# Initial Velocity Effect on Acceleration Fall of a Spherical Particle through Still Fluid

Authors: Zegao Yin; Zhenlu Wang; Bingchen Liang; Li Zhang
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

A spherical particle’s acceleration fall through still fluid was investigated analytically and experimentally using the Basset-Boussinesq-Oseen equation. The relationship between drag coefficient and Reynolds number was studied, and various parameters in the drag coefficient equation were obtained with respect to the small, medium, and large Reynolds number zones. Next, some equations were used to derive the finite fall time and distance equations in terms of certain assumptions. A simple experiment was conducted to measure the fall time and distance for a spherical particle falling through still water. Sets of experimental data were used to validate the relationship between fall velocity, time, and distance. Finally, the initial velocity effect on the total fall time and distance was discussed with different terminal Reynolds numbers, and it was determined that the initial velocity plays a more important role in the falling motion for small terminal Reynolds numbers than for large terminal Reynolds number scenarios.


---

## Body

## 1. Introduction

Since the middle of the 19th century, a solid particle falling in gases or liquids has been investigated in many fields, such as hydraulics and chemistry. Considerable attention has been given to the steady-state motion of a spherical particle falling in incompressible Newtonian fluids, especially for the relationship between drag coefficientCD and Reynolds number R. Stokes [1] neglected the inertia item in the Navier-Stokes equations and proposed an inversely proportional relationship between CD and R with R<0.1 scenarios for the first time. Then, the inertia item was taken into account by Oseen [2, 3], and an improved relationship was proposed for R<0.4 scenarios. Then, the perturbation theory was used to obtain a CD formula for R<1 scenarios [4, 5]. A more recent study on CD correlations with R can be found in the works of Chester et al., Clift et al., and Liao and Julien [6–9]. It should be noted that most of these aforementioned correlations had little deviations [10].In contrast to the steady-state fall of a particle, an unsteady-state fall has an acceleration process before it reaches the terminal fall velocity. Chang and Yen [11] discussed the importance of added mass and the Basset force of a sphere falling for a low sphere-to-fluid density ratio and low R. Jalaal et al. [12] used the homotopy perturbation method to derive the fall velocity, acceleration, and distance equations of a spherical particle falling unsteadily in a Newtonian fluid, ignoring the effect of the Basset force. Guo [13] analysed the acceleration fall behaviour of a sphere through still fluid for arbitrary Reynolds numbers and presented a simple solution for fall distance and velocity. Yin et al. [14] neglected the Basset force effect and experimentally investigated a spherical particle acceleration fall through still water with large R.The objective of our study was to investigate the initial velocity effect on a spherical particle acceleration fall through still fluid analytically and experimentally. In this study, the relationship betweenCD and R was investigated, and different parameters in the CD equation were obtained with respect to small, medium, and large R zones. Then, the finite fall time and distance equations were derived in terms of some assumptions by using the results reported by Guo [13]. Furthermore, a simple experiment was conducted to measure the fall time and distance for a spherical particle falling through still water. In addition, sets of experimental data were used to validate the relationship between fall velocity, time, and distance. The initial velocity effect on fall time and distance was discussed with different terminal Reynolds numbers. Finally, the conclusions are summarized.


## 2. Kinetic Equation of an Acceleration Fall

### 2.1. Equation Derivations

Concerning a spherical particle falling with acceleration through still fluid, the classic Basset-Boussinesq-Oseen (BBO) equation has often been used to depict the behaviour of the fall [3, 11–13, 15–17].(1)msdwdt=ms-mg-CD2ρw2πD24-CA0mdwdt-CBD2πρμ1/2∫0tdwdτdτt-τ1/2,where the left-hand side of (1) is the inertia of a spherical particle. The first, second, third, and fourth terms on the right-hand side are its submerged weight, drag, added mass force, and Basset history force, respectively. ms is the spherical particle mass, w is the fall velocity, t is the time, m is the mass of fluid for the same volume of spherical particle, g is the gravitational acceleration, ρ is the fluid density, D is the diameter of spherical particle, CD is the drag coefficient, CA0 is the classic added mass coefficient, CB is the classic Basset force coefficient, CB=3/2, μ is the dynamic viscosity of fluid, and τ is the dummy variable for integration.The submerged weight is given by subtracting the buoyancy force from the particle weight, and it is well accepted that this term plays an important role in the total force [9]. The contribution of drag force is decided by CD to a great extent. Considerable attention has been given to its value with different R (R=wD/ν), in which ν is the kinematic viscosity of fluid [1–3, 7, 11–13]. In terms of R, it can be classified as small, medium, and large numbers, and the corresponding critical values are assumed to be Rcr1=wcr1D/ν=0.1 and Rcr2=wcr2D/ν=1000 [13, 18], where wcr1=0.1ν/D is defined as the first critical velocity between small and medium R zones, wcr2=1000ν/D is defined as the second critical value between medium and large R zones, and their corresponding times were defined as tcr1 and tcr2, respectively. The relationship between CD and R can be expressed as the following general expression with Rubey’s drag law [19]: (2)CD=βR+α.For smallR in a laminar zone (R<0.1), α=0 and β=24 [1]. For large R in a turbulent zone (1000<R<105), α=0.44±0.06 and β=24, which agreed well with the experimental data over a wide range of Reynolds numbers [13]. As for medium R in a transitional zone (0.1<R<1000), Lapple and Shepherd [20] stated that CD decreases with increasing R: (3)CD=241+0.125R0.72R.The range ofCD deviation in (3) is −8% to 5% [7]. Using (3), results for R range from 0.1 to 1000 at a step of 0.1 were obtained, and (3) was fitted as(4)CD=25.057R+0.5.The correlation coefficient is 0.997. Therefore, (4) can be used to depict the relationship between CD and R in a transitional zone without causing major error.Concerning the added mass force term, it deserves a brief description. One important property of the solid-fluid interaction is that the solid is not entirely free to move within the fluid. As the solid particle moves from one position to the other, an equal volume of fluid must move in the opposite direction. So if the mass of solid particle is accelerated, the mass of an equal volume of fluid must also be accelerated. And this concept is referred to as the added mass [9]. In (1), CA0 is equal to 1/2 for the creeping motion after the integration of the potential unsteady pressure over the sphere’s surface [21]. Hamilton and Lindell [22] also concluded that CA0 is consistently experimentally close to 0.5 for R values up to 35000, by neglecting the sphere acceleration effect. In contrast, CA0 increases with R because of flow separation [9]. Odar [23] further correlated CA0 using an instantaneous acceleration modulus; nevertheless, the modulus is arduous to obtain explicitly.The Basset force contribution to total force was determined by the density ratioρ/ρs to a considerable extent, where ρs is the density of the particle. It was found that the Basset force is always important in the calculations for ρ/ρs>1 scenarios. For some heavy particles in air such as ρ/ρs<0.004, the Basset force plays an insignificant role in the total force. The Basset force contribution for 0.004<ρ/ρs<1 depends on the type of particles and on the frequency of the fluid velocity fluctuations [24]. Particular attention has been devoted to the importance of the Basset force compared to the other hydrodynamic forces [25]. Lawrence and Weinbaum [26] investigated the unsteady force on a sphere at low Reynolds number and pointed out that the Basset force for an arbitrary velocity contained a new memory integral whose kernel differed from the classical behaviour derived by Basset [15]. Sobral et al. [27] considered the unsteady motion of a rigid spherical particle in fluid flows and found that its motion is significantly affected by the Basset force on the early stages of the motion and on the approach to the steady state. Due mainly to the complexity of the Basset force, a series of numerical methods were employed to resolve it, such as the Laplace transform method [28], the “window model” [29], and fractional-derivative approach [30]. However, its computation was far more expensive than other items in the BBO equation because of the dramatic demands of computational time and memory. In contrast to the numerical solutions, its analytical solution has advantages including simplicity and reliability. However, the analytic solutions of a spherical particle falling in still water were still not found. Attempts to resolve this dilemma were made for the theoretical solution of the Basset force and BBO equation with a simplified method. Guo [13] assumed dw/dτ to be a constant to separate it from the definite integral and D2(πρμ)1/2∫0t(dτ/(t-τ)1/2) as m in the Basset force item. He empirically combined the added mass force and Basset force into one integrated term, CAm(dw/dt) (CA is the integrated added mass coefficient), and the fall velocity was expressed as (5)w=2c-bw0+kw0coth⁡kt-t0/2b+2aw0+kcoth⁡kt-t0/2,where a=(3α/4)(1/Δ+CAD), b=(3β/4)(v/Δ+CAD2), c=(Δ-1)g/(Δ+CA), k=b2+4ac,  Δ=ρs/ρ, t0 is the initial time, and w0 is the initial velocity. The terminal fall velocity, wT, can be deduced with (5) for t→∞: (6)wT=k-b2a.Using (5) with wcr1=0.1ν/D or wcr2=1000ν/D, tcr1 and tcr2 can be expressed as (7a)tcr1=1kln⁡bν+0.2aD+kνb+2aw0-kbν+0.2aD-kνb+2aw0+k+t0(7b)tcr2=1kln⁡bν+2000aD+kνb+2aw0-kbν+2000aD-kνb+2aw0+k+t0.The fall distance was expressed as(8)s=1alnk-b-2aw0e-kt-t0+k+b+2aw02k+k-bt-t02a.Assuming an initial fall velocity of zero, Guo [13] derived the following relationship between fall distance and fall velocity:(9)s=12aln⁡cc-bw-aw2-2bktanh-1⁡kw2c-bw,where tanh-1 is the inverse hyperbolic function.


### 2.2. Finite Values of Acceleration Fall Time and Distance

Fort→∞, the fall distance in (8) approaches an infinite value. In order to simplify the study, the finite values of fall time and distance were expressed with the following assumptions.Forw0<0.99wT scenarios, the acceleration fall process was assumed to be completed when the fall velocity reached 99% of the terminal fall velocity. Applying (6) and w=0.99wT to (5) gives the corresponding time as follows:(10)tS=1kln⁡1.99k+0.01bb+2aw0-k0.01b-0.01kb+2aw0+k+t0.Substituting (10) into (8) gives (11)sS=k+b2aln⁡b+2aw0+k1.99k+0.01b+k-b2akln⁡b+2aw0-k0.01b-0.01k.Forw0>1.01wT scenarios, the acceleration fall process was assumed to be completed when the fall velocity reached 101% of the terminal fall velocity. Applying (6) and w=1.01wT to (5) gives the corresponding time as follows: (12)tL=1kln⁡2.01k-0.01bb+2aw0-k0.01k-0.01bb+2aw0+k+t0.Substituting (12) into (8) gives (13)sL=k+b2aln⁡b+2aw0+k2.01k-0.01b+k-b2akln⁡b+2aw0-k0.01k-0.01b.Note that the scenarios of0.99wT⩽w0⩽wT or wT⩽w0⩽1.01wT are not addressed in our study because they are very close to the terminal fall velocity.


## 2.1. Equation Derivations

Concerning a spherical particle falling with acceleration through still fluid, the classic Basset-Boussinesq-Oseen (BBO) equation has often been used to depict the behaviour of the fall [3, 11–13, 15–17].(1)msdwdt=ms-mg-CD2ρw2πD24-CA0mdwdt-CBD2πρμ1/2∫0tdwdτdτt-τ1/2,where the left-hand side of (1) is the inertia of a spherical particle. The first, second, third, and fourth terms on the right-hand side are its submerged weight, drag, added mass force, and Basset history force, respectively. ms is the spherical particle mass, w is the fall velocity, t is the time, m is the mass of fluid for the same volume of spherical particle, g is the gravitational acceleration, ρ is the fluid density, D is the diameter of spherical particle, CD is the drag coefficient, CA0 is the classic added mass coefficient, CB is the classic Basset force coefficient, CB=3/2, μ is the dynamic viscosity of fluid, and τ is the dummy variable for integration.The submerged weight is given by subtracting the buoyancy force from the particle weight, and it is well accepted that this term plays an important role in the total force [9]. The contribution of drag force is decided by CD to a great extent. Considerable attention has been given to its value with different R (R=wD/ν), in which ν is the kinematic viscosity of fluid [1–3, 7, 11–13]. In terms of R, it can be classified as small, medium, and large numbers, and the corresponding critical values are assumed to be Rcr1=wcr1D/ν=0.1 and Rcr2=wcr2D/ν=1000 [13, 18], where wcr1=0.1ν/D is defined as the first critical velocity between small and medium R zones, wcr2=1000ν/D is defined as the second critical value between medium and large R zones, and their corresponding times were defined as tcr1 and tcr2, respectively. The relationship between CD and R can be expressed as the following general expression with Rubey’s drag law [19]: (2)CD=βR+α.For smallR in a laminar zone (R<0.1), α=0 and β=24 [1]. For large R in a turbulent zone (1000<R<105), α=0.44±0.06 and β=24, which agreed well with the experimental data over a wide range of Reynolds numbers [13]. As for medium R in a transitional zone (0.1<R<1000), Lapple and Shepherd [20] stated that CD decreases with increasing R: (3)CD=241+0.125R0.72R.The range ofCD deviation in (3) is −8% to 5% [7]. Using (3), results for R range from 0.1 to 1000 at a step of 0.1 were obtained, and (3) was fitted as(4)CD=25.057R+0.5.The correlation coefficient is 0.997. Therefore, (4) can be used to depict the relationship between CD and R in a transitional zone without causing major error.Concerning the added mass force term, it deserves a brief description. One important property of the solid-fluid interaction is that the solid is not entirely free to move within the fluid. As the solid particle moves from one position to the other, an equal volume of fluid must move in the opposite direction. So if the mass of solid particle is accelerated, the mass of an equal volume of fluid must also be accelerated. And this concept is referred to as the added mass [9]. In (1), CA0 is equal to 1/2 for the creeping motion after the integration of the potential unsteady pressure over the sphere’s surface [21]. Hamilton and Lindell [22] also concluded that CA0 is consistently experimentally close to 0.5 for R values up to 35000, by neglecting the sphere acceleration effect. In contrast, CA0 increases with R because of flow separation [9]. Odar [23] further correlated CA0 using an instantaneous acceleration modulus; nevertheless, the modulus is arduous to obtain explicitly.The Basset force contribution to total force was determined by the density ratioρ/ρs to a considerable extent, where ρs is the density of the particle. It was found that the Basset force is always important in the calculations for ρ/ρs>1 scenarios. For some heavy particles in air such as ρ/ρs<0.004, the Basset force plays an insignificant role in the total force. The Basset force contribution for 0.004<ρ/ρs<1 depends on the type of particles and on the frequency of the fluid velocity fluctuations [24]. Particular attention has been devoted to the importance of the Basset force compared to the other hydrodynamic forces [25]. Lawrence and Weinbaum [26] investigated the unsteady force on a sphere at low Reynolds number and pointed out that the Basset force for an arbitrary velocity contained a new memory integral whose kernel differed from the classical behaviour derived by Basset [15]. Sobral et al. [27] considered the unsteady motion of a rigid spherical particle in fluid flows and found that its motion is significantly affected by the Basset force on the early stages of the motion and on the approach to the steady state. Due mainly to the complexity of the Basset force, a series of numerical methods were employed to resolve it, such as the Laplace transform method [28], the “window model” [29], and fractional-derivative approach [30]. However, its computation was far more expensive than other items in the BBO equation because of the dramatic demands of computational time and memory. In contrast to the numerical solutions, its analytical solution has advantages including simplicity and reliability. However, the analytic solutions of a spherical particle falling in still water were still not found. Attempts to resolve this dilemma were made for the theoretical solution of the Basset force and BBO equation with a simplified method. Guo [13] assumed dw/dτ to be a constant to separate it from the definite integral and D2(πρμ)1/2∫0t(dτ/(t-τ)1/2) as m in the Basset force item. He empirically combined the added mass force and Basset force into one integrated term, CAm(dw/dt) (CA is the integrated added mass coefficient), and the fall velocity was expressed as (5)w=2c-bw0+kw0coth⁡kt-t0/2b+2aw0+kcoth⁡kt-t0/2,where a=(3α/4)(1/Δ+CAD), b=(3β/4)(v/Δ+CAD2), c=(Δ-1)g/(Δ+CA), k=b2+4ac,  Δ=ρs/ρ, t0 is the initial time, and w0 is the initial velocity. The terminal fall velocity, wT, can be deduced with (5) for t→∞: (6)wT=k-b2a.Using (5) with wcr1=0.1ν/D or wcr2=1000ν/D, tcr1 and tcr2 can be expressed as (7a)tcr1=1kln⁡bν+0.2aD+kνb+2aw0-kbν+0.2aD-kνb+2aw0+k+t0(7b)tcr2=1kln⁡bν+2000aD+kνb+2aw0-kbν+2000aD-kνb+2aw0+k+t0.The fall distance was expressed as(8)s=1alnk-b-2aw0e-kt-t0+k+b+2aw02k+k-bt-t02a.Assuming an initial fall velocity of zero, Guo [13] derived the following relationship between fall distance and fall velocity:(9)s=12aln⁡cc-bw-aw2-2bktanh-1⁡kw2c-bw,where tanh-1 is the inverse hyperbolic function.


## 2.2. Finite Values of Acceleration Fall Time and Distance

Fort→∞, the fall distance in (8) approaches an infinite value. In order to simplify the study, the finite values of fall time and distance were expressed with the following assumptions.Forw0<0.99wT scenarios, the acceleration fall process was assumed to be completed when the fall velocity reached 99% of the terminal fall velocity. Applying (6) and w=0.99wT to (5) gives the corresponding time as follows:(10)tS=1kln⁡1.99k+0.01bb+2aw0-k0.01b-0.01kb+2aw0+k+t0.Substituting (10) into (8) gives (11)sS=k+b2aln⁡b+2aw0+k1.99k+0.01b+k-b2akln⁡b+2aw0-k0.01b-0.01k.Forw0>1.01wT scenarios, the acceleration fall process was assumed to be completed when the fall velocity reached 101% of the terminal fall velocity. Applying (6) and w=1.01wT to (5) gives the corresponding time as follows: (12)tL=1kln⁡2.01k-0.01bb+2aw0-k0.01k-0.01bb+2aw0+k+t0.Substituting (12) into (8) gives (13)sL=k+b2aln⁡b+2aw0+k2.01k-0.01b+k-b2akln⁡b+2aw0-k0.01k-0.01b.Note that the scenarios of0.99wT⩽w0⩽wT or wT⩽w0⩽1.01wT are not addressed in our study because they are very close to the terminal fall velocity.


## 3. Validation by Experimental Data

In order to validate (9), two data sets of a spherical particle acceleration falling through still fluid from Allen [31] and Moorman [32] were used. Allen [31] measured a steel sphere falling in a rectangular water tank; its depth, length, and width were 28 cm, 11.5 cm, and 3 cm, respectively. The water temperature was 17.8°C, Δ=7.82, D=3.18 mm, w0=0, wT=83 cm/s, and the terminal Reynolds number RT=wTD/ν=2440. Moorman’s (1955) tests were conducted in an oil tank with D=12.7 mm, ρs=7780 kg/m3, ρ=876 kg/m3, ν=3.54×10-5 m2s−1, w0=0, wT=174.8 cm/s, and RT=627. The fall velocities and distances of the experiments were plotted, and their comparisons with (9) were shown in Figure 1. Note that α=0 and β=24 for R<0.1, α=0.5 and β=25.507 for 0.1<R<1000, CA=0.5 for R<1000, and α=0.39, β=24, and CA=2 for 1000<R<105 as reported by Guo [13] because of its good validation with Allen’s [31] and Moorman’s [32] test data. Figure 1 shows that the results from (9) for Allen’s [31] scenario and the experimental data agreed well. As can be seen from Figure 1, the results from (9) are slightly overestimated according to the experiment with Moorman's scenarios [32].Figure 1
Fall velocities and distance relationships for experimental data and computational results.In order to further validate (8), a series of experiments was conducted in the Hydraulic Lab at Ocean University of China. A rectangular glass tank was used with a height of 2.0 m, a width of 0.3 m, and a length of 0.3 m. It was filled with tap water to a water depth of 1.8 m. The water temperature was measured with a thermometer to be 20.7°C. The spherical particles were made of plastic or glass. The diameters of the spherical particles were measured by Vernier calipers, and the mass of the spherical particles was measured by a balance scale. As a result, the densities of the spherical particles were calculated. The scale paper was pasted on the outside of the tank to obtain its distance. The spherical particle was released on the surface of the water. A video camera was used to record the falling process at a speed of 30 frames per second. Video player software was used to read the video frame by frame, and the corresponding time and distance of each frame were obtained. Table 1 presents the detailed scenarios of the fall experiments.Table 1
Experimental scenarios.
Scenario number D (mm) ρ s (kgm−3) R T 1 4.5 1125.6 520 2 5.1 1236.7 880 3 5.3 1445.6 1458 4 14.4 1165 4020 5 15.7 1165 4586 6 16.6 1165 4982 7 17.5 1165 5391Figure2 illustrates the relationship between fall distance and time for scenarios 1 to 4 with w0=0. Note that the fall in a laminar zone was not measured because of the limitation of camera performance and its very small value. Figures 2(a) and 2(b) show the data only in the transitional zone, and the horizontal coordinates of t=0.113 s and t=0.074 s are the critical values between transitional and turbulent zones as shown in Figures 2(c) and 2(d), respectively. It was observed that results from (8) agree well with experimental data, confirming that the determination of CA, α, and β is reliable to compute the falling motion.Figure 2
Fall distance relationship with time forw0=0.
(a)
Scenario 1 (b)
Scenario 2 (c)
Scenario 3 (d)
Scenario 4


## 4. Different Initial Velocity Effects on Total Fall Time and Distance

Previous research shows that the initial velocity plays an important role in the fall time and distance [12, 13, 31, 32], and it deserves detailed comments. In terms of w0 and RT, the total fall time T and the total acceleration fall distance sT were discussed in the following nine scenarios.


### 4.1.w0⩽wcr1 for RT<0.1

Forw0<0.99wT scenarios, the fall velocity increases from w0 to wT. Therefore, (14a)T=tSα=0,β=24,CA=0.5(14b)sT=sSα=0,β=24,CA=0.5.For w0>1.01wT scenarios, the fall velocity decreases from w0 to wT. Therefore,(14c)T=tLα=0,β=24,CA=0.5(14d)sT=sLα=0,β=24,CA=0.5.


### 4.2.wcr1<w0⩽wcr2  for RT<0.1

The fall velocity decreases fromw0 to wcr1 and wT, and the corresponding time increases from t0 to tcr1 and tL. Therefore, (15a)T=tcr1-t0+tL-tcr1(15b)sT=sα=0.5,β=25.057,CA=0.5,t=tcr1+sLα=0,β=24,CA=0.5,w0=wcr1,where tL can be obtained with α=0, β=24, CA=0.5, w0=wcr1, and t0=tcr1 using (12).


### 4.3.w0>wcr2 for RT<0.1

The fall velocity decreases fromw0 to wcr2, wcr1, and wT, and the corresponding time increases from t0 to tcr2, tcr1, and tL. Therefore, (16a)T=tcr2-t0+tcr1-tcr2+tL-tcr1(16b)sT=sα=0.39,β=24,CA=2,t=tcr2+sα=0.5,β=25.057,CA=0.5,w0=wcr2,t0=tcr2,t=tcr1+sLα=0,β=24,CA=0.5,w0=wcr1,where tL can be obtained with α=0, β=24, CA=0.5, w0=wcr1, and t0=tcr1 using (12).


### 4.4.w0⩽wcr1 for 0.1<RT<1000

The fall velocity increases fromw0 to wcr1 and wT, and the corresponding time increases from t0 to tcr1 and tS. Therefore, (17a)T=tcr1-t0+tS-tcr1(17b)sT=sα=0,β=24,CA=0.5,t=tcr1+sSα=0.5,β=25.057,CA=0.5,w0=wcr1,where tS can be obtained with α=0.5, β=25.057, CA=0.5, w0=wcr1, and t0=tcr1 using (10).


### 4.5.wcr1<w0⩽wcr2 for 0.1<RT<1000

Forw0<0.99wT scenarios, the fall velocity increases from w0 to wT. Therefore, (18a)T=tSα=0.5,β=25.057,CA=0.5(18b)sT=sSα=0.5,β=25.057,CA=0.5.For w0>1.01wT scenarios, the fall velocity decreases from w0 to wT. Therefore, (19a)T=tLα=0.5,β=25.057,CA=0.5(19b)sT=sLα=0.5,β=25.057,CA=0.5.


### 4.6.w0>wcr2 for 0.1<RT<1000

The fall velocity decreases fromw0 to wcr2 and wT, and the corresponding time increases from t0 to tcr2 and tL. Therefore, (20a)T=tcr2-t0+tL-tcr2(20b)sT=sα=0.39,β=24,CA=2,t=tcr2+sLα=0.5,β=25.057,CA=0.5,w0=wcr2,where tL can be obtained with α=0.5, β=25.057, CA=0.5, w0=wcr2, and t0=tcr2 using (12).


### 4.7.w0⩽wcr1 for RT>1000

The fall velocity increases fromw0 to wcr1, wcr2, and wT, and the corresponding time increases from t0 to tcr1, tcr2, and tS. Therefore, (21a)T=tcr1-t0+tcr2-tcr1+tS-tcr2(21b)sT=sα=0,β=24,CA=0.5,t=tcr1+sα=0.5,β=25.057,CA=0.5,w0=wcr1,t0=tcr1,t=tcr2+sSα=0.39,β=24,CA=2,w0=wcr2,where tS can be obtained with α=0.39, β=24, CA=2, w0=wcr2, and t0=tcr2 using (10).Table2 shows the total acceleration fall distance comparison between (21b) and experimental data with w0=0. In Table 2, sTT is the sT value of (21b), sTE is the sT value from the experiments, and the relative error was defined as E=((sTT-sTE)/sTE)×100%. It was found that the all of the relative errors between them were smaller than 5.0%, confirming that (21b) is reliable to compute sT for w0⩽wcr1 with RT>1000.Table 2
Eq. (21b) and experimental distance of total acceleration with w0=0.
Scenario number S T T (m) S T E (m) E (%) 4 0.299 0.290 3.1 5 0.330 0.315 4.8 6 0.347 0.357 - 2.8 7 0.368 0.377 - 2.4


### 4.8.wcr1<w0⩽wcr2 for RT>1000

The fall velocity increases fromw0 to wcr2 and wT, and the corresponding time increases from t0 to tcr2 and tS. Therefore, (22a)T=tcr2-t0+tS-tcr2(22b)sT=sα=0.5,β=25.057,CA=0.5,t=tcr2+sSα=0.39,β=24,CA=2,w0=wcr2,where tS can be obtained with α=0.39, β=24, CA=2, w0=wcr2, and t0=tcr2 using (10).Figure3 shows the relationship between T, sT, and w0 for wcr1<w0⩽wcr2 and RT>1000. Note that EV. is the abbreviation for the experimental values in Figures 3(b) and 4(b). Figure 3(a) illustrates that T decreases with increasing w0, and its reduction rate is higher for small RT than for large RT. Figure 3(b) shows that sT decreases slightly with increasing w0, and its reduction rate increases very slightly with decreasing RT, although it appears to remain constant to a great extent. It was also found that results from (22b) are consistent with the experimental values, as shown in Figure 3(b).Figure 3
Relationship betweenT, sT, and w0 for wcr1<w0⩽wcr2 and RT>1000.
(a)
T and w0 (b)
s T and w0Figure 4
Relationship betweenT, sT, and w0 for w0>wcr2 and RT>1000.
(a)
T and w0 (b)
s T and w0


### 4.9.w0>wcr2 for RT>1000

Forw0<0.99wT scenarios, the fall velocity increases from w0 to wT. Therefore, (23a)T=tSα=0.39,β=24,CA=2(23b)sT=sSα=0.39,β=24,CA=2.For w0>1.01wT scenarios, the fall velocity decreases from w0 to wT. Therefore, (24a)T=tLα=0.39,β=24,CA=2(24b)sT=sLα=0.39,β=24,CA=2.For w0>wcr2 and RT>1000 scenarios, the relationship between T, sT, and w0 is shown in Figure 4. Figure 4 indicates that T and sT increase with increasing w0, and the increase rates are higher for small RT than for large RT. Figure 4(b) shows that the results from (24b) agree with the experimental values. Consequently, the initial velocity plays a more important role in the acceleration motion for small RT than for large RT scenarios as shown in Figures 3 and 4.


## 4.1.w0⩽wcr1 for RT<0.1

Forw0<0.99wT scenarios, the fall velocity increases from w0 to wT. Therefore, (14a)T=tSα=0,β=24,CA=0.5(14b)sT=sSα=0,β=24,CA=0.5.For w0>1.01wT scenarios, the fall velocity decreases from w0 to wT. Therefore,(14c)T=tLα=0,β=24,CA=0.5(14d)sT=sLα=0,β=24,CA=0.5.


## 4.2.wcr1<w0⩽wcr2  for RT<0.1

The fall velocity decreases fromw0 to wcr1 and wT, and the corresponding time increases from t0 to tcr1 and tL. Therefore, (15a)T=tcr1-t0+tL-tcr1(15b)sT=sα=0.5,β=25.057,CA=0.5,t=tcr1+sLα=0,β=24,CA=0.5,w0=wcr1,where tL can be obtained with α=0, β=24, CA=0.5, w0=wcr1, and t0=tcr1 using (12).


## 4.3.w0>wcr2 for RT<0.1

The fall velocity decreases fromw0 to wcr2, wcr1, and wT, and the corresponding time increases from t0 to tcr2, tcr1, and tL. Therefore, (16a)T=tcr2-t0+tcr1-tcr2+tL-tcr1(16b)sT=sα=0.39,β=24,CA=2,t=tcr2+sα=0.5,β=25.057,CA=0.5,w0=wcr2,t0=tcr2,t=tcr1+sLα=0,β=24,CA=0.5,w0=wcr1,where tL can be obtained with α=0, β=24, CA=0.5, w0=wcr1, and t0=tcr1 using (12).


## 4.4.w0⩽wcr1 for 0.1<RT<1000

The fall velocity increases fromw0 to wcr1 and wT, and the corresponding time increases from t0 to tcr1 and tS. Therefore, (17a)T=tcr1-t0+tS-tcr1(17b)sT=sα=0,β=24,CA=0.5,t=tcr1+sSα=0.5,β=25.057,CA=0.5,w0=wcr1,where tS can be obtained with α=0.5, β=25.057, CA=0.5, w0=wcr1, and t0=tcr1 using (10).


## 4.5.wcr1<w0⩽wcr2 for 0.1<RT<1000

Forw0<0.99wT scenarios, the fall velocity increases from w0 to wT. Therefore, (18a)T=tSα=0.5,β=25.057,CA=0.5(18b)sT=sSα=0.5,β=25.057,CA=0.5.For w0>1.01wT scenarios, the fall velocity decreases from w0 to wT. Therefore, (19a)T=tLα=0.5,β=25.057,CA=0.5(19b)sT=sLα=0.5,β=25.057,CA=0.5.


## 4.6.w0>wcr2 for 0.1<RT<1000

The fall velocity decreases fromw0 to wcr2 and wT, and the corresponding time increases from t0 to tcr2 and tL. Therefore, (20a)T=tcr2-t0+tL-tcr2(20b)sT=sα=0.39,β=24,CA=2,t=tcr2+sLα=0.5,β=25.057,CA=0.5,w0=wcr2,where tL can be obtained with α=0.5, β=25.057, CA=0.5, w0=wcr2, and t0=tcr2 using (12).


## 4.7.w0⩽wcr1 for RT>1000

The fall velocity increases fromw0 to wcr1, wcr2, and wT, and the corresponding time increases from t0 to tcr1, tcr2, and tS. Therefore, (21a)T=tcr1-t0+tcr2-tcr1+tS-tcr2(21b)sT=sα=0,β=24,CA=0.5,t=tcr1+sα=0.5,β=25.057,CA=0.5,w0=wcr1,t0=tcr1,t=tcr2+sSα=0.39,β=24,CA=2,w0=wcr2,where tS can be obtained with α=0.39, β=24, CA=2, w0=wcr2, and t0=tcr2 using (10).Table2 shows the total acceleration fall distance comparison between (21b) and experimental data with w0=0. In Table 2, sTT is the sT value of (21b), sTE is the sT value from the experiments, and the relative error was defined as E=((sTT-sTE)/sTE)×100%. It was found that the all of the relative errors between them were smaller than 5.0%, confirming that (21b) is reliable to compute sT for w0⩽wcr1 with RT>1000.Table 2
Eq. (21b) and experimental distance of total acceleration with w0=0.
Scenario number S T T (m) S T E (m) E (%) 4 0.299 0.290 3.1 5 0.330 0.315 4.8 6 0.347 0.357 - 2.8 7 0.368 0.377 - 2.4


## 4.8.wcr1<w0⩽wcr2 for RT>1000

The fall velocity increases fromw0 to wcr2 and wT, and the corresponding time increases from t0 to tcr2 and tS. Therefore, (22a)T=tcr2-t0+tS-tcr2(22b)sT=sα=0.5,β=25.057,CA=0.5,t=tcr2+sSα=0.39,β=24,CA=2,w0=wcr2,where tS can be obtained with α=0.39, β=24, CA=2, w0=wcr2, and t0=tcr2 using (10).Figure3 shows the relationship between T, sT, and w0 for wcr1<w0⩽wcr2 and RT>1000. Note that EV. is the abbreviation for the experimental values in Figures 3(b) and 4(b). Figure 3(a) illustrates that T decreases with increasing w0, and its reduction rate is higher for small RT than for large RT. Figure 3(b) shows that sT decreases slightly with increasing w0, and its reduction rate increases very slightly with decreasing RT, although it appears to remain constant to a great extent. It was also found that results from (22b) are consistent with the experimental values, as shown in Figure 3(b).Figure 3
Relationship betweenT, sT, and w0 for wcr1<w0⩽wcr2 and RT>1000.
(a)
T and w0 (b)
s T and w0Figure 4
Relationship betweenT, sT, and w0 for w0>wcr2 and RT>1000.
(a)
T and w0 (b)
s T and w0


## 4.9.w0>wcr2 for RT>1000

Forw0<0.99wT scenarios, the fall velocity increases from w0 to wT. Therefore, (23a)T=tSα=0.39,β=24,CA=2(23b)sT=sSα=0.39,β=24,CA=2.For w0>1.01wT scenarios, the fall velocity decreases from w0 to wT. Therefore, (24a)T=tLα=0.39,β=24,CA=2(24b)sT=sLα=0.39,β=24,CA=2.For w0>wcr2 and RT>1000 scenarios, the relationship between T, sT, and w0 is shown in Figure 4. Figure 4 indicates that T and sT increase with increasing w0, and the increase rates are higher for small RT than for large RT. Figure 4(b) shows that the results from (24b) agree with the experimental values. Consequently, the initial velocity plays a more important role in the acceleration motion for small RT than for large RT scenarios as shown in Figures 3 and 4.


## 5. Conclusions

The acceleration motion of a spherical particle falling through still fluid was analytically and experimentally investigated using the BBO equation. The results of this study can be summarized as follows.(1)
The relationship between the drag coefficient and Reynolds number was studied, and different parameters in the drag coefficient equation were obtained with respect to the small, medium, and large Reynolds number zones. In terms of some assumptions, the finite fall time and distance equations were derived using the results reported by Guo [13].(2)
A simple experiment was conducted in order to measure a spherical particle fall process through still water. Literature and experimental data were used to validate the equations between fall time, velocity, and distance, confirming the determination of a drag force coefficient and an integrated added mass coefficient for different Reynolds number zones.(3)
The different initial velocities and terminal Reynolds numbers were taken into account, and the total fall time and distance equations were obtained. They fitted well with experimental data forRT>1000 scenarios. It was found that the initial velocity plays a more important role in the falling motion for small RT than for large RT scenarios.(4)
The results of this study are possibly useful in hydraulic, environmental, and chemical engineering fields, such as particle size analysis, sediment deposition in water treatment engineering, and chemical reactions of particles in a fluidized bed.It should be noted that this study experimentally measured fall data only in medium and largeRT zones. We failed to measure the corresponding data in small RT zones because of camera performance limitations, spherical particle material, and spherical particle size; this might deserve further experimental evaluation in future work.



---

*Source: 9795286-2017-02-06.xml*