ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR A METHOD FOR CALCULATING THE RATE OF EVAPORATION AND THE CHANGE IN DROP SIZE DISTRIBUTION FOR PURE SPRAYS INJECTED INTO UNSATURATED AIR By MYRON TRIBUS JOHN KLEIN JOHN REMBOWSKI Project M992-C WRIGHT AIR DEVELOPMENT CENTER, U. S. AIR FORCE CONTRACT NO. AF 18(600)-51, E. O. No. 462 Br-1 May 1952

SUMMARY In this report the equations for the evaporation of a single water drop, a spray of uniform drops and a spray of nonuniform drops are presented and solved. It is shown that when wet-bulb depression is used as a parameter, considerable simplification occurs and the solutions may be obtained in closed form for the first two cases and may be found rapidly by a graphical method for the third. It is also shown that, given an initial drop-size spectrum, the successive spectra may be deduced almost immediately, without reference to the evaporation equations. Sample calculations are given for water sprays. Limitations and extensions of the analyses are discussed. ii

SYMBOLS AND NOMENCLATURE B = barometric pressure, lbs/ft2 D = diffusion coefficient for mass transfer ft2/hr k =thermal conductivity of the air or gas through which diffusion occurs, Btu/hr-ft2 (F/ft) Lv = heat of vaporization, Btu/lb M = molecular weight, lbs/mol N = drop distribution function defined in Eq. 15, ft-2 n = drop distribution function defined in Eq. 12, ft-1 P = pressure, lbs/ft2 r = drop radius, ft R = universal gas constant, ft-lbs/mol-~R S = r2, ft2 T = temperature, ~R W = weight of spray, lbs = radius ratio p = drop density, lbs/ft3 9 = time, hr Subscripts s = adiabatic saturation wb = wet bulb db = dry bulb = liquid v = vapor o = initial value at 9 = 0. max = largest drop iii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A METHOD FOR CALCULATING TEE RATE OF EVAPORATION AND THE CHANGE IN DROP SIZE DISTRIBUTION FOR PURE SPRAYS INJECTED INTO UNSATURATED AIR INTRODUCTION The impetus to this study was given by a problem arising in the prevention of ice accretions on aircraft. In the simulation of icing conditions it is customary to spray water droplets into the air ahead of the wind-tunnel inlet. To reduce the turbulence level in the tunnel, the drops are sprayed into the air a good distance upstream. During the travel from the nozzles to the tunnel it is necessary to know the evaporation from the spray in order to control both humidity and drop size in the tunnel test section. The solutions resulting from this study are applicable to a number of problems of technical interest. For example problems of firefighting, insecticide spraying, and paint spraying all require knowledge of the evaporation of droplets. Spray cooling towers utilize water-drop evaporation. In the case where the spray is not a pure liquid, the data of Reference 1 should be incorporated to allow for the effect of dissolved materials on the vapor pressure of the solvent. Only pure liquids are treated in the present report. FORMJLATION OF THE PROBLEM As shown in References 1 and 2, the following idealizations are accurate representations of evaporation from small drops:

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1. For small Reynolds and Grashof Moduli, the evaporation and heat transfer occur by diffusion. 2. During evaporation the drops are at the wet-bulb temperature. 3. The steady-state diffusion equation describes the heat and mass transfer. In Reference 3 it is shown that when the air is saturated surface-tension effects are appreciable and cause small drops to evaporate and condensation to occur on the larger drops. When the air is not saturated, surface tension has negligible effect on evaporation. When a small water drop is projected into still air, its velocity falls very rapidly to a low value. Therefore, for drops less than 100 microns in diameter, such as occur in icing problems and fogs, the major portion of the droplet life will be at a low Reynolds Modulus. For steady-state diffusion from a drop, a mass balance yields: _4rr2 dr = 4tDMr d P ( dQ BBR TTd In cases where the heat of vaporization is large, the omission of the thermal capacitance of the drop yields negligible error. For such a case,an energy balance yields: 4rr2pLvdr/dg = 4itkr(Td-T,).(2) The simultaneous solution of Eqs. 1 and 2 yields the droplet radius, temperature, and vapor pressure as functions of time and the psychrometric conditions far from the droplet. The solution of Eqs. 1 and 2, however, leads to the conventional equation for wet-bulb depression, and considerable simplification of the calculations is effected if Postulate 2 above is adopted and Eq. 2 utilized alone. A SINGLE DROP IN A LARGE ATMOSPHERE If the ambient temperature and humidity are unchanged, integration of Eq. 2 leads to: r2 - r2 = 2k (Twb - Tdb) (5)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Letting - = r/ro ^2=1- ^h-i. ^b. (4) 2 _ 2k(Tab-Twb) (4) d L r2 v o Figs. 1, 2, and 3 show the drop radius as a function of time for several conditions. Note the changed time scale (ten to one) for the higher temperatures. The right-hand scale of each figure is a cubic scale which shows the percentage of droplet volume remaining. In Reference 1, part II, Fig. 8 is a graph of the measured diameter of a single water drop plotted against time. The drop was initially over 103 microns in diameter, and therefore did not have a negligibly small value of Grashof Modulus. Nevertheless, the drop size squared plotted against time yielded a straight line whose slope was only 10 per cent larger than predicted by Eq. 4. A UNIFORM SPRAY OF DROPS IN A LIMITED ATMOSPHERE During the evaporation of a spray of drops, the ambient humidity and temperature will change. The integration of 2 then requires that the variation in wet-bulb depression be specified. Consider a spray of uniform drops having radius ro. Let t = r/r 0 Then if W~ is the initial spray density, the spray density at any time is given by Wp = wo5. (5) If W is the initial water-vapor density, the water-vapor vyo density at any time is given by W = Wv, + W, (1 - 3) * (6) It is convenient to define = Wo + Wvs - Wvo (7) S —.. —----------

LIFE OF A WATER DROP IN A.50 T /75~F. _ LARGE VOLUME OF UNSATU- 100 RATED AIR. 0 0 2610246 20 w I SQ_ Q2 0.4 Q6___ ______ \HUMIDITY- 0.8_____________________ 10 4 22 0'cI. TIME IN SECONDS FIGURE I

| | Tc 230 F. | l l l l l l | | LIFE OF A WATER DROP IN A 50 --- LARGE VOLUME OF UNSATU- 100 RATED AIR. 40 X S- w 50 0 zr. HUMIDITY <4 20 \ 28 32 3 4 3 ~ ~ ~ ~ ~ ~ ~ TMINSCN0SFIUE 20 ___- \_ — --- - - - - - -- -- - B\ \ ~o 0 0 4 0 4 8 12 16 20 24 28 32 36 40 TIME IN SECONDS FIGURE 2

ToO=-4~F. LIFE OF A WATER DROP IN A LARGE VOLUME OF UNSATURATED AIR. 450 0 __ _________ ______ __ ) 0 0 20 Q 40.4 0.6 6 HUMIDITY-0.8 1 0 o a.s,4s s ~Tk I ON F 0 20 40 60 80 100 120 140 160 180 200 TIME IN SECONDS FIGURE 3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where Wv = water vapor density at adiabatic saturation and radius ratio required to make W = W in Eq. 6. An inspection of the psychrometric charts shows that Twb and Tdb vary linearly with the absolute humidity during adiabatic saturation. (For water the variation in Twb is almost zero.) Thus one may write: Tdb = Tdbo + (Tdb,s Tdb o) (8) and Twb =Twb + (Twb,s Twb.,o) 93 (9) Substitution of Eqs. 8 and 9 into 2, noting that Twb,s = Tdb,s, yields d!= _ k(Tdb,o - Twb,o) (10) The integral of Eq. 10 (for = 1 at Q = O) is: k(Tdb,o - Twb.o) 1 - 3 1 ( 2+s +2)l - )2 _ ______...., _ ____ 5)n r02 p L 2s (s+ + 1)( s)2 - tan-1 S2 + ftan-1 s (11) - tan_ vs+ tan A graph of Eq. 11 is shown in Fig. 4. The parameter on each curve | is (l - S3)l, or the ratio of the weight of initial spray to the weight of water required for saturation. Figs. 5 and 6 show specific solutions to spray equations of interest in icing work. 7 - -

1.4 — J J WEIGHT OF SPRAY 2 I.I WEIGHT OF WATER REQUIRED FOR SATURATION (I) z.0. 0.9 0.4 r/r0 FIGURE 4 Oo~ ~~~~~~~~~~~~^' FIUR 4

DROP SIZE VARIATION WITH TIME FOR SPRAY INJECTED ___ INTO UNSATURATED AIR AT _75F. AND RELATIVE HUM- _ IDITY 0.6 AND 0.2. h=.6 T 750 __ 5 h0~ s^^ ___ ___ ___ ___I- WATER SPRAY - Igm/ m3 UNIFORM DROP SIZE= 50/ 40 \ =s 4 h=. 2 3 %Th=.661 T =73.30 z 0 o-_ h=.258 h ___ =:.696 T 72.20 o 30 ^ 20 --- --- --- --- --- <h 269 -—: —---— h:.7 11 T:71.81.20. I0 \h=.272,'\h=.717 T=71.6_~ 0 1 2 3 4 5 6 7 8 9 10 TIME IN SECONDS FIGURE 5

DROP SIZE VARIATION WITH TIME FOR SPRAY INJECTED INTO AIR AT 23~F., RELATIVE HUMIDITY 0.6 8 0.2. _ h=.60 T=230F.__ ___ 50 h=i\^^,_______, |_ _!_ IWATER SPRAY - I gm/m3 UNIFORM DROP SIZE= 50O, 0 h= 37 2h=.80 T=21.3~ _ 40 04 3 40 N -- -- -- -- N - -- - -- -- -- -- - -- -- -— TIM E1560 SEC. ~' "'"'- V-V —-- o0 a. 20 __ ___ __ ___ h=.5695 0 4 8 12 16 20 24 28 32 36 40 TIME IN SECONDS FIGURE 6

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A NONUNIFORM SPRAY OF DROPS IN A LIMITED ATMOSPHERE The conventional method for the presentation of information concerning sprays of droplets is to graph the spray distribution function n(r) (Reference 4). The distribution function has the characteristic that n(r) dr is the number of drops having drop radii between r and r + dr. From the drop distribution function, one may calculate: 00 Number of drops = n(r)dr (12) oo Total droplet surface = 4r2n(r)dr (13) 00 Total volume = 4 jt43n(r)dr (14) Fig. 7 shows a typical graph of n(r) for a commercial spray nozzle. As will be shown later, it is convenient to define a new parameter S = r2 and a new drop distribution function N(S) such that N(S) dS is the number of drops having values of r2 between S and S + dS. The spray characteristics are then computed as: 00 Number of drops = N(S)dS (15) 0 Total drop surface = j 4ASN(S)dS (16) 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ ^ -11-_ —-----------------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Total drop volume = O _ s3/2N(S)dS (17) 3 Now Eq. 2 applies to any two drops, say drops which are identified by subscripts 1 and 2. Rewriting Eq. 2 in terms of S, dSl dS2 -2k(Tdb - Twb) --- ~~= =..........,~~ ~ (18) d9 dd9 Pd Lv Since the right-hand term of Eq. 18 is the same function of time for all drops (if the spray is well mixed), it follows that dS/dQ is the same for all drops. Fig. 8 is a replot of Fig. 7, wherein the distribution function N(S) is shown. It follows from the conclusion above that all SUCCESSIVE DROP DISTRIBUTIONS ARE OBTAINED FROM FIG. 8 BY A LINEAR SHIFT ALONG THE S AXIS. IN THE CASE OF COMPLETE EVAPORATION OF SOME DROPS THE CURVE TO THE LEFT OF THE ORIGIN IS OMITTED (S<O). In view of the above conclusion, the rate of evaporation from the spray may be computed by calculating the variation in drop size of the largest drop and relating all other drop to this maximum drop radius. 2 For example, let S = r for the largest drop initially max o 0 present in the spray. Then by graphical integration one may obtain the weight of spray as a function of Smax: Smax W (Sm ) = N( S)S S/2dS. (19) max 3 0 A graph of Wf versus Smax for the data of Fig. 7 is shown in Fig. 9. The variation in wet-bulb depression as before is a linear function of the weight of spray which has evaporated; i.e., T -T (T - T,s -,oW T wv, -,,o (20) db wb wb,o dbo W - W 12

GRAPH OF n(d) FOR DISTILLED ___ ___ ___ ___ _ WATER SPRAYED FROM A COMMERCIAL SWIRL CHAMBER NOZZLE,2000 Tu __ I _ I_ I_ __ _ ___ I 1 psia INTO STILL AIR. 0CD ~O 10 — - --- - -- -- - -- - TOTAL NUMBER OF n(d)dd DROPS PER cc. J E 9/ (, 8 z i NOTE: THESE DATA WERE OBTAINED *0 4___~~~ __l _ l l t \ l lTHROUGH THE COURTESY OF MR. J. A. _ | | l l~ l~ l I I CONSIGLIO AND ARE FROM HIS UNPUBLISHED DOCTORAL THESIS. 0? \o _I _ _ _ _.... I,/ 0 2 4 6 8 10 12 14 16 20 22 24 26 28 30 32 34 36 3840 DIAMETER, MICRONS FIGURE 7

DATA OF FIGURE 7 REPLOTTED AGAINST S(=r2). C T10 — -- - -- - -- -- - -- - -- - TOTAL NUMBER OF /N(s)d OF I 0 DROPS. M Is F 0 c) e7 (I) w4 2: 0 40 80 120 160 200 240 280 320 360 S, MICRONS2 FIGURE 8

WEIGHT OF SPRAY NOT YET EVAPORATED AS A FUNCTION OF Smox (SQUARE OF LARGEST DROP RADIUS). 1.0 (L / E E.6 a. cn' iL0.4 INITIAL SPRAY WEIGHT- I gram/meter3.2 0 40 80 120 160 200 240 280 320 360 Snx, MICRONS2 FIGURE 9

VARIATION IN WET BULB DEFRESSION WITH Smox FOR I GRAM SPRAY / METER3 ___ I0 ~,-T = 75 F Co uj~_ ___ -— 6 -- --— 60% RELATIVE HUMIDITY _T ic a. 05 CD -J 34 2 — -- -- -- -- -- -- -- -- -- -- -- -- — ~T0230 F _ t _ __^ 60% RELATIVE -- --- -- -- -- --- -- --- --- -- -- --- ^- HUM IDITY 0 40 80 120 160 200 240 280 320 Smox, MICRON2 FIGUR 10 FIGUREZ I0

28 —! — I I I I VARIATION IN MAXIMUM DROP SIZE(S=r2) WITH TIME FOR I GRAM SPRAY/METER3. 24';20.. 230F. S \60% R.H. w -J z w 16 E I ^. 750 F. <N 8 --- --- -- --- -- 60 oR.H.4 0.1.2.3.4.5.6.7.8.9 1.0 Smax /Smaxo FIGURE II

SPRAY DISTRIBUTION FUNCTION VARIATION WITH TIME I __ -— 9:0 Sec. --.125 l" To=75 F.______- -- 25 t" ho=60%.5 e W = 1.0 gm/m3 9 7 ---:1.0. 9/ I \ "\ " I | —— 9o0 Sec. 7 =- - -0 —-^ —- - -— _8:o.ss5 T023~F - 6 /___ ___I 1'_^\ --- ~=0.85"i ho=60% /9 1 X - -- 3.0 " w =1.0 g/m3,' \ \ ^ 3 I 4I - _ 2.___ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 r FIGURE 12

VOLUME DISTRIBUTION OF THE SPRAY AS A FUNCTION OF TIME. 12C ---- =O _jjj__ _____ __ I __ __ ____ __ 12 = —-820 f,'' -,'\. —. 8=.125 / l -,, ro'750OI -—' —8 =.25 =75O /89=.5 h060% I I - ^ l-^ -.5 —-... TI 23 —- -— 8 =85 20 __h = 60% O4 _ --- _:1. / _0 _ i _ -- Ir, MICRONS FIGURE 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The wet-bulb depression can therefore be presented as a function of S as shown in Fig. 10. Graphical integration of Eq. 18 in the form: max S max d s 2 k 9 (21) ax(T db Twb) Pd v yields the results shown in Fig. 11. The spray distribution function can then be read from Fig. 8 by displacing the origin an amount Smax-Smax, The drop distribution function in the conventional representation, n(r), is shown for various times in Fig. 12. Fig. 13 shows the volume distribution function 4/35n(r)r3 versus r at successive times. DISCUSSION The analysis presented above does not include the effects of surface tension or electrical charge on the vapor pressure of the droplet. These effects are negligible except in cases where the drops exist in saturated or slightly supersaturated air. For such cases, Langmuir (Reference 3) has shown that the small drops tend to evaporate, while the large drops increase in size. Eventually only a few large drops would be produced by this process, were it not for other effects such as falling out of the large drops under the influence of gravity or breaking up under stresses produced by air currents. When the weight of spray introduced is sufficient to saturate the air, the problem may be divided into two parts. In the first part of the evaporation process, the analysis given here may be applied. This analysis leads to a spray distribution in saturated air. Using this latter distribution one may then proceed, with the method of analysis by Langmuir (Reference 3), to calculate the shift in droplet spectrum. Another case often occurs in which the droplets do not consist of pure liquid but are composed of a volatile solvent and an involatile solute, as for example in paint spraying. Reference 1 shows how the appropriate wet-bulb depression must be found for each concentration of solute, if the solute serves to lower the vapor pressure of the droplet. The method of attack in this case is not as clear, since although each droplet starts at the same concentration of solute, after a short while the concentration is different for each drop. 20

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This work was carried out for the Wright Air Development Center, U.S. Air Forces under contract with the Flight Research Laboratory, whose support and permission to publish this report are appreciated. 21

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN REFERENCES 1. Ranz, W. E., and Marshall, W. R., Jr., "Evaporation from Drops" Chem. Eng. Prog. Part I, March 1952. Part II April 1952 (p.173). 2. Kinzer, G., and Gunn, Ross, "The Evaporation, Temperature and Thermal Relaxation Time of Freely Falling Water Drops" Jour. Meteor., V. 8. No. 2., 71-83(April, 1951). 3. Langmuir, Irving, "Super-Cooled Water Droplets in Rising Currents of Cold Saturated Air" Parts I and II, General Electric Co., Report No. RL-223, October 1943, August, 1944. 4. Mugele, R. A., and Evans, H. D. "Droplet Size Distribution in Sprays" Ind. and Eng. Chem., 1317-1324 (June 1951). 22