Flow Dynamics Measured and Simulated Inside a Single Levitated

A Parallel Strategy for a Level Set Simulation of Droplets Moving in a Liquid ... Hybrid Distributed-/Shared-Memory Parallelization For Re-initializin...
0 downloads 0 Views 549KB Size
Download PDF
416

Ind. Eng. Chem. Res. 2006, 45, 416-423

Flow Dynamics Measured and Simulated Inside a Single Levitated Droplet Edwin Gross-Hardt,† Andrea Amar,‡ Siegfried Stapf,‡ Bernhard Blu1 mich,‡ and Andreas Pfennig*,† Department of Chemical Engineering, Thermal Unit Operations, RWTH Aachen UniVersity, Wu¨llnerstrasse 5, 52056 Aachen, Germany, and Department of Macromolecular Chemistry, ITMC RWTH Aachen UniVersity, Worringerweg 1, 52074 Aachen, Germany

The flow dynamics inside a single liquid droplet of toluene levitated by a countercurrent of deuterated water (D2O) is investigated noninvasively and quantitatively by nuclear magnetic resonance (NMR) imaging techniques for the first time and compared with computational fluid-dynamics (CFD) calculations. For this purpose, a unique device is developed meeting the requirements for long period measurements (namely, a permanently stable droplet position and shape) and for measurements on series of droplets (namely, a high reproducibility of generation and position of droplets). The measurement cell of this device is manufactured with low tolerances regarding axial symmetry and positioned vertical and central in the NMR magnet with high precision enabling axially symmetric flow dynamics. The measured flow field shows good agreement with 2D axially symmetric CFD calculations with the finite element (FEM) code SEPRAN. A simple interface model with one free parameter accounts for the rigid cap encountered in the measurements. This free parameter is estimated by matching the velocity fields of the simulation with those determined experimentally. 1. Introduction The knowledge of the behavior of single droplets is the basis for the design of solvent extraction columns.1-4 Drop behavior is also important in other processes such as spray drying, fuel atomization, and spray coating. Thus, literature on single droplets can be found in several fields of science and engineering.5 In liquid-liquid extraction, the single-droplet behavior is determined by mass transfer and droplet sedimentation, which take place simultaneously and influence each other.1-3,5 Although theoretical, numerical, and experimental investigations on single droplets have been conducted in the past (e.g., a broad literature survey can be found in refs 2 and 5), sedimentation velocities (Figure 1b) and mass-transfer rates (Figure 1a) cannot be predicted a priori; experimental data can only be matched by additional empirical parameters.2,5 In Figure 1b, this is done by interpolating between the model of a droplet with ideally mobile interfacial region and the model of a rigid sphere. On the basis of the field for creeping flow found by Hadamard and Rybzinski, mass transfer rates are underestimated by orders of magnitude.7 The analytically found maximal mass transfer of Kronig and Brink, which is based on this field, is shown in Figure 1a. But also flow fields found with 2D axisymmetric CFD simulations for nondeformable droplets with an ideally mobile interfacial region lead to an underestimation.9,10 On the other hand, the sedimentation velocities are overestimated with the same flow fields as shown in Figure 1b. This is especially true for droplets with small diameters, which tend to behave like rigid spheres.2,5,10 Experimental investigations with single-droplet cells usually employ integral measurements of sedimentation and mass transfer.11-14 However, the exact fluid dynamics inside the droplet are crucial for precise modeling of sedimentation and mass transfer as discussed above. Particle-tracer methods have been used to visualize the flow pattern inside of droplets.15 However, the optical accessibility * Corresponding author. E-mail: [email protected]. † RWTH Aachen University. ‡ ITMC RWTH Aachen University.

Figure 1. (a) Mass transfer at single droplets1,6 (points) in comparison to the analytical model of Kronig and Brink7 (line) and to the model of a droplet with ideally mobile interfacial region1 (dotted line). (b) Sedimentation velocities of single droplets with and without mass transfer1 (points) in comparison with CFD models with rigid and ideally mobile interfacial region1,8 (dotted lines) and empirical correlation of measurement values1 (lines).

of the droplet in a measuring cell is very limited, frequently monitoring only motion in one suitable section within the droplet. Furthermore, they represent strictly speaking an invasive technique that can compromise the accuracy of the derived results on the fluid flow field. While the interactions of bulk phase and particles are usually negligible, the interaction of the interfacial region and the particles is significant, since exactly

10.1021/ie0506015 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/10/2005

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 417 Table 1. List of Substances

chemical formula purity/deg of deuteration [wt/wt] manufacturer density [kg/m3] dynamical viscosity [10-3 Pa‚s]

toluene

deuterated water

C6H5CH3 99.5% Aldrich 998,1 0,6

D2O 99,8% Aldrich 1107,0 1,3

the distinct influence of the behavior of the interfacial region is responsible for details of droplet fluid dynamics. For instance, it is known that they are very sensitive to traces of impurities in the system, which tend to accumulate at the interface.2,5,16,17 Thus, the experimentally validated modeling of single droplets strictly demands a noninvasive and quantitative measurement technique. Pulsed field gradient (PFG) nuclear magnetic resonance (NMR) appears to be an exceptionally suitable technique since it offers not only the possibility of noninvasively monitoring the droplet’s internal fluid dynamics and its change with time but also optical accessibility to every section within the droplet. Thus, a suitable experimental setup for stabilizing a sedimenting droplet in a countercurrent of surrounding liquid is developed. This device is embedded into the NMR apparatus, and NMR measurements on the stable levitated droplet are conducted. To model the flow field inside the droplet and in the surroundings, computational fluid dynamics (CFD) is used. Besides the Navier-Stokes equations a model for the interfacial region has also been implemented. The free parameter of the interface model has been estimated by matching experiment and simulation. This is an inverse problem where the cause (here the interfacial stress and velocity distribution) must be derived from its effects (here the flow field of the adjoining phases). Since the thickness of the interfacial region is of microscopic scale in comparison to the macroscopic scale of the droplet, velocity and stress in the interface cannot be directly measured, and this inverse approach is the only viable way.18 2. Experimental Methods A suitable experimental setup for a droplet stabilized in position by a countercurrent of surrounding liquid has been built into a NMR-apparatus. The choice of substances (section 2.1), the development of a levitation device (section 2.2), the use of suitable NMR hardware and pulse sequences (section 2.3), and the experimental procedure (section 2.4) are described. 2.1. Substances. The substances chosen should have 1H NMR spectra that can be separated from each other and properties appropriate for solvent extraction, namely, mutually insoluble liquids with a significant difference in density. Furthermore, the droplets composed thereof should have an interfacial region that is not completely rigid so that internal fluid dynamics can be measured. On the basis of the standard test systems for physical solvent extraction of the European Federation of Chemical Engineering (EFCE),19 a toluene droplet in a countercurrent of deuterated water was chosen (Table 1). By choosing D2O instead of water, the measurement signal of the surrounding phase was suppressed below the detectability limit. Frequency-selective radiofrequency (RF) pulses are used to isolate the methyl line of the multiline spectrum of toluene (see section 2.3). From the measurement of sedimentation velocities of toluene droplets in water with a diameter larger than 2 mm in comparison to the sedimentation velocities predicted for droplets with ideally mobile and rigid interfaces (compare Figure 1b),2 it can be expected that the interfacial region is not rigid.

Figure 2. Experimental setup for NMR measurements on levitated single droplets with a magnification of the cell geometry facilitating permanently stable droplet positions.

2.2. Experimental Setup. The experimental setup must meet the requirements for long period measurements (namely, a permanently stable droplet position and shape) and for measurements on series of droplets (namely, a high reproducibility of generation and position of droplets). Furthermore, the flow dynamics should be axially symmetric to facilitate comparison with 2D axial symmetric simulations. Starting from the single-droplet measurement device for mass transfer used at the Department of Chemical Engineering, Thermal Unit Operations of the RWTH Aachen University,1 a suitable computer-controlled experimental setup has been developed and is shown in Figure 2 (data see Table 2). Here, a countercurrent of surrounding phase of D2O flows from top to bottom of the cell. The droplet is produced by a precision injector where volume and injection speed are chosen such as to avoid daughter-drop generation. The droplet then rises until vertical force equilibrium is reached in the conical part, which is positioned in the magnetic center of the apparatus. This is ensured by adjusting the vertical position of the cell in initial experiments. To reduce the time for the droplet to reach equilibrium position in the magnetic center, the counter flow is switched on just before the drop arrives there. Then the velocity distribution inside the droplet is measured by NMR imaging techniques. After the measurement, the countercurrent is switched off, so that the droplet rises to the top where it does not effect the flow any more and can be removed. A glass cell manufactured with low tolerances regarding axial symmetry (see Figure 2 for dimension and assembly) was employed and fixated precisely in three points to facilitate vertical and central alignment of the cell and to avoid torsion. The geometry follows the experiences from a geometry opti-

418

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006

Table 2. List of Devices NMR hardware

gear pump for D2O precision injector for toluene droplet computer software measurement data logging and control

Bruker DSX 500 spectrometer with a field strength of 11.7 T (500 MHz 1H Larmor frequency) standard Bruker microimaging hardware with a maximum gradient strength of 1.0 T/m birdcage resonator of 10 mm i.d. Verder VG015 with VG1000 digit Hamilton PSD/2 precision syringe pump with 2500 µL syringe, injection needle with 0,7 mm i.d. PC 133 MHz, USB, RS232, OS MS Windows95 Visual Designer Intelligent Instrumentation UDAS-1001/E-4 with 5B-modules for 4-20 mA

mization with the objective to maximize the stability of droplet position, in particular the cell has a small inner diameter and a double cone with an inclination of less than 3° to avoid uncontrolled backflow. 2.3. NMR Basics, Hardware, and Pulse Sequences. The image and velocity information in an NMR experiment both rely on the same principle of spatially dependent magnetic field strengths provided by PFGs. The Larmor frequency (ω) can generally be written as

ω(r) ) γ|B(r)| ) γ(|B0| + g‚r) where γ is the gyromagnetic ratio of the proton, g is the first derivative of the z component of the magnetic field (B) with respect to space, and B0 is the constant main field. This provides one possibility to generate image information by acquiring the signal in the presence of a gradient so that the Fourier transform of the signal corresponds to the one-dimensional projection of the object onto the gradient axis, assuming that the intrinsic Larmor frequency in the constant field, ω0(r) ) γ|B0|), is identical for all spins. The second approach exploits the phase information that is acquired with the complex NMR signal. Applying a pulsed gradient (gphase) for a duration (δ) generates a phase shift

φ(r) ) [ω(r) - ω0]δ ) 2πkr relative to a reference value, where the wave vector is defined as k ) (1/2π)γgδ. The total signal intensity S(k)snormalized by its value in the absence of a gradientscan then be written as an integral over all spins in the sample:

S(k) )

∫P(r) exp[i2πk‚r] dr

Scanning k space evenly allows the reconstruction of the spindensity function P(r) following an inverse Fourier transformation. The scheme can be combined to obtain three-dimensional images, and a wide range of techniques have been developed that reduce the acquisition time of a full image considerably by either repeated refocusing of the signal or sectioning of the magnetization. Velocity (V) or rather displacement (R) during an interval (∆) is encoded in much the same way, by applying a pair of gradient pulses of opposite sign but identical area. This gradient pulse pair gives rise to a phase shift (φ(R) ) 2πkR) that is proportional to displacement and can be used for individual encoding schemes as well as in combination with the mentioned imaging sequences. In analogy to the phase encoded imaging, the distribution function of velocities, P(V), can be reconstructed. The conflicting requirements for the NMR pulse sequences applied to single-droplet dynamics are high resolution in both spatial and velocity dimensions as well as a short total measurement time for possibly transient phenomena. For steadystate flow dynamics in a permanently levitated droplet, multiple

acquisition techniques can be employed according to the internal velocities to be resolved. For the investigations of the levitated toluene droplets, a Bruker DSX 500 spectrometer with a supraconducting magnet with a field strength of 11.7 T (500 MHz 1H Larmor frequency) was used with a standard Bruker microimaging hardware providing a maximum magnetic gradient strength of 1.0 T/m and a birdcage resonator of 10 mm i.d. (compare Figure 2, data in Table 2). Figure 3 shows the measurement pulse sequence for the velocity map (Figure 3a). The sequence consists of a velocity encoding module (termed “PGSE”) and an imaging module (“spin-echo”). The first module prepares magnetization in such a way that a phase shift is assigned to each spin proportional to its velocity ((2) see above). The module acts on the whole sample, but only the methyl line of toluene is excited in order to avoid artifacts due to a superposition of information from the different spectral lines (1). The prepared magnetization is subsequently used as the input to the imaging module; here, the combination of a phase encoding PFG (3.1) and a frequency encoding step (3.3) produce a 2D image, which is restricted to all spins within a slice due to the frequency-selective 180° pulse used (3.2). The final image is reconstructed from 128 phase encoding steps. Velocity images were generated by comparing one pair of measurements with different values of the velocity encoding gradients for each direction (i.e., parallel and perpendicular to the flow axis of the continuous phase). The total duration of such an experiment was 83 min. The sequences for all other NMR measurements can be derived by omitting parts of the complete sequence: while velocity distributions without spatial resolution were obtained by executing only the first (PGSE) module with variation of the gradient strength, the positional stability of the drops was verified from the spinecho sequence, omitting the phase encoding step for the second dimension. Further methodical details can be found in Amar et al.20 The time of the velocity encoding (∆t1 ) ∆) and the time between velocity encoding and spatial mapping of velocity (∆t2) are chosen depending on the flow dynamics to avoid possible erroneous measurements, which are depicted in Figure 3b,c. If the time of the velocity encoding is going to zero, the exact velocity should be encoded. However, then displacements due to diffusion will dominate so that the correct velocity cannot be derived. Since the time for the velocity encoding is finite, only an average velocity can be measured. A reasonable upper limit of the time for the velocity encoding is given by Figure 3b. Here a droplet held in position by a countercurrent is shown along with the streamlines of the flow field. An exemplary fluid element in the droplet is depicted with a start point when the first gradient pulse is applied, the path along the streamlines covered between the pulses and the end point when the second gradient pulse is applied. If the time span is expanded even more, the packet of spins are changing the direction more than 90° as compared to the starting direction, and the measured velocities will become monotonically smaller, even if the spin

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 419

2.4. Experimental Proceeding. Besides the mutual saturation of the two phases, all components of the experimental setup being in contact with either liquid were thoroughly cleaned before each series of experiments to avoid detrimental effects on the droplet boundary mobility. This included glass parts, PTFE pipes, and connectors as well as the pump head itself. The droplet generation and the establishing of the droplet levitation is described in section 2.2. The average sedimentation velocity as a sensitive indicator for the interfacial mobility of the droplet (compare Figure 1b) is measured repeatedly between measurements. For this, the time the droplet needs to rise to its stable position in the resonator (without countercurrent) is recorded. The average sedimentation velocity compares well with the final sedimentation velocity as the time of acceleration is negligible as compared to the total rising time. If the sedimentation velocity changes dramatically between measurements, the substances are probably degraded, so that the device must be cleaned and the substances must be substituted. Now NMR measurements on the stability and steadiness and the reproducibility are conducted. First the stability of droplet position and shape and the steadiness of the flow dynamics can be confirmed with the first levitated droplet: (i) The stability of the droplet position and shape is verified with fast one-dimensional profiles in short succession along the vertical (z) and at least one of the horizontal (x, y) axes. (ii) The droplet shape is determined by two- and threedimensional images. (iii) The distribution function of the velocity in three orthogonal directions inside the droplet is measured over the whole droplet volume for different times of the levitation. In this way the steadiness of the flow dynamics can be checked. Then, with a series of droplets, the reproducibility of the droplet position, droplet shape and the internal flow dynamics are confirmed: (i) The positions of different droplets is checked with onedimensional profiles. In this way the reproducibility of the droplet position can be checked. (ii) The three distribution functions of the velocity are measured for different droplets. In this way the reproducibility of the flow dynamics can be checked. After confirming stability, steadiness, and reproducibility, the velocity maps are measured with a final droplet on individual slices or averaged over the dimension normal to the slices. Slices and velocities always lie in the same plane (e.g., Vx-Vy velocities are shown in a xy plane). Figure 3. (a) NMR measurement sequence for velocity imaging, (b) possible measurement error of the velocity encoding depending on ∆, and (c) possible error of the spatial allocation of the velocities depending on ∆t2.

is not decelerating. In our experiments, with measured velocities smaller than 40 mm/s the maximum displacement of spins during the encoding time (∆) of 2.5 ms was less than 0.1 mm, which is small as compared to the topology of the flow field. Furthermore, if the spins are to be spatially allocated in a volume element (voxel), the upper limit for the time of the beginning of velocity encoding until the end of spatial mapping is shown in Figure 3c. If the velocity encoded spin can leave the voxel within the time between velocity encoding and spatial mapping, the spatial allocation will be erroneous. The time between velocity and spatial encoding was about 5 ms, so that with measured velocities smaller than 40 mm/s the maximum change of position was considerably less than 0.2 mm in our experiments.

3. Theoretical Calculations We apply theoretical calculations to derive and validate a physically founded model for the droplet behavior. The complete model consists of the standard flow equations and a model for the interfacial region. The latter is to be derived in an inverse approach, that is, if the results of complete model and measurement coincide within the error of measurement, the model for the interfacial region is validated. The validation concerns the structure of the model as well as its estimated free parameters. The standard model for the interfacial region of a droplet is the ideally mobile interface. Here tangential velocity and stress are continuous when crossing the interface. However, in the literature (e.g., refs 5 and 15), it is discussed that a rigid cap can form on the droplet backside regarding the flow. Here the tangential velocity seems to be zero. A very simple model to account for this behavior is to model part of the interfacial region as ideally mobile and part as rigid.

420

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006

Figure 4 shows the procedure of the theoretical calculation which is described in the following. The given values are the volume flux, droplet volume, and shape. Thus, the inflow velocity (Vmax) and the diameters (D1 and D2) are given. The angle of the rigid cap (Rcap) has to be estimated by comparison of the flow field inside the droplet given by simulation and measurement. The droplet position (zdrop) is derived if the vertical force balance of the droplet is fulfilled and inertia forces are zero. This droplet position can be compared with the measurement of the position in the cell. The central part is the calculation that comprises the geometry creation and discretization, the mesh-independent solution of the physical model, and the derivation of the vertical forces exerted on the droplet. The physical model and its numerical solution are presented in the following sections. 3.1. Physical Model. Since no dimensionless parameter correlation is determined here, all equations are given in their general and not in their dimensionless form. The computational domains are denoted c for the surrounding continuous phase and d for the droplet. The incompressible continuity and the Navier-Stokes equations without turbulence model for newtonian fluids are solved on both domains (i.e., inside the droplet with density (Fd ) Ftoluene) and dynamic viscosity (ηd ) ηtoluene) and in the surrounding of the droplet with Fc ) FD2O and ηc ) ηD2O) in steady state with 2D axial symmetry, where p is the pressure:

divV b)

[

F Vr

]

∂rVr ∂Vz + )0 r ∂r ∂z

(1)

(( ) )

∂ 2V r ∂Vr ∂Vr ∂p ∂ ∂rVr )- +η + 2 + Vz ∂r ∂z ∂r ∂r r ∂r ∂z

((

(2)

)

)

∂Vz ∂r ∂Vz ∂Vz ∂2Vz ∂r ∂p F Vr + Vz + 2 + g (3) )- +η ∂r ∂z ∂z r ∂r ∂r

[

]

The stress (τ) at the interface of the droplet that is needed to calculate the vertical forces exerted on the droplet (see section 3.2) is given in local coordinates on the droplet surface, where σ is the interfacial tension, by (compare Figure 4 right)

(

τtt ) ηd

[

) ( ] [

)

∂ut ∂un ∂ut ∂un ) ηc + + ∂n ∂t d ∂n ∂t

]

c

∂un ∂un 2σ ) pc - 2ηc + . ∂n ∂n R

τnn ) pd - 2ηd

Figure 4. Flowchart for the estimation of the free parameter of the simple interfacial model. On the right, a coarse version of the discretization of the computational domain along with a schematic representation of the simple interface model is shown.

The point of transition between the two parts given by Rcap is a free parameter that is estimated as described before (see Figure 4). Since the deformation of the droplet is not changing with time, the interfacial tension needs not to be taken into account. Boundary conditions are formulated for the inlet and outlet of the continuous phase, the wall, and the symmetry line. At the walls, the no-slip condition is applied:

Vr ) Vz ) 0

(11)

(4)

At the symmetry line, the radial velocity and the tangential stress are zero:

(5)

Vr ) 0

(12)

∂Vz )0 ∂r

(13)

The droplet is modeled non-deformable and spherical and with an interface consisting of two different parts (compare Figure 4 right). For the part that is modeled as an ideally mobile interface, the boundary conditions in local coordinates on the droplet surface are

At the inlet, a parabolic profile of a completely developed laminar inflow for the velocity is given:

Vt,d ) Vt,c

(6)

Vr ) 0

τtt,d ) τtt,c

(7)

Vn,d ) Vn,c ) 0

(8)

And for the part that is modeled rigid, the boundary conditions are

Vt,d ) Vt,c ) 0

(9)

Vn,d ) Vn,c ) 0.

(10)

(14)

( ( ))

Vz ) Vmax 1 -

r Rin

2

(15)

At the outlet the flow field is undisturbed:

Vr ) 0

(16)

∂Vz )0 ∂z

(17)

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 421

A zero reference pressure is set arbitrarily at the center of the droplet. For post-processing purposes, the lines of constant stream function (ψ) are shown, which are tangent to the local velocity vectors. The stream function (ψ) is calculated by

ψ ) 2π

∫0z rVr dz* ) 2π∫0r rVz dr*

(18)

3.2. Numerical Solution. The physical model is solved with the general-purpose finite-element (FEM) package SEPRAN,21 which allows analysis of two-dimensional, axially symmetric and three-dimensional steady-state or transient simulations in complex geometries. The package is used in a wide variety of engineering applications including laminar flows of incompressible liquids and is developed at the Ingenieursbureau SEPRA and Delft University of Technology. As a first step of the numerical solution, the geometry is generated and discretized. Here, the nodal points must be densely distributed where high gradients of flow variables are expected and coarser where gradients are small with a smooth transition between these regions. The resulting elements must nowhere be distorted. In our case triangles are used for FEM. Here, it is necessary that the sinus of at least one angle in the triangle is noticeably different from zero (e.g., one angle should be noticeably different from 0° and 180°).22 After fulfilling these requirements, it was checked that the result of the simulation is not dependent on the discretization of the geometry (e.g., the mesh was adequately refined). A coarse version of the employed mesh is shown in Figure 4 (right). Then the flow equations are given on the discretized geometry and solved. With the used CFD code this is done with finite elements and the Galerkin method. The continuity equation is decoupled, and the pressure is a derived quantity when using the penalty method on Crouzeix-Raviart-triangle elements. The linear problem is solved directly, since the penalty method leads to a dense nonsymmetrical profile matrix. The nonlinear convection is solved iteratively with the locally quadratically converging Newton-Raphson method. A starting value is found with the creeping-flow solution. Then several linear converging Picard iterations are applied before switching to the NewtonRaphson method to facilitate convergence. The boundary integrals for the reaction forces are solved with the Simpson method. A detailed description of the methods as well as a survey of literature can be found in ref 21.

Figure 5. Time-dependent density profile of toluene obtained from a series of one-dimensional profiles along the vertical (z) and the horizontal (x) axis. 512 profiles each were acquired with a separation of 200 ms. Measurement of the profiles was started immediately after the arrival of the droplet in the center of the cell.

4. Results 4.1. Stability and Reproducibility. To verify the stability of position in all three directions, one-dimensional profiles of the toluene droplets were acquired every 200 ms. The experiments consisted of a conventional spin-echo sequence with a read gradient applied in either of the three orthogonal directions, employing one scan each to avoid averaging. The experiments were begun immediately after the drop had reached its final position at the center of the resonator and had come to rest. The spatial resolution in these one-dimensional images was 20 µm. It was found that once the drops reached their equilibrium position, they stabilized immediately, and from the projections, time-invariant position and size could be deduced with an error of less than 1% (i.e., a possible variation in position along any of the axes was less than the spatial resolution of the profiles of 20 µm). The drops were ellipsoidal in shape, with a ratio between the axial and sagittal diameters of 1.19. Figure 5 shows series of 512 projections acquired along the x and the z axis, respectively, covering intervals of 102 s each. Figure 5

Figure 6. Probability densities of velocities in three orthogonal directions for one drop and two different drop ages (left) and for two different drops and one drop age (right).

demonstrates with certainty the permanent stability of the droplet in position and shape while levitated. The droplet position is exactly in the center of the resonator. Since the measurement cell is fixated precisely central in the resonator (compare Figure 2) the droplet must be in the center of the measuring cell as well. The reproducibility and stability of the velocity fields can be proven in a similar way, by repeatedly measuring the velocity distribution function in all three directions in short succession. Figure 6 shows the velocity distribution averaged over the whole droplet, acquired at two different drop ages. The highest velocities occurring were in excess of 30 mm/s in z direction, in which an asymmetry is clearly observed, as compared to

422

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006

Figure 8. Absolute velocity inside the droplet in the direct vicinity of the interfacial region as a function of the angle regarding the center of the droplet.

Figure 7. Vector plots of the internal velocities in a toluene droplet measured in a slice of thickness of 0.5 mm (left) in comparison with a CFD simulation including a new interface model.

almost symmetric velocity distribution functions of Vx and Vy . For all directions, the average velocity, obtained from integrating over the functions in Figure 6, vanishes within experimental error. Although the mobile interface of the toluene droplet is expected to be susceptible to accumulated impurities, a significant change of the shape and width of the velocity distribution functions was not observed for experiments carried out over a total time of 8 h. Moreover, no deviation could be determined between the velocity distributions obtained from several independent drops. One can thus confirm that it is possible to generate drops with a pre-defined volume that possess identical velocity function. 4.2. Measured Velocity Fields and Comparison with CFD Calculations. In Figure 7 on the left, the vector representation of the velocity field measured by NMR is shown in a vertically oriented slice of 0.5 mm thickness. The circulation (vortex) pattern is restricted to approximately the upper half of the drop, while typically 10 times smaller velocities occur in the bottom half. They do not show a particular feature and appear to be random. The comparison with CFD simulations as described in section 3 is presented on the right of Figure 7. The angle of the rigid cap in the simulation has been estimated by comparing the center and extension of the vortex pattern with Rcap ) 139°. The agreement of measured and simulated droplet position is within experimental error. In Figure 8, the absolute velocity inside the droplet in the direct vicinity of the interfacial region is shown as a function of the angle regarding the center of the droplet. It is possible in the CFD simulation to get very precise values of the tangential velocities directly on the interface of the droplet by interpolating the velocity field in the direct vicinity. However, in the NMR measurement only the flow field inside the droplet is known at discrete points of space and the velocity can only be given in the direct vicinity of the interface. Thus, for the simulation, the velocities were interpolated on those discrete points in space and were then analyzed in the same manner as the velocities measured by NMR. For this proceeding the comparison is enhanced. The NMR measurement represented by points is in agreement with the CFD simulation represented by the line. There seems

to be a small shift regarding the angle β. That is, the peak of the velocities in the measurement is between 20° and 30° while the peak of the velocities in the simulation is at about 35°. Unfortunately, just between 20° and 30° there are only few measurement points. 5. Discussion and Conclusion 5.1. Experimental Setup. The measurement device together with the NMR-imaging technique is a promising approach to obtain noninvasive, spatially resolved experimental data of the internal flow dynamics in a levitated single droplet. The droplet position and shape is stable for hours and days (compare Figure 5). The position is central and precisely in the measurement volume. Furthermore, the velocity distribution inside the droplet is not changing with time (compare Figure 6), a strong evidence that also the velocity field is steady. Thus, a time integrating measurement and a steady-state simulation are suitable for the flow field of the droplet. Droplet position and shape as well as the velocity distribution inside the droplet are reproducible, so it is possible to get the full measurement information by integrating over different droplets instead of integrating over time. This has important implications for future studies on mass transfer where transient conditions might require the reconstruction of the desired image information from a series of individual drops. That is, instead of integrating the full measurement information over a long period of time to get the full measurement information, the information is constructed by superposing short measurements on different droplets of a series. A similar approach has been successfully applied for falling droplets by Han.23 5.2. Measured Velocity Fields. The measurement shows axially symmetric flow field with a clearly defined torus and a rigid cap in the back part. They are in agreement with experimental results found by Savic.15 He used aluminum particles in water drops of 10-20 mm diameter falling through castor oil to visualize the flow pattern inside the droplets. Because of refractive index differences, the generated data are only qualitative. He found vortex pattern inside the droplets and rigid caps in the backpart regarding the oncoming flow. Also droplets without visible rigid caps were observed. This is in agreement with the observation that such large droplets normally tend to sediment like droplets with ideally mobile interfacial region. The common theory for the development of rigid caps are the presence of surface active substances which tend to accumulate at the interface between two fluids. These contaminants are swept to rear by the surrounding flow leaving the front region uncontaminated. In the rear they lower the surface tension

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 423

and restrict the mobility of the rear interfacial region. Even with considerable effort in purification of the fluids, it seems impossible to completely eliminate the possibility of surface active contaminants in the fluids. 5.3. Comparison with CFD Calculations. The obtained measurement data can be used to discriminate different interface models and to estimate their free parameters. So far a simple model including a mobile and a rigid part of the interface with one free parameter concerning the point of transition of the two parts has been compared with the measurement data. The agreement of the measured and simulated flow field as shown in Figure 7 is promising, although there was only one parameter to fit the measurement data. In Figure 8 a small shift of the peak of velocities in the vicinity of the interfacial region is encountered. There are three possible sources for the deviation. First the scatter of the points is due to stochastic errors of the measurement is making the exact allocation of the peak difficult. Second the shift of measurement and simulation can be an effect of the systematic measurement error as described in section 2.3, since here the velocities are much higher as in the center of the vortex, which was used for adjustment of the model parameter Rcap. Thus, for a more precise comparison it will be necessary to incorporate a model for these effects. At last, the model might be only an approximation of the reality representing very well the streamlines encountered but not the absolute velocities at the interface. This is a good prospect for future model discrimination. 5.4. Future Prospects. The model parameter will depend on the experimental conditions and on the substances employed. Thus, more research has to be done to predict the angle Rcap for a given droplet volume and interfacial properties. In the future, a more sophisticated three-dimensional transient simulation with a free interfacial interface will be employed. Here, physically founded interface models such as the Newtonian surface fluid model25 will be implemented and tested with the experimental data. To account for measurement errors (compare Figure 5), a model for the transformation of the physical flow field into the measured flow field will be implemented. Model discrimination will be enhanced by the model-based approach, which is described in a survey paper of Beck and Woodbury.24 Here, the inverse problem to derive an interfacial model is solved rigorously along with the derivation of a characteristic statistical number for the model performance. Furthermore, the experimental degrees of freedom are optimized with a model-based experimental design resulting in more valuable measurement information. Acknowledgment This research is partially supported by the Deutsche Forschungsgesellschaft (DFG) within SFB 540 “Model-based experimental analysis of kinetic phenomena in fluid multi-phase reactive systems” at RWTH Aachen University, Germany. We thank Professor Guus Segal from the Delft University of

Technology for continual modifications of the SEPRAN source code to meet our needs. We are grateful to Song-I Han and E.G.-H. for planning and performing the first NMR experiments on levitated drops. Literature Cited (1) Henschke, M.; Pfennig, A. AIChE J. 1999, 45, 2079-2086. (2) Henschke, M. Auslegung pulsierter Siebboden-Extraktionskolonnen; Shaker Aachen: Aachen, Germany, 2004. (3) Gross-Hardt, E.; Henschke, M.; Klinger, S.; Pfennig, A. Design of pulsed extraction columns based on lab-scale experiments with a small number of drops. In Proceedings of the International SolVent Extraction Conference, ISEC 2002; Sole, K. C., Cole, P. M., Preston, J. S., Robinson, D. J., Eds.; 2002. (4) Bart, H.-J.; Modes, G.; Bro¨der, D. Chem. Ing. Tech. 1999, 71, 246249. (5) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978. (6) Hoting, B. Untersuchung zur Fluiddynamik und Stoffu¨bertragung in Extraktionskolonnen mit strukturierten Packungen; Fortschr.-Ber. VDI, Reihe III 439; VDI-Verlag: 1996. (7) Kronig, R.; Brink, J. C. Appl. Sci. Res. 1950, A2, 142-154. (8) Waheed, M. A.; Henschke, M.; Pfennig, A. Int. J. Heat Mass Transfer 2002, 45, 4507-4514. (9) Gross-Hardt, E.; Henschke, M.; Pfennig, A. AIChE J. 2003, 49, 1611. (10) Waheed, M. A. Fluiddynamik und Stoffaustausch bei freier und erzwungener Konvektion umstro¨mter Tropfen. Thesis, RWTH Aachen, 2001. (11) Modigell, M. Untersuchung der Stoffu¨bertragung zwischen zwei Flu¨ssigkeiten unter Beru¨cksichtigung von Grenzfla¨chenpha¨nomenen. Thesis, RWTH Aachen, 1981. (12) Schro¨ter, J.; Ba¨cker, W.; Hampe, M. J. Chem. Ing. Tech. 1998, 70, 279-283. (13) Schu¨gerl, K.; Ha¨nsel, R.; Schlichting, E.; W., H. Chem. Ing. Tech. 1986, 58, 308-317. (14) Otto, W.; Schu¨gerl, K. Chem. Ing. Tech. 1973, 45, 563-566. (15) Savic, P. Circulation and distortion of liquid drops falling through a viscous medium. Technical Report MT-22; National Research Council of Canada: 1953. (16) Mersmann, A. Chem. Ing. Tech. 1980, 52, 933-942. (17) Wesselingh, J. A. Chem. Eng. Process. 1987, 21, 9-14. (18) Slattery, J. C. Interfacial Transport Phenomena; Springer: Berlin, 1990. (19) Misek, T., Berger, R., Schro¨ter, J., Eds. Standard Test Systems For Liquid Extraction; European Federation of Chemical Engineering (EFCE), Publication Series 46; Institution of Chemical Engineers: Rugby, Warwickshire, England, 1985. (20) Amar, A.; Gross-Hardt, E.; Khrapitchev, A. A.; Stapf, S.; Pfennig, A.; Blu¨mich, B. J. Magn. Reson. 2005, 177, 75-85. (21) Segal, G. “Sepra Analysis”, Ingenieursbureau SEPRA, Park Nabij 3, 2491 EG Den Haag, The Netherlands, 2004. (22) van Kan, J. J. I. M.; Segal, A. Numerik partieller Differentialgleichungen; Teubner: 1995. (23) Han, S. I. Correlation of Position and Motion by NMR: Pipe Flow, Falling Drop, and Salt Water Ice. Thesis, RWTH Aachen University, 2001. (24) Beck, J. V.; Woodbury, K. A. Meas. Sci. Technol. 1998, 9, 839847. (25) Scriven, L. E. Chem. Eng. Sci. 1960, 12, 98-108.

ReceiVed for reView May 20, 2005 ReVised manuscript receiVed September 26, 2005 Accepted October 12, 2005 IE0506015