Intermezzo 1 - Settling velocity of solid particle in a liquid
Intermezzo I. SETTLING VELOCITY OF SOLID PARTICLE IN A LIQUIDI.1 TERMINAL SETTLING VELOCITY OF A SPHERICALPARTICLE, vtsA balance of the gravitational, buoyancy and drag forces on the submerged solidbody determines a settling velocity of the body.Figure 1.7. Force balance on a solid body submerged in a quiescent liquid.I.1.1 General equation for settling velocityThe terminal settling velocity of a spherical particle, vts, in a quiescent liquid is aproduct of a balance between the submerged weight of a solid particle in the liquid thedrag force of the liquid acting onto the settling particle in the direction opposite to thesettling velocity vector:GRAVITY FORCE - BUOYANCY FORCE = DRAG FORCE, i.e. msg - mfg SUBMERGED WEIGH = DRAG FORCE Volume.ρsg - Volume.ρfg πd 3 (ρs − ρf )g = C D πd 2 v 2 ρf ts (I.1) 6 8and the terminal settling velocity is I.1
I.2 INTERMEZZO I 4 (ρs − ρ f ) gd v ts = (I.2) 3 ρf CD vts terminal settling velocity of a spherical solid particle [m/s] ρs density of solid particle [kg/m3] ρf density of liquid [kg/m3] g gravitational acceleration [m/s2] d diameter of a particle [m] CD drag coefficient of flow round settling particle [-]The drag coefficient CD is sensitive to a regime of the liquid flow round the settlingsolid particle and this can be expressed by a relationship CD = fn(Rep). Rep is the v dparticle Reynolds number Re p = ts . νfI.1.2 Application of the general equation to different regimes of particle settlingIn a laminar regime (obeying Stokes law and occurring for Rep < 0.1, i.e.sand-density particles of d < 0.05 mm approximately) the relation is hyperbolic, CD =24/Rep, so that (ρ − ρf ) gd 2 v ts = s (I.3). 18ρ f νfIn a turbulent regime (obeying Newtons law and occurring for Rep >500, i.e.sand-density particles of d > 2 mm approximately) the drag coefficient is no longerdependent on Rep because the process of fast settling of coarse particles is governedby inertial rather than viscous forces. CD = 0.445 for spherical particles and theterminal velocity is given by (ρs − ρf ) v ts = 1.73 gd (I.4). ρfThese two regimes are connected via a transition regime. In this regime a CD valuemight be determined from the empirically obtained curve CD = fn(Rep) (see Figure1.8) or its mathematical approximation (Turton & Levenspiel, 1986) 24 0.657 + 0.413 CD = 1 + 0.173 Re p (I.5). Re p 1 + 1.63x104 Re −1.09 pThe determination of vts for the transition regime requires an iteration [vts = fn(CD)is an implicit equation]. Grace (1986) proposed a method for a determination of vtswithout necessity to iterate.
SETTLING VELOCITY OF SOLID PARTICLE IN A LIQUID I.3I.1.3 The Grace method for the solution of the settling-velocity equationThe Grace method uses two dimensionless parameters ρ (ρ − ρ f )g- the dimensionless particle size: d* = d.3 f s 2 µf 2 ρf- the dimensionless settling velocity: v * = v ts 3 ts µ f (ρs − ρ f )g Figure I.1. Dimensionless terminal velocity, v*ts, as a function of dimensionless particle diameter, d*, for rigid spheres, after Grace (1986).
I.4 INTERMEZZO IThose are mutually related as shown in Figure I.1. Thus using the curve andrearranging gives directly the velocity vts as a function of particle diameter d. Noiteration is required.The curve in Figure I.1. can be also described by analytic expressions (Table I.1.)appropriate for a computational determination of vts according to Grace method. Table I.1. Correlations for vts* as a function of d*, after Grace (1986). Range Correlation d* ≤ 3.8 vts* = (d*)2/18 - 3.1234x10-4(d*)5 + 1.6415x10-6(d*)8 3.8 < d* ≤ 7.58 logvts* = -1.5446 + 2.9162(logd*) - 1.0432(logd*)2 7.58 < d* ≤ 227 logvts* = -1.64758 + 2.94786(logd*) - 1.0907(logd*)2 + 0.17129(logd*)3 227 < d* ≤ 3500 logvts* = 5.1837 - 4.51034(logd*) + 1.687(logd*)2 - 0.189135(logd*)3I.2 TERMINAL SETTLING VELOCITY OF A NON-SPHERICAL PARTICLE, vtThe above equations are suitable for the spherical particles. The particles of dredgedsolids are not spherical. The non-spherical shape of a particle reduces its settlingvelocity.I.2.1 Shape factor of a solid particleThe reduction of settling velocity due to a non-spherical shape of a solid particle isquantified by the velocity ratio vt ξ= (I.6) v ts ξ shape factor of a solid particle [kg/m3] vt terminal settling velocity of a non-spherical solid particle [m/s] vts terminal settling velocity of a spherical solid particle [m/s].
SETTLING VELOCITY OF SOLID PARTICLE IN A LIQUID I.5I.2.1.1 The Grace method for a determination of the shape factorξ is a function of the volumetric form factor K (K=0.26 for sand, gravel) and the ρ (ρ − ρ f )gdimensionless particle diameter d* = 3 f s .d (see Figure I.2.). The 2 µfterminal velocity for sand particles is typically 50-60 % of the value for the sphere ofthe equivalent diameter (see Table I.3 in paragraph I.4). Figure I.2. The shape factor, ξ, as a function of the dimensionless particle diameter d* and the volumetric form factor K.The curves in Fig. I.2 can be analytically approximated by the following correlation − 0.045+ 0.05K−0.6 − 0.0287(55000)K −0.524log10ξ = −0.55 + K − 0.0015K− 2 + 0.03(1000)K −0.524 + cosh 2.55log10d* − 1.114 This equation takes a simpler form for sand-shape particles (K=0.26) 0.0656 log10 ξ = −0.3073 + (I.7). cosh 2.55log10 d* − 1.114 I.2.1.2 The estimation of the shape factor value for sand and gravelA different method suggests the ξ value of about 0.7 for sand and gravel so that vt issimply vt = 0.7vts.
I.6 INTERMEZZO II.2.2 Empirical equations for settling velocity of sand and gravel particlesThe above methods for a determination of the terminal velocity of spherical andnon-spherical particles are supposed to be valid for all sorts of solids settling in anarbitrary fluid. In a dredging practice the terminal settling velocity is often determinedusing simple equations found specifically for sand particles. These equations need noiteration.Warning: in the following equations the particle diameter, d, is in mm, and thesettling velocity, vt, in mm/s.For the laminar region, considered for sand particles smaller than 0.1 mm, the Stokesequation is written as v t = 424 (Ss − Sf ) d 2 (I.8). SfFor the transition zone, 0.1 mm < d < 1 mm, the Budryck equation is used 8.925 (Ss − Sf ) 3 vt = 1 + 95 d − 1 (I.9). d Sf The turbulent regime of settling is estimated to occur for sand particles larger than 1mm. This is described by the Rittinger equation (Ss − Sf ) v t = 87 d (I.10). Sf Ss specific density of solids, ρs/ρw [-] Sf specific density of liquid, ρf/ρw [-]
SETTLING VELOCITY OF SOLID PARTICLE IN A LIQUID I.7 500 Terminal settling velocity, vt [mm/s] 200 100 50 20 10 0.1 0.2 0.5 1 2 5 10 Particle diameter, d [mm] Figure I.3a. Terminal settling velocity of sand & gravel particles using Stokes, Budryck and Rittinger equations.
I.8 INTERMEZZO I 0,1valsnelheid in water v (mm/s) 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 klasse 1 2,0 3,0 4,0 5,0 . 8925 6,0 v= ⋅ 1 − 95 ⋅ ( 2.65 − 1) ⋅ d3 − 1 [Budryck] 7,0 d 8,0 9,0 10,0 v= 87 ⋅ ( 2,6 klasse 2 20,0 5− 1) ⋅ d [Ri 30,0 ttin ger 40,0 ] 50,0 60,0 70,0 80,0 90,0 v= 100,0 klasse 3 4 24 ⋅ (2 200,0 ,6 5 − 1) 300,0 ⋅d 400,0 6 500,0 [ St o 600,0 700,0 ke s 800,0 900,0 ] 1000,0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00 10,00 korrelgrootte d (mm) Figure I.3b. Terminal settling velocity of sand & gravel particles using Stokes, Budryck and Rittinger equations.
SETTLING VELOCITY OF SOLID PARTICLE IN A LIQUID I.9I.3 HINDERED SETTLING VELOCITY OF A PARTICLE, vthWhen a cloud of solid particles is settling in a quiescent liquid additional hinderingeffects influence its settling velocity. These are the increased drag caused by theproximity of particles within the cloud and the upflow of liquid as it is displaced bythe descending particles. The hindering effects are strongly dependent on thevolumetric concentration of solids in the cloud, Cv, (Richardson & Zaki, 1954) v th = v t (1 − C v )m (I.11),in which vth is the hindered settling velocity of solid particle, vt is the terminalsettling velocity of the solid particle and m is the empirical exponent related to theparticle Reynolds number Rep (see Table I.2). Table I.2. Values for the Richarson-Zaki index m vd Re p = t m νf Rep ≤ 0.2 4.6 0.2 < Rep < 1 4.4Rep-0.03 1 < Rep < 500 4.4Rep-0.1 500 ≤ Rep 2.4Wallis (1969) suggested determining the index m by the approximation 4.71 + 0.15 Re0.687 p m= (I.12) 0.687 1 + 0.253 Re pcovering all ranges of Rep values.Theoretically, the validity of the Richardson & Zaki equation is limited by themaximum solids concentration that permits solids particle settling in a particulatecloud. This maximum concentration corresponds with the concentration in anincipient fluidized bed (Cv of about 0.57). Practically, the equation wasexperimentally verified for concentrations not far above 0.30.
I.10 INTERMEZZO II.4. TYPICAL VALUES OF PARAMETERS DESCRIBING A SETTLING PROCESS FOR SAND PARTICLE Table I.3. Values of parameters associated with a particle settling for quartz sand particles (ρs = 2650 kg/m3, shape factor K = 0.26) of various sizes in water at room temperature. d [mm] vts [mm/s] Rep ξ m Grace Table I.2. (1986) 0.074 4.75 0.35 0.51 4.54 0.149 15.6 2.32 0.53 4.04 0.297 41.0 12.1 0.57 3.43 0.595 96.7 57.4 0.59 2.93 1.000 160 160 0.55 2.65 2.000 276 551 0.52 2.40I.5. REFERENCESGrace, J.R. (1986). Contacting modes and behaviour classification of gas-solid andother two-phase suspensions. Can. J. Chem. Eng., 64, 353-63.Richardson, J.F. & Zaki, W.M. (1954). Sedimentation and fluidisation: Part I,Trans. Instn Chem. Engrs, 32, 35.Turton, R. & Levenspiel, O. (1986). A short note on the drag correlation for spheres.Powder Technol., 47, 83-6.Wallis, G.B. (1969). One Dimensional Two Phase Flow. McGraw-Hill.I.6. RECOMMENDED LITERATUREClift, R., Grace, J.R. and Weber, M.E. (1978). Bubbles, Drops and Particles.Academic Press.Coulson, J.M., Richardson, J.F., Backhurst, J.R. & Harker, J.H. (1996).Chemical Engineering. Vol. 1: Fluid Flow, Heat Transfer and Mass Transfer.Butterworth-Heinemann.Wilson, K.C., Addie G.R., Sellgren, A. & Clift, R. (1997). Slurry Transport UsingCentrifugal Pumps. Chapman & Hall, Blackie Academic & Professional.
SETTLING VELOCITY OF SOLID PARTICLE IN A LIQUID I.11CASE STUDY ITerminal and hindered settling velocities of grains of sand and gravel.Estimate the terminal settling velocity, vt, and hindered settling velocity, vth, forgrains of fine sand (grain size d = 0.120 mm) and gravel (d = 6.0 mm) in thequiescent column of water at room temperature (for calculations take water density ρf= 1000 kg/m3 and kinematic viscosity νf = 10-6 m2/s). For both sand and gravelgrains the following values are typical:solids density ρs = 2650 kg/m3volumetric shape factor K = 0.26.For hindered settling consider a cloud of particles settling in a water column. Settlingparticles occupy 27% per cent of the total volume of the column so that thevolumetric concentration of solids Cv = 0.27.Solution:a. Terminal settling velocity according to GraceThe Grace fitting correlations (Table I.1.) give for the terminal settling velocity (vts)of a sphere of d = 0.120 mm the value vts = 11.21 mm/s while for a sphere of d = 6mm the value vts = 554.13 mm/s. A very similar result is reached if the Grace graph(Figure I.1.) is used: 1000x1650x 9.81d* = 0.0001203 = 3.04 gives v*ts = 0.44 for the 0.120 mm sand 0.000001and 1000x1650x 9.81d* = 0.0063 = 152 gives v*ts = 22 for the 6 mm gravel. 0.000001 0.001x1650x 9.81Then v ts = v* 3 ts that gives 1000 2vts = 11 mm/s for fine sand and vts = 557 mm/s for medium gravel.The velocity ratio ξ is determined using the fitting correlation (I.7). This gives ξ =0.522 for the fine sand and ξ = 0.503 for the studied gravel. Again, very similarresults should be obtained using the ξ graph (Figure I.2.).Using vt = ξvts givesfor fine sand of d = 0.120 mm the velocity vt = 5.85 mm/s andfor medium gravel of d = 6 mm the velocity vt = 278.52 mm/s.
I.12 INTERMEZZO Ib. Terminal settling velocity according to Budryck, Rittinger respectivelyAccording to this method (see Eqs. I.8-I.10) the terminal settling velocity of a solidgrain vt is calculated directly, thus without intermediation of the velocity vts. Theequations 8.925 2650 − 1000 vt = 1 + 95 0.1203 − 1 = 9.47 mm/s for the fine sand of 0.120 1000 diameter d = 0.120 mm (Budryck, Eq. I.9) and 2650 − 1000v t = 87 6 = 273.74 mm/s for the medium gravel of diameter d = 6 mm 1000(Rittinger, Eq. I.10).For gravel the result obtained from the Rittinger equation is very similar to that givenby Grace. For fine sand, however, the Grace method predicts considerably lowerterminal settling velocity than the Budryck equation.c. Hindered settling velocity according to Richardson & ZakiIf volumetric concentration of solids is known, the hindered settling velocity can bedetermined using the Richardson & Zaki equation (Eq. I.11). The index m in thisequation is related to the particle Reynolds number Rep. A use of the vt valuesaccording to Budryck and Rittinger (see above) givesRep = 120 x 10-6 x 9.47 x 10-3 / 10-6 = 1.14 for our fine sand andRep = 6 x 10-3 x 0.274 / 10-6 = 1642 for our gravel.Wallis correlation (Eq. I.12) provides m = 4.286 (fine sand) and m = 2.832 (gravel).The Richardson-Zaki equation givesfor fine sand: vth = 9.47(1-0.27)4.286 = 2.46 mm/s andfor medium gravel: vth = 273.74(1-0.27)2.832 = 112.27 mm/s.Summary of the results:Fine sand (d = 0.12 mm):terminal settling velocity: vt = 9.47 mm/shindered settling velocity (Cv = 0.27): vth = 2.46 mm/sMedium gravel (d = 6.00 mm):terminal settling velocity: vt = 273.74 mm/shindered settling velocity (Cv = 0.27): vth = 112.27 mm/s