p}= 500 s to t

_{p}= 2000 s for I

_{m}= 2.0 m

^{3}s

^{−1}did not affect the value of the sensitivity index S

_{p}= f(t

_{p}, λ), while a change to the peak flow at its maximum in the range from I

_{m}= 2.0 m

^{3}s

^{−1}to I

_{m}= 4.0 m

^{3}s

^{−1}reduced the sensitivity index S

_{tp}by 14 × 10

^{6}to 8 × 10

^{6}(43%).

_{p}= 500 ÷ 2000 s (Figure 5c) reduced the sensitivity index S

_{λ}= f(t

_{p}, λ). Nevertheless, it should be noted that in the range of I

_{m}= 0.0 ÷ 3.0 m

^{3}s

^{−1}, the values of S

_{λ}for t

_{p}= 500 s and t

_{p}= 1500 s changed from −4200 to −600 and −11,000 to −600, respectively, whereas for I

_{m}> 3.0 m

^{3}s

^{−1}, the change of S

_{λ}= f(t

_{p}, λ) in the above-mentioned ranges did not exceed 10% and 14%, respectively. The performed calculations (Figure 5d) indicated that, for η ≈ 0 in the analyzed range I

_{max}, there was a sharp change in the sensitivity index S

_{η}, whereas for η > η

_{gr}, the values of the sensitivity index S

_{η}were constant and equaled −0.3 × 10

^{9}.

**Figure 5.**Variability of the sensitivity index (

**a**) S

_{Im}= f(I

_{max}= I

_{m}, t

_{p}), (

**b**) S

_{tp}= f(I

_{max}= I

_{m}, t

_{p}), (

**c**) S

_{λ}= f(λ, t

_{p}), (

**d**) S

_{η}= f(η, I

_{max}= I

_{m}).

_{max}, λ) on the reservoir area was not analyzed. In turn, the paper by Guo [66] presented an analytical–graphical method supported by several computational variants. This paper concerned two methodologies: a very precise one allowing errors lower than 2% and another, simplified one, for which the difference between the obtained results and the reference data was at the level of 4%.

#### 3.2. Influence of the Influent Hydrograph Parameters and Tank Characteristics on the Errors in the Prediction of Its Volume

_{V}) committed using the developed graph were determined with Formula (22). Table 1 lists only the minimum and maximum values of ε

_{V}.

_{Vmax}) was observed for β = 0.1 and its extreme values ranged from −7.01% to 3.77%. On the contrary, the greatest error noted was for β = 0.9 and varied from −36.33% to 22.13%. This means that the relative error values related to the tank volume prediction were significant, thus confirming the need for their estimation. In the performed work, ε

_{V}was first determined using the multiple stepwise regression method because the variables with negligible effect on the relative error of the retention chamber volume were omitted during the calculations. Thus, the following formula was obtained:

_{m}), the time of the peak flow (t

_{p}), the diameter of the outlet device (D

_{or}), and the diameter of the retention chamber (D

_{ret}). The parameters of the proposed mathematical models are presented in Table 2, while Figure 6 and Figure 7 show a comparison of the ε

_{v}values, both calculated and determined on the basis of the numerical simulations and the developed nomogram η = f(ω, β). The performed calculations (Table 2) indicated a comparable agreement of the calculations and measurements of the relative errors (ε

_{V}) obtained by the means of an artificial neural network with 6 neurons in the hidden and exponential activation functions, and using genetic programming. This was confirmed by the values of the particular fitting parameters (Table 2) since in the case of multiple regressions, ANN, and GP (ω, β), the correlation coefficient and Akaike information criterion amounted to 0.6851 and −3553; 0.9310 and −3010 as well as 0.9010 and −3236, respectively.

**Figure 8.**Comparison of the error values obtained on the basis of the developed nomogram ε

_{V[mod]}, calculated using multiple regression ε

_{V [prog]}.

**Figure 9.**Comparison of the error values obtained on the basis of the developed nomogram ε

_{V[mod]}, calculated using genetic programming ε

_{V [prog]}.

**Figure 10.**Comparison of the error values obtained on the basis of the developed nomogram ε

_{V[mod]}, calculated using ANN ε

_{V [prog]}.

_{V[mod]}< −0.10.

_{max}= 3.34 m

^{3}/s t = 15 min and V = 15,000 m

^{3}i.e., ω = 3.34 × 15 × 60/15,000 = 0.20 (according Equation (11)).

^{2}) × (19.62 × 3.5)

^{0.5}= 0.10 (according Equation (13)).

_{or}= 0.57 × 0.785 × 0.3

^{2}× (19.62

^{0.5}) = 0.313 (according Equation (13)).

_{ret}= [0.313 × 15 × 60/(3.5

^{0.21}× 15,000

^{0.65}× 0.04)]

^{2.857}= 815.3 m (according Equation (10)).

_{ret}= 0.011 was obtained, which gave a final result of L

_{ret}= 823.5 m.

_{ret}calculation error values are relatively small, and amount to 1.1%.

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A schematic presentation of the algorithm for establishing the parameters of the pipe tank.

**Table 1.**Maximum values of relative errors (ε

_{Vmax}) for the peak flow reduction coefficients β = 0.1 ÷ 0.9.

β | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|

ε_{Vmax} | 3.77 | 5.75 | 6.16 | 6.31 | 9.45 | 11.19 | 12.92 | 18.44 | 22.13 |

ε_{Vmin} | −7.01 | −7.28 | −9.79 | −12.07 | −18.08 | −20.51 | −22.93 | −25.03 | −36.33 |

**Table 2.**Fitting measures of the obtained models predicting the relative error of tank volume (ε

_{v}).

Method | RRE | MPE | r | AIC |
---|---|---|---|---|

Regression | 0.0442 | 47.51 | 0.6851 | −3553 |

ANN | 0.0272 | 5.60 | 0.9310 | −3010 |

GP(ω, η) | 0.0345 | 8.02 | 0.8872 | −3321 |

GP(ω, β) | 0.0312 | 7.89 | 0.8953 | −3299 |

GP(ω, β) | 0.0296 | 7.21 | 0.9010 | −3236 |

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**MDPI and ACS Style**

Szeląg, B.; Kiczko, A.; Musz-Pomorska, A.; Widomski, M.K.; Zaburko, J.; Łagód, G.; Stránský, D.; Sokáč, M. Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments. *Water* **2021**, *13*, 1035.
https://doi.org/10.3390/w13081035

**AMA Style**

Szeląg B, Kiczko A, Musz-Pomorska A, Widomski MK, Zaburko J, Łagód G, Stránský D, Sokáč M. Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments. *Water*. 2021; 13(8):1035.
https://doi.org/10.3390/w13081035

**Chicago/Turabian Style**

Szeląg, Bartosz, Adam Kiczko, Anna Musz-Pomorska, Marcin K. Widomski, Jacek Zaburko, Grzegorz Łagód, David Stránský, and Marek Sokáč. 2021. "Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments" *Water* 13, no. 8: 1035.
https://doi.org/10.3390/w13081035