Simulation of Permanent Movement in Collectors Non-Standardized Sewerage

Abstract

Sewage collectors for domestic waste water and rainwater have flow sections that are standardized in shape and size. The collector with vertical dimension H > 1.50 m is admitted to be visitable and for H ≤ 0.80 m it is not visitable. Sewage collectors can be made in demanding situations with non standardized flow sections. A series of natural and anthropogenic action can change the geometric and hydraulic parameters of the flow section over time. Thus, the flow section no longer respects the initial geometric and hydraulic parameters and becomes a non-standardized section. Knowing the hydraulic parameters at such a flow section becomes difficult for monitoring the mining process. The circular, ovoid, circular bell, parabolic bell type flow section transforms into sections consisting of straight lines and circular arcs or only circular arcs. The factors that produce are erosion, siltation and cementation of transported material, repair and rehabilitation works and others. The erosion phenomenon also causes a change in the roughness on the watered perimeter of the section. In order to obtain the hydraulic operating parameters of the visitable sever collectors with non-standard flow section, several calculation programs have been elaborated on the forms of permanent water movement.

1. Introduction

The transport of waste water required the definition of flow cross-sections with simple geometric shapes, which would achieve high transport velocities towards the lower part of the section. The most used geometric shapes for sewer collectors are circle, ovoid and bell with various variants (circular, elliptical and parabolic). Hydraulic calculation methods were developed for these geometric shapes, which were standardized and included in design regulations [1].

The hydraulic calculation for dimensioning and verification of sewer collectors has evolved over time according to the contribution of scientific research. From the grapho-analytical calculation used about 50 years ago [2, 3, 4], we have reached the use of special calculation programs based on the classic geometric shapes of sewer collectors [5, 6].

Current hydraulic research on sewer manifolds is focused on several areas of interest. One area of research analyzes the velocity profile in the flow section and its optimization to increase the sediment transport velocity by using the maximum entropy principle, CFD or numerical methods for special situations and so on [7, 8]. Another field of research analyzes ways to increase the flow rate in the flow section taking into account the geometric and hydraulic parameters [9, 10]. Research is mainly carried out on closed and free-flowing channel sections (mainly circular and ovoid) [10, 11].

An important field of research lately is represented by the monitoring of the functional parameters of the sewerage networks and especially the hydraulic parameters (minimum allowed speeds, flow rates, water depths, degree of filling and others). Modern investigation methods such as CFD [11], fuzzy logic [12], with improved reliability of hydraulic data [13] and concrete applications on urban sewage networks are used in the research. The processes of washing the sediments deposited in the sewage collectors is a field of research targeted in the last period of time [5, 14].

The creation of new sewage systems and the rehabilitation and modernization of the existing ones led to the appearance of pipes with different sections than the classic ones. Also, the degradation over time of the geometry of the flow section, through the transformation from a shape made of arcs of a circle to a shape with arcs of a circle + straight line segments imposed a new field of research, namely that of collectors with sections of non-standard flow [15].

For now, this field is less addressed, but it has become necessary considering the monitoring of the operation of these collectors, which occupy the main positions in an urban sewage network. A series of researches have been carried out for galleries where the flow is free-level, and the circular section has been transformed into a section consisting of arcs of a circle and line segments [16]. Accurate knowledge of geometric and hydraulic operating parameters in various flow regimes requires a more detailed approach to hydraulic parameters.

2. Sewer collectors with non-standard sections

The sewage system consists of a sequence of closed channels (sewage collectors) in which the flow of water generally takes place with a free level. Sewer collectors can be classified according to their position and role in the urban sewer scheme as follows: service, secondary, main and discharge. The main and exhaust collectors have large dimensions and are executed on site or by using prefabricated sections. The geometric form of the main collectors is standardized for the typification of the hydraulic calculation and the technological process of execution. The most commonly used standardized geometric shapes are, ovoid, bell and circular (Figure 1). The construction material of the main collectors of the old type was stone and brick, while the modern ones use reinforced concrete and composite structures. On the route of the collectors there are a series of constructions (fireplaces, overflow chambers, basins, pumping stations, underpasses and others) that ensure a correct exploitation process. The correct operation of the collector is achieved by corroborating all its structural components to the temporal variation of the transported flow [1].

Figure 1.

The profile and the geometrical parameters of drawing the standardized sewerage collectors: (a) Ovoid type; (b) semi-eliptical bel type.

Depending on the height H of the collector section, the following are allowed in Romania: non-visitable collectors with H ≤ 0.80 m; semi-visitable collectors with 0.80 m ≤ H < 1.50 m; visitable collectors with H > 1.50 m [2, 15].

The natural and anthropogenic factors in the site intervene on the collectors with a complex of actions, which influence their behavior over time. Among the natural factors can be listed the geotechnical and hydrogeological characteristics of the site, seismic action, land subsidence and so on. Anthropogenic factors acting on the collector show an evolution over time as influence and impact. They influence the behavior of the sewer collector through the design concept, the execution technology, the degree of application of the maintenance works and annual repairs and so on. The most influential anthropogenic factors that degrade the flow section are wastewater concentration, chemical parameters, transport speed, protection material of the flow section and absence of maintenance works and so on. Some of the natural factors in conjunction with the anthropogenic ones cause the appearance of continuous degradation processes of the collector structure and change its functional parameters over time [16, 17].

Non-standard flow sections are determined by the following situations:

• the production by some production companies of pipes that do not respect the standardized dimensions;

• degradation of the flow section of the collector through phenomena of hydrodynamic erosion and clogging: in certain situations cementation of the clogging materials occurs; these phenomena substantially change the geometric shape of the flow section;

• the repair works, but especially those for the rehabilitation of the flow section, determine the modification of the geometric shape; the execution technologies and the machines used may require a certain shape of the flow section in the lower area;

• the transformation of the main collectors with unitary operation (domestic waste water + rainwater) into collectors with separate operation; this mode of operation, imposed by the EU regulations of recent years, determines the geometric modification of the flow section and implicitly the appearance of a non-standard shape.

Hydrodynamic erosion and clogging of the flow section change the initial circular, ovoid or bell-type geometric shape of the collector into a new shape with non-standard parameters. The new shape presents curves differentiated from the original ones, but also the presence of straight lines in the flow section. The research carried out in situ, in a gallery with a circular section, where the water flow was at a free level, highlighted the modification of the geometric shape by the flattening of the lower area and the curvature of the watered perimeter. At the same time, the phenomenon of hydrodynamic erosion and the execution works changed the roughness of the watered perimeter, causing the appearance of three to four areas of roughness variation starting from the radius and continuing on the wall (Figure 2a) [16].

Figure 2.

Standardized circular section for a free-level flow modified by hydrodynamic erosion and technological repair processes: (a) General view of the registered changes; (b) Modification scheme of the lower area (blue line – standard section, red line – modification areas of the section), BA – Reinforced concrete, BP prestressed concrete [16].

Thus, the flow section transforms geometrically over time, where the shape made of circular arcs changes into a mixed shape (arcs + straight line segments) (Figure 2b) [16].

The research carried out [17] in the sewage networks in Romania showed that the flow section of the visitable collectors was substantially modified compared to the designed one. Thus, an ovoid section can become a polycentric section through the erosion processes of the flood bed and the walls (Figure 3a). The same situation also occurs at the bell-type sections (Figure 3b), where the erosion phenomena and the consolidation of the deposited sediments have substantially transformed the flow section.

Figure 3.

Scheme of modification sewer of standardized section at the visitable sewage collectors: (a) ovoid section (blue line – standard section, red line – Modification areas of the section); (b) circular bell section [17].

The application of some rehabilitation works to the collectors can geometrically transform the flow section by creating a transport or support area for the execution machinery. Monitoring of flow parameters (flow rate, minimum speed and degree of filling) in non-standardized sections can no longer be performed correctly. In some cases, the geometric modification of the section determined the reduction of the speed and the transported flow by about 15–28% [17].

3. Hydraulic calculation of non-standard sewer collectors

3.1 Types of wastewater movement in the sewage network

The movement of waste water with a free level in a sewer network with a closed section can be of non-permanent and permanent type. The operating safety of a sewage system is most affected by the non-permanent (transient) flow regime. This, depending on the speed of variation of the hydraulic parameters - quantitatively, the flow Q, and qualitatively, the depth, h, or the free water level share, z - can be considered as being slowly variable, or rapidly variable.

The non-permanent regime can be simulated by using complex hydraulic-mathematical models formed, in general, from: 1 - motion equations; 2 - continuity equations; 3 - initial conditions; 4 - contour conditions. The initial conditions are represented by a permanent flow regime, which can be considered of varied gradual non-uniform type. To be able to approach any hydraulic simulation, the basic data of the sewer network must be obtained, namely geometric and hydraulic parameters of the uniform permanent movement for any form of collector (standardized and non-standard).

To generalize the analysis of the main sewer collectors in the flow regimes existing in practice and for all types of collector sections (standardized and non-standardized) used, their geometric and hydraulic characteristics

$$\mathit{A}={\mathit{f}}_{\mathrm{A}}\left(\mathit{h}\right),\mathit{P}={\mathit{f}}_{\mathrm{P}}\left(\mathit{h}\right),{\mathit{R}}_{\mathrm{h}}={\mathit{f}}_{\mathrm{Rh}}\left(\mathit{h}\right),\mathit{B}={\mathit{f}}_{\mathrm{B}}\left(\mathit{h}\right),{\mathit{z}}_{\mathrm{G}}={\mathrm{f}}_{\mathrm{zG}}\left(\mathit{h}\right),\mathit{W}={\mathit{f}}_{\mathrm{W}}\left(\mathit{h}\right),\mathit{K}={\mathit{f}}_{\mathrm{k}}\left(\mathit{h}\right)$$E1

were considered by the matrix M, of the form:

$$\mathrm{M}=\left[\mathrm{h}\mathrm{A}\mathrm{P}\mathrm{B}{\mathrm{R}}_{\mathrm{h}}{\mathrm{Z}}_{\mathrm{G}}\mathrm{W}\mathrm{K}\right].$$E2

The M matrix was determined based on a series of mathematical models and computer programs developed for the characteristics of sewer components in various flow regimes. The elements of the matrix M are NC-dimensional column vectors, where NC represents the number of distinct values assigned to the value pairs

$${\left\{h,B,A,R_h,\right.\left.z_G,W,K\right\}}_i,i=1,2,\dots ,N_C$$E3

In carrying out the simulations along the length of a collector, it is more convenient to perform the substitution

$$h=z-z_a$$E4

where z_a is the apron elevation of the channel and z - the elevation of the free water level. Thus, in the characteristics (Eq. (1)) similar characteristics will be generated, but which presents the z rate as an independent variable,

$$\mathit{A}={\mathit{f}}_{\mathrm{A}}\left(\mathit{z}\right),\mathit{P}={\mathit{f}}_{\mathrm{P}}\left(\mathit{z}\right),{\mathit{R}}_{\mathrm{h}}={\mathit{f}}_{\mathrm{Rh}}\left(\mathit{z}\right),\mathit{B}={\mathit{f}}_{\mathrm{B}}\left(\mathit{z}\right),{\mathit{z}}_{\mathrm{G}}={\mathit{f}}_{\mathrm{zG}}\left(\mathit{z}\right),\mathit{W}={\mathit{f}}_{\mathrm{W}}\left(\mathit{z}\right),\mathit{K}={\mathit{f}}_{\mathrm{k}}\left(\mathit{z}\right)$$E5

The main sewer collectors were considered on the analysis sections as prismatic/cylindrical beds, with a constant slope. The connection of collector sections with different shapes and/or sizes, as well as the confluence with lower-order collectors, is achieved by using connecting or intersection manholes. In this case, adequate contour conditions must be allowed in the inspection manhole section.

3.2 Hydraulic calculation elements for sewer collectors

Geometric parameters of the flow section at a sewer collector are as follows (Figures 1 and 4): section shape, constructive height (D, H), constructive width (B), radii of curvature (r), angles, geodesic slope of apron collector (I) section and others. Hydraulic parameters considered in the calculation are the following: total flow (Q) and specific (q), flow modulus (K), average speed (v), calculation degree of filling (a) and maximum imposed (a_max), minimum (v_min) and maximum imposed speed (v_max), hydraulic slope (J), the wetted area of the flow section (A), wetted perimeter (P), hydraulic radius (R or R_h), roughness (k) or roughness coefficient (n - Manning), depth of the centre of gravity of the watered area (z_G) and others.

Figure 4.

Parameters of the flow section at sewer collectors: (a) ovoid; (b) circular bell; (c) parabolic bell.

The calculation of dimensioning and verification of sewerage collectors goes through the two forms of movement of water specific to permanent movement: a – uniform movement; b – non-uniform movement with gradually varied non-uniform and rapidly varied non-uniform types [1, 3].

The calculation relations used for uniform permanent motion are as follows:

$$Q=\mathit{AC}\sqrt{\mathit{RI}}$$E6

$$v=C\sqrt{\mathit{RI}}$$E7

In the case of sewer pipes with partially filled sections, the analytical expressions for each of the functions presented in Eq. (1) are used in the calculation and are customized for each geometric shape of the collector section.

The solution methods are analytical, with the use of numerical calculation programs and graph analytic, with the use of graphs and tables specialized on section shapes in the case of calculations with relatively low precision.

The calculation relation of parameters for non-uniform gradually varied permanent motion is as follows [4]:

$$\mathit{dl}=\frac{1-\frac{\propto Q^{2}B\left(h\right)}{g{A}^3\left(h\right)}}{I-\frac{Q^2}{K^2\left(h\right)}}\mathit{dh}$$E8

$$l_2-l_1=\frac{1}{I}{\int }_{h_1}^{h_2}f\left(h\right)\mathit{dh}$$E9

In the case of closed channels and with special shapes of the section (circular, ovoid and bell), the determination of the geometric and hydraulic parameters (A, P, R, C, K, Q, v, and so on) is carried out depending on the degree of filling defined by the ratio a = h/H. By introducing a correlation of the type Q/Q_p = K/K_p = f(a), where Q_p and K_p are the flow, respectively the flow modulus, at the full section, the equation of the gradually varied non-uniform movement can be written in the form [4]:

$$\frac{\left|I\right|L}{H}=\frac{f^2\left(a\right)\left(1-\frac{\propto Q^{2}B}{g{A}^3}\right)}{f^2\left(a\right)\pm \frac{Q^2}{Q_p^2}}\mathit{AC}\sqrt{\mathit{Ri}}$$E10

and by integration over a section of length L and in two sections 1–1 and 2–2 we obtain:

$$\frac{\left|I\right|L}{H}=\phi \left(a_1,\frac{Q}{Q_p}\right)-\phi \left(a_2,\frac{Q}{Q_p}\right)-\frac{\alpha Q^2}{g{H}^5}\left[\mathrm{\Psi }\left(a_1,\frac{Q}{Q_p}\right)-\mathrm{\Psi }\left(a_{21},\frac{Q}{Q_p}\right)\right]$$E11

The functions φ and ψ present different integration domains, according to the sign of the slope and the ratio between the slope of the bottom of the bed i and the hydraulic slope J. To solve practical problems, the functions φ and ψ are integrated by different methods. For calculations with a relatively good precision, the functions φ and ψ were integrated and presented in tabular or graphical form [4].

Rapidly varying non-uniform permanent motion in free-flowing sewer collectors presents special calculation features determined by the magnitude of the second conjugate depth of the hydraulic jump. The hydraulic jump calculation relation has the form:

$$\frac{{\alpha }_0{Q}^2}{g{A}_1}+h_{G1}{A}_1=\frac{{\alpha }_0{Q}^2}{g{A}_2}+h_{G2}{A}_2,$$E12

where h_G1, h_G2, are the depths of the centre of gravity corresponding to the watered areas A₁ and A₂.

On a collector section with diameter D, or height H, the second conjugate depth of the hydraulic jump, depending on its size, can impose a free-level or pressurized flow downstream on a certain bed length. This aspect determines the differentiation of hydraulic calculation problems.

In the first case, for the ratio h₂/D < 1 (h₂ – second conjugate depth), the flow in the collector will be aerated, and the movement of water downstream of the jump will be free level (Figure 5a). The hydraulic jump develops as in an open bed, and the conjugate depths are determined by solving the equation (Eq. (12)).

Figure 5.

Analysis scheme of the hydraulic jump in closed channels and flow with free level: (a) case h2/D < 1; (b) case h2/D > 1 [4].

In the second case, when h₂/D > 1, respectively h₂ > D, the collector enters under pressure downstream (after the jump) and at the same time introduces a vacuum upstream of the jump (Figure 5b). If the air flow is not compensated by a process of aerating the collector, instability of the movement occurs, manifested by the oscillation of the parameters of the jump and of the liquid current.

3.3 Hydraulic calculation elements for non-standard sewer collectors from uniform movement

The monitoring of the operation of the sewer collectors is carried out by taking the characteristic hydraulic parameters from the imposed control sections. Simulating the operation of a collector, and especially the main ones, requires the calculation of hydraulic parameters for permanent and non-permanent movement. For a collector with a non-standard section and with a variable roughness on the wetted perimeter, it is necessary to go through several stages of analysis and calculation, respectively:

Stage I. Field analysis and retrieval of geometric and hydraulic data from the flow section of the collector. Carrying out a topographical study along the length of the simulation (longitudinal profile and transverse profiles) [16, 17]. Carrying out an in situ research on the way of roughness variation on the perimeter of the flow section. Data processing and creation of a database specific to each collector.

Stage II. Determination of geometric and hydraulic parameters for uniform permanent movement by using calculation programs specific to sections formed by curves and straight lines. The results obtained will be included in a database specific to each analyzed collector.

Stage III. Determination of geometric and hydraulic parameters for non-uniform gradual varied and rapidly varied permanent motion by using calculation programs. The results obtained will be included in a database specific to each analyzed collector.

The flow sections in sewer collectors geometrically made of curved lines were treated as “single bed” [18], where they are composed only of circular arcs (Figure 6a). Also, some sections were considered “multi-bed” being composed of straight line segments only. In a particular study case [16], a “single bed” composed of both circular arcs and straight segments was analyzed (the case of gallery-type collectors formed by a rectangle and a semicircular area on the apron (Figure 6b).

Figure 6.

Calculation schemes for non-standard collector sections: (a) semicircular bell; (b) rectangular collector with semicircular groove.

Later, a generalized analysis model was developed for „single bed” sections composed of both curved lines (generally circular arcs) and straight line segments). In the analysis, it was considered that the tube that forms the collector has symmetry along a longitudinal vertical plane (Figure 6a,b). It follows that the contour of the cross-sectional half of the tube is completely determined by an odd number of points, N_M, conveniently established by a yOz coordinate system. The number of points is defined by the relationship:

$${\left(y_C,z_C\right)}_{k}k=1,2,\dots ,N_M$$E13

The coordinates (Eq. (13)) are related to a Cartesian system of coordinate axes, yOz, defined as follows: the Oz axis, vertical included in the longitudinal plane of symmetry; the origin O at the point on the minimum height tube eraser, and the horizontal axis Oy, normal to the axis Oz, included in the transverse half-section of the tube.

The hydraulic calculation model of sewer collectors for the permanent and non-permanent flow regime uses the following hydraulic characteristics depending on the water depth, h or the elevation of the free water level, z: B(h) or B(z), A(h) or A(z), P(h) or P(z), R_h(h) or R_h(z), z_G(h) or z_G(z), W(h) or W(z) and K(h) or K(z).

In sewer collectors with sections composed of N_a ≥ 2 arcs of a circle and/or line segments, the analytical expressions for each of the functions A(h), P(h), R_h(h), z_G(h), W(h) and K(h) must be rendered by N_a functions defined on portions of the perimeter, respectively:

$$\begin{array}{l}{\mathit{A}}_{\mathrm{i}}\left(\mathit{h}\right),{\mathit{P}}_{\mathrm{i}}\left(\mathit{h}\right),{\mathit{R}}_{\mathrm{hi}}\left(\mathit{h}\right),{\mathit{B}}_{\mathrm{i}}\left(\mathit{h}\right),{\mathit{z}}_{\mathrm{Gi}}\left(\mathit{h}\right),{\mathit{W}}_{\mathrm{i}}\left(\mathit{h}\right),{\mathit{K}}_{\mathrm{i}}\left(\mathit{h}\right),\text{with}{\mathit{z}}_{\mathrm{i}}-{\mathit{z}}_{\text{C0}}\le \mathit{h}\le \left({\mathit{z}}_{\mathrm{i}+1}–{\mathit{z}}_{\text{C0}}\right),\\ \mathit{i}=1,\dots ,{\mathit{N}}_{\mathrm{a}}\end{array}$$E14

where z_C0 (usually z_C0 = 0) is the height of the tube eraser; z_i and z_i + 1 ordinates of the points that delimit the lower, respectively upper, circle arc or line segment i.

To determine the functions (Eq. (14)) the following steps must be solved:

1. Deduction of analytical expressions for circles and/or supporting lines of N_A circle arcs and/or line segments.

2. Determination of the N_A + 1 points that border the lower and upper N_A circle arcs and/or straight line segments, (y_i, z_i), (i = 1, 2, …, N_A + 1).

3. Deduction of analytical expressions, by portions, for the hydraulic characteristics of sewer collectors.

For the analytical expressions describing circles and support lines of circle arcs and line segments, the N_M coordinate points (Eq. (13)) were considered. From these were formed N_A groups of three successive points - M_i1, M_i2, M_i3 - of coordinates:

$${\left(y_i,z_i\right)}_j,\text{with}\left(j=1,2,3\right),\left(i=1,2,\dots ,N_A\right)\text{and}{z}_{i1}\le z_{i2}\le z_{i3}.$$E15

Each group of three such points determines:

• the circle arc i, centre C_i (b_i, a_i) and radius R_i – when the points M_i1, M_i2, M_i3 are non-collinear;

• the line segment i, with the coefficients of the line (a_i, b_i, c_i) - when the points M_i1, M_i2, M_i3 are collinear (the line segment i can be considered an arc of a circle with radius R_i tending to infinity).

Based on the mentioned, the equations describing the circle arcs and the straight line segments that form the geometric figure of the studied collector were deduced.

To determine the coordinates of the points bordering the N_A circle arcs and straight line segments, proceed as follows:

The N_A distinct geometric shapes (circle arcs and/or straight line segments) that structure the cross-section of the tube are bordered by N_A + 1 points, coordinates

$$\left(y_i,z_i\right)\left(i=1,\dots ,N_A+1\right),$$E16

For extreme points, coordinates of the type (16) are selected directly from the set of coordinates (13)

$$\left(y_{1,}{z}_1\right)={\left(y_{C,}{z}_C\right)}_1=\left(y_{C1,}{z}_{C1}\right),\left(y_{N_A+1},z_{N_A+1}\right)={\left(y_{C,}{z}_C\right)}_{N_M}=\left(y_{{\mathit{CN}}_M,}{z}_{{\mathit{CN}}_M}\right)$$E17

while the coordinates

$$\left(y_{i+1},z_{i+1}\right)\left(i=1,\dots ,N_A-1\right),$$E18

correspond to the intersection points between each grouping of circles or lines related to adjacent geometric elements i and i + 1.

Depending on the geometric shape (arc of a circle or line segment) of elements i and i + 1, four distinct analysis situations are distinguished:

1. Both elements, i and i + 1, are arcs of a circle.

2. Element i is an arc of a circle and element i + 1 is a line segment.

3. Element i is a line segment and element i + 1 is an arc of a circle.

4. Both elements, i and i + 1, are line segments.

For the four analysis situations, the equations were deduced that allow obtaining the coordinates of the points bordering the N_A circle arcs and straight line segments for the geometric shape of the studied collector.

The hydraulic-mathematical model is customized according to the shape and structure of the non-standard section. In the case of a bell-type collector with a section modified by a new manufacturing technology, or by rehabilitation works, with the simulation scheme shown in Figure 6a, the following calculation steps were carried out:

Stage 1. Definition and calculation of plot coordinates of geometric parameters (y, z) for the modified cross-section. It is assumed that the bell-shaped section (Figure 6a) is formed at the bottom by a circle segment with radius R1, with centre O₁ (0, R₁), a chord of length B and height ζ. Also, the bell-type section is described at the top (dome) by a semicircle with radius R₂ = B/2 from the centre O₂ (0, ζ) (Figure 6b). The way of working for this type of section is as follows:

I.1. Determining the plotting coordinates of the flow area.

I.2. Determination of the drawing coordinates of the hydraulic radius.

I.3. Determining the drawing coordinates of the centre of gravity of the flow area and so on.

The constructive parameters ζ and R₁ are evaluated according to the representative sizes B and H of the flow section with the relations:

$$\zeta =H-B/2,R_1=\frac{B\left[{\beta }^2+{\left(2-\beta \right)}^2\right]}{4\beta \left(2-\beta \right)},\text{with}\beta =\frac{B}{H}$$E19

The semicircular geometric profile of the collector is completely determined by five coordinate points (y_C, z_C).

$$\left(y_C,z_C\right)\in \left[\left(0,0\right),\left(\sqrt{\zeta \left(4R_1-\zeta \right)}/2,\zeta /2\right),\left(B/2,\zeta \right),\left(B\sqrt{3}/4,B/4+\zeta \right),\left(O,H\right)\right].$$E20

Stage 2. Mathematical modeling of hydraulic parameters specific to the non-standard collector (modified geometric shape). The analytical expressions for the hydraulic characteristics of the sewer pipes were derived on Na portions, on which the water depth h satisfies the condition z_i < h ≤ z_i + 1, (i = 1, …, N_A).

The non-uniform distribution of the roughness on the wetted perimeter of the tube was taken into consideration by assigning a specific value to the roughness coefficient (the Manning coefficient was accepted), n_i, on each circle arc element or right segment i, i = 1, …, N_A. A series of roughness coefficient values have been determined by research on main sewer collectors in operation.

The expressions of the geometric parameters proposed to be evaluated for the circular bell type collector are the following (z_c0 = 0 was considered):

2.1. Hydraulic parameters of the flow section: A(h) or A(z), P(h) or P(z), R_h(h) or R_h(z), B(h) or B(z) and z_G(h) or z_G(z).

2.2. Mathematical modeling of the hydraulic characteristics for the collector with modified geometric shape: the velocity module, W(h) or W(z) and the flow modulus, K(h) or K(z).

To solve the mathematical model, a numerical calculation program Collector.Non-Standard_Uniform Motion_Unique Bed (Col.NS_UM_UB), was developed in the MATLAB programming environment.

Each type of non-standard collector geometry has an identification code attached to it in the uniform motion simulation program. Through this code, the computer selects the menu and work subroutines.

The program entry data are as follows: E.1 - Work menu (Specific code); E.2 – General data (Dt_H, Dt_B, respectively the specific calculation distance in the Cartesian system [calculation step, (m)]: E.3 – Geometric characteristic data of the section (D, B, H); E.4 - Roughness coefficient values on characteristic perimeter lengths, n_i.

The output data from the program is as follows:

O.1. The geometry of the section represented by y, z - the coordinates of the connection points on the contour of the non-standard collector.

O.2. Geometric and functional characteristics of the collector section: data for partially filled section (h, B, A, R, z_G, W and K) and full section (h_p, B_p, A_p, R_hp, z_Gp, W_p and K_p). The calculated data are represented in a table depending on the length gap chosen.

O.3. Graphic representations for contour geometry and hydraulic characteristics:

the characteristic curves of the geometry of the collector section B = f_B(h), z_G = f_zG(h);

the hydraulic characteristics of the flow section of the collector A = f_A(h), P = f_P(h), R_h = f_Rh (h).

the hydraulic-functional characteristics of the collector W = f_W (h) and K = f_K (h).

the hydraulic characteristics of partial filling of the collector Q/Q_p = K/K_p = f_Q (h/H) and v/v_p = W/Wp = f_v (h/H).

The simulation results are presented in the form of customized tables on the operating domains and on the calculation sections of the collector, as well as by graphical representations of the calculated functions.

3.4 Hydraulic calculation in non-uniform permanent motion

The hydraulic parameters (flow Q and depth h, or elevation z) are a function only of the one-dimensional spatial coordinate l in the gradually varied permanent motion. The hydraulic calculation model in the varied gradual non-uniform permanent movement consists of: 1 - the differential equation of the varied gradual movement; 2 - contour conditions. A synthesis of the developed hydraulic model is presented next.

A calculation section of the main sewerage collector, which is delimited by two successive homes with a feeding role, is considered as a prismatic/cylindrical bed and with a constant slope. On this section, a permanent flow regime (Q = const) can be admitted and the fundamental equation of the varied gradual non-uniform movement has the form (Eq. (8)).

Eq. (8) is a first-order differential equation with partial derivatives, in which case it can be transposed in the form:

$$\mathrm{\Psi }\left(h\right)\mathit{dh}=\mathit{dl}$$E21

where:

$$\mathrm{\Psi }\left(h\right)=\left(1-\frac{\alpha Q^2}{g}\bullet \frac{B\left(h\right)}{A^3\left(h\right)}\right)/\left(1-\frac{Q^2}{I-\frac{Q^2}{K^2\left(h\right)}}\right).$$E22

Equations of the type (Eqs. (8) and (22)) are generally solved with the following simple boundary condition:

$$l=l^0;h=h^0,$$E23

where l⁰ and h⁰ represent known values for the coordinate l and the depth h, respectively.

In this case, the Cauchy problem defined by the relations [Eqs. (21)and (23)] presents the following particular solution explained in relation to the variable l by the relation:

$$l=l^0+\mathrm{\Phi }\left(h\right)-\mathrm{\Phi }\left(h^0\right)$$E24

where the function F(h) has the expression,

$$\mathrm{\Phi }\left(h\right)=\int \mathrm{\Psi }\left(h\right)\mathit{dh}$$E25

and represents a primitive of the function Y(h).

The solution of the equation (Eq. (24)) is of the form

$$l=f_s\left(h,h^0,l^0\right)$$E26

and can be explained later in the h variable in the form

$$h=f_l^{-1}\left(l,l^0,h^0\right)$$E27

The main sewer collectors have complex geometric shapes, where the characteristics A(h), B(h) and K(h) are expressed by complicated functions, which are defined by portions, or graphically or tabular. The primitive function (Eq. (25)) can be determined by numerical methods, with the use of quadrature formulas or Taylor series expansions.

The concrete analytical expressions for the contour conditions in the permanent regime are established on the basis of the hydraulic-functional characteristics of the accessory constructions existing along the length of the collector (manholes with various functions). For some types of manholes, the hydraulic characteristic was determined taking into account their structural form and the way of access and evacuation of waste water.

To solve the mathematical model, a numerical calculation program named Collector_Movement_Permanent_Gradual_Variate (Col_MPGV) was developed in the MATLAB programming environment.

The results obtained are represented by hydraulic parameters of the movement along the collector and in imposed calculation sections, as well as the graphic representation of the curves of the free surface of the water. Thus, the water depth h or free level elevation z, degree of filling a = h/H, water speed v and others can be checked along the collector.

4. Results and discussion

The Col.NS_UM_UB program was tested and validated for the „standard semicircular bell” profile with parameters: B/H = 600/380 mm, uniform roughness on the perimeter n = 0.0135 and geodesic slope I = 0.0004. The calculation steps for height H and width B were Dt_H = 0.02 m and Dt_B = 0.02 m, respectively. The number of points considered was N_M = 5.

The data obtained were compared with those obtained by using the classical calculation method for the standard semicircular bell section. The results obtained (Table 1) and analyzed comparatively show a good agreement between the values calculated with the Col.NS_UM_UB program and the values calculated with the classic model used in the design of standardized sewer collectors.

Type	Ap, (m^−2)	Rhp, (m)	Qp, (m^3s^−1)	vp, (ms^−1)
CC	0.1738	0.1104	0.591	0.3416
CP	0.1735	0.1107	0.594	0.3406

Table 1.

Comparative analysis of calculation program verification data.

CC, classical calculation; CP, calculation program.

The calculation program allows obtaining a high accuracy of the obtained data and a significant reduction of the calculation time. The depths corresponding to the maximum values for the hydraulic radius R_h and the velocity module W are identical, because the roughness is uniform throughout the perimeter of the sewer collector.

In the next stage, the uniform movement was simulated for a sewer collector with non-standardized semicircular bell-type profile (Figure 6a), for which the geometric and hydraulic parameters were calculated. The section of the collector is bell type with dimensions B/H = 2.800/2.400 m and is made of reinforced concrete. The roughness on the perimeter is uneven in the range of values n = 0.015–0.019, specific to collectors made of reinforced concrete and in operation for a long period of time.

The input data to the program (I.1, I.2, I.3, I.4) in a synthetic presentation were: calculation code for non-standardized semicircular bell; calculation steps Dt_H = 0.02 m, Dt_B = 0.02 m, B = 2.800 m, H = 2.400 m, n = 0.017–0.019 (Manning) and N_M = 5.

The exit data from the program, in a synthetic presentation, were as follows:

O.1. Contour geometry: NA = 2, where the coordinates of the connection points y, z, on the collector contour are shown in bold characters in Table 2; code with analysis specification, Kod = 2_circle arcs; a = [1.4800, 1.0000]: ordinates of the centres of the support circles of the arcs, (m); b = [0.0000, 0.0000]: the abscissas of the centres of the support circles of the circle arcs, (m); R = [1.4800, 1.4000]: radii of the support circles of the circular arcs, (m).

Coordinates	1	2	3	4	5
yC, z (m)	0	1.109	1.400	1.212	0
yC, z (m)	0	0.500	1.000	1.700	2.400
n = [0.0190–0.0150]

Table 2.

The coordinates of the representative points on the contour non-standardized semicircular bell type collector.

Five representative points N_M of coordinate’s yC, z (Table 2) were considered on the contour of the collector to define the shape. The spikes were evaluated with the equation (Eq. (20)).

O.2. The geometric and hydraulic-functional characteristics: N_C = 120. The N_C elements of each of the vectors hC, BC, AC, RhC, zGC, WC and KC, which constitute the columns of the matrix M, are tabulated (Table 3). Also, the values of the parameters for the full section h_p, B_p, A_p, R_hp, z_Gp, W_p and K_p are presented in Table 3, last line in bold. The results presented in Table 3 are used as a database for the simulation of the gradually varied movement in the collector.

hC; (m)	BC; (m)	AC; (m^2)	RhC; (m)	zGC; (m)	WC; (ms^−1)	KC; (m^3s^−1)
0.010	0.343	0.002	0.007	0.006	1.863	0.004
0.050	0.764	0.025	0.033	0.030	5.435	0.139
0.291	1.763	0.349	0.185	0.174	17.098	5.977
0.512	2.239	0.795	0.313	0.304	24.282	19.318
0793	2.262	1.483	0.460	0.467	31.392	46.570
1074	2.796	2.253	0.590	0.627	37.172	83.768
1.496	2.618	3.405	0.726	0.849	43.515	148.167
1.918	2.114	4.417	0.778	1.044	46.234	204.224
2.359	0,666	5.106	0.690	1.187	43.382	218.791
2.380	0.473	5.117	0.674	1.190	42.752	218.791
2.400	0.000	5.124	0.635	1.192	41.223	211.225

Table 3.

Matrix elements M = [hC BC AC RhC zGC WC KC] obtained for the non-standard semicircular bell collector (extracted from 120 values).

The absolute and relative maximum values of the hydraulic parameters W, K and for the absolute water depth h and relative h/H at which these parameters are recorded in the circular bell collector are centralized and presented in Table 4. Due to the roughness coefficient n, which is uneven on the perimeter of the collector, the depths corresponding to the maximum values for the hydraulic radius R_h and the velocity modulus W are different.

Parameters		Geometric parameters			Hydraulic parameters
		A (m^2)	B (m)	zG (m)	P (m)	Rh (m)	W (m/s)	K (m^3/s)
1		2	3	4	5	6	7	8
Maximum; values	Abs.	2.800	5.124	1.193	8.070	0.778	46.304	225.68
	Rel.	—	—	—	—	1.226	1.123	1.068
h, (m)		1.014	2.400	2.400		1.938	1.998	2.259
h/H, (−)		0.422	1.000	1.000	1.000	0.807	0.832	0.941

Table 4.

Maximum values of the geometric and hydraulic parameters obtained for the non-standard ovoid type section B/H = 2.800/2.400 m.

O.3. Graphic representations for the contour geometry and the hydraulic and functional characteristics of the analyzed collector: graphs for the geometric characteristic curves, the hydraulic characteristic curves and the hydraulic-functional characteristic curves.

In Figure 7 the geometric characteristic curves A = f_A(h), B = f_B(h), z_G = f_zG(h) and the hydraulic ones P = f_P(h), R_h = f_Rh (h) are presented.

Figure 7.

Geometric and hydraulic characteristics A = f_A(h), B = f_B(h), z_G = f_zG(h) P = f_P(h), R_h = f_Rh (h) at the non-standardized semicircular bell collector B/H = 2.80/2.40 m.

Figure 8a shows the hydraulic-functional characteristics W = f_W (h) and K = f_K (h). Figure 8b shows the hydraulic-functional characteristics when the collector is partially filled, respectively f_Q (h/H) = Q/Q_p = K/K_p and f_v (h/H) = v/v_p = W/W_p.

Figure 8.

Hydraulic and functional characteristics at the non-standardized semicircular bell collector B/H = 2.80/2.40 m: a – For W = f_W (h) and K = f_K (h); hydraulic flow characteristics f_Q (h/H) = Q/Q_p = K/K_p and f_v (h/H) = v/v_p = W/W_p at partial filling.

Performing a simulation of the operation of a collector with standard or non-standard section in permanent movement requires the serial use of the three simulation programs made for uniform movement, for the hydraulic characteristics of the inspection chambers and for non-uniform movement.

The results obtained by running the program and which define the parameters of the permanent uniform movement at this sewer collector can be used in the analysis of the non-uniform permanent movement, respectively the non-permanent movement. The Col.NS_UM_UB program was also run for a collector with a section consisting of circle arcs + straight line segments (Figure 6b). The obtained results were used to simulate the non-permanent movement in the case of the transport of rainwater on a collector with dimensions 3.450/2.250 m and length 2416 m [15, 19].

The separation of domestic wastewater from rainwater in some urban sewerage networks in Romania has determined the modification of the geometric shape of the main collectors. The modernization principle consists in creating a pipe inside the collector, which will separately transport household waste water. The modified remainder of the collector section will carry storm water only. The monitoring of the hydraulic and functional parameters of the collector with the modified geometric shape, especially during the evacuation of torrential rains, requires the use of special simulation programs. Thus, the research carried out and the creation of a set of programs for simulating the permanent and non-permanent movement of waste water in non-standard sewerage collectors contribute to the technical basis of the monitoring system.

5. Conclusions

The main sewerage collectors of domestic and rainwater with a “visitable” type of construction have a long operating life, and the action of natural and anthropogenic factors in the site determines the evolution of complex processes of degradation of the flow section. Physico-chemical degradation processes cause the flow section to change to a shape different from the designed one.

The flow section of the visible sewer collectors can be modified as a geometric shape over time by hydrodynamic erosion and clogging phenomena, but also by technological repair and rehabilitation works. Degradation processes determine a new geometric shape consisting of curves and lines that change the value of the hydraulic parameters in the flow section.

The monitoring of the hydraulic parameters of the collector during the exploitation process requires a correct measurement and interpretation of them with high precision, a situation that cannot be achieved with collectors that present geometric shapes modified over time or non-standardized sections.

The calculation programs developed for uniform and non-uniform permanent movement have the advantage that they allow obtaining the geometric and hydraulic parameters of the sewer collector with non-standardized section with high accuracy and in a relatively short working time. The developed calculation programs have the advantage of taking into account the roughness variation on the wetted perimeter of the flow section.

The application of the developed calculation programs requires the presence of a database, which must be achieved through measurements and data retrieval from the inside of the visitable sewage collector. The database must contain a cross-sectional topographic survey and a detailed longitudinal profile, as well as an analysis of the values and variation of the roughness on the perimeter of the collector.

The hydraulic-mathematical model created can be generalized to obtain the geometric and hydraulic characteristics [A = f_A(h), P = f_P(h), R_h = f_Rh(h), B = f_B(h), z_G = f_zG(h), W = f_W(h) and K = f_K (h)] for single-bed closed channel sections consisting of curves, circular arcs and straight line segments. The simulations carried out and on concrete cases existing in practice confirm the viability of the calculation program.

Nomenclature

h (m)

water depth

z (m)

free water level share

A (m²)

cross sectional area of flow

P (m)

wetted perimeter

R_h (m)

hydraulic radius

B (m)

width of water surface

b (m)

width at the bottom of the channel

C (m^0,5s⁻¹)

Chézy coefficient

g (m s⁻²)

gravitational acceleration

n (s/m^1/3)

roughness coefficient (Manning)

k (m)

roughness

v (m s⁻¹)

average speed

Q (m³s⁻¹)

flow

a (−)

degree of filling

W (m s⁻¹)

speed module

K (m³s⁻¹)

flow module

J (−)

hydraulic slope

z_a (m)

apron share of the collector

D (m)

collector diameter

H (m)

collector height

r (m)

radii of curvature

l (m)

calculation length

L (m)

total length

v_p (m s⁻¹)

speed at full section

Q_p (m³s⁻¹)

flow at full section

z_G, h_G (m)

ordinate to the centre of gravity of the watered area