Spherical cap bubbles in fluidized beds and bubble columns: A new formula for added mass

International Journal of Multiphase Flow 120 (2019) 103098

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International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow

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Spherical cap bubbles in fluidized beds and bubble columns: A new formula for added mass M. Puncochar∗, M.C. Ruzicka, M. Simcik Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502 Prague, Czech Republic

a r t i c l e

i n f o

Article history: Received 26 June 2019 Revised 15 August 2019 Accepted 25 August 2019 Available online 30 August 2019

a b s t r a c t A simple approximate formula was found on heuristic basis for the added mass of both rigid and deformable cap bodies. Its prediction agrees with the published CFD data. It can be used as a closure for numerical studies. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Cap bubble Added mass Fluidized bed Bubble column

1. Introduction In the multiphase flow systems, the bubbles shapes are various (Grace 1973; Darton et al., 1977; Clift et al., 1978). The spherical cap geometry occurs in both the fluidized beds (FB) and the bubble columns (BC). In gas-solid fluidization, the bubbles are voids (gas pockets) of low concentration of the dispersed particles, with no sharp interphase boundary since the emulsion phase has no surface tension. The cap is typically bigger than the hemisphere with cap angle α less than 160° (e.g. Schügerl et al., 1961, Rowe 1971). The experimental data on FB obtained by Rowe and Partridge (1965) show the values α within about 110–150°. In gas-liquid systems, the bubbles are well defined entities demarcated by the liquid surface. The cap is typically smaller than the hemisphere with α over 200° (e.g. Clift et al., 1978). The experimental data on BC obtained by Davies and Taylor (1950) show the values α about 230–270°. This shape difference between the cap bubbles in FB and BC has been known for a long time. The experimentally observed cap bubbles at steady rise were studied thoroughly, the speed obeys the basic scaling U ∼ (gD)1/2 where g - gravity, D - bubble diameter (Davies and Taylor 1950; Jackson 20 0 0; Puncochar et al., 2016). However, their unsteady behaviour has been rarely investigated and there are only few studies on the added mass effect at the vertical rise. Based on the published numerical data, our goal is to obtain a simple closed for∗

Corresponding author. E-mail addresses: [email protected], [email protected] (M. Puncochar). URL: http://www.icpf.cas.cz (M. Puncochar)

https://doi.org/10.1016/j.ijmultiphaseflow.2019.103098 0301-9322/© 2019 Elsevier Ltd. All rights reserved.

mula for the added mass coefficient C that can be used as a closure for CFD simulations and for building low-order physical models of multiphase flows. The definition sketch of the cap body is in Fig. 1. The dimensionless cap volume h = Vc /V0 is a geometric simplex that relates to the cap angle α ,

h = 1/4(2 − cos(α /2 ) )(1 + cos(α /2 ) )2

(1.1)

where α = 0° has h = 1 (full sphere) and α = 360° has h = 0 (vanishingly small cap, infinitesimal flat disk). 2. Available data for rigid and deformable cap bodies The added mass concept is well known in the fluid mechanics (e.g. Lamb 1932; Birkhoff 1960; Batchelor 1967). The inertial resistance reaction of a fluid with respect to an accelerating submerged body can be expressed by the inertia mass tensor M that relates the body acceleration vector a to the inertia force vector Fa ,

Fa = −M.a.

(2.1)

The mass tensor can be expressed as M = ρ Vb C, where ρ is the fluid density, Vb is the body volume, and C is the inertia coefficient tensor. In the case with sufficient symmetry, e.g. sphere, spherical cap, the tensor C becomes a mere scalar C termed added mass coefficient. It expresses how much fluid would experience the same acceleration as the body, if it would stick to it and move with it, making the body ’heavier’. In reality, the fluid does not stick to the body, but this demonstrative interpretation is dynamically equivalent to the actual flow field generated by the body. It is defined as C = Va /Vb , where Va is the virtual ‘added volume of

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M. Puncochar, M.C. Ruzicka and M. Simcik / International Journal of Multiphase Flow 120 (2019) 103098

List of symbols a b C d D F g h m M U V

acceleration [m/s2 ] correction factor [−] added mass coefficient [−] differential operator in derivation [−] diameter [m] force [N] gravity [m/s2 ] geometric simplex, dimensionless ratio [−] mass [kg] inertia tensor [kg] bubble speed [m/s] volume [m3 ]

Greek

α  ϕ ρ

Indexes 0 1, 2 a b c w ∗

+

cap angle [°] difference, increment [−] value of Golden Ratio, 1.618… [−] density, continuous phase density [kg/m3 ] original sphere fluidized bed (FB), bubble column (BC) added mass bubble cap shape, bubble, body wake deformable cap shape overestimated value of C

Abbreviations BC bubble column, bubbles in gas-liquid systems FB fluidized bed, bubbles in gas-solid systems

fluid’ travelling with the body. Despite Va is a virtual quantity, it can be used for making simple engineering models of the complex physical reality of dispersed systems (e.g. Zuber 1964; Simcik et al., 2014, various cell-models etc.). For an unidirectional axial translation of a cap body, the retarding added mass force is Fa = - a.ma , ma = ρ Va , Va = CVb , where a - body acceleration, ma - added mass. We want to know the relation between the added mass coefficient C and the cap shape geometry expressed e.g. by α , or h or others. For a rigid cap body, there are not many results in the literature on the relation C(α ). To our knowledge, its first estimate was made by Kendoush (2003) with an approximate analytical theory based on the potential flow, the value here denoted as C+ . Later on, Simcik et al. (2014) suggested an ad hoc engineering heuristic approach yielding the same result. The near-equivalence of these two different methods was shown in Ruzicka et al. (2015). Finally, Simcik et al. (2014) performed the direct numerical simulations

Fig. 1. Cap body. Original sphere (index 0) of diameter D is divided by cap angle (α ) into cap (c) and wake (w) behind it. The volumes obey: V0 = Vc + Vw .

Fig. 2. Available data. Rigid C+ , bold line - approximation by Kendoush (2003) and Simcik et al. (2014). Rigid C, full circles - CFD by Simcik et al. (2014), full line Eq. (3.4). Deformable C∗ , empty circles - CFD by Simcik et al. (2014), dotted line Eq. (3.5).

with CFD of the incompressible inviscid flow past a rigid cap body and determined the added mass, denoted as C. This value C is considered as the correct one and is considerably lower than C+ . For a deformable cap body, there are even less results. The only one we are aware of is by Simcik et al. (2014), who computed the multiphase flow over a fluid cap-shaped body with CFD using the Volume of Fluid method (VOF). The initial body acceleration was determined from a short time interval (∼ 10−5 s) where the body itself could manage to deform. From the acceleration, the added mass was calculated, denoted as C∗ . The results are summarized in Fig. 2, where we see the ordering of the coefficient values: C+ > C > C∗ . The output of our previous efforts is a set of complicated discontinuous correlations valid only within certain intervals of the cap angle that are not suitable for practical usage (Simcik et al., 2014). Here we offer a new simple formula covering a broad range of cap shapes. 3. The new formula for added mass Here we try to find a relation between the cap shape and its added mass in a formal way. We consider a hypothetical case of the cap formation from the original sphere. Consider the following mental experiment depicted in Fig. 3. The original sphere of a fixed size (volume V0 ) is divided into two parts, upper cap (Vc ) of a given angle α and lower wake (Vw ). The added mass (ma ) is schematically portrayed as the added volume (Va ) around the cap, Fig. 3a. We put this volume (Va ) inside the cap volume (Vc ), filling it from below and leaving its top part empty (Vc − Va ), Fig. 3b.

Fig. 3. Cap formation, added volume is (a) outside, (b) inside.

M. Puncochar, M.C. Ruzicka and M. Simcik / International Journal of Multiphase Flow 120 (2019) 103098

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Fig. 4. New formula: rigid cap C. (a) Solution of Eq. (3.3), Va /Vc = C. (b) Eq. (3.4) (line) agrees with numerical data (marks) from Fig. 2.

Now we slightly increase the cap angle by α , which makes the cap flatter and smaller by Vc . By flattening the cap, its inertial resistance rises, expressed by an increment Va .

We now postulate a relation between the two volume increments Vc and Va in the form:

(3.1)

This relation can be considered as a differential equation1 for the unknown function Va = f(Vc ),

dVa /dVc = Va /Vc − 1

(3.2)

with the initial condition Vc = V0 , Va = (1/2)V0 , which solves to

Va = Vc (1/2 + ln(V0 /Vc ))

(3.3)

In Fig. 4a, the graph line starts at 0, passes through the maximum (dashed line Va = Vc ) and drops to 0.5. The cap formation starts from the full sphere at [1, 0.5], follows the red arrow to the maximum [0.61, 0.61], and then declines to zero, the limit of vanishing cap Vc → 0. Eq. (3.3) can be written using h,

C = 1/2 + ln(1/h ),

factor < 1 at the logarithmic term in the formula Eq. (3.4),

C ∗ = 1/2 + b.ln(1/h )

3.1. Rigid cap C

−Vc /Vc = Va /(Vc − Va )

Fig. 5. New formula: deformable cap C∗ . (a) Values of b calculated from CFD data by Eq. (3.6) - marks, their average b = 0.501 - line. (b) Eq. (3.5) (line) agrees with numerical data (marks) from Fig. 2.

(3.4)

giving the new formula for rigid cap, see Fig. 4b. 3.2. Deformable cap C∗ The numerical data show that deformable shapes have a lower inertial resistance, C∗ ≤ C. This can be reflected by a correction 1 Eq. (3.2), dy/dx = y/x - 1, belongs to the class of similarity equations and is invariant with respect to the map (x, y) → (α x, α y) for any α = 0.

(3.5)

Its value can be obtained by merging Eqs. (3.4) and (3.5), by eliminating the logarithmic term,

b = (C ∗ − 1/2 )/(C − 1/2 )

(3.6)

The current values of b were calculated from the CFD data over a range of α in Fig. 2 (equivalently, range of h) and are plotted against h in Fig. 5a. In the intermediate α range of 10 0°−30 0°, b is close to 0.5 with a weak parabolic trend b = 0.502 − 0.173 h + 0.245 h2 (correl. coeff. Rxy = 0.980). Neglecting this trend and considering b near-constant, we can use its mean value 0.501 as a practical engineering estimate. With this value, the model equation Eq. (3.5) agrees well with the CFD data for C∗ , see Fig. 5b. Outside the practical range, in the limit of the full sphere (α = 0°, h = 1), b tends to unity because both the rigid and deformable spheres share the same value C = C∗ = 0.5. In the limit of the vanishing cap (flat disc, α = 360°, h = 0), we lack the CFD data to evaluate b. 3.3. Special case of C = C∗ = 1 For both the rigid and the deformable caps, the values of C with increasing cap angle α starts from ½ (full sphere) and diverges to infinity (flat disk), see Fig. 2. At the particular value of C = 1, the added volume equals the cap volume (Va = Vc ). If the cap body is a void in a fluid (gas bubble in liquid), its effective dynamic mass is the added mass. The bubble, virtually filled with the surrounding liquid, is like a material body of mass ma . The void dynamically disappears and the surrounding liquid stays intact, reclaiming its mechanical inviolacy. Thus a freely accelerating cap bubble with

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M. Puncochar, M.C. Ruzicka and M. Simcik / International Journal of Multiphase Flow 120 (2019) 103098

round b off to ½ and eliminate the logarithmic term ln(1/h) from Eqs. (3.4) and (3.5), we obtain a simple approximate relation between the rigid cap C and the deformable cap C∗ in the vicinity of unity where C = C∗ = 1,

C ∗ = 1/2 ( 1/2 + C )

(3.9) C∗

Here, the deformable cap equals the arithmetic mean of the full sphere (½) plus the rigid cap (C). 4. Conclusions

Fig. 6. Cap bubbles in experiments. (a) fluidized beds (FB), (b) bubble columns (BC).

C = 1 in inviscid liquid would be physically equivalent to a freely falling solid body in a vacuum.2 It follows from Fig. 2, that added mass = 1 corresponds to two different values of the cap angle, α 1 ≈ 160° and α 2 ≈ 200°. The former value corresponds to the rigid cap (C) that is larger than the hemisphere and reminds the cap bubbles typically observed in fluidized beds (FB, gas-solid system, index 1). The latter value corresponds to the deformable cap (C∗ ) that is smaller than the hemisphere and reminds the cap bubbles typically observed in bubble columns (BC, gas-liquid system, index 2). The both cases are shown in Fig. 6. The two cap shapes in Fig. 6 have certain symmetry features. The cap volume (Vc1 ) in FB is the wake volume (Vw2 ) in BC and vice versa, Vc1 = Vw2 and Vc2 = Vw1 . If we calculate the volume ratio of the original sphere to its bigger spherical segment, and, the volume ratio of the bigger spherical segment to the smaller spherical segment, we obtain the value ≈ 1.62,

V0 /Vc1 = Vc1 /Vw1 ≈ 1.62,

(FB )

(3.7)

V0 /Vw2 = Vw2 /Vc2 ≈ 1.62.

(BC )

(3.8)

This ratio is close to the well-known Golden Ratio, (whole/bigger) = (bigger/smaller), that takes the universal value ϕ = 1.618… . The Eqs. (3.7), (3.8) can be written in terms of the cap simplexes h1 and h2 , where h2 = 1 − h1 , to obtain e.g. h1 ≈ 1/ϕ ≈ 0.62 and h2 ≈ 2 − ϕ ≈ 0.38. If we insert C∗ = 1 and h2 = 0.38 into Eq. (3.5), we obtain the value b ≈ 0.51. If we

2 In Sec. 1, the long-term steady rise of a cap bubble in a real FB under the force equilibrium (net buoyancy = total drag) where the acceleration is zero, as it is observed in experiments, was mentioned, leading to the steady speed estimate U ∼ (gD)1/2 . On the contrary, in Sec. 3.3, the short-term unsteady rise of an accelerating cap-shaped void with C = 1 is considered, in a virtual mental experiment, where only the inertia and the gravity interplay: (added mass x acceleration) = net buoyancy, i.e. (ρ .Va ).a = g.ρ .Vc . Since Va = Vc , we have a = g, which is the free-fall law for a rigid body in the vacuum.

In this contribution, a new formula was obtained for the dependence of added mass coefficient C on the cap body shape in a broad range of the cap flatness. It agrees well with the published numerical data by CFD, for both the rigid and the deformable caps. This simple formula can be used instead of the set of complicated discontinuous correlations valid only within certain intervals of the cap angle reported previously (Simcik et al., 2014). Acknowledgements The work was supported by GACR project No. 19-09518S. Mr. A.A. Dear helped at correcting the English language. References Batchelor, G.K., 1967. An Introduction to Fluid Dynamics. Cambridge University Press. Birkhoff, G., 1960. Hydrodynamics: A study in logic, Fact and Similitude. Princeton University Press, New Jersey. Clift, R., Grace, J., Weber, M.E., 1978. Bubbles, Drops and Particles. Academic Press, NY. Darton, R.C., LaNauze, R.D., Davidson, J.F., Harrison, D., 1977. Bubble growth due to coalescence in fluidised beds. Trans. I. Chem. E. 55, 274–280. Davies, R.M., Taylor, G., 1950. The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. Roy. Soc. A 200, 375–390. Grace, J.R., 1973. Shapes and velocities of bubbles rising in infinite fluids. Trans. I. Chem. E. 51, 116–120. Jackson, R., 20 0 0. The Dynamics of Fluidized Particles. Cambridge University Press, UK. Kendoush, A.A., 2003. The virtual mass of a spherical-cap bubble. Phys. Fluids 15, 2782–2785. Lamb, H., 1932. Hydrodynamics. Cambridge University Press ISBN: 0 12 205550 0. Puncochar, M., Ruzicka, M.C., Simcik, M., 2016. Bubble swarm rise velocity in fluidized beds. Chem. Eng. Sci. 152, 84–94. Rowe, P.N., 1971. Experimental properties of bubbles. In: Davidson, J.F., Harrison, D. (Eds.), Fluidization. Academic Press, London, pp. 121–191 and New York. Rowe, P.N., Partridge, B.A., 1965. An X-ray study of bubbles in fluidised beds. Trans. I. Chem. Eng. 43, T157–T175. Ruzicka, M.C., Simcik, M., Puncochar, M., 2015. How to estimate added mass of a spherical cap body: two approaches. Chem. Eng. Sci. 134, 308–311. Schügerl, K., Merz, M., Fetting, F., 1961. Ermittlung der Dichteverteilung in gasdurchströmten Fliessbettsystemen durch Röntgenstrahlen. Chem. Eng. Sci 15 (, 1–2 ), 39–74. Simcik, M., Puncochar, M., Ruzicka, M.C., 2014. Added mass of a spherical cap body. Chem. Eng. Sci. 118, 1–8. Zuber, N., 1964. On the dispersed 2-phase flow in the laminar flow regime. Chem. Eng. Sci. 19, 897–917.