Effect of Applied Vibration on Silo Hopper Design

Transcription

1 Effect of Applied Vibration on Silo Hopper Design Th. Kollmann and J. Tomas Mechanical Process Engineering, Department of Process Engineering, The Otto-von-Guericke- University of Magdeburg, P.O. Box 412, D-3916 Magdeburg, Germany Abstract The vibration promoted flow of cohesive particulate solids was investigated for and Titania powder, each with an average particle size of about 1 µm. The flow behaviour of fine cohesive powders can be improved by the application of harmonic vibrations, as reflected in the flow properties. Based on the work of Roberts et al., a shear testing technique has been established that allows the measurement of the effect of vibrations on the flow properties of particulate solids. The experimental results show that the vibration application leads to a significant reduction of the shear strength. The unconfined yield strength decreases with increasing vibration velocity, whereas the angle of internal friction is nearly independent of applied vibration. Also, the unconfined yield strength and the wall friction angle ϕ W can be reduced by applying mechanical vibration. Based on the lab-scale shear tests, the hopper half angle and the unconfined yield strength were estimated. For the given example the conical outlet width is reduced from.88 m to about.4 m the maximum hopper angle increases from 5 up to approximately 2 due to the vibration application. 1 INTRODUCTION Recently, the production of very fine particles has gained considerable importance in many powder technology applications. Finer particles coincide with increasing adhesion forces and make higher demands on reliable powder storage and handling equipment. Static silo hopper design often reach limit when hopper half angles of 1 and below combined with wide outlet dimensions are required. bin vibratory excitor flexible connection vibratory excitor (a) internal baffle discharge (b) Figure 1 Examples for vibrational flow promoting devices: a) vibrator at the hopper wall (left) and b) vibrating hopper (right)

2 When trouble-free powder discharge from silos due to gravity seems impossible, mechanical vibration are widely used to promote and control gravity flow. Figure 1 shows two typical applications of mechanical vibrations in flow promoting and bin discharging: a) vibrator, e.g. unbalanced motor or pneumatic vibrator, outside of the bin hopper for vibration of the hopper wall, b) vibrating hopper (also called bin activator or vibrational disc feeder). A great deal of practical experience for apparatus design has been gained but these recommendations are often not applicable beyond the empirical situation from which they were derived (Bell, 1999, Thomson, 1997). The present work strives to contribute to better understanding of the flow behaviour of very fine, cohesive powders in the presence of harmonic vibrations. The study focuses on the gravity powder flow that is initiated and promoted by the application of harmonic vibrations. Dumbaugh (1984) proposed the term vibration induced gravity flow, an appropriate description for such applications. Based on the work of Roberts et al. (1978, 1984), a test technique has been established and expanded that allows the measurement of the effect of vibrations on the flow properties of particulate solids. Test apparatus and testing technique are based on the wide-spread direct shear test according to Jenike (1964). The Jenike-method has been proven in industrial practice and so the results of the vibrated shear testing are expected to be applicable for storage and handling equipment design as well. 2 TEST EQUIPMENT AND TESTING TECHNIQUE For testing the vibration induced powder flow behaviour, a vibrating direct shear tester was designed and constructed, which was based on the test apparatus developed by Roberts et al. (1978, 1984). Two test arrangements are possible to carry out shear tests in the presence of vibrations (Figure 2): The a) the top half of the shear cell is vibrated, b) the whole shear cell is vibrated. Normal force F N Shear force F S Normal force F N Shear force F S Shear ring Shear base Vibrating plate Excitation rod a e a r ae (a) f, F e (b) Leaf spring f, F e Figure 2 Vibrating direct shear tester: (a) Top half is vibrated, (b) Vibration of the whole cell a) Vibration of the top half of the shear cell (Figure 2 (a)): The shear base is fixed and the vibrations are applied to the horizontal plane, perpendicular to the shear direction, Figure 2 (a). This arrangement allows the measurement of powder flow properties needed for storage and handling equipment design, including wall friction angle, as functions of the vibration parameters. b) Vibration of the whole shear cell (Figure 2 (b) and Figure 3):

3 Here, the whole shear cell is located on a vibrating plate, which is mounted on vertical leaf springs. On the shear base and the shear ring, one piezoelectric accelerometer is located on each for measuring the base- and the response- vibration acceleration a e and a r respectively. The ratio a r /a e can be used to determine resonance and damping behaviour of powders and the influence of resonance on shear strength. The experimental results reported here were obtained using the arrangement shown in Figure 2 (a), the top half of the shear cell is vibrated with sinusoidal vibration excitation Figure 3: Vibration of the whole shear cell: 1) shear cell, 2) frame for normal load, 3) shear force sensor, 4) accelerometers, 5) leaf spring, 6) electrodyn. vibrator, 7) excitation rod Table 1 Procedures for vibrated shear testing Method Vibration Excitation Application, Examples A 1 during shear pulsed vibration e.g. for bridge breaking and discharging B C 1 D during preshear und shear during pre-consolidation between preshear and shear 1 see Roberts et al. (1978, 1984) continuous vibration during discharge, e.g. vibrating hopper undesirable vibration during silo filling and storage time without discharging undesirable vibration during storage, e.g. transportation by truck and train (equivalent to time consolidation ) To carry out vibrational shear tests, several test procedures are possible, as shown in Table 1. For the first Method, called A, pre-consolidation and preshear stages are carried out according to Standard Shear Testing Technique SSTT (1989). The vibrations are applied only during shear. This procedure is for simulation of pulsed vibration excitation, which is frequently used for bridge breaking. If continuous vibrations are applied during discharge, for instance by vibrating hoppers, the steady state

4 flow of powders will be influenced by the vibrations as well. This procedure can be simulated by method B. Here, the vibrations are already applied to the powder sample during preshear. Further procedures are explained in Table 1. However, the focus of present study was only on methods A and B. One - and one Titania-powder, each with median particle size of about 1 µm, were used to carry out vibrated shear tests. Figure 4 depicts particle size distributions of the test powders complemented with selected powder properties. The testing conditions are summarized in the Appendix. Cumulative Volume in % TiO 2 median particle size.6 µm particle density 3.86 g/cm 3 moisture content.4 % CaCO 3 median particle size 1.3 µm; particle density 2.65 g/cm 3 moisture content.4 %,1 1 1 Particle size in µm Figure 4 Particle size distribution, particle density (true density) and moisture content (dry based) of the test powders 3 RESULTS 3.1 Flow properties for incipient flow (vibrated shear) Typical recorded shear force curves for a powder are drawn in Figure 5. The left curve illustrates the preshear stage without vibrations. The second curve shows the shear force during shear, without vibrations as well. The third curve depicts the shear force during vibrated shear for a new sample and the same consolidation and preshear conditions. The vibrations lead to a lower shear force maximum at the same normal stress level. In addition, the fourth example shows the vibrated shear response. However, after the shear force maximum had been reached, the vibration excitation was switched off, marked by S in Figure 5. After that the shear force increased again, nearly up to the unvibrated peak value. It seems, that the powder is only activated by applying mechanical vibration. Measured vibrated shear stresses for various vibration frequencies are each plotted against vibration displacement, vibration velocity and vibration acceleration in Figures 6-9. From these plots, the ratio of shear stress τ vibr to the unvibrated shear stress τ shows good agreement with various parameters. Nevertheless, the vibrated shear stress correlates best to the maximum vibration velocity v max. The shear stress ratio τ vibr /τ declines with increasing maximum vibration velocity and approaches asymptotically a minimum value.

5 3 Pre-shear ( = 4. kpa) Shear (σ = 2.4 kpa) 25 unvibrated Shear Force F S in N particle size 1.3 µm frequency Hz acceleration 8 m/s² vibrated S Shear Displacement s in mm Figure 5 Typical shear force vs. time curves for unvibrated and vibrated shear tests (method A, vibrated shear), shear velocity 2 mm/min These vibrated shear tests were carried out for various vibration frequencies. The results show, that the effect of applied vibration can be determined with help of the vibration velocity. All of the following test results (except wall friction) were derived by the variation of v max at constant vibration frequency of Hz (see the Appendix). Roberts (1984) introduced a failure criterion concerning the vibrational powder flow. The shear stress in the presence of vibrations τ vibr depends directly on the maximum vibration velocity v max. This correlation is given in dimensionless terms by Eq. ( 1). τ vibr τ β v max = 1 1 exp ( 1) τ γ τ vibr shear stress in the presence of vibrations τ shear stress unvibrated v max maximum vibration velocity β, γ material constants The parameter β indicates the maximum possible shear stress reduction. The parameter γ represents a characteristic vibration velocity. The plotted curve in Figure 9 corresponds to Eq. ( 1), which is fitted to the experimental data. In this example, the characteristic vibration velocity γ was found to be 14 mm/s and the parameter β to be 1.1 kpa.

6 Hz 5 Hz Hz 2 Hz 3 Hz 1. x max.1 mm.14 mm.28 mm τ vibr / τ τ vibr / τ.8.6 = 4. kpa σ = 2.4 kpa.6 = 4. kpa σ = 2.4 kpa Maximum Vibration Displacement x max in mm Figure 6 Shear stress ratio vs. vibration displacement Frequency f in Hz Figure 8 Shear stress ratio vs. vibration frequency Hz 5 Hz Hz 2 Hz 3 Hz Hz 5 Hz Hz 2 Hz 3 Hz τ vibr / τ τ vibr / τ.6.6 = 4. kpa σ = 2.4 kpa.4 = 4. kpa σ = 2.4 kpa Maximum Vibration Acceleration a max in m/s² Maximum Vibration Velocity v max in m/s Figure 7 Shear stress ratio vs. vibration acceleration Figure 9 Shear stress ratio vs. vibration velocity

7 Shear Stress τ in kpa pre-shear shear unvibrated vibrated (v max = 2 mm/s) 95 % confidence interval = 4. kpa = 8. kpa SYL ϕ st σ c,vibr σ c,vibr -1 σ σ c Normal Stress σ in kpa Figure 1 Vibrated and unvibrated yield loci for powder (a max = 12 m/s²; f = Hz) σ c As previous experimental investigations have shown, the characteristic vibration velocity γ is independent of the consolidation stress level and the applied normal stress during shear (Kollmann and Tomas, 2, 21a, 21b). The parameter β depends only on the consolidation stress level, that can be characterised by the centre stress σ = ( σ + σ 2 ) / 2 of cohesive steady-state flow (Tomas, 1991, M,st ). The ratio β/σm,st remains also nearly constant for various average stress levels during consolidation pre-history (Kollmann and Tomas, 2, 21a, 21b). Figure 1 shows two examples of yield loci for powder, unvibrated (solid lines) and in the presence of vibration (dotted lines). The maximum vibration velocity v max was 2 mm/s. For one yield locus, the applied vibration reduces the shear stress τ for each applied normal stress σ by the same amount τ, according to Eq. ( 2). Hence, the angle of internal friction ϕi remains constant and the yield loci are shifted parallel towards smaller shear stresses. v τ = β 1 exp max ( 2) γ While the angle of internal friction ϕ i is nearly independent of the vibration application, the unconfined yield strength σ c decreases with increasing vibration velocity as shown for several consolidation stress levels in Figure 11. Just like the shear stress τ, the unconfined yield strength σ c declines asymptotically and approaches a minimum value. For linear yield locus approximation, the correlation can be expressed by analogy with Eq. ( 1).

8 Unconfined Yield Strength σ c in kpa 25 ϕ i = 34 ± σ c = 16.3 kpa = 8. kpa = 4. kpa = 1.9 kpa Maximum Vibration Velocity v max in m/s Angle of Internal Friction ϕ i in Unconfined Yield Strength σ c in kpa ϕ i = 3 ± 2 Titania = 1.9 kpa = 4. kpa = 8. kpa Maximum Vibration Velocity v max in m/s Angle of Internal Friction ϕ i in Figure 11 Effect of vibration during shear on the flow properties ϕ i and σ c The unconfined yield strength is 1+ sin ϕ sin ϕ i i σ c = 2 τc = 2 σ t. ( 3) cosϕi 1 sin ϕi Considering the vibration-independent angle of internal friction ϕ i, the vibrated unconfined yield strength σ c,vibr is given by: 1+ sin ϕ sin ϕ i i σ c,vibr = 2 τc,vibr = 2 σ t,vibr ( 4) cosϕi 1 sin ϕi τ c,vibr expressed by Eq. ( 1) gives:

9 σ c,vibr 1+ sin ϕ i β v max = 2 τc 1 1 exp ( 5) cosϕi τc γ The tensile strength σ t in terms of the angle of internal friction ϕ i is expressed as: σ t,vibr = τ c,vibr / tan ϕ i ( 6) Combining Eqs. ( 5) and ( 6) leads to: σ c,vibr sin ϕ i β v max = 2 σ t 1 1 exp. ( 7) 1 sin ϕi σ t tan ϕi γ The curves for the unconfined yield strength in Figure 11 are given by Eq. ( 7) and the material parameters β and γ. 3.2 Flow properties for steady-state flow (Vibration during preshear and shear) Method B, which involves vibrated shear and preshear stages, is summarised. Typical shear force versus time curves are plotted in Figure 12. Curves a) and c) are for preshear and shear, respectively. These curves illustrate preshear without vibration and shear with vibration, as already discussed in section 3.1. Curves b) and d) each show the shear force during vibrating shear tests according to method B. As expected, the measured preshear force is reduced due to the vibrations, compared to the unvibrated preshear curve a). However, the results for vibrated shearing (constant normal stress) after unvibrated and vibrated preshearing are nearly equivalent. These results add further weight to the argument that the powder is only activated by the applied vibrations. The excitation energy input during shear does not change powder properties permanently. 3 unvibrated pre-shear Shear Force F S in N a) vibrated b) = 4. kpa σ = 2.4 kpa f = Hz a = 8 m/s 2 vibrated shear c) d) Shear Displacement s in mm Figure 12 Shear force vs. time curves for vibrated preshearing and vibrated shearing, (method B), shear velocity 2 mm/min

10 ϕ st σ ϕ st in ϕ i σ in kpa Maximum Vibration Velocity v max in m/s.25 Figure 13 Vibration effect on the stationary angle of internal friction ϕ st and the isostatic tensile strength σ The shear stress τ depends on the angle of internal friction ϕ i and the tensile strength σ t. i ( σ + σ t τ = tan ϕ ) ( 8) The tensile strength σ t is influenced by the stressing pre-history, characterised by the centre stress of the Mohr circle for cohesive steady state flow σ M,st (Figure 1), the stationary angle of internal friction ϕ st and the isostatic tensile strength σ of the unconsolidated powder (Tomas, 1991, 1999, 2). sin ϕ st sin ϕst = tan ϕ i σ + 1 σ M,st + σ ( 9) sin ϕi sin ϕi τ While σ remains nearly constant with increasing vibration velocity, Figure 13, a distinct decline in ϕ st due to the vibration velocity was found. But, as shown in section 3.1 the angle of internal friction ϕ i remains constant. Because of the relation between ϕ i and ϕ st (Tomas, 1991, 1999, Molerus, 1975) tan ϕ st = ( 1+ κ) tan ϕi ( 1) the stationary angle of friction ϕ st cannot be less than the angle of internal friction ϕ i. The possible shear stress reduction is limited by the stationary yield locus (SYL) with ϕ. st ϕ i σ R,st = sin ϕst ( σm,st + σ ) ( 11) Where, σ R,st is the radius stress of the Mohr circle for cohesive steady state flow. 3.3 Wall friction The kinematic friction between a powder and hopper wall material can be reduced by applying mechanical vibrations equivalent to the reduction of the internal friction (Roberts, 1984). The wall shear stress τ W decreases with increasing vibration velocity to approach a minimum value. Each plotted against vibration displacement, vibration velocity, vibration acceleration and vibration

11 frequency, the wall shear stress ratio τ W,vibr /τ W correlates best with the vibration velocity as was shown for the shear stress ratio τ vibr /τ in Figures 6-9. The dependence of wall shear stress ratio τ W,vibr /τ W in dependence on the maximum vibration velocity v max is plotted in Figure 14 for powder on stainless steel τ W,vibr / τ W vs. stainless steel σ W = 5 kpa 5 Hz Hz 2 Hz Maximum Vibration Velocity v max in m/s Figure 14 Wall shear stress ratio vs. maximum vibration velocity Drawing an analogy with Eq. ( 1), the wall shear stress ratio holds: τ W,vibr τ W β W v max = 1 1 exp. ( 12) τ W γ W The parameters β W and γ W were observed as constant for given combination of powder and wall material. 35 Wall Friction Angle ϕ W in vs. stainless steel σ W = 5 kpa Maximum Vibration Velocity v max in m/s Figure 15 Wall friction angle vs. maximum vibration velocity ( / stainless steel)

12 The decline of wall shear stress coincides with the decrease in the wall friction angle ϕ W. In Figure 15, the wall friction angle ϕ W is plotted against the maximum vibration velocity v max for on stainless steel. Because of tan ϕ W = τ W / σ W ( 13) and Eq. ( 12) the following relationship for vibrated wall friction angle ϕ W,vibr holds: τ W β W v max tan ϕ W,vibr = 1 1 exp. ( 14) σ W τ W γ W 3.4 Consequences for hopper design Mass flow bin design includes two characteristic hopper parameters: minimum outlet diameter b min hopper angle Θ The minimum outlet diameter mainly depends on the unconfined yield strength σ c. The lower the unconfined yield strength, the smaller is the minimum outlet diameter. Figure 16 shows the unconfined yield strength for various vibration velocities against the major principal stress during consolidation σ 1. The dashed lines show the range of effective wall stresses σ 1 = σ 1 /ff of a cohesive powder arch for common values of the flow factor ff. The intersection point of σ c and σ 1 delivers the so-called critical unconfined yield strength σ c,crit. Generally, the minimum outlet diameter to avoid bridging in a mass flow hopper b min is directly proportional to σ c,crit. Unconfined yield strength σ c in kpa unvibrated v max = 5 mm/s v max =1 mm/s v max =2 mm/s v max =3 mm/s curve fit ff= Major consolidating stress σ 1 in kpa Figure 16 Unconfined yield strength vs. major principal stress during consolidation for powder at various vibration velocities The critical outlet diameter b min to avoid bridging in mass flow hoppers decreases with increasing vibration velocity, as shown in Figure 17. For this example, the conical outlet width is reduced from

13 .88 m to about.4 m. However, at vibration velocities for more than 1 mm/s, b min remains approximately constant. While the minimum outlet diameter mainly depends on the unconfined yield strength, the hopper angle to ensure mass flow Θ is essentially influenced by the wall friction angle ϕ W. The smaller the wall friction angle the shallower is the hopper angle. For the example illustrated in Figure 17, the maximum hopper angle increases from 5 up to approximately vs. stainless steeel Θ b min 25 b min in m Θ in.2 Θ mass flow b min conical hopper Maximum Vibration Velocity v max in m/s Figure 17 Critical outlet diameter and mass flow hopper slope vs. vibration velocity ( powder / stainless steel) 4 SUMMARY AND CONCLUSIONS The vibration-promoted flow of cohesive particulate solids was investigated for and Titania powder, each with a median particle size of about 1 µm. The flow behaviour of fine cohesive powders can be improved by the application of harmonic vibrations, as reflected in the flow properties. The vibration application leads to a significant reduction of the shear strength. To quantify the vibration intensity the maximum vibration velocity should be used, as already proposed by Roberts (1984). The shear stress declines with increasing vibration velocity and approaches asymptotically a minimum value. This correlation can be described by two material parameters, the vibration velocity coefficient γ and the stress ratio β/σ M,st. The possible shear stress reduction β depends on the consolidation stress level, here characterised by the centre stress σ M,st. The ratio β/σ M,st is nearly constant for one powder and was found for the investigated powders to be approximately.2. That means, the shear strength can be reduced by 2% of the amount of the centre stress σ M,st. The velocity γ was found to be 9 mm/s for the Titania powder and 14 mm/s for the powder. The coefficient γ represents the maximum vibration velocity, where 67 % of the maximum shear stress reduction β is reached. The exponential decay of the shear strength coincides with the decrease in the unconfined yield strength (with increasing vibration velocity). whereas the angle of internal friction is nearly independent of applied vibration. (For the investigated and Titania powders, the angle ϕ i are 34 and 3 respectively. For the first time, the effect of vibration on the steady state flow was quantified. The influence of vibration velocity can be observed with help of the stationary angle of internal friction ϕ st which was

14 reduced by vibration application by approximately 1. The shear stress during steady state flow is reduced just as described for the shear stress during incipient flow. However, vibration during preshear only slightly changes the powder properties after preshearing. The vibration does not change the powder properties permanently. The powder is only dynamically activated by means of the additional energy input. Equivalent to the internal friction of powders, the kinematic friction between powder and hopper wall material can be lowered. The wall shear stress also correlates best to the maximum vibration velocity. The wall shear stress reduction β W was about 2 kpa (independent on the wall normal stess) for both Titania and. The experimental results show that the unconfined yield strength σ c as well as the wall friction angle ϕ W can be reduced by applying mechanical vibration. The minimum outlet diameter of silo hoppers mainly depends on the unconfined yield strength. The hopper angle to ensure mass flow Θ is essentially influenced by the wall friction angle ϕ W. Hence, smaller outlet diameters and shallower mass flow hopper angles are possible. Based on the lab-scale shear tests the hopper half angle and the unconfined yield strength were estimated. For the given example, the conical outlet diameter is reduced from.88 m to about.4 m the maximum hopper angle increases from 5 up to approximately 2. 5 ACKNOWLEDGEMENT The research described in this paper has been sponsored by the Deutsche Forschungsgemeinschaft (DFG). We would like to acknowledge the DFG for their support. We are also indebted to Ammar Al- Hilo, Elisaveta Georgieva Shopova, Aimo Haack and Guido Kache, who derived many experimental results. They are much appreciated. 6 NOMENCLATURE Symbols Indices a vibration acceleration, m/s² c compressive b min critical outlet diameter, m crit critical f frequency, Hz e exciting F force, N M centre v vibration velocity, m/s max maximum value β maximum shear stress reduction, kpa n normal γ characteristic velocity, m/s pre preshear Θ hopper angle r response κ particle contact consolidation coefficient S shear σ normal stress, kpa st stationary σ c unconfined yield strength, kpa vibr vibration σ M,st centre stress for steady-state flow, kpa σ t tensile strength, kpa σ isostatic tensile strength, kpa σ 1, σ 2 principle stresses, kpa τ shear stress, kpa τ c cohesion, kpa ϕ i angle of internal friction, deg stationary angle of internal friction, deg ϕ st

15 7 REFERENCES Bell, T.A., Industrial Needs in Solids Flow for the 21 st Century, Powder Handling & Processing, 11, Thomson, F.M., Storage and flow of particulate solids, M.E. Fayed and L. Otten (eds.): Handbook of Powder Science and Technology, 2 nd ed., Chapman & Hall, New York, Dumbaugh, G.D., The Induced Vertical Flow of Bulk Solids from Storage. Bulk Solids Handling, 4: Roberts, A.W. and Scott, O.J., An Investigation into the Effects of Sinusoidal and Random Vibrations on the Strength and Flow Properties of Bulk Solids, Powder Technology, 21: Roberts, A.W., Vibration of Powders and its Application, M.E. Fayed and L. Otten (eds.), Handbook on Powder Science and Technology, Van Nostrand, Jenike, A.W., Storage and flow of Solids, Bul. 123, University of Utah, Utah Engineering Station. Standard shear testing technique for particulate solids using the Jenike shear cell, Institution of Chemical Engineers, European Federation of Chemical Engineering. Kollmann, Th. and Tomas, J., 2. On the vibrational flow of fine powders, The 3 rd Israeli Conference for Conveying and Handling of Particulate Solids, The Dead Sea, Israel, Kollmann, Th. and Tomas, J., 21a. Vibrational flow of fine powders, Handbook of Conveying and Handling of Particulate Solids, Kalman, H. and Levi, A., Elsevier, Kollmann, Th. and Tomas, J., 21b. The Influence of Applied Vibration on the Friction of and Titania Powders. J.H. Adair, V.M. Puri, K.S. Haris and C.C. Huang (eds.): Fine Powder Processing 1, Penn State University, PA, USA, Tomas, J., Modellierung des Fließverhaltens von Schüttgütern auf der Grundlage der Wechselwirkungskräfte zwischen den Partikeln und Anwendung bei der Auslegung von Bunkeranlagen, Habilitation, TU Bergakademie Freiberg. Tomas, J., Particle Adhesion Fundamentals and Bulk Powder Consolidation, Reliable Flow of Particulate Solids III, Porsgrunn, Tomas, J., 2. Particle Adhesion Fundamentals and Bulk Powder Consolidation, KONA, No 18: Molerus, O., Theory of yield of cohesive powders, Powder Technology, 12:

16 APPENDIX TESTING CONDITIONS Powder Test procedure in kpa A Titania A B Titania B 1.9 / stainless steel σ in kpa.8 / 1.1 / / 2.4 / / 4.8 / / 9.7 / / 1.1 / / 2.4 / / 4.8 / wall friction for each normal load σ * for each vibration frequency No. of vibrated shear tests , * > 1 + > 1 + > > 1 * > 1 > 1 Frequency in Hz 25/5//2/3 5 / / 2