survey of shape factor effects in fall velocity of individual particles in to sedimentary flows

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1 International Research Journal of Applied and Basic Sciences 13 Available online at ISSN X / Vol, 4 (3): Science Explorer Publications survey of shape factor effects in fall velocity of individual particles in to sedimentary flows roozbeh riazi 1, mahmood shafai bejestan 2 1. roozbeh riazi, islamic azad university, dehdash branch, dehdasht, iran 2. mahmood shafai bejestan, professor, faculty of water science engineering, shahid chamran university, ahwaz, iran * Corresponding Author roozbehriazi@gmail.com ABSTRACT: the fall velocities of sedimentary particles are important variants in estimate of sediment rate of wash load and bed load. this parameter has an important role in designing and physical modeling of settling pools, so that estimating the exact rate of it, is very important and if the exact estimating not to be done, it can be caused of much errors to estimate sediment stilling or designing of sedimentary pool. in this research, with classification of sediments in four different shape factors (sf=1, 0.7 sf 0.8, 0.5 sf 0.6, 0.2 sf 0.3) and three size of sedimentary particle (1 inch, 0.5 inch, 3/8 inch), the effect of shape factor on fall velocity obtained with measuring of fall velocity of particles in certain denseness classification (dense 2.18 gram in a liter, dense 5.32 gram in a liter and dense gram in a liter). then with exploration of dimensionless equations and with obtaining the results of fall velocity for different categories of particles, profiles and tables (charts) plotted and finally relations for fall velocity of sedimentary particles were obtained. key words: dense rate, fall velocity, shape factor, sedimentary particles, wash load, bed load INTRODUCTION mechanic of spheres falling in water can be surveyed simply for a sphere which falls with constant velocity in static water. considering the forces exerted on a sphere falling in the static water, it can be written as following (ning et al., 1999): 3 πd m = ( γ s γ ) 6 (1) πd 2 ρω 2 F = CD (2) 4 2 in which is fall velocity and c d is drag coefficient and other parameters include d the particle size, and s are the real specific weight of water and particle respectively, m is particle's weight and f is drag force. in equilibrium conditions and stable velocity it can be written as: γ s γ ω =.. gd 3 C γ D (3) the value of c d is a function of reynolds number (d/). if the reynolds number is less than 0.1, initial orientation of spheres is constant and the particles will fall constantly. in the values of larger reynolds number, the particle fluctuates and begins to spin falling and may never reach to a fixed direction. experiments showed that if r e is in the limit of 10 to 10 5, the particles will fall along the c axis (the smallest axis of particle) (ning et al., 1999). in formulas of fall velocity, the confine is usually considered to be without boundary, but this situation is often not close to reality. especially in experimental researches on fall velocity, the container's size is small; then it is necessary for the boundary effect on fall velocity to be obtained. if the ratio of sedimentary particles' diameter to cylinder diameter (d/d 0 ) is small, then resistance effect will be reduced. increasing of d/d 0 in the range of high reynolds in stockes area causes to increase effect. in

2 summarized we can say that if the particles' distance to container's body is between 2 to 3 cm, then the boundary existence can be ignored for small particles in stockes relation (ning et al., 1999). the effect of concentration on fall velocity if the water contains a large number of sedimentary particles, sedimentation of each particle will be affected by surrounding particles. the effect of these changes is equivalent to increasing viscosity; so fall velocity of particles will decreases (alger et al., 1968). in stockes range, two analyses can be applied for sedimentation in low concentration. both types of analyses produce the following formula essentially with ignoring more forms (shafaei, 1994): ω0 D = 1+ k ω S (4) in which is fall velocity of sedimentary particles in s v concentration and 0 is fall velocity of sedimentary particles in zero concentration. s is also the effective distance between contiguous particles. the ratio of d/s is attributed to volume density (s v ) : SV D = (5) S different scholar are considered different values for coefficient of k which is shown in the table 1: Table 1. The Values Of K Coefficient In Equality Of / 0 author burgers smoluchowski cai cunningham mcnown uchida k the effect of shape factor on fall velocity shape factor is the smallest axis of the particle multiplied by the square root of the mean axis and the major axis of the particle. (shafaei, 1994) c = (6) a. b vanoni (1975) presented the relationship between particle shape factor and fall velocity graphically figure 1. relationship between particle shape factor and fall velocity vanoni (1975) dimensional analysis according to involved parameters in fall velocity of particle () and using of buckingham's theory, then obtained results are as follows: gd µ C f (, Gs,, ) = 2 ω ρωd ρ (7) 584

3 in this equation gd/ 2 is inverse to square of froude number and /d is also inverse to reynolds number. this equation can be written as follows: ω C = f (, G s,, Re ) gd ρ (8) in which sf is the shape factor. further, g s is specific density of sedimentary particles and c is suspense load density and is also mass density. considering the size of particles used in this research (table2), the value of re in this research, is very large and therefore its effect can be nonsignificants. MATERIALS AND METHODS the hydraulic model was used for the experiment and natural samples were collected from zohreh river and spherical samples were compared with natural samples as for instance. three sizes were considered for the experiment; therefore, the preparation of spherical sample included these same sizes. three considered sizes including the particles collected on the sieve were 1 and half and also three-eighths inches. the model includes sedimentation cylinder made of plexiglas with height of 289 cm and cm diameter. also sediment storage made of iron and as it is shown in figure 2, the storage is a cylinder with diameter of 78 cm and a bottom having moderate slope with discharge valve. figure 2. view of model for exploration effect of shape factor, samples were categorized to specific shapes factors. according to formula of schulz et al.(c/ ab), the value of the shape factor (sf) were defined for these samples. the range and changes include the shape factor, particles' sizes, and also experimented densities which is shown in table 2. Table 2. shape factor, particle size and experimented densities particle size (inch) shape factor density (gr/lit) sf sf /8 0.7 sf sf= for each experiment, the samples were placed in direction of its largest diameter and fall time of each particle was measured and the value of fall velocity was calculated with having distance of fall's height. then fall velocity was obtained. for doing each experiment, at first considered velocity were made in sediment storage and was mixed well with the mixer installed above the storage tank. after ensuring from complete mixing, the flow was entered into the test cylinder. then sedimentary particles were dropped individually into the tank, the time was measured with a stopwatch. then sedimentation with other density was prepared and the mentioned experiment was repeated. RESULTS AND DISCUSSION 585

4 the examples shows that the fall velocity of particles was increase with increasing of shape factor and also the fall velocity was decrease with decreasing of density. in order to study of the froude number of particle, / gd in figure (3), (4) and (5) are plotted against any dimensionless parameters of equitation (8)(sf, gs-1, c/). figure (3) shows the changes of particle's froude number against c/. as can be seen, the value of particle's froude number decreased with increasing of density. then it will place on a constant value. / gd c/ 0.3=<=> =<=> =<=>0.7 1= figure 3. particle's froude number variation against c/ / gd c/p= 2.18c/p= figure 4. particle's froude number variation against sf / gd Gs-1 0C= 14.24C= figure 5. particle's froude number variation against g s -1 for each density, variation of fall velocity was plotted against the shape factor in figure (6), (7), (8) and (9). as can be seen, the value of fall velocity was decrease with decreasing of diameter. also the fall velocity was decrease with decreasing of density. 586

5 Pure Water figure 6. fall velocity variation against sf for c=0 gr/lit C=2.18 gr/lit 170 figure 7. fall velocity variation against sf for c=2.18 gr/lit C=5.32 gr/lit 70 figure 8. fall velocity variation against sf for c=5.32 gr/lit C=14.24 gr/lit figure 9.fall velocity variation against sf for c=14.24 gr/lit 587

6 in figure 6 to 9 we can deduction that the fall velocity was increase with increasing of shape factor and maximum value of the fall velocity was occur in sf=1. this means that if shape of a particle more spherical, the drag force is lower and fall velocity is more than the other shapes. finally, the data obtained with spss statistical software were reviewed and variation of / gd to c/, sf and gs -1 were obtained and the final formula was obtained from output of software as follows: this formula was influence for fall velocity of sedimentary particles in sedimentary flows. the r square is ω C = 0.02 ( ) (9) gd( G s 1) ρ comparison of above formula with results of experiment shows in figure 10. as can be seen in the figure, the results show relatively well correspondence with the experimental results. observe and calculate fall velocity is closer for sf=1. this is because of the assumption of average shape factor for each interval (for example, shape factor of 0.75 is used in calculation of shape factor between 0.7 to 0.8) /gd(gs-1)=0.02***1.066*(c/)** R**2=0.754 Observe Calculate Figure 10. Comparison of calculated data from formula with results of experiment Conclusion According to mentioned items and presented figures, the following results are obtained from the discussed research: The fall velocity of particles will increase with increasing of shape factor. Velocity variation toward shape factor will increase with increasing density. In constant diameters, the value of sphere's fall velocity will decrease with closing value of sf=1. The effect of density will decrease with its increasing. The fall velocity will decrease with decreasing of density. The fall velocity will decrease with decreasing of sphere diameter for each particular shape factor. The effect of density on fall velocity will be high in lower densities. REFERENCES alger gr, simmons db fall velocity of irregular shaped particles. hydraulic eng div j. 94: ning q, zhaohui w mechanics of sediment transport. in: american society of civil engineers, 1 st edn. new york. riazi r. 06. survey of sediment denseness effects in fall velocity of individual particles. msc thesis. islamic azad university. science and research branch, ahwaz, iran. (in farsi). schultz ef, wilde rh, albertson ml influence of shape on fall velocity of sedimentary particles. report for the missouri river div. corps of engineers, u.s. army, through colorado research foundation. fort collins co. shafaei bajestan m sediment hydraulic. in: shahid chamran university press, 1 st edn. ahwaz, iran. shafaei bajestan m. 05. physical and hydraulical modeling. in: shahid chamran university press, 1 st edn. ahwaz, iran. vanoni va sedimentation engineering. in: american society of civil engineers, 1 st edn. new york. 588