Incipient motion of tire rubber grains bed

Movimento incipiente do leito de grãos de borracha de pneu

ABSTRACT

In the present work, it was verified that the use of rubber grains from tires, obtained by recycling, could be a viable alternative for use in mobile-bed river models. In this way, it was experimentally verified whether the rubber grains met the incipient motion criteria proposed by Shields. So, a mobile-bed of tire rubber grains was placed inside a rectangular channel, with 3 m long, 14 cm wide and with a slope of 0.001 m/m. Water flowed over the mobile-bed with different discharge increments until the incipient motion of the particles at the bottom was verified. Video image analysis was used to establish a criterion for the incipient motion. Two diameter classes of rubber particles were evaluated (3.00 mm to 2.00 mm and 2.00 mm to 1.70 mm). The experimental critical values of the Shields parameter of the rubber particles were satisfactory because the values were within the dispersion range of the experiments by Shields diagram.

Keywords:
Incipient motion; Rubber grains; Physics model

RESUMO

No presente trabalho verificou-se que a utilização de grãos de borracha de pneus, obtidos por reciclagem, pode ser uma alternativa viável para utilização em modelos de leito móvel de rios. Desta forma, foi verificado experimentalmente se os grãos de borracha atendiam aos critérios de movimento incipiente propostos por Shields. Então, foi colocado um leito móvel de grãos de borracha de pneu dentro de um canal retangular, com 3 m de comprimento, 14 cm de largura e inclinação de 0,001 m/m. A água escou sobre o leito móvel com diferentes incrementos de descarga até que se verificasse o movimento incipiente das partículas no fundo. A análise de imagens de vídeo foi utilizada para estabelecer um critério para o movimento incipiente. Foram avaliadas duas classes de diâmetro de partículas de borracha (3,00 mm a 2,00 mm e 2,00 mm a 1,70 mm). Os valores críticos experimentais do parâmetro Shields das partículas de borracha foram satisfatórios porque os valores estavam dentro da faixa de dispersão dos experimentos do diagrama de Shields.

Palavras-chave:
Movimento incipiente; Grãos de borracha; Modelos físicos

INTRODUCTION

According to the ASCE (ASCE Task Committee on Hydraulic Modeling, 2000), the physical modeling method for incipient bottom movement commonly derives from the Shields parameter (dimensionless shear stress), it also is termed particle mobility number (Yalin, 1989), flow intensity (Einstein, 1942), particle Froude number $F{r}_{*}$ and densimetric Froude number (Sharp, 1981). After applying a dimensional analysis, there is a relation of the incipient movement of a particle that follows the critical value of the Shields parameter, ${\tau }_{*c}$, with others parameters as defined by Equation 1,

${\tau }_{*c}=\frac{{\tau }_o}{\left({\gamma }_s-{\gamma }_f\right)d_s}=\frac{\rho u_{*}^2}{\left({\gamma }_s-{\gamma }_f\right)d_s}=f\left(tan\varnothing ,\frac{u_{*}{d}_s}{{\vartheta }_f},\frac{lift}{drag},shape\dots \right)$ (1)

Where ${\tau }_o$ is the shear stress at the bottom, $d_s$ is the average sediment diameter, ${\gamma }_s$ is the specific weight of the sediment, ${\gamma }_f$ is the specific weight of the fluid, $\rho$ is the mass density of the fluid, $u_{*}$ is the shear rate, and ${\nu }_f$ is the kinematic viscosity of the fluid. Julien (2010) mentions that Duboys (1879) was first derived the Shields parameter, and defined its proportionality with tan ($\varphi$). Shields (1936) defined the relationship to the grain shear Reynolds number, $R{e}_{*}=\frac{u_{*}{d}_s}{{\vartheta }_f}$.

Later studies have allowed establishing a new relation between the Shields parameter and dimensionless grain diameter (Julien, 2010),

$d_{*}=d_s{\left(\frac{\left(G-1\right)g}{{\nu }_f^2}\right)}^{\frac{1}{3}}$ (2)

Where $G=\frac{{\gamma }_s}{{\gamma }_f}$

Thus, the grain shear Reynolds number can be replaced by the dimensionless grain diameter, resulting in the in the modified Shields diagram, Julien (2010) proposed the equation, considering the angle of repose ($\varphi$):

${\tau }_{*c}\approx 0.3e^{\frac{-d_{*}}{3}}+0.06\tan \varphi \left(1-e^{\frac{-d_{*}}{20}}\right)$ (3)

Therefore, considering the angle of repose in the incipient movement is more important in very angular grains (crushed material) than in naturally rounded grains.

The mobile-bed river models are useful when sediment transport is import, for examples drop structures, local scours, erosion below spillways, locks and dams, reservoir sedimentation, etc. The mobile-bed river modeling process consists in establishing the similarity criteria to four parameters: Froude number; Manning-Strickler coefficient; critical values of Shields parameter; dimensionless particle diameter; referring to the prototype and the model; this means $F{r}_p=F{r}_m$, $n_p=n_m$, ${}_{*cp}={}_{*cm}$ and $d_{*p}=d_{*m}$ (the subscripts p and m designate the prototype and the model, respectively). The $d_s$ of the model may not follow the geometric scale established, since diameters smaller than 0.22 mm show interactions between flow and sediment, causing changes in the behavior of sediment transport on the model scale (Heller, 2011). To maintain the similarity of dimensionless sediment diameter, it is sometimes necessary to reduce grain density and increase model diameter. The use of lighter materials (low density) and larger diameters depends on the physical and chemical characteristics of the material. Materials present problems (Julien, 2002), for example: polystyrene floats after some time in water, nut shells have little durability, PVC presents hydrophobia, acrylic and river sand have dirt mixed in them, coal has dirt and non-homogeneity of specific density, Bakelite is porous and tends to rot, etc. In the present work, it was verified whether the use of rubber grains from tires, obtained by recycling, could be a viable alternative for use in mobile-bed river models (Martinez et al., 2023; Teixeira et al., 2020). In this way, it was experimentally verified whether the rubber grains met the incipient motion criteria proposed by Shields.

MATERIAL AND METHODS

The material used is crushed tire rubber. It was washed with detergents in tanks to remove impurities from the rubber, then it was left to rest, and part of the rubber that floated in the water was removed, leaving the rubber with more density than water. After drying in open air, the rubber was sieved to obtain particle size range of retained material between the diameters 1.18 mm to 3.00 mm, representing approximately 65% of the material's particle size distribution. In the other particle size ranges, there was not enough material accumulation to make the experiment of the incipient motion. The sand was collected from the reservoir of the Small Hydroelectric Power Plant of Salto de Paraopeba, from the banks, the adduction channel and in deposits formed close to the dam. The samples selected for analysis and particle size classification were those that represented the material from the adduction channel. Afterwards, the material went through the washing and drying process again, and separated into the particle size range between diameters of 0.125 mm to 0.50 mm, obtaining almost 80% of the material. In the other particle size ranges, there was not enough material accumulation, preventing its use in experiments at the incipient movement.

The angle of repose ($\varphi$) of sand and tire rubber was obtained experimentally using a funnel completely filled with the sample material. Then, the funnel was raised at a constant speed without contact with the deposited material and the funnel exit, thus forming a material deposition cone (Method C1444-00 - American Society of Testing and Materials, 2001; Chik & Vallejo, 2005). This process was performed three times for each particle size class of materials used in experiments. After the formation of the deposition cone, a photographic camera was used. It was adjusted to the horizontal plane of the stacked material. A graduated ruler was positioned orthogonally to the horizontal plane to record and measure results. The rest angle was determined by the software AUTOCAD (2016) as seen in Figure 1.

Figure 1
Determination of the angle of repose: a) design of the deposition cone; b) sand cone class S1.

The experiments were carried out for the rubber particle size classes R1 (1.70 mm to 2.00 mm) and R2 (2.00 mm to 3.00 mm), and for sand particle size classes S1 (0.125 mm to 0.250 mm) and S2 (0.250 mm to 0.500 mm). The particle size class of rubber within the range of 1.18 mm to 1.7 mm was not considered in the experiment, as it presented very low flow rates that were difficult to reproduce in the pumping system.

The tests to determine the incipient movement of the tire rubber particles were conducted in a rectangular section of an acrylic channel with 0.14 m of base, 0.20 m high, and 3 m long. The original slope of the bed in the reservoir prototype was 0.001 m/m. The rubber grains were used to model the movement of the natural sediment. There was a need to keep the rubber particles humidified in containers separated by particle size classes to fill the pores of the material with water, thus avoiding a change in density with the incorporation of air. It is remarkable that, without this technique, it would not be possible to determine the beginning of its movement. To avoid the effects of excavation of the bottom due to the entrance and exit of the flow in the flume, a fixed bottom of approximately 15 cm in length was glued to the entrance and another one was fixed at the bottom with 5 cm in length. Both bottoms had a roughness similar as that of the tested material, and a thickness of 1 cm. At the exit, an acrylic sill was placed at a height lower than the flow sheet to obtain the largest stretch of uniform flow in the canal. The analysis material was disposed along the channel and distributed evenly using a spatula and a trowel. The thickness of the material in the canal bed was 1 cm.

For sand, five experiments were carried out for each of the particle size class (S1 and S2). For rubber, 19 experiments were carried out for the particle size class R1 and 21 experiments for the particle size class R2. In laboratory, the sand mass density of 2,850 kg/m³ and the tire rubber mass density of 1,140 kg/m³ were determined (Associação Brasileira de Normas Técnicas, 1986).

To evaluate the incipient movement of particles, preliminary experiments were carried out to observe flow and sediment behavior along the channel. The results showed that the movement process, for flows much lower than Shields' theoretical flow, already occurred close to the downstream boundary condition. After the increase in flow, the movement of particles occurred from downstream to upstream, quickly and with each increase in flow. However, particle movement was limited to approximately one meter from the downstream boundary condition, and upstream of this point the channel remained without movement. From this stage onwards, with the increase in flow, the movement in these two regions of the channel did not change significantly until a certain flow range, from which movement began in the rest of the channel.

Based on what was previously observed, three regions were established in the channel: Region A, characterizing the zone of influence of the downstream boundary condition; Region B, corresponding to the range for observing and defining the incipient movement of particles, acquiring images and measuring hydraulic depths; and, Region C, subject to the influence of the upstream boundary condition, Figure 2 illustrates the location of these regions for analysis of incipient movement. In addition, in Region B was defined an incipient motion zone, between 90 cm and 165 cm of the total length of the canal, where the flow had already fully developed without the influence of boundary conditions. The video camera used in the tests was positioned at 120 cm from the channel entrance, and it was adjusted so as to form an approximate angle of 30º to 45º with the channel bed, then the camera focused the bottom of the channel in the measure zone of the incipient motion.

Figure 2
Sketch of the channel and location of the regions of influence of the canal, image acquisition, and evaluation of the beginning of movement.

In the zone of incipient movement, a sub-zone called the movement measurement zone was defined, with an area of 280 cm², in this sub-zone the video images were analyzed. As a criterion for incipient movement, it was established that the frequency of movement between particles must have occur in a time interval of less than 1s. If this criterion was not met, the flow rate was increased until particle movement was close to the established movement frequency criterion. The analysis of the images considering this criterion made it possible to observe the behavior of the incipient movement of the material for each flow rate, as well as predict whether, for flows close to the theoretical value, the entire section of the channel had already started moving, whether the bed still remained stable or whether found himself on the verge of moving.

RESULTS AND DISCUSSIONS

Table 1 shows the values of angle of repose. By using the graph developed by Simons (1957), which relates rest angles of non-cohesive materials, angle of repose of the sand were determined considering it as a very rounded sediment. In the case of rubber, an angular shape was considered.

Table 1
Average rest angles of rubber and sand.

Class	Measure Number	$\varphi$ (degrees) a	$\varphi$(degrees) Simons (1957)
S1	3	28.8	29
S2	3	28.8	29.5
R1	3	43.0	36
R2	3	42.3	37.5

• ^a
  Rest angle measure by AUTOCAD (2016).

The values obtained for the angle of repose of S1 and S2 were close to the values determined in the graph of Simons (1957). The formation of the pile cone for classes R1 and R2 showed an upper curvature (Figure 3), because the angle of repose values were higher than those determined by the Simons graph.

Figure 3
Cone formation for rubber, classes R1 and R2.

The tables 2, 3 and 4 show the experimental results of the incipient movement for the sand and tire rubber, respectively. Q is the rate flow, y_h is the flow height, R_h is the hydraulic radius, τ_{c exp} is the critical shear stress at the bottom, τ_{*c exp} is the experimental critical value of the Shield parameter, and slope of 0.001 m/m.

Table 2
Results of the incipient movement of sand S1 (0.125 mm to 0.250 mm) and S2 (0.250 mm to 0.500 mm).

Experiments S1	Q (l/s)	yh (cm)	Rh (m)	τcexp (N/m2)	τ*c exp	u* (m/s)	Re*	d*
1	0.65	1.98	0.0154	0.151	0.044	0.0123	2.30	4.94
2	0.80	1.93	0.0151	0.148	0.043	0.0122	2.28	4.94
3	0.65	1.73	0.0139	0.136	0.039	0.0117	2.18	4.94
4	0.45	1.49	0.0123	0.120	0.035	0.0110	2.05	4.94
5	0.45	1.41	0.0117	0.115	0.033	0.0107	2.00	4.94
Experiments S2
1	0.46	2.02	0.0157	0.153	0.022	0.0124	4.63	9.88
2	0.47	1.92	0.0151	0.147	0.021	0.0122	4.54	9.88
3	0.43	1.75	0.0140	0.137	0.020	0.0117	4.38	9.88
4	0.41	1.54	0.0126	0.123	0.018	0.0111	4.16	9.88
5	0.45	1.60	0.0130	0.127	0.018	0.0113	4.22	9.88

Table 3
Results of the incipient movement of rubber R1 (2.00 mm to 1.70 mm).

Experiments R1	Q (l/s)	yh (cm)	Rh (m)	τc exp (N/m2)	τ*c exp	u* (m/s)	Re*	d*
1	0.072	0.84	0.0075	0.073	0.028	0.0086	15.60	20.44
2	0.063	0.87	0.0077	0.076	0.029	0.0087	15.84	20.44
3	0.060	0.96	0.0084	0.083	0.032	0.0091	16.55	20.44
4	0.052	0.90	0.0080	0.078	0.030	0.0088	16.08	20.44
5	0.090	1.00	0.0088	0.086	0.033	0.0093	16.85	20.44
6	0.080	1.06	0.0092	0.090	0.035	0.0095	17.28	20.44
7	0.080	1.09	0.0094	0.092	0.036	0.0096	17.49	20.44
8	0.071	1.06	0.0092	0.090	0.035	0.0095	17.28	20.44
9	0.081	1.05	0.0091	0.089	0.035	0.0095	17.21	20.44
10	0.060	0.96	0.0084	0.083	0.032	0.0091	16.55	20.44
11	0.072	0.95	0.0084	0.082	0.032	0.0091	16.47	20.44
12	0.072	0.91	0.0081	0.079	0.031	0.0089	16.16	20.44
13	0.090	1.07	0.0093	0.091	0.035	0.0095	17.35	20.44
14	0.092	1.07	0.0093	0.091	0.035	0.0095	17.35	20.44
15	0.081	0.96	0.0084	0.083	0.032	0.0091	16.55	20.44
16	0.061	0.87	0.0077	0.076	0.029	0.0087	15.84	20.44
17	0.071	0.96	0.0084	0.083	0.032	0.0091	16.55	20.44
18	0.084	0.99	0.0087	0.085	0.033	0.0092	16.77	20.44
19	0.075	0.91	0.0081	0.079	0.031	0.0089	16.16	20.44

Table 4
Results of the incipient movement of rubber R2 (2.00 mm to 1.70 mm).

Experiments R2	Q (l/s)	yh (cm)	Rh (m)	τc exp (N/m2)	τ*c exp	u* (m/s)	Re*	d*
1	0.173	1.33	0.0112	0.109	0.031	0.0105	25.73	27.62
2	0.150	1.38	0.0115	0.113	0.032	0.0106	26.13	27.62
3	0.157	1.40	0.0117	0.114	0.033	0.0107	26.29	27.62
4	0.180	1.42	0.0118	0.115	0.033	0.0108	26.45	27.62
5	0.192	1.44	0.0119	0.117	0.034	0.0108	26.60	27.62
6	0.193	1.46	0.0121	0.118	0.034	0.0109	26.75	27.62
7	0.190	1.48	0.0122	0.119	0.034	0.0109	26.90	27.62
8	0.160	1.45	0.0120	0.117	0.034	0.0108	26.68	27.62
9	0.140	1.43	0.0119	0.116	0.033	0.0108	26.52	27.62
10	0.162	1.46	0.0121	0.118	0.034	0.0109	26.75	27.62
11	0.150	1.41	0.0117	0.115	0.033	0.0107	26.37	27.62
12	0.180	1.56	0.0128	0.125	0.036	0.0112	27.49	27.62
13	0.180	1.50	0.0124	0.121	0.035	0.0110	27.05	27.62
14	0.150	1.46	0.0121	0.118	0.034	0.0109	26.75	27.62
15	0.161	1.49	0.0123	0.120	0.035	0.0110	26.98	27.62
16	0.191	1.65	0.0134	0.131	0.038	0.0114	28.13	27.62
17	0.194	1.50	0.0124	0.121	0.035	0.0110	27.05	27.62
18	0.221	1.58	0.0129	0.126	0.036	0.0112	27.63	27.62
19	0.182	1.50	0.0124	0.121	0.035	0.0110	27.05	27.62
20	0.173	1.50	0.0124	0.121	0.035	0.0110	27.05	27.62

Experiments demonstrated that determiner the critical values of the Shields parameter may vary significantly depending on what degree of bed mobility is set as the threshold of incipient motion (Shvidchenko & Pender, 2000).

The Figure 4 presents the experimental results of Delft Hydraulics (Rijn, 1989) . natural non-cohesive sediment: curve A. occasional particle movement at some locations; curve B. frequent particle movement at some locations; curve C. frequent particle movement at many locations; curve D. frequent particle movement at nearly all locations; curve E. frequent particle movement at all locations; curve F. permanent particle movement at all locations; curve G. general transport (initiation of ripples). and Shields curve (Shields, 1936). In fact. Shields did not derive a model or an equation (Miedema. 2010). and published his findings as a graph preserving the representation of the data by a wide band. but Rouse (1939, 1949) subsequently fitting a line through the data (Buffington, 1999).

Figure 4
The experimental result of Delft Hydraulics [after Rijn, 1989] and incipient movement results for sand and tire rubber sediments.

The results of class sand. S1 (0.125 mm to 0.250 mm) and S2 (0.250 mm to 0.500 mm). were between curves A and D. these results are expected. as our criterion of incipient movement is almost similar to those adopted for curves A and D.

On the other hand. the results of the rubber R1 (2.00 mm to 1.70 mm) and R2 (2.00 mm to 1.70 mm) were between curves E and G. These results were due to the lower density of the rubber sediment. and experimentally a strong sensitivity of the increase in the movement of the rubber sediments with the increase in shear stress (increase in flow rate) was observed. quickly changing from an occasional movement (similar to curve A) to a general movement (similar to curve G).

Another criterion of incipient movement was adopted by Graf & Pazis (1977) for natural non-cohesive sediment. where in an area the number of moving particles is counted. 1. 10. 100 and 1000. Figure 5. The results for the sand class. S1 (0.125 mm to 0.250 mm) and S2 (0.250 mm to 0.500 mm). were outside curves 1 and 10. One explanation is that the experiments by Graf and Pazt were carried out for coarser sediment. The results for rubber. R1 (2.00 mm to 1.70 mm) and R2 (2.00 mm to 1.70 mm). were between curves 100 and 1000. but it must be considered that the experiment by Graf and Pazt was made with natural sediment and the rubber experiment used a lower density material. so a lower density sediment offers less resistance to the shear stress of flow

Figure 5
The experimental result of Graf & Pazis (1977) and incipient movement results for sand and tire rubber sediments.

Comparing the incipient movement of the rubber sediment with other sediments. Graf (2010) presents materials with different specific gravity and authors. Figure 6: Amber (1.05. Shields.); Lignite (1.27. Shields); Glass beads (2.49. Vanoni); Barite (4.25. Shields); Granite (2.70. Shields); Steel shot (7.90. White); Sand in air (2.10. White); Sand (2.51. White) and de others sand (2.65. Casey. Vanoni. Gilbert. Kramer. and USWES).

Figure 6
Shields’ diagram. Dimensionless shear stress vs Shear Reynolds number (Graf, 2010).

The results plotted in the Shields diagram (Graf, 2010) were compared with the results of sand and rubber sediments. It can be seen that the results of S1 and S2 were below the Shields curve. in the case of the results of rubber sediments (R1 and R2) values are around the Shields curve.

One of the objections to the use of the Shields diagram that the dependent variables appear in both ordinate and abscissa parameters. Depending on the nature of the problem. the dependent variable can be critical shear stress or grain size (Yang, 1996).

A second parameter is used (Vanoni, 1975). that only depends on the grain diameter. $d_{*}=\frac{d}{\vartheta }{\left[0.1\left(G-1\right)gd\right]}^{1/2}$. Afterwards. Julien (2010) proposed. $d_{*}=d_s{\left(\frac{\left(G-1\right)g}{{\nu }_f^2}\right)}^{\frac{1}{3}}$.

Figure 7 is the Shields diagram modified by Julien. The critical values of the Shields parameter for sand (S1 and S2) were below the Shields curve. the critical values of the Shields parameter for rubber (R1 and R2) were slightly above the Shields curve. but between the critical values of Shields and other authors.

Figure 7
Modified Shields diagram (Julien, 2010).

Using Equation 3 (Julien, 2010). the theoretical critical values of the Shields parameter for sand and rubber were calculated (Table 5). The critical value of the Shields parameter for sand presents values within the data cloud (Figure 7). but the critical values of the Shields parameter for rubber presented values (0.036 and 0.0409) inside the range of τ_*c.

Table 5
Critical Valore of Shields Parameter by Julien´s equation.

Sediment	d (mm)	d *	Angle of repose (°)	τ*c ( Julien, 2010 )	τ*c exp (Average)
S1	0.19	4.94	28.80	0.065	0.039
S2	0.38	9.88	28.80	0.025	0.020
R1	1.85	20.44	43.00	0.0361	0.032
R2	2.50	27.62	42.30	0.0409	0.034

When critical values of Shields parameters and dimensionless particle diameters in the model and in the prototype match (Sharp, 1981). it is possible to establish a scale relation between particle size and particle apparent weight.

$\frac{{\rho }_{sm}-{\rho }_{water}}{{\rho }_{sp}-{\rho }_{water}}={\left(\frac{d_{sp}}{d_{sm}}\right)}^3$ (4)

Where $d_{sm}$ = sediment diameter in the model; $d_{sp}$ = sediment diameter in the prototype; ${\rho }_{water}$ = water density; ${\rho }_{sm}$ = sediment density in the model. in this case rubber – 1.140 kg/m³; and ${\rho }_{sp}$= density of the natural sediment in the prototype. whose values can vary between 2.000 kg/m³ and 2.900 kg/m³ depending on the origin of the material under study. Considering the model diameter as of class R1 (average diameter of 1.85 mm). the sediment range of the prototype is 0.77 mm to 0.93 mm. For class R2 (average diameter of 2.5 mm). the sediment range of the prototype is 1.048 mm to 1.25 mm.

CONCLUSIONS

The experiments to determine the incipient movement of sand particles (S1 and S2) presented critical values of Shields parameters (dimensionless shear stress) slightly below the Shields and Julien curves. this is due to the incipient movement criterion adopted in this study. which tends to underestimate the critical value of dimensionless shear stress. when considering that incipient movement occurs when there is an interval of one second between the movement of one particle and another.

The experimental critical values of Shields parameters for rubber sediments R1 (3.00 mm a 2.00 mm) and R2 (2.00 mm to 1.70 mm to mm) were within the dispersion range of the experiments by Shields diagram. Then. it was experimentally verified that the rubber grains meet the incipient movement criteria proposed by Shields. Finally. it is possible to confirm the potential use of rubber sediments in the study of sediment transport in mobile-bed river models.

ACKNOWLEDGEMENTS

To the Minas Gerais Research Support Foundation (FAPEMIG) and Companhia Energética de Minas Gerais (CEMIG) for funding the research.

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Edited by

• Editor in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Edson Cezar Wendland

Publication Dates

• Publication in this collection
  25 Apr 2025
• Date of issue
  2025

History

• Received
  17 Dec 2024
• Reviewed
  12 Feb 2025
• Accepted
  20 Feb 2025