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Mass transfer correlations for dissolution of cylindrical additions in liquid metals with gas agitation
International Journal of Heat and Mass Transfer 97 (2016) 767–778
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Mass transfer correlations for dissolution of cylindrical additions in liquid metals with gas agitation Mehran Seyed Ahmadi a, Markus Bussmann b,⇑, Stavros A. Argyropoulos a a b
Department of Materials Science and Engineering, University of Toronto, ON, Canada Department of Mechanical and Industrial Engineering, University of Toronto, ON, Canada
a r t i c l e
i n f o
Article history: Received 25 June 2015 Received in revised form 27 November 2015 Accepted 15 February 2016
a b s t r a c t The results of an experimental study of the effect of gas agitation on mass transfer from cylindrical Si specimens to molten Al were used to deduce mean mass transfer coefficients in a two-phase flow. Then the three-dimensional gas-agitated liquid was simulated using the commercial software FLOW-3D, to obtain predictions of bubble distribution and estimates of the velocity field within the opaque liquid metal, to be used to predict mass transfer rates. An existing mass transfer correlation for a gas-agitated liquid without an external bulk flow was used to fit the experimental data within 5%. With a bulk velocity, a correlation for mass transfer from a vertical cylinder in cross flow was modified to incorporate the effect of gas injection, and it predicted the measured mass transfer coefficients within 11%. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction A major challenge to the production of Al–Si alloys is the slow rate of dissolution of solid Si in molten Al, which leads to significant energy and material losses [1]. To accelerate the alloying process, either higher bulk liquid velocities (e.g. by means of rotating impellers) or agitation of the liquid (e.g. by means of gas injection) can be used. We recently reported the rate of dissolution of vertical cylindrical Si additions in molten Al with and without imposing a bulk flow, and with and without gas injection [2,3]. In the current study, computational fluid dynamics (CFD) was used to predict the three-dimensional velocity field in the gas-agitated tank, and these velocities are used to extend the results of the dissolution experiments by developing correlations for mass transfer. Dimensionless correlations for convective heat and mass transfer from a solid to a liquid are based on boundary layer theory. When fluid motion is generated by an external source (as opposed to natural convection caused by thermal and density gradients), the classical mass transfer correlations (e.g. the Ranz–Marshall correlation [4,5]) are functions of Re and Sc (or Pr) [6]:
Sh ðor NuÞ ¼ fðSc ðor PrÞ; ReÞ
ð1Þ
These types of correlations are only applicable when the turbulence intensity in the external bulk flow is low. Following this approach, in a recent study [2] we fitted experimental data of the dissolution of Si in Al in a rotating tank without agitation with a dimensionless correlation for cylinders in cross flow [7], as a ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.02.043 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
function of only Re and Sc. However, ladle metallurgical processes involve the agitation of liquid at large scales that produce flow with high levels of free stream turbulence [8]. It is customary to take the effect of such turbulence into account by introducing another ~ lc =m, into the correlations, based on rms Reynolds number, ReT ¼ u velocity fluctuations, a characteristic length lc, and the kinematic viscosity m. This is similar to what is done for heat and mass transfer phenomena in flows with grid-generated turbulence [9–11]:
Sh ðor NuÞ ¼ fðSc ðor PrÞ; Re; ReT Þ
ð2Þ
~ represents the root In the definition of ReT, the velocity scale u mean square (rms) velocity fluctuations [12,13]. In a cylindrical coordinate system, this is calculated as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 02 u02 r þ uh þ uz ~¼ u 3
ð3Þ
Instead of ReT, the turbulence intensity can also be used [14,15]:
~ u Tu ¼ V
ð4Þ
represents a mean value of the local fluid motion near a where V solid, which would be calculated by taking into account the influence of multi-dimensional flow as in a gas-agitated system [13–15]:
h; z; tÞ ¼ Vðr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r þ u 2h þ u 2z u
ð5Þ
Note that this definition of Tu is different than that for gridgenerated turbulence, where the velocity component in the flow direction is dominant. In that case, the velocity fluctuations are
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Nomenclature English symbols C concentration, kg m3 d diameter of a cylindrical Si specimen, m d0 initial diameter of a cylindrical Si specimen, m db bubble diameter, m 3 Grm,l mass transfer Grashof number ½¼ gð@ q=@C Si ÞT l bulk 2 ðC saturation C Þ= q m Si Si H liquid height inside the tank, m Kl liquid kinetic energy per unit mass of liquid, m2 s2 km mean combined convection mass transfer coefficient, m s1 N k mean natural convection mass transfer coefficient for a m vertical cylinder, m s1 F k mean mass transfer coefficient for a vertical cylinder in m the gas-agitated bath with or without bulk velocity, m s1 l length of cylindrical Si specimen, m characteristic length, m lc M total number of data used in time-averaging Nu Nusselt number NV number of cells in a volume representing a Si cylinder atmospheric pressure, Pa P0 Pr Prandtl number [=m/a] Q gas flow rate at the lance exit, m3 s1 QS gas flow rate at standard temperature and pressure (273.15 K and 100 kPa), m3 s1 r, h, z Cylindrical coordinates Ro outer diameter of the bath, m Re Reynolds number ReT Reynolds number based on rms velocity fluctuations Sc Schmidt number [=m/D] ShF mean Sherwood number in the gas-agitated bath with or without bulk velocity normalized based on a unidirectional bulk flow velocity. On the other hand, the movement of discrete bubbles in a gas-agitated liquid generates both a three-dimensional mean flow and agitation (corresponding to mean and fluctuating velocities, respectively) even in the absence of an external bulk flow. Defining the turbulence intensity using the more general Eq. (4) allows for quantifying the degree of turbulence in a bubble-induced flow. Various transport correlations have been developed for gasagitated systems. These correlations require flow velocities, which can be determined by solving the Navier–Stokes equations in conjunction with a turbulence model. CFD is especially useful in high temperature liquid metals where it is very difficult to measure velocity. Several studies have utilized air–water systems to correlate heat or mass transfer from solid additions to the mean and fluctuating velocities of a recirculating flow generated by a bubble plume. In a study of the melting rate of horizontal ice rods [16], the heat transfer coefficients were correlated to the computed local mean velocity and turbulence intensity [17] based on the mean velocity at the centerline of the ladle, and Mazumdar et al. [14] proposed a correlation for the dissolution of solid benzoic acid cylinders in a gas-agitated water ladle taking into account computed local mean and fluctuating velocities. In high temperature liquid metals, Szekely et al. [15] developed an axisymmetric numerical model for a cylindrical plume to compute a velocity profile in an Ar-stirred steel ladle. Applying the bulk velocity values to the modified Lavender and Pei correlation [18] for cylinders, and assuming a constant turbulence intensity of 0.3, they demonstrated satisfactory agreement with the measured mass transfer from graphite cylinders. Following a similar approach,
T Tu t Ub
temperature, K turbulence intensity time, s bulk velocity of liquid normal to the centerline of a vertical cylinder or an injection lance, m s1 ~ u rms velocity fluctuations, m s1 ur, uh, uz velocity components in cylindrical coordinates, m s1 h , u z time-averaged mean velocity components in cylindrical r , u u coordinates, m s1 0 0 0 ur , uh , uz velocity fluctuation components in cylindrical coordinates, m s1 V mean velocity of liquid, m s1 v volume, m3 Greek symbols a thermal diffusivity, m2 s1 u general quantity h relative angular position with respect to Si specimen, deg hc contact angle, deg m kinematic viscosity, m2 s1 q density, kg m3 Subscripts Al molten Al rms root mean square Si solid Si Other || hi
absolute value spatial averaging
but this time prescribing a conical-shaped plume, Mazumdar et al. [19] predicted the local mean velocities and rms velocity fluctuations in a 25 kg gas-agitated melt, and showed that their previously-derived correlation for a cold system predicted the experimental data of Wright [20] reasonably well. In the above CFD studies [14–16,19], simple quasi-single phase models were developed based on a specified axisymmetric plume, with the buoyancy force acting on the surrounding liquid. For flows with larger discrete bubbles, such quasi-single phase models cannot be used to predict the flow in the plume [21]. This is because quasi-single phase models are based on a variable density formulation which does not incorporate bubbles or bubble dynamics, but rather are simply based on a core of reduced density fluid that takes into account buoyancy. In addition, none of the above studies superimposed gas injection onto a constant bulk flow. A recent study [2,3] examined the effects of bulk velocity and of gas agitation on the dissolution of vertical cylindrical Si additions in molten Al. In the current paper, corresponding CFD results of the fluid dynamics of those experiments are presented. The predicted velocities are used to develop correlations for mass transfer from solids in a gas-agitated liquid, that are compared to the experimental measurements of the dissolution rates of the Si cylinders in molten Al.
2. Summary of the experiments Hundreds of experiments were conducted to measure the dissolution rate of vertical cylindrical Si specimens in molten Al as a function of bulk flow velocity and the rate of gas injection [2,3].
M. Seyed Ahmadi et al. / International Journal of Heat and Mass Transfer 97 (2016) 767–778
A brief summary of the experiments, and of the flow behavior of the bubbles, is presented here.
2.1. Dissolution experiments The experiments were carried out in a Revolving Liquid Metal Tank (RLMT) of radius R0 = 17.5 cm and liquid height H = 15 cm, containing 35 kg of molten Al. For each experiment, a Si cylinder (d0 = 1.87 cm, l = 9–10 cm) was immersed vertically at a radial distance of 13.5 cm from the center of the tank. To agitate the molten Al, N2 was delivered to the top of a straight Ti lance (0.437 cm ID and 0.625 cm OD) immersed10 cm below the molten Al surface. For each experiment, once the Al reached the desired temperature and rotational speed, the lance was inserted into the melt, the gas was turned on, and N2 bubbles began to form. Then a Si cylinder was immersed 8 cm into the liquid for 3 min, at the same radial distance as the injection lance. Each partially-dissolved Si specimen was weighed to determine the amount dissolved. Three experiments were conducted at each experimental condition; the dissolved fraction data were used to calculate a mean mass transfer coefficient.
2.2. Bubble behavior Molten Al is opaque and so does not allow for the direct observation of the formation and trajectory of bubbles. Instead, we measured the pressure fluctuations in the gas delivery line, and video recorded the free surface of the molten Al, to determine the characteristics of the individual bubbles as they rose from the bottom of the lance. The pressure transducer data clearly indicated the bubble frequency. The average single bubble size was estimated by knowing the gas flow rate and assuming spherical bubbles. The estimated bubble diameters for four gas flow rates (0.25, 0.50, 0.75, 1.00 SLPM) without tank rotation were in the range of 0.9–1.5 cm.
(a)
769
When rotating the RLMT at 7 cm s1, the bubbles were 3–5% smaller than in the non-rotating tank. Information on the trajectory of the bubbles was obtained by observing the melt free surface (Fig. 1). Regardless of tank rotation rate, the bubbles rose along the lance and broke the free surface adjacent to the lance, as was expected due to the large contact angle between molten Al and the boron nitride coating that was applied to the lance (hc = 160° [22,23]). 3. Numerical methodology The commercial software FLOW-3D (version 10) was used to predict the bubble dynamics within the molten Al, and estimate the induced liquid velocity field in the tank. This section presents the model setup, the initial and boundary conditions, and the assumptions and simplifications. 3.1. Simulation setup The Navier–Stokes equations were solved with the standard two equation k–e model to account for turbulence. Bubbles and the liquid free surface were represented using the Volume of Fluid (VoF) method [24], and surface tension was accounted for. The simulations were transient as a steady state model would not predict discrete bubble formation and rise, and consequently bubbleinduced velocity fluctuations. The interaction of bubbles with the lance was taken into account by specifying a molten Al/boron nitride contact angle of 160°. The simulations were run using the ‘‘one-fluid” model that solves for flow within the liquid phase, but not within the air above the free surface. 3.1.1. Domain The computational domain included the lance and the tank, with enough room above the liquid to allow for free surface fluctuations (Fig. 2). The Si cylinder was not modeled. Rather, the results of the simulations were used to assess mean and fluctuating
(b)
Fig. 1. Bubble arrival at the liquid Al surface (a) bulk velocity = 0, (b) bulk velocity = 3.5 cm s1 (gas flow rate = 0.50 SLPM, the coated lance diameter is approximately 0.725 cm).
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M. Seyed Ahmadi et al. / International Journal of Heat and Mass Transfer 97 (2016) 767–778 Table 1 Effect of the number of cells on the predicted velocities (bulk velocity = 0, gas flow rate = 0.5 SLPM). Number of mesh cells 6
1.7 10 3 106 6.1 106
(cm s1) hVi
~ i (cm s1) hu
1.53 1.78 1.86
0.34 0.46 0.54
Table 2 List of simulations.
Fig. 2. Cylindrical domain for the FLOW-3D simulations of the N2–Al system. All dimensions are in cm.
Bulk velocity of liquid (cm s1)
Standard gas flow rate (SLPM)
Actual gas flow rate (LPM)
Non-rotating tank
0
0.50 0.75 1.00
1.76 2.57 3.24
Rotating tank
1.4 3.5
0.50 0
1.76 0
0.50 1.00 0.50 1.00
1.76 3.24 1.76 3.24
velocities at the various positions up and downstream of the lance where Si cylinders were positioned in the experiments. A similar approach was adopted by other studies (e.g. [14–16,19,25,26]) in which flow was predicted about an immersed melting or dissolving specimen. 3.1.2. Mesh The computational domain was discretized with a 3D cylindrical mesh, with the radial, angular and vertical coordinates denoted by r, h, and z, respectively. The mesh was selectively refined around the lance to capture bubble formation. As FLOW-3D solves the advection term explicitly, time steps are proportional to the control volume size, and so very small cells adversely affect runtime. To eliminate the very small cells at the center of the cylindrical domain (r = 0), a core of 4 cm radius (corresponding to 5.2% of the liquid volume) at the center of the domain was not meshed, and the mesh size near the core was gradually increased to avoid very small cells. 3.1.3. Initial conditions For the non-rotating tank, the liquid velocity field was initialized to zero. For the rotating tank, the velocity was initialized to a solid body rotation. 3.1.4. Boundary conditions The boundary conditions were specified as per the experimental conditions. To take into account the gas volume expansion due to heat exchange between injected gas and liquid metal, the standard gas volume flow rates (SLPM at 273.15 K and 100 kPa) were corrected to account for the melt temperature and static pressure at the lance exit. No-slip boundary conditions were imposed at the bottom of the tank (z = 0) and on the tank walls. When simulations were run of a rotating tank, appropriate velocities were specified at those walls. The pressure at the melt surface was specified as atmospheric pressure, P0. Although the flow was modeled in the entire tank (0° 6 h 6 360°), FLOW-3D nevertheless requires that one define a pair of boundaries in the angular direction (i.e. a pair of rz-planes). For the non-rotating tank, a periodic boundary condition was specified at these rz-planes, so that the liquid flow that left one plane re-entered the other. Both boundaries were located 180° from the lance to minimize the influence on bubble generation and on the flow in the vicinity of the lance. For the rotating tank, a solid body rotation velocity profile was imposed as an inflow condition at the rz-plane 90° upstream of the
7.0
Table 3 Comparison of experimental and CFD predictions of the equivalent bubble diameter at the lance exit (bulk velocity = 0). QS (SLPM)
0.50 1.00
db (cm) Experimental (estimated from pressure transducer data)
Predicted (counting bubbles at the lance exit)
1.16 ± 0.01 1.47 ± 0.02
1.26 ± 0.15 1.59 ± 0.20
lance, and an outflow boundary condition was specified at the same position 270° downstream. This is a small simplification of the experiments, by assuming that the solid body velocity profile is recovered far downstream of the lance. 3.1.5. Stopping criterion Full domain simulations were run with a mesh of about 3 106 cells and time steps of 106–105 s. Each simulation ran for one to two months on an 8 core Intel i7-4770 machine running at 3.40 GHz, with 32 GB of RAM. The simulations were run until a quasi-steady state was reached, evaluated by monitoring a sliding l. 1 s average of the liquid kinetic energy per unit mass of liquid, K Simulations were stopped when jdK l =dtj was reduced by a factor of 5 from its initial value. This occurred at about 9 s of simulation time for the non-rotating tank; for consistency, the simulations in the rotating tank were run for the same duration. The rate of l =dtj at 9 s was small, implying that a more stringent decrease of jdK stopping criterion would require far longer run times. 3.2. Temporal- and spatial-averaged velocities A pragmatic approach was adopted to estimate mean velocities and turbulence intensities at various positions in the domain, corresponding to where the Si cylinders were immersed into molten Al. The results were averaged over time to capture the effect of many bubbles, while taking into account the transient response of the system. Results were averaged spatially over a volume equivalent to that of a Si cylinder. As the simulations were transient and started from the onset of bubble injection, the mean velocities were time-dependent. Timeaveraged mean velocity components and the rms values of fluctuating components in each direction were calculated as follows:
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(a)
771
(b)
Fig. 3. Predicted instantaneous bubble distributions at gas flow rates of (a) 0.50 SLPM and (b) 1.00 SLPM (bulk velocity = 0).
r ðr;h;z;tÞ ¼ u
M M M 1X 1X 1X h ðr;h;z;tÞ ¼ z ðr;h;z;tÞ ¼ uir ; u uih ; u ui M i¼1 M i¼1 M i¼1 z
3.3. Mesh and time step independence
ð6Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XM ðui u r Þ2 r ; i¼1 M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffi XM ðui u z Þ2 z u02 z ¼ i¼1 M qffiffiffiffiffiffi u02 r ¼
qffiffiffiffiffiffi u02 h ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XM ðui u h Þ2 h ; i¼1 M ð7Þ
where M was the number of liquid velocity temporal data. Substituting Eq. (6) into Eq. (5) and Eq. (7) into Eq. (3) yielded mean and rms velocity fluctuations u ~. velocity V Time-averaging must be performed over a duration that is sufficiently longer than that of any velocity fluctuation [27]. Therefore, a sliding temporal average was calculated for a period of 1 s, during which about 30 bubbles were released, based on data at every 0.005 s (i.e. M = 200). Sampling the data every 0.05 s for one of the cases yielded very similar time-averaged results. In the experiments, the overall dissolution rate of each Si cylinder was measured, and so spatial averages of mean and rms velocity fluctuations were calculated within the volume that would encompass a Si specimen. Assuming an equal weight for each cell in the volume:
hui ¼
P
vu
Nv
ð8Þ
where u represents a time-averaged mean velocity or a rms value of the velocity fluctuations, and Nv is the number of cells (15,000–45,000) where the cylinder would be.
Treating fluid advection explicitly results in smaller time steps on a finer mesh and so ensures time step independency. To determine an acceptable mesh size, three meshes: 1.7 106, 3 106, and 6.1 106 cells were evaluated for the case of a non-rotating tank and a gas flow rate of 0.5 SLPM. The spatially-averaged mean velocity and rms velocity fluctuations at h = 30° from the lance were compared. The results are tabulated in Table 1. The selection of an acceptable mesh size is a balance between speed and accuracy. For example, increasing the number of cells from 3 106 to hu ~ i, and the predicted mass transfer coeffi6.1 106 changed hVi, cients (see Section 5) by 4%, 19%, and 7% respectively, but increased the solution time by a factor of 2.2. A mesh of 3 106 cells was used for all of the simulations presented in this paper. 4. Results and discussion Table 2 lists the nine simulations that were run. Simulation results of the bubble dynamics, and of the time and spatial averages of mean velocities and velocity fluctuations are presented at various gas flow rates, with and without tank rotation. We only consider a Si specimen immersed 30° downstream of the lance, as this was the closest distance between the specimen and the lance, and because the change in the dissolution rate enhancement due to gas agitation as a function of angular position was in the range of uncertainty of the experimental data [3]. 4.1. Non-rotating tank Simulations in a non-rotating tank were run at gas flow rates of 0.50, 0.75 and 1.00 SLPM. Given the uncertainties: the contact
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Fig. 4. Spatially-averaged (a) absolute mean velocity and (b) rms velocity fluctuations in the r, h, and z directions (bulk velocity = 0, gas flow rate = 0.50 SLPM).
Fig. 5. Comparison of spatially-averaged (a) absolute mean velocity and (b) rms velocity fluctuations at various gas flow rates (bulk velocity = 0).
angle, the temperature-adjusted gas flow rate at the lance exit, and the uncertainties of transport properties (density, viscosity, and surface tension) at high temperatures, the FLOW-3D results are surprisingly good. The predicted bubble size was 8% higher than the values estimated from the experiments, as indicated in Table 3. The simulation at a gas flow rate of 0.50 SLPM was also run with the contact angle set to 150° and 170°. As expected, average bubble size increased and decreased, respectively. At 170° and a gas flow rate of 0.50 SLPM, the average size was 1.22 ± 0.18 cm, just slightly larger than the measured size. Fig. 3 shows sample results for gas flow rates of 0.50 and 1.00 SLPM, that illustrate bubbles rising along the lance, and that can be compared to Fig. 1. At 0.50 SLPM, bubbles rise with a velocity of 0.63 ± 0.05 m s1, higher than that of unhindered bubbles of a similar size [28,29]. The attachment of ascending bubbles to a poorly wetted lance has been previously reported [30,31] in water
model studies, and Watanabe and Iguchi [31] showed that such bubbles rise more quickly than those away from a lance. Fig. 4 illustrates the spatial average of the absolute mean velocity (Eq. (5)) and the rms velocity fluctuations (Eq. (3)) in the r, h, and z directions for a Si cylinder 30° downstream of the lance, at a gas flow rate of 0.50 SLPM. Fig. 4(a) illustrates the multidimensionality of the mean flow: the velocity components are of the same order of magnitude, although the mean tangential velocity component is larger than the vertical and radial components. This result confirms that assuming an axisymmetric flow relative to the lance, and neglecting flow perpendicular to the plane that passes through the solid addition and the lance [14,15,26], would underestimate the mean velocity. Fig. 4(b) shows that the velocity fluctuations are appreciable in all directions, although the tangential and vertical components are greater than the radial one, which implies a deviation from isotropic turbulence, as per the results of [32].
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Fig. 6. Spatially-averaged absolute mean velocity in the r, h, and z directions (bulk velocity = 3.5 cm s1, gas flow rate = 0 SLPM).
Fig. 7. Spatially-averaged rms velocity fluctuations and turbulence intensity (bulk velocity = 3.5 cm s1, gas flow rate = 0 SLPM).
Fig. 5 shows the mean velocities and rms velocity fluctuations at various gas flow rates in a non-rotating tank. Increasing the gas flow rate from 0.50 to 1.00 SLPM increases the velocity fluctu indicating that a ~ i much more than the mean velocity hVi, ations hu higher gas flow rate induces more liquid agitation than a stronger mean flow. This observation is in agreement with the experimental findings of [33].
4.2. Rotating tank Tank rotation generates a liquid bulk velocity. As discussed in Section 2, this causes a small decrease in the bubble size, but the bubbles continue to rise along the lance, as confirmed by Fig. 1. FLOW-3D predicts these behaviors. By increasing the bulk velocity from 0 to 3.5 cm s1, at gas flow rates of 0.50 SLPM and 1.00 SLPM, the average bubble size decreases slightly from 1.26 cm to 1.24 cm,
Fig. 8. Spatially-averaged (a) absolute mean velocity and (b) rms velocity fluctuations in the r, h, and z directions (bulk velocity = 3.5 cm s1, gas flow rate = 0.50 SLPM).
and from 1.49 cm to 1.48 cm, respectively. These reductions are negligible compared to the standard deviations associated with the equivalent bubble diameter; a similar conclusion was drawn from the bubble frequency measurements. Before examining mean and fluctuating velocities due to gas agitation in the rotating tank, it is instructive to consider the influence of the lance on the flow field without gas injection, because the simplifying assumption of solid body rotation is invalid near the lance. Such a simulation offers an estimate of the velocity fluctuations due to pure shear-induced turbulence. Note that the rotating tank simulations were initiated from a solid body rotation velocity field similar to the experiments; this choice of initial condition did not affect the quasi-steady state values predicted by the simulations. Fig. 6 shows absolute mean velocities at h = 30° averaged over the cylinder volume. The tangential velocity decreases from the
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Fig. 9. Comparison of spatially-averaged absolute mean velocities in the (a) r, (b) h, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s1).
Fig. 10. Comparison of spatially-averaged rms velocity fluctuations in the (a) r, (b) h, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s1).
Table 4 Predicted mass transfer coefficients for the non-rotating tank, and a comparison with experimental values. Gas flow rate (SLPM)
Re based on initial specimen diameter
Mean transfer coefficient 105, m s1
hVi
~i hu
Red0 based on hVi
ReT based ~i on hu
Experimental
Predicted via Eq. (9)
Modified using combined convection correlation, Eq. (11)
correlationExperimental Error (%) = Combined convection 100 Experimental
– 1.78 1.90
– 0.46 0.81
– 637 680
– 166 290
5.17 6.04 6.48
– 4.40 5.35
– 5.75 6.26
– 5 3
Table 5 Predicted mass transfer coefficients for the rotating tank, and a comparison with experimental values. Gas flow rate (SLPM)
0.50 0.50 1.00 0.50 1.00
Ub, Bulk velocity (cm s1)
1.4 3.5 3.5 7.0 7.0
Bubbleinduced velocities, cm s1 (average at steady state)
Re based on initial specimen diameter
Mean mass transfer coefficient 105, m s1
hVi
~i hu
Red0 based on hVi
ReT based ~i on hu
Experimental
Predicted via Eq. (13)
Modified using combined convection correlation, Eq. (11)
correlationExperimental Error ð%Þ ¼ Combined convection 100 Experimental
1.70 3.43 3.46 6.91 7.02
0.72 0.76 0.91 0.96 1.06
609 1227 1236 2473 2511
259 272 326 342 379
6.77 8.36 9.08 11.06 11.52
5.28 7.58 7.70 11.00 11.23
6.21 7.96 8.06 11.19 11.35
8 5 11 +2 1
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0 0.50 1.00
Bubble-induced velocities, cm s1 (average at steady state)
775
776
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Table 6 Dimensionless parameters involved in combined natural and forced convection from a Si vertical cylinder in liquid Al cross flow without gas agitation. l (cm) d0 (cm) T (°C) Sc Grm,l 8
1.87
700
Ub (cm s1) Red0
42 5.5 108 1.4 3.5 7.0
4
ðd0 =lÞ ðGrm;l =Re2d0 Þ
510 6.5 1260 1.0 2530 0.2
5. Correlations The mean mass transfer coefficients obtained from the experimental results are compared to predictions from a correlation for mass transfer from a cylindrical addition in a gas-agitated liquid (i.e. non-rotating tank), and in the cross flow of a turbulent liquid (i.e. rotating tank), where the predicted velocities of Section 4 are used to evaluate the various dimensionless groups. 5.1. Gas agitation without bulk velocity
initial 3.5 cm s1 to a steady 2.5 cm s1 due to the drag force exerted by the lance. There are also small radial and vertical velocities that are an order of magnitude less than the tangential velocity due to the flow disturbances induced by the lance. Fig. 7 shows the rms velocity fluctuations and turbulence intensity. The spatially-averaged turbulence intensity hTui is about 11% at steady state. This value is in the expected range for single-phase flows [32,34]. For the remainder of the discussion of the effect of tank rotation, only the data for a bulk velocity of 3.5 cm s1 are presented. Data at other rotation rates yielded conclusions similar to those presented here. Fig. 8(a) shows the mean velocity components at a gas flow rate of 0.50 SLPM. The results demonstrate radial and vertical velocities of the same order of magnitude, albeit much smaller than the tangential velocity in the bulk flow direction. This is a crucial distinction from previous studies on heat and mass transfer from vertical cylinders in cross flow with grid-generated turbulence [9,10,18,35,36]. These experiments were conducted in wind tunnels where the mean velocity was one-dimensional and constant along the test section; correlations derived from such experiments would be difficult to apply to a system with gas agitation as the source of velocity fluctuations. Fig. 8(b) indicates that the rms velocity fluctuations are significant in all directions, and that fluctuations in the tangential direction are slightly higher than in the vertical and radial directions because of the interaction of the bulk and bubble-induced flows. These results further confirm the deviation from isotropic gridgenerated turbulence. Figs. 9 and 10 illustrate a comparison of mean velocities and rms velocity fluctuations, respectively, at three gas flow rates. As the gas flow rate increases from 0.50 SLPM to 1.00 SLPM, larger bubbles form at the lance exit (Table 3), displacing a larger volume of liquid that increases the mean flow and induces a higher degree of agitation (and thus increases the turbulence intensity). As demonstrated in Fig. 9(b), the mean tangential velocity decreases downstream of the lance due to the effect of larger bubble wakes, but the absolute mean velocities increase in the radial (Fig. 9(a)) and in particular the vertical directions (Fig. 9(c)). Increasing the gas flow rate from 0.50 to 1.00 SLPM, the mean tangential velocity (at quasi-steady state) decreases by about 10%, while the absolute mean radial and vertical velocities increase 30% and 50%, respectively. This implies that for transport phenomena taking place in bubble-induced flows, neglecting the three-dimensionality of the velocity field may lead to inconsistencies when correlating the operating parameters (e.g. gas flow rate) to the mass transfer rate. Nevertheless, the net effect of the aforementioned changes is a negligible change to the mean velocity hVi. Finally, Fig. 10 shows that increasing the gas flow rate from 0.50 SLPM to 1.00 SLPM increases the turbulence intensity in all directions. The components of rms velocity fluctuations increase in the radial, tangential and vertical directions by approximately 30%, 20%, and 20%, respectively, and so the overall rms velocity fluctua~ i also increase by 20%. This demonstrates that an increase tions hu in mass transfer rates at higher gas flow rates is primarily associated with the enhanced velocity fluctuations.
For the non-rotating tank, the results of the dissolution experiments with gas injection [3] are compared to the correlation of Mazumdar et al. [14,19] for mass transfer from vertical benzoic acid cylinders in a gas-stirred water tank [14], and vertical steel cylinders in a gas-stirred carbon saturated iron bath [19], both inside and outside the two-phase region:
ShF ¼ 0:73ðRed Þ0:25 ðReT Þ0:32 ðScÞ0:33
ð9Þ
Red and ReT are based on local mean and rms velocity fluctuations, respectively. The simulation predictions of mean velocity and rms velocity fluctuations, and the corresponding values of Red and ReT, are tabulated in Table 4. Eq. (9) underestimates the mean mass transfer coefficients by 27% and 17% at gas flow rates of 0.50 and 1.00 SLPM, respectively. Why? Eq. (9) assumes that mass transfer due to gas agitation is dominant compared to natural convection mass transfer with no gas agitation. This may be valid in [14,19] because both and u ~ at the location of the dissolving specimens are much V < 32 cm s1, higher (benzoic acid cylinders in water: 5 cm s1 < V ~ < 7.5 cm s1 [14]; steel cylinders in carbon saturated 1.1 cm s1 < u < 43 cm s1, 1.3 cm s1 < u ~ < 5.0 cm s1 [19]) iron: 18 cm s1 < V than the values predicted in the current study (Fig. 5 and Table 4). As a result, natural convection effects were less important in those studies. To take into account natural convection, the Mazumdar correlation values were combined with a measured natural convection (no gas injection, no bulk velocity) mass transfer coefficient (5.17 105 m s1), or with a correlation for natural convection mass transfer proposed in [2]. Using the general combining law similar to that developed in [2] yields a much better agreement with the experimental data. A general combining law for transfer processes which vary uniformly between two known limiting solutions is given by [37]:
Z ¼ ðX p þ Y p Þ
1=p
ð10Þ
where X and Y are the limiting solutions for asymptotically small and large values of an independent variable such as agitation or external velocity. For the current problem the mass transfer coeffiN . With cient with no rotation and no gas agitation is given by Y ¼ k m a high degree of gas agitation and/or an external bulk velocity, F represents the asymptotic solution. As in [2], we set p = 4. X¼k m Combining the mass transfer coefficient for gas-agitated flow (Eq. (9)), and a correlation for natural convection, Eq. (10) becomes:
h i1=4 m ¼ ðk N Þ4 þ ðk F Þ4 k m m
ð11Þ
Table 4 shows that Eq. (11) predicts the experimental results within 5% and 3% at gas flow rates of 0.50 and 1.00 SLPM, respectively. Finally, note that the uncertainty of the velocity field predictions associated with mesh resolution can explain some of the discrepancy between the correlation predictions and the experimental results. Using a finer mesh (6.1 M compared to 3 M cells, as presented in Table 1) for the case of a gas flow rate of 0.5 SLPM with no tank rotation, velocities increased slightly, and
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those increases would increase the mass transfer coefficient by 7%, yielding even better agreement between correlation predictions and experimental results. Unfortunately we were unable to run all simulations at this resolution in a reasonable timeframe. 5.2. Gas agitation with bulk velocity The Mazumdar correlation was also applied to the rotating tank (bulk velocity = 3.5 cm s1, for which the gas-agitated velocity field was predicted in Section 4). However, even when considering natural convection, the correlation underpredicts the experimental results by 19% and 23% at gas flow rates of 0.50 SLPM and 1.00 SLPM, respectively. This is because the Mazumdar fluid flow configuration with respect to the cylindrical specimen [14,19] is different than in the rotating tank. For this reason, an effort was made to compare the experimental results to a more appropriate turbulent flow configuration. There are many correlations that predict heat and mass transfer from spheres and cylinders in cross flow, but usually in singlephase air flow characterized by low turbulence intensities (e.g. [18]). In studies involving two-phase flow (i.e. air/water), Iguchi et al. [38] and Szekely et al. [16] proposed correlations for heat transfer based on the melting rate of ice spheres and horizontal cylinders in a bottom-stirred cylindrical water tank, respectively. Iguchi et al. [38] defined Tu based on axial mean and rms velocity fluctuations along the plume centerline; Szekely et al. [16] normalized the local velocity fluctuations based on gas entry velocity. As a result, these correlations are not based on local mean velocities at the position of the addition, and so their general use is not possible. To the best of the authors’ knowledge, only Sandoval-Robles et al. [39] proposed a correlation for mass transfer from solid spheres in a liquid phase cross flow with high turbulence intensities. In that case, the turbulence was generated by a porous plate. They used the mean and rms velocity fluctuations at the location of the solid to calculate Re and ReT (330 < Red < 1720 and 0.04 < Tu < 0.3, in the range of Red and Tu of the current study). The effect of mean velocity was taken into account by Re1=2 d , and the turbulence effect was accounted for by Re0:066 . T This suggests that for a turbulent bulk flow, ReT with the same exponent could be incorporated into the Churchill and Bernstein correlation [7] for heat and mass transfer from cylinders in cross flow, written here for mass transfer: 1=2
Sh ¼ 0:3 þ F
0:62Red Sc1=3 ½1 þ ð0:4=ScÞ2=3
"
1=4
5=8 #4=5 Red 1þ 282; 000
ð12Þ
The predictions of this correlation when combined with natural convection were in satisfactory agreement with experiments without gas injection [2]. Therefore, Eq. (12) can be adapted as follows: 1=2
Sh ¼ 0:3 þ F
0:62Red Sc1=3 ½1 þ ð0:4=ScÞ2=3
1=4
"
5=8 #4=5 Red 1þ Re0:066 T 282; 000
velocities tested, which implies that natural convection should be taken into account to evaluate the mass transfer from the dissolving specimens at these velocities. The agreement between the predictions of the combined correlation and the experimental values is much better than with the Mazumdar correlation, yet the experimental data are underpredicted at lower velocities and overpredicted at higher velocities. The underprediction may be due to the use of the initial diameter of the cylindrical Si specimens for calculating the mass transfer coefficients. The initial diameter underpredicts the mass transfer coefficient as the dissolution progresses. As shown in [2], as the radius of the specimen decreases, the rate of decrease of cylinder radius increases, indicating a higher mass transfer coefficient. On the other hand, the correlation may overpredict the experimental data because the effect of gas agitation in Eq. (13) is represented only by the magnitude of velocity fluctuations, irrespective of the bulk velocity. There are studies that suggest that when predicting transport rate enhancement in a turbulent flow, the velocity fluctuations must be normalized by the liquid bulk velocity or its square root. This reduction of dissolution enhancement at higher bulk velocities is observed experimentally [3], but is not incorporated in the correlation. 6. Summary and conclusions CFD simulations were run to predict 3D velocity fields within a revolving liquid metal tank, with and without top lance gas injection. In the rotating tank, time and spatial averaging of the velocity field reveals that the introduction of gas increases both mean velocity and turbulence intensity. However, increasing the gas flow rate by a factor of two mainly affects the rms velocity fluctuations (which increase by 20%), while the mean velocity increases much less. The predicted velocity fields were used to assess and develop correlations for the dissolution of solid additions in gas-agitated liquid metals. The two main contributions of this work are the following: In a non-rotating tank, with only gas injection, a correlation for mass transfer from vertical solid cylinders: ShF ¼ 0:73ðRed Þ0:25 ðReT Þ0:32 ðScÞ0:33 [14,19] (for 637 < Red < 680, 166 < ReT < 290, and 29 < Sc < 42), combined with a correlation for natural convection [2], yields predictions of experimental mass transfer coefficients within 5%. In a rotating tank, a correlation was developed for vertical cylinders in cross flow that takes into account the turbulence in the 5=8 4=5 1=2 0:62Re Sc1=3 Red d bulk flow: ShF ¼ 0:3 þ 1 þ 282;000 Re0:066 T 2=3 1=4 ½1þð0:4=ScÞ
(for 609 < Red < 2511, 259 < ReT < 379, and 29 < Sc < 42). Combining this equation with a correlation for natural convection yields mass transfer coefficients within 11% of experimental values.
ð13Þ Table 5 presents a comparison of mass transfer coefficients obtained from Eq. (13) (based on the initial diameter of the cylinder, with and without adding the natural convection effect) with experimental values. The values of Grm,l and Red0 for the experiments of this study are presented in Table 6. An asymptotic solution for buoyancy effects on a boundary layer along an infinite cylinder in a cross flow [40] suggests that the strength of the two convection regimes on heat or mass transfer is determined by 4
the magnitude of ½ðd=lÞ ðGrm;l =Re2d Þ. Notice that the values of 4
½ðd=lÞ ðGrm;l =Re2d Þ are larger than 0.1 and even unity at the
Acknowledgment The authors wish to acknowledge the generosity of the Natural Sciences and Engineering Research Council of Canada (NSERC) for the support provided for this project through a Strategic Grant. References [1] D. Doutre, Personal Communication, Novelis Global Technology Center, Kingston, Ontario, 2013. [2] M. Seyed Ahmadi, S.A. Argyropoulos, M. Bussmann, D. Doutre, Comparative study of solid silicon dissolution in molten aluminum under different flow
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass transfer correlations for dissolution of cylindrical additions in liquid metals with gas agitation Mehran Seyed Ahmadi a, Markus Bussmann b,⇑, Stavros A. Argyropoulos a a b
Department of Materials Science and Engineering, University of Toronto, ON, Canada Department of Mechanical and Industrial Engineering, University of Toronto, ON, Canada
a r t i c l e
i n f o
Article history: Received 25 June 2015 Received in revised form 27 November 2015 Accepted 15 February 2016
a b s t r a c t The results of an experimental study of the effect of gas agitation on mass transfer from cylindrical Si specimens to molten Al were used to deduce mean mass transfer coefficients in a two-phase flow. Then the three-dimensional gas-agitated liquid was simulated using the commercial software FLOW-3D, to obtain predictions of bubble distribution and estimates of the velocity field within the opaque liquid metal, to be used to predict mass transfer rates. An existing mass transfer correlation for a gas-agitated liquid without an external bulk flow was used to fit the experimental data within 5%. With a bulk velocity, a correlation for mass transfer from a vertical cylinder in cross flow was modified to incorporate the effect of gas injection, and it predicted the measured mass transfer coefficients within 11%. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction A major challenge to the production of Al–Si alloys is the slow rate of dissolution of solid Si in molten Al, which leads to significant energy and material losses [1]. To accelerate the alloying process, either higher bulk liquid velocities (e.g. by means of rotating impellers) or agitation of the liquid (e.g. by means of gas injection) can be used. We recently reported the rate of dissolution of vertical cylindrical Si additions in molten Al with and without imposing a bulk flow, and with and without gas injection [2,3]. In the current study, computational fluid dynamics (CFD) was used to predict the three-dimensional velocity field in the gas-agitated tank, and these velocities are used to extend the results of the dissolution experiments by developing correlations for mass transfer. Dimensionless correlations for convective heat and mass transfer from a solid to a liquid are based on boundary layer theory. When fluid motion is generated by an external source (as opposed to natural convection caused by thermal and density gradients), the classical mass transfer correlations (e.g. the Ranz–Marshall correlation [4,5]) are functions of Re and Sc (or Pr) [6]:
Sh ðor NuÞ ¼ fðSc ðor PrÞ; ReÞ
ð1Þ
These types of correlations are only applicable when the turbulence intensity in the external bulk flow is low. Following this approach, in a recent study [2] we fitted experimental data of the dissolution of Si in Al in a rotating tank without agitation with a dimensionless correlation for cylinders in cross flow [7], as a ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.02.043 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
function of only Re and Sc. However, ladle metallurgical processes involve the agitation of liquid at large scales that produce flow with high levels of free stream turbulence [8]. It is customary to take the effect of such turbulence into account by introducing another ~ lc =m, into the correlations, based on rms Reynolds number, ReT ¼ u velocity fluctuations, a characteristic length lc, and the kinematic viscosity m. This is similar to what is done for heat and mass transfer phenomena in flows with grid-generated turbulence [9–11]:
Sh ðor NuÞ ¼ fðSc ðor PrÞ; Re; ReT Þ
ð2Þ
~ represents the root In the definition of ReT, the velocity scale u mean square (rms) velocity fluctuations [12,13]. In a cylindrical coordinate system, this is calculated as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 02 u02 r þ uh þ uz ~¼ u 3
ð3Þ
Instead of ReT, the turbulence intensity can also be used [14,15]:
~ u Tu ¼ V
ð4Þ
represents a mean value of the local fluid motion near a where V solid, which would be calculated by taking into account the influence of multi-dimensional flow as in a gas-agitated system [13–15]:
h; z; tÞ ¼ Vðr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r þ u 2h þ u 2z u
ð5Þ
Note that this definition of Tu is different than that for gridgenerated turbulence, where the velocity component in the flow direction is dominant. In that case, the velocity fluctuations are
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Nomenclature English symbols C concentration, kg m3 d diameter of a cylindrical Si specimen, m d0 initial diameter of a cylindrical Si specimen, m db bubble diameter, m 3 Grm,l mass transfer Grashof number ½¼ gð@ q=@C Si ÞT l bulk 2 ðC saturation C Þ= q m Si Si H liquid height inside the tank, m Kl liquid kinetic energy per unit mass of liquid, m2 s2 km mean combined convection mass transfer coefficient, m s1 N k mean natural convection mass transfer coefficient for a m vertical cylinder, m s1 F k mean mass transfer coefficient for a vertical cylinder in m the gas-agitated bath with or without bulk velocity, m s1 l length of cylindrical Si specimen, m characteristic length, m lc M total number of data used in time-averaging Nu Nusselt number NV number of cells in a volume representing a Si cylinder atmospheric pressure, Pa P0 Pr Prandtl number [=m/a] Q gas flow rate at the lance exit, m3 s1 QS gas flow rate at standard temperature and pressure (273.15 K and 100 kPa), m3 s1 r, h, z Cylindrical coordinates Ro outer diameter of the bath, m Re Reynolds number ReT Reynolds number based on rms velocity fluctuations Sc Schmidt number [=m/D] ShF mean Sherwood number in the gas-agitated bath with or without bulk velocity normalized based on a unidirectional bulk flow velocity. On the other hand, the movement of discrete bubbles in a gas-agitated liquid generates both a three-dimensional mean flow and agitation (corresponding to mean and fluctuating velocities, respectively) even in the absence of an external bulk flow. Defining the turbulence intensity using the more general Eq. (4) allows for quantifying the degree of turbulence in a bubble-induced flow. Various transport correlations have been developed for gasagitated systems. These correlations require flow velocities, which can be determined by solving the Navier–Stokes equations in conjunction with a turbulence model. CFD is especially useful in high temperature liquid metals where it is very difficult to measure velocity. Several studies have utilized air–water systems to correlate heat or mass transfer from solid additions to the mean and fluctuating velocities of a recirculating flow generated by a bubble plume. In a study of the melting rate of horizontal ice rods [16], the heat transfer coefficients were correlated to the computed local mean velocity and turbulence intensity [17] based on the mean velocity at the centerline of the ladle, and Mazumdar et al. [14] proposed a correlation for the dissolution of solid benzoic acid cylinders in a gas-agitated water ladle taking into account computed local mean and fluctuating velocities. In high temperature liquid metals, Szekely et al. [15] developed an axisymmetric numerical model for a cylindrical plume to compute a velocity profile in an Ar-stirred steel ladle. Applying the bulk velocity values to the modified Lavender and Pei correlation [18] for cylinders, and assuming a constant turbulence intensity of 0.3, they demonstrated satisfactory agreement with the measured mass transfer from graphite cylinders. Following a similar approach,
T Tu t Ub
temperature, K turbulence intensity time, s bulk velocity of liquid normal to the centerline of a vertical cylinder or an injection lance, m s1 ~ u rms velocity fluctuations, m s1 ur, uh, uz velocity components in cylindrical coordinates, m s1 h , u z time-averaged mean velocity components in cylindrical r , u u coordinates, m s1 0 0 0 ur , uh , uz velocity fluctuation components in cylindrical coordinates, m s1 V mean velocity of liquid, m s1 v volume, m3 Greek symbols a thermal diffusivity, m2 s1 u general quantity h relative angular position with respect to Si specimen, deg hc contact angle, deg m kinematic viscosity, m2 s1 q density, kg m3 Subscripts Al molten Al rms root mean square Si solid Si Other || hi
absolute value spatial averaging
but this time prescribing a conical-shaped plume, Mazumdar et al. [19] predicted the local mean velocities and rms velocity fluctuations in a 25 kg gas-agitated melt, and showed that their previously-derived correlation for a cold system predicted the experimental data of Wright [20] reasonably well. In the above CFD studies [14–16,19], simple quasi-single phase models were developed based on a specified axisymmetric plume, with the buoyancy force acting on the surrounding liquid. For flows with larger discrete bubbles, such quasi-single phase models cannot be used to predict the flow in the plume [21]. This is because quasi-single phase models are based on a variable density formulation which does not incorporate bubbles or bubble dynamics, but rather are simply based on a core of reduced density fluid that takes into account buoyancy. In addition, none of the above studies superimposed gas injection onto a constant bulk flow. A recent study [2,3] examined the effects of bulk velocity and of gas agitation on the dissolution of vertical cylindrical Si additions in molten Al. In the current paper, corresponding CFD results of the fluid dynamics of those experiments are presented. The predicted velocities are used to develop correlations for mass transfer from solids in a gas-agitated liquid, that are compared to the experimental measurements of the dissolution rates of the Si cylinders in molten Al.
2. Summary of the experiments Hundreds of experiments were conducted to measure the dissolution rate of vertical cylindrical Si specimens in molten Al as a function of bulk flow velocity and the rate of gas injection [2,3].
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A brief summary of the experiments, and of the flow behavior of the bubbles, is presented here.
2.1. Dissolution experiments The experiments were carried out in a Revolving Liquid Metal Tank (RLMT) of radius R0 = 17.5 cm and liquid height H = 15 cm, containing 35 kg of molten Al. For each experiment, a Si cylinder (d0 = 1.87 cm, l = 9–10 cm) was immersed vertically at a radial distance of 13.5 cm from the center of the tank. To agitate the molten Al, N2 was delivered to the top of a straight Ti lance (0.437 cm ID and 0.625 cm OD) immersed10 cm below the molten Al surface. For each experiment, once the Al reached the desired temperature and rotational speed, the lance was inserted into the melt, the gas was turned on, and N2 bubbles began to form. Then a Si cylinder was immersed 8 cm into the liquid for 3 min, at the same radial distance as the injection lance. Each partially-dissolved Si specimen was weighed to determine the amount dissolved. Three experiments were conducted at each experimental condition; the dissolved fraction data were used to calculate a mean mass transfer coefficient.
2.2. Bubble behavior Molten Al is opaque and so does not allow for the direct observation of the formation and trajectory of bubbles. Instead, we measured the pressure fluctuations in the gas delivery line, and video recorded the free surface of the molten Al, to determine the characteristics of the individual bubbles as they rose from the bottom of the lance. The pressure transducer data clearly indicated the bubble frequency. The average single bubble size was estimated by knowing the gas flow rate and assuming spherical bubbles. The estimated bubble diameters for four gas flow rates (0.25, 0.50, 0.75, 1.00 SLPM) without tank rotation were in the range of 0.9–1.5 cm.
(a)
769
When rotating the RLMT at 7 cm s1, the bubbles were 3–5% smaller than in the non-rotating tank. Information on the trajectory of the bubbles was obtained by observing the melt free surface (Fig. 1). Regardless of tank rotation rate, the bubbles rose along the lance and broke the free surface adjacent to the lance, as was expected due to the large contact angle between molten Al and the boron nitride coating that was applied to the lance (hc = 160° [22,23]). 3. Numerical methodology The commercial software FLOW-3D (version 10) was used to predict the bubble dynamics within the molten Al, and estimate the induced liquid velocity field in the tank. This section presents the model setup, the initial and boundary conditions, and the assumptions and simplifications. 3.1. Simulation setup The Navier–Stokes equations were solved with the standard two equation k–e model to account for turbulence. Bubbles and the liquid free surface were represented using the Volume of Fluid (VoF) method [24], and surface tension was accounted for. The simulations were transient as a steady state model would not predict discrete bubble formation and rise, and consequently bubbleinduced velocity fluctuations. The interaction of bubbles with the lance was taken into account by specifying a molten Al/boron nitride contact angle of 160°. The simulations were run using the ‘‘one-fluid” model that solves for flow within the liquid phase, but not within the air above the free surface. 3.1.1. Domain The computational domain included the lance and the tank, with enough room above the liquid to allow for free surface fluctuations (Fig. 2). The Si cylinder was not modeled. Rather, the results of the simulations were used to assess mean and fluctuating
(b)
Fig. 1. Bubble arrival at the liquid Al surface (a) bulk velocity = 0, (b) bulk velocity = 3.5 cm s1 (gas flow rate = 0.50 SLPM, the coated lance diameter is approximately 0.725 cm).
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M. Seyed Ahmadi et al. / International Journal of Heat and Mass Transfer 97 (2016) 767–778 Table 1 Effect of the number of cells on the predicted velocities (bulk velocity = 0, gas flow rate = 0.5 SLPM). Number of mesh cells 6
1.7 10 3 106 6.1 106
(cm s1) hVi
~ i (cm s1) hu
1.53 1.78 1.86
0.34 0.46 0.54
Table 2 List of simulations.
Fig. 2. Cylindrical domain for the FLOW-3D simulations of the N2–Al system. All dimensions are in cm.
Bulk velocity of liquid (cm s1)
Standard gas flow rate (SLPM)
Actual gas flow rate (LPM)
Non-rotating tank
0
0.50 0.75 1.00
1.76 2.57 3.24
Rotating tank
1.4 3.5
0.50 0
1.76 0
0.50 1.00 0.50 1.00
1.76 3.24 1.76 3.24
velocities at the various positions up and downstream of the lance where Si cylinders were positioned in the experiments. A similar approach was adopted by other studies (e.g. [14–16,19,25,26]) in which flow was predicted about an immersed melting or dissolving specimen. 3.1.2. Mesh The computational domain was discretized with a 3D cylindrical mesh, with the radial, angular and vertical coordinates denoted by r, h, and z, respectively. The mesh was selectively refined around the lance to capture bubble formation. As FLOW-3D solves the advection term explicitly, time steps are proportional to the control volume size, and so very small cells adversely affect runtime. To eliminate the very small cells at the center of the cylindrical domain (r = 0), a core of 4 cm radius (corresponding to 5.2% of the liquid volume) at the center of the domain was not meshed, and the mesh size near the core was gradually increased to avoid very small cells. 3.1.3. Initial conditions For the non-rotating tank, the liquid velocity field was initialized to zero. For the rotating tank, the velocity was initialized to a solid body rotation. 3.1.4. Boundary conditions The boundary conditions were specified as per the experimental conditions. To take into account the gas volume expansion due to heat exchange between injected gas and liquid metal, the standard gas volume flow rates (SLPM at 273.15 K and 100 kPa) were corrected to account for the melt temperature and static pressure at the lance exit. No-slip boundary conditions were imposed at the bottom of the tank (z = 0) and on the tank walls. When simulations were run of a rotating tank, appropriate velocities were specified at those walls. The pressure at the melt surface was specified as atmospheric pressure, P0. Although the flow was modeled in the entire tank (0° 6 h 6 360°), FLOW-3D nevertheless requires that one define a pair of boundaries in the angular direction (i.e. a pair of rz-planes). For the non-rotating tank, a periodic boundary condition was specified at these rz-planes, so that the liquid flow that left one plane re-entered the other. Both boundaries were located 180° from the lance to minimize the influence on bubble generation and on the flow in the vicinity of the lance. For the rotating tank, a solid body rotation velocity profile was imposed as an inflow condition at the rz-plane 90° upstream of the
7.0
Table 3 Comparison of experimental and CFD predictions of the equivalent bubble diameter at the lance exit (bulk velocity = 0). QS (SLPM)
0.50 1.00
db (cm) Experimental (estimated from pressure transducer data)
Predicted (counting bubbles at the lance exit)
1.16 ± 0.01 1.47 ± 0.02
1.26 ± 0.15 1.59 ± 0.20
lance, and an outflow boundary condition was specified at the same position 270° downstream. This is a small simplification of the experiments, by assuming that the solid body velocity profile is recovered far downstream of the lance. 3.1.5. Stopping criterion Full domain simulations were run with a mesh of about 3 106 cells and time steps of 106–105 s. Each simulation ran for one to two months on an 8 core Intel i7-4770 machine running at 3.40 GHz, with 32 GB of RAM. The simulations were run until a quasi-steady state was reached, evaluated by monitoring a sliding l. 1 s average of the liquid kinetic energy per unit mass of liquid, K Simulations were stopped when jdK l =dtj was reduced by a factor of 5 from its initial value. This occurred at about 9 s of simulation time for the non-rotating tank; for consistency, the simulations in the rotating tank were run for the same duration. The rate of l =dtj at 9 s was small, implying that a more stringent decrease of jdK stopping criterion would require far longer run times. 3.2. Temporal- and spatial-averaged velocities A pragmatic approach was adopted to estimate mean velocities and turbulence intensities at various positions in the domain, corresponding to where the Si cylinders were immersed into molten Al. The results were averaged over time to capture the effect of many bubbles, while taking into account the transient response of the system. Results were averaged spatially over a volume equivalent to that of a Si cylinder. As the simulations were transient and started from the onset of bubble injection, the mean velocities were time-dependent. Timeaveraged mean velocity components and the rms values of fluctuating components in each direction were calculated as follows:
M. Seyed Ahmadi et al. / International Journal of Heat and Mass Transfer 97 (2016) 767–778
(a)
771
(b)
Fig. 3. Predicted instantaneous bubble distributions at gas flow rates of (a) 0.50 SLPM and (b) 1.00 SLPM (bulk velocity = 0).
r ðr;h;z;tÞ ¼ u
M M M 1X 1X 1X h ðr;h;z;tÞ ¼ z ðr;h;z;tÞ ¼ uir ; u uih ; u ui M i¼1 M i¼1 M i¼1 z
3.3. Mesh and time step independence
ð6Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XM ðui u r Þ2 r ; i¼1 M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffi XM ðui u z Þ2 z u02 z ¼ i¼1 M qffiffiffiffiffiffi u02 r ¼
qffiffiffiffiffiffi u02 h ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XM ðui u h Þ2 h ; i¼1 M ð7Þ
where M was the number of liquid velocity temporal data. Substituting Eq. (6) into Eq. (5) and Eq. (7) into Eq. (3) yielded mean and rms velocity fluctuations u ~. velocity V Time-averaging must be performed over a duration that is sufficiently longer than that of any velocity fluctuation [27]. Therefore, a sliding temporal average was calculated for a period of 1 s, during which about 30 bubbles were released, based on data at every 0.005 s (i.e. M = 200). Sampling the data every 0.05 s for one of the cases yielded very similar time-averaged results. In the experiments, the overall dissolution rate of each Si cylinder was measured, and so spatial averages of mean and rms velocity fluctuations were calculated within the volume that would encompass a Si specimen. Assuming an equal weight for each cell in the volume:
hui ¼
P
vu
Nv
ð8Þ
where u represents a time-averaged mean velocity or a rms value of the velocity fluctuations, and Nv is the number of cells (15,000–45,000) where the cylinder would be.
Treating fluid advection explicitly results in smaller time steps on a finer mesh and so ensures time step independency. To determine an acceptable mesh size, three meshes: 1.7 106, 3 106, and 6.1 106 cells were evaluated for the case of a non-rotating tank and a gas flow rate of 0.5 SLPM. The spatially-averaged mean velocity and rms velocity fluctuations at h = 30° from the lance were compared. The results are tabulated in Table 1. The selection of an acceptable mesh size is a balance between speed and accuracy. For example, increasing the number of cells from 3 106 to hu ~ i, and the predicted mass transfer coeffi6.1 106 changed hVi, cients (see Section 5) by 4%, 19%, and 7% respectively, but increased the solution time by a factor of 2.2. A mesh of 3 106 cells was used for all of the simulations presented in this paper. 4. Results and discussion Table 2 lists the nine simulations that were run. Simulation results of the bubble dynamics, and of the time and spatial averages of mean velocities and velocity fluctuations are presented at various gas flow rates, with and without tank rotation. We only consider a Si specimen immersed 30° downstream of the lance, as this was the closest distance between the specimen and the lance, and because the change in the dissolution rate enhancement due to gas agitation as a function of angular position was in the range of uncertainty of the experimental data [3]. 4.1. Non-rotating tank Simulations in a non-rotating tank were run at gas flow rates of 0.50, 0.75 and 1.00 SLPM. Given the uncertainties: the contact
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Fig. 4. Spatially-averaged (a) absolute mean velocity and (b) rms velocity fluctuations in the r, h, and z directions (bulk velocity = 0, gas flow rate = 0.50 SLPM).
Fig. 5. Comparison of spatially-averaged (a) absolute mean velocity and (b) rms velocity fluctuations at various gas flow rates (bulk velocity = 0).
angle, the temperature-adjusted gas flow rate at the lance exit, and the uncertainties of transport properties (density, viscosity, and surface tension) at high temperatures, the FLOW-3D results are surprisingly good. The predicted bubble size was 8% higher than the values estimated from the experiments, as indicated in Table 3. The simulation at a gas flow rate of 0.50 SLPM was also run with the contact angle set to 150° and 170°. As expected, average bubble size increased and decreased, respectively. At 170° and a gas flow rate of 0.50 SLPM, the average size was 1.22 ± 0.18 cm, just slightly larger than the measured size. Fig. 3 shows sample results for gas flow rates of 0.50 and 1.00 SLPM, that illustrate bubbles rising along the lance, and that can be compared to Fig. 1. At 0.50 SLPM, bubbles rise with a velocity of 0.63 ± 0.05 m s1, higher than that of unhindered bubbles of a similar size [28,29]. The attachment of ascending bubbles to a poorly wetted lance has been previously reported [30,31] in water
model studies, and Watanabe and Iguchi [31] showed that such bubbles rise more quickly than those away from a lance. Fig. 4 illustrates the spatial average of the absolute mean velocity (Eq. (5)) and the rms velocity fluctuations (Eq. (3)) in the r, h, and z directions for a Si cylinder 30° downstream of the lance, at a gas flow rate of 0.50 SLPM. Fig. 4(a) illustrates the multidimensionality of the mean flow: the velocity components are of the same order of magnitude, although the mean tangential velocity component is larger than the vertical and radial components. This result confirms that assuming an axisymmetric flow relative to the lance, and neglecting flow perpendicular to the plane that passes through the solid addition and the lance [14,15,26], would underestimate the mean velocity. Fig. 4(b) shows that the velocity fluctuations are appreciable in all directions, although the tangential and vertical components are greater than the radial one, which implies a deviation from isotropic turbulence, as per the results of [32].
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Fig. 6. Spatially-averaged absolute mean velocity in the r, h, and z directions (bulk velocity = 3.5 cm s1, gas flow rate = 0 SLPM).
Fig. 7. Spatially-averaged rms velocity fluctuations and turbulence intensity (bulk velocity = 3.5 cm s1, gas flow rate = 0 SLPM).
Fig. 5 shows the mean velocities and rms velocity fluctuations at various gas flow rates in a non-rotating tank. Increasing the gas flow rate from 0.50 to 1.00 SLPM increases the velocity fluctu indicating that a ~ i much more than the mean velocity hVi, ations hu higher gas flow rate induces more liquid agitation than a stronger mean flow. This observation is in agreement with the experimental findings of [33].
4.2. Rotating tank Tank rotation generates a liquid bulk velocity. As discussed in Section 2, this causes a small decrease in the bubble size, but the bubbles continue to rise along the lance, as confirmed by Fig. 1. FLOW-3D predicts these behaviors. By increasing the bulk velocity from 0 to 3.5 cm s1, at gas flow rates of 0.50 SLPM and 1.00 SLPM, the average bubble size decreases slightly from 1.26 cm to 1.24 cm,
Fig. 8. Spatially-averaged (a) absolute mean velocity and (b) rms velocity fluctuations in the r, h, and z directions (bulk velocity = 3.5 cm s1, gas flow rate = 0.50 SLPM).
and from 1.49 cm to 1.48 cm, respectively. These reductions are negligible compared to the standard deviations associated with the equivalent bubble diameter; a similar conclusion was drawn from the bubble frequency measurements. Before examining mean and fluctuating velocities due to gas agitation in the rotating tank, it is instructive to consider the influence of the lance on the flow field without gas injection, because the simplifying assumption of solid body rotation is invalid near the lance. Such a simulation offers an estimate of the velocity fluctuations due to pure shear-induced turbulence. Note that the rotating tank simulations were initiated from a solid body rotation velocity field similar to the experiments; this choice of initial condition did not affect the quasi-steady state values predicted by the simulations. Fig. 6 shows absolute mean velocities at h = 30° averaged over the cylinder volume. The tangential velocity decreases from the
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Fig. 9. Comparison of spatially-averaged absolute mean velocities in the (a) r, (b) h, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s1).
Fig. 10. Comparison of spatially-averaged rms velocity fluctuations in the (a) r, (b) h, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s1).
Table 4 Predicted mass transfer coefficients for the non-rotating tank, and a comparison with experimental values. Gas flow rate (SLPM)
Re based on initial specimen diameter
Mean transfer coefficient 105, m s1
hVi
~i hu
Red0 based on hVi
ReT based ~i on hu
Experimental
Predicted via Eq. (9)
Modified using combined convection correlation, Eq. (11)
correlationExperimental Error (%) = Combined convection 100 Experimental
– 1.78 1.90
– 0.46 0.81
– 637 680
– 166 290
5.17 6.04 6.48
– 4.40 5.35
– 5.75 6.26
– 5 3
Table 5 Predicted mass transfer coefficients for the rotating tank, and a comparison with experimental values. Gas flow rate (SLPM)
0.50 0.50 1.00 0.50 1.00
Ub, Bulk velocity (cm s1)
1.4 3.5 3.5 7.0 7.0
Bubbleinduced velocities, cm s1 (average at steady state)
Re based on initial specimen diameter
Mean mass transfer coefficient 105, m s1
hVi
~i hu
Red0 based on hVi
ReT based ~i on hu
Experimental
Predicted via Eq. (13)
Modified using combined convection correlation, Eq. (11)
correlationExperimental Error ð%Þ ¼ Combined convection 100 Experimental
1.70 3.43 3.46 6.91 7.02
0.72 0.76 0.91 0.96 1.06
609 1227 1236 2473 2511
259 272 326 342 379
6.77 8.36 9.08 11.06 11.52
5.28 7.58 7.70 11.00 11.23
6.21 7.96 8.06 11.19 11.35
8 5 11 +2 1
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0 0.50 1.00
Bubble-induced velocities, cm s1 (average at steady state)
775
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Table 6 Dimensionless parameters involved in combined natural and forced convection from a Si vertical cylinder in liquid Al cross flow without gas agitation. l (cm) d0 (cm) T (°C) Sc Grm,l 8
1.87
700
Ub (cm s1) Red0
42 5.5 108 1.4 3.5 7.0
4
ðd0 =lÞ ðGrm;l =Re2d0 Þ
510 6.5 1260 1.0 2530 0.2
5. Correlations The mean mass transfer coefficients obtained from the experimental results are compared to predictions from a correlation for mass transfer from a cylindrical addition in a gas-agitated liquid (i.e. non-rotating tank), and in the cross flow of a turbulent liquid (i.e. rotating tank), where the predicted velocities of Section 4 are used to evaluate the various dimensionless groups. 5.1. Gas agitation without bulk velocity
initial 3.5 cm s1 to a steady 2.5 cm s1 due to the drag force exerted by the lance. There are also small radial and vertical velocities that are an order of magnitude less than the tangential velocity due to the flow disturbances induced by the lance. Fig. 7 shows the rms velocity fluctuations and turbulence intensity. The spatially-averaged turbulence intensity hTui is about 11% at steady state. This value is in the expected range for single-phase flows [32,34]. For the remainder of the discussion of the effect of tank rotation, only the data for a bulk velocity of 3.5 cm s1 are presented. Data at other rotation rates yielded conclusions similar to those presented here. Fig. 8(a) shows the mean velocity components at a gas flow rate of 0.50 SLPM. The results demonstrate radial and vertical velocities of the same order of magnitude, albeit much smaller than the tangential velocity in the bulk flow direction. This is a crucial distinction from previous studies on heat and mass transfer from vertical cylinders in cross flow with grid-generated turbulence [9,10,18,35,36]. These experiments were conducted in wind tunnels where the mean velocity was one-dimensional and constant along the test section; correlations derived from such experiments would be difficult to apply to a system with gas agitation as the source of velocity fluctuations. Fig. 8(b) indicates that the rms velocity fluctuations are significant in all directions, and that fluctuations in the tangential direction are slightly higher than in the vertical and radial directions because of the interaction of the bulk and bubble-induced flows. These results further confirm the deviation from isotropic gridgenerated turbulence. Figs. 9 and 10 illustrate a comparison of mean velocities and rms velocity fluctuations, respectively, at three gas flow rates. As the gas flow rate increases from 0.50 SLPM to 1.00 SLPM, larger bubbles form at the lance exit (Table 3), displacing a larger volume of liquid that increases the mean flow and induces a higher degree of agitation (and thus increases the turbulence intensity). As demonstrated in Fig. 9(b), the mean tangential velocity decreases downstream of the lance due to the effect of larger bubble wakes, but the absolute mean velocities increase in the radial (Fig. 9(a)) and in particular the vertical directions (Fig. 9(c)). Increasing the gas flow rate from 0.50 to 1.00 SLPM, the mean tangential velocity (at quasi-steady state) decreases by about 10%, while the absolute mean radial and vertical velocities increase 30% and 50%, respectively. This implies that for transport phenomena taking place in bubble-induced flows, neglecting the three-dimensionality of the velocity field may lead to inconsistencies when correlating the operating parameters (e.g. gas flow rate) to the mass transfer rate. Nevertheless, the net effect of the aforementioned changes is a negligible change to the mean velocity hVi. Finally, Fig. 10 shows that increasing the gas flow rate from 0.50 SLPM to 1.00 SLPM increases the turbulence intensity in all directions. The components of rms velocity fluctuations increase in the radial, tangential and vertical directions by approximately 30%, 20%, and 20%, respectively, and so the overall rms velocity fluctua~ i also increase by 20%. This demonstrates that an increase tions hu in mass transfer rates at higher gas flow rates is primarily associated with the enhanced velocity fluctuations.
For the non-rotating tank, the results of the dissolution experiments with gas injection [3] are compared to the correlation of Mazumdar et al. [14,19] for mass transfer from vertical benzoic acid cylinders in a gas-stirred water tank [14], and vertical steel cylinders in a gas-stirred carbon saturated iron bath [19], both inside and outside the two-phase region:
ShF ¼ 0:73ðRed Þ0:25 ðReT Þ0:32 ðScÞ0:33
ð9Þ
Red and ReT are based on local mean and rms velocity fluctuations, respectively. The simulation predictions of mean velocity and rms velocity fluctuations, and the corresponding values of Red and ReT, are tabulated in Table 4. Eq. (9) underestimates the mean mass transfer coefficients by 27% and 17% at gas flow rates of 0.50 and 1.00 SLPM, respectively. Why? Eq. (9) assumes that mass transfer due to gas agitation is dominant compared to natural convection mass transfer with no gas agitation. This may be valid in [14,19] because both and u ~ at the location of the dissolving specimens are much V < 32 cm s1, higher (benzoic acid cylinders in water: 5 cm s1 < V ~ < 7.5 cm s1 [14]; steel cylinders in carbon saturated 1.1 cm s1 < u < 43 cm s1, 1.3 cm s1 < u ~ < 5.0 cm s1 [19]) iron: 18 cm s1 < V than the values predicted in the current study (Fig. 5 and Table 4). As a result, natural convection effects were less important in those studies. To take into account natural convection, the Mazumdar correlation values were combined with a measured natural convection (no gas injection, no bulk velocity) mass transfer coefficient (5.17 105 m s1), or with a correlation for natural convection mass transfer proposed in [2]. Using the general combining law similar to that developed in [2] yields a much better agreement with the experimental data. A general combining law for transfer processes which vary uniformly between two known limiting solutions is given by [37]:
Z ¼ ðX p þ Y p Þ
1=p
ð10Þ
where X and Y are the limiting solutions for asymptotically small and large values of an independent variable such as agitation or external velocity. For the current problem the mass transfer coeffiN . With cient with no rotation and no gas agitation is given by Y ¼ k m a high degree of gas agitation and/or an external bulk velocity, F represents the asymptotic solution. As in [2], we set p = 4. X¼k m Combining the mass transfer coefficient for gas-agitated flow (Eq. (9)), and a correlation for natural convection, Eq. (10) becomes:
h i1=4 m ¼ ðk N Þ4 þ ðk F Þ4 k m m
ð11Þ
Table 4 shows that Eq. (11) predicts the experimental results within 5% and 3% at gas flow rates of 0.50 and 1.00 SLPM, respectively. Finally, note that the uncertainty of the velocity field predictions associated with mesh resolution can explain some of the discrepancy between the correlation predictions and the experimental results. Using a finer mesh (6.1 M compared to 3 M cells, as presented in Table 1) for the case of a gas flow rate of 0.5 SLPM with no tank rotation, velocities increased slightly, and
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those increases would increase the mass transfer coefficient by 7%, yielding even better agreement between correlation predictions and experimental results. Unfortunately we were unable to run all simulations at this resolution in a reasonable timeframe. 5.2. Gas agitation with bulk velocity The Mazumdar correlation was also applied to the rotating tank (bulk velocity = 3.5 cm s1, for which the gas-agitated velocity field was predicted in Section 4). However, even when considering natural convection, the correlation underpredicts the experimental results by 19% and 23% at gas flow rates of 0.50 SLPM and 1.00 SLPM, respectively. This is because the Mazumdar fluid flow configuration with respect to the cylindrical specimen [14,19] is different than in the rotating tank. For this reason, an effort was made to compare the experimental results to a more appropriate turbulent flow configuration. There are many correlations that predict heat and mass transfer from spheres and cylinders in cross flow, but usually in singlephase air flow characterized by low turbulence intensities (e.g. [18]). In studies involving two-phase flow (i.e. air/water), Iguchi et al. [38] and Szekely et al. [16] proposed correlations for heat transfer based on the melting rate of ice spheres and horizontal cylinders in a bottom-stirred cylindrical water tank, respectively. Iguchi et al. [38] defined Tu based on axial mean and rms velocity fluctuations along the plume centerline; Szekely et al. [16] normalized the local velocity fluctuations based on gas entry velocity. As a result, these correlations are not based on local mean velocities at the position of the addition, and so their general use is not possible. To the best of the authors’ knowledge, only Sandoval-Robles et al. [39] proposed a correlation for mass transfer from solid spheres in a liquid phase cross flow with high turbulence intensities. In that case, the turbulence was generated by a porous plate. They used the mean and rms velocity fluctuations at the location of the solid to calculate Re and ReT (330 < Red < 1720 and 0.04 < Tu < 0.3, in the range of Red and Tu of the current study). The effect of mean velocity was taken into account by Re1=2 d , and the turbulence effect was accounted for by Re0:066 . T This suggests that for a turbulent bulk flow, ReT with the same exponent could be incorporated into the Churchill and Bernstein correlation [7] for heat and mass transfer from cylinders in cross flow, written here for mass transfer: 1=2
Sh ¼ 0:3 þ F
0:62Red Sc1=3 ½1 þ ð0:4=ScÞ2=3
"
1=4
5=8 #4=5 Red 1þ 282; 000
ð12Þ
The predictions of this correlation when combined with natural convection were in satisfactory agreement with experiments without gas injection [2]. Therefore, Eq. (12) can be adapted as follows: 1=2
Sh ¼ 0:3 þ F
0:62Red Sc1=3 ½1 þ ð0:4=ScÞ2=3
1=4
"
5=8 #4=5 Red 1þ Re0:066 T 282; 000
velocities tested, which implies that natural convection should be taken into account to evaluate the mass transfer from the dissolving specimens at these velocities. The agreement between the predictions of the combined correlation and the experimental values is much better than with the Mazumdar correlation, yet the experimental data are underpredicted at lower velocities and overpredicted at higher velocities. The underprediction may be due to the use of the initial diameter of the cylindrical Si specimens for calculating the mass transfer coefficients. The initial diameter underpredicts the mass transfer coefficient as the dissolution progresses. As shown in [2], as the radius of the specimen decreases, the rate of decrease of cylinder radius increases, indicating a higher mass transfer coefficient. On the other hand, the correlation may overpredict the experimental data because the effect of gas agitation in Eq. (13) is represented only by the magnitude of velocity fluctuations, irrespective of the bulk velocity. There are studies that suggest that when predicting transport rate enhancement in a turbulent flow, the velocity fluctuations must be normalized by the liquid bulk velocity or its square root. This reduction of dissolution enhancement at higher bulk velocities is observed experimentally [3], but is not incorporated in the correlation. 6. Summary and conclusions CFD simulations were run to predict 3D velocity fields within a revolving liquid metal tank, with and without top lance gas injection. In the rotating tank, time and spatial averaging of the velocity field reveals that the introduction of gas increases both mean velocity and turbulence intensity. However, increasing the gas flow rate by a factor of two mainly affects the rms velocity fluctuations (which increase by 20%), while the mean velocity increases much less. The predicted velocity fields were used to assess and develop correlations for the dissolution of solid additions in gas-agitated liquid metals. The two main contributions of this work are the following: In a non-rotating tank, with only gas injection, a correlation for mass transfer from vertical solid cylinders: ShF ¼ 0:73ðRed Þ0:25 ðReT Þ0:32 ðScÞ0:33 [14,19] (for 637 < Red < 680, 166 < ReT < 290, and 29 < Sc < 42), combined with a correlation for natural convection [2], yields predictions of experimental mass transfer coefficients within 5%. In a rotating tank, a correlation was developed for vertical cylinders in cross flow that takes into account the turbulence in the 5=8 4=5 1=2 0:62Re Sc1=3 Red d bulk flow: ShF ¼ 0:3 þ 1 þ 282;000 Re0:066 T 2=3 1=4 ½1þð0:4=ScÞ
(for 609 < Red < 2511, 259 < ReT < 379, and 29 < Sc < 42). Combining this equation with a correlation for natural convection yields mass transfer coefficients within 11% of experimental values.
ð13Þ Table 5 presents a comparison of mass transfer coefficients obtained from Eq. (13) (based on the initial diameter of the cylinder, with and without adding the natural convection effect) with experimental values. The values of Grm,l and Red0 for the experiments of this study are presented in Table 6. An asymptotic solution for buoyancy effects on a boundary layer along an infinite cylinder in a cross flow [40] suggests that the strength of the two convection regimes on heat or mass transfer is determined by 4
the magnitude of ½ðd=lÞ ðGrm;l =Re2d Þ. Notice that the values of 4
½ðd=lÞ ðGrm;l =Re2d Þ are larger than 0.1 and even unity at the
Acknowledgment The authors wish to acknowledge the generosity of the Natural Sciences and Engineering Research Council of Canada (NSERC) for the support provided for this project through a Strategic Grant. References [1] D. Doutre, Personal Communication, Novelis Global Technology Center, Kingston, Ontario, 2013. [2] M. Seyed Ahmadi, S.A. Argyropoulos, M. Bussmann, D. Doutre, Comparative study of solid silicon dissolution in molten aluminum under different flow
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