### Abstract

- Top of page
- Abstract
- Introduction
- Theory
- Experiment
- Results and Discussion
- Conclusions
- Notation
- Literature Cited
- Appendix
- Appendix B

The equations describing fluid flow and mass transfer around a sphere buried in a packed bed are presented, with due consideration given to the processes of transverse and longitudinal dispersion. Numerical solution of the equations was undertaken to obtain point values of the Sherwood number as a function of the Peclet and Schmidt numbers over a wide range of values of the relevant parameters. A correlation is proposed that describes accurately the dependence found numerically between the values of the Sherwood number and the values of Peclet and Schmidt numbers. Experiments on the dissolution of solid spheres (benzoic acid and 2-naphthol) buried in packed beds of sand, through which water was forced, at temperatures in the range 293 K to 373 K, gave values of the Sherwood number that were used to test the theoretical results obtained. Excellent agreement was found between theory and experiment, including the large number of available data for naphthalene–air, and this helps establish the proposed correlation as “general” for mass transfer between a buried sphere and the fluid (liquid or gas) flowing around it. © 2004 American Institute of Chemical Engineers *AIChE J*, 50: 65–74, 2004

### Introduction

- Top of page
- Abstract
- Introduction
- Theory
- Experiment
- Results and Discussion
- Conclusions
- Notation
- Literature Cited
- Appendix
- Appendix B

In several situations of practical interest a large solid mass interacts with the liquid flowing around it through the interstices of a packed bed of inerts. Examples are the leaching of buried rocks and the contamination of underground waters by compacted buried waste. The dissolution of a slightly soluble sphere buried in a packed bed of sand, through which water flows, is a useful model for such processes, and it is considered in the present work.

This study may be seen as a natural extension of work made by our team (Coelho and Guedes de Carvalho, 1988b; Guedes de Carvalho and Alves, 1999) on mass transfer around a sphere buried in a packed bed through which gas flows, which is important in the analysis of char combustion in fluidized beds. From a fundamental point of view, working with water has great interest, since it allows a considerable variation in the value of the Schmidt number, *Sc*. Indeed, whereas *Sc* is of order 1 for most gas mixtures, it is possible to cover the approximate range 50 < *Sc* < 1000 by working with common solutes in water, at temperatures between 290 and 373 K.

In a recent study on transverse dispersion in liquids, Delgado and Guedes de Carvalho (2001) showed that there is a significant dependence between *D*_{T} and *Sc*, in the range *Sc* < 550. Since the rate of mass transfer around a buried sphere exposed to a flowing fluid is strongly determined by *D*_{T} (see Guedes de Carvalho and Alves, 1999), it may be expected that this mass transfer will show a significant dependence on *Sc*.

The only data available on mass transfer between a sphere immersed in a packed bed of inerts and the liquid flowing around it seem to be those of Guedes de Carvalho and Delgado (1999), who worked with water near ambient temperature. Their results showed that the approach of Coelho and Guedes de Carvalho (1988b) could be extended to predict the rates of mass transfer in liquids at high values of the Peclet number, *Pe*′ (based on the diameter of the sphere).

The present work was undertaken in an effort to both widen the range of application of the theoretical analysis and provide experimental data over a substantial range of values of *Sc*.

### Theory

- Top of page
- Abstract
- Introduction
- Theory
- Experiment
- Results and Discussion
- Conclusions
- Notation
- Literature Cited
- Appendix
- Appendix B

In terms of analysis, we consider the situation of a slightly soluble sphere of diameter *d*_{1} buried in a bed of inert particles of diameter *d* (with *d* ≪ *d*_{1}), packed uniformly (void fraction ϵ) around the sphere. The packed bed is assumed to be “infinite” in extent, and a uniform interstitial velocity of liquid, *u*_{0}, is imposed at a great distance from the sphere.

Darcy's law, *u* = −*K**grad**p*, is assumed to hold, and if it is coupled with the continuity relation for an incompressible fluid, div *u* = 0, Laplace's equation ∇^{2}ϕ = 0 is obtained for the flow potential, ϕ = *Kp*, around the sphere.

In terms of spherical coordinates (*r*, θ), the potential and stream functions are, respectively (see Currie, 1993)

- (1)

- (2)

and the velocity components are

- (3)

- (4)

The analysis of mass transfer is based on a steady-state material balance on the solute crossing the borders of an elementary volume, limited by the potential surfaces ϕ and ϕ + δϕ, and the stream surfaces ψ and ψ + δψ. The resulting equation is (see Guedes de Carvalho and Alves, 1999)

- (5)

where ω is the distance to the flow axis, and *D*_{L} and *D*_{T} are the longitudinal and transverse dispersion coefficients, respectively.

The boundary conditions to be observed in the integration of Eq. 5 are: (1) the solute concentration is equal to the background concentration, *c*_{0}, far away from the sphere; (2) the solute concentration is equal to the equilibrium concentration, *c* = *c**, on the surface of the sphere; and (3) the concentration field is symmetric about the flow axis.

The procedure followed in the numerical solution of Eq. 5, with the appropriate boundary conditions, is an improved version of the method described by Guedes de Carvalho and Alves (1999), and it is detailed in Appendix A.

The numerical solution of Eq. 5 gives the concentration field, and from it the instant rate of dissolution of the sphere, *n*, is obtained by integrating the diffusion/dispersion flux over the surface *r* = *R*

- (6)

In the analysis of the results of the numerical computations, it is convenient to consider two separate ranges of *Pe*(= *u*_{0}*d*/*D*).

- 1
Low

*Pe* (say,

*Pe* < 0.1). For very low

*Pe*, dispersion is the direct result of molecular diffusion, with

*D*_{T} =

*D*_{L} =

*D*, and the numerical solution obtained by Guedes de Carvalho and Alves (

1999) applies. Those authors suggest that their results are well approximated (with an error of less than 1% in

*Sh*′) by

- (7)

where

*Pe*′ =

*u*_{0}*d*_{1}/

*D* is the Peclet number for the sphere.

- 2
Intermediate to high

*Pe* (say,

*Pe* > 0.1). As

*Pe* is increased above 0.1, values of

*D*_{T} and

*D*_{L} start to deviate from

*D*, and this has to be taken into account in the integration of Eq.

5. For gas flow, it is generally accepted that good approximate values are given by

- (8)

- (9)

for the entire range of values of

*Pe*, with

*Pe*_{L}(∞) = 2 and

*Pe*_{T}(∞) = 12, for flow through beds of approximately isometric particles (see Wilhelm,

1962; Coelho and Guedes de Carvalho,

1988a).

The numerical solution of Eq. 5, with *D*_{L} and *D*_{T} given by Eqs. 8 and 9, was worked out by Guedes de Carvalho and Alves (1999), who observed that the values of *Sh*′ obtained from the numerical solution are well represented (within 2% in *Sh*′) by

- (10)

Now, for liquids, since Eqs. 8 and 9 are still good approximations at low to intermediate values of *Pe* (say, up to *Pe* ≈ 1, for *D*_{L}, and up to *Pe* ≈ 10, for *D*_{T}), Eq. 10 will be adequate, but only for a narrow range of *Pe*. Beyond that range, more accurate values of *D*_{T} and *D*_{L} are required, in the numerical solution of Eq. 5.

Values of *D*_{T} for liquid flow have been reported recently by Delgado and Guedes de Carvalho (2001) and Guedes de Carvalho and Delgado (2000) in what seems to be the only available study on the influence of *Sc* on *D*_{T}. Their data for the range of values of *Pe* of interest in the present study are shown in Figure 1, where it is possible to observe the dependence of *D*_{T} on the Schmidt number for the range *Sc* ≤ 550. An empirical correlation was found to describe the measured data of *D*_{T} for *Sc* ≤ 550 (shown in Figure 1 as dashed lines)

- (11)

For *Sc* > 550 the transverse dispersion coefficient is found to be independent of the Schmidt number, and the correlation reduces to

- (12)

As for *D*_{L}, it is fortunate that its value is not needed with accuracy, since for *Pe* > 1, the boundary layer for mass transfer around the sphere is thin, provided that the approximate condition *d*_{1}/*d* > 10 is observed. Indeed, for *Pe*′(= *Pe**d*_{1}/*d*) > 10, the boundary layer is thin and the term with *D*_{L} in Eq. 5, can be neglected (Coelho and Guedes de Carvalho, 1988b); numerical computations were undertaken in the present work that confirm the insensitivity of *Sh*′ to *D*_{L}, for *Pe*′ > 10.

For each value of *Sc* indicated in Figure 1, Eq. 5 was solved numerically, with the point values of *D*_{T} given by the corresponding fitted curve (Eqs. 11 and 12). From the numerical simulations, plots of *Sh*′/ϵ vs. *Pe*′ were prepared for given values of *d*/*d*_{1} in a similar fashion to what was done by Guedes de Carvalho and Alves (1999); in the present case, a set of plots had to be made for each value of *Sc*. The results of the numerical computations are shown as points in Figure 2a–2d, and an expression was sought to describe the functional dependence observed, with good accuracy. The following equation is proposed for *Sc* ≤ 550

- (13)

For *Sc* > 550, the value of *Sh*′ is independent of *Sc*, since *D*_{T}/*D* is independent of *Sc*. Substituting *Sc* = 550 in Eq. 13 leads to

- (14)

and this can be expected to predict mass-transfer coefficients for *Sc* ≥ 550. In the plots in Figure 2, the solid lines represent either Eq. 13 or Eq. 14, and it may be seen that they describe the results of the numerical computations with very good accuracy.

The second term (with square brackets) on the righthand side of Eqs. 13 and 14 is therefore the “enhancement factor” due to convective dispersion. It will be noticed that the enhancement factor is independent of *Sc* for high values of this parameter and dependent on *Sc* for *Sc* ≤ 550. This is because mass-transfer rates around the sphere depend strongly on *D*_{T} and the value of *D*_{T}/*D* is independent of *Sc* only for *Sc* > 550, as shown recently by Delgado and Guedes de Carvalho (2001) in a detailed study on dispersion in liquids.

A convenient way of presenting the result given by Eqs. 13 and 14 is then

- (16)

with

- (17A)

and

- (17B)

It can be seen that η approaches unity as *Pe* is reduced, and for gas flow, with *Sc* ≅ 1, Eq. 17b is virtually coincident with η = (1 + *Pe*/9)^{1/2}, which was the expression suggested by Guedes de Carvalho and Alves (1999). The values of η obtained from the numerical solution of Eq. 5 are represented as points in Figure 3, alongside the lines corresponding to Eqs. 17a,b. The scale is linear and it can be seen that the agreement is excellent.

Equations 16 and 17 therefore can be seen as a general result derived from first principles for mass transfer between a sphere of diameter *d*_{1} buried in a packed bed of inerts of diameter *d*, and the fluid flowing through the interstices of the bed with velocity *u*_{0} (at some distance from the sphere). Experiments were performed to test this result over a wide range of temperatures, as described in the next section.

### Experiment

- Top of page
- Abstract
- Introduction
- Theory
- Experiment
- Results and Discussion
- Conclusions
- Notation
- Literature Cited
- Appendix
- Appendix B

Experiments were performed on the dissolution of single-spheres of both benzoic acid and 2-naphthol, buried in beds of sand (0.219 mm or 0.496 mm average particle diameter) through which water was steadily forced down at temperatures in the range 293 K to 373 K.

Figure 4 shows the experimental rig (not to scale). The distilled water in the feed reservoir was initially dearated under vacuum to avoid freeing gas bubbles in the rig at high temperature. The test column was a stainless steel tube (100 mm ID and 500 mm long), and it was immersed in a silicone oil bath kept at the desired operating temperature by means of a thermosetting bath head (not represented in the figure). The copper tubing feeding the pressurized water to the column at a constant metered rate was partly immersed in a preheater, and it had a significant length immersed in the same thermosetting bath as the test column; the copper tubing leaving the test column was immersed in a chiller to cool the outlet stream before reaching the UV analyzer.

In order to replace the sphere buried in the sand, valves B1 and B2 were closed, the lid of the test column was removed, and a source of distilled water was connected near N to direct water “backwards” to the test column. Valve B2 was then opened gently to reach incipient fluidization of the sand; the soluble sphere was then replaced, the water flow was immediately stopped, and the lid of the test column was placed back in position. The downflow of water through the test column was then initiated at a very low flow rate, while the column was immersed in the silicone oil bath and the temperature was allowed to rise (slowly) to the value intended for the experiment. The flow rate of water was then adjusted to the required value, v, and the concentration of solute in the outlet stream was continuously monitored by means of a UV/VIS Spectrophotometer (set at 274 nm for 2-naphthol, and at 226 nm for benzoic acid). The instant rate of dissolution of the sphere was calculated from the steady-state concentration of solute at the exit, *c*_{e}, as *n* = *vc*_{e}.

The spheres of benzoic acid and 2-naphthol were prepared from *p.a.* grade material, which was molten and then poured into moulds made of silicone rubber to give spheres with approximate diameters of 11, 15, 19, and 25 mm. If any slight imperfections showed on the surface of the spheres, they were easily removed by rubbing with fine sandpaper. Vernier callipers were used to make three measurements of the diameter of each sphere, along three perpendicular directions. The spheres would be discarded if the difference between any two measurements exceeded 1 mm; otherwise, they would be kept for the experiments and taken to have a diameter equal to the average of the three measurements.

The silica sand used in the experiments was previously washed, dried, and sieved in closely sized batches. Table 1 gives the corresponding size distribution, as determined by a laser diffraction technique.

Table 1. Properties of the Sand BedsNominal Size (mm) | ϵ | τ | Particle Diameter (< μm) |
---|

10% vol. | 20% vol. | 30% vol. | 40% vol. | 50% vol. | 60% vol. | 70% vol. | 80% vol. | 90% vol. | 100% vol. |
---|

0.219 | 0.40 | 1.41 | 161 | 180 | 195 | 210 | 225 | 241 | 259 | 281 | 313 | 476 |

0.496 | 0.34 | 1.41 | 277 | 340 | 389 | 434 | 480 | 530 | 588 | 660 | 756 | 900 |

### Conclusions

- Top of page
- Abstract
- Introduction
- Theory
- Experiment
- Results and Discussion
- Conclusions
- Notation
- Literature Cited
- Appendix
- Appendix B

The present work shows that a theory for mass transfer between a sphere buried in a packed bed of inerts and the fluid flowing past it may be derived from first principles, which is valid for any value of *Sc*. The numerical solution of the equation representing the theory gives the “exact” values of *Sh*′/ϵ, and these are well represented by Eqs. 13 and 14 (or, alternatively, by Eqs. 16 and 17a,b).

The large number of experimental points reported, covering a wide range of the relevant parameters, provide strong support for the theory developed.

In Figure 9 predictions of the theory presented in this work are compared with those given by equations proposed by other authors (see Table 2). Figure 9 refers only to one value of *d*/*d*_{1} (similar curves are obtained for other values of *d*/*d*_{1}), and the two “extreme” values of *Sc* = 1 and *Sc* = 980, corresponding approximately to gas flow and cold water flow, respectively, are considered. With hot water (or other low-viscosity liquids) or supercritical fluids, the intermediate range of *Sc* is covered and the lines giving the different correlations are situated between the two extremes.

Table 2. Equations Available in Literature for Mass Transfer from Large Active Particles in Beds of Small InertsReference | Model | Equation |
---|

La Nauze et al. (1984) | (gas flow) | 19 |

Prins et al. (1985) | | 20 |

| | 20a |

| | 20b |

Coelho and G. Carvalho (1988b) | | 21 |

Agarwal et al. (1988) | | 22 |

| | 22a |

| | 22b |

| | 22c |

| | 22d |

G. Carvalho and Alves (1999) | | 23 |

The large number of experimental values reported in the present work, together with the many available for gas flow [see Guedes de Carvalho and Alves (1999)], confirms the validity of Eqs. 13 and 14 over the entire range of *Sc*. A small sample of data points for naphthalene-air and 2-naphthol–water at ambient temperature are represented in Figure 9. It can be seen that the other equations give very inaccurate predictions over a wide range of values of *Pe*.

### Notation

- Top of page
- Abstract
- Introduction
- Theory
- Experiment
- Results and Discussion
- Conclusions
- Notation
- Literature Cited
- Appendix
- Appendix B

*c*_{0} = bulk concentration of solute

*c** = saturation concentration of solute

*c*_{e} = concentration in the outlet stream

*C* = dimensionless solute concentration (as defined in Eq. A1)

*d* = diameter of inert particles

*d*_{1} = diameter of active sphere

*D*_{L} = longitudinal dispersion coefficient

*D*_{m} = molecular diffusion coefficient

*D* = effective molecular diffusion coefficient (= *D*_{m}/τ)

*D*_{T} = transverse (radial) dispersion coefficient

*K* = permeability in Darcy's law

*k* = average mass-transfer coefficient

*Pe*_{L}(∞) = asymptotic value of *Pe*_{L} when *Re*_{p} ∞

*Pe*_{T}(∞) = asymptotic value of *Pe*_{T} when *Re*_{p} ∞

ℜ = dimensionless spherical radial coordinate (= *r*/*R*)

*r* = spherical radial coordinate (distance to the center of the soluble sphere)

*U* = dimensionless interstitial velocity (= *u*/*u*_{0})

*u* = absolute value of interstitial velocity

*u* = interstitial velocity (vector)

*u*_{0} = absolute value of interstitial velocity far from the active sphere

*u*_{r}, *u*_{θ} = components of fluid interstitial velocity

#### Greek letters

Φ = dimensionless potential function (as defined in Eq. A4)

ϕ = potential function (defined in Eq. 1)

η = enhancement factor due to convective dispersion (defined in Eq. 16)

θ = spherical angular coordinate

ω = cyclindrical radial coordinate (distance to the axis)

Ψ = dimensionless stream function (as defined in Eq. A5)

ψ = stream function (defined in Eq. 2)

#### Dimensionless groups

*Pe*′ = Peclet number based on diameter of active sphere (= *u*_{0}*d*_{1}/*D*)

*Pe* = Peclet number based on diameter of inert particles (= *u*_{0}*d*/*D*)

*Re*_{p} = Reynolds number based on diameter of inert particles (= ρ*ud*/μ)

*Sc* = Schmidt number (= μ/ρ*D*_{m})

*Sh*′ = Sherwood number (= *kd*_{1}/*d*)

#### Subscripts and superscripts

*i, j* = grid node indices (see Figure A1)