Absorption of Gases by Liquid Droplets Design of Simple Spray

cyclone scrubbers, Johnstone and Kleinschmidt (14) pro- posed a theory by which the rate of absorption of highly soluble gases could be determined whe...
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Absorption of Gases by Liquid Droplets Design of Simple J

Spray Scrubbers Measurements on the rates of absorption of several gases by falling drops of various sizes and at different gas velocities are reported. The results confirm the theoretical equations of Johnstone and Kleinschmidt (14). The effect of the deformation of droplets and drag resistance on the theory is discussed. On the basis of the size distribution of droplets from a commercial spray nozzle and the integrated equations of motion, the volume absorption coefficients are calculated for a simple countercurrent scrubber. With nozzles properly designed and operated, relatively high coefficients may be obtained. The equations may be applied to the design of spray humidifiers.

H. F. JOHNSTONE AND G. C. WILLIAMS University of Illinois, Urbana, 111.

After the results given here were obtained, Froessling (6) reported some measurements on the rates of evaporation of drops of water, aniline, and nitrobenzene suspended on a thin glass rod in a current of air. The rates, measured photographically, were correlated by an empirical equation on the basis of dimensionless groups containing the velocity of the air, the size of the drop, and the properties of the diffusing vapors. The equation has been shown to reduce to that derived below (IS). Most of the data reported on rates of absorption by actual sprays have resulted from measurements made on small ex-

I

N A RECENT paper on the absorption of gases in wet

cyclone scrubbers, Johnstone and Kleinschmidt (14) proposed a theory by which the rate of absorption of highly soluble gases could be determined when the size of the droplets and their interfacial velocities through the gas were known. While the derived equations agreed closely with the experimental results of Whitman, Long, and Wang (23) on the rate of absorption of ammonia by falling drops, additional evidence of their general application is desirable. I n this paper, measurements on the rate of absorption of several gases by falling drops of various shes and at different air velocities are reported which further substantiate the theoretical equations. On the basis of some data on size distribution from a commercial spray nozzle, the interfacial velocities of droplets have been calculated for a simple countercurrent spray system. The application of the theory then predicts the performance of such a scrubber and yields information on the effect of the fineness of atomization, the spacing of the spray banks, the pressure under which the nozales operate and the gas velocity through the scrubber.

Previous Work The measurements of Whitman, Long, and Wang ($3) were made on single drops falling through a tower of fixed height and a t varying rates of drop formation. By extrapolation to zero time of formation, the absorption during the falling period was calculated. Similar measurements were made by Hatta, Ueda, and Baba (IO) with towers of va.rying height. Since no data were given by which correction could be made for the absorption during the time of drop formation, correlation of the results with the earlier measurements, except in a qualitative way, is impossible. Both sets of data, however, show the existence of high absorption coefficients for freely falling drops.

Courtesy, Schutte & Koerting Company

A 36-INCH STAINLESSSTEEL OBNOXIOUS-VAPOR CONDENSER IN A LARGJII RENDERING PLANT 993

994

INDUSTRIAL AND ENGINEERING CHEMISTRY

perimental apparatus. Because of the size of the equipment, absorption on the walls has contributed to the total absorption measured and consequently precludes correlation of the data. Another factor which prevents correlation is the difference in the atomization produced by different spray nozzles. Kowalke, Hougen, and Watson (15) obtained values of the over-all coefficient, &a, for the absorption of ammonia by water in a 16-inch tower 3.8 feet high, ranging from 0.71

VOL. 31, NO. 8

proximately proportional to the liquid rate and inversely proportional to the square root of the tower height. The correlation of the data was only fair. No consideration was given to the velocity of the droplet, which in the case of countercurrent flow should be added to that of the gas to determine the interfacial velocity. Furthermore, it is evident that the liquid film resistance, especially in sulfur dioxide absorption, cannot be neglected.

FIGURE 1. DIAGRAM OF APPARATUS

to 3.06 pound moles per hour per cubic foot per atmosphere when water was sprayed from five “Vermorel” sprays a t a rate of 450 pounds per hour per square foot. Other liquid rates were studied. The coefficient was found to be proportional to the 0.7 power of the gas rate over a velocity range of 40 to 120 pounds per hour per square foot. For ammonia the over-all coefficient is approximately the same as the individual gas film coefficient. Haslam, Ryan, and Weber (8)found values of &a between 0.45 and 0.68 for the absorption of sulfur dioxide by water in a small spray tower 8 inches in diameter and 30 inches high. Their data show the importance of both liquid and gas film resistances for the absorption of sulfur dioxide. At a gas velocity of 0.3 foot per second the liquid film resistance was 74 per cent of the over-all resistance. The results indicated that the gas film coefficient varies approximately as the 0.8 power of the gas velocity. This relation has been well established for absorption on wetted walls when the flow of gas is turbulent (6). Hixson and Scott (11) attempted to isolate the wall effect from the absorption by the droplets by collecting the liquid sample; the rate of absorption was determined from a central core of the spray. They used a tower 2 I/s inches in diameter, varying in length from 19 to 54 inches, and equipped with a perforated plate for distributing the water. They measured rates of absorption of ammonia and sulfur dioxide by water, and of benzene by straw oil for several gas and liquid rates and for three different tower heights. Values of K,a for ammonia were from 0.5 to 4.8 a t gas velocities of 108 to 164 pounds per hour per square foot. They correlated their data by assuming that the effect of gas velocity was the same as that usually found for all mechanisms of transfer through the so-called “stagnant” films in fluids in turbulent flow-that is, that the coefficient varies directly as the 0.8 power of the mass gas flow. On this basis the coefficients were found to be ap-

The passage of a drop through a gas results in distinctly different surface conditions from those a t the boundary of a fluid flowing in turbulence through tubes or about plates. Deviations from Stokes’ law for flow past rigid spheres occur a t Reynolds numbers as low as unity, apparently owing to the existence of vortex rings which become unstable a t a Reynolds number of about 150. Well-defined turbulence probably does not set in, however, until a Reynolds number of 130,000 is reached. I n the case of large droplets, deformation from the spherical shape also occurs a t a low Reynolds number with an added complication in the flow (4). For these reasons it seems doubtful that the usual concept of gas films in turbulent flow is applicable to absorption by spray droplets.

Theory It is assumed that each drop makes contact with a tube of gas described by the trace of its periphery and of thickness This thickness represents the distance that the gas molecules can move by diffusion while the particular layer is in contact with the drop surface. Increase in interfacial velocity between the drop and gas will decrease the thickness of this layer. At very high velocities 2 will not decrease indefinitely but will approach a minimum value comparable with the molecular mean free path of the gas. This layer does not correspond to the so-called stagnant film in turbulent flow, for here the movement of the inert gas in the film is no different from that outside of the layer. For steady-state diffusion of one gas through a second stagnant gas, the rate is represented by 2.

For the present pa, will be considered zero. The time of contact of the droplet with a point will be taken as the time

AUGUST, 1939

INDUSTRIAL AND ENGINEERING CHEMISTRY

it is traveling one diameter-i. e., D,/u where u is the velocity of the drop. Integrating between the limits of zero and Ds/u, N = - AbP -RTx

PA, ~

R

D. M u

Since it is assumed that all the molecules are absorbed that are within the distance x of the drop surface,

Equating,

(4)

The rate of absorption by a droplet then becomes (5)

995

method similar to that used by Whitman, Long, and Wang, and gave similar calibration curves. Drops weighing from 0.056 to 0.11 gram were obtained from glass capillaries of different diameters. In order to obtain smaller drops, steel syringe needles, ground and polished flat on the ends, were used. The velocity of the auxiliary air through tube L was approximately twice that of the main gas stream through the tower. The concentration of gas in the path of the drop was explored by means of tube T . Samples were drawn at half-inch intervals, beginning at the capillary tip and proceeding downward until the concentration was the same as that of the inlet gas. The exploring tube was moved out of the path of the drops before the flow started. The gas concentration was integrated graphically against time, assuming the gravitational law with no resistance due to the gas in order to find the average concentration. A plot of one of the concentration-distance and corresponding concentration-time curves is shown in Figure 2. From the average concentration thus found, the area of the drop taken as a rigid sphere, and the amount of absorption, coefficient ko was calculated.

where and

A

=

TD,L

(7)

For a drop falling from rest through a quiescent gas, integration of Equation 5 over the time of fall shows that the average value of k , over any distance L is

For gases moving upward a t a constant velocity V,, the average value of the coefficient for the falling drop becomes CONCENTRATION

OF

NHlt

PERCENT

OF CURVES USEDFOR FIGURE 2. EXAMPLE INTEGRATION OF GAS CONCENTRATION

If the integration is performed over the time of contact of the droplet with a point, that is (9)

where et = total time of travel of the droplet

Absorption Experiments Tests were made on dilute air-gas mixtures to determine the effect on the rate of absorption by drops of interfacial velocity, drop diameter, diffusivity, and, in several cases, the effect of the concentration of the absorbing reagent. Variation in drop size was obtained by the use of capillary tubes of different diameters. Interfacial velocity changes were produced by changing the gas velocity. Ammonia, hydrogen chloride, sulfur dioxide, hydrogen sulfide, and carbon dioxide were used as the solute gases in order to show the effect of the diffusivity and other properties of the gas absorbed. The experimental method was similar to that used by Whitman, Long, and Wang with the modification of preventing absorption during the formation of the drop by surrounding the capillary tip with an atmosphere of the inert gas. The apparatus is shown in Figure 1. The absorption tube was 168 cm. long and 2.86 cm. in diameter. Compressed air was passed through flowmeter B and saturator D into the mixing chamber. The gas from a cylinder passed through meter C and into the mixing chamber. The gases then passed through a Venturi meter into the bottom of the tower, and out through a side tube at the top. The absorbent was fed from storage bottle F to a constant-head box, G, from which it flowed through capillary tube J * The drops were collected under a layer of paraffin oil in receiver H , from which the solution was drawn off for analysis through a side tube. The capillaries used for forming the drops were prepared by a

Gas analyses for ammonia and hydrogen chloride were made by drawing a measured quantity of the gas through standard acid or base, respectively, and titrating the excess reagent. Standard iodine solution was used for sulfur dioxide and hydrogen sulfide. Carbon dioxide was absorbed in standard base to which had been added barium chloride; the excess base was titrated with acid, using phenolphthalein as the indicator. All of the scrubbing solutions were standard solutions of acid or base. Analyses of the solution to determine the absorption of ammonia and hydrogen chloride were made simply by titrating the excess reagent. In the case of sulfur dioxide, neutral hydrogen peroxide was added to the sample after absorption in order to obtain a better end point. For the absorption of hydrogen sulfide, peroxide was added to the hydroxide before standardization and this solution was used as the absorbent. To determine the amount of carbon dioxide absorbed, barium chloride was added to a measured sample of the solution, and the titration was made with standard acid, as in the gas analysis. In all cases, blank tests were run with air alone. The experimental data obtained are shown in Table I. The percentage deviation of the observed value of the absorption coefficient from that calculated by Equation 8 is given in the last column. The theory as outlined above is obviously a simplification of conditions that actually exist. It is well to consider some factors that might cause deviation from the original assumptions.

Resistance within the Drop It is obvious that liquid film resistance would decrease the rate of absorption. This means that any liquid-phase reactions that occur must be fast if the absorption is to follow Equation 5. The absorption of sulfur dioxide by sodium hydroxide solution demonstrates the effect of a chemical reaction in the liquid film on the rate of absorption of a gas of intermediate

INDUSTRIAL AND ENGINEERING CHEMISTRY

996

VOL. 31, NO. 8

TABLEI. ABSORPTIONOF GASESBY FALLING DROPS Run No.

Temp. of Satd. Air O

c.

Barometer M m . hg

%

% A.

14 15 16 17 18 19 35 36 37 38 39

25' 25 25 25 25 25 18 17 17 18 16

Ds,

Solute Gas ComPn. Entering Av.

740 740 743 744 748 745 743 743 743 743 743

Density of Liquid

Drop Diam.

Amount of Absorption

Normality of Absorbing Soh.

Air Velooity

Moles oas/mole

G./oc. Ha0 X 102 Absorption of Ammonia by Sulfuric Acid Solutions Mm.

1.23 1.23 1.23 2.10 2.12 1.83 3.07 3.10 2.68 1.65 3.14

1.50 1.40 1.44 0.905 0.925 0.904 1.61 1.75 1.32 0.782 1.70

Cm./eec.

1.4 1.4 1.4 1.4 1.4 1.4 305 305 196 125 305

k--

-a

Obsvd. Calod. Lb. moZe/(hr.) (sq. ft.)

Deviation from Equation 8

%

(atm.)

3.83 4.23 4.16 2.89 2.92 2.98 3.62 3.99 3.55 3.13 3.69

4.16 4.14 4.04 2.91 2.91 3.11 4.46 4.29 3.86 3.52 4.30

- 7.9 + 2.2 +- 3.0 0.7 +- 0.3 4.2

1.10 1.38 1.00 0.98 2.45 2.35 2.81 2.57 2.88 3.16

2.34 2.43 3.10 3.23 2.43 2.24 3.12 2.74 3.02 3.36

-53.0 -43.2 -67.8 -69.7 0.8 3.5 9.9 6.2 4.6 6.0

+ +-

1.66 2.81 2.65 3.38

2.72 2.72 2.53 3.48

-39.0 3.3 ++- 3.0 4.7

2.05 1.68 1.54 1.50

3.41 2.39 2.39 2.39

-39.9 -29.7 -35.6 -37.2

2.46 2.06 2.06 1.77 2.43 0.80 0.083

2.27 2.27 2.27 2.27 2.27 2.54 2.54

---18.8 7.0 8.1 -11.1

-14.2

B. Absorption of Sulfur Dioxide by Sodium Hydroxide Solutions 21 22 23 24 25 26 27 40 41 42

25 25 25 25 25 25 25 18 16 16

740 743 753 736 740 746 746 740 743 743

31 32 33 34

25 25 25 25

743 746 742 742

2.90 2.12 2.74 2.74

28 48 49 50

25 Dry Dry Dry

742 746 746 746

2.06 1.76 1.76 1.76

0.0985 0.0985 0.0985 0.0985 2.0 2.0 2.0 2.0 2.0 2.0

1.4 1.4 1.4 1.4 1.4 1.4 1.4 125 196 305

--

C. Absorption of Hydrogen Sulfide by Sodium Hydroxide-Hydrogen Peroxide (1 N) Solutions

2.36 1.72 2.46 2.46

4.75 4.75 5.46 2.90

1.008 1.006 1.021 1.040

0.670 0.800 0.725 0.226

0.266b 0.167 0.341 0.951

1.4 1.4 1.4 1.4

D. Absorption of Hydrogen Chloride by Sodium Hydroxide Solutions 1.70 1.73 1.73 1.73

E. 43 44 45 46 47 51 57

25 25 25 25 25 25 20

745 745 745 745 745 745 742

2.26 2.15 2.19 2.19 2.10 2.35 4.75

2.94 5.97 5.97 5.97

.

1.001 1.006 1.009 1.006

0.996 0.410 0.372 0.375

0.0996 0.201 0.400 0.201

1.4 1.4 1.4 1.4

Absorption of Carbon Dioxide by Sodium Hydroxide Solutions

2.26 2.15 2.19 2.19 2.10 2.35 4.75

5.97 5.97 5.97 5.97 5.97 4.75 4.75

1.086 1.075 1.067 1.060 1.082 1.006 1.000

0.761 0.637 0.625 0.750 0.537 0.310 0.029

2.27 1.82 1.48 1.03 2.27 0.212 0

1.4 1.4 1.4 1.4 1.4 1.4 1.4

+- 9.3 8.4

+

9.3 -22.0 7.1 -68.6

....

0 Runs at 2 5' C. were made with a jaoketed thermostatted oolumn. The gases, saturated with water vapor, and the absorbent were brought t o this temperature before entering the column. In runs at other temperatures the column was not jacketed. b Peroxide, 0.5 N .

solubility for which the liquid film resistance is ordinarily not negligible. An increase in the concentration of hydroxide results in a decrease in the liquid film resistance and a net increase in the over-all rate of absorption. While not enough data were secured to show clearly the relation between concentration of hydroxide in the absorbing liquid and the concentration of sulfur dioxide in the gas mixture, a threshold concentration of hydroxide necessary to reduce the liquid film resistance to a negligible quantity is indicated for each gas composition. Difficulty was encountered with the experimental procedure for hydrogen sulfide when standard hydroxide alone was used as the absorbent. Analyses by iodine did not give check results. It is probable that some of the hydrogen sulfide was oxidized before the analysis could be run. The data reported were obtained by adding hydrogen peroxide to the caustic scrubbing solution and titrating the excess caustic after absorption of hydrogen sulfide. The data on carbon dioxide show that, over the range of gas concentrations considered , the liquid film resistance is an important factor, and that a sufficient increase in caustic concentration will reduce this film resistance to a negligible quantity. Hatta (9) assumes that the reaction between carbon dioxide and hydroxide occurs rapidly and is not necessarily preceded by hydration but may take place directly as: COZ

+ 2KOH

-+.

KzCOa

+ H20

The subject of hydration of carbon dioxide and its effect on the rate of absorption by hydroxide and carbonate solutions is important in both theory and practice. Undoubtedly this gas is abnormal in that its equilibrium hydration, which must take place before ionization, is very small. If no hydroxide is present and consequently such direct reactions as suggested by Hatta cannot take place, it may be expected that the rate of absorption must be controlled by the rate of hydration. Some evidence exists for this in the low rate of absorption by carbonate solutions. The possibility that the hydration reaction itself can be catalysed to speed up the entire process has been suggested several times. One of the catalysts is an enzyme which occurs in the blood and hastens the release of carbon dioxide from the lungs (17). The results reported in Table IE indicate that a t a concentration of about 2 normal hydroxide the rate of reaction between the dissolved carbon dioxide and the hydroxide is sufficiently rapid to eliminate the liquid film resistance.

Effect of Fog Formation The absorption of hydrogen chloride presents a phenomenon similar to that observed by Chambers and Sherwood (2) in the absorption of nitric oxide. Variations in the concentration of caustic seem to have little effect on the absorption rate. I n running the gas analyses, it was observed that a mist formed in the absorption bottle if the gas sample was

IXDUSTRIAL AND ENGINEERING CHEMISTRY

AUGUST, 1939

bubbled through rapidly. A mist was probably also formed in the absorption tower when run 28 was made. Three runs (48-50) were carried out in which the air-hydrogen chloride mixture was dried by bubbling the gas through concentrated sulfuric acid. Here, although the alkali concentration was increased to as much as four times that used in the previous runs, the observed absorption coefficients remained below the theoretical. Since drying the gas exerted little effect on the absorption, it seems evident that a fog, if formed, must result

where CD = drag coefficient p a=V 2 kinematic pressure 2 m - 2 = projected area of the drop Then the acceleration,

let

then By means of a graphical integration, the velocity and the distance traveled were calculated for the four drop sizes used in these experiments, assuming no drop deformation. The results showed that there would be only a 3 per cent increase in the time of fall of spherical drops over that calculated for unresisted fall. I n the same manner the velocity and time of fall of drops, allowing for the deformation, can be calculated by Equation 12 if data are available on the magnitude of the deformation. The terminal velocities of freely falling drops were determined FIGURE3. DROP DEFORMATION AS CALCULATED FROM TERMINAL VELOCITIES

from reaction of hydrogen chloride and water vapor from the drops. Since these particles would no longer have molecular dimensions, i t would not be expected that the diffusivity calculated for a hydrogen chloride-air mixture would be applicable.

Effect of Resistance to Motion of Falling Drops I n addition to the fact that falling drops do not follow either the gravitational law or that of Stokes over the entire distance of fall, they also do not remain perfect spheres. An approximation of these effects can be made. Saito (21) demonstrated mathematically that, for the case of water droplets falling through air, the forces acting on the drops tend to deform them into spheroids, oblate normal to the direction of motion. It is interesting to note that the theoretical treatment for mercury droplets leads t o prolate spheroids. Lenard and Hochswender (16) showed that raindrops which are initially spherical take the shape of inverted cups a t velocities from 6 to 8 meters per second and finally burst into fragments. The disrupting velocity varies from 20 to 10 meters per second for drops from 2.5 to 6 mm. in diameter, respectively Others (1, 19) have shown that droplets falling by gravity reach a terminal velocity much less than that predicted from Stokes' law when the drop is more than 2 mm. in diameter. Using the observed terminal velocities for a given size of drop, it is possible to calculate the maximum deformation that may occur when there is a flattening normal to the direction of motion. Diehl (3, 24) showed that for cylinders with the ratio of height to diameter less than unity, the drag coefficients are approximately the same as those for disks. A comparison of the curves of Prandtl (20) and of Squires and Squires (22) shows that for disks the coefficients closely approximate those for spheres for the same Reynolds number. By means of these approximations and the terminal velocities mentioned above, the ratio of the deformed diameter to the original diameter will now be estimated. Consider a drop falling under the influence of gravity and subjected to the retarding force of the air through which it falls. The equation of the force system acting on the body will be as follows:

, FIGURE4. CALCULATED TIME OF FALL FOR DROPS6 MM. I N DIAMETER experimentally by several investigators (1, 7, 16). The data cover a range of drops from 0.1 to 6 mm. in diameter. When a drop has reached its terminal velocity, the forces acting upon it are in equilibrium. Then,

Solving for De and substituting for pl, pa, and g,

Solution of this equation for D, is accomplished by trial and error. The drag coefficient changes slowly with Reynolds number in the range where the diameter change is largest. Table I1 shows the results of such a calculation of the maximum,deformation based on observed terminal velocities. The ratio of the deformed diameter to the original diameter is plotted against the size of the drop in Figure 3.

INDUSTRIAL AND ENGINEERING CHEMISTRY

998

TABLE11. DEFORMATION OF FREELY FALLINQ DROPS ---Velocity, Drop Diam., Da, Mm. 0.1 0.2 0.4 1.0 2.0 2.5 3.0 4.5 5.5 6.0

Stokes' law 30 121 464 3,030 12,100 18,900 27,200 61,500 89,800 109,000

Cm./Sec.Calcd. Re Diam. of for no from Cvlinder ii,rro-m de- Observed Obforma- V and Equation 13, served tion DS CD Mm. 32 32 2 14.00 0.1 130 125 16 3.20 0.2 180 1so 45 1.50 0.4 395 0.73 395 246 1.0 620 0.52 577 723 2.21 715 640 1000 0.49 2.92 822 0.44 3.76 692 1300 946 0.40 800 2250 6.25 800 2720 1080 0.39 8.48 800 1300 0.38 3000 9.88

If we assume that within the range of velocities of interest here, the deformation, D,- D,,is proportional to the square of the velocity, the proportionality constant can be evaluated for any size drop from the curve in Figure 3. On this basis an estimate of the time of fall, corrected for deformation and drag, was made for a drop 6 mm. in diameter. The results are shown in Table 111. A comparison of the time of fall for the three methods of calculation is shown in Figure 4. Although the deformation becomes quite large near the bottom of the tower, the increase in time of fall through a distance of 168 cm. over that required by the gravitational law is about 10 per cent.

VOL. 31, NO. 8

Other Effects Causing Deviation from Simple Theory The simple theory does not take into account the convection currents present in the wake of a sphere moving through a gas. Nisi and Porter (18) showed the existence of these and demonstrated qualitatively that the velocity of the gas at this point is different from the apparent interfacial velocity. The data available permit only the statement that these currents exist and that they should increase the absorption over that predicted by the simple theory. Another factor which would cause deviation from the simple theory is the change in velocity near the surface of the drop. This effect can be estimated only qualitatively. A larger number of gas molecules will be brought into contact with the drop than would be predicted by theory, as a result of the increase in both velocity and pressure a t the sides of the droplet. This effect is probably negligible a t low velocities but might become appreciable as the interfacial velocity becomes great.

Conclusions

From the foregoing discussion, if the 6-mm. drop is deformed into a true cylinder, the calculated amount of absorption should be multiplied by 1.1,owing to the longer time reowing to the quired to fall through the tower, and by deformation, and divided by diT,owing to the smaller averagevelocity. The net correction factor is 0.86. It is probable that the actual correction factor is nearer unity since the shape is an oblate TABLE 111. ESTIMATED VELOCITIES OF A FREELY FALLINQ DROP, spheroid rather than a true cylinder. 6 MM. IN DIAMETER, ALLOWING FOR DEFORMATION The object here has not been to calculate the Velocity of Spheric$ Drops_ exact correction factors to be applied to the Velocity at End of Total No simple theory, but rather to show that these Diameter, Av. Time of Time InDistance resistEquaD~ Velocity Fall Re CD terval Traveled anoe tion 12 factors are relatively small and possibly of a Mm. Cm./sec. Sec. Cm./sec. Cm. Cm./sec. compensating nature. The important fact is 6.00 49 0.1 180 0.81 96 4.8 98 98 that the calculated values, on the basis of the 196 196 189 19.0 518 0.59 147 0.2 6.04 simple theory, agree closely with those actually 294 286 282 42.0 940 0.49 238 0.3 6.30 6.60 331 0.4 1370 0.42 0.46 368 74.5 392 379 observed in the case of four of the gases studied. 490 469 1810 447 104.0 417 0.5 6.90 7.25 492 0.6 2260 0.39 505 151 .O 588 550 For the other one, the formation of a mist out7.46 512 0.7 2410 0.39 550 203.0 686 ... side of the drop caused considerable deviation.

Assuming that the deformation of the droplets causes them to assume cylindrical shapes, there will be a direct effect on the rate of absorption other than that due to a decrease in velocity. Let the deformed diameter be D, and the height of the cylinder be h. Then, -

for cylinders: for spheres:

bgc =

b,

=

&& & dg

(14)

Since the volume of the drop does not change,

or

D, =

4%

then Thus, if the sphere is deformed into a true cylinder, regardless of the ratio of the diameter of the cylinder to that of the sphere, the absorption by the deformed droplet will be 81.5 per cent of the amount absorbed if the droplet had remained a sphere and fallen through the same distance.

m

Practical Application of Results The experimental data obtained in this work may be considered as a confirmation of the theory of absorption of highly

TABLE IV. DISTRIBUTION OF SIZES OF DROPLETS FROM NOZZLE Diameter of Drop,a Microns

No. Drops Measured 878 460 190 89 53 33 22 16 13 11 10 8 7

No. Drops per Cc. of Sprayb 97,250 51,000 21,000 9,880 5,870 3,650 2,440 1,770 1,440 1,220 1,107 886 776 664 554 443 332 221 110

7

of Vofume

A

Area per Cc. SPrrtY. Sq. Cm. 1.9 4.0 6.6 6.9 7.4 7.2 6.9 6.8 7.2 7.8 8.7 8.4 8.8 8.8 8.5 7.8 6.7 5.0 2.8 0

Of

0.072 0.334 1.11 1.75 2.46 3.00 3.46 3.97 4.83 5.84 7.25 500 7.74 550 8.79 600 9.59 6 650 9.68 5 700 9.84 4 750 8.94 3 800 7.13 2 850 4.22 1 900 0 0 0 950 0 0 0 0 1200 128.2 1811 200,583 100.00 Total a Each size group includes drops in a range equal to the interval groups, centering about the nominal size-i. e., the 100 group drops from 76-124 microns. b Number of drops of each size produced b y each CC. of spray. 25 50 100 150 200 250 300 350 400 450

SPRAY %

Of

Area 1.48 3.12 5.15 5.38 5.78 5.61 5.38 5.30 5.61 6.09 6.79 6.55 6.87 6.87 6.64 6.08 5.22 3.90 2.18 0

0 -

100.00 between includes

AUGUST, 1939

INDUSTRIAL AND ENGINEERING CHEMISTRY

soluble gases by moving droplets, particularly in view of the agreement already noted with the work of Whitman, Long, and Wang, and of Froessling. Immediate application to a problem of practical importance may be made in the design of simple absorbers in which a spray of liquid is directed into a moving stream of gas. A number of types of simple spray scrubbers are in use. Spray devices are quite common as humidifiers. They may be considered as deabsorbers, for which the theory should hold as well as for absorption. It is believed that a theoretical discussion of simple sprays would be of interest and possibly would throw further light on the proper design of both scrubbers and humidifiers. Information on two items is essential: (a) the distribution of drop size in the spray and ( b ) the interfacial velocity of the drops. It is evident that the former is an important characteristic of any spray nozzle. Different designs of nozzles give vastly different size distribution, even when the orifices are the same diameter and when the sprays are operated under the same pressure. Only a few data on the size distribution of droplets from nozzles are available. Since the technique for making these measurements has been worked out ( l a ) ,it is suggested that manufacturers could obtain valuable information on their nozzles and possibly improve their efficiency by making such studies. As an illustration of the application of the absorption theory, an estimate will be made for a simple spray nozzle, for which the size distribution of the droplets is shown in orifice. Table IV (14). This is a lava nozzle with a 3/16-in~h It is known that this nozzle is not highly efficient in producing fineness of atomization, but the extremely large number of particles produced per cubic centimeter of water is interesting. It will be evident from the calculations that the particle size range from 200 to 600 microns in diameter is the most important. The spray is directed downward into an ascending stream of gas. I n order to show the effect of pressure and of gas velocity on the performance of the spray, calculations will

DISTANCE,

FEET

999

Courtesy, Buffalo Forge Company

SPRAY OF

AN

AIR WASHER

be made on pressures of 35 and 65 pounds per square inch and gas velocities of 5 and 10 feet per second. The assumption will be made that the atomization produced by the nozzle is the same a t the higher pressure as a t 35 pounds per square inch, a t which the distribution study was made.

VOL. 31, NO. 8

INDUSTRIAL AND ENGINEERING CHEMISTRY

1000

The absorption coefficients are calculated for the absorption of sulfur dioxide by a highly buffered or alkaline solution in which the liquid film resistance is eliminated. Estimates of the absorption coefficients of other gases, or for the evaporation of water, can be made by multiplying by the ratio of the square roots of the diffusivities. The assumed liquid rate is one gallon per minute per square foot of tower cross section. All the droplets are considered to be projected from the nozzles a t velocities resulting from perfect energy conversion. At 35 pounds per square inch the initial velocity is 72.2 feet (2200 em.) per second; a t 65 pounds per square inch the initial velocity is 98.4 feet (3000 em.) per second. With these initial conditions the velocities of the various sizes of drops were calculated by graphical integration of Equation 12. Values of the drag coefficient, CD, were taken from the data of Zahm (24) for rigid spheres on the assumption that the small droplets are not deformed. The velocity-time curves and their graphically integrated distance-time curves are shown for 35 pounds per square inch nozzle pressure and 5 feet per second counter gas velocity in Figure 5. From these two sets of curves the velocity-distance function was found for each drop size. The absorption coefficient was then calculated by means of Equation 6. The calculated absorption coefficient decreases linearly with distance from the nozzle a t a rate practically independent of the nozzle pressure and of the gas velocity. Within the limits of the error of the graphical calculations, the coefficient for any drop size and for any initial interfacial velocity, uo, is given by the empirical equation :

IC,

= 57.4

42- -

1.7 X 105LD-1.75

The fraction of the drops of diameter dl that succeed in passing through any short increment of distance (in which ni may be considered constant) without coalescence with any of the larger drops is given by

where nl’/nl

ratio of number of drops of diameter 1 leaving the section to that entering di, ni, ABi = diameter, number, and time of passage, respectively, of the size group of order igreater than 1 AB = quotient of the increment of distance by the average velocity =

From the velocity-distance curves, therefore, the coalescence of the drops may be estimated by a step-by-step method. From the total number of collisions and the probability of collision with each group size, the number of drops growing into a larger group size would be estimated when the coalescence results in a substantial increase in size. After such rearrangement of the size distribution for a short increment o€ the distance, a second step was calculated as before until the distribution was found for the entire tower. Fortunately,

(19)

The units in which the constants are given are for k, in pound moles per hour per square foot per atmosphere, ~0 in feet per second, drop diameter D in microns, and distance from the nozzle, L, in feet. The volume absorption coefficient, koa,could then be determined if the interfacial area per unit volume of tower was known. This was obviously a function of the number of drops present a t each point in the tower and of their linear velocity. Examination of the curves in Figure 5 showed that careful consideration must be given to the coalescence of the drops. It was apparent that, even a t the lower gas velocity, the small sizes travel only a short distance from the nozzle before their direction ,is rewrsed and they, are carried back by the gas. Only those drops larger than 500 to 700 microns, depending on the gas velocity, penetrate more than 8 feet. Actually there is a critical size, between 400 and 600 microns for the conditions studied, which reaches a terminal velocity equal to the upward velocity of the gas. These drops would tend, therefore, to accumulate at a distance of about 5.5 feet from the nozzles if coalescence did not take place. Since this was obviously not the case, it was evident that the disappearance of the smaller drops and the growth of the larger drops could not be neglected in the calculation of a. Without information, which is not available at present, the estimation of the coalescence of the drops can be made only by calculating the number of collisions between drops of different sizes moving a t different velocities, assuming that each collision results in coalescence and that the number of drops of various sizes leaving the nozzle is actually that given in Table IV. The calculation of the number of collisions which was somewhat tedious and inexact because of a stepby-step integration, was made as follows: The probability of collision of the drops of a given diameter, d,, with the drops of any given larger diameter, d ~within , the distance d L , is

01 0

1

I

1

2

I

I

3 4 DISTANCE FROM

J

I

I

1

I

5

6

7

8

NOZZLE, FEET

COEFFICENTS FOR FIGURE 7. CALCULATED ABSORPTION NOZZLES

9

SPRAY

the calculations indicated that the chances of collision with the smaller drops was significant only just before these drops reached their turning point in the tower. It was not necessary, therefore, to consider the collision with drops moving in the direction opposite to the spray. As a matter of fact, experience indicates that this is true as the percentage of the liquid actually driven upward by a gas stream opposing a spray nozzle is relatively small, unless the spray is extremely fine and uniform. A few of the drop distribution curves are given in Figure 6 for the case of a nozzle pressure of 35 pounds per square

AUGUST, 1939

INDUSTRIAL AND ENGINEERING CHEMISTRY

inch and a counter gas velocity of 5 feet per second. In these calculations only the initial group sizes of round hundreds of microns were considered. The distribution shown in Table IV was changed so that each size represented a range equal to the interval between groups, centering about the nominal size; i. e., the 100-micron group included those from 51 to 149 microns. It did not greatly affect the accuracy of the final curves. All drops below 51 microns were neglected, since it was apparent that they could not travel more than a few inches from the nozzle even a t high initial velocities. After the determination of the drop size distribution in the tower, coefficient a for each drop size was found by the equation :

Acknowledgment The clerical assistance of several students working under grants from the National Youth Administration is gratefully acknowledged.

Nomenclature Consistent units are used in the development of the general equations. The final numerical values of the absorption coefficient are expressed in English units. A a

=

=

volume of solution crossing unit area in unit time

This is a constant, equal in these calculations to one gallon per minute per square foot. The product (k,a)i was then determined for each drop size, and a summation was made for the total volume coefficient of absorption as a function of distance from the nozzle. The final values of a and koa for the four combinations of conditions are shown in Figure 7. It appears from the curves that, for the chosen conditions, while the total number of drops decreases rapidly with distance, in front of the nozzle, the interfacial area for absorption actually increases for a distance of about 2 feet and then decreases slightly. This effect may be traced to the great decrease in the velocity of the drops, resulting in an increase in their effective concentration. The value of koa, however, decreases with distance almost linearly beyond 2 feet, owing to the decreasing interfacial velocity. The highest absorption coefficients are obtained with the lower nozzle pressure and for the higher counter gas velocity. The effect of neither pressure nor gas velocity is great, however. Hixson and Scott (11) found the volume coefficient to be inversely proportional to the square root of the distance. However, the droplets used by these authors were much larger (1800 to 2000 microns in diameter) than those obtained from a spray nozzle. In this range there is an acceleration of the drops, even a t high nozzle pressures, and the value of a would decrease with distance more than k , would increase. At low initial velocities, such as were used by Hixson and Scott, this effect would be even greater. For large drops, for which the resistance is small compared to the accelerating force, it can be shown that koa is proportional to the -0.25 power of the distance traveled. Verification of the shape of the curves given in Figure 7 by experimental data would be highly desirable. Although this cannot be given a t present, one of the authors has available some results on the absorption of sulfur dioxide from flue gases by a solution of ammonium sulfite-bisulfite in a 42-inch spray tower. The nozzles used were Spraco No. 2B, which gave a hollow cone spray. Ten of these nozzles were arranged uniformly on a circle 21 inches in diameter. The average liquid rate was 0.57 gallon per minute per square foot. The nozzle pressure was 55 pounds per square inch. A few sectional values of the absorption coefficient are shown by the indicated points in Figure 7. The crosses represent measurements made when the gas velocity was approximately 2 feet per second; the circle is the result of one measurement made a t 1 foot per second. Although the results vary among themselves, they agree in order of magnitude with the predicted values. This indicates that these commercial spray nozzles probably give nearly the same effective size distribution of drops as those on which the calculations were. made.

area of tube described by the trace of the droplet

= surface area per unit volume of scrubber; subscript

indicates value for drops of diameter of order i = diffusivity of absorbed gas

i

b C D = drag coefficient

= diameter; subscript s refers to spheres, c to cylinders, i

D

where w

1001

= h = k, = K, = g

L

=

m

= = = =

N

n

P

p~ ~ T

R r

B

= = M

= =i

T

= = = =

u

V

= =

w

=

p

e

t o group i in size distribution in a spray acceleration of gravity height of cylindrical drop gas film absorption coefficient over-all absorption coefficient length of path of drop mass moles of gas absorbed number of drops per unit volume of spray liquid total pressure partial pressure of absorbed gas mean partial pressure of absorbed gas ratio of circumference to diameter gasconstant radius of droplet density; subscript a refers to gas, 2 to liquid. absolute temperature time; subscript d refers to time required for droplet to travel one diameter, t to total time of travel interfacial velocity; subscript o refers to initial value linear velocity; subscript o refers t o initial velocity, g to velocity of gas, t to terminal velocity volume of spray per unit time per unit area

Literature Cited Arnold, H. D., Phil. Mag., 22, 755 (1911). Chambers, F. S., and Sherwood, T. K., IND. ENQ. CHEM.,29, 1415 (1937).

Diehl, W. S.,“Engineering Aerodynamics,” p. 265, New Ywk, Ronald Press, 1936. Dryden, H. R., Murnaghan, F. D., and Bateman, H., Natl. Research Council, Bull. 84, 300 (1932). Froessling, N., Gerlanda Beitr. Geophys., 52, 170 (1938). Gilliland, E. R., and Sherwood, T. K., IND. ENQ. CHEM.,26, 517 (1934).

Goldstein, S., Proc. Roy. SOC.(London), 123, 225 (1929). Haslam, R. T.,Ryan, W. P., and Weber, H. C., Trans. Am. I n s t . Chem. Engrs., 15, 177 (1923). Hatta, S., Tech. Repts. Tohoku Imp. Univ., 10,613 (1932). Hatta. S.,Ueda, T., and Baba, A., J . SOC. Chem. Ind. Japan, 37, 383 (1934). (11) Hixson, A. W., and Scott, C. E., IND. ENQ.CHEM.,27,307(1935)(12) Houghton, H.G., Physics, 2, 467 (1932). (13) Jakob, M., Trans. Am. Inst. Chem. Engrs., 34, 587 (1938). (14) Johnstone, H. F.,and Kleinschmidt, R. V., Ibid., 34, 187 (1938). (15) Kowalke, 0.L., Hougen, 0. A., and Watson, K. M., Univ. Wisconsin Eng. Expt. Sta., Bull. 68 (June, 1925). (16) Lenard, P.,Ann. Physik, 65, 629 (1921). (17) Meldrum, N. U.,and Roughton, F. J. W., J. Physiol., 80, 113 (1933). (18) Nisi, H., and Porter, A. W., Phil. Mag., 46, 754 (1923). (19) Oseen, C.W., Arkiv Mat. Astron. Fysik, 9,1 (1913).

(20) Prandtl, L., “Ergebnisse der aerodynamische Versuchsanstalt zu Gottingen,” 11, p. 28, Munich, R. Oldenberg, 1923. (21) Saito, S.,Science Repts. Tohokulmp. Univ., 2, 179 (1913). (22) Squires, L.,and Squires, W., Trans. Am. Inst. Chem. Engrs., 33, 9 (1937). (23) Whitman, W. G., Long, L., and Wang, H. W., IND. ENQ.CHEM., 18,363 (1926). (24) Zahm, A. F., Natl. Advisory Comm. Aeronaut., Tech. Report 253 (1926). PRE~ENTED before the Division of Industrial and Engineering Chemistry a t the 96th Meeting of the American Chemioal Sooiety, Milwaukee, Wis.