Motion analysis of a spherical solid particle in plane Couette Newtonian fluid flow

Powder Technology 274 (2015) 186–192

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Motion analysis of a spherical solid particle in plane Couette Newtonian fluid flow A.S. Dogonchi a, M. Hatami b, G. Domairry c,⁎ a b c

Mechanical Engineering Department, Mazandaran Institute of Technology, P.O. Box 747, Babol, Iran Mechanical Engineering Department, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran Mechanical Engineering Department, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran

a r t i c l e

i n f o

Article history: Received 15 November 2014 Received in revised form 27 December 2014 Accepted 8 January 2015 Available online 14 January 2015 Keywords: Solid spherical particle Plane Couette fluid flow Newtonian fluid Velocity alterations Differential Transformation Method (DTM)

a b s t r a c t In this article, the motion of a spherical particle in a plane Couette Newtonian fluid flow is studied. The governing equations of a spherical solid particle's motion in the plane Couette fluid flow are investigated using the Differential Transformation Method (DTM) and a Padé approximant which are an analytical solution technique. For validation of the analytical solution, the governing equation is solved numerically. The horizontal and vertical velocities of spherical solid particle are shown for different fluids and values of the embedding parameters. The DTM–Padé results indicate that the horizontal and vertical velocities of spherical solid particle in water fluid are higher than the glycerin and ethylene-glycol fluids. Also the horizontal and vertical velocities increase with an increase in the particle radius. Comparison of the results (DTM and numerical) was shown that the analytical method and numerical data are in a good agreement with each other. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The fluid–solid flow particles are occurring in various industrial and natural applications. Knowledge of the behavior of solid particles allows the treatment and improvement of several industrial processes in the field of process engineering, e.g. drying, agitation, mixing and separation as well as aerospace and, etc. The transport of arbitrarily shaped particles is of great importance also in several biomedical applications: particles of various shapes, e.g. spheres, disks and rods have been developed for controlling and improving the systemic administration of therapeutic and contrast agents. Owing to the importance of the aforesaid applications, considerable attention has been devoted to the study of the accelerated motion of a sphere in a fluid, and an excellent account of the theoretical developments in this area has been given by R. Clift et al. [1] for Newtonian fluids. More recently, several studies were performed on spherical and non-spherical solid particles. M. Hatami et al. [2] applied the Multi-step Differential Transformation Method (Ms-DTM) on particle's motion in Couette fluid flow with considering the rotation and shear effects on lift force and neglecting gravity. M. Hatami and D.D. Ganji [3] solved the equation of a particle's motion on a rotating parabolic surface with the Multi-step Differential Transformation Method (Ms-DTM) and achieved comparable results to the

⁎ Corresponding author. Tel./fax: +98 915 718 63 74. E-mail addresses: [email protected] (M. Hatami), [email protected] (G. Domairry).

http://dx.doi.org/10.1016/j.powtec.2015.01.018 0032-5910/© 2015 Elsevier B.V. All rights reserved.

numerical ones. The motion of a particle in a fluid forced vortex was studied by M. Hatami and D.D. Ganji [4]. They applied the Differential Transformation Method (DTM) and Differential Quadrature Method (DQM) to solve governing equations. The unsteady settling behavior of solid spherical particles falling in the water as a Newtonian fluid was investigated by R. Nouri et al. [5]. They applied three different analytical methods to analyze the characteristics of particle motion. M. Hatami and G. Domairry [6] studied the unsteady settling behavior of a soluble spherical particle falling in a Newtonian fluid media by DTM–Padé. They discussed about the influence of the solubility parameter on velocity profile. M. Jalaal and D.D. Ganji [7] studied the unsteady motion of a spherical particle rolling down an inclined plane submerged in a Newtonian environment using a drag of the form given by R.P. Chhabra and J.M. Ferreira [8], for a wide range of Reynolds numbers by the homotopy perturbation method (HPM). They observed that the settling velocity, acceleration duration and displacement are proportional to the inclination angle, while for a constant inclination angle, the settling veloc-ity and acceleration duration are decreased by increasing the fluid viscosity. M. Jalaal et al. [9] applied the HPM to obtain exact analytical solutions for the motion of a spherical particle in a plane Couette flow. M. Jalaal et al. [10] applied the variational iteration method (VIM) on the acceleration motion of a non-spherical particle in an incompressible Newtonian environment for a wide range of Reynolds numbers using a novel drag coefficient as defined by S.F. Chien [11]. In [12,13] the unsteady motion of a spherical particle falling in a Newtonian fluid was analyzed using the HPM. M. Torabi et al. [14] applied HPM–Padé approximant method to obtain exact analytical

A.S. Dogonchi et al. / Powder Technology 274 (2015) 186–192

solutions for acceleration motion of a vertically falling spherical particle. S.M. Hamidi et al. [15] applied HPM–Padé approximant method to obtain exact analytical solutions for the motion of a spherical solid particle in the plane Couette fluid flow. The concept of Differential Transformation Method (DTM) was first introduced by J.K. Zhou [16] and it was used to solve both linear and nonlinear initial value problems. This method can be applied directly to linear and nonlinear differential equation without requiring linearization, discretization, or perturbation and this is the main benefit of this method. T. Abbasov, A.R. Bahadir [17] employed DTM to obtain approximate solutions of the linear and non-linear equations related to engineering problems and they showed that the numerical results are in good agreement with the analytical solutions. M.M. Rashidi et al. [18] solved the problem of mixed convection about an inclined flat plate embedded in a porous medium by DTM; they applied the Padé approximant to increase the convergence of the solution. S. Ghafoori et al. [19] used the DTM for solving the nonlinear oscillation equation. I.H. Abdel-Halim Hassan [20] has applied the DTM for different systems of differential equations and he has discussed the convergency of this method in several examples of linear and non-linear systems of differential equations. The goal of this study is obtaining an analytical solution for governing equations of a spherical solid particle's motion and particle motion analysis in the plane Couette fluid flow. Also the Differential Transformation Method (DTM)-Padé is applied to solve linear problems analytically. To validate analytical results, the obtained DTM–Padé results are compared with numerical data. 2. Problem description T.J. Vander Werff [21] proposed a constructive mathematical formulation for the motion of a spherical particle in a plane Couette flow. T.J. Vander Werff [21] assumed a two-dimensional velocity profile incompressible Newtonian flow. Considering the rotation of the particle, it was assumed that the particle will rotate with a constant angular velocity Ω given by one-half the curl of the fluid motion. Generally, the particle's motion is determined by the combined effects of inertia, drag and lift. Gravity and buoyancy effects will be assumed negligible [21]. Subsequently, the governing equations are obtained as: 8
 4 3 1 3 > > < πr ρs €x ¼ πr ρ f α y −6πμr x −αy 3 2    4 3 1 3 > 2 1=2 1=2
> €¼ : πr ρs y αy− x −6πμry πr ρ f α þ 6:46r ρ f α ν 3 2 

187

Fig. 1. Comparison of horizontal velocity (u) obtained by DTM and DTM–Padé with numerical solution when A = B = C = α = u0 = v0 = 1.

injection of the particle into the fluid or statistical fluctuations: Alternatively, nonzero values of x or y could have been chosen where no generality is lost by specifying the particular (and physically more meaningful) conditions above. Eq. (1) can be rewritten in the following forms: (


 €x−A y þB x −αy ¼ 0
 € þ B y þðA þ C Þ x −αy ¼ 0 y 







ð2Þ







ð1Þ

where r, ρs and ρf represent the particle radius, particle density and fluid density, respectively. Also, μ and ν indicate the dynamic fluid viscosity and kinematic fluid viscosity, respectively and α is a positive proportionality constant with dimensions of inverse time. Moreover, the dots represent differentiation with respect to time. The relative velocities of the particle and fluid are considered small enough for the Stokes law of drag [21]. For this system to possess a nontrivial solution, nonzero initial conditions must be specified. The following might represent either

Table 1 The fundamental operations of the differential transform method. Original function w(t) = αu(t) ± βv(t) m

wðt Þ ¼ d dtumðtÞ w(t) = u(t)v(t)

Transformed function W(k) = αU(k) ± βV(k) Þ! W ðkÞ ¼ ðkþm U ðk þ mÞ k! k

w(t) = tm

W ðkÞ ¼ ∑ U ðlÞV ðk−lÞ l¼0  1; if k ¼ m W ðkÞ ¼ δðk−mÞ ¼ 0; if k≠m

w(t) = exp(t)

W ðkÞ ¼ k!1

Fig. 2. Comparison of vertical velocity (v) obtained by DTM and DTM–Padé with numerical solution when A = B = C = α = u0 = v0 = 1.

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Table 2 The results of the DTM and DTM–Padé and NM for horizontal and vertical velocities. t

u (m/s) DTM − Padé[10/10]

NM

Error

v (m/s) DTM − Padé[10/10]

NM

Error

0 1 2 3 4 5 6 7 8 9 10

1.0000000000 0.7357588825 0.4060058496 0.1991482733 0.0915781942 0.0404276817 0.0173512648 0.0072950527 0.0030191431 0.0012339934 0.0004989797

1.0000000000 0.7357589112 0.4060058788 0.1991482970 0.0915782073 0.0404276801 0.0173512487 0.0072950309 0.0030191407 0.0012340738 0.0004993769

0.0000000000 0.0000000287 0.0000000292 0.0000000237 0.0000000131 0.0000000016 0.0000000161 0.0000000219 0.0000000024 0.0000000804 0.0000003973

1.0000000000 -0.0000000001 -0.1353352829 -0.0995741367 -0.0549469162 -0.0269517879 -0.0123937608 -0.0054712890 -0.0023482182 -0.0009871721 -0.0004081722

1.0000000000 -0.0000000439 -0.1353353334 -0.0995741854 -0.0549469585 -0.0269518219 -0.0123937804 -0.0054712930 -0.0023482314 -0.0009872667 -0.0004085863

0.0000000000 0.0000000437 0.0000000505 0.0000000488 0.0000000423 0.0000000340 0.0000000196 0.0000000040 0.0000000132 0.0000000946 0.0000004141

where coefficients A to C are defined as: A¼

   3α ρ f ρs 8 

B¼

9ν 2r 2

ð3Þ

3. Fundamental of Differential Transformation Method

  ρf ρs

α 1=2 ν1=2 C ¼ 1:542 r

ð4Þ !  ρf ρs

ð5Þ

Eq. (2), defining the motion of the particle in plane Couette fluid flow, are linear and could be solved with different mathematical methods. As mentioned in the text, the nonzero initial conditions of equations of motion could be different for unlike situations. The following might represent either injection of the particle into the fluid or statistical fluctuations: 

x¼0; y¼0;



x¼ u0 y¼ v0 

In the rest of this paper, couple equations of a spherical solid particle's motion in plane Couette fluid flow are solved and analyzed by using DTM–Padé technique.

at at

t¼0 t¼0

ð6Þ

Fig. 3. Comparison between the DTM–Padé and numerical solution for horizontal velocity for different values of [L/M] when A = B = C = α = u0 = v0 = 1.

Let x(t) be analytic in a domain D and let t = ti represent any point in D. The function x(t) is then represented by one power series whose center is located at ti. The Taylor series expansion function of x(t) is in the form of: " # ∞ X ðt−t i Þk dk xðt Þ xðt Þ ¼ ∀ t∈D k! dt k t¼t k¼0

ð7Þ

i

The particular case of Eq. (7) when ti = 0 is referred to as the Maclaurin series of x(t) and is expressed as: " # ∞ k X t dk xðt Þ xðt Þ ¼ k! dt k k¼0

∀ t∈D

ð8Þ

t¼0

Fig. 4. Comparison between the DTM–Padé and numerical solution for vertical velocity for different values of [L/M] when A = B = C = α = u0 = v0 = 1.

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Table 3 Physical properties of materials. Material

Density (kg/m3)

Viscosity (kg/m s)

Water Glycerin Ethylene-glycol Aluminum

996.51 1259.90 1111.40 2702.00

0.001 0.799 0.0157 –

As explained in [16,22] the differential transformation of the function x(t) is defined as follows: X ðkÞ ¼

" # k k ∞ X H d xðt Þ k! dt k k¼0

ð9Þ

t¼0

Where x(t) is the original function and X(k) is the transformed function. The differential spectrum of X(k) is confined within the interval t ∈ [0, H], where H is a constant. The differential inverse transform of X(k) is defined as follows: xðt Þ ¼

∞  k X t k¼0

H

X ðkÞ

ð10Þ

It is clear that the concept of the differential transformation is based upon the Taylor series expansion. The values of function X(k) at values of argument k are referred to as discrete, i.e. X(0) is known as the zero discrete, X(1) as the first discrete, etc. the more discrete available, the more precise it is possible to restore the unknown function. The function x(t) consists of T function X(k), and its value is given by the sum of the T-function with (t/H)k as its coefficient. In real applications, at the right choice of constant H, the larger values of argument k the discrete of spectrum reduce rapidly. The function x(t) is expressed by a finite series and Eq. (10) can be written as: xðt Þ ¼

n  k X t k¼0

H

X ðkÞ

ð11Þ

Mathematical operations performed by differential transform method are listed in Table 1.

Fig. 5. Horizontal velocity profile of the particle when r = 0.001, α = 1 for different fluids.

Fig. 6. Vertical velocity profile of the particle when r = 0.001, α = 1 for different fluids.

4. Padé approximants A Padé approximant is the ratio of two polynomials constructed from the coefficients of the Taylor series expansion of a function u(x). The [L/M] Padé approximants to a function y(x) are given by [23]   L P ðxÞ ¼ L M Q M ðxÞ

ð12Þ

where PL(x) is polynomial of the degree of at most L and Q M(x) is a polynomial of the degree of at most M. The formal power series yðxÞ ¼

∞ X

ai x

i

ð13Þ

i¼0

Fig. 7. Horizontal velocity profile of the particle when α = 1 for different values of particle radius in water.

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Fig. 8. Vertical velocity profile of the particle when α = 1 for different values of particle radius in water.

and yðxÞ−



P L ðxÞ LþMþ1 ¼O x Q M ðxÞ

ð14Þ

determine the coefficients of PL(x) and Q M(x) by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave [L/M] unchanged, we imposed the normalization condition Q M ð0Þ ¼ 1:0

ð15Þ

Fig. 9. Horizontal velocity profile of the particle when r = 0.001 for different values of α in water.

8 a ¼ p0 ; > > > 0 > < a1 þ a0 q1 ¼ p1 ; a2 þ a1 q1 þ a0 q2 ¼ p2 ; > > ⋮ > > : aL þ aL−1 q1 þ … þ a0 qL ¼ pL

To solve these equations, we start with (17), which is a set of linear equations for all the unknown q's. Once the q's are known, then (18) gives an explicit formula for the unknown p's, which complete the solution. If (17) and (18) are nonsingular, then we can solve them directly and obtain (19) where (19) holds, and if the lower index on a sum exceeds the upper, the sum is replaced by zero [24,25]: 2

Finally, PL(x) and Q M(x) need to include noncommon factors. If the coefficient of PL(x) and Q M(x) is written as (

2

ð16Þ

hLi M

and using Eqs. (15) and (16), we may multiply (14) by Q M(x), which linearizes the coefficient equations. Eq. (14) can be presented in more details as 8 þ aL q1 þ … þ aL−Mþ1 qM ¼ 0 ; a > > < Lþ1 aLþ2 þ aLþ1 q1 þ … þ aL−Mþ2 qM ¼ 0 ; ð17Þ ⋮ > > : aLþM þ aLþM−1 q1 þ … þ aL qM ¼ 0 ;

¼

j¼M

⋯ ⋱ ⋯

aL−Mþ2 ⋮ aLþ1 L X j a j−Mþ1 x

aL−Mþ1 6 ⋮ 6 6 aL det6 L 6X j 4 a j−M x

L

P L ðxÞ ¼ p0 þ p1 x þ p2 x þ … þ pL x ; 2 M Q M ðxÞ ¼ q0 þ q1 x þ q2 x þ … þ qM x ;

ð18Þ

2

⋯

j¼M−1

aL−Mþ1 6 ⋮ det6 4 aL M x

aLþ1 ⋮ aLþM L X j a jx

3 7 7 7 7 7 5

j¼0

aL−Mþ2 ⋮ aLþ1 M−1 x

⋯ ⋱ ⋯ ⋯

3 aLþ1 ⋮ 7 7 aLþM 5 1

ð19Þ To obtain a diagonal Padé approximants of a different order such as [2/2], [4/4], or [6/6], the symbolic calculus software Maple is used.

5. Analytical solution The constants for the considered system including, Eqs. (2), (6), simply calculated dependent on physical conditions and no generality is lost by specifying the particular (and physically more meaningful) conditions [21]. Thus, to simplify the solution, constants dependent on physical properties of solid–fluid combination are considered to be A = B = C = α = u0 = v0 = 1. Now we apply the Differential Transformation Method for the Eq. (2) and taking the differential transform of Eq. (2) with respect to t according Table 1, gives: 

ðk þ 1Þðk þ 2ÞX ðk þ 2Þ þ ðk þ 1ÞX ðk þ 1Þ−ðk þ 1ÞY ðk þ 1Þ−Y ðkÞ ¼ 0 ðk þ 1Þðk þ 2ÞY ðk þ 2Þ þ ðk þ 1ÞY ðk þ 1Þ þ ðk þ 1ÞX ðk þ 1Þ−Y ðkÞ ¼ 0

ð20Þ

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191

From a process of inverse differential transformation, it can be shown that the solutions of each sub-domain take n + 1 term for the power series like Eq. (11), i.e. 8 n  k X > t > > X i ðkÞ; 0≤t ≤H i > xi ðt Þ ¼ < H i k¼0 ð21Þ   n k X t > > > Y i ðkÞ; 0≤t ≤H i > : yi ðt Þ ¼ Hi k¼0 Where k = 0, 1, 2, …, n represents the number of term of the power series, i = 0, 1, 2, … expresses the ith sub-domain and Hi is the sub-domain interval. From initial condition in Eq. (6), that we have it in point t = 0 and t = 1, and exerting transformation  X ð0Þ ¼ 0 ð22Þ Y ð0Þ ¼ 0  X ð1Þ ¼ 1 ð23Þ Y ð1Þ ¼ 1 Accordingly, from a process of inverse differential transformation, in this problem we calculated X(k + 2) and Y(k + 2) from Eq. (20) as following: 8 8 1 1 > >  < X ð3Þ ¼ − < X ð4Þ ¼ X ð2Þ ¼ 0 6 12 Y ð2Þ ¼ −1 > > : Y ð3Þ ¼ 1 : Y ð4Þ ¼ − 1 2 6 8 8 ð24Þ 1 > 1 > < X ð5Þ ¼ − < X ð6Þ ¼ 40 180 ; … > > : Y ð5Þ ¼ 1 : Y ð6Þ ¼ − 1 24 120 The above process is continuous. Substituting Eqs. (22)–(24) into the main equation based on DTM, it can be obtained that the closed form of the solutions is: xðt Þ ¼

yðt Þ ¼

22 X

1 3 1 4 1 5 1 6 1 7 1 8 1 1 1 1 1 K 9 10 11 12 13 X ðK Þt ¼ t− t þ t − t þ t − t þ t − t þ t − t þ t − t 6 12 40 180 1008 6720 51840 453600 4435200 47900160 566092800 K¼0 1 1 1 1 1 1 14 15 16 17 18 19 þ t − t þ t − t þ t − t 7264857600 100590336000 1494484992000 23712495206400 400148356608000 7155594141696000 1 1 1 20 21 22 þ t − t þ t ð25Þ 135161222676480000 2688996956405760000 56200036388880384000 22 X

1 3 1 4 1 5 1 6 1 7 1 8 1 1 1 1 1 K 2 9 10 11 12 13 Y ðK Þt ¼ t−t þ t − t þ t − t þ t − t þ t − t þ t − t þ t 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 1 1 1 1 1 1 14 15 16 17 18 19 − t þ t − t þ t − t þ t 6227020800 87178291200 1307674368000 20922789888000 355687428096000 6402373705728000 1 1 1 20 21 22 − t þ t − t ð26Þ 121645100408832000 2432902008176640000 51090942171709440000

K¼0

As it is obvious, solution of terms vary periodically and in each step more duration of particle motion is covered. By increasing series terms, the accuracy of DTM solution is improved and a larger period of acceleration motion of the particle is covered. Basically, by estimating the constants A − C for each selected combinations of solid–fluid, results can be derived easily. After obtaining the result of 22th iteration for DTM, it is seen from graphs and tables that the DTM for this problem don't have exact solution therefore we will apply the Padé approximation for variations of velocities of the particle as follows 

uðt Þ½4=4 ¼x ðt Þ½4=4         31 1537 47 3539 67 2 3 4 − t þ t − t þ t 58060800 8128512000 1354752000 1463132160000 217728000         ¼ 31 67 211 43 143 2 3 4 þ tþ t þ t þ t 58060800 217728000 2709504000 4064256000 209018880000

ð27Þ



vðt Þ½4=4 ¼y ðt Þ½4=4         1457 2089 6899 481 1763 2 3 4 − tþ t − t þ t 870912000 870912000 8128512000 3657830400 209018880000         ¼ 1457 11 1901 1 907 2 3 4 þ tþ t þ t þ t 870912000 11612160 8128512000 32659200 487710720000

ð28Þ

Therefore, we are able to give an accurate approximate solution of the considered problem. 6. Results and discussion An analytical solution for the motion of a spherical particle in a plane Couette Newtonian fluid flow by DTM–Padé approximant

was obtained. The results are compared with DTM and numerical solution. Figs. 1 and 2 depict the horizontal and vertical velocities versus time. It is observed that the DTM–Padé approximant solution is more accurate than DTM. Comparing DTM–Padé [4/4] and DTM–Padé [10/10],

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worth noting that the DTM which is improved by Padé approximants is an effective, simple and quite accurate tool for handling and solving linear and nonlinear equations and it is predicted that the Differential Transformation Padé technique will be found widely applications in engineering problems. For the specific conditions used in the numerical computations, the following conclusions are drawn. 1- The particle magnitude velocities are reduced gradually with increasing the liquid viscosity. 2- As the radius of the particle increases the magnitude velocity of the particle increases. References

Fig. 10. Vertical velocity profile of the particle when r = 0.001 for different values of α in water.

DTM–Padé[10/10] gives closer results to numerical solution. This fact is more pronounced for large values of time i.e. t = 10. Moreover, this interesting agreement between DTM–Padé approximant and numerical solution is tabulated in Table 2. As noted previously, DTM–Padé[10/10] gives closer results to numerical solution; therefore, it will produce more acceptable results regarding velocities of the particle. This is confirmed by the curves in Figs. 3 and 4. Mentioned method was applied in real combination of solid–liquid. Required physical properties of selected materials are given in Table 3. Figs. 5 and 6 depict the horizontal and vertical velocity profiles of the particles versus time for different fluids. The velocity profiles show that the particle magnitude velocities reduced gradually with an increasing the liquid viscosity. As it can be seen from these figures for AluminumGlycerin system the magnitude velocity is lower than both Aluminum– Ethylene-glycol and Aluminum–Water systems. In these figures, it should be noted that the Aluminum–Water system has the highest magnitude velocity among others. The effect of particle radius (r) on velocity in water fluid is shown in Figs. 7 and 8, for both horizontal velocity and vertical velocity, respectively. It is illustrated that the magnitude of velocity is increased with increasing the particle radius (r) that this increase in horizontal velocity is higher than vertical velocity. In addition, in Figs. 9 and 10, the effect of α on horizontal and vertical velocities in water fluid is shown, respectively. 7. Conclusion In this paper, the DTM–Padé approximation was applied successfully to solve couple equations governing the motion of a particle in the plane Couette fluid flow. As DTM solution quickly diverges for high values of t, especially in this case study, Padé technique is applied to increase the approximate solution's accuracy and increase the convergence domain. It has been revealed that the obtained analytical results are in excellent agreement with those obtained by numerical methods. Moreover, the proposed algorithm is employed without using linearization, discretization, transformation, or restrictive assumptions. It is

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