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The Effect of Thermal Diffusion on

Decarburization Kinetics

Paul Wu, Yindong Yang, Mansoor Barati and Alex McLean

Version Post-print/Accepted Manuscript

Citation (published version)

Wu, P., Yang, Y., Barati, M. et al. Metall and Materi Trans B (2014) 45: 1974. https://doi.org/10.1007/s11663-014-0211-z

Publisher’s statement This is a post-peer-review, pre-copyedit version of an article published in Metallurgical and Materials Transactions B. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11663-014-0211-z

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1

The Effect of Thermal Diffusion on Decarburization Kinetics

Paul Wu, Yindong Yang, Mansoor Barati and Alex McLean

Department of Materials Science and Engineering, University of Toronto, 184 College Street,

Toronto, Ontario M5S 3E4 Canada

ABSTRACT: The influence of thermal diffusion on the kinetics of decarburization of Fe-Cr-C

droplets with CO2-Ar gas mixtures was investigated. With incorporation of the effect of thermal

diffusion, a new correlation has been proposed to express the decarburization kinetics of

levitated droplets for flows in the range of Reynolds number between 2 to 100. A thermal

diffusion factor of 0.228 was evaluated for CO2-Ar gas mixtures at 1873K (1600°C).

KEY WORDS: Decarburization, mass transfer; carbon dioxide; thermal diffusion; levitated

droplets; stainless steel refining.

INTRODUCTION

In a recent paper by the current authors[1], results were reported for the decarburization of Fe-Cr-

C levitated droplets with CO2-Ar gas mixtures, using the equipment shown in Figure 1. It was

found that mass transfer correlations such as the Ranz-Marshall[2] and Steinberger-Treybal[3]

equations, while widely employed, do not describe the data obtained from the droplet

experiments. Based on analysis of the rate data a new correlation was proposed that accounts for

2

contributions from both natural and forced convection and is appropriate for mass transfer

involving levitated droplets:

𝑆ℎ = 2 − 0.317(𝐺𝑟′𝑆𝑐)14 + 3𝑅𝑒0.415𝑆𝑐

13 [1]

where 𝑆ℎ, 𝐺𝑟′, 𝑆𝑐 and 𝑅𝑒 correspond to the Sherwood, Mean Grashof, Schmidt and Reynolds

numbers respectively.

While it is true that natural convection may assist with mass transport from the droplet to the

surroundings in a shrinking-sphere experiment as in the case of the Steinberger-Treybal

experiments, it may hinder the process, as indicated by the negative sign in Eq. [1], if the

concentration gradient and the temperature gradient are in opposite directions.

(Insert Fig. 1)

The presence of a steep temperature gradient near the gas-metal interface, an inherent feature of

the levitation technique, gives rise to variation in gas properties and flow patterns across the

boundary layer and as a result, mass flux due to thermal diffusion becomes significant. The

objective of the present discussion is to provide an alternative approach to rationalize the

reported discrepancy between the existing models and the experimental measurements by

considering the implications of the effect of thermal diffusion on reaction kinetics in the presence

of large temperature differences between the two phases.

3

THE EFFECTS OF THERMAL DIFFUSION ON MASS TRANSFER

The work by Sain and Belton[4] suggests that interfacial reactions are faster than both liquid

transport and gas transport. In the high carbon regime, the transfer of carbon is faster than the

flux of oxidant species arriving at the gas-metal interface. Therefore at high carbon content the

experimental reaction rate is controlled by gas diffusion. The reaction geometry associated with

the levitation technique, consists of gas mixtures flowing over a liquid sphere of surface area A

with weight W. The decarburization rate can be related to the flux of arriving gaseous oxidant,

𝐽𝐶𝑂2, using the expression:

𝑑(𝑤𝑡 𝑝𝑐𝑡 𝐶)

𝑑𝑡=

−1200𝐴

𝑊𝐽𝐶𝑂2

=−1200𝐴

𝑊

𝑆ℎ𝐷𝐴𝐵𝑃

𝑑𝑝𝑅𝑇𝑓(𝑋𝐶𝑂2

𝑏 − 𝑋𝐶𝑂2

𝑖 ) [2]

where 𝐷𝐴𝐵 is the mutual diffusion coefficient, P is the total pressure, 𝑑𝑝 is the diameter of metal

droplet, R is the gas constant and 𝑇𝑓 is the film temperature, XCO2

b is the mole fraction of CO2 in

the bulk gas at the edge of the boundary layer and 𝑋𝐶𝑂2

𝑖 is the mole fraction of CO2 at the surface

of the liquid sphere. In the gas phase controlled regime, the mole fraction of CO2 at the gas-metal

interface is negligibly low.

In a levitated droplet experiment, thermal equilibrium is reached when the heat dissipation

through radiation and convection equals the heat gain through induction by the applied

electromagnetic field. The steep temperature gradient between the droplet surface and the

incoming reaction gas may cause mass flux by thermal diffusion. Therefore, the net flux of a

4

species has contributions from molecular diffusion, thermal diffusion and convection, which may

be written as:

𝐽𝐶𝑂2=

𝑆ℎ𝐷𝐴𝐵𝑃

𝑑𝑝𝑅𝑇𝑓𝑋𝐶𝑂2

𝑏 + 𝐽𝑇 [3]

where 𝐽𝑇 represents the contribution to the net flux due to thermal diffusion effects. The process

of thermal diffusion depends greatly on the gas composition. This dependence is reflected in the

expression for the thermal diffusion term defined[5] as:

𝐽𝑇 = 𝑘𝑇∇𝑙𝑛𝑇 = (𝐷𝐴𝐵𝐶𝛼𝑋𝐶𝑂2

𝑏 𝑋𝐴𝑟𝑏

𝑇

𝑑𝑇

𝑑𝑟)

𝑖

[4]

where 𝑘𝑇 is the thermal diffusion ratio, C is the total molar concentration in the gas phase, 𝛼 is

the binary thermal diffusion factor, which is independent of the mole fractions of the binary gas

mixture and r is the radial distance from the droplet surface. Combining Eqs. [2], [3], and [4]:

𝑊

−1200𝐴

𝑑(𝑤𝑡 𝑝𝑐𝑡 𝐶)

𝑑𝑡=

𝑆ℎ𝐷𝐴𝐵𝑃

𝑑𝑝𝑅𝑇𝑓𝑋𝐶𝑂2

𝑏 + (𝐷𝐴𝐵𝐶𝛼𝑋𝐶𝑂2

𝑏 𝑋𝐴𝑟𝑏

𝑇

𝑑𝑇

𝑑𝑟)

𝑖

[5]

In order to utilize Eq. [5] to assess the decarburization data described in the current work, 𝑆ℎ and

dT/dr must both be known. Following the approach of El-Kaddah and Szekely[6] the dT/dr term

can be expressed as:

5

𝑑𝑇

𝑑𝑟= −

𝑁𝑢

𝑑𝑝

(𝑇𝑖 − 𝑇𝑏) −𝜎𝑆𝐵𝜀

𝑘(𝑇𝑖

4 − 𝑇𝑏4) [6]

where 𝑁𝑢 is the Nusselt number, 𝜎𝑆𝐵 is the Stefan-Boltzmann constant, 𝜀 is the emissivity of the

metal drop and k is the thermal conductivity of the gas phase. The Nusselt number can be

expressed in terms of natural and forced convection using the Grashof number for heat transfer,

GrH, the Prandtl number Pr and the Reynolds Number:

𝑁𝑢 = 0.78(𝐺𝑟𝐻𝑃𝑟)14 + 0.8𝑅𝑒

12𝑃𝑟

13 [7]

EVALUATION OF THE THERMAL DIFFUSION FACTOR

The thermal diffusion factor is used to express the ratio of contributions from the mutual

diffusion coefficient (DAB) and the thermal diffusion coefficient (DT), in non-equilibrium steady

state:

𝛼𝑖 = 𝑇𝐷𝑇

𝐷𝐴𝐵= −

𝑇

𝑥𝑖(1 − 𝑥𝑖)

∇𝑥𝑖

∇𝑇 [8]

𝛼𝑖 is often used to estimate DT, which is otherwise difficult to measure. Although DT shares

some similarity with DAB, DT also depends on composition whereas DAB does not.

Over the past five decades, there have been a number of studies on the measurement of the

thermal diffusion factor, 𝛼 for Ar-CO2 gas mixtures. Perhaps due to experimental limitations,

6

none of these studies have reported values of 𝛼 for temperatures greater than 700K (427°C).

Shashkov et al.[7] reported values for several temperatures averaged at 326K (53°C) from

measurements made in a two-bulb apparatus. Recent work by McCourt[8] proposed an expression

for temperature dependence of 𝛼 using experimental data by Taylor and Pickett[9], up to 550K

(277°C). Sunderland et al.[5] suggested the levitation melting technique could be an appropriate

method for measurements of 𝛼 at temperatures of about 1900K (1627°C).

Experimental data[1] obtained using CO2-Ar gas mixtures at very low flow rates (0.1L min-1),

with Reynolds numbers in the range 0.3~0.6, were considered in order to determine the

contribution from thermal diffusion. In this regime where the Archimedes number (Ar) >>1, the

predominant mode of mass transfer is natural convection and the contribution of forced

convection is minimal. The thermal diffusion factor 𝛼 can be experimentally deduced by varying

the gas composition.

(Insert Table I)

From Table I, it can be seen that the experimentally determined Sherwood numbers are

consistently lower than those calculated, Sh(S-T), from the Steinberger-Treybal equation using

only the first two terms which account for molecular diffusion and natural convection

respectively:

𝑆ℎ = 2 + 0.569(𝐺𝑟′𝑆𝑐)0.25 + 0.347𝑅𝑒0.62𝑆𝑐0.31 [9]

7

More importantly, the observed Sherwood numbers are even smaller than the minimum expected

value of 2 from molecular diffusion. The effect of thermal diffusion may account at least in part

for this difference. The observed decarburization rate may be used to determine the thermal

diffusion factor by rearranging Eq. [5]:

𝑊

−1200𝐴

𝑑(𝑤𝑡 𝑝𝑐𝑡 𝐶)

𝑑𝑡−

𝑆ℎ𝐷𝐴𝐵𝑃

𝑑𝑝𝑅𝑇𝑓𝑋𝐶𝑂2

𝑏 = (𝐷𝐴𝐵𝐶𝛼𝑋𝐶𝑂2

𝑋𝐴𝑟

𝑇

𝑑𝑇

𝑑𝑟)

𝑖 [10]

When evaluated at the film temperature, molecular diffusion and natural convection terms in the

Steinberger-Treybal equation can be used to estimate the Sherwood number; allowing 𝛼 to be

found when dT/dr is calculated using Eqs. [6] and [7]. The results are plotted against

composition in Figure 2 and it is evident that the mole fraction of CO2 has little influence on the

value of the thermal diffusion factor. At 1873K (1600°C), 𝛼 is found to have an average value of

0.228.

(Insert Fig. 2)

DEVELOPMENT OF A NEW MASS TRANSPORT CORRELATION

As mentioned in the previous work[1], the inability to utilize Ranz-Marshall or Steinberger-

Treybal correlations to describe the mass transport behavior associated with the levitated droplets

is largely influenced by the difference in experimental conditions. For the conditions prevailing

in the development of Ranz-Marshall and Steinberger-Treybal correlations, where the

temperature difference between the reacting phases was about 200 degrees, the contribution from

8

thermal diffusion would be insignificant compared to natural and forced convection. In contrast

however, analysis of the rate data from levitated droplet studies, suggests thermal diffusion

effects should not be ignored in systems characterized by large temperature gradients, such as the

present system with ∆T≈1550 degrees. Thus when assessing transport phenomena within a

particular system, it is important to take into consideration the specific conditions in order to

apply the most appropriate correlation model.

In the case of the rate data[1] taken from higher gas flow rate regimes (0.385, 1, 3, and 12.2L min-

1), conditions where the Archimedes number Ar <<1, the predominant mode of mass transfer is

forced convection. The Sherwood number takes the general form, based on the Frössling

equation[10]:

𝑆ℎ = 2 + 𝛽𝑅𝑒𝑚𝑆𝑐𝑛 [11]

The Schmidt number Sc takes account of the fluid transfer characteristics and remains largely

unchanged given the gas compositions and a fixed effective temperature in the boundary layer.

Therefore, the Schmidt number is not a variable in this investigation. For values of Sc less than

250, the coefficient n is commonly taken as 1/3. In the present study, the contribution of the

Reynolds number to the Sherwood number pertains to the relative velocity between the droplet

and the gas stream. In the case of stationary levitated droplets this can be directly related to the

gas flow rate. Combining Eqs. [3] and [11]:

(𝐽𝐶𝑂2− 𝐽𝑇)

𝑑𝑝𝑅𝑇𝐸

𝐷𝐴𝐵𝑃𝐶𝑂2

= 2 + 𝛽𝑅𝑒𝑚𝑆𝑐13 [12]

9

It should be noted that effective film temperature 𝑇𝐸 is used in place of the common film

temperature 𝑇𝑓 for reasons detailed in the previous work[1]. With the Reynolds number and

Schmidt number determined from experimental conditions on the right hand side of Eq. [12]

and the Sherwood number, taking into account the effect of thermal diffusion, on the left hand

side of the equation, coefficients β and m can be readily determined. Figure 3 shows a

logarithmic plot of the relationship represented by Eq. [12] where the linear slope corresponds

to the constant m and the y-intercept value can be used to calculate the constant β. Upon

evaluation, Eq. [11] becomes:

𝑆ℎ = 2 + 4.32𝑅𝑒0.363𝑆𝑐13 [13]

(Insert Fig. 3)

As shown in Figure 4, good agreement was found between data available in the literature[11,12] for

the rate of CO2 decarburization of molten steel droplets and Eq. [13] from the present

investigation. As described in more detail in the previous work[1], these two earlier studies were

also conducted using the electromagnetic levitation method. The work by Lee and Rao[11]

involved decarburization of Fe-C droplets with CO-CO2 gas mixtures while Sun and Pehlke[12]

carried out decarburization with N2-CO2 gas mixtures, in each case with 𝑃𝐶𝑂2 ranging from 0.1 to

0.2 atm and flow rates between 200 and 2000 𝑚𝐿 𝑚𝑖𝑛−1. It is evident from Figure 4 that the data

from past studies with Reynolds numbers as low as 2, are well characterized by the correlation

derived during the present investigation.

10

(Insert Fig. 4)

For comparison purposes the Sherwood number predicted by the Ranz-Marshall equation using

an average Schmidt number of 0.7, is included in Figure 4. It is evident that there is a significant

difference between the relationship represented by the Ranz-Marshall equation and the published

experimental data which are in accord with the proposed model from the present work. A

noticeable difference between the two correlations is the weaker dependence of Sh on Re in the

presence of a large ∆T, i.e. metal–gas systems presented here. This may be expected noting that

the steeper temperature gradient between the two phases promotes turbulence in the boundary

layer, resulting in an enhanced effective Re in the vicinity of the droplet, while the apparent Re

calculated based on the average gas velocity is constant. It is suggested that the temperature

gradient, ΔT, between the gas stream and the liquid droplet has a strong influence on models

describing and predicting mass transfer numbers such as Sh. It is therefore speculated that the

experimentally derived coefficients in Eq. [11], β and m, are dependent on the temperature

gradient. To validate this suggestion, additional droplet experiments using systems with different

temperature gradients would be of value.

SUMMARY AND CONCLUSIONS

11

Electromagnetic levitation is a useful technique for measuring thermal diffusion effects at

elevated temperatures. In this study, a forced convection model that incorporates the effects of

thermal diffusion has been developed to describe the decarburization kinetics of levitated iron-

alloy droplets for Reynolds numbers between 2 and 100. The thermal diffusion factor for CO2-Ar

gas mixtures at 1873K (1600°C) was found to have a value of 0.228. Both Sherwood number

expressions, 𝑆ℎ = 2 + 4.32𝑅𝑒0.363𝑆𝑐1

3 and 𝑆ℎ = 2 − 0.317(𝐺𝑟′𝑆𝑐)1

4 + 3𝑅𝑒0.415𝑆𝑐1

3 [1], were

found to be in excellent agreement with rate data from other decarburization studies using a

similar experimental approach involving levitated droplets. However, there was a significant

difference between the relationships represented by the proposed models and those

corresponding to the Ranz-Marshall and the Steinberger-Treybal equations. It is therefore

recommended that such mass transfer correlations should only be used after taking into account

the specific experimental conditions. It is speculated that the experimentally determined

coefficients in the different mass transfer relationships have temperature gradient dependence.

Quantification of this aspect requires further investigation.

ACKNOWLEDGEMENTS

Appreciation is expressed to the Natural Sciences and Engineering Research Council of Canada

for provision of funding in support of this project.

APPENDIX

Assuming an ideal gas phase, the density is calculated from the relationship:

12

𝜌𝑔 =𝑀𝑃

𝑅𝑇

Data for viscosity, inter-diffusivity, heat capacity and thermal conductivity are available from

AspenONE Engineering Suite – Heat Exchanger Design.

Tf=1079K (806°C); DAB=1.520 cm2/s

CO2 mole

fraction

𝜌𝑔, Density at

Tf

𝜇𝑔, Viscosity

at Tf

𝐶𝑝, Heat

capacity at Tf

𝑘, Thermal conductivity

of gas at Tf

2 4.52 x10-4 5.58 x10-4 0.532 4.60 x10-4

6 4.54 x10-4 5.61 x10-4 0.560 4.64 x10-4

10 4.56 x10-4 5.52 x10-4 0.600 4.61 x10-4

15 4.58 x10-4 5.60 x10-4 0.636 4.62 x10-4

20 4.60 x10-4 5.37 x10-4 0.679 4.64 x10-4

25 4.63 x10-4 5.29 x10-4 0.717 4.70 x10-4

30 4.65 x10-4 5.22 x10-4 0.755 4.82 x10-4

TE=896K (623°C); DAB=1.247 cm2/s

CO2 mole

fraction

𝜌𝑔, Density at

TE

𝜇𝑔, Viscosity

at TE

𝐶𝑝, Heat

capacity at TE

𝑘, Thermal conductivity

of gas at TE

10 5.47 x10-4 4.90 x10-4 0.594 3.93 x10-4

30 5.58 x10-4 4.62 x10-4 0.738 4.11 x10-4

TABLE OF SYMBOLS

13

𝐴 Area of droplet surface(𝑐𝑚2)

Ar Archimedes number (=𝐺𝑟′

𝑅𝑒2)

𝐶 Total molar concentration in gas phase (𝑚𝑜𝑙 𝑐𝑚−3)

𝐶𝑝 Heat capacity of gas ( 𝐽 𝑔−1𝐾−1)

𝐷𝐴𝐵 Mutual diffusion coefficient in gas phase(𝑐𝑚2𝑠−1)

𝐷𝑇 Thermal diffusion coefficient (𝑐𝑚2𝑠−1)

𝑑𝑝 Diameter of the droplet (𝑐𝑚)

𝐺𝑟′ Mean Grashof number (= 𝐺𝑟𝑚 + 𝐺𝑟𝐻(𝑆𝑐/𝑃𝑟)0.5)

𝐺𝑟𝑚 Grashof number for mass transfer (=𝜌𝑔𝑔𝑑𝑝

3(𝐶𝑖−𝐶𝑏)

𝜇𝑔2 )

𝐺𝑟𝐻 Grashof number for heat transfer (=𝑔𝑑𝑝

3(𝑇𝑖−𝑇𝑏)

𝑇𝑓𝜇𝑔2 )

ℎ Heat transfer coefficient (𝐽 𝑐𝑚−2𝑠−1𝐾−1)

𝐽𝑖 Flux of diffusion species i (𝑚𝑜𝑙 𝑐𝑚−2𝑠−1)

𝑘 Thermal conductivity of gas ( 𝐽 𝑐𝑚−1𝑠−1𝐾−1)

𝑘𝑇 Thermal diffusion ratio

m Coefficient in Eq. [11]

n Coefficient in Eq. [11]

𝑁𝑢 Nusselt number (=𝑑𝑝ℎ

𝑘)

𝑃 Total pressure (𝑎𝑡𝑚)

𝑃𝑟 Prandtl number (=𝜇𝑔𝐶𝑝

𝑘)

𝑅 Gas constant (𝑐𝑚3𝑎𝑡𝑚 𝑚𝑜𝑙−1𝐾−1)

r Radial distance from the droplet surface (cm)

14

REFERENCES

Re Reynolds number (=𝑑𝑝𝑣𝜌𝑔

𝜇𝑔)

Sc Schmidt number (=𝜇𝑔

𝜌𝑔𝐷𝐴𝐵)

Sh Sherwood number (=𝑑𝑝𝑘𝑔

𝐷𝐴𝐵)

t Time (s)

𝑇𝐸 Effective temperature of gases (K) (= 0.83𝑇𝑖+𝑇𝑏

2)

𝑇𝑓 Film temperature of gases (K) (=𝑇𝑖+𝑇𝑏

2)

𝑇𝑏 Bulk gas temperature (𝐾)

𝑇𝑖 Gas-metal interface temperature (𝐾)

𝑣 Relative velocity between gas and droplet (𝑐𝑚 𝑠−1)

𝑊 Mass of the droplet (𝑔)

𝑋𝐶𝑂2

𝑏 Mole fraction of CO2 in the bulk gas

𝑋𝐶𝑂2

𝑖 Mole fraction of CO2 on the gas-metal interface

𝑥𝑖 Mole fraction of component i

α Binary thermal diffusion factor

𝛼𝑖 Thermal diffusion factor of component i

β Coefficient in Eq. [11]

𝜎𝑆𝐵 Stefan-Boltzmann constant (= 5.67037 × 10−12𝐽 𝑐𝑚−2𝑠−1𝐾−4)

𝜀 Emissivity of metal

𝜇𝑔 Gas viscosity ( 𝑔 𝑐𝑚−1𝑠−1)

15

1. P. Wu, Y. Yang, M. Barati and A. McLean: Metall. Mater. Trans. B, 2014, DOI

10.1007/s11663-014-0126-8.

2. W.E. Ranz and W.R. Marshall: Chem. Eng. Prog., 1952, vol. 48, pp. 141-146.

3. R.L. Steinberger and R.E. Treybal: AIChE J., 1960, vol. 6 (2), pp. 227-232.

4. D.R. Sain and G.R. Belton: Metall. Trans. B, 1976, vol. 7B, pp. 235-244.

5. M. Sunderland, A.E. Hamielec, W.K. Lu and A. McLean: Metall. Trans. B, 1973, vol. 4B, pp.

575-583.

6. N. El-Kaddah and J. Szekely: Metall. Trans. B, 1983, vol. 14B, pp. 401-410.

7. A.G. Shashkov, A.F. Zolotukhina, T.N. Abramenko, B.P. Mathur and S.C. Saxena: J. Phys.

B: At. Mol. Phys., 1979, vol. 12 (21), pp. 3619-3630.

8. F.R.W. McCourt: Mol. Phys., 2003, vol. 101 (21), pp. 3223-3229.

9. W.L. Taylor and P.T. Pickett: Int. J. Thermophys., 1986, vol. 7 (4), pp. 837-849.

10. N. Frössling: Beitraege zur Geophysik, 1938, vol. 52, pp. 170-216.

11. H.G. Lee and Y.K. Rao: Metall. Trans. B, 1982, vol. 13B, pp. 403-409.

12. H. Sun and R.D. Pehlke: Metall. Mater. Trans. B, 1995, vol. 26B, pp. 335-344.

List of captions

Fig. 1− Schematic diagram of the electromagnetic levitation equipment.[1]

16

Fig. 2− Evaluation of α for the Ar-CO2 system at 1873K (1600°C).

Fig. 3− Linear regression of the data expressed in terms of Eq. [12].

Fig. 4− Comparison of correlation data derived from previous decarburization studies[11,12] with

Eq. [13] from the present work.

Table I. Experimental results[1] for decarburization using CO2-Ar at flow rate of 0.1L/min

17

Fig. 1− Schematic diagram of the electromagnetic levitation equipment.[1]

Fig. 2− Evaluation of α for the Ar-CO2 system at 1873K (1600°C).

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4

α

mole fraction CO2

18

Fig. 3− Linear regression of the data expressed in terms of Eq. [12].

Fig. 4− Comparison of correlation data derived from previous decarburization studies[11,12] with

Eq. [13] from the present work.

y = 0.363x + 0.636

R² = 0.9744

0

0.5

1

1.5

2

0 0.5 1 1.5 2

log (

Sh

-2)

-1/3

log S

c

log Re

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2

log (

Sh

-2)

-1/3

log S

c

log Re

Lee and Rao [11]

Sun and Pehlke [12]

Ranz-Marshall

Equation 13

19

Table I. Experimental results[1] for decarburization using CO2-Ar at flow rate of 0.1L/min

[Cr]

(wt pct)

Composition

(pct CO2)

Droplet

mass (g)

-d[C]/dt

(wt pct/sec)

Sh

(Observed)

Sh

(S-T)

α

10 2 0.72 0.00112 1.08 4.00 0.205

10 6 0.72 0.00312 1.00 3.96 0.216

10 10 0.56 0.00605 0.988 3.77 0.230

10 15 0.75 0.00663 0.877 3.84 0.237

10 15 0.64 0.00912 1.09 3.77 0.227

10 20 0.56 0.0117 0.957 3.59 0.247

10 20 0.73 0.0116 1.13 3.69 0.222

10 20 0.68 0.0118 1.10 3.66 0.227

17 6 0.49 0.00396 0.988 3.78 0.231

17 10 0.49 0.00759 1.13 3.71 0.222

17 15 0.51 0.0106 1.09 3.67 0.234

17 15 0.48 0.0106 1.04 3.64 0.240

17 30 0.72 0.0184 1.19 3.38 0.225

17 30 0.58 0.0244 1.36 3.29 0.220

20 6 0.71 0.00291 0.930 3.95 0.222

20 15 0.55 0.00943 1.02 3.70 0.237

20 20 0.76 0.00946 0.947 3.71 0.235

20 25 0.76 0.0145 1.16 3.58 0.225

20 25 0.73 0.0144 1.12 3.56 0.230

20 25 0.79 0.0141 1.15 3.60 0.223