# Determining the Flow Resistance of Racks and the Resulting Flow Dynamics in the Channel by Using the Saint-Venant Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods, Experimental Investigation

#### 2.1. Experimental Setup

**free outflow (unsubmerged flow)**: the weir provides a maximum cross-section with a net width of nearly the channel width. This condition occurs in real applications where a specific downstream flow depth is undesired or irrelevant, particularly in irrigation systems or culverts.**medium submerged flow**: the weir was closed stepwise (resulting in increasing downstream flow depth) until a downstream flow velocity between $0.6\phantom{\rule{3.33333pt}{0ex}}{\mathrm{ms}}^{-1}\le {v}_{2}\le 0.8\phantom{\rule{3.33333pt}{0ex}}{\mathrm{ms}}^{-1}$ was achieved. Such velocities are usually desired in channels of water treatment plants behind the rack and before the grit chamber.**full submerged flow**: the downstream flow depth was marginally lower than the upstream one, and the difference between approach velocity and the downstream velocity was nearly negligible. This flow condition corresponds to most of the experiments described in literature.

#### 2.2. Measurement Results

## 3. Bernoulli’s Energy Principle with Sudden Losses

#### 3.1. Determining the Flow Resistance of a Rack

#### 3.2. Empirical Approaches Based on Bernoulli’s Energy Principle

**Table 1.**Selection of empirical formulas which are most relevant in this study to determine the flow depth difference $\Delta h$ or the hydraulic head loss ${h}_{V}$ along racks.

Reference | Formula | Remarks and Particularites |
---|---|---|

Kirschmer [23] | $\Delta h=\beta {\left(\right)}^{\frac{s}{e}}\frac{4}{3}$ | flow depth difference related to approach velocity in upstream channel ${v}_{1}$ |

Wahl [29] | $\Delta h=\left(\right)open="("\; close=")">1.45-0.45(1-p)-{(1-p)}^{2}$ | flow depth difference related to velocity through net flow area ${v}_{s}$, regardless of bar shape and inclination angle, considers blockage ratio |

Meusburger [5] | ${h}_{V}=\beta {\left(\right)}^{\frac{p}{1-p}}\frac{3}{2}{p}^{-1.4tan\delta}{k}_{V}sin\alpha \xb7\frac{{v}_{1}^{2}}{2g}$ | ${k}_{V}\left(p\right)$ considers sectional blockage of the rack, ${k}_{V}=1$ represents no sectional blockage |

Clark et al. [50] | $\Delta h=7.43\beta (1+2.44{tan}^{2}\delta ){p}^{2}\frac{{v}_{1}^{2}}{2g}$ | investigated submerged trash racks in pressurised conduits |

Raynal et al. [6] | ${h}_{V}=\beta {\left(\right)}^{\frac{p}{1-p}}1.6{\left(\right)}^{\frac{1-p}{p}}3$ | $\beta =2.89,{\beta}_{i}=1.69$ for rectangular and $\beta =1.70,{\beta}_{i}=2.78$ for hydrodynamic bars |

Metcalf Eddy Inc. [1] | coarse screens: $\Delta h=\frac{1}{{C}_{c}}\frac{{v}_{s}^{2}-{v}_{1}^{2}}{2g}$ fine screens: $\Delta h=\frac{{v}_{s}^{2}}{2g{C}_{f}^{2}}$ | consider an empirical discharge coefficient to account for turbulence and eddy losses: ${C}_{c}\approx 0.7$, ${C}_{f}\approx 0.6$ for clean screens |

Tsikata et al. [7] | ${h}_{V}={\beta}_{T}\left(\right)open="("\; close=")">1+2.5{\left(\right)}^{\frac{s}{d}}0.1\left(\right)open="("\; close=")">1+(1000e-4.6){\left(\right)}^{tan}2$ | bar shape coefficient ${\beta}_{T}$: 3.4 for square nose and tail, 2.23 for round nose, 1.93 for streamlined bar shape |

Zayed et al. [9] | $\Delta h=7.88{(sin\alpha )}^{3.5}{p}^{1.68}{\left(\right)}^{\frac{qg}{{v}_{1}^{3}}}0.37$ | experiments on circular bars in subcritical flow conditions, combination of experiments and Π-theorem |

#### 3.3. Evaluating the Channel’s Hydraulic Capacity by Predicting the Upstream Flow Depth

- For lower Froude numbers, the two positive roots range between $0\le {h}_{2}/{h}_{1}\le 1$. Even though both solutions are plausible (${h}_{2}\le {h}_{1}$), determining one flow depth from the given other flow depth remains ambiguous.
- For a certain Froude number (depending on the value of $\zeta $), these two roots unify and increase with increasing ${\mathsf{Fr}}_{\mathsf{1}}$. Despite sharing the same real part, both roots have imaginary parts.
- After a particular threshold for ${\mathsf{Fr}}_{\mathsf{1}}$, the roots become ${h}_{2}/{h}_{1}>1$. That would imply that the flow depth increases after passing the rack. This solution contradicts the flow’s physics.

- The first root is negative ${h}_{2}/{h}_{1}<0$, revealing a non-physical process. The solution is valueless.
- The second root is greater than one: ${h}_{2}/{h}_{1}>1$, which would imply that the flow depth increases after passing the rack. This solution also contradicts the flow’s physics.
- The third root is a purely real positive number. It is smaller or equal than one: $0<{h}_{2}/{h}_{1}\le 1$ and hence the only reasonable solution. For very small downstream velocities, such as in fully submerged conditions, the flow depth before the obstacle becomes similar to the flow depth behind it. On the contrary, the upstream flow depth will take very high values for rapid downstream flows, as in free outflow conditions.

#### 3.4. Evaluating the Application of Bernoulli’s Energy Principle and the Empirical Formulas

**Validity:**

**The Undisturbed Cross-Section:**

**Measurements in the Downstream Cross-Section:**

**Ambiguous Evaluation of the Experiments:**

## 4. Result of the Study: Applying the Saint-Venant Equations to Determine the Flow Resistance of Racks

#### 4.1. The Saint-Venant Equations including Local Resistances

#### 4.2. Solving the Saint-Venant Equations in Matlab^{®}

^{®}solves parabolic (or elliptical) partial differential equation systems numerically by using the pdepde function [54]. For this, the pdepde algorithm expects the partial differential equations presented in a particular standard form

**x**represents the independent spatial variable and

**t**is the independent time variable, respectively. The dependent variable

**u**is differentiated with respect to

**x**and

**t**, and $\partial \mathbf{u}/\partial \mathbf{x}$ is the partial spatial derivative. The coefficients

**c**,

**f**, and

**s**represent the coefficients in the partial differential equations and are coded in terms of the input variables

**x**,

**t**,

**u**, and $\partial \mathbf{u}/\partial \mathbf{x}$. The constant

**m**indicates whether the equations are written in Cartesian ($m=0$, as in this study), in cylindrical ($m=1$), or in spherical coordinates ($m=2$) [54].

**u**can consist of several entries. For Equations (15) and (16), the solution vector consists of the cross-section A (and consequently the flow depth h) and the flow velocity v. The coefficients

**c**,

**f**, and

**s**must be defined in such a way that Equations (15) and (16) are obtained. In Appendix B, we present and explain an exemplary source code to implement the 1D-Saint-Venant equations in the pdepe function for a rectangular cross-section, including the local resistance due to the rack and continuous flow resistances. Additionally, we provide a detailed executable example in the Supplementary Material.

#### 4.3. Model Calibration

## 5. Discussion

- Option 1: The experiments from the literature will be repeated, but this time by measuring the longitudinal flow depth profile in the channel instead of the flow depth in just two cross-sections. Then, following our workflow, the $\zeta $-values are determined by calibrating the Saint-Venant model with the measurement results. Although option 1 will give a reliable and comprehensive data set, realising this requires diligence due to numerous necessary experiments and possible combinations of rack parameters.
- Option 2: The experimental conditions (setup, flow properties, rack characteristics) are generally well-described in the corresponding literature. The measured flow depths are usually documented in (at least) two cross-sections (in front of and behind the rack). The Saint-Venant model can be applied to reproduce the flow dynamics in the channel by identifying the most appropriate $\zeta $-value such that the simulated flow depth profile superimposes best with the (at least) two available flow depth values. Finding the $\zeta $-values with this option is more convenient than option 1 but offers a higher uncertainty due to less empirical data. However, the effect of altering the clear distance between the bars or the bar thickness can be directly studied in the Saint-Venant model since e and s form the blockage ratio p, which determines the net width of the rack.

## 6. Conclusions

- As a first step, well-documented experiments under controlled conditions should provide detailed flow depth profiles under varying discharges and backwater situations. For similar flow conditions, rack and channel properties, literature data from previous experiments can also be used as a substitute (above-mentioned option 2).
- Afterwards, the Saint-Venant model is calibrated to the measured data by determining the rack’s loss coefficient $\zeta $. The provided executable example in the Supplementary Materials can be adapted for that purpose.
- Then, the Saint-Venant equations can simulate the flow depth profile (and the velocity) for an arbitrary application considering an installed rack. As a result, the respective flow velocities and the freeboard are available at any point in the channel.

^{®}environment for conducting this study. Hence, the executable example provided in the Supplementary Materials performs best for Matlab

^{®}applications. Nevertheless, implementing the Saint-Venant model into other computing platforms is also feasible. Since, besides Matlab

^{®}, Octave, Python

^{TM}, Maple, and Mathematica are widespread in engineering practice, the proposed approach from this study might contribute to determining the flow resistance of racks and the resulting flow dynamics more reliable and in a more contemporary way without adding complexity to the engineering application.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

^{®}source code of the Saint-Venant model is available in the Supplementary Material.

## Acknowledgments

## Conflicts of Interest

## Notation

d | bar depth [m] |

${d}_{Hyd}$ | hydraulic diameter [m] |

e | clear distance between the bars [m] |

$g=9.81$ | gravitational acceleration [${\mathrm{ms}}^{-2}$] |

h | flow depth [m] |

${h}_{1}$ | upstream flow depth (in front of the rack) [m] |

${h}_{2}$ | downstream flow depth (behind the rack) [m] |

${h}_{crit}$ | critical flow depth [m] |

${h}_{N}$ | normal water depth [m] |

${h}_{V}$ | hydraulic head loss [m] |

$\Delta h$ | flow depth difference [m] |

${k}_{s}$ | equivalent sand roughness [m] |

$p=s/(s+e)$ | blockage ratio [1] |

$q=Q/B$ | specific discharge [${\mathrm{m}}^{3}{\mathrm{s}}^{-1}{\mathrm{m}}^{-1}$] |

s | bar thickness [m] |

t | time [s] |

${v}_{1}$ | approach (upstream) velocity [${\mathrm{ms}}^{-1}$] |

${v}_{2}$ | downstream velocity [${\mathrm{ms}}^{-1}$] |

${v}_{s}$ | flow velocity through the openings of the rack [${\mathrm{ms}}^{-1}$] |

x | longitudinal coordinate [m] |

y | lateral coordinate [m] |

z | vertical coordinate [m] |

${z}_{Bi}$ | bottom coordinates [m] |

A | area of cross-section [${\mathrm{m}}^{2}$] |

B | channel width [m] |

$\mathsf{Fr}$ | Froude number [1] |

J | channel slope [1] |

L | length [m] |

${L}_{S}$ | length of sudden loss [m] |

Q | discharge [${\mathrm{m}}^{3}{\mathrm{s}}^{-1}$] |

$\mathsf{Re}$ | Reynolds number [1] |

$\alpha $ | inclination of the bar from the horizontal [${}^{\circ}$] |

$\beta $ | bar shape coefficient [1] |

$\delta $ | angle of the approach flow [${}^{\circ}$] |

$\zeta $ | sudden loss coefficient [1] |

$\lambda $ | Weisbach friction coefficient [1] |

## Appendix A. Experimental Data

**Table A1.**Data for flow depth and critical flow depth for the three results from Figure 2.

Long. Direction | Free Outflow | Medium Submerged | Full Submerged | |||
---|---|---|---|---|---|---|

x[m] | h[m] | ${\mathit{h}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}$ [m] | h[m] | ${\mathit{h}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}$ [m] | $\mathit{h}$ [m] | ${\mathit{h}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}$ [m] |

2.47 | 0.793 | 0.231 | 0.502 | 0.133 | 0.772 | 0.189 |

2.97 | 0.796 | 0.231 | 0.503 | 0.133 | 0.771 | 0.189 |

3.47 | 0.796 | 0.231 | 0.504 | 0.133 | 0.772 | 0.189 |

3.97 | 0.795 | 0.231 | 0.505 | 0.133 | 0.773 | 0.189 |

4.47 | 0.798 | 0.231 | 0.507 | 0.133 | 0.773 | 0.189 |

4.97 | 0.798 | 0.231 | 0.508 | 0.133 | 0.775 | 0.189 |

5.47 | 0.799 | 0.231 | 0.508 | 0.133 | 0.774 | 0.189 |

5.97 | 0.802 | 0.231 | 0.510 | 0.133 | 0.777 | 0.189 |

6.47 | 0.806 | 0.231 | 0.511 | 0.133 | 0.778 | 0.189 |

6.97 | 0.805 | 0.231 | 0.512 | 0.133 | 0.780 | 0.189 |

7.47 | 0.804 | 0.231 | 0.514 | 0.133 | 0.781 | 0.189 |

7.97 | 0.806 | 0.231 | 0.515 | 0.133 | 0.783 | 0.189 |

8.47 | 0.807 | 0.231 | 0.516 | 0.133 | 0.783 | 0.189 |

8.97 | 0.808 | 0.231 | 0.517 | 0.133 | 0.786 | 0.189 |

9.47 | 0.811 | 0.231 | 0.518 | 0.133 | 0.787 | 0.189 |

9.97 | 0.813 | 0.231 | 0.521 | 0.133 | 0.791 | 0.189 |

10.47 | 0.814 | 0.231 | 0.523 | 0.133 | 0.787 | 0.189 |

10.97 | 0.815 | 0.231 | 0.523 | 0.133 | 0.792 | 0.189 |

11.47 | 0.809 | 0.231 | 0.525 | 0.133 | 0.791 | 0.189 |

11.97 | 0.821 | 0.231 | 0.526 | 0.133 | 0.792 | 0.189 |

12.47 | 0.821 | 0.231 | 0.528 | 0.133 | 0.796 | 0.189 |

12.97 | 0.822 | 0.231 | 0.529 | 0.133 | 0.797 | 0.189 |

13.47 | 0.820 | 0.231 | 0.531 | 0.133 | 0.798 | 0.189 |

13.97 | 0.824 | 0.231 | 0.531 | 0.133 | 0.798 | 0.189 |

14.47 | 0.824 | 0.231 | 0.533 | 0.133 | 0.800 | 0.189 |

14.97 | 0.827 | 0.231 | 0.534 | 0.133 | 0.800 | 0.189 |

15.47 | 0.828 | 0.231 | 0.535 | 0.133 | 0.801 | 0.189 |

15.47 | 0.826 | 0.231 | 0.536 | 0.133 | 0.801 | 0.189 |

16.47 | 0.830 | 0.231 | 0.538 | 0.133 | 0.804 | 0.189 |

16.97 | 0.830 | 0.231 | 0.539 | 0.133 | 0.806 | 0.189 |

17.47 | 0.831 | 0.231 | 0.540 | 0.133 | 0.807 | 0.189 |

17.97 | 0.833 | 0.231 | 0.541 | 0.133 | 0.808 | 0.189 |

18.47 | 0.833 | 0.231 | 0.542 | 0.133 | 0.809 | 0.189 |

18.97 | 0.834 | 0.231 | 0.543 | 0.133 | 0.811 | 0.189 |

19.47 | 0.835 | 0.231 | 0.544 | 0.133 | 0.811 | 0.189 |

19.97 | 0.836 | 0.231 | 0.545 | 0.133 | 0.814 | 0.189 |

20.47 | 0.840 | 0.231 | 0.547 | 0.133 | 0.813 | 0.189 |

20.97 | 0.839 | 0.231 | 0.547 | 0.133 | 0.814 | 0.189 |

21.07 | 0.839 | 0.231 | 0.547 | 0.133 | 0.812 | 0.189 |

21.17 | 0.835 | 0.231 | 0.546 | 0.133 | 0.812 | 0.189 |

21.27 | 0.830 | 0.236 | 0.544 | 0.136 | 0.809 | 0.193 |

21.28 | 0.832 | 0.239 | 0.543 | 0.138 | 0.810 | 0.195 |

21.29 | 0.829 | 0.242 | 0.543 | 0.139 | 0.809 | 0.197 |

21.30 | 0.835 | 0.245 | 0.543 | 0.141 | 0.809 | 0.200 |

21.31 | 0.829 | 0.248 | 0.543 | 0.143 | 0.809 | 0.202 |

21.32 | 0.830 | 0.251 | 0.542 | 0.144 | 0.809 | 0.205 |

21.33 | 0.828 | 0.254 | 0.542 | 0.146 | 0.807 | 0.207 |

21.34 | 0.827 | 0.257 | 0.542 | 0.148 | 0.807 | 0.210 |

21.35 | 0.827 | 0.261 | 0.541 | 0.150 | 0.808 | 0.213 |

26.00 | 0.175 | 0.231 | 0.301 | 0.133 | 0.635 | 0.189 |

25.30 | 0.156 | 0.231 | 0.300 | 0.133 | 0.636 | 0.189 |

26.70 | 0.118 | 0.231 | 0.301 | 0.133 | 0.636 | 0.189 |

## Appendix B. Matlab^{®} Source Code

^{®}version R2020b Update 1 (9.9.0.1495850).

**u**can consist of several entries. For Equations (15) and (16), the solution vector consists of the cross-section A (and consequently the flow depth h) (source code line 2) and the flow velocity v (source code line 3). The coefficients

**c**,

**f**, and

**s**must be defined in such a way that Equations (15) and (16) are obtained (source code line 8–16). Line 4 calculates the flow depth from the determined rectangular cross-section, while lines 5, 6, and 7 calculate the turbulent viscosity, hydraulic diameter, and the Weisbach friction factor.

`function [c,f,s] = stvenant(x,t,u,DuDx) % function signature`

`A = u(1); % cross-section A as first solution of the partial diff. equ.`

`v = u(2); % velocity v as second solution of the partial diff. equ.`

`h = A/B; % calculating flow depth from cross-section and width`

`nut = abs(6*0.41/log(12*h/ks)*v*h); % approach for turbulent viscosity`

`dhyd=4*A/(B + 2*h); % hydraulic diameter`

`lambda=colebrook_white(v*dhyd/1.e-6,ks/dhyd); % Weisbach friction`

`% c,f,s are elements in Matlab standard form for pdepe`

`c = [1;1];`

`f = [-v*A;nut*DuDx(2)-v^2/2-g*h-g*zbpp(x)];`

`s = [0;-lambda/dhyd*v*abs(v)/2];`

`zeta = 5.2; % sudden loss coefficient, local resistance of the rack`

`x1 = 21.25; x2 = 21.25 + 1.28; % start and end of the rack on the grid`

`if x >= x1 && x <= x2 % consider rack only between x1 and x2`

`s = [0;-lambda/dhyd*v*abs(v)/2-zeta/(x2-x1)*v*abs(v)/2];`

`end`

`end`

## References

- Metcalf & Eddy Inc. Wastewater Engineering: Treatment and Resource Recovery, 5th ed.; McGraw-Hill Education: New York, NY, USA, 2014. [Google Scholar]
- Benn, J.; Kitchen, A.; Kirby, A.; Fosbeary, C.; Faulkner, D.; Latham, D.; Hemsworth, M. Culvert, Screen and Outfall Manual: CIRIA C786. 2019. Available online: https://www.ciria.org/CIRIA/News/CIRIA_news2/Culvert_screen_and_outfall_manual_C786_PR.aspx (accessed on 1 June 2022).
- James, C.S. Hydraulic Structures; Springer International Publishing: Cham, Switzerland, 2020. [Google Scholar] [CrossRef]
- Chow, V.T. Open-Channel Hydraulics; Blackburn Press: Caldwell, NJ, USA, 2009. [Google Scholar]
- Meusburger, H. Energieverluste an Einlaufrechen von Flusskraftwerken; Mtteilungen der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie Nr. 179; Eidgenössische Technische Hochschule: Zürich, Switzerland, 2002. [Google Scholar]
- Raynal, S.; Chatellier, L.; Courret, D.; Larinier, M.; David, L. An experimental study on fish-friendly trashracks—Part 2. Angled trashracks. J. Hydraul. Res.
**2013**, 51, 67–75. [Google Scholar] [CrossRef] - Tsikata, J.M.; Tachie, M.F.; Katopodis, C. Open-channel turbulent flow through bar racks. J. Hydraul. Res.
**2014**, 52, 630–643. [Google Scholar] [CrossRef] - Josiah, N.R.; Tissera, H.P.S.; Pathirana, K.P.P. An Experimental Investigation of Head loss through Trash Racks in Conveyance Systems. Eng. J. Inst. Eng.
**2016**, 49, 1. [Google Scholar] [CrossRef] - Zayed, M.; El Molla, A.; Sallah, M. An experimental study on angled trash screen in open channels. Alex. Eng. J.
**2018**, 57, 3067–3074. [Google Scholar] [CrossRef] - Salah Abd Elmoaty, M. An experimental investigation of the impact of aquatic weeds trash racks on water surface profile in open channels. Water Sci.
**2019**, 33, 84–92. [Google Scholar] [CrossRef] [Green Version] - Walczak, N.; Walczak, Z.; Nieć, J. Assessment of the Resistance Value of Trash Racks at a Small Hydropower Plant Operating at Low Temperature. Energies
**2020**, 13, 1775. [Google Scholar] [CrossRef] [Green Version] - Gisen, D. Modeling Upstream Fish Migration in Small-Scale Using the Eulerian-Lagrangian-Agent Method (ELAM): BAW Dissertationen Nr. 1. Ph.D. Thesis, Universität der Bundeswehr München, Neubiberg, Germany, 2018. [Google Scholar]
- Beck, C. Fish Protection and Fish Guidance at Water Intakes Using Innovative Curved-Bar Rack bypass Systems: VAW Mitteilungen Nr. 257. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 2020. [Google Scholar]
- Lemkecher, F.; Chatellier, L.; Courret, D.; David, L. Experimental study of fish-friendly angled bar racks with horizontal bars. J. Hydraul. Res.
**2022**, 60, 136–147. [Google Scholar] [CrossRef] - Zayed, M. Velocity Distributions around Blocked Trash Racks. J. Irrig. Drain. Eng.
**2021**, 147, 04021037. [Google Scholar] [CrossRef] - Zayed, M.; Farouk, E. Effect of blocked trash rack on open channel infrastructure. Water Pract. Technol.
**2021**, 16, 247–262. [Google Scholar] [CrossRef] - Waldy, M.; Gabl, R.; Seibl, J.; Aufleger, M. Alternative Methoden für die Implementierung von Rechenverlusten in die 3D-numerische Berechnung mit FLOW-3D. Österr. Wasser-Abfallwirtsch.
**2015**, 67, 64–69. [Google Scholar] [CrossRef] - Krzyzagorski, S.; Gabl, R.; Seibl, J.; Böttcher, H.; Aufleger, M. Implementierung eines schräg angeströmten Rechens in die 3D-numerische Berechnung mit FLOW-3D. Österr. Wasser-Abfallwirtsch.
**2016**, 68, 146–153. [Google Scholar] [CrossRef] [Green Version] - Lučin, I.; Čarija, Z.; Grbčić, L.; Kranjčević, L. Assessment of head loss coefficients for water turbine intake trash-racks by numerical modeling. J. Adv. Res.
**2020**, 21, 109–119. [Google Scholar] [CrossRef] [PubMed] - Droste, R.L.; Gehr, R. Theory and Practice of Water and Wastewater Treatment, 2nd ed.; John Wiley & Sons Incorporated: Newark, NJ, USA, 2018. [Google Scholar]
- Weisbach, J.L. Die Experimental-Hydraulik. Eine Anleitung zur Ausführung Hydraulischer Versuche im Kleinen, Nebst Beschreibung der Hierzu nöthigen Apparate…; J.G. Engelhardt: Freiberg, Germany, 1855. [Google Scholar]
- Colebrook, C.F. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng.
**1939**, 11, 133–156. [Google Scholar] [CrossRef] - Kirschmer, O. Untersuchungen über den Gefällsverlust an Rechen: Mitteilungen des Hydraulischen Instituts der Technischen Hochschule München, Heft 1. Ph.D. Thesis, Technische Hochschule, München, Germany, 1926. [Google Scholar]
- Spangler, J. Untersuchung über den Gefällsverlust an Rechen bei schräger Zuströmung: Mitteilungen des Hydraulischen Instituts der Technischen Hochschule München, Heft 2. Ph.D. Thesis, Technischen Hochschule München, München, Germany, 1928. [Google Scholar]
- Spangler, J. Investigation of the loss through trash racks inclined obliquely to the stream flow. In Hydraulic Laboratory Practice; Freeman, J.R., Ed.; American Society of Mechanical Engineers: New York, NY, USA, 1929; pp. 461–470. [Google Scholar]
- Mosonyi, E. Wasserkraftwerke: Band 1; Wasserkraftwerke, VDI-Verlag: Düsseldorf, Germany, 1966. [Google Scholar]
- Osborn, J.F. Rectangular-bar trashrack and baffle headlosses. J. Power Div.
**1968**, 94, 111–123. [Google Scholar] [CrossRef] - Zimmermann, J. Widerstand Schräg Angeströmter Rechengitter: Mitteilungen Heft 157. Ph.D. Thesis, Universität Fridericana Karlsruhe, Karlsruhe, Germany, 1969. [Google Scholar]
- Wahl, T.L. Trash Control Structures and Equipment: A Literature Review and Survey of Bureau of Reclamation Experience: R-92-05; US Bureau of Reclamation: Washington, DC, USA, 1992. [Google Scholar]
- Bozkus, Z.; Çakir, P.; Ger, A.M. Energy dissipation by vertically placed screens. Can. J. Civ. Eng.
**2007**, 34, 557–564. [Google Scholar] [CrossRef] - Tsikata, J.M.; Katopodis, C.; Tachie, M.F. Experimental study of turbulent flow near model trashracks. J. Hydraul. Res.
**2009**, 47, 275–280. [Google Scholar] [CrossRef] - Fellenius, W.; Lindquist, E. Loss of head in protecting racks at hydraulic power plants. In Hydraulic Laboratory Practice; Freeman, J.R., Ed.; American Society of Mechanical Engineers: New York, NY, USA, 1929; pp. 533–538. [Google Scholar]
- Scimemi, E. Rilievi sperimentali sul funzionamento idraulico dei grandi impianti industriali. L’Energ. Elettr.
**1933**, 10, 47. [Google Scholar] - Escande, L. Pertes de Charge a la Traversèe des Grilles: Toulouse: Complements d’Hydraulique 1, Publications de l’Institut Electronique et de l’Institut Mecanique des Fluides l; Université de Toulouse: Toulouse, France, 1947. [Google Scholar]
- Levin, L. Problèmes de perte de charge et de stabilité des grilles de prise d’eau. Houille Blanche
**1967**, 53, 271–278. [Google Scholar] [CrossRef] [Green Version] - United States Army Corps of Engineers. Open channel flow trash rack losses. In Hydraulic Design Criteria; United States Army Corps of Engineers, Ed.; United States Army Corps of Engineers: Vicksburg, MS, USA, 1987; p. MS. Sheet 010–7. [Google Scholar]
- Sinniger, R.O.; Hager, W.H. Constructions Hydrauliques: Ecoulements Stationnaires; Traite de Genie Civil; Presses Polytechniques Romandes: Lausanne, Switzerland, 1989. [Google Scholar]
- Hager, W.H. Abwasserhydraulik: Theorie und Praxis; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar] [CrossRef]
- Förster, G.; Hoff, S. Beitrag zur Bemessung von Feinrechen. Korresp. Abwasser Abfall
**1999**, 46, 1607–1610. [Google Scholar] - Giesecke, J.; Mosonyi, E. Wasserkraftanlagen: Planung, Bau und Betrieb, 5th ed.; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Uckschies, T. Feinrechen in der Abwasserreinigung: Planung und Störungsfreier Betrieb für Kommunale Kläranlagen; Wasser, Springer Vieweg: Wiesbaden, Germany, 2017. [Google Scholar] [CrossRef]
- Katopodis, C.; Ead, S.A.; Standen, G.; Rajaratnam, N. Structure of Flow Upstream of Vertical Angled Screens in Open Channels. J. Hydraul. Eng.
**2005**, 131, 294–304. [Google Scholar] [CrossRef] - Tsikata, J.M.; Tachie, M.F.; Katopodis, C. Particle Image Velocimetry Study of Flow near Trashrack Models. J. Hydraul. Eng.
**2009**, 135, 671–684. [Google Scholar] [CrossRef] - Chatellier, L.; Wang, R.W.; David, L.; Courret, D.; Larinier, M. Experimental characterization of the flow across fish-friendly angled trashrack models. In Proceedings of the 34th IAHR World Congress, Brisbane, Australia, 26 June–1 July 2011; pp. 1–8. [Google Scholar]
- Kriewitz-Byun, C.R. Leitrechen an Fischabstiegsanlagen: Hydraulik und Fischbiologische Effizienz: VAW Mitteilungen Nr. 230. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 2015. [Google Scholar] [CrossRef]
- Berger, C. Rechenverluste und Auslegung von (Elektrifizierten) Schrägrechen Anhand Ethohydraulischer Studien. Ph.D. Thesis, Technische Universität Darmstadt, Darmstadt, Germany, 2017. [Google Scholar]
- Khan, L.A.; Wicklein, E.A.; Rashid, M.; Ebner, L.L.; Richards, N.A. Computational fluid dynamics modeling of turbine intake hydraulics at a hydropower plant. J. Hydraul. Res.
**2004**, 42, 61–69. [Google Scholar] [CrossRef] - Naudascher, E. Hydrodynamic Forces: IAHR Hydraulic Structures Design Manuals 3, 1st ed.; IAHR Design Manual; CRC Press: Rotterdam, The Netherlands, 1991; Volume V. [Google Scholar]
- Idelčik, I.E.; Ginevskiĭ, A.S. (Eds.) Handbook of Hydraulic Resistance, 4th ed.; Begell House: Redding, CT, USA, 2007. [Google Scholar]
- Clark, S.P.; Tsikata, J.M.; Haresign, M. Experimental study of energy loss through submerged trashracks. J. Hydraul. Res.
**2010**, 48, 113–118. [Google Scholar] [CrossRef] - Malcherek, A. Fließgewässer: Hydraulik, Hydrologie, Morphologie und Wasserbau; Springer Vieweg in Springer Fachmedien Wiesbaden GmbH: Wiesbaden, Germany, 2019. [Google Scholar]
- Hager, W. Wastewater Hydraulics: Theory and Practice, 2nd ed.; Springer: Berlin, Germany; New York, NY, USA, 2010. [Google Scholar]
- Samuels, P.G. Backwater lengths in rivers. Proc. Inst. Civ. Eng.
**1989**, 87, 571–582. [Google Scholar] [CrossRef] - MathWorks. Pdepe: Solve 1-D Parabolic and Elliptic PDEs. 2022. Available online: https://de.mathworks.com/help/matlab/ref/pdepe.html (accessed on 1 June 2022).
- Raynal, S.; Courret, D.; Chatellier, L.; Larinier, M.; David, L. An experimental study on fish-friendly trashracks—Part 1. Inclined trashracks. J. Hydraul. Res.
**2013**, 51, 56–66. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The experimental setup consists of a rectangular channel with glass walls and an installed rack. The water is pumped in from a reservoir and passed an electromagnetic flowmeter before entering the channel. A lamellae weir controls the water level downstream of the rack.

**Figure 2.**Exemplary profiles for the measured flow depth in longitudinal direction (x-direction) for the three investigated backwaters: (

**a**) free outflow, (

**b**) medium submerged flow, and (

**c**) full submerged flow for an exemplary discharge each. The grey and white box marks the location of the rack and the weir, respectively. The flow direction is from left to right. The profile of the critical flow depth (dotted line) depends on the channel width of the actual cross-sections and reveals the predominant flow regime (supercritical flow: $h<{h}_{crit}$, subcritical flow: $h>{h}_{crit}$).

**Figure 3.**Schematic illustrations for the two theoretical approaches: (

**a**) The Bernoulli equation compares the energy budgets between two cross-sections (Equation (2)). The area in-between contains the rack but remains as a hydraulic black box. (

**b**) The Saint-Venant equations (Section 4) do not require fixed cross-sections and cover the longitudinal profile entirely. Besides the bottom slope, they consider the bed, wall, and viscous friction.

**Figure 4.**The loss coefficients were determined by evaluating the measurement data from this study with Bernoulli’s energy principle (Equation (6)). The loss coefficient $\zeta $ is highly affected by the backwater situation and depends on the flow velocity. Furthermore, Bernoulli’s energy principle overestimates the more reliable results from the Saint-Venant approach (Section 4).

**Figure 5.**The determined loss coefficients from this study as a function of the discharge compared to the empirical formulas from Table 1: The results are shown for the three investigated backwater conditions: (

**top left**) free outflow with flow depth difference formulas, (

**top right**) free outflow with hydraulic head loss formulas, (

**bottom left**) medium submerged, and (

**bottom right**) full submerged.

**Figure 6.**The solution of the flow depth ratio (here with $\zeta =1$) depends on the velocity for the sudden loss and the Froude number. (

**a**) The ratio of flow depths as a function of the upstream Froude number ${\mathsf{Fr}}_{\mathsf{1}}$ (Equation (10)) yields ambiguous and complex solutions, and is thus incorrect. (

**b**) The ratio of the flow depths as a function of the downstream Froude number ${\mathsf{Fr}}_{\mathsf{2}}$ (Equation (12)) gives ambiguous but real solutions.

**Figure 7.**Hybrid workflow in this study to determine the loss coefficient of the rack, the entire flow dynamics, and the channel freeboard. The flowchart starts in the top left corner by selecting the properties of the rack, the channel, and the flow.

**Figure 8.**Comparison of measured and simulated flow depth for (

**a**) the free outflow, (

**b**) the medium submerged flow, and (

**c**) the full submerged flow for the exemplary results from Figure 2. Using the Saint-Venant equations enables to simulate the flow depth profile from the experiments very accurately. The normal water depths listed in the subscriptions are calculated from Equation (13). The bed-parallel normal water depth profile intersects with the simulated water level several hundred meters ahead of the rack at the end of the backwater curve, the transition point between the backwater curve and the undisturbed cross-section.

**Figure 9.**The simulation area from Figure 2c was extended 300 m upstream to reveal the entire backwater curve and to pinpoint the upstream undisturbed cross-section. On the contrary, Bernoulli’s energy principle and the empirical approaches do not consider stretches of true undisturbed cross-sections upstream of the impoundment.

**Table 2.**Measurement results from this study for the three investigated backwater conditions and measured specific discharges q. The flow depth and velocity data belong to a cross-section directly in front of the rack (median values at $20\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}<x<21\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$) and to the median values from the three stationary measurement locations behind the rack. The third and fourth column express the flow depth differences and the velocity head differences, respectively. The fifth column presents the sudden loss coefficients $\zeta $ determined from the Saint-Venant model based on the experiments from this study.

Backwater | q$\left[{\mathbf{m}}^{3}{\mathbf{s}}^{-1}{\mathbf{m}}^{-1}\right]$ | ${\mathit{h}}_{1}-{\mathit{h}}_{2}\phantom{\rule{0.166667em}{0ex}}[\mathbf{m}]$ | $({\mathit{v}}_{1}^{2}-{\mathit{v}}_{2}^{2})/\left(2\mathit{g}\right)\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | $\mathit{\zeta}$ |
---|---|---|---|---|

free outflow | 0.102 | 0.362 | −0.106 | 7.4 |

0.152 | 0.451 | −0.177 | 6.0 | |

0.201 | 0.521 | −0.212 | 5.3 | |

0.256 | 0.587 | −0.234 | 4.7 | |

0.316 | 0.652 | −0.249 | 4.3 | |

0.347 | 0.687 | −0.260 | 4.1 | |

medium submerged flow | 0.104 | 0.191 | −0.005 | 8.0 |

0.152 | 0.245 | −0.009 | 6.6 | |

0.203 | 0.306 | −0.014 | 5.7 | |

0.255 | 0.360 | −0.020 | 4.9 | |

0.310 | 0.424 | −0.029 | 4.4 | |

0.359 | 0.486 | −0.040 | 4.1 | |

full submerged flow | 0.102 | 0.081 | −0.001 | 8.0 |

0.154 | 0.096 | −0.001 | 6.5 | |

0.204 | 0.159 | −0.002 | 5.9 | |

0.257 | 0.179 | −0.003 | 5.2 | |

0.311 | 0.266 | −0.008 | 4.6 | |

0.352 | 0.346 | −0.014 | 4.4 |

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**MDPI and ACS Style**

Baselt, I.; Malcherek, A.
Determining the Flow Resistance of Racks and the Resulting Flow Dynamics in the Channel by Using the Saint-Venant Equations. *Water* **2022**, *14*, 2469.
https://doi.org/10.3390/w14162469

**AMA Style**

Baselt I, Malcherek A.
Determining the Flow Resistance of Racks and the Resulting Flow Dynamics in the Channel by Using the Saint-Venant Equations. *Water*. 2022; 14(16):2469.
https://doi.org/10.3390/w14162469

**Chicago/Turabian Style**

Baselt, Ivo, and Andreas Malcherek.
2022. "Determining the Flow Resistance of Racks and the Resulting Flow Dynamics in the Channel by Using the Saint-Venant Equations" *Water* 14, no. 16: 2469.
https://doi.org/10.3390/w14162469