VAPORIZATION OF SMALL SULFUR DROPLETS H. F. JOHNSTONE AND DAVID K.EADS University of Illinois. Urbana, Ill.
An equation based on €rSssling's correlation of ~ a ~ transfer data from evaporating drops is preaented for predicting the time required for aomplete vaporiaation of liquid droplets. By application of these relationships to experimental studies, t h e d i h i v i t i e a of di-n-butylphthalate and sulfur vapola were determined. The vaporization rates of droplets of sulfur modified by the addition of bromine and iodine were slightly greater than those for pure sulfur, within the range of the diameters investigated.
with a velocity equal to that in the undisturbed flow. Their solution obviously does not hold a t high Reynolds numbers. Frassling has shown that a partial solution of Equation 3 may be obtained if it is assumed that the principal change of relative velocity from the value at the surface of the sphere to that in the undisturbed flow takes place over a distance which is small compared to the dimensions of the sphere. Then
l surface l
Nu' = 9 (So, Re'h)
(4 1 To verify this relationship, he measured the rates of evaporation
of spheres of naphthalene and droplets of several liquids in flowing gas streams. His results were expressed by means of the equation
T
HIS work waa undertaken in conjunction with a study of the combustion of aulfur in spray-type burners in which the rate of vaporieation was aonsidewd to be one of the controlling factors (4). Since the behavior of sulfur in the liquid and vapor states is complex, i t was advisable to measure the vaporization rates experimentally, so that the proper diffusivity could be determined, rather than to attempt to predict the diffusivity from the usual correlations. Furthermore, because of the pronounced effect of bromine and iodine on the properties of liquid sulfur, the effectof these modifying agents on the vaporization rates of sulfur droplets was of interest.
Nu' = 2
+ 0.552 Sc'/a Re1/¶
(5)
which is the relationship that will be used in this work (6).
INTEGRATION OF RATE EQUATION When Equation 1 is combined with Equation 5, we obtain
For a spherical droplet therefore,
RATE EQUATIONS The steady state diffusion of a vapor from a spherical droplet is yiven by
dm de
The quantity,
c
Z nZ.-.D.PMap -
When it is assumed that the temperature of the droplet is constant, which is equivalent to assuming that the vapor pressure does not change with time,
RTpBM
Z/X,is analogous to the Nusselt group, Nu
for heat transfer and is designated by Nu'. When the velocity of the dropais zero and when
=
hZ
T,
P is esaenPBM
tislly unity for materials of low vapor pressure NU'
Z
=2
Also, if the relative velocity of the droplet does not change with time, the term
(2 )
The validity of Equation 2 a t ordinary pressures and for droplets larger than 10 microns in diameter was established by Frassling ( B ) , Houghton (8),and others (IS). For relative velocities greater than 5ero, the equation that must be solved is
With theso substitutions, Equation 7 becomes:
Integration of this equation gives
where
'
de
- is the concentration gradient and dF is a unit surface bn
vector. The integral is evaluated over the entire surface of the sphere. Because of the complicated nature of the flow lines around a sphere, an exact solution for this equation is not pot+ sible. Johnstone el at. (IO) obtained an approximate solution to this equation by neglecting the diffusion in the direction of flow and assuming that the streamlines are everywhere parallel to the 1
Present address. Melloa Inntitub of Indurtrisl Researoh, Pittaburgh, Pa.
When
e =
0, corresponding to Bero relative velocity,
ye
=
21
2
For convenience Equation 11 may be written in the form Y
Equation 12 gives the time required for complete vaporization of e droplet of initial diameter 2. Since numerical calculations using 2293
2294
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 42, No. 11
surface temperature (wet bulb temperature) and the temperature of the flowing gas. Values of the Nusselt group were obtained by utilizing the principle of mathematical similarity between heat and mass transfer. Thus, the Nusselt group was obtained from the relationship Nu = 2
Figure 1.
Data of Table I
this equation are tedious, a table of values of f ( Z , e ) has been prepared (Table I), and the function is presented graphically in Figure 1. The vaporization of a falling droplet can be calculated from Figure 1 by means of a stepwise procedure regarding t,he velocity of fall as constant over short ranges of the diameter.
Table I. Values of the Function t(2,e) Z.Cm. 0.08
0.10 ..Crn.-111 3 4 5 6 7 8 9
10
20 30 40
50 60 70 80 YO
100
-J(Z,e)
1837 1667 1542 1434 1354 1258 1185 1121 727.7 539.7 429.1 356.3 304.6 266.1 236.2 212.4 192.9
1193 1106 1026 980,5 900.0 848.9 802.5 761 .O 502.8 376.1 300.6 250.4 214.7 187.9 167.0 150.3 136.7
0.06 0.04 X lOS, s q . m c-. 700.5 323.1 303.8 649.0 607.0 286.8 271.6 570.7 257.9 538.1 247.7 511.6 240.9 490.7 224.3 460.0 156.8 311.0 120.8 235.4 98.27 189,5 158.6 82.88 136.4 71.89 63.14 119.9 56.43 106.6 51.02 96.13 46.55 87.53
0.02
... ... .,. ... ... ...
:
64 55 47.14 37.63 31.43 26.89 23.62
... ... .*.
...
CALCULATION OF SURFACE TEMPERATURE OF EVAPORATING DROPLET 1x1 order to estimate the temperature of the surface of an evaporating drop, it is assumed that all the heat of vaporization is supplied by convection. In the experimental work, precautions were taken to prevent heat transfer by radiation. The rate of heat transfer is then equal to the rate of the latent heat demand, or
+ 0.552 Prl/s Re1/*
When using Equation 15 in analyzing the experimental data, it was found that the ratio Nu'lNu, which contains the Reynolds number in both the numerator and denominator, remains essentially constant over the range of diameter studied. The variation was from 1.45 a t a diameter of 0.066 cm. to 1.34 a t a diameter of 0.022 cm. in the calculations on run P-7, di-n-butyl phthalate, and from 1.27 a t a diameter of 0.083 cm. to 1.23 a t a diameter of 0.038 cm. in run P-8, pure sulfur. An average value was used in all cases. Since the order of magnitude of At is small---that is, 0.5 to 2' C.-the assumption as to the constancy of droplet temperature is not inaccurate.
EXPERIMENTAL
A schematic diagram of the apparatus used is shown in Figure 2. Nitrogen gas was fed from the cylinder through an orifice and then through a small copper cooler which served to stabilize the temperature of the incoming gas and to prevent heat conduction from the heaters to the orifice. The gas was then assed through a preheater to bring it up to the experimentar temperature Closely coupled to the preheater was the observation tube in which the droplet to be observed was suspended. This was equipped with a heater to maintain the wall temperature close to that of the incoming gas so that temperature error8 due to radiation were eliminated. The droplet was suspended from a small fiber in the center of the tube and wm photographed through windows in the wall. A glass fiber was used in the measurements on di-n-butyl phthalate and a tungsten fiber in those on sulfur droplets. The observation tube was a S/leinch wall, '/ginch inside diameter stainless steel tube, 11 inches long; it was suspended inside a 2l/pinch copper tube by means of Transite rings at the ends. A radiant heater of Chrome1 ribbon was wound in the annular space. The wall temperature was measured by means of four No. 36 copper-constantan thermocouples. The thermocouple wells were drilled tangentially into the wall to a depth of 0.5 inch so as to eliminate errors in temperature measurement due to heat conduction along the leads.
y 0 CAMERA
COOLER
-
NITROGEN
I L
PREHEATER
I ,
i!
AH, = RT2
blnp (w)
Combining Equations 1, 13, and 14 and solving for At,
D PapT ( k (%)) (%) This equation expresses the difference between the droplet
I PITOT we€ I
._C=Il=.
OBSERVATION
The heat of vaporization may be calculated from the vapor pressure by means of the Clausius-Clapeyron equation
(16)
L$IT SOURCE
Figure 2. Schematic Diagram of Apparatue The droplets to be observed were placed on a small carrier (Figure 3) which could be fitted into the tube from the side. This was essentially a tapered plug fitted with two thermocouples and with the fiber sus ended between two tungsten wires. Two such plugs were provi&d; one was a dumm with no fiber which was used while the system was being brouglt to tem erature equilibrium. The droplet carrier was exchanged for tge dummy after equilibrium was reached. The plugs were inrh thick at the
INDUSTRIAL AND ENGINEERING CHEMISTRY
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The prehester WBS eonstmeted in the m e manner as the ob=mation tube except thst the inner tube was made of O.&inch capper tubing. The oonnection to the observation tube wa4 made by means of B copper buahing tapered to obtain B tight fit in hoth tubes. Illumination of the drops waa provided by B 3W-wstt spotlight. The light besm w w filtered through 1 cm. of saturated copwr suliate solution. A Bausch snd Iamb micrographic camera, fitted with an adapter for 3S.mm. film, and a 7.5-mm. microsoope objective were wed. Satisfactory definition was obtained at. B shutter speed of aeoond. Exposures were made on regular Eastman Super XX Psnehmmatic film which waa developed in an extreme high contrast developer. Figure 4 show a typical series a i photographs. Droplet diameter8 were measured from the original negatives under a low power micrwcope equipped with B filar miammeter eyepiece. .411 measurements were made slang a diameter at right. mgles to the suspending fiber or in the direction of gas flow. The droplets were appreoiably distorted
and figure 3. Arrangsment of Trurrmocoupl~ S-mion fiber
center, 0.5 inch in diameter st the inner wall, snd were cut with B taper of 'heinch er inch They were provided with a key for alignment, and wRen in dosition the ends were Bush with the inner wall. Their heat cepaoity was emell and ioterchsnge of plum did not ~p reeiably didxrb thwmal equilibrium. The carrier t~ermoooupleswere mede ai No. 36 copper-constantan wires calibrated against staoderd thermometers. The leads extending into the gas stream were left bare for B length of about I cm. The droolet sumension frame was made bv silver solderinz
could be tied mmSs them, pe endieu1;tr to the tube Call. Tension on the fiber wss a d j u s d b y adjuating t,he position of the brass tubea. When a gl- fiber w w used, it wa4 cemented a e r w the frame with a saturated solution of sodi,im dithionatn. The
Tabla 11. Experimental Data on Evaporation of Small Droplets Run P-8.
e
