A new method of eddy diffusivity calculation for droplets of a venturi scrubber.

Rahimi, A. ; Fathikalajahi, J. ; Taheri, M. 等

An analytical model using eddy diffusivity is applied for predicting droplet concentration distribution and liquid film formation in a Venturi scrubber. By comparing experimental data of film formation reported in literature (Viswanathan et al., 1984) and the results obtained from this model, a semi-empirical correlation for liquid droplets eddy diffusivity is obtained. The validity of this correlation is confirmed by obtaining good agreement between theoretical and experimental data of droplet concentration distribution and film formation in a Venturi scrubber (Viswanathan, 1998; Viswanathan et al., 1984).

On a applique un modele analytique utilisant la diffusivite turbulente pour la prediction de la distribution de concentration des gouttelettes et de la formation de film liquide dans un laveur a Venturi. En comparant les donnees experimentales de formation de film presentees dans la litterature scientifique (Viswanathan et al., 1984) et les resultats obtenus a partir de ce modele, on obtient une correlation semi-empirique pour la diffusivite turbulente de gouttelettes de liquide. La validite de cette correlation est confirmee en obtenant un bon accord entre les donnees theoriques et experimentales sur la distribution de concentration des gouttelettes et la formation de film dans un laveur a venturi (Viswanathan, 1998; Viswanathan et al., 1984).

Keywords: venturi scrubber, eddy diffusivity, liquid film

The simplicity of Venturi scrubbers has led to their emergence as one of the most widely used pieces of equipment in removing particulate and gaseous pollutants from a gas stream. The polluted gas stream is accelerated to the desired maximum throat velocity in the converging entrance section of the Venturi scrubbers. Scrubbing liquid is atomized at the throat inlet by the high velocity gas stream to produce a spectrum of drop sizes. Particulate matter that travels essentially at the speed of the carrier gas is removed by impaction and interception mechanisms with the slower moving liquid drops. The collection efficiency of particulate and gaseous pollutant is dependent upon the liquid to gas ratio, liquid jet penetration, droplets size distribution, droplet concentration distribution, and gas velocity.

The efficiency of the particulate and gaseous pollutant collection was calculated by Taheri and Hains (1969), Taheri and Sheih (1975), Fathikalajahi et al. (1995, 1996), Talaie et al. (1997) and Viswanathan (1997), and some improvements were made in the mathematical modelling of the Venturi scrubbers. In the above works the effect of liquid film formation on Venturi walls was not considered.

The overall pressure drop is also significant for a successful design of the Venturi scrubbers. Momentum gained by drops through the accelerating zone and wall losses define the magnitude of pressure drop. The continuous deposition and entrainment of drops along the Venturi scrubber length also contributes to the overall pressure drop of the system.

Visual observations have indicated the existence of an annular mist two-phase, two-component flow having a thin layer of liquid on the walls and a high-velocity gas stream carrying the droplets in the core (Taheri and Hains, 1969; Viswanathan et al., 1983).

Several attempts have been made in recent years to predict pressure drop (Boll, 1973; Azzopardi and Govan 1984; Viswanathan et al., 1985; Viswanathan, 1998). Before 1980 the principal limitations of models were the assumption of an existence of only a homogeneous core with drops and the absence of film flow on the wall. Since then, new generations of pressure drop models have been proposed. Azzopardi and Govan (1984) redefined the overall momentum balance around the scrubber to include the momentum losses due to the acceleration of droplets entrained from the film flow and the interfacial drag between the fast moving core and the slower moving liquid film flow. Viswanathan et al. (1985) analyzed the deviation between the experimental values and the earlier model predictions and attributed them to realistic multiphase losses and developed an annular flow pressure model. They assumed that the fraction of injected liquid flowing as a film flow is constant. Recently Viswanathan (1998) examined the liquid film characteristics and analyzed its effect on the pressure drop over the range of Venturi scrubber operations encountered in industrial applications. He modified an annular flow model to include the variation in liquid film thickness using empirical correlation.

It is clear that the capability of the theoretical models in predicting the film flow rate will be a good improvement in prediction of the removal efficiency and the pressure drop in the Venturi scrubbers. Amount of liquid film is a complex function of gas velocity, liquid to gas ratio, liquid jet penetration and geometrical configuration of Venturi scrubber, and eddy diffusivity of droplets.

In this study the rate of droplets transfer from the bulk of gas stream to the walls of a Venturi scrubber will be treated mathematically in a manner that will yield a relation for predicting film flow rate. By matching experimental data of film formation and theoretical values, one can determine eddy diffusivity of droplets. Also a dimensionless correlation for eddy diffusivity of droplets is obtained as a function of operating variables.

THEORY

Consider a Venturi scrubber in which a jet of water is injected perpendicular to gas stream (Figure 1). The droplets formed by atomization will be accelerated or decelerated by drag force of the gas flow until their velocities approach that of the gas stream. The droplets are usually dispersed non-uniformly over cross section of the Venturi scrubber. In addition to some initial momentum droplet suspended in a turbulent stream receives impulses from various directions. The impulses are not entirely random, since the turbulent shear stress depends upon the degree of correlation between transverse and longitudinal velocity fluctuations. However, the overall effect is assumed to be governed by a diffusion law. In other words, a diffusion law can predict the net rate of droplet transfer to the wall.

[FIGURE 1 OMITTED]

Our model is based on a two-dimensional dispersion of droplets by convection and eddy diffusivity. Because of symmetry only one half of the flow cross-section in the y direction needs to be considered. Since there was a small separation distance between liquid injection orifices along the z direction, a negligible variation in the drop concentration was assumed in this direction. Viswanathan et al. (1984) has shown this matter experimentally. These simplifications lead to cross-sectional area equal to one half the distance between adjacent nozzles.

By taking a mass balance on a volume element having a length [delta]x, a height [delta]y and width w, the following mass conservation equation is readily obtained:

[partial derivative] ([V.sub.dx][C.sub.d])/[partial derivative]x = [[alpha].sub.d] [[partial derivative].sup.2][C.sub.d]/[partial derivative][y.sup.2] (1)

In Equation (1), the droplets are convected in x direction, while they are dispersed in y direction.

It is assumed that the velocity of the droplets is substantially uniform over the cross section and equal to the average gas velocity. Although for particle removal efficiency calculation this assumption is not valid, for the calculation of the droplet concentration distribution and liquid film flow rate is justified. This is true because for regions down stream close to the water injection point, the velocity of water droplets is nearly equal to gas velocity. This matter can be shown easily by solving equation of motion of liquid droplets. For those cases where the only important external force is the drag force, the equation of motion for a single spherical droplet can be expressed as follows (Boll, 1973):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [V.sub.dx] is the axial velocity of droplets. In this equation the modified drag coefficient can be calculated by using the following equation (Talaie et al., 1997):

[C.sub.DN] = 18.65 [Re.sup.0.16.sub.d] (3)

The droplet diameter in Equation (2) is obtained by correlation proposed by Nukiyama and Tanasawa (1938). Their equation, which gives the average drop diameter, is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where L is liquid flow rate, G is gas flow rate, Vr is relative velocity of droplets to gas, [[sigma].sub.1] and [[rho].sub.1] are the surface tension of liquid and liquid density respectively.

Equation (2) can be solved by using the Runge-Kutta 4th order method for obtaining droplet velocity. In Figure 2 the droplet velocity has been shown. It is found that the droplet velocity reaches the gas velocity rapidly in a very short length. Thus, for distances that we apply our model, we can assume same velocity for water droplets and gas stream and Equation (1) changes as follows:

[V.sub.G] [partial derivative][C.sub.d]/[partial derivative]x = [[alpha].sub.d] [[partial derivative].sup.2][C.sub.d]/ [partial derivative][y.sup.2] (5)

where [[alpha].sub.d], is the droplets eddy diffusivity. This parameter also takes into account the effect of initial momentum of droplets in y direction. This is true because the experimental data reported for liquid film formation on the wall is used for obtaining the corresponding value of eddy diffusivity.

[FIGURE 2 OMITTED]

When water is injected to a gas stream the initial distribution of droplets concentration is mostly affected by injection velocity and liquid jet penetration length. In this analysis we assume a line source of droplets at x=0 from atomizing point (liquid jet penetration length) to central axis of Venturi throat. Also observations and measurements have shown that, droplet concentration falls rapidly toward zero near the wall (Viswanathan et al., 1984). Thus the boundary conditions are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the above mentioned boundary conditions, Equation (5) is solved by using the Separation-of-Variables method. The following solution can be obtained for Equation (5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [L.sup.**] is the penetration length of liquid jet before atomization, which is calculated by using a modified form of correlation by Viswanathan et al. (1983) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The normalized concentration for drops can be obtained by using Equation (6) and uniform concentration definition as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The mass velocity of the liquid, which is still in suspension at any section along the length of Venturi scrubber, is given by the integral:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

By using this equation the fraction of injected liquid flowing on the wall as a film can be obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

By using Equation (11), [[alpha].sub.d] can be calculated for a specified gas velocity, jet penetration length and axial distance and a measured film flow rate. Equation (11) and experimental data presented by Viswanathan et al. (1984) were used for calculation of [[alpha].sub.d] for each operating condition. Then by using these values a correlation has been developed for eddy diffusivity as a function of operating parameters such as gas throat velocity, liquid jet penetration length and axial distance from injection point. Viswanathan et al. (1984) has measured liquid droplets core and wall film flow rate in a pilot plant scale Venturi scrubber of Pease-Anthony design for liquid to gas flow ratios varying between 0.00041 and 0.00195 [m.sup.3][H.sub.2]O/[m.sup.3]Air. The gas throat velocity were 45.4, 60.6 and 75.6 m/s. Liquid core and wall film flow rate have been measured at the distance of 0.13, 0.53 and 0.83 m from injection point. In this experimental work it is observed that for L/G less than 0.00041, droplet concentration distribution scheme is different essentially from other conditions. In these cases because of small penetration length of liquid jet a fraction of liquid can not be atomized and move as a film layer on Venturi throat walls. Since the main mechanism of film layer formation in this case is not based on a mass transfer phenomenon, we don't consider this data in this analysis.

By using these measurements [[alpha].sub.d] values are calculated by Equation (11) and the results are summarized in Table 1.

As pointed out before, the fraction of liquid that moves as a film layer is a complex function of operating variables. The effect of all of these variables in Equation (5) is represented by [[alpha].sub.d]. By using calculated data in Table 1 a dimensionless correlation was considered for eddy diffusivity of droplets as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

In Equation (12) [Re.sub.th] and [D.sub.H] are the throat Reynolds number and the hydraulic diameter of throat respectively.

By using a least square curve fit of data points (36 series) presented in Table 1, values of C1, a, b, c calculated as follows:

[C.sub.1] = 7.242 x [10.sup.-3] a = -0.497 b = -0.1 c = -0.957

By using these values Equation (12) can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

In Equation (13) liquid jet penetration length can be calculated by Equation (8).

Equation (13) provides an estimated value for eddy diffusivity of droplets for a wide range of operating conditions that usually are used in an industrial Venturi scrubber:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is clear that by using Equations (13), (11) and (9) respectively, liquid film percent as a fraction of total injected liquid and the normalized flux of droplets in each cross section of Venturi length can be calculated.

RESULTS AND DISCUSSION

For confirming the validity of the present model and Equation (13), the predicted values of liquid film percent and droplet concentration distribution have been compared with a different set of experimental data presented by Viswanathan et al. (1984) and Viswanathan et al. (1997). Also, predicted values of eddy diffusivity of droplets by Equation (10) have been compared with values obtained by other correlations too.

Liquid Film and Droplet Concentration Distribution Prediction

In Figure 3 the predicted values of liquid film as a fraction of total injected liquid have been compared with the experimental data. This figure provides a good validation for the model and Equation (13). It should be noted that experimental data for this comparison is for orifice diameter of 2.565 mm, which is different than the orifice diameter of 2.1 used for developing Equation (13).

[FIGURE 3 OMITTED]

Also, in Figures 4 and 5 the predicted values of normalized flux for droplets have been compared with the experimental data. The model results were obtained by using Equation (9) for the given operational conditions. The agreement in Figure 4 between experimental and model is very good. Also, in Figure 5 this agreement is very good except in only a one point. This deviation about this set of experiments also observed in our detailed numerical 3-D model (Rahimi, 2002; Rahimi et al., 2005). It seems that experimental errors cause this deviation.

[FIGURES 4-5 OMITTED]

Droplet Eddy Diffusivity Prediction

As pointed out before several attempts have been made to calculate particulate and gaseous pollutant collection efficiency in Venturi scrubbers (Taheri and Hains, 1969; Taheri and Sheih, 1975; Fathikalajahi et al., 1996; Talaie et al., 1997; Viswanathan, 1997). Most of these studies take into account the effect of non-uniform droplet concentration distribution on the particulate or gaseous pollutants removal efficiency. For this aim, mostly a two- or three-dimensional dispersion model is considered and it is assumed that the mixing of droplets across the scrubber is the result of the eddy diffusivity of droplets. For calculating eddy diffusivity of droplets, droplet mixing length is computed by using the equation of motion. The droplet motion is considered due to drag force of gas fluctuation velocity. For solving the equation of motion in most cases it is assumed that the initial velocity of droplets is zero. By using this idea it is assumed that the droplet trajectory is depended on gas eddy diffusivity only and initial momentum of droplets has no effect on droplets distribution. Based on this method some correlations presented by investigators that relate eddy diffusivity of droplets to gas eddy diffusivity. For instance the following equation is used frequently for calculating eddy diffusivity of droplets. (Viswanathan et al., 1984; Fathikalajahi et al., 1995; Taheri and Sheih, 1975).

[[alpha].sub.d]'[[alpha].sub.g] = [b.sup.2]/[[ohm].sup.2] + [b.ssup.2]

1/Pe = [[alpha].sub.g]/[V.sub.Gth][D.sub.H] (14)

where b is the stokes number and [omega] is the frequency of the fluctuation of the turbulent air in radians per second.

In previous studies of this subject various values are assumed for Pe number in order to obtain the values of gas and droplets eddy diffusivities. Viswanathan et al. (1984) proposed a constant value of 100 for Pe while Fathikalajahi et al. (1995) proposed a constant value of 130 for Pe. Recently Goncalves et al. (2000) proposed Pe=70. The different values of Pe number (different eddy diffusivity of droplets) comes from the assumption that the initial momentum of droplets is not considered in their theory. Also, the different correlations used for droplet diameter calculation causes large difference between adjusted Pe number. For most cases the droplets diameter predicted by two famous correlation, Nukiyama and Tanasawa (1938) and Boll (1973), are different largely. The Boll (1973) correlation often predicts droplets diameter 40-60 percent larger than Nukiyama and Tanasawa (1938) equation. For small liquid to gas ratio (L/G) the droplets diameter is smaller and the effect of initial momentum of droplets on droplet distribution concentration can be neglected. In this case the droplet mixing is mostly affected by gas fluctuation velocity and the predicted values of droplet eddy diffusivity by Equation (14) are in good agreement with experimental data. But for large diameter of droplets (high L/G) predicted values of eddy diffusivity by Equation (14) for Pe=100 or higher, have large differences with experimental data. Goncalves et al. (2000) proposed a smaller Pe number (Pe=70) in order to fit models results with experimental data.

This matter has been shown in Table 2. Eddy diffusivity of droplets predicted by Equations (13) and (14) with Pe=100 are compared in Table (2).

As one can see by increasing liquid to gas ratio (larger drops) the difference between eddy diffusivity of droplets predicted by Equations (13) and (14) becomes larger. Only for small droplet diameters this difference is negligible (row 4) and there is a good agreement between results of Equations (13) and (14). In these cases the stopping distance for small droplets is very small and droplets mixing is mostly occurred by gas fluctuation velocity.

However, for larger droplets the initial momentum of droplets in y direction may have had a significant effect and Equation (14) predicts a smaller value for droplets eddy diffusivity. This equation is based only on transfer by diffusion. However, in correlation developed in this study the effect of initial momentum of droplets in y direction is indirectly considered by using the actual experimental data.

The results in Figures 3, 4 and 5 and Table 2 all together confirm that the semi-empirical correlation presented in this study can be used to predict eddy diffusivity in Venturi type scrubber. The advantage of this semi-empirical correlation to previous methods is that one does not have to assume a constant Pe number or drop diameters in order to calculate eddy diffusivity of droplets. The operating parameters used actually take into account the effect of drop diameter.

CONCLUSION

Based on an analytical model and using experimental data reported in literature a semi-empirical correlation has been obtained for calculation of eddy diffusivity of droplets in a Venturi scrubber. By using this semi-empirical correlation and relations that have been obtained theoretically the amount of liquid film on Venturi scrubber walls and droplet concentration distribution can be calculated directly. The advantage of this semi-empirical correlation to previous methods is that liquid film percent and eddy diffusivity of liquid droplets can be calculated based on operating parameters without assuming a constant Peclet number.

ACKNOWLEDGMENT

This research was supported by the Shiraz University Research Council.

NOMENCLATURE

b (18[mu]/[[rho].sub.D][d.sup.2.sub.d])
[C.sub.DN] modified drag coefficient
[C.sub.d] drop concentration (number/[m.sup.3])
[C.sub.d0] drop initial concentration (number/[m.sup.3])
[D.sub.d] drop diameter (m)
[d.sub.0] orifice diameter (m)
[D.sub.H] hydraulic diameter (m)
[D.sub.d] drop diameter (micron)
f liquid film fraction (dimensionless)
G gas volumetric flow rate ([m.sup.3]/s)
H half of throat height (m)
[L.sub.G] mass or number flux of drops (number/s)
L volumetric flow rate of liquid ([m.sup.3]/s)
[L.sup.**] jet penetration length (m)
n integer number
N number of Nozzles
Pe Peclet number (dimensionless)
[V.sub.dx] axial velocity of drops (m/s)
[V.sub.Grth] gas velocity in throat (m/s)
w width of Venturi throat (m)
x length (m)
y height (m)
z width (m)

Greek Symbols

[[alpha].sub.d] drop eddy diffusivity ([m.sup.2]/s)
[[alpha].sub.G] gas eddy diffusivity ([m.sup.2]/s)
[[rho].sub.L] liquid density (kg/[m.sup.3])
[[rho].sub.G] gas density (kg/[m.sup.3])
[mu] gas viscosity (kg/m.s)

REFERENCES

Azzopardi, B. J. and A. H. Govan, "The Modeling of Venturi Scrubbers," Filtration Sep. 23, 196 (1984).

Boll, R. H., "Particle Collection and Pressure Drop in Venturi Scrubbers," Ind. Eng. Chem. Fundam. 12, 40 (1973).

Fathikalajahi, J., M. Taheri and M. R. Talaie, "Theoretical Study of Liquid Droplets Dispersion in a Venturi Scrubber," J. Air Waste Manage. Assoc. 45, 181-185 (1995).

Fathikalajahi, J., M. Taheri and M. R. Talaie, "Theoretical Study of Non-Uniform Droplet Concentration Distribution on Venturi Scrubber Performance," Part. Sci. Tech. 14, 153 (1996).

Goncalves, J. A. S., M. A. Martines Costa, K. L. Serra and J. R. Coury, "Droplet Dispersion in a Rectangular Ventrui Scrubber," 14th Int. Cong. Chem. Proc. Eng., 27-31, Praha, Czech Republic, August (2000).

Nukiyama, S. and Y. Tanasawa, "An Experiment on the Atomization of Liquid by Means of an Air Stream," Trans. Soc. Mech. Eng. 4, 86 (1938).

Rahimi, A., M. Taheri and J. Fathikalajahi, "Mathematical Modelling of Non-Isothermal Venturi Scrubbers," CJChE 83, 401-408 (2005).

Rahimi, A., "Simulation of Air Pollutants Removal in a Non-Isothermal Venturi Scrubber," PhD Thesis, Chemical Engineering Department, Shiraz University, Shiraz, Iran (2002).

Taheri, M. and G. F. Hains, "Optimization of Factors Affecting Scrubber Performance," J. Air. Pollut. Control. Assoc. 19, 427 (1969).

Taheri, M. and Ch. Sheih, "Mathematical Modeling of Atomizing Scrubber," AIChE J. 21, 153 (1975).

Talaie, M. R., J. Fathikalajahi and M. Taheri, "Mathematical Modeling of SO2 Absorption in a Venturi Scrubber," J. Air Waste Manage. Assoc. 47, 1211-1215 (1997).

Viswanathan, S., W. A. Gnyp and C. St. Pierre, "Jet Penetration Measurement in a Venturi Scrubber," Can. J. Chem. Eng. 61, 504 (1983).

Viswanathan, S., W. A. Gnyp and C. St. Pierre, "Examination of Gas Liquid in a Venturi Scrubber," Ind. Eng. Chem. Fundam. 23, 303-308 (1984).

Viswanathan, S., W. A. Gnyp and C. St. Pierre, "Annular Flow Pressure Drop Model for Pease-Anthony-Type Venturi Scrubbers," AIChE J. 31(12), 1947-1958 (1985).

Viswanathan, S., "Modeling of Venturi Scrubber Performance," Ind. Eng. Chem. Res. 36, 4308-4317 (1997).

Viswanathan, S., W. A. Gnyp and C. St. Pierre, "Estimating Film Flow Rate in a Venturi Scrubber," Particle. Sci. Tech. J. 15, 65-76 (1997).

Viswanathan, S., "Examination of Liquid Film Characteristics in the Prediction of Pressure Drop in a Venturi Scrubber," Chem. Eng. Sci. 53(17), 3161-3175 (1998).

Manuscript received September 1, 2004; revised manuscript received October 14, 2005; accepted for publication December 9, 2005.

A. Rahimi (1), J. Fathikalajahi (2 *) and M. Taheri (2)

(1.) Chemical Engineering Department, College of Engineering, Isfahan University, Isfahan, Iran

(2.) Department of Chemical Engineering, School of Engineering, Shiraz University, Shiraz, Iran

* Author to whom correspondence may be addressed.

E-mail address: zeglda@succ.shirazu.ac.ir

Table 1. Calculated values of [[alpha].sub.d] from Equation (11)
and experimental data Viswanathan et al. (1984)

[V.sub.Gth] L/G L ** [f.sub.x1=0.13] [f.sub.x2=0.534]
(m/s) (m) (exp.) (exp.)

45.4 9.54E-4 0.0188 0.06 0.08
45.4 1.23E-3 0.0243 0.05 0.07
45.4 1.5E-3 0.0296 0.04 0.07
45.4 1.9E-3 0.0375 0.03 0.08
60.6 9.54E-4 0.0188 0.06 0.07
60.6 1.23E-3 0.0243 0.04 0.065
60.6 1.5E-3 0.0296 0.03 0.06
60.6 1.9E-3 0.0375 0.025 0.03
75.6 9.54E-4 0.0188 0.05 0.05
75.6 1.23E-3 0.0243 0.04 0.04
75.6 1.5E-3 0.0296 0.02 0.03
75.6 1.9E-3 0.0375 0.01 0.01

[V.sub.Gth] [f.sub.x3=0.838] [[alpha].sub.x1](cal)
(m/s) (exp.) Eq.11

45.4 0.11 0.044
45.4 0.1 0.054
45.4 0.1 0.0595
45.4 0.1 0.0621
60.6 0.075 0.059
60.6 0.07 0.07
60.6 0.06 0.0772
60.6 0.035 0.0817
75.6 0.05 0.0706
75.6 0.04 0.0874
75.6 0.03 0.0937
75.6 0.005 0.0993

[V.sub.Gth] [[alpha].sub.x2](cal) [[alpha].sub.x3](cal)
(m/s) Eq.11 Eq.11

45.4 0.0117 0.0084
45.4 0.0141 0.0099
45.4 0.0158 0.011
45.4 0.0173 0.0117
60.6 0.015 0.0098
60.6 0.0185 0.012
60.6 0.0205 0.0131
60.6 0.0202 0.013
75.6 0.0172 0.011
75.6 0.0213 0.0136
75.6 0.0235 0.015
75.6 0.0239 0.015

Table 2. Comparison between eddy diffusivities calculated from
Equations (9) and (10)

[V.sub.Gth] L/G [D.sub.d] ([mu]) b [[alpha].sub.g]
(m/s) from Nukiyama and ([m.sup.2]/s)
 Tanasawa (1938)
 correlation

45.4 0.000954 62.4 92.5 0.0454
45.4 0.00123 73 67.5 0.0454
45.4 0.0019 110 32 0.0454
75.6 0.000954 46 170 0.0756
75.6 0.00123 57 113 0.0756
75.6 0.0019 95 40 0.0756

[V.sub.Gth] [[alpha].sub.g] [[alpha].sub.g]
(m/s) ([m.sup.2]/s) ([m.sup.2]/s)
 from Equation (14) from Equation (13)

45.4 0.008 0.021
45.4 0.005 0.026
45.4 0.00112 0.03
75.6 0.031 0.033
75.6 0.0183 0.04
75.6 0.016 0.046