# Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Establish the Formulas

#### 2.1. Basic Hydraulic Parameters and Formulas

^{3}/s), A is the discharge section area (m

^{2}) corresponding to the critical depth, T is the water width (m) corresponding to the critical depth, g is the acceleration due to gravity (9.8 m/s

^{2}), and α is the velocity distribution coefficient (generally 1.0).

#### 2.1.1. Circular Sections

#### 2.1.2. Arched Sections

#### 2.1.3. Egg-Shaped Sections

_{i}(i = 1, 2, 3) is the central angle of the corresponding various water depths in rad and y is the water depth in m. On substituting Equations (9)–(11) into Equation (1), the governing equations for the critical depths of egg-shaped sections are obtained:

#### 2.2. Functional Model Construction

#### 2.2.1. Circular Sections

_{c}is the critical depth in a circular section, m.

_{c}), so Equation (8) can be worked out as follow:

_{c}= f(η

_{c}), consider the endpoints of the curve of ε

_{c}= f(η

_{c}). Obtain the limit via Equation (16):

#### 2.2.2. Arched Sections

_{a}is the critical depth in an arched section, m.

_{a}− 1), so Equation (8) can be written as:

_{a}= 0, ε

_{a}= 0; at the same time, when ε

_{a}= 0, the derivation of 1/f(η

_{a}) is 1. For Equation (20), when the left endpoint is η

_{a}= 0, we get ε

_{a}= 0; when the right endpoint is η

_{a}= 2, the value of η

_{a}is greater than the critical point of the transient mixed free-surface-pressure flow in the process of water diversion, so neglect η

_{a}= 2. Meanwhile, when η

_{a}= 0, the derivation of ε

_{a}= f(η

_{a}) is 1. According to the properties of the inverse function, the following is obtained:

#### 2.2.3. Egg-Shaped Sections

_{e}is the significant depth in the egg-shaped section, m.

_{i}(i = 1, 2, 3) and y, Equation (8) can be rewritten as follows:

_{e}and η

_{e}functions of an egg-shaped section is shown in Figure 6.

#### 2.3. The Fitting Parameter to Establish the Formula

_{1}and c

_{2}are learning factors; rand

_{1}and rand

_{2}are random numbers between 0 and 1; $Pbes{t}_{id}^{k}$ is a position at which a particle I is located at the individual extremum point of the d-th dimension; and $Gbes{t}_{d}^{k}$ is the position of the global extremum of the entire population in the d-th dimension.

#### 2.3.1. Circular Sections

_{c}and characteristic parameter ε

_{c}of the overwater section of the circular section. The minimum value of η

_{c}is 0 in theory, but when the water depth is 0, the calculation of the critical depth is meaningless. Thus, the dimensionless relative critical depth η

_{c}is limited to 0.005 to facilitate the application of the formula to small flow calculations. Based on the research of Swamee [3], a maximum value of η

_{c}of 1.0 is adopted. On substituting η

_{c}into Equation (16), obtain the range of ε

_{c}as 0–+∞.

_{c}and ε

_{c}in the set range. Repeatedly implement Equations (28) and (29), and when the algorithm stagnates to the optimization range, the particles are released using Equation (30). Test and compare the three constructed functional models of circular section in Equation (18). The absolute value of the maximum relative error obtained using models I and II is more than 1%, which does not meet the functional requirements of the project. The maximum error of the optimal solution obtained using model III is only 0.182%, and the parameters are a = 1.6692, b = 0.7690, c = 0.6602, d = 0.8990, and e = 1.0000. So we select model III as the final formula form of circular section.

#### 2.3.2. Arched Sections

_{a}and the characteristic parameters of the cross-section of conduit ε

_{a}are scoped. The minimum value of η

_{a}is 0 in theory, but the calculation of the critical depth is meaningless when the water depth is 0. In this article, let the lowest limit of η

_{a}be 0.01. In theory, the maximum value of η

_{a}is 2. However, to prevent the occurrence of transient mixed free-surface-pressure flow in the process of water diversion, it is necessary that the clearance area above the free surface of the conduit be greater than 15% of the total cross-section area. Take the starting point η

_{a}of the transient mixed free-surface-pressure flow as 1.64 in the arched section, and thus set the fitting upper limit of η

_{a}as 1.64. Therefore, the range of the dimensionless water depth η

_{a}is 0.01–1.64. On substituting η

_{a}into Equation (21), obtain the range of ε

_{a}as 0.01–1.64.

#### 2.3.3. Egg-Shaped Sections

_{e}and characteristic parameter ε

_{e}of the overwater section. In theory, the maximum value of η

_{e}is 1. However, to avoid a transient mixed free-surface-pressure flow in the process of the water diversion, it is necessary that the clearance area above the free surface of the conduit be greater than 15% of the total cross-section area. Therefore, set the fitting upper limit of η

_{e}as 0.82. The minimum η

_{e}is 0 in theory, but the calculation of a critical depth is meaningless when the water depth is 0. In this article, the lowest limit of η

_{e}is determined to be 0.05. Therefore, the range of dimensionless water depth η

_{e}is 0.05–0.82. On substituting η

_{e}into Equation (26), we obtain the range of ε

_{e}as 0.02–3.03.

_{e}and ε

_{e}. Repeatedly implement Equations (28) and (29), and when the algorithm stagnates to the optimization range, the particles are released using Equation (30). Finally, the optimal solution combination of the model parameters is as follows: a = 0.324, b = 0.755, c = 0.22, d = 3.241, e = −0.02, f = 1.162, g = −0.262, h = 1.152, I = 0.788, and j = 2.144. The formula can be expressed as:

## 3. Results

## 4. Discussion

#### 4.1. Formula Evaluation

#### 4.1.1. Circular Sections

_{c}and then substitute these values into Equation (15). To calculate the value of ε

_{c}, substitute ε

_{c}into Equation (18). To obtain the approximate calculation value of η

_{c}, set the exact value of the relative water depth η

_{c}as ${\eta}_{c}^{\ast}$ and the relative error $\Delta =\left({\eta}_{c}-{\eta}_{c}^{\ast}\right)/{\eta}_{c}^{\ast}\times 100\%$. Figure 3 shows the entire distribution of the errors of Equation (31) in the range of application. According to the error analysis, the relative error absolute value of Equation (31) in the commonly used engineering range is less than 0.182%, and the accuracy of the formula fully meets the actual engineering requirements.

#### 4.1.2. Arched Sections

_{a}, and then substitute these numerical values into Equation (21) to calculate the value of ε

_{a}. Then, substitute ε

_{a}into Equation (32) to obtain the approximate value of η

_{a}. Set the exact value of the relative water depth η

_{c}as ${\eta}_{a}^{\ast}$ and the relative error $\Delta =\left({\eta}_{a}-{\eta}_{a}^{\ast}\right)/{\eta}_{a}^{\ast}\times 100\%$. Figure 8 shows the entire distribution of the errors of Equation (32) in the range of application. According to the error analysis, the relative error absolute value of Equation (32) in the commonly used engineering range (the ratio of the critical depth to the radius of the crown is between 0.01 and 1.64) is less than 0.06%, the accuracy of the formula fully meets the actual engineering requirements, and the upper limit of the application of the formula can be extended to 0.85 when the maximum error remains unchanged.

#### 4.1.3. Egg-Shaped Sections

_{e}and then take substitute these values into Equation (33). To obtain the approximately calculated value of η

_{e}, set the exact value of the relative water depth η

_{e}as ${\eta}_{e}^{\ast}$ and the relative error $\Delta =\left({\eta}_{e}-{\eta}_{e}^{\ast}\right)/{\eta}_{e}^{\ast}\times 100\%$. Figure 9 shows the entire distribution of the errors of Equation (33) in the application range.

#### 4.2. Engineering Verification

#### 4.2.1. Circular Sections

_{1}= 0.08 m

^{3}/s and Q

_{2}= 200 m

^{3}/s, the critical depth can be calculated using the following procedures.

_{1}= 0.08 m

^{3}/s, substitute the factors into Equation (13), which yields:

_{c}= η

_{c}× D = 0.0862 m.

_{c}= 0.0862 m, and the relative error Δ = −0.0573%.

_{2}= 200 m

^{3}/s, similarly,

_{c}= 0.4510. Therefore, through multiple iterations, we obtain the critical depth as y

_{c}= η

_{c}× D = 4.5101 m.

#### 4.2.2. Arched Sections

_{1}= 8 m

^{3}/s and Q

_{2}= 50 m

^{3}/s; the critical depth can be calculated using the following procedures.

_{1}= 8 m

^{3}/s, substituting the factors into Equation (19) yields:

_{a}= η

_{a}× r = 0.7418 m, the exact solution y

_{c}is 0.7418 m, and the relative error is 0%.

_{2}= 50 m

^{3}/s, one can obtain that:

_{a}= 1.2483 is obtained. Thus, through multiple iterations, we obtain the critical depth as y

_{a}= η

_{a}× r = 2.4965 m, the exact solution y

_{a}as 2.4956 m, and the relative error as 0.0382%.

^{3}/s and 50 m

^{3}/s, the critical depth is, respectively, located in the wall and crown of the conduit; this demonstrates that the critical depth in Equation (32) is common to the wall and crown of the conduit in arched sections and has high accuracy.

#### 4.2.3. Egg-Shaped Sections

_{1}= 0.5 m

^{3}/s, Q

_{2}= 10 m

^{3}/s, and Q

_{3}= 100 m

^{3}/s, the critical depth can be calculated using the following procedures.

_{1}= 0.5 m

^{3}/s, substitute the factors into Equation (24), which yields:

_{e}into Equation (33):

_{e}= η

_{e}× 3r = 0.3273 m, and through multiple iterations, obtain that y

_{e}= 0.3275 m and that the relative error Δ = −0.0449%.

_{2}= 10 m

^{3}/s, substitute the factors into Equation (24), which yields:

_{e}into Equation (33); similarly, obtain that η

_{e}= 0.2569 and the critical depth y

_{e}= η

_{e}× 3r = 1.5412 m, and through multiple iterations, obtain that y

_{n}= 1.5424 m and the relative error Δ = −0.0783%.

_{3}= 100 m

^{3}/s, substitute the factors into Equation (16), which yields:

_{e}into Equation (33); similarly, obtain that the value of η

_{e}is 0.8115, so the critical depth y

_{e}= η

_{e}× 3r = 4.8691 m, and through multiple iterations, obtain that y

_{e}= 4.8644 m and the relative error Δ = 0.8107%.

^{3}/s, 10 m

^{3}/s, and 100 m

^{3}/s, the critical depth is, respectively, located in the three partitions of the conduit; this demonstrates that the critical depth in Equation (33) is familiar to all partitions of the conduit in egg-shaped sections and has high accuracy.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | discharge section area; |

a | model parameter; |

b | model parameter; |

c | model parameter; |

D | channel diameter; |

d | model parameter; |

e | model parameter; |

f | model parameter; |

g | acceleration due to gravity (9.8 m/s^{2}); |

j | model parameter; |

Q | flow discharge; |

r | radius of crown; |

T | water width (m); |

y | water depth in conduit; |

y_{c} | critical depth in circular sections; |

y_{a} | critical depth in arched crossing-sections; |

y_{e} | critical depth in egg-shaped sections; |

α | velocity distribution coefficient |

ε_{c} | characteristic parameter of overwater section in circular sections; |

ε_{a} | characteristic parameter of overwater section in arched crossing sections; |

ε_{e} | characteristic parameter of overwater section in egg-shaped sections; |

γ | central angle of the wet perimeter; |

θ | half central angle of the wet perimeter in circular sections; |

θ_{i} | central angle of different water depths corresponding to egg-shaped sections; |

${\eta}_{c}$ | dimensionless relative water depth in circular sections; |

${\eta}_{a}$ | dimensionless relative water depth in arched crossing sections; |

${\eta}_{e}$ | dimensionless relative water depth in egg-shaped sections. |

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**Figure 3.**Cross-section of an egg-shaped conduit ((

**a**) 0 ≤ y ≤ r/5, (

**b**) r/5 ≤ y ≤ 2r, (

**c**) 2r ≤ y ≤ 3r).

Type of Sections of Conduits | Formula Form | Range (η_{n}) | Maximum Error/% |
---|---|---|---|

Circular section | ${\eta}_{c}={(1+3.83{{\epsilon}_{c}}^{-2.1454}-3.2{{\epsilon}_{c}}^{-2.1})}^{-0.115}$ | 0.005–1 | 0.182 |

Arch section | ${\eta}_{a}=\frac{{\epsilon}_{a}+0.0602{{\epsilon}_{a}}^{6.5426}}{1+0.0605{{\epsilon}_{a}}^{5.7161}}$ | 0.01–1.64 | 0.06 |

Egg-shaped section | ${\eta}_{e}=\frac{0.324{{\epsilon}_{e}}^{0.755}+0.22{{\epsilon}_{e}}^{3.241}-0.02{{\epsilon}_{e}}^{4.162}}{1-0.262{{\epsilon}_{e}}^{1.152}+0.788{{\epsilon}_{e}}^{2.144}}$ | 0.05–0.82 | 0.17 |

Formula | Formula Form | Range (η_{n}) | Maximum Relative Error/% |
---|---|---|---|

Vatankhah and Easa (2011) | ${\eta}_{c}={(1+0.77{\epsilon}^{-3})}^{-0.085}$ | [0.02,1.0] | 1.464 |

Vatankhah (2012) | ${\eta}_{c}=\frac{0.9584{\epsilon}^{0.25}}{{(1+0.0106{\epsilon}^{0.26}-0.0132{\epsilon}^{1.863})}^{-10.022}}$ | [0,0.92] | 0.249 |

Proposed | ${\eta}_{c}={(1+3.83{\epsilon}^{-2.1454}-3.2{\epsilon}^{-2.1})}^{-0.115}$ | [0.005,1] | 0.182 |

Formula | Formula Form | Range (η_{c}) | Maximum Relative Error/% |
---|---|---|---|

Wang (1998) | ${\eta}_{c1}=0.580{{\epsilon}_{1}}^{0.4}-{\epsilon}_{1}/54-0.935$ | 1.10–1.80 | 0.50 |

Vatankhah (2012) | ${\eta}_{c}=0.63{{\epsilon}_{1}}^{((0.3333+0.0046{{\epsilon}_{1}}^{1.607})/(1+0.013{{\epsilon}_{1}}^{1.656}))}$ | 0.01–1.64 | 0.07 |

Proposed | ${\eta}_{a}=\frac{{\epsilon}_{a}+0.0602{{\epsilon}_{a}}^{6.5426}}{1+0.0605{{\epsilon}_{a}}^{5.7161}}$ | 0.01–1.64 | 0.06 |

Formula | Formula Form | Range (η_{n}) | Maximum Error/% |
---|---|---|---|

Wu | $\{\begin{array}{ll}{\eta}_{c}=0.1682{\epsilon}_{1}^{0.2562}& (0.0667\le {\eta}_{c}\le 0.6667)\\ {\eta}_{c}=0.1944{\epsilon}_{1}^{0.2289}& (0.6667\le {\eta}_{c}\le 0.8500)\end{array}$ | 0.0667–0.85 | 0.91 |

Teng | ${\eta}_{c}=1.497{\mathrm{sin}}^{0.775}(0.15\epsilon )$ | 0.05–0.80 | 0.649 |

Proposed | ${\eta}_{e}=\frac{0.324{{\epsilon}_{e}}^{0.755}+0.22{{\epsilon}_{e}}^{3.241}-0.02{{\epsilon}_{e}}^{4.162}}{1-0.262{{\epsilon}_{e}}^{1.152}+0.788{{\epsilon}_{e}}^{2.144}}$ | 0.05–0.82 | 0.17 |

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## Share and Cite

**MDPI and ACS Style**

Shang, H.; Xu, S.; Zhang, K.; Zhao, L. Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full. *Water* **2019**, *11*, 2124.
https://doi.org/10.3390/w11102124

**AMA Style**

Shang H, Xu S, Zhang K, Zhao L. Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full. *Water*. 2019; 11(10):2124.
https://doi.org/10.3390/w11102124

**Chicago/Turabian Style**

Shang, Haixin, Song Xu, Kuandi Zhang, and Luyou Zhao. 2019. "Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full" *Water* 11, no. 10: 2124.
https://doi.org/10.3390/w11102124