Erosion of sand under high flow velocities

Transcription

1 Delft University of Technology Faculty of Mechanical, Maritime and Materials Engineering Department of Offshore Engineering Erosion of sand under high flow velocities Author: Juneed Sethi MSc Thesis Thesis Committee: Prof. Dr.ir. C. van Rhee Ir. F. Bisschop Dr.ir.P.J. Visser Dr. ir S.A. Miedema

2 Abstract The existing functions for the erosion of sand overestimate the erosion rate during breaching of a dike and dredging of sand. The existing functions (like Van Rijn, 1993) are based on experiments with flow velocities of up to 2 m/s, while the flow velocity during breaching and jetting are higher. During breaching of dikes the flow velocity will be up to 10 m/s. In the dredging industry trailing suction hopper dredgers are used for the reclamation of sand. By using water jets granular sediment is loosened. Here the typical flow velocities are around 30 to 60 m/s (Van Rhee, 2007, 2010). The existing erosion functions (like Van Rijn, 1993) are pick-up functions of single sand particles. Due to turbulence near the bed the particles are picked-up by the flow from the bed and injected into the flow. At low flow velocities the particles are picked up individually. With increasing flow and erosion velocity the sand particles are picked up in layers. Bisschop et al. (2010) and van Rhee (2010) concluded that a lower permeability of the sand bed causes a lower erosion rate. To evaluate this theory there was limited data available of erosion experiments with sand at flow velocities of more than 2 m/s. In this thesis the physical process during the erosion of sand at high flow velocities is described. In this context erosion experiments with flow velocities of more than 2 m/s were executed. The experiments were done in the laboratory in the Delft University of Technology, faculty Mechanical, Maritime and Materials Engineering (3mE). With the new collected data the physical process during erosion of sand is analyzed. After analyzing the collected data the conclusion is made that the function of Van Rijn is overestimated. Course sand has a higher erosion rate than fine sand and the pick-up flux is independent of density of the sand bed. 2

3 List of figures Figure 1.1: Vertical displacement of an eroding sand bed Figure 2.1: Empirical pick-up function of Van Rijn (1993) Figure 2.2: Van Rijn: Erosion rate vs Bed Shear stress Figure 2.3: Contractant behaviour: decrease of porosity during shearing of loosely packed sand (Lubking, 2004) Figure 2.4: Dilatant behaviour: increase of porosity during shearing of densely packed sand (Lubking, 2004) Figure 2.5: Equilibrium of particles on a sloping surface (Van Rhee, 2007) Figure 2.6: Influence of the concentration Figure 2.7: Comparison Van Rijn & Van Rhee: Erosion rate vs Bed Shear stress Figure 2.8: Observed erosion velocities Figure 2.9:Dimensionless erosion rate Figure 2.10: Winterwerp: Erosion rate vs Bed Shearstress Figure 2.11: Mastbergen: Erosion rate vs Bed Shear stress Figure 2.12: Layout of the research station Figure 2.13:Amount of eroded sand versus flow rate Figure 2.14: Comparison Erosion Rate Figure 3.1: Sketch test set-up Figure 3.2: Overview measurement section Figure 3.3: Conductivity probes at lexanplate Figure 3.4a: Front view high speed camera Figure 3.4b: Side view high speed camera Figure 3.4: Pressure Calibrator Figure 3.5: water column Figure 3.6: Discharge meter 1: flow vel. at 0 m/s vs time Figure 3.7:Discharge meter 2: flow vel. at 0 m/s vs time Figure 3.8: Comparison discharge meters Figure 3.9: Calibration RA-Density meter Figure 3.10: Calibration RA-Density meter Figure 3.11: Overview height gauge radioactive concentration meter Figure 3.12: Height gauge RA-meter vs potential difference Figure 3.13: EMS meter old situation Figure 3.15: result erosion test Density as a function of Time Figure 3.16: Signal of conductivity probes is circular Figure 3.17: calculation expansion lexan plate Surface vs Linear on the basis of the measured density change of test nr Figure 3.18: location measurement points lexan plate Figure 3.19: Expansion lexan plate at different flow velocities Figure 3.20: Expansion lexan plate in vertical distance Figure 3.21: Expansion lexan plate in horizontal distance Figure 3.22: Comparison between derived and measured expansion lexan plate Figure 3.24: result erosion test Density as a function of time after installing clamps Figure 4.1: New location of connected conductivity probe from test Figure 4.2: Calibration conductivity probes (phase B) Figure 4.3: Calibration conductivity probes (phase B) Figure 4.4: Comparison methods of densification sand bed Figure 4.5: Calibration conductivity probes (phase D) Figure 4.6: Calibration conductivity probes (phase D) Figure 4.7: Result calibration constants probe A to D Figure 4.8: Result calibration constants probe E to H

4 Figure 4.9: Result calibration constants probe I to L Figure 4.10: Result calibration constants probe M to P Figure 4.11: Average flow velocity and pressure loss (test 26) Figure 4.12: Test 26 Density vs Time Erosion Figure 4.13: Erosion moments test Figure 4.14: Flow velocity vs erosion rate (D 50 =262µm) Figure 4.15: Flow velocity vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.16: Flow velocity vs erosion rate (D 50 =262 µm Figure 4.17:Flow velocity vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.18:Flow velocity vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.19: Flow velocity vs erosion rate (D 50 =125µm) Figure 4.20: Flow velocity vs erosion rate with relative density sand bed (D 50 =125µm) Figure 4.21: Flow velocity vs erosion rate (D 50 =125µm) Figure 4.22:Flow velocity vs erosion rate with relative density sand bed (D 50 =125µm) Figure 4.23: Flow velocity vs erosion rate with relative density sand bed (D 50 =125µm) Figure 4.24: Comparison of pressure losses Figure 4.25: Bed Shear Stress vs erosion rate (D 50 =262µm) Figure 4.26: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.27: Bed Shear Stress vs erosion rate (D 50 =262µm) Figure 4.28: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.29: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.30: Bed Shear Stress vs erosion rate (D 50 =125µm) Figure 4.31: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =125µm) Figure 4.32: Bed Shear Stress vs erosion rate (D 50 =125µm) Figure 4.33: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =125µm) Figure 4.34: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) Figure 4.35: Influence grain size on erosion rate Figure 4.36: Influence grain size on erosion rate with relative density [0-0.20] Figure 4.37: Influence grain size on erosion rate with relative density [ ] Figure 4.38: Influence grain size on erosion rate with relative density [ ] Figure 4.39: Influence concentration (C b ) on pick-up flux (relative density [0-0.30]) Figure 4.40: Influence concentration (C b ) on pick-up flux (relative density [0-0.30]) Figure 4.41: Influence density sand bed on pick-up flux (grain size D 50 : 262 µm) Figure 4.42: Influence density sand bed on pick-up flux (grain size D 50 : 262 µm) Figure 4.43: Influence density eroding flow on pick-up flux (grain size D 50 : 262 µm) Figure 4.44: Influence density eroding flow on pick-up flux (grain size D 50 : 262 µm) Figure 4.45: Influence concentration (C b ) on pick-up flux (relative density [0-0.20]) Figure 4.46: Influence concentration (Cb) on pick-up flux (relative density [0-0.20]) Figure 4.47: Influence density sand bed on pick-up flux (grain size D 50 : 125 µm) Figure 4.48: Influence density sand bed on pick-up flux (grain size D 50 : 125 µm) Figure 4.49: Influence density eroding flow on pick-up flux (grain size D 50 : 125 µm) Figure 4.50: Influence density eroding flow on pick-up flux (grain size D 50 : 125 µm) Figure 4.51: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Figure 4.52: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Figure 4.53 Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Figure 4.54: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Figure 4.55: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Figure 4.56: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) Figure 4.57: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) Figure 4.58: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) Figure 4.59: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm)

5 Figure 4.60: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) Figure 4.61: Bed shear stress vs pick-up flux Figure 4.62: Images test Figure 4.63: Density vs Time (test 27) Figure 4.64: Images test Figure 4.65: Density vs Time (test 31) Figure 4.66: Concentration vs time: test 27 and test 31 (CB: 30 mm above sand bed) Figure 4.67: Erosion moments with density sand bed: high vs low (test 27 vs test 31) Figure 4.68: Images test Figure 4.69: Density vs Time (test 36) Figure 4.70: Images test Figure 4.71: Density vs Time (test 41) Figure 4.72: Concentration vs time: test 36 and test 41 (CB: 30 mm above sand bed) Figure 4.73: Erosion moments with density eroding flow: 1000 kg/m 3 vs 1400 kg/m 3 (test 36 vs test 41) Figure 4.74: Flow velocity vs Erosion Rate (test 36 vs test 41) Figure 4.75: Images test Figure 4.77: Concentration vs time: test 36 and test 41 (CB: 30 mm above sand bed) Figure 4.78: Erosion moments with grain size (D 50 ): 262 m vs 125 m (test 27 vs test 44) Figure 4.79: Time vs Flow Velocity with grain size (D 50 ): 262 m vs 125 m (test 27 vs test 44) Figure AA-1: Calibration of HDP Figure AA-2: Calibration of HDP Figure AA-3: Calibration of HDP Figure AA-4: Calibration of VDP Figure AA-5: Calibration of VDP Figure AA-6: Calibration of VDP Figure AA-7: Calibration of DP Figure AA-8: Calibration of DP Figure AB-1: Influence concentration (Cb) on pick-up flux (relative density [ ]) Figure AB-2: Influence concentration (Cb) on pick-up flux ( Figure AB-3: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-4: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-5: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-6: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-7: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-8: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-9: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-10: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-11: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-12: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-13: Influence concentration (C b ) on pick-up flux (relative density [0-0.20]) Figure AB-14: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AB-15: Influence concentration (C b ) on pick-up flux (relative density [0-0.20]) Figure AB-16: Influence concentration (C b ) on pick-up flux (relative density [ ]) Figure AC-1: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) 139 Figure AC-2: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) 139 Figure AC-3: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) 140 Figure AC-4: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) 140 Figure AC-5: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) 141 Figure AC-6: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) 141 Figure AC-7: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) 142 5

6 Figure AC-8: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) 142 Figure AC-9: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) 143 Figure AC-10: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) Figure AC-11: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) Figure AC-12: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) Figure AC-13: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) Figure AC-14: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) Figure AC-15: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm above SB) Figure AC-16: Comparison pick-up flux measured with pick-up flux functions (C b : 10 mm SB) Figure AC-17: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) Figure AC-18: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) Figure AC-19: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB) Figure AC-20: Comparison pick-up flux measured with pick-up flux functions (C b : 20 mm above SB)

7 List of tables Table 2.1: Overview values used parameters Table 2.2: Overview erosion formulae Table 2.3: Influence of D 50, b and k on erosion rate Table 3.1:Explanation sketch test set-up Table 3.2: Location of DP meters Table 3.3: Location connected conductivity probes Table 3.4: Conversion factor water column Table 3.5: Selected calibration constants DP meters Table3.6: Calibration constants Table 3.7:Average potential difference discharge meter 1 and 2 at a flow velocity of 0 m/s Table 3.8:Average discharges according to different calibration constants Table 3.9: Derivation of density water Table 3.10: Derivation of density glass-water mixture Table 3.11: Calibration RA-Density meter Table 3.12: average potential difference in relation with different height of height gauge RA-meter 46 Table 3.13: Calibration constants height gauge RA-meter Table 3.14: expansion lexan plate at different flow velocities Table 3.15: Expansion lexan plate in vertical distance (meters) Table 3.16: Expansion lexan plate in horizontal distance (meters) Table 3.17:Expansion lexan plate before and after installing clamps Figure 3.23: Expansion lexan plate before and after installing clamps Table 4.1: Sand types Table 4.2: Approximate height of density measurement with RA-meter and related conductivity probes in phase B from test Table 4.3: New location of connected conductivity probe from test Table 4.4: Calculated density at the height of the conductivity probes (phase B) Table 4.5: Approximate height of density measurement with RA-meter and related concentration meters in phase D from test Table 4.6: Calculated density at the height of the conductivity probes (phase D) Table 4.7: Result calibration constants probes A to P Table 4.8: Test 26 Erosion velocities Table 4.9: compared images high speed camera Table 4.10: Transition moment where the density of eroding flow increases Table 4.11: Transition moment where the density of eroding flow increases Table AA-1: Calibration of HDP Table AA-2: Calibration of HDP Table AA-3: Calibration of HDP Table AA-4: Calibration of VDP Table AA-5: Calibration of VDP Table AA-6: Calibration of VDP Table AA-7: Calibration of DP Table AA-8: Calibration of DP Table AA-9: Calibration constants of HDP Table AA-10: Calibration constants of HDP Table AA-11: Calibration constants of HDP Table AA-12: Calibration constants of HDP Table AA-13: Calibration constants of VDP Table AA-14: Calibration constants of VDP Table AA-15: Calibration constants of DP

8 Table AA-16: Calibration constants of DP

9 Content Abstract... 2 Content Introduction Objective Setup of thesis Literature Study Introduction Van Rijn (1993) Van Rhee (2007, 2010) Winterwerp et al. (1992) D.R. Mastbergen (2003) Erosion and pore pressure gradients Comparison of the erosion functions Laboratory experiments Description of test set-up Instruments measurement section Calibration of instruments Differential pressure meters Discharge meter Radioactive meter Height gauge of the radioactive concentration meter Determination of height EMS meter Analysis increase of density Analysis results laboratory experiments Description of test program and analysis Influence flow velocity Grain size (D 50 ) 262 m Grain size (D 50 ) 125 m Influence bed shear stress Grain size (D50) 262 m Grain size (D 50 ) 125 m Influence grain size

10 4.5 Pick-up flux Comparison of functions with measured data Description of images high speed camera Conclusions and recommendations Recommendations References Appendix A: Calibration of differential pressure meters Appendix B: Influence concentration (C b ) on pick-up flux Appendix C: Comparison pick-up flux measured with pick-up flux functions Appendix D: List of symbols

11 1 Introduction The existing functions for the erosion of sand overestimate the erosion rate during breaching of a dike and dredging of sand. The existing functions are based on experiments with flow velocities of up to 2 m/s, while the flow velocity during breaching and jetting are higher. During breaching of dikes the flow velocity will be up to 10 m/s. In the dredging industry trailing suction hopper dredgers are used for the reclamation of sand. By using water jets granular sediment is loosened. Here the typical flow velocities are around 30 to 60 m/s (Van Rhee, 2007, 2010). The existing erosion functions (like Van Rijn, 1993) are pick-up functions of single sand particles. Due to turbulence near the bed the particles are picked-up by the flow from the bed and injected into the flow. At low flow velocities the particles are picked up individually. With increasing flow and erosion velocity the sand particles are picked up in layers. The top layer of the sand bed is sheared. The porosity decreases during shearing when sand particles are loosely packed. This is known as contractant behaviour. When the particles are densely packed the porosity will increase during shearing. This is called dilatant behaviour. Due to the increase in pore volume the pore pressure decreases in the sheared sand layer causing an inward hydraulic gradient of water. The hydraulic gradient pushes the top layer against the bed and hinders erosion. This gradient will increase with the erosion rate and decreases with the permeability of the sand. Bisschop et al. (2010) and van Rhee (2010) concluded that a lower permeability causes a lower erosion rate. The lower the permeability the higher the extra downward force due to the extra hydraulic gradient. To evaluate this theory there is limited data available of erosion experiments with sand at flow velocities of more than 2 m/s. Bisschop et al. (2010) evaluated the theory on the basis of data gathered during a large scale breaching experiment in the Zwin Channel in 1994 (Visser, 1998). 1.1 Objective The objective of this master thesis is to describe the physical process during the erosion of sand at high flow velocities. In this context erosion experiments were executed. The experiments were done in the laboratory in the Delft University of Technology, faculty Mechanical, Maritime and Materials Engineering (3mE). With the new collected data the physical process during erosion of sand will be analyzed and described. 1.2 Setup of thesis This thesis is started with a literature study where the present empirical formulae and theories related to the erosion of sand are described and compared (chapter 2). The test equipment is prepared for the execution of the erosion experiments. All the instruments used during the experiments are calibrated and tested (chapter 3). After execution of the experiments all the collected data is analysed (chapter 4). 11

12 2 Literature Study 2.1 Introduction The first step of this master thesis is a literature study. The aim of this literature study is to describe the present empirical formulae and theories related to the erosion of sand. This chapter describes the following theories related to the erosion of sand (section 2.2 up until 2.5) - Van Rijn (1993) - Van Rhee (2007,2010) - Winterwerp et al. (1992) - Mastbergen (2003) Besides these theories the effect of pore pressure gradients on the erosion of sand was studied by Jacobsen and Magda (1988). The results of this research were studied to improve the understanding of the influence of pore pressure gradients in the sand bed on the erosion process (section 2.6). Section 2.7 compares different erosion functions and describes the differences between these functions. An overview of the studied literature of the existing erosion functions is given in the references. Erosion of sand is a process in which grains of sand are picked up from the sand bed and transported by the flow. This thesis comprises the results of the research of the erosion of sand due to the flow of water or a sand-water mixture. Erosion is defined as the vertical displacement of an eroding sand bed (Figure 1.1) as function of time and is called the erosion rate (v e ). In the opposite way the vertical displacement of a settling sand bed as function of time is called the sedimentation rate (V sed ). Figure 1.1: Vertical displacement of an eroding sand bed The mass of the eroding grains per unit area and time (kg/sm 2 ) is called the pick-up flux (E). In the opposite way the mass of the settling grains per unit area and time(kg/sm 2 ) is called the settling flux (S). The erosion rate can be written as the result from the settling and pick-up flux: In which V e = vertical displacement of the an eroding sand bed as function of time (m/s) (erosion rate) [1] 12

13 V sed = vertical displacement of a settling sand bed as function of time (m/s) (sedimentation rate) E = pick-up flux in mass per unit area and time (kg/sm 2 ) S = settling flux in mass per unit area and time (kg/sm 2 ) ρ s = density of sediment [kg/m³] = in-situ porosity [-] c b = sand concentration near the sand bed [-] During erosion of sand the settling flux is equal to zero (S=0) and during sedimentation of sand the pick-up flux is equal to zero (E=0). Erosion will take place when the bed shear stress exceeds a certain value, the critical Shields parameter (θ cr ): θ ρ ρ [2] In which: = critical bed-shear stress [N/m²] ρ w = density of water [kg/m³] g = acceleration of gravity [m/s²] D 50 = grain size at which 50% of the total mass of particles has a smaller diameter [m] The critical Shields parameter is expressed as function of the dimensionless particle diameter D *, but other expressions were also possible: [3] [4] In which: = kinematic viscosity of water [m 2 /s] = relative density difference: (ρ s- ρ w )/ρ w [-] 13

14 2.2 Van Rijn (1993) The sediment erosion formula of Van Rijn (Van Rijn, 1993) is a pick-up function of single particles. Van Rijn performed experiments to determine the pick-up rate of particles in the range of 130 to 1500 μm (Van Rijn, 1993). Tests were performed with mean flow velocities in the range of 0.5 to 1.0 m/s. The data of the experiments are presented in figure 2.1. Figure 2.1: Empirical pick-up function of Van Rijn (1993) Analysis of the experimental data yielded the following empirical pick-up function: [5] [6] In which: = pick-up rate in mass per unit area and time [kg/sm 2 ] = excess bed-shear stress parameter [-] = bed-shear stress [N/ ] = Shields parameter [-] = critical Shields parameter [-] 14

15 The erosion formula is simplified and substituted to show the influence of the bed shear stress ( ) and the median grain size (D 50 ) solely. The values of other relevant parameters are given in table 2.1. These values are used in the rest of this chapter. Parameter Value Dimension Meaning parameter ρ s 2650 kg/m 3 density of sediment ρ w 1000 kg/m 3 density of water g 9,81 m/s 2 acceleration of gravity 1,65 - Relative density difference n 0 0,4 - in-situ porosity 1,00E-06 m 2 /s Kinematic viscosity Table 2.1: Overview values used parameters ρ θ θ θ θ θ ρ ρ 15

16 Erosion rate (V e ) [m/s] In figure 2.2 the erosion rate ( ) is plotted as a function of bed shear stress ( ). This figure presents the influence of the grain size (D 50 ) and the bed shear stress ( ) on the erosion rate ( ). The erosion rate is plotted for grain sizes of 100 m, 200 m and 300 m. 4,0 VAN RIJN 3,5 3,0 2,5 2,0 1,5 1,0 Van Rijn 100 μm Van Rijn 200 μm Van Rijn 300 μm 0,5 0, Bed Shear Stress (τb) [Pa] Figure 2.2: Van Rijn: Erosion rate vs Bed Shear stress According to the theory of van Rijn the erosion rate is increasing exponentially (power: 1.5) with the increase of the bed shear stress. The effect of the grain size on the erosion rate is less. The erosion rate increases with a power of 0.26 of the grain size. The theory of van Rijn predicts for coarse sand higher erosion rates than for fine sand at a bed shear stress of more than 1 Pa. 16

17 2.3 Van Rhee (2007, 2010) At low flow velocities the particles are picked up individually. At increasing erosion rate the sand particles are picked up in layers. The top layer of the sand bed is sheared. The porosity decreases during shearing when sand particles are loosely packed. This is known as contractant behaviour (Figure 2.3). When the particles are densely packed the porosity increases during shearing. This is called dilatant behaviour (Figure 2.4). During erosion the sand is mainly densely packed causing dilatant behaviour during shearing. Figure 2.3: Contractant behaviour: decrease of porosity during shearing of loosely packed sand (Lubking, 2004) Figure 2.4: Dilatant behaviour: increase of porosity during shearing of densely packed sand (Lubking, 2004) Due to the increase in pore volume the pore pressure decreases in the sheared sand layer causing an inward hydraulic gradient of water. The hydraulic gradient pushes the top layer on the bed and hinders erosion. The hydraulic gradient ) over the top layer is: In which: = permeability [m/s] = porosity sheared layer [-] [7] The hydraulic gradient ) increases by increase of erosion rate and decreases by increase of permeability. The hydraulic gradient has an influence on the stability of the particles. Figure 2.5 presents the equilibrium of the particles on a sloping surface. In this figure the slope angle is shown 17

18 as β and the angle of internal friction of the sediment as φ. On the sediment particles the forces of gravity G, the flowing water F s and an inward directed hydraulic gradient F i are acting. The equilibrium limit is reached when: φ φ φ [8] Figure 2.5: Equilibrium of particles on a sloping surface (Van Rhee, 2007) The hydraulic gradient has an influence on the stability of a slope. This is studied by van Rhee and Bezuijen (1992). An inward directed hydraulic gradient exerts an extra resisting force on the soil particles, an outward directed hydraulic gradient results in a lower resisting force. This is approached in two different perspectives. In case of an inward gradient a single particle mode is approached and in case of an outward directed gradient a continuum mode is approached. In the single particle mode, the limiting stability of a particle on top of an arrangement of circular particles is analyzed while in the continuum mode the stability of a small block of soil located at the surface is considered. The forces due to gravity and a hydraulic gradient are defined as follows: Single particle: Continuum: [9] [10] [11] [12] For a horizontal bed the stability criterion without a hydraulic gradient is F s G tanφ, which is commonly represented as F s /G tanφ θ cr. The stability criterion on a sloping surface in presence of a hydraulic gradient is as follows: where [13] 18

19 in which A=3/4 for single particles and A= (1/ (1-n 0 )) in case of a continuum approach. The erosion rate can be written as the result from sedimentation and a pick-up flux: In which [14] c b = sand concentration near the sand bed [-] The pick-up flux is related to the turbulence near the bed. Bursts of turbulence pick up particles from the bed and inject them into the flow. It is assumed that turbulent eddies pick up particles at the bed where a concentration of is present. If an eddy transports a volume of water and sediment from the bed, due to continuity the same volume of water must be transported back to the bed surface. The settling flux reads: [15] In which = settling velocity of a single grain in still water [m/s] = settling velocity of one particle [m/s] = hindered settling exponent [-] The settling velocity is affected by the concentration. This is known as hindered settling (Richardson and Zaki 1954). The exponent is a function of particle size and varies between 2.3 and 5, depending on the particle size (Richardson and Zaki 1954). Rowe achieved a smooth function of the hindered settling exponent using the following function: [16] In which is Reynolds number for the particles, defined as: [17] The concentration influences the erosion rate. When the concentration of the erodoing flow increases the effective erosion rate will decrease. This is depicted in figure 2.6 where the grain size D 50 is 200 m. 19

20 Erosion rate (V e ) [m/s] van Rhee 0,0140 0,0120 0,0100 0,0080 0,0060 0,0040 0,0020 Van Rhee 200 μm van Rhee-200 μm (c = 0.10) van Rhee-200 μm (c = 0.20) van Rhee-200 μm (c = 0.30) 0, Bed shear stress (τb) [Pa] Figure 2.6: Influence of the concentration Van Rhee (2007, 2010) derived for the erosion rate the following equation: [18] In which for the van Rijn pick-up function is used, but other pick-up functions were also possible: [19] The erosion rate is present on both side of the equation since the adapted critical Shields parameter is also a function of the erosion rate. It is not possible to solve this equation analytically for.bisschop (2010) simplified this equation. The analytical derived approximation of Bisschop (2010) is as follows: In which [20] [21] [22] The permeability is defined as (hazen equation) [23] In which for natural sand is: 20

21 [24] This approximation gives us a better understanding of the mechanism of hindered erosion. The erosion formula of Bisschop (2010) is simplified and substituted as follow: Where In figure 2.7 the erosion velocity ( ) is plotted as a function of bed shear stress ( ). This figure presents a comparison of the functions of van Rijn, van Rhee and the simplification of van Rhee by Bisschop. The erosion rate is plotted for grain sizes of 100 m, 200 m and 300 m. The erosion depends mainly on the permeability and bed shear stress (both parameters with a power 0.6). In theory two types of sand with the same permeability but different grain size will show more or less the same erosion rate. However this will not be the case in nature. Mainly coarse sand has a higher permeability than fine sand, so the grain size influences the erosion rate via the permeability. The conclusion can be made that the function of van Rijn overestimates the erosion rate at shear stress of more than 1 Pa. 21

22 Erosion rate (V e ) [m/s] 1,0000 Van Rijn vs Van Rhee vs Bisschop Van Rijn 100 μm 0,1000 0,0100 0,0010 Van Rijn 200 μm Van Rijn 300 μm Van Rhee 100 μm Van Rhee 200 μm Van Rhee 300 μm Bisschop 100 μm Bisschop 200 μm 0, Bed shear stress (τb) [Pa] Bisschop 300 μm Figure 2.7: Comparison Van Rijn & Van Rhee: Erosion rate vs Bed Shear stress According to the theory of Van Rhee (2007, 2010) the conclusion can be made that a lower permeability causes a lower erosion rate, meaning that finer sand will have a lower erosion rate in comparison with coarse sand. For material with relatively large in-situ porosity (low ), a relatively small increase of the porosity is necessary, resulting in lower hydraulic gradient and thus a higher erosion rate. Increase of the bed shear stress leads to an increase of the erosion rate. 22

23 2.4 Winterwerp et al. (1992) Field surveys and experiments in a tilting flume on hyper concentrated sand-water mixture flows were conducted by Winterwerp et al. (1992). At slopes slightly smaller than the equilibrium slope, a rapid development of sand bars propagating upstream was observed. The erosion process on the bar is limited by the high sand concentrations and flow velocities on the steep slopes. As a result the erosion rate was more moderate than predicted by the pickup functions used in classical sediment transport theories. The results of the tests showed that the netto sedimentation rate ) was found to vary between 0.1 mm/s and 2 mm/s and the sedimentation flux between 0.16 kg/m 2 s and 3.2 kg/m 2 s. This sedimentation flux S is defined as: and the sedimentation rate is defined as: [25] [26] During the test it was observed that the maximum value of the erosion rate is limited by the sand concentration of the sand-water mixture. This is illustrated in figure 2.8, showing all data of the observed erosion rates. This phenomenon can be indicated as hindered erosion. This is different hindered erosion then described in the theory of Van Rhee. According to the theory of Van Rhee the erosion is hindered by the hydraulic gradient ( ). In this case the erosion is hindered by the sand concentration of the sand-water mixture. Figure 2.8: Observed erosion velocities 23

24 The dimensionless form of the pickup flux is: ρ [27] From the difference between the observed sedimentation and pick-up flux, the erosion rate can be determined. A new empirical function was derived from the experiments for the erosion of the lee side of the bar. The more traditional pickup functions were not applicable, because the observed pick-up fluxes were much lower than predicted by these classical formulas. The new formula reads: θ θ [28] in which in analogy with the pickup function of Van Rijn (1984) the dimensionless grain parameter is (equation 4): and the dimensionless bed shear stress parameter is: [29] In which: = friction velocity [m/s] = mean flow velocity [m/s] f 0 = local Darcy-Weisbach bottom friction factor [-] The friction velocity represents the shear stress in terms of the unit velocity: [30] In contrast to the traditional pickup function of Van Rijn the erosion rate is proportional to instead of as a result of the hindered erosion process described by winterwerp. Figure 2.9 provides the erosion function of the shear-stress parameter together with the experimental data. The data used by Van Rijn to establish his classical pickup function are also presented. From the pick-up flux, the maximum propagation rate v bar of the sand bars is determined from the observed erosion rate v e : ρ [31] in which: = slope of the lee side of the bar 24

25 Figure 2.9:Dimensionless erosion rate The erosion formula of Winterwerp et al (1992) is simplified and substituted as follows: In case of erosion of a horizontal bed ( 0) the erosion is: 25

26 Erosion rate (V e ) [m/s] 0,030 WINTERWERP 0,025 winterwerp 100 μm 0,020 0,015 0,010 winterwerp 200 μm winterwerp 300 μm 0,005 0, Bed shear stress (τb) [Pa] Figure 2.10: Winterwerp: Erosion rate vs Bed Shearstress According to the theory of Winterwerp (1992) the erosion rate increase with the grain size D 50 to the power 0.3 (figure 2.10). The erosion rate increases with the square root of the bed shear stress. 26

27 2.5 D.R. Mastbergen (2003) Mastbergen and Van den Berg (2003) explained the observation of gradual retrogressive failure of very steep slopes in fine non-cohesive sands. This process, which is called breaching, may produce large failures in sand bars or river banks. When the sand is loosely packed it can only erode when the pore volume is increased. This will cause shear dilatancy and negative pore pressures. When the permeability is low, these negative pressures will be released slowly and the rate of erosion will be retarded. Mastbergen and Van den Berg (2003) provide a quantitative analysis of the breaching process and the resulting density under flow. The mechanism of breaching is related to the generation of negative pore pressures retarding erosion due to flow or gravity. According to Winterwerp et al. (1992), the erosion rate can be described as (see paragraph 1.3): A ( ) m D n [32] cr * in which: A = coefficient (about 0.018) m = shear stress power in erosion function (m=1.5) n = grain size power in erosion function (n=0.3) in which is the dimensionless pick-up flux according to equation 27: ρ For high erosion rates or fine sand with relatively low permeability, dilatancy effects play a role. Mastbergen and Van den Berg defined the erosion rate as follows: θ θ [33] The erosion formula is simplified and substituted to present the influence of the bed shear stress ( ) and the median grain size (D 50 ) assuming that the sedimentation flux (S) is negligible relative to the pick-up flux (E): θ θ Where θ 27

28 Erosion rate (V e ) [m/s] (Hazen formula) 0,045 Mastbergen 0,040 0,035 0,030 0,025 0,020 Mastbergen 100 μm Mastbergen 200 μm Mastbergen 300 μm 0,015 0,010 0,005 0, Bed Shear Stress (τb) [Pa] Figure 2.11: Mastbergen: Erosion rate vs Bed Shear stress Figure 2.11 shows that the erosion rate increases with increasing bed shear stress. The erosion rate also increases when the grain size is larger. This is the result of the influence of the permeability. 28

29 2.6 Erosion and pore pressure gradients H.Moust Jacobsen and W. Magda (1988) deal in their paper with the problem of simultaneous water current along seabed and up- or downwards seepage across the seabed. When a wave train passes a sandy seabed cyclic alternating shear stresses are developed and pore pressures develop. Cyclic mobility and even liquefaction may take place in certain soils. If the seabed surface is influenced by water currents erosion may also occur. In this paragraph the experimental test set-up is described as well as the conclusion of the performed tests. A flow of water into a permeable seabed results in a downward seepage force on the bed particles due the hydraulic gradient necessary for the flow through the voids. This is balanced by the gravity of the particles and the contact pressures between the particles. This is the same process which is considered to cause hindered erosion. Model tests on the seabed stability have been conducted in a water discharge flume (a). A Layout of the research station is shown below. The complete test facility consisted of many individual parts and units. Figure 2.12: Layout of the research station The flume is 15 m long and it has a closed water circuit (see figure 2.12).The flume is equipped with an electric centrifugal pump where the water inflow can be controlled manually. Box (b) homed the sandy bed sample. Box (c) played a role of a trap of sand eroded from box (b) during the test execution. The slope and the horizontal approaching bottom (d) created convenient conditions for the water jet flow over tested element of the sandy bed in the box (b). The main goal of it was to minimize the flow turbulence of the water passing over the box (b) and (c). The constant head tank (f) with overflow served as a water supply reservoir for water supply into and water drainage out of the sand bed inside the box (b)the propeller hydrometric current meter (h) and the current meter counter measured the horizontal discharge flow velocity in the flume. 29

30 The bed material used was sand Lund no 00 with a specific density ρ s of The D 50 of the material was 0.20 mm, the uniformity coefficient (U = d 60 / d 10 ) 1.40 and the coefficient of curvature (C c = (d 30 ) 2 / (d 10 d 60 ))= The first test series shows the transportation of sand, when no pore water movement takes place. The weight of transported sand per 30 minutes at different flow rates is shown in figure 2.13 as curve A. The effect of pore water flowing out of the sandy bed with a gradient of 0.73 is shown in curve B. The influence of the hydraulic gradient seems to be rather small. It affects the threshold flow rate to some extent, but for higher flow rates, when the sand weight per 30 minutes exceeds 0.6 N, the two curves are nearly identical. Seepage into the seabed prevents erosion by increasing the threshold flow rate and decreasing the inclination of the curve C. It is caused by the boundary shear stress as well as the seepage force both of which depends on the horizontal water flow and the seepage. This result can give positive confirmation of the idea of designing seashore protection against erosion and transport of sand along the line by creating an artificial seepage into a porous seabed by means of submerged wells. Figure 2.13:Amount of eroded sand versus flow rate 30

31 2.7 Comparison of the erosion functions In this paragraph the erosion functions which are derived by Van Rijn, Van Rhee, Winterwerp and Mastbergen are compared. Table 2.2 gives an overview of the simplified and substituted erosion formulae of the different theories. Theory Van Rijn Van Rhee / Bisschop Erosion Formula - Winterwerp et al. - Mastbergen Table 2.2: Overview erosion formulae Table 2.3 and figure 2.14 present the influence of the grain size (D 50 ), bed shear stress ( b ), and permeability (k) on the erosion rate( ). In table 2.3 also the influence of the grain size including the permeability is given, where for the relation between the permeability and grain size the Hazen equation (equation 23 and 24) is used, based on the assumption that D 10 is approximately equal to 2 D 50. The erosion rate is plotted with as function of the bed shear stress for different grain sizes (100 m, 200 m and 300 m). According to all theories the erosion rate increases with the grain size (D 50 ). In the theory of Van Rhee the grain size (D 50 ) has the highest influence on the erosion rate comparing to the other mentioned theories, when the effect of the permeability is translated in an extra effect of the permeability (table. 2.3). The bed shear stress ( b ) has the highest influence on the theory of Van Rijn, while the effect of the bed shear stress for the other theories is more or less comparable. Figure 2.14 shows that the theories of Van Rhee, Winterwerp et al (1992) and Mastbergen (2003) are agree more or less to each other, however the effect of the bed shear stress, permeability and grain size on the erosion rate is not exactly the same (see table 2.3). The functions of van Rhee (2010) and Mastbergen (2003) agree with respect to the effect of the permeability, while the effect of the grain size on the erosion rate is according to the function of van Rhee (2010) negligible in comparison to the function of Mastbergen (2003). The influence of permeability is not incorporated in the function of Winterwerp et al (1992). The conclusion can be made that according to the traditional theory of Van Rijn the erosion rate is a factor 10 to 100 higher in comparison with the theories of Winterwerp et al (1992), Mastbergen (2003) and van Rhee (2007, 2010). Theory D 50 b k D 50 (incl. k) Van Rijn (1993) - Van Rhee (2007,2010) / Bisschop (2010) Winterwerp et al. (1992) - Mastbergen (2003) Table 2.3: Influence of D 50, b and k on erosion rate 31

32 Erosion rate (V e ) [m/s] 10,0000 Comparison Erosion Rate 1,0000 0,1000 0,0100 0,0010 0, Bed Shear Stress (τb) [Pa] Van Rijn 100 μm Van Rijn 200 μm Van Rijn 300 μm Mastbergen 100 μm Mastbergen 200 μm Mastbergen 300 μm winterwerp 100 μm winterwerp 200 μm winterwerp 300 μm Bisschop 100 μm Bisschop 200 μm Bisschop 300 μm Van Rhee 100 μm Van Rhee 200 μm Van Rhee 300 μm Figure 2.14: Comparison Erosion Rate 32

33 3 Laboratory experiments Laboratory experiments were executed to analyze the physical process during erosion of sand at high flow velocities. In this chapter the test set-up is described, including a description of the instruments which are used. 3.1 Description of test set-up Figure 3.1 presents an overview of the test set-up. The system consists of a recirculation circuit (4) and (5) through which water and sand can be pumped by an electric centrifugal pump (2). The sand silo (3) is used as storage of the sand. The sand is added to the circulation circuit by pulling up a coneshaped valve in the silo and open valve 1 (6). In the recirculation circuit a measurement section (11) is constructed including a by-pass. The by-pass enables sedimentation of sand in the measurement section, while the sand-water mixture can flow via the by-pass and the remaining part of the recirculation circuit at a velocity above the critical velocity. This is necessary to avoid sedimentation of sand in the circulation circuit and clogging of the recirculation circuit. The measurement section is approximately 5 meter long, 0.3 meter high and 0.1 meter wide. This section is used to study the erosion process. By opening valve 4 (8) and closing valve 2 (7) the sand-water mixture can flow back to the sand silo. When the sand-water mixture flows back to the silo the flow velocity is kept high enough to avoid clogging of the system and low enough to avoid overflow in the silo. A brief description of the test program is given in paragraph 3.5. The background of the design of the test set-up is described by De Jong and Van Diepen (2010). Figure 3.1: Sketch test set-up 33

34 No Part in test set-up Function 1 Electric generator Drives the centrifugal pump 2 Centrifugal pump Generates pressure in the test set-up 3 Sand silo Storage of sand-water mixture 4 Vertical loop Measurement average density of mixture 5 Main line Pipeline where the discharge flows through 6 Valve 1 Sand-water mixture from sand silo can enter the system by opening this valve 7 Valve 2 Open/close the main line to lead sand-water mixture back to sand silo 8 Valve 4 Open/close sand silo to lead sand-water mixture back to sand silo 9 Bypass Pipeline where the remaining discharge can flow when measurement section is closed 10 Butterfly valve Valve which can open or close bypass and measurement section 11 Measurement section Rectangular part where all the parameters are measured related to erosion of sand 3.2 Instruments measurement section Table 3.1:Explanation sketch test set-up In the measurement section instruments are installed to measure the erosion process under high flow velocities. The following instruments are installed: Differential pressure meters(hdp1, HDP2 and HDP3); Discharge meters; Radioactive concentration meter (RA meter); Thermometer; Conductivity probes; Electro magnetic flow velocity meter (EMS meter); High speed camera (in front of glass window); Figure3.2 gives an overview of the measurement section and the installed instruments. 34

35 Figure 3.2: Overview measurement section Differential pressure meter The differential pressure meters measure the pressure difference between two distances. In total eight differential pressure meters are installed in the test set-up. These differential pressure meters are used to determine the: hydraulic gradient in the measurement section; waterpressure in the sand; average density of the mixture flowing through the test set-up. To determine the hydraulic gradient three differential pressure meters are placed in the measurement section in horizontal direction which are called HDP1, HDP2 and HDP3. These differential pressure meters measure the pressure difference at three locations from the same reference point. These measurements are used to determine effective bed shear stress during erosion. In vertical direction the water pressure in the sand bed during erosion is measured with the differential pressure meters VDP4, VDP5 and VDP6. To calculate the average density in the whole test set-up two differential meters are placed in the vertical loop of the test set-up (DP18 and DP19). The differential pressure meters give a potential difference in volt which is converted with a calibration constant into pressure difference in Pa. The calibration constants of the differential pressure meters are given in paragraph Table 3.2 and figure 3.2 present the location of the differential pressure meters. 35

36 Pressure meter Function Location x [mm] y [mm] HDP1 Measurement of hydraulic gradient horizontal distance HDP2 Measurement of hydraulic gradient horizontal distance HDP3 Measurement of hydraulic gradient horizontal distance z [mm] VDP4 Measurement water pressure vertical distance from point A (figure 3.2) VDP5 Measurement water pressure vertical distance from point A (figure 3.2) VDP6 Measurement water pressure vertical distance from point A (figure 3.2) DP18 DP19 Measurement density mixture Measurement density mixture Table 3.2: Location of DP meters Vertical Loop (upward direction) Vertical Loop (downward direction) Discharge meter The discharge meters are used to: Determine the flow velocity during erosion(discharge meter 2); Check flow velocity is above critical (discharge meter 1 and 2). For this purpose two electromagnetic discharge meters are installed in the test set-up. Discharge meter 1 is installed in the vertical loop of the test set-up and discharge meter 2 is installed in the measurement section (Figure 3.2). The electromagnetic discharge meters give a potential difference in volt which is converted with a calibration constant into a flow velocity in m/s. The calibration constants of the discharge meters are given in paragraph The average flow velocity of the mixture is a critical parameter during the preparation and execution of the erosion tests. If the flow velocity in the system comes below critical sand will settle. This can clog the system and tests cannot be performed. Conductivity probe The concentration of the sand bed or sand-water mixture during the erosion tests is measured by using conductivity probes. The conductivity probes are installed in anon conducting lexan plate (Figure 3.3) which is 600 mm wide, 288 mm high and 50 mm thick. During the erosion tests the conductivity probes are used to derive the erosion rate and the density of the sand bed and sandwater mixture. The conductivity probes are calibrated with a radio-active density meter. When erosion takes place the sand bed height decreases. This results in a lower concentration which is measured by the conductivity probes. In total 56 conductivity probes are installed in a sloping matrix on the lexan plate from which 16 probes are connected to the data acquisition system. The 16 conductivity probes are named a to p which are given in figure 3.3 and table 3.3. For optimal resolution the vertical distance of the probes is 5 mm and horizontal distance is 50 mm. 36

37 Figure 3.3: Conductivity probes at lexanplate Concentration meter location Concentration meter location Concentration meter location Concentration meter location A 3 E 11 I 19 M 27 B 5 F 13 J 21 N 29 C 7 G 15 K 23 O 35 D 9 H 17 L 25 P 45 Table 3.3: Location connected conductivity probes Thermometer The thermometer measures the temperature in the measurement section. The temperature influences the viscosity of the fluid. When the temperature increases, the viscosity will also increase. This will influence the erosion process. Radioactive concentration meter (RA-meter) The RA-meter is used to calibrate the conductivity probes at the lexan plate. The RA-meter is adjustable in vertical direction so it can be set at the same height with the conductivity probes at the lexan plate. The RA-meter gives a potential difference in volt which is converted with a calibration constant into density in kg/m 3. The calibration constant is given in paragraph EMS meter The electromagnetic flow velocity meter (EMS) measures the flow velocity at one location. The EMS is placed just above the sand bed to measure the flow velocity. With this velocity the shear stress can be determined. The EMS meter gives a potential difference in counts which is converted with a calibration constant into m/s. Due to technical problems the EMS meter is not used during the tests. High speed camera A high speed camera is used to capture the erosion process on video. This camera was hired from Deltares during the execution of the tests. The camera delivers images up to 12 seconds with 500 frames a second. This camera is placed in front of the glass window (figure 3.4a and 3.4b). 37

38 Figure 3.4a: Front view high speed camera Figure 3.4b: Side view high speed camera 3.3 Calibration of instruments To determine the deviations of the instruments it is important to calibrate all the instruments before starting with the test program. This is also important for the reliability of the measurement results. In this paragraph is explained how the instruments are calibrated Differential pressure meters Two methods are used to calibrate the differential pressure meters. The first method is by connecting the meters to a pressure calibrator (Figure 3.4). With the pressure calibrator the pressure in the differential pressure meters is increased. A certain pressure gives a certain potential difference, which can be read from the pressure calibrator. By exerting different pressures on the different meters and reading the resulting potential difference a calibration equation is derived for the differential pressure meters. Figure 3.4: Pressure Calibrator 38

39 The second method of the calibration of the differential pressure meters was with the help of a water column (Figure 3.5) and a potential meter. In April 2012 and May 2012 the calibration of the differential pressure meters was executed by using the pressure calibrator and in July 2012 the water column was used. The results with both methods were approximately the same (Appendix B). Figure 3.5: water column The difference in water height in this column is proportional to the pressure difference. The differential pressure meter is connected to a potential meter. A certain pressure in the water column gives an output in potential difference. The conversion factor which is used to convert the height of the water column to a certain pressure is given in table 3.4. By exerting different water levels in the water column and reading the resulting potential difference from the potential meter a calibration equation is derived for the differential pressure meters. These calibrations were compared with the calibration from the factory. conversion factor water column pressure [cm] [Pa] Table 3.4: Conversion factor water column The calibration equation is as follows: [36] in which: = pressure [Pa] = potentialdifference [Volt] a,b = calibration constant [-] 39

40 The differential pressure meters HDP1, HDP2 and HDP3 are calibrated with the pressure calibrator and by using the water column. The calibration with the water column is repeated three times. The differential pressure meters VDP4, VDP5, VDP6, DP18 and DP19 were only calibrated with the pressure calibrator. This calibration is repeated three times. The results of the calibration are given in appendix A. Tables AB-1 to AB-8 depict the exerting pressure with the resulting voltage and tables AA-9 to AA-16 give the resulting calibration constants of each calibration. In figures AA-1 to AA-8 a comparison is made between the calibration formulas. The chosen calibration constants are given in table 3.5. These are the calibration constants which are used for the analysis of the executed tests in July Pressure meter Chosen calibration constants 2012 a b HDP HDP HDP VDP VDP VDP DP DP Table 3.5: Selected calibration constants DP meters 40

41 3.3.2 Discharge meter In 2012 different calibration constants of the two discharge meters were already known. By comparing these constants with the potential differences of the discharge meter at zero flow velocity the best calibration constants were analyzed. The calibration constants are called respectively 2010 (used during tests in 2010), OLD-l, Labview (already set in the laboratory computer) and OLD-ll. In table 3.6 the values of the constants are shown Old-I Labview Old-II Gain Offset Gain Offset Gain Offset Gain Offset Discharge Discharge Table3.6: Calibration constants The potential differences of the discharge meters at zero flow velocity are measured (centrifugal pump is switched off) during 20 seconds. With the calibration constants from table 3.6 a flow velocity of 0 m/s should be derived. Table 3.7 present the average potential differences of the discharge meters during 20 seconds at a flow velocity of 0 m/s. Table 3.8 presents the average discharge according to the different calibration constants. Average pot. Diff. [Volt] Discharge Discharge Table 3.7:Average potential difference discharge meter 1 and 2 at a flow velocity of 0 m/s Average discharge 1 Average discharge 2 [m/s] [m/s] Old-I Labview Old-II Table 3.8:Average discharges according to different calibration constants In figures 3.6 and 3.7 for both discharge meters the flow velocity of 0 m/s is plotted as function of time. These figures show that the best calibration constant for discharge meter 1 is 2010 and for discharge meter 2 is Old-ll. According to these calibration constants the flow velocity is the closest to 0 m/s. 41

42 Flow velocity 0 m/s [m/s] Flow velocity 0 m/s [m/s] 0,20 Discharge meter 1 0,15 0,10 0,05 0,00-0,05-0, Old-I Labview Old-II -0,15-0, Time [sec] Figure 3.6: Discharge meter 1: flow vel. at 0 m/s vs time 0,20 Discharge meter 2 0,15 0,10 0,05 0,00-0,05-0, Old-I Labview Old-II -0,15-0, Time [sec] Figure 3.7:Discharge meter 2: flow vel. at 0 m/s vs time Figure 3.8 shows the flow velocity through the measurement section (by-pass closed) given by both discharge meters with the chosen calibration constants. Both discharge meters give the same values of velocity at low flow velocities. The differences of the discharge meters start increasing with the velocity. 42

43 Flow Velocity (m/s) 11,00 10,00 9,00 8,00 7,00 6,00 5,00 4,00 3,00 2,00 1,00 0,00 Comparison Discharge Meters Time (sec) Discharge Meter 1 Discharge Meter 2 Figure 3.8: Comparison discharge meters Radioactive meter The calibration of the radioactive concentration meter (RA-meter) is executed by calibration with two different materials (water and glass-water mixture). A certain density will be related to a certain potential difference. From two different densities with the corresponding potential difference a calibration constant is derived. The RA-meter is installed on a square container which has the same dimensions as the measurement section (Figure 3.9). First the container is filled with water and the potential difference of water is measured using the RA-meter. This measurement is repeated three times. The results are given in table 3.9. Figure 3.9: Calibration RA-Density meter The density of the water is theoretically derived as function of the water temperature by McCutcheon S.C., Martin J.L., Barnwell T.O. (1993): 43

44 in which: ρ [37] = Density of water as a function of temperature [kg/m^3] T = Temperature of water [ 0 C] Voltage Density Water (theory) Temperature [Volt] [kg/m 3 ] C Table 3.9: Derivation of density water Secondly a block of glass of which the density is known is placed in the container and the potential difference of the glass-water mixture is measured. The density of the glass-water mixture is theoretically derived as follows: ρ ρ ρ [38] in which: ρ = Density of glass-water mixture [kg/m^3] ρ = Density of water [kg/m^3] ρ = Density of glass [kg/m^3] = Thickness of water [m] = diameter inner side of container [m] = Thickness of glass [m] The result is given in table 3.10 Thickness Density [Meter] [kg/m 3 ] glass water Measurement section Glass-Water mixture Table 3.10: Derivation of density glass-water mixture With the density of water and glas-water mixture and their corresponding average potential difference a calibration constant of the RA-Density meter is derived. The results are given in figure 3.10 and table

45 Density [kg/m 3 ] Calibration RA-Density Meter y = 206,09x + 418, ,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 Voltage [Volt] Figure 3.10: Calibration RA-Density meter Gain Offset RA-Density meter Table 3.11: Calibration RA-Density meter Height gauge of the radioactive concentration meter The height gauge of the radioactive concentration meter give a potential difference in volt which is converted with a calibration constant into height in meter. The calibration of the height gauge of the radioactive concentration meter is executed by measuring the potential difference at different heights. A certain height gives a certain output in volt. By setting the height gauge at different heights and measuring the potential differences a calibration constant is derived. The height gauge is set at five different heights (Figure 3.11) where the potential difference is measured by the laboratory computer during 3 seconds. Table 3.12 gives the average potential difference (Volt) in relation with the different heights of the height gauge. 45

46 Figure 3.11: Overview height gauge radioactive concentration meter Potential difference Height RA-meter above inner lower part ms. below ruler [Volt] [Volt 2 ] [mm] [mm] Table 3.12: average potential difference in relation with different height of height gauge RA-meter The results of the average potential differences in relation with the different heights are plotted in figure In table 3.13 the calibration constants of the height gauge of the radioactive meter are given. 46

47 Height RA-meter above bottom inner part measurement section [mm] Height gauge RA-meter vs potential difference y = -12,496x 2-43,43x + 262,45 0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 Potential difference [Volt] Figure 3.12: Height gauge RA-meter vs potential difference Regression: height RA-meter versus potential difference a b c Table 3.13: Calibration constants height gauge RA-meter 3.4 Determination of height EMS meter The EMS measures the flow velocity during calibration of the conductivity probes and during erosion of the sand bed. The EMS meter is installed in a cylindrical tube above the measurement section (Figure 3.13 and Figure 3.14). The EMS meter is clamped at the top and bottom of the cylindrical tube so that it does not cause vibrations during high flow velocities. To measure the accurate flow velocity during calibration of the conductivity probes the sensor of the EMS meter need to be placed in the middle of the height of the measurement section. During erosion of the sand bed the EMS meter need to be placed in the middle of the sand bed. This is about at a fourth part of the height of the measurement section from the bottom of the measurement section. The construction of the cylindrical tube was constructed in such a way that it was not possible to clamp the EMS meter at the top and bottom of cylindrical tube (Figure 3.12). To prevent vibrations of the EMS meter the construction of the cylindrical tube was adjusted to clamp the EMS meter at both locations. 47

48 Figure 3.13: EMS meter old situation In the new situation the cylindrical tube is mounted 45 mm higher and is 75 mm shortened (Figure 3.13). Figure 3.14: EMS meter new situation 48

49 3.5 Analysis increase of density The erosion of the sand bed is measured in the measurement section of the test setup. The sand is eroded by opening the valve to the measurement section. After opening this valve water flows into the measurement section, leading to erosion of the sand. During all tests performed in 2010 a difference in density is measured before and after opening of the valve to the measurement section. The density is measured with the conductivity probes at the lexan plate. Figure 3.15 presents the result of an erosion test performed in The density of the sand bed during erosion in relation with time is presented in this figure. The density of the sand bed decreases before it starts eroding at the moment the valve is opened. An explanation for the difference in density before and after opening the valve to the measurement section is that there is an expansion of the lexan plate during the test due to the pressure build-up in the system. Due to the expansion of the lexan plate a water film arises between the sand bed and the conductivity probes which influence the measurement. Figure 3.15: result erosion test Density as a function of Time Because of the expansion of the lexan plate a waterfilm is formed between the sand bed and the expanded lexan plate. It is assumed that the expansion of the lexan plate is equal to the thickness of the waterfilm and that the signal of the conductivity probes is circular (Figure 3.16). The relation between the expansion of the lexan plate and the imaginary change of the measured density is analysed on two different ways: The signal of the conductivity probes is linear following the circumference of a half circle; The signal of the conductivity probes is defined by the surface of a half circle. 49

50 Figure 3.16: Signal of conductivity probes is circular In the case that the signal of the conductivity probes follows the circumference of a half circle the theoretical influence of the expansion of the lexan plate on the measured density is derived as follows: in which: ρ 1 = density of sediment before opening valve [kg/m³] ρ 2 = density of sediment after opening valve [kg/m³] ρ w = density of water [kg/m³] r = radius of circle (half distance between conductivity probes) [m] = 0,004 [m] d = expansion of lexanplate (thickness of waterfilm) [m] In the case that the signal of the conductivity is defined by the surface of a half circle the theoretical influence of the expansion of the lexan plate on the measured density is derived as follows: 50

51 Expansion lexan plate [meters] During the first series of test (2010) the erosion was measured at different flow velocities. One of the flow velocities during these tests was 1.5 m/s(test nr.1). Based on the measured densities with the conductivity probes the expansion of the lexan plate is calculated with the equations presented above. Figure 3.17presents the calculated expansion of the lexan plate as function of the distance above the bottom of the measurement section.densityρ 1 andρ 2 are based on the results of test nr. 1. Figure 3.17shows that according to both calculation methods the expansion of the lexan plate increases with the vertical distance. The reference point in the vertical direction is the bottom of the measurement section where z = 0 [m]. The vertical distance is positive in the upward direction. The reference point in horizontal direction is where the test setup is located at the split of the by-pass and measurement section where x = 0 [m]. The horizontal direction is positive in the direction of the measurement section. The centre of the lexan plate in vertical distance is at z = m. 0,0003 0,00025 Flow velocity 1,5 m/s 0,0002 0, ,0001 Surface Linear 0, ,01 0,03 0,05 0,07 0,09 0,11 0,13 Vertical distance [meters] Figure 3.17: calculation expansion lexan plate Surface vs Linear on the basis of the measured density change of test nr. 1 The expansion of the lexan plate is also measured manually. This is done by using a dial gauge. The expansion of the lexan plate is measured during different flow velocities and at different distances in vertical and horizontal direction. Figure3.18 depicts the location of the measurement points at the lexan plate. All the measurements are made on the right side of the lexan plate. 51

52 Figure 3.18: location measurement points lexan plate First the expansion of the lexan plate is measured in the middle of the lexan plate (z = m) with different flow velocities. This is at the intersection of the blue lines in figure 3.18.First the water flow is led through the by-pass of the test set-up and the valve which leads the flow to the measurement section is kept closed. The expansion of the lexan plate is measured before opening the valve to the measurement section. After opening the valve to the measurement section the expansion of the lexan plate is measured again. The difference in expansion of the lexan plate before and after opening the valve to the measurement section is considered as the expansion during the erosion tests. The results of the measured expansion at different flow velocities are presented in table 3.14 and figure Q 1 in table 3.14 is the flow velocity measured by the discharge meter outside of the measurement section. Q 2 is the flow velocity measured by the discharge meter in the measurement section. Figure 3.18 depicts that higher flow velocities, causing an increase of the pressure loss increases, lead to a larger expansion of the lexan plate. Flow Velocity flow through by-pass Flow velocity Flow velocity Q 1 Q 2 Location: X= m, Y= m, Z= m Expansion flow through MS/by-pass Flow velocity Flow velocity Expansion Q 1 Q 2 Expansion difference [m/s] [m/s] [m/s] [m] [m/s] [m/s] [m] [m] Table 3.14: expansion lexan plate at different flow velocities 52

53 Expansion lexan plate [meters] 0,012 0,010 0,008 0,006 0,004 0,002 0,000 Vertical distance: 0,144 m Flow Velocity [m/s] Figure 3.19: Expansion lexan plate at different flow velocities The expansion of the lexan plate is also measured at different vertical and horizontal locations on the lexan plate. The water flow velocity is kept constant at 4 m/s. The results are presented in table 3.15 and 3.16 and in figure 3.20and figure The expansion of the lexan plate increases in the direction of the flow. This is depicted by figure Flow velocity: 4 m/s Location: X= m, Y= m Vertical distance flow through by-pass flow through MS/by-pass Flow Flow velocity Flow velocity Flow velocity Expansion z-coordinate Expansion Expansion velocity Q 1 Q 2 Q 1 Q 2 difference [m] [m/s] [m/s] [m] [m/s] [m/s] [m] [m] Table 3.15: Expansion lexan plate in vertical distance (meters) 53

54 Expansion lexan plate [meters] Expansion lexan plate [meters] x-coordinate flow through by-pass Flow velocity Flow velocity Expansion Q 1 Q 2 Flow velocity: 4 m/s Location: Y= m, Z= m flow through MS/by-pass Flow velocity Flow velocity Expansion Q 1 Q 2 Expansion difference [mm] [m/s] [m/s] [m] [m/s] [m/s] [m] [m] Table 3.16: Expansion lexan plate in horizontal distance (meters) Flow velocity: 4 m/s 1,20E-04 1,00E-04 8,00E-05 6,00E-05 4,00E-05 2,00E-05 0,00E+00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,17 0,22 Vertical distance [meters] Figure 3.20: Expansion lexan plate in vertical distance 1,20E-04 1,00E-04 8,00E-05 6,00E-05 4,00E-05 2,00E-05 Flow velocity: 4 m/s 0,00E+00 4,97 4,99 5,03 5,05 5,07 5,09 5,11 5,13 5,17 Horizontal distance [meters] Figure 3.21: Expansion lexan plate in horizontal distance 54

55 Expansion lexan plate [meters] In figure 3.22the derived and measured expansion of the lexan plate are compared. The derivation of the expansion of the lexan plate is done where the flow velocity before erosion was set at 5 m/s (test nr. 3). During the manually measured expansion the flow velocity was set at 4m/s. When the vertical distance increase the expansion of the lexan plate increases in the same proportion. The pressure surges on the lexan plate are not fully comparable, because during test number 3 the measurement section was filled with a sand bed and during the manually measured expansion with water. The results of the derived expansion, where it is assumed that the signal of the conductivity probes is defined by the surface of a half circle, are close to the results of the measured expansion. From these results we can validate the assumption that difference in density which is measured before and after opening the valve on the measurement section is caused by the expansion of the lexan plate. Also can be concluded that the signal of the conductivity probes is defined by the material in the area defined by the surface of a half circle between both probes. 0,0004 0, ,0003 0, ,0002 0, ,0001 0, ,000 0,050 0,100 0,150 0,200 0,250 Vertical distance [meters] Manual Measured Linear Surface Figure 3.22: Comparison between derived and measured expansion lexan plate The expansion of the lexan plate is decreased by use of clamps. Clamps are pressing the lexan plate in such a way that the lexan plate does not expand by opening the valve of the measurement section during erosion of sand. After installing clamps the expansion of the lexan plate is measured again. This measurement is repeated two times (measurement 1 and measurement 2) at the concentration meters which are used during the tests. The results of this test are given in table 3.17 and figure In this table and figure a comparison is made between the expansion of the lexan plate before and after installing clamps. In figure 3.23 is shown that the expansion of the lexan plate is decreased by use of clamps. In the test performed in 2012 there is less difference in density before and after opening the valve to the measurement section (figure 3.24). 55

56 Expansion Lexanplate [meters] Conc. meter Height conc. Before installing clamps Flow vel. 4 m/s After installing clamps Flow vel. 5 m/s measurement 1 measurement 2 [-] [meter] [meter] [meter] A B C D E F G H I J K L M N O P Table 3.17:Expansion lexan plate before and after installing clamps 0,0004 0, ,0003 0,00025 Linear Surface 0,0002 0, ,0001 0, ,000 0,050 0,100 0,150 0,200 0,250 Vertical distance [meters] Figure 3.23: Expansion lexan plate before and after installing clamps Manual Measured (before installing clamps) measurement 1 (after installing clamps) measurement 2 (after installing clamps) 56

57 Density [kg/m 3 ] Test 30 : Density vs. time (erosion) ,0 2,0 4,0 6,0 8,0 10,0 12,0 14,0 16,0 18,0 Time [sec] Figure 3.24: result erosion test Density as a function of time after installing clamps 57

58 4 Analysis results laboratory experiments In 2010 the first series of the laboratory experiments 17 tests were performed (test number 1 to test number 17). Before the second series of experiments, which were performed in July 2012, the laboratory set-up was prepared. To check all the instruments and sensors are working correct a series of tests were performed (test number 18 to test number 23). The final test series of July 2012 consist of 36 tests (test number 24 to test number 60). In this chapter the results of the laboratory experiments are discussed. First a description of the test program and an explanation of how the results are analysed will be given (paragraph 4.1). The influences of the flow velocity, density, bed shear stress and grain size on the erosion of sand are discussed in paragraph 4.2 to paragraph 4.4. In paragraph 4.5 the pick-up flux is discussed. In paragraph 4.6 a comparison is made between the theories mentioned in chapter 2 and the measured data. In paragraph 4.7 a description of images of the high speed camera is made. The laboratory experiments are executed with two kinds of sand with different grain size (D 50 ). Table 4.1 gives an overview of the types of sand their properties. Type of sand Grain Size Density Porosity [-] D 50 D 60 D 10 D 60 /D 10 ρ dmin ρ dmax n max n min [mm] [mm] [mm] [-] [kg/m 3 ] [kg/m 3 ] [-] [-] Geba Zilverzand M Description of test program and analysis Table 4.1: Sand types Before performing tests the silo is filled with sand. This is done adding sand in the silo while water flows through the circuit and silo till the density of the sand-water mixture has reached approximately 1500 kg/m 3. The test program is set-up in five phases and the phases are called A till E. The first (A), second (B) and fourth (D) phase are used to calibrate the conductivity probes. In the third (C) phase a sand bed is created by sedimentation and in the fifth (E) phase the erosion of sand takes place. The results of all the phases (phase A to phase E) of the laboratory experiment will be described on basis of one test (test number 26). The rest of the tests are analyzed in the same way. Phase A During phase A the potential difference is measured with water in the measurement section and density with RA-meter at a level of mm above the bottom of the measurement section. This is executed with a flow velocity of 0 and 4 m/s. In this situation all the conductivity probes on the lexan plate measure the same density. Before executing the tests in this phase the following actions on the test equipment are done: density of water is approximately 1000 kg/m 3 by-pass is closed measurement section is open valve 1 is open (silo) valve 2 is open (system) valve 4 is closed (to silo) turn on RA meter 58

59 wait 3 minutes after turning on RA-meter check if signals conductivity probes are working check position of RA-meter The measurement computer gives an indication of the out coming values. The measurement frequency of the tests is 10 Hz and the log time is set on 30 seconds. The measurement computer logs the measured files as ascii-files. The names of the logged files in this phase are 26A1 and 26A2. The first digits in the filename indicate the test number and the letter indicates the test phase. The second digit indicates the part in the test phase. Phase B During phase B the potential difference is measured while flowing with a sand-water mixture of approximately 1400 kg/m 3. The measurement of the potential difference of the conductivity probes is based on the average of 10 tests and the measurement of the density with the RA-meter at 10 different heights (same height as 10 concentration meters). The conductivity probes which are used from test 27 to calibrate during this phase are given in table 4.2. Before test 27 other heights were used. (Figure 4.1 and table 4.3) Location RA-meter Conc Height Ruler RA-meter Height Bottom Innerside MS Location RA-meter Conc Height Ruler RA-meter Height Bottom Innerside MS [nr.] [nr.] [mm] [mm] [nr.] [nr.] [mm] [mm] 1 A I B K D N E O G P Table 4.2: Approximate height of density measurement with RA-meter and related conductivity probes in phase B from test 27 Before executing the tests in this phase sand is added to the system and the following actions were executed: Open the measurement section Close the bypass Set flow velocity on 4 m/s Open Valve 1 (silo) Open Valve 4 (to silo) Add Sand to the system Open Valve 2 (system) Close Valve 4 (silo) The measurement frequency of the tests in phase B is set on 5 HZ with log time of 30 seconds. The logged files are named from 26B1 to 26B10. The results of test 26 are given in figure 4.2 where the density at the same height as conductivity probe A is much higher than the average density of 1400 kg/m 3. An explanation of difference in 59

60 Density [kg/m3] density is that the measurements with the RA-meter at the corresponding height of conductivity probe A are influenced by the thickness and shape of the measurement section. Therefore conductivity probe A is not taken into account for the calibration of the sand-water mixture (figure 4.2). From test 27 the location of probe A during phase B and phase D is changed to 19 mm from the bottom of inner side measurement section (figure 4.1 and table 4.3). Figure 4.1: New location of connected conductivity probe from test 27 Location RA-meter Conc Height Ruler RA-meter Height Bottom Innerside MS [nr.] [nr.] [mm] [mm] 1 A Table 4.3: New location of connected conductivity probe from test A Calibration Conductivity Probes (phase B) B D E G I K N O P Height [mm] Figure 4.2: Calibration conductivity probes (phase B) 60

61 Density [kg/m3] B Calibration Conductivity Probes (Phase B) D E G I K N O P Height [mm] Figure 4.3: Calibration conductivity probes (phase B) On the basis of a regression equation between the measured density and height of the RA-meter the density at the height of all conductivity probes was calculated. For test 26 the regression equation for the test with the sand-water mixture was: in which y = -1E-08x 5 + 7E-06x x x x [39] y = density of the sand-water mixture (kg/m 3 ) x = height from bottom inside measurement section (m) The density at the height of the conductivity probes A to P were derived with equation number39. The results are given in table

62 Conductivity Probe Height Conductivity Probe Density [-] [mm] [kg/m 3 ] A B C D E F G H I J K L M N O P Table 4.4: Calculated density at the height of the conductivity probes (phase B) Phase C Test 24 and 25 are used to determine the effect of the different methods for sedimentation and densification on the density of the sand bed. During these tests sand with grain size (D 50 ) 262 m was used. During test 24 the sand is sedimented with flow velocity of 0.75 m/s and the density of the sand bed is measured. After this the sand bed is densified with a hammer by slashing ten times on the measurement section after every 0.1 meter in horizontal direction. During test 25 the same procedure is repeated where the flow velocity during sedimentation is 0 m/s. In figure 4.4 is shown that the different methods for sedimentation and densification give difference of density of the send bed. During execution of tests with grain size (D 50 ) 262 m these different methods for sedimentation and densification of the sand bed are used. During tests with grain size (D 50 ) 125 m the sand bed is sedimented with flow velocity of 0.5 m/s. 62

63 Height [mm] Relative density after sedimentation (D 50 = 262 m) Flow velocity 0.75 m/s Flow velocity 0.75 m/s incl. hammering Flow velocity 0 m/s Flow velocity 0 m/s incl. hammering Figure 4.4: Comparison methods of densification sand bed In phase C the sand bed is created by (partly) closing the butterfly valve to the measurement section. The butterfly valve to the bypass remains open. Three different methods were used to create a difference of the density of the sand bed for different tests. For the sand type Zilverzand M62 the following methods for sedimentation and densification were used: sedimentation at a flow velocity of 0 m/s (butterfly valve fully closed); 0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Relative density [-] sedimentation at a flow velocity of 0.75 m/s (butterfly valve partly closed); sedimentation at a flow velocity of 0.75 m/s including hammering on the measurement section. For the sand type Geba the following methods for sedimentation are used: sedimentation at a flow velocity of 0 m/s (butterfly valve fully closed); sedimentation at a flow velocity of 0.5 m/s (butterfly valve partly closed); sedimentation at a flow velocity of 0.5 m/s including hammering on the measurement section. Before executing the tests in phase C the following actions on the test equipment are executed: open the Bypass set flow velocity by pass at 3 m/s open valve 1 (silo) open valve 2 (system) close valve 4 (to silo) close measurement section check position of RA meter The measurement frequency of the tests in phase C is set on 10 Hz with a log time of 60 seconds. The logged file is named as 26C. In this thesis the sedimentation process is not analyzed. 63

64 Phase D After sedimentation the remaining sand is removed from the system. This is executed by opening valve 4 to silo and closing valve 2 to the system. In phase D the potential difference of the sedimented sand bed is measured. The measured potential difference of all concentration meters is based on the average of 10 tests and the measurement of the density with the RA-meter at 10 different heights (same height as 10 concentration meters). In table 4.5 the used concentration meters in this phase are given from test 27. Before test 27 other heights were used. (Figure 4.1 and table 4.3) Location RA-meter Conc Height Ruler RA-meter Height Bottom Innerside MS Location RA-meter Conc Height Ruler RA-meter Height Bottom Innerside MS [nr.] [nr.] [mm] [mm] [nr.] [nr.] [mm] [mm] 1 A H B J D L E N G O Table 4.5: Approximate height of density measurement with RA-meter and related concentration meters in phase D from test 27 Before executing the tests in phase D the following actions on the test equipment are done: open bypass open valve 1 (from silo) open valve 4 (to silo) close valve 2 (system) remove sand from system turn off pump close valve 4 (to silo) open valve 2 (system) The measurement frequency of the tests in phase D is set on 5 Hz with a log time of 30 seconds. The logged files are named from 26D1 to 26D10. The results are presented in figure 4.5. The results at the same height as conductivity probes A, L, N and O differ from the average sand bed density of 1900 kg/m 3. An explanation of difference in density is that the measurements with the RA-meter at the corresponding height of conductivity probe A are influenced by the thickness and shape of the measurement section. The measurements at the height of conductivity probes L, N and O were executed above the sand bed. These heights are not taken into account for the calibration of the sand bed (figure 4.6). 64

65 Density [kg/m3] Density [kg/m3] Calibration Conductivity Probes (Phase D) A B D E G H J L N O Height [mm] Figure 4.5: Calibration conductivity probes (phase D) Calibration Conductivity Probes (Phase D) B D E G H J Height [mm] Figure 4.6: Calibration conductivity probes (phase D) On the basis of figure 4.6 a regression equation is determined for the relation between the density and the height in the measurement section. This equation is used to calculate the density for each conductivity probe: y = -5E-07x x x x x [40] in which y = density of the sand bed (kg/m 3 ) x = height from bottom inside measurement section (m) 65

66 Table 4.6 presents the calculated density for each conductivity probe. In this case the regression equation could only be used up to conductivity probe J, because these probes were covered by the sand bed in this test. Conductivity Probe Height Conductivity Probe Density [-] [mm] [kg/m3] A B C D E F G H I J K L M N O P Table 4.6: Calculated density at the height of the conductivity probes (phase D) During phase A, B and D the potential difference (volt) of the conductivity probes and densities at the height of the conductivity probes with the RA-meter are measured. With these potential difference and the (partly calibrated) measured densities a relation between the density and measured potential difference for each conductivity probe could be derived (figures 4.7 to 4.10). The conductivity probes M to P were not in the sand bed during phase D. To calibrate the conductivity probes M to P only two points are used. Table 4.7 gives an overview of the calibration constants of conductivity probes A to P. 66

67 Density [kg/m3] Density [kg/m3] Density [kg/m3] Calibration Constants Probe A to D a = x b = x c = x d = x a b c 1200 d Voltage concentration meter [Volt] Figure 4.7: Result calibration constants probe A to D Calibration Constants Probe E to H e= x f = x g = x h= x e f g 1200 h Voltage concentration meter [Volt] Figure 4.8: Result calibration constants probe E to H Calibration Constants Probe I to L i = x j = x k = x l = x i j k 1200 l Voltage concentration meter [Volt] Figure 4.9: Result calibration constants probe I to L 67

68 Density [kg/m3] Calibration Constants Probe M to P m = x n = x o = x p = x m n o 1200 p Voltage concentration meter [Volt] Figure 4.10: Result calibration constants probe M to P Conductivity Probe Calibration Constant [-] Gain Offset A B C D E F G H I J K L M N O P Table 4.7: Result calibration constants probes A to P Phase E In phase E the sand bed is eroded. During the test program tests were executed with different flow velocities and different flow densities of 1000 kg/m 3, 1200 kg/m 3 and 1400 kg/m 3. First the flow passes the bypass to reach a certain flow velocity. The erosion test is executed by opening the butterfly valve to the measurement section. Now the flow passes the sand bed and erosion takes place. The measurement instruments in the measurement section collect all the data. The measurement frequency and log time depends on the expected erosion rate during the test. 68

69 Flow velocity [m/s] Pressure loss [Pa] During test 26 the sand bed is eroded with density of 1000 kg/m 3 and flow velocity of approximately 2 m/s. The measurement frequency was set on 50 Hz with a log time of 120 seconds. Figure 4.11 depicts the average flow velocity and the pressure loss in the measurement section. After approximately 3 seconds the valve to the measurement section is opened and the flow velocity starts increasing. The flow velocity in the measurement section accelerates till the velocity of approximately 2 m/s is reached in 20 seconds. On the secondary vertical axes the pressure losses in the measurement section measured by HDP1 and HDP2 are given. HDP1 and HDP2 measure respectively the pressure loss at the distance of 0.89 meter and 2.07 meter (see chapter 3.2). Test 26: Flow velocity vs. Density vs. time (ero.) 2, ,00 1,50 1,00 0, ,00 Average flow velocity (measurement section) [m/s] HDP1 HDP2-0, ,0 20,0 40,0 60,0 80,0 100,0 120,0 Time [sec] Figure 4.11: Average flow velocity and pressure loss (test 26) 0 With the calibration constants from table 4.6 the density at each conductivity probe is derived. The density at the conductivity probes in relation with time is plotted in figure In this figure is shown that the density at conductivity probes A to E remains approximately the same after 120 seconds. This means that the sand bed is not fully eroded. The possible cause is that the flow velocity remained below critical at conductivity probes A to E. The density at conductivity probe F is decreasing rapidly after approximately 16 seconds and is increasing again after approximately 27 seconds. The possible cause is that the sand is sedimented again due to decrease of the flow velocity below critical at the height of conductivity probe F. Conductivity probes L to P remained above the sand bed before erosion. The sand bed is only eroded at conductivity probes G to K. The erosion velocity at each conductivity probe is manually determined. The criteria for the erosion moment is set manually were the density starts decreasing and is still above the wet density of the sand bed (dots in figure 4.12 and figure 4.13). The difference in height between the conductivity probes is known. The erosion velocity is determined by dividing the height difference of the conductivity probes by the difference in time of the erosion moments at both probes. For the analysis of the result the average effective flow velocity is used which is based on the total height of the measurement section. The effective flow velocity is calculated as follows: 69

70 Density [kg/m 3 ] [41] In which = average effective flow velocity (m/s) U av = average flow velocity (m/s) H ms = height measurement section (m) H Con(av) = average height conductivity probe (m) The results are given in table 4.8. Test 26: Density vs. time (ero.) ,0 20,0 40,0 60,0 80,0 100,0 120,0 Time [sec] a b c d e f g h i j k l m n o p ρmin Figure 4.12: Test 26 Density vs Time Erosion 70

71 Density [kg/m 3 ] Test 26: Density vs. time (ero.) ,0 14,0 16,0 18,0 20,0 Time [sec] a b c d e f g h i j k l m n o p ρmin Figure 4.13: Erosion moments test 26 conductivity probe Time eroded Height Conductivity Probe Erosion velocity (V e ) [-] [sec] [meter] [m/s] a b c d e f g h i j k l m n o p Flow velocity (average) (U av ) [m/s] Effective Flow velocity (average) (u eff(av) ) [m/s] Table 4.8: Test 26 Erosion velocities 71

72 Erosion Rate (Ve) [m/s] 4.2 Influence flow velocity In this paragraph the influence of the flow velocity (U)on the erosion rate (Ve) with different densities of the sand bed and eroding flow is discussed. In paragraph the results with grain size (D 50 ) 262 m are presented and in paragraph the results with grain size (D 50 ) 125 m Grain size (D 50 ) 262 m Figure 4.14 shows the influence of the flow velocity with different methods of densification of the sand bed. In each method of densification the results are based on different tests. The grain size (D50) is 262 m and density eroding flow 1000 kg/m3. The erosion rate increase with the flow velocity. The method of densification of the sand bed influences the erosion rate. In figure 4.14 the influence of the density above the sand bed (c b ) is not included which probably cause a large spread in the results. The influence of the density above the sand bed (c b ) will be discussed from paragraph ,09 Flow velocity vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 262 µm 0,08 0,07 0,06 0,05 0,04 0,03 Method of densification: Flow velocity 0 m/s Method of densification: Flow velocity 0.75 m/s Method of densification: Flow velocity 0.75 m/s and hammering 0,02 0,01 0,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 Flow Velocity (Vs) [m/s] Figure 4.14: Flow velocity vs erosion rate (D 50 =262µm) In figure 4.15 the influence of the flow velocity and relative density on the erosion rate is plotted for different relative densities of the sand bed: [0-0.30] [ ] [ ] According to the results in figure 4.15 the conclusion can be made that with increase of relative density of the sand bed the erosion rate decreases. 72

73 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] 0,08 Flow velocity vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 262 µm 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 Flow Velocity (Vs) [m/s] Relative Density [0-0.30] [ ] [ ] Figure 4.15: Flow velocity vs erosion rate with relative density sand bed (D 50 =262µm) In figure 4.16 the influence of the flow velocity on the erosion rate is shown with different densities of the eroding flow. The erosion rate increases with the flow velocity. At flow velocities between 2.5 m/s and 4.0 m/s the erosion rate is decreasing with the density of the eroding flow. 0,09 Flow velocity vs Erosion Rate Method of densification: flow velocity 0.75 m/s Grain size D 50 = 262 µm 0,08 0,07 0,06 0,05 0,04 0,03 0,02 Density Eroding flow: 1000 kg/m3 Density Eroding flow: 1200 kg/m3 Density Eroding flow: 1400 kg/m3 0,01 0,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 Flow Velocity (Vs) [m/s] Figure 4.16: Flow velocity vs erosion rate (D 50 =262 µm) In figures 4.17 and 4.18 the influence of density of the eroding on the erosion rate is plotted again where the density of the sand bed is standardized in the following relative densities: 73

74 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] [ ] [ ] [ ] According to the results in figure 4.17 and4.18 the conclusion can be made that the erosion rate is decreasing with density of the eroding flow. 0,10 Flow velocity vs Erosion Rate Method of densification: flow velocity 0.75 m/s Grain size D 50 = 262 µm 0,08 0,06 0,04 0,02 0,00 2 2,5 3 3,5 4 4,5 5 5,5 6 Flow Velocity (Vs) [m/s] Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 [ ] Relative [ ] [ ] [ ] Figure 4.17:Flow velocity vs erosion rate with relative density sand bed (D 50 =262µm) 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 Flow velocity vs Erosion Rate Method of densification: flow velocity 0.75 m/s Grain size D 50 = 262 µm 0,00 2,00 2,50 3,00 3,50 4,00 4,50 Flow Velocity (Vs) [m/s] 5,00 5,50 6,00 Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 Relative [ ] [ ] [ ] Figure 4.18:Flow velocity vs erosion rate with relative density sand bed (D 50 =262µm) 74

75 Erosion Rate (Ve) [m/s] Grain size (D 50 ) 125 m Figure 4.19 shows the influence of the flow velocity with different methods of densification of the sand bed. The grain size (D50) is 125 m and density eroding flow 1000 kg/m3. The erosion rate increase with the flow velocity. The method of densification of the sand bed influences the erosion rate. 0,09 Flow velocity vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 Flow Velocity (Vs) [m/s] Figure 4.19: Flow velocity vs erosion rate (D 50 =125µm) Method of densification: Flow velocity 0 m/s Method of densification: Flow velocity 0.5 m/s Method of densification: Flow velocity 0.5 m/s and hammering In figure 4.20 the influence of the flow velocity on the erosion rate is plotted again where the density is standardized in the following relative densities: [0-0.20] [ ] [ ] [ ] According to the results in figure 4.20 the conclusion can be made that the erosion rate increases with flow velocity and decreases with relative density of the sand bed. 75

76 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 Flow velocity vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm 0,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 Flow Velocity (Vs) [m/s] Relative Density [0-0.20] [ ] [ ] [ ] Figure 4.20: Flow velocity vs erosion rate with relative density sand bed (D 50 =125µm) In figure 4.21 the influence of the flow velocity on the erosion rate is shown with different densities of the eroding flow. The erosion rate increases with the flow velocity where density of eroding flow is 1000 kg/m 3 and 1200 kg/m 2. The erosion rate remains the same with eroding flow of 1400 kg/m 3. 0,05 0,05 0,04 0,04 0,03 0,03 0,02 0,02 0,01 0,01 Flow velocity vs Erosion Rate Method of densification: flow velocity 0.5 m/s Grain size D 50 = 125 µm 0,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 Flow Velocity (Vs) [m/s] Figure 4.21: Flow velocity vs erosion rate (D 50 =125µm) Density Eroding flow: 1000 kg/m3 Density Eroding flow: 1200 kg/m3 Density Eroding flow: 1400 kg/m3 In figures 4.22 and 4.23 the influence of density of the eroding on the erosion rate is plotted again where the density of the sand bed is standardized in the following relative densities: 76

77 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] [0-0.20] [ ] According to the results in figures 4.22 and 4.23 the conclusion can be made that the erosion rate is decreasing with density of the eroding flow. The erosion rate remains the same where density eroding flow is 1400 kg/m 3. 0,050 0,045 0,040 0,035 0,030 0,025 0,020 0,015 0,010 0,005 Flow velocity vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm 0,000 1,50 2,00 2,50 3,00 3,50 4,00 4,50 Flow Velocity (Vs) [m/s] 5,00 5,50 6,00 Relative Density 1400 Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 [0-0.20] [0-0.20] [0-0.20] Figure 4.22:Flow velocity vs erosion rate with relative density sand bed (D 50 =125µm) 0,050 0,045 0,040 0,035 0,030 0,025 0,020 0,015 0,010 0,005 Flow velocity vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm 0,000 1,50 2,00 2,50 3,00 3,50 4,00 4,50 Flow Velocity (Vs) [m/s] 5,00 5,50 6,00 Relative Density Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 [ ] [ ] [ ] Figure 4.23: Flow velocity vs erosion rate with relative density sand bed (D 50 =125µm) 77

78 Pressure Loss/meter [Pa]/m 4.3 Influence bed shear stress In this paragraph the influence of the bed shear stress on the erosion velocity is discussed. The bed shear stress is determined on the basis of the measured pressure loss in the measurement section and the measured flow velocity. The pressure loss in the measurement section is measured by three differential pressure meters HDP1, HDP2, HDP3 (paragraph 3.2). Figure 4.24 depicts the pressure losses measured during execution of tests with water at different flow velocities. In this figure is shown that the results measured by HDP2 are the closest to the calculated pressure loss. For the calculation of the bed shear stress the results from pressure meter HDP2 are taken into account. In paragraph the results with grain size (D 50 ) 262 m are given and in paragraph the results with grain size (D 50 ) 125 m HDP1 HDP2 HDP3 Pressure loss calculated Flow velocity squared [m2/s2] Figure 4.24: Comparison of pressure losses The bed shear stress is calculated as follows (Cheng, 2005): ρ [42] [43] ρ ρ [44] ρ [45] 78

79 ρ [46] ρ ρ ρ ρ In which = average bed shear stress [Pa] = average bed friction factor based on the pressure loss measurements [-] ρ = density of water [kg/m 3 ] = average overall friction factor for the whole measurement section [-] = average flow height in the measurement section [m] = width measurement section [m] = theoretical friction factor according to Colebrook, C.F. (February 1939) ρ = pressure loss due to friction [Pa] measured [N] = average hydraulic diameter [m] = measured distance differential pressure meter HDP2 = average force loss due to friction [N] = total force acting on the fluid [N] = pressure loss due to the density gradient in time [N] = pressure loss due to the velocity gradient in time [N] = pressure loss due to the change in flow height in time [N] = force acting on the fluid over the length over which the pressure loss is 79

80 Erosion Rate (Ve) [m/s] Grain size (D50) 262 m Figure 4.25 shows the influence of the bed shear stress with different methods of densification of the sand bed. The grain size (D50) is 262 m and density eroding flow 1000 kg/m3. The erosion rate increase with the bed shear stress. 0,09 Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 262 µm 0,08 0,07 0,06 0,05 0,04 0,03 Method of densification: Flow velocity 0 m/s Method of densification: Flow velocity 0.75 m/s Method of densification: Flow velocity 0.75 m/s and hammering 0,02 0,01 0, Bed Shear Stress (τb) [Pa] Figure 4.25: Bed Shear Stress vs erosion rate (D 50 =262µm) In figure 4.26 the influence of the bed shear stress on the erosion rate is plotted again where the density is standardized in the following relative densities: [0-0.30] [ ] [ ] According to the results in figure 4.26 the conclusion can be made that the erosion rate increases with bed shear stress and decreases with relative density of the sand bed. 80

81 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] 0,08 Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 262 µm 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0, Bed Shear Stress (τb) [Pa] Relative Density [0-0.30] [ ] [ ] Figure 4.26: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) In figure 4.27 the influence of the bed shear stress on the erosion rate is shown with different densities of the eroding flow. The erosion rate increases with the bed shear stress. 0,09 Bed Shear Stress vs Erosion Rate Method of densification: flow velocity 0.75 m/s Grain size D 50 = 262 µm 0,08 0,07 0,06 0,05 0,04 0,03 0,02 Density Eroding flow: 1000 kg/m3 Density Eroding flow: 1200 kg/m3 Density Eroding flow: 1400 kg/m3 0,01 0, Bed Shear Stress (τb) [Pa] Figure 4.27: Bed Shear Stress vs erosion rate (D 50 =262µm) In figures 4.28 and 4.29 the influence of density of the eroding flow on the erosion rate is plotted again where the density of the sand bed is standardized in the following relative densities: 81

82 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] [ ] [ ] [ ] According to the results in figures 4.28 and 4.29 the conclusion can be made that the erosion rate is increasing with bed shear stress and decreasing with density of the eroding flow. 0,10 Bed Shear Stress vs Erosion Rate Method of densification: flow velocity 0.75 m/s Grain size D 50 = 262 µm 0,08 0,06 0,04 0,02 0, Bed Shear Stress (τb) [Pa] Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 [ ] Relative [ ] [ ] [ ] Figure 4.28: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0,00 Bed Shear Stress vs Erosion Rate Method of densification: flow velocity 0.75 m/s Grain size D 50 = 262 µm Bed Shear Stress (τb) [Pa] Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 Relative [ ] [ ] [ ] Figure 4.29: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) 82

83 Erosion Rate (Ve) [m/s] Grain size (D 50 ) 125 m Figure 4.30 shows the influence of the bed shear stress with different methods of densification of the sand bed. The grain size (D50) is 125 m and density eroding flow 1000 kg/m3. The erosion rate increase with the bed shear stress. The method of densification of the sand bed influences the erosion rate. 0,09 Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 Method of densification: Flow velocity 0 m/s Method of densification: Flow velocity 0.5 m/s Method of densification: Flow velocity 0.5 m/s and hammering 0, Bed Shear Stress (τb) [Pa] Figure 4.30: Bed Shear Stress vs erosion rate (D 50 =125µm) In figure 4.31 the influence of the bed shear stress on the erosion rate is plotted again where the density is standardized in the following relative densities: [0-0.20] [ ] [ ] [ ] According to the results in figure 4.31 the conclusion can be made that the erosion rate increases with bed shear stress and decreases with relative density of the sand bed. 83

84 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0,00 Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm Bed Shear Stress (τb) [Pa] Relative Density [0-0.20] [ ] [ ] [ ] Figure 4.31: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =125µm) In figure 4.32 the influence of the bed shear stress on the erosion rate is shown with different densities of the eroding flow. The erosion rate does not change with density of eroding flow. 0,05 Bed Shear Stress vs Erosion Rate Method of densification: flow velocity 0.5 m/s Grain size D 50 = 125 µm 0,04 0,03 0,02 0,01 Density Eroding flow: 1000 kg/m3 Density Eroding flow: 1200 kg/m3 Density Eroding flow: 1400 kg/m3 0, Bed Shear Stress (τb) [Pa] Figure 4.32: Bed Shear Stress vs erosion rate (D 50 =125µm) In figures 4.33 and 4.34 the influence of density of the eroding flow on the erosion rate is plotted again where the density of the sand bed is standardized in the following relative densities: [0-0.20] 84

85 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] [ ] According to the results in figures 4.33 and 4.34 the conclusion can be made that the erosion rate is decreasing with density of the eroding flow. The erosion rate remains the same where density eroding flow is 1400 kg/m3. Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm 0,050 0,045 0,040 0,035 0,030 0,025 0,020 0,015 0,010 0,005 0, Bed Shear Stress (τb) [Pa] Relative Density Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 [0-0.20] [0-0.20] [0-0.20] Figure 4.33: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =125µm) 0,045 0,040 0,035 0,030 0,025 0,020 0,015 0,010 0,005 0,000 Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Grain size D 50 = 125 µm Bed Shear Stress (τb) [Pa] Relative Density Density eroding flow: Density eroding flow: Density eroding flow: 1000 kg/m kg/m kg/m3 [ ] [ ] [ ] Figure 4.34: Bed Shear Stress vs erosion rate with relative density sand bed (D 50 =262µm) 85

86 Erosion Rate (Ve) [m/s] 4.4 Influence grain size Figure 4.35 shows the influence of the grain sizes (D 50 =262 m and D 50 =125 m) on the erosion rate. Method of densification is 0 m/s and density of eroding flow is 1000 kg/m 3. From figure 4.35 no difference on the erosion rate can be observed between the grain sizes. 0,100 0,090 0,080 0,070 0,060 0,050 0,040 0,030 0,020 0,010 0,000 Influence Grain Size Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m Bed Shear Stress (τb) [Pa] Figure 4.35: Influence grain size on erosion rate Grainsize D50 = 2.62E-04 µm Grainsize D50 = 1.25E-04 µm In figure 4.36 to figure 4.38 the influence of grain size on the erosion rate is plotted again where the density of the sand bed is standardized in the following relative densities: [0-0.20] [ ] [ ] The conclusion can be made that course sand (D 50 =262 m) has a higher erosion rate than fine sand (D 50 =125 m). 86

87 Erosion Rate (Ve) [m/s] Erosion Rate (Ve) [m/s] 0,100 0,090 0,080 0,070 0,060 0,050 0,040 0,030 0,020 0,010 0,000 Influence Grain Size Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Relative density: [0-0.20] Bed Shear Stress (τb) [Pa] Grainsize (D 50 ) 2.62E-04 µm 1.25E-04 µm Figure 4.36: Influence grain size on erosion rate with relative density [0-0.20] 0,100 0,090 0,080 0,070 0,060 0,050 0,040 0,030 0,020 0,010 0,000 Influence Grain Size Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Relative density: [ ] Bed Shear Stress (τb) [Pa] Grainsize (D 50 ) 2.62E-04 µm 1.25E-04 µm Figure 4.37: Influence grain size on erosion rate with relative density [ ] 87

88 Erosion Rate (Ve) [m/s] 0,100 0,090 0,080 0,070 0,060 0,050 0,040 0,030 0,020 0,010 0,000 Influence Grain Size Bed Shear Stress vs Erosion Rate Density eroding flow: 1000 kg/m3 Relative density: [ ] Bed Shear Stress (τb) [Pa] Grainsize (D 50 ) 2.62E-04 µm 1.25E-04 µm Figure 4.38: Influence grain size on erosion rate with relative density [ ] 88

89 4.5 Pick-up flux The erosion rate can be written as the result from settling flux (S) and pick-up flux (E) (equation 14 and 15): => S [47] [48] In which: = measured erosion rate [m/s] = measured concentration above sand bed [-] = settling velocity of a single grain in still water [m/s] = settling velocity of one particle [m/s] = measured density sand-water mixture above sand bed [kg/m 3 ] = density of water [kg/m 3 ] = density of sediment [kg/m 3 ] = hindered settling exponent [-] (equation 16) = shape factor (0,7) The concentration (c b ) is defined as the concentration just above the sand bed (Rhee, 2010), however the concentration of the eroding flow depends on the exact height above the sand bed. The influence of the concentration is studied by analysing the influence of the concentration on the calculated pick-up flux at three different levels above the eroding sand bed. These heights are respectively 10 mm, 20 mm and 30 mm. In figure 4.39, 4.40 and appendix AB-1 to AB-4 a comparison is made between the different concentrations above the sand bed where the density is divided in the following relative densities: [0 0.30] [ ] [ ] The density of the incoming eroding flow is 1000 kg/m 3 and the grain size (D 50 ) is 262µm. In figure 4.39 the pick-up flux (E) is plotted as function of the flow velocity (v s )and in figure 4.40 as function of the bed shear stress ( ). From these figures it can be observed that when the pick-up flux is corrected for the concentration of the flow at a level of 30 mm above the sand bed the least variation in the relation between the pick-up flux and flow velocity and bed shear stress appears. The tests with a sand bed with density of the incoming eroding flow of 1200 kg/m 3 and 1400 kg/m 3 give the same results (Appendix AB-5 to AB-8). For the incoming eroding flow of 1400 kg/m 3 the pick-up flux remains the same when the flow velocity increases. A possible cause is that the eroding flow is saturated with sand. The C b with 30 mm above sand bed is taken for further analysis in this chapter. 89

90 Pick-up Flux (E) [kg/sm 2 ] Pick-up Flux (E) [kg/sm 2 ] 100,00 Flow velocity vs Pick-up Flux Density eroding flow: 1000 kg/m3 Grainsize D50 = 262 µm 10,00 CB: 30 mm 1,00 0,10 2,00 2,50 3,00 3,50 4,00 4,50 Flow Velocity (Vs) [m/s] Relative Density CB 10 mm above SB CB 20 mm above SB CB 30 mm above SB [0-0.30] [0-0.30] [0-0.30] Figure 4.39: Influence concentration (C b ) on pick-up flux (relative density [0-0.30]) 100,00 Bed Shear Stress vs Pick-up Flux Density eroding flow: 1000 kg/m3 Grainsize D50 = 262 µm 10,00 CB: 30 mm CB: 20 mm 1,00 CB: 10 mm 0, Bed Shear Stress (τb) [Pa] Relative Density CB 10 mm above SB CB 20 mm above SB CB 30 mm above SB [0-0.30] [0-0.30] [0-0.30] Figure 4.40: Influence concentration (C b ) on pick-up flux (relative density [0-0.30]) 90

91 Pick-up Flux (E) [kg/sm 2 ] In figures 4.41 and 4.42 the influence of density of the sand bed on the pick-up flux is shown. The grain size (D 50 )is 262 µm. The density of the sand bed is divided in the following relative densities: [0 0.30] [ ] [ ] According to the results in figure 4.41 and 4.42 the conclusion can be made that the pick-up flux is independent of the density of sand bed. For relative density of [ ] and [ ] the pickup flux increases with flow velocity and bed shear stress. According to figures 4.41 and 4.42 the pickup flux remains the same where relative density of the sand bed is [0-0.30]. 100,00 Flow velocity vs Pick-up Flux CB above sand bed: 30 mm; Density eroding flow: 1000 kg/m3 Grainsize D50 = 262 µm 10,00 1,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 Flow Velocity (Vs) [m/s] Relative Density [0-0.30] [ ] [ ] Figure 4.41: Influence density sand bed on pick-up flux (grain size D 50 : 262 µm) 91

92 Pick-up Flux (E) [kg/sm 2 ] 100,00 Bed Shear Stress vs Pick-up Flux CB above sand bed: 30 mm; Density eroding flow: 1000 kg/m3 Grainsize D50 = 262 µm 10,00 1, Bed Shear Stress (τb) [Pa] Relative Density [0-0.30] [ ] [ ] Figure 4.42: Influence density sand bed on pick-up flux (grain size D 50 : 262 µm) In figures 4.43 and 4.44 the influence of density of the eroding flow on the pick-up flux is shown. The grain size (D 50 )is 262 µm. The following densities of the eroding flow are compared: 1000 kg/m kg/m kg/m 3 According to the results in figures 4.43 and 4.44 the conclusion can be made that the pick-up flux decreases with density of eroding flow and is independent of the relative density. For the eroding flow of 1200 kg/m 3 with relative density of [ ] the pick-up flux decreases with bed shear stress (figure 4.44). The influence of the density of eroding flow is the most visible where the density of the eroding flow is 1400 kg/m 3. 92

93 Pick-up Flux (E) [kg/sm 2 ] Pick-up Flux (E) [kg/sm 2 ] 100,00 Flow velocity vs Pick-up Flux CB above Sand Bed: 30 mm Grainsize D50 = 262 µm 10, kg/m3 [ ] 1,00 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 6,50 Density eroding flow: 1000 kg/m3 Density eroding flow: 1200 kg/m3 Density eroding flow: 1400 kg/m3 Flow Velocity (Vs) [m/s] Relative Density Relative Density [ ] [ ] [ ] [ ] [ ] Figure 4.43: Influence density eroding flow on pick-up flux (grain size D 50 : 262 µm) 100,00 Bed Shear Stress vs Pick-up Flux CB above Sand Bed: 30 mm Grainsize D50 = 262 µm 10,00 1, Bed Shear Stress (τb) [Pa] Relative Density Relative Density Density eroding flow: 1000 kg/m3 [ ] [ ] Density eroding flow: 1200 kg/m3 [ ] [ ] Density eroding flow: 1400 kg/m3 [ ] Figure 4.44: Influence density eroding flow on pick-up flux (grain size D 50 : 262 µm) 93

94 Pick-up Flux (E) [kg/sm 2 ] In figure 4.45, 4.46 and appendix AB-9 to AB-12 a comparison is made between the different concentrations above the sand bed where the grain size (D 50 ) is 125 µm and density of the incoming flow is 1000 kg/m 3. The density is divided in the following relative densities: [0 0.20] [ ] [ ] In figure 4.45 the pick-up flux (E) is plotted as function of the flow velocity (v s )and in figure 4.46 as function of the bed shear stress ( ). From these figures and appendix AB-9 to AB-12 it can be observed that the concentration level of 30 mm above the sand bed gives the least variation in the results. The tests with a sand bed with density of the incoming eroding flow of 1200 kg/m 3 and 1400 kg/m 3 give the same results (Appendix AB-13 to AB-16). For the incoming eroding flow of 1400 kg/m 3 the pick-up flux remains the same when the flow velocity increases. A possible cause is that the eroding flow is saturated with sand. For grain size (D 50 ) of 125µm the C b with 30 mm above sand bed is taken for further analysis in this chapter. 100,00 Flow velocity vs Pick-up Flux Density eroding flow: 1000 kg/m3 Grainsize D50 = 125 µm 10,00 1,00 0,10 1,50 2,00 2,50 3,00 3,50 4,00 4,50 5,00 Flow Velocity (Vs) [m/s] CB 10 mm above SB CB 20 mm above SB CB 30 mm above SB Relative Density [0-0.20] [0-0.20] [0-0.20] Figure 4.45: Influence concentration (C b ) on pick-up flux (relative density [0-0.20]) 94

95 Pick-up Flux (E) [kg/sm 2 ] Pick-up Flux (E) [kg/sm 2 ] 100,00 Flow velocity vs Pick-up Flux Density eroding flow: 1000 kg/m3 Grainsize D50 = 125 µm 10,00 1,00 CB: 10 mm 0, Bed Shear Stress (τb) [Pa] Relative Density CB 10 mm above SB CB 20 mm above SB CB 30 mm above SB [0-0.20] [0-0.20] [0-0.20] Figure 4.46: Influence concentration (Cb) on pick-up flux (relative density [0-0.20]) In figures 4.47 and 4.48 the influence of density of the sand bed on the pick-up flux is shown. The grain size (D 50 )is 125 µm. The density of the sand bed is divided in the following relative densities: [0 0.20] [ ] [ ] According to the results in figure 4.47 and 4.48 the conclusion can be made that the pick-up flux is independent of density of sand bed. 100,00 Flow velocity vs Pick-up Flux CB above sand bed: 30 mm; Density eroding flow: 1000 kg/m3 Grainsize D50 = 125 µm 10,00 1,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 6,00 Flow Velocity (Vs) [m/s] Relative Density [0-0.20] [ ] [ ] Figure 4.47: Influence density sand bed on pick-up flux (grain size D 50 : 125 µm) 95

96 Pick-up Flux (E) [kg/sm 2 ] 100,00 Bed Shear Stressvs Pick-up Flux CB above sand bed: 30 mm; Density eroding flow: 1000 kg/m3 Grainsize D50 = 125 µm 10,00 1, Bed Shear Stress (τb) [Pa] Relative Density [0-0.20] [ ] [ ] Figure 4.48: Influence density sand bed on pick-up flux (grain size D 50 : 125 µm) In figures 4.49 and 4.50 the influence of density of the eroding flow on the pick-up flux is shown. The grain size (D 50 )is 125 µm. The following densities of the eroding flow are compared: 1000 kg/m kg/m kg/m 3 According to the results in figures 4.49 and 4.51 the conclusion can be made that the pick-up flux decreases with density of eroding flow and is independent of the relative density. The influence of the density of eroding flow is the most visible where the density of the eroding flow is 1400 kg/m 3. The influence of the density of eroding flow is more visible with fine sand (D 50 =125 µm) than with coarse sand (D 50 =262 µm). This is because the transport capacity with fine sand is higher than with coarse sand. 96

97 Pick-up Flux (E) [kg/sm 2 ] Pick-up Flux (E) [kg/sm 2 ] 100,00 Flow velocity vs Pick-up Flux CB above Sand Bed: 30 mm Grainsize D50 = 125 µm 10,00 1,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50 5,00 5,50 Density eroding flow: 1000 kg/m3 Density eroding flow: 1200 kg/m3 Density eroding flow: 1400 kg/m3 Flow Velocity (Vs) [m/s] Relative Density [0-0.20] Relative Density [ ] [0-0.20] [ ] [0-0.20] [ ] Figure 4.49: Influence density eroding flow on pick-up flux (grain size D 50 : 125 µm) 100,00 Bed Shear Stress vs Pick-up Flux CB above Sand Bed: 30 mm Grainsize D50 = 125 µm 10, kg/m3 [ ] 1, Bed Shear Stress (τb) [Pa] Relative Density Relative Density Density eroding flow: 1000 kg/m3 [0-0.20] [ ] Density eroding flow: 1200 kg/m3 [0-0.20] [ ] Density eroding flow: 1400 kg/m3 [0-0.20] [ ] Figure 4.50: Influence density eroding flow on pick-up flux (grain size D 50 : 125 µm) 97

98 Pick-up Flux measured (E) [kg/sm 2 ] 4.6 Comparison of functions with measured data In figures 4.51 to 4.55 the pick-up flux as calculated with the function of Van Rijn, Winterwerp, Van Rhee and Mastbergen are compared with the measured pick-up flux. The grain size (D 50 ) is 262 µm and the concentration C b is taken from 30 mm above sand bed (concentration C b taken from 10 mm and 20 mm above sand bed is given in appendix C). Figures 4.51 to 4.52 differ in density of the sand bed. Figures 4.54 and 4.55 differ in density of the eroding flow. Figure 4.51 shows that the function of van Rijn is a factor 10 to 100 higher in comparison with the measured results. The conclusion can be made that the function of van Rijn is overestimated, therefore this function is not shown in the following figures. From the figures in this paragraph the conclusion can be made that the function of Winterwerp is the most consistent with the measured pick-up flux. At high densities of the eroding flow (1400 kg/m 3 ) the functions do not agree with the results. The functions need to be corrected for high densities of the eroding flow. Pick-up flux measured vs Pick-up flux functions Density eroding flow: 1000 kg/m3; Density sand bed: ±1900 kg/m3 Grainsize D50 = 262 µm; Cb: 30 mm above sand bed ,00 10,00 100, , , ,00 Pick-up Flux functions (E) [kg/sm 2 ] van Rijn Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.51: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) 98

99 Pick-up Flux measured (E) [kg/sm 2 ] Pick-up Flux measured (E) [kg/sm 2 ] Pick-up flux vs Erosion Rate Density eroding flow: 1000 kg/m3; Density sand bed: ±1950 kg/m3 Grainsize D50 = 262 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.52: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Pick-up flux vs Erosion Rate Density eroding flow: 1000 kg/m3; Density sand bed: ±1970 kg/m3 Grainsize D50 = 262 µm; Cb: 30 mm above sand bed Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.53 Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) 99

100 Pick-up Flux measured (E) [kg/sm 2 ] Pick-up Flux measured (E) [kg/sm 2 ] Pick-up flux vs Erosion Rate Density eroding flow: 1200 kg/m3; Density sand bed: ±1950 kg/m3 Grainsize D50 = 262 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.54: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) Pick-up flux vs Erosion Rate Density eroding flow: 1400 kg/m3; Density sand bed: ±1950 kg/m3 Grainsize D50 = 262 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.55: Comparison pick-up flux measured with pick-up flux functions (D 50 : 262 µm) 100

101 Pick-up Flux measured (E) [kg/sm 2 ] In figures 4.56 to 4.60 the pick-up flux of the functions are compared with the measured pick-up flux where the grain size (D 50 ) is 125 µm and the concentration C b is taken from 30 mm above sand bed (concentration C b taken from 10 mm and 20 mm above sand bed is given in appendix C). Figures 4.54 to 4.58 differ in density of the sand bed. Figures 4.59 and 4.60 differ in density of the eroding flow. Figure 4.56 shows that the function of van Rijn is a factor 10 to 100 higher in comparison with the measured results. The conclusion can be made that the function of van Rijn overestimates the pickup flux, therefor this function is not shown in the following figures. At high densities of the eroding flow (1400 kg/m 3 ) the functions do not agree with the results. The functions need to be corrected for high densities of the eroding flow. From this paragraph the conclusion can be made the function of Winterwerp is the most consistent with the measured pick-up flux. The function of Winterwerp gives a reasonable estimation of the pick-up flux at low relative densities and low densities of the eroding flow Pick-up flux measured vs Pick-up flux functions Density eroding flow: 1000 kg/m3; Density sand bed: ±1850 kg/m3 Grainsize D50 = 125 µm; Cb: 30 mm above sand bed ,00 10,00 100, , , ,00 Pick-up Flux functions (E) [kg/sm 2 ] van Rijn Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.56: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) 101

102 Pick-up Flux measured (E) [kg/sm 2 ] Pick-up Flux measured (E) [kg/sm 2 ] Pick-up flux measured vs Pick-up flux functions Density eroding flow: 1000 kg/m3; Density sand bed: ±1870 kg/m3 Grainsize D50 = 125 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.57: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) Pick-up flux measured vs Pick-up flux functions Density eroding flow: 1000 kg/m3; Density sand bed: ±1900 kg/m3 Grainsize D50 = 125 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.58: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) 102

103 Pick-up Flux measured (E) [kg/sm 2 ] Pick-up Flux measured (E) [kg/sm 2 ] Pick-up flux measured vs Pick-up flux functions Density eroding flow: 1200 kg/m3; Density sand bed: ±1870 kg/m3 Grainsize D50 = 125 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.59: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) Pick-up flux measured vs Pick-up flux functions Density eroding flow: 1400 kg/m3; Density sand bed: ±1870 kg/m3 Grainsize D50 = 125 µm; Cb: 30 mm above sand bed ,00 10,00 100,00 Pick-up Flux functions (E) [kg/sm 2 ] Van Rhee/Bisschop Winterwerp Mastbergen Figure 4.60: Comparison pick-up flux measured with pick-up flux functions (D 50 : 125 µm) 103

104 Pick-up Flux (E) [kg/sm 2 ] 4.7 Description of images high speed camera A high speed camera is used to capture the erosion process on video. The high speed camera was placed in front of the glass window (paragraph 3.2). In this paragraph the erosion process on video of five erosion tests are described (table 4.9). A comparison of tests is made between high and low density of the sand bed (test 27 and test 31) and between density of eroding flow of 1000 kg/m 3 and 1400 kg/m 3 (test 36 and 41) and between grain size (test 27 and 44). In each video the start of erosion process (t video =0) is chosen where the sand bed starts decreasing in vertical direction. This is not the same as in the measurements. During the measurements the start of the erosion process is chosen when the valve is opened to the measurement section. In figure 4.61 is for the different test the pick-up flux plotted as function of the bed shear stress. Test number Flow velocity Grain Size (D 50 ) Density eroding flow Rel. Density sand bed [-] m/s m Kg/m 3 [-] (Low) (High) (Medium) (Medium) (Low) Table 4.9: compared images high speed camera 100,00 Bed Shear Stress vs Pick-up Flux CB above sand bed: 30 mm 10,00 1, Bed Shear Stress (τb) [Pa] Test 27 test 31 Test 36 Test 41 Test 44 Figure 4.61: Bed shear stress vs pick-up flux The relative density of the sand bed during test 27 was relative low: relative density 0.2. Before the erosion process starts (t=0) the height of the sand bed during test 27 is approximately 133 mm from the bottom of the measurement section (figure 4.62). After opening the valve to the measurement section water flows into the measurement section leading to erosion of the sand bed. In the video images can be seen that during the first 1.48 seconds the sand bed decreases constantly in vertical direction. After approximately t video =1.48s. the density of the eroding flow increases due to a sand bed between the valve and measurement section (figure 4.63) which means that the sand bed erodes with a sand-water mixture despite of water. The density of the eroding flow increases due to erosion of the sand bed between the valve to the measurement section and lexan plate where the 104

105 data is measured. At t video =1.48s the flow velocity (v s ) is approximately 2,31 m/s and the shear stress ( b ) 240 Pa. In the video images is observed that the entire sand bed is shifted in horizontal direction. The grains are picked up in layers and a part of the grains are sedimented again. After t video = 1.80 s. a large part of the sand bed is eroded. Test 31 is executed with a high density of the sand bed: relative density 0.6. The height of the sand bed at t video =0 is approximately 180 mm from the bottom of the measurement section (figure 4.64). According to the video images the sand bed starts eroding in layers and decreases irregular in vertical direction. After t video =3.06 seconds the density of the eroding flow increases due to a sand bed between the valve and measurement section (figure 4.65) which means that the sand bed erodes with a sand-water mixture despite of water. The density of the eroding flow increases due to a sand bed between the valve to the measurement section and lexan plate where the data is measured. At t video =3.06 s the flow velocity (v s ) is approximately 3,34 m/s and the bed shear stress ( b ) 395 Pa. After t video =4.80 s. a large part of the sand bed is eroded. Table 4.10 depicts the transition moment where the density of eroding flow increases. In figure 4.66 is for test 27 and test 31 the concentration plotted as function of the time. The sand bed with low density (test 27) has an irregular erosion than sand bed with high density (test 31). Sand bed with low density has a lower bed shear stress and higher erosion rate then the sand bed with high density. The higher erosion rate makes that the concentration of the sand bed is higher for sand bed with low density. Test number Rel. Density sand bed Time after start decreasing Sand Bed (t video ) Flow velocity (v s ) Bed Shear Stress (τ b ) [-] [-] Sec. m/s Pa (Low) 1,48 2, (High) 3,06 3, Table 4.10: Transition moment where the density of eroding flow increases 105

106 Density [kg/m 3 ] Figure 4.62: Images test 27t=0 s. t=1.48 s. t=1.80 s. height in mm Test 27: Density vs. Time t video = 0 t video = 1,48 t video = 1, ,00 1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00 10, Time [sec] 224 Figure 4.63: Density vs Time (test 27) 106

107 Density [kg/m 3 ] Figure 4.64: Images test 31t=0 s. t=3.06 s. t=4.80 s. height in mm Test 31: Density vs. time t video = 0 t video = 3,06 t video = 4, ,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0 14,0 15,0 174 Time [sec] 224 Figure 4.65: Density vs Time (test 31) 107

108 Density [kg/m 3 ] Concentration [-] 0,35 Time vs Relative Density CB: 30 mm above SB 0,30 0,25 0,20 0,15 0,10 0,05 0, Time [sec] Grain Size (D 50 ) 262 µm (Low Density SB: Test 27) 262 µm (High Density SB: Test 31) Figure 4.66: Concentration vs time: test 27 and test 31 (CB: 30 mm above sand bed) Figure 4.67 depicts the erosion moments of test 27 (low density sand bed) and test 31 (high density sand bed) by the conductivity probes at the lexan plate. Although the conductivity probes are at a different place in the measurement section than the high speed camera figure 4.65 represents the same erosion process as the video images. The erosion rate with low density of sand bed is higher than with high density of sand bed Density sand bed: High vs Low Density Sand Bed: Low Density Sand bed: High ,0 5,0 10,0 15,0 20,0 25,0 30,0 35,0 Time [sec] Figure 4.67: Erosion moments with density sand bed: high vs low (test 27 vs test 31) 108

109 Test 36 is executed with eroding flow of 1000 kg/m 3. In test 36 the height of the sand bed at t video =0 is approximately 184 mm (figure 4.68). The first 1.52 seconds the sand bed decreases constantly in vertical direction and the sand grains are picked up individually. At t video =1.52s. the flow velocity (v s ) is approximately 3,87 m/s and the shear stress ( b ) 1064 Pa. After t video =1.52s. the top layer of the sand bed is loosened and the grains are picked up in layers. The density of the eroding flow increases (figure 4.69) which means that the sand bed is eroded with a sand-water mixture. In test 41 the height of the sand bed at t video =0 is approximately 184 mm (figure 4.70). The density of the eroding flow is 1400 kg/m 3. In comparison with test 36 the height of the sand bed is still 184 mm at t video =1.52s. In the video is observed that the entire sand bed is shifted in horizontal direction before it starts decreasing in vertical direction. At t video =2.30 s the sand bed starts decreasing in vertical direction. The flow velocity (v s ) is approximately 4,02 m/s and the shear stress ( b ) 560 Pa. The sand bed decreases in parts. This means that each time small parts are eroded from the sand bed (figure 4.70). Table 4.11 depicts the transition moment where density of the eroding flow increases. The density of the eroding flow remains relatively constant at approximately 1400 kg/m 3 (figure 4.71).In comparison with test 36 the erosion rate is lower in test 41. In figure 4.70 is for test 36 and test 41 the concentration plotted as function of the time. The concentration of test 36 is irregular compared with test 41. Test number Density eroding flow Time after start decreasing Sand Bed (t video ) Flow velocity (v s ) Bed Shear Stress (τ b ) [-] Kg/m 3 Sec. m/s Pa ,52 3, ,30 4, Table 4.11: Transition moment where the density of eroding flow increases 109

110 Density [kg/m 3 ] Figure 4.68: Images test 36t=0 s. t=1.52 s. t=2.72 s Test 36: Density vs. time t video = 0 t video = 1,52 t video = 2, ,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 Time [sec] Figure 4.69: Density vs Time (test 36) height in mm

111 Density [kg/m 3 ] Figure 4.70: Images test 41t=0 s. t=1.52 s. t=3.36 s. height in mm Test 41: Density vs. time t video = 0 t video = 1,52 t video = 3, ,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0 14,0 174 Time [sec] 224 Figure 4.71: Density vs Time (test 41) 111

112 Density [kg/m 3 ] Concentration [-] 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 Time vs Relative Density CB: 30 mm above SB 0,00 0,00 2,00 4,00 6,00 8,00 10,00 12,00 14,00 16,00 18,00 Time [sec] Grain Size (D 50 ) Density eroding flow:1000 kg/m3(test 36) Density Eroding flow:1400 kg/m3(test 41) Figure 4.72: Concentration vs time: test 36 and test 41 (CB: 30 mm above sand bed) Figure 4.73 depicts the erosion moments of test 36 and test 41 measured by the conductivity probes where density of the eroding flow is respectively 1000 kg/m 3 and 1400 kg/m 3. The erosion rate with density of eroding flow of 1000 kg/m 3 is higher than with 1400 kg/m 3 (figure 4.74) Density Eroding Flow: 1000 kg/m3 vs 1400 kg/m ,0 5,0 10,0 15,0 20,0 Time [sec] Density Eroding Flow: 1000 kg/m3 Density Eroding Flow: 1400 kg/m3 Figure 4.73: Erosion moments with density eroding flow: 1000 kg/m 3 vs 1400 kg/m 3 (test 36 vs test 41) 112

113 Erosion Rate (Ve) [m/s] 0,05 Flow velocity vs Erosion Rate 0,04 0,03 0,02 0,01 0,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50 Flow velocity (Vs) [m/s] Test 36 Test 41 Figure 4.74: Flow velocity vs Erosion Rate (test 36 vs test 41) Test 44 is executed with grain size (D 50 ) of 125 m and density of the eroding flow is 1000 kg/m 3. This test is compared with test 27 (grain size (D 50 ): 262 m). The height of the sand bed in test 44 is at t video =0 approximately 154 mm (figure 4.75). At t video =1.48s. did the height of the sand bed not decrease in comparison with test 36. At t video =1.48s. the density of the eroding flow increases which leads to hinder of erosion. The sand bed is eroded in layers. The flow velocity (v s ) is approximately 2,36 m/s and the shear stress ( b ) 403 Pa. In figure 4.77 is for test 27 and test 44 the concentration plotted as function of the time. The concentration of fine sand (test 27) is higher than concentration of coarse sand (test 44). 113

114 Density [kg/m 3 ] Figure 4.75: Images test 44t=0 s. t=1.48 s. t=2.00 s. height in mm Test 44: Density vs. time t video = 0 t video = 1,48 t video = 2, ,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0 14,0 174 Time [sec] 224 Figure 4.76: Density vs Time (test 44) 114