literature Cited
Abegg, C. F., et al., A.I.Ch.E. J . 14, 118 (1968). Beck, W. F., Sc.D. thesis (Ch.E.), hlassachusetts Institute of Technology, Cambridge, Mass., 1964. Belyustin, A. V., Portnov, V. N,, Soviet Phys. Crystallog. 7, 214 (1962). \ - - - - I -
Beiinema, P., “Crystal Growth,” H. Peiser, ed., p. 413, Pergamon Press, Sew York, 1967. Bennema, P., Ph.D. thesis, University of Delft, Delft, Holland, 1O f i i
Han. C. D.. Skinnar. R.. 61st A.1.Ch.E. National ?\leetine. -, Hbuston, Tex., February 1967. Keenen, F. G., France, W. G., J . A m . Cerurn. Soc. 10, 821 (1927). Komarova, T. A., Figurovski, N. A., Zh. Fiz. Khirn. 28, 1774 (1954). Kossel, W., Nachr. Ges. Wiss. Gottingen, Math. Phys. Klasse 1927, 135. McCabe, W. L., Stevens, R. P., Chem. Eng. Progr. 47, 168 (1951). RIullin, J. W., Garside, J., Trans. Inst. Chern. Engrs. 45, T285 (1967a). hlullin, J. W., Garside, J., Trans. Inst. Chern. Engrs. 45, T291
RIasq.. - , 1968. ~ - - -
Frank, F. C., Discussions Faraday S O ~5,. 48 (1949). Furman, N. H., “Standard Methods of Chemical Analysis,” p. 1007, Van Nostrand, Princeton, N. J., 1962. Gunther, O., cited by Van Hook, A., in “Crystallization,” p. 186, Reinhold, Iiew York, 1961.
-.
(1967hl ~
Beii;yek., P., Phys. Status Solidi 17, 555 (1966a). Bennema, P., Phys. Slatus Solidi 17, 563 (196813). Bennema, P., et al., Phys. Status Solidi 19, 211 (1967). Booth, A. H., Buckley, H. E., Kature 169 (4296), 367 (1952). Bransom, S. H., Brit. Chem. Eng. 5,838 (1960). Brice, J. C., J . Crystal Growth 1, 218 (1967). Buckley, H. E., 2.Krist. 73,443 (1930). Burton, U’.K., Cabrera, K., Frank, F. C., Phil. Trans. Roy. SOC. London A243, 299 (1951). Cahn, J. W., Hillig, W.B., Acta Met. 14, 564 (1966). Denk, E. G., Jr., hl.S. thesis (Ch.E.), Tufts University, Medford,
1
1,.
Nielsen, A. E., “Kinetics of Precipitation,” p. 46, Pergamon Press, New York, 1964. Paine, P. A., France, W. G., J . Phys. Chern. 39,425 (1935). Portnov, V. N., Soviet Phys. Crystallog., 11, 774 (1967). Portnov, V. N., Belyustin, A. V., Soviet Phys. Crystallog. 10, 291 (1965). Randolph, A. D., Larson, &I.A., A.I.Ch.E. J . 8 , 639 (1962). Spangenberg, K., 2. Krist. 61, 189 (1925). Stranski, I. N., 2.Physik. Chem. 136, 259 (1928). for review May 9, 1969 RECEIVED ACCEPTED February 27, 1970
Study supported by the National Science Foundation through Grant GK 1015. The financial assistance is gratefully acknowledged. Paper presented a t 64th National Rleeting, American Institute of Chemical Engineers, New Orleans, La., March 1969.
Rates of Evaporation and Condensation between Pure Liquids and Their Own Vapors Jer Ru Maa Distillation Research Laboratory, Rochester Institute of Technology, Rochester, -V. Y . 14623
In the process of evaporation and condensation, the random motions of the vapor molecules are distorted by their group streaming. The effect of this mass vapor movement on the rate of phase change was examined by using the jet stream tensimeter and Schrage’s theory was found satisfactory over the tested range. An approximate method i s suggested for considering the motion of the phase boundary due to the loss and gain of surface liquid in the heat transfer calculation. The agreement between the theoretical rates of evaporation and condensation so computed and those observed experimentally confirms with new certainty the unity of the evaporation and condensation coefficients of common liquids.
T H E EVAPORATIOK or condensation coefficient of ordinary liquids is unity or nearly so and the rates of these processes depend on two factors (Hickman, 1954, 1965; Alaa, 1967, 1969), the effective pressure of the vapor and the true temperature of the liquid surface. The effective pressure of the vapor is complicated by the mass movement of vapor molecules to or from the liquid surface. Direct readings of surface temperature cannot yet be made because of steep thermal gradients beneath the liquid surface. During the process of evaporation (or condensation), there is always a mass movement of vapor molecules from (or toward) the vapor-liquid interface. Schrage (1953) considered the effect of this mass vapor movement and derived a correction factor for the calculation of the net rate of phase change. This factor was adopted by many workers in this field
(Bonacci and Eagleton, 1966; Jamieson, 1965; Maa, 1967, 1969; Mills, 1965; Nabavian and Bromley, 1963). However, some of the assumptions used in his derivation were disputed by others (Standart and Cihla, 1958). Because of the importance of this method of correction to the calculated rates of evaporation and condensation, it is desirable t o examine and verify it by simple experiments. Our first objective here is to do so by using the jet stream tensimeter. Calculations for thermal gradients in the liquid have recently been offered for a liquid stream in laminar flow exposed t o vapor for -0.001 second (Maa, 1967, 1969). The yields observed experimentally for many common liquids including water agree with values calculated for a n evaporation coefficient of unity with, however, increasing divergence as the total rate of evaporation or condensation is increased-that Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
283
-
Pm pb
c
I I
0.29 A
,lll
2
4
6
8
Figure 1.
2
0 t ,
4
6
8
sec. x 10-4
W/wb ratio as function of exposure time
0 Experimental data Jet length. 0.307 0.715 cm 0.1 05 cm Average jet diameter. 0.1 0 3 Average Tb. 6.3'C Experimenting time. 4 3 4 2 - 1 4 8 2 5 sec Curves Calculated values, Tb = 6.30°C A. Using Schrage's correction B. Using = 1
-
-
r
is, the evaporation rate is greater than calculated and the condensation rate is correspondingly less. A correction that has not been applied and one that should operate in opposite and appropriate directions in evaporation and condensation is for the attrition of the evaporation layer and for the buildup of the condensed layer, the former facilitating heat transfer and the latter impeding it. This paper suggests a method of correction for these changes in thermal skin thickness and demonstrates that the theoretically deduced curves so adjusted agree closely with the experimental readings. Examining Method of Vapor Correction
Since the evaporation coefficient of ordinary liquids is unity, the rate of molecule exchange between a free liquid surface and its vapor a t equilibrium is the same as the striking rate of vapor molecules a t the interface given by the
4s; 7
kinetic theory of gases, which is p m
When the
system is not a t equilibrium, the net rate of condensation or evaporation is described by the modified Hertz-Knudsen equation = plr
4 F - d+ ART 17
rpm
TRT,
(1)
where the factor I? recognizes the effect of the mass movement of the vapor. The striking rate is increased (or decreased) because of the net velocity of vapor toward (or from) the liquid surface. This correction was derived by Schrage (1953). It is
r
= ~~p
(-
M 2RT,
u'> -
-
where U is the average velocity of vapor molecules from the phase interface. A method of calculating the liquid surface temperature and the rates of evaporation and condensation based on Equations 1 and 2 was given previously (Maa, 1967, 1969). This method is further refined in this paper. 284 Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
The experiments here were designed to examine Schrage's method of computing the effect of mass vapor movement on the rate of phase change. Working Fluid. The measurement of the surface temperature of an evaporating liquid is difficult and the correctness of the theoretically calculated values depends largely on how close the assumed mathematical model is to reality. To study the effect of the mass movement of the vapor molecules on the rate of evaporation exclusively, this uncertainty on the surface temperature is substantially reduced by evaporating a liquid of relatively low vapor pressure. The liquid used for this purpose was p-cymene, which has a vapor pressure of 0.39 mm of Hg a t 6.3"C. This vapor pressure is high enough to give sufficient amount of distillate a t a reasonable duration of measurement (1 to 4 hours), and low enough to give slow evaporation rate and consequently low surface cooling, so that the uncertainty in the thermal gradient calculation causes little error in the computed rate of distillation. This is shown by the small changes of the theoretically calculated E/wb ratios with respect to time a t exposure times longer than 0.2 X lo-* second in Figure 1. Any significant discrepancy between the experimentally measured rates of evaporation from the theoretical curves is due to the method of correction for the mass movement of the vapor used in the calculation. This is shown by the large distances between the curves calculated using Schrage's correction factor and those without, as shown in Figure 1. Apparatus. The experimental apparatus used is the jet stream tensimeter (Hickman, 1954, 1965; Maa, 1967, 1969). The heart of the system is the jet assembly shown in Figure 2. The liquid from a large reservoir is pumped through a temperature controller before issuing from the ejecting orifice, e. Air is excluded from the system by a vacuum pump. The pressure in vapor space a is controlled by pinchclamp c , and is measured by an inclined oil manometer through port d. The pressure of vapor in space 6 is also indicated by manometer and is controlled to be equal to that of vapor space a by clamp f , so that vapor leakage through the annular gap between the liquid jet stream and the receiving orifice, r, is
1
E L
I
a
Bl-
2"
0.2
0
-
0
L
4
temperature, "C
Figure 3.
Figure 2.
reduced to a minimum. The distillate from the jet stream is collected in a cold trap situated beyond clamp c and its amount is measured by a buret a t the base of the trap. The inclined manometer used for measuring p,, the pressure in vapor space a, was set at an angle of approximately 6' to the horizontal. This angle was checked before each experiment by calibrating its reading against a vertical manometer. The fluid used in the manometers is dioctyl phthalate with a density a t room temperature determined to be 0.985 gram per cm8. Hence each centimeter on this inclined manometer is equivalent to a pressure difference of about 0.0725 mm of Hg. The reference vacuum for the manometer system was provided by a diffusion pump. The effective length of the liquid jet stream, or the distance between the ejecting orifice e and the receiving orifice r , was measured by a cathetometer during the run. The diameter of the liquid jet of p-cymene issued from the glass nozzle used in this work was approximately 1.05 mm. The exact values of jet diameter at different temperatures and velocities were determined by focusing its image, along with two glass rods of similar and known diameter on both sides, through a lens to a distant screen, which enlarged the dimensions more than 40 times. Vapor Pressure of p-Cymene. The p-cymene used in this work was redistilled in the laboratory. Since the saturation vapor pressure is one of the most important factors to the rate of evaporation, its values within the temperature range of interest of this work were redetermined also using the jet tensimeter. The speed of the liquid jet in the vapor pressure determination was 2060 cm per second, with an exposed length of 0.74 cm. Air was excluded from the system and clamp c in Figure 2 closed completely. Since there was no net evaporation or condensation, the surface of the liquid jet stream was in equilibrium with the vapor in space a, and its temperature was the same as that of the bulk liquid, which was measured by a thermometer upstream. The jet housing of this work was not thermally insulated and there was a temperature gradient in vapor space a. It was assumed that the effective vapor temperature at the vapor liquid interface is the same as that of the liquid. The pressure of vapor space a-the saturation vapor pressure of the jet liquid-was measured by the inclined manometer at various liquid jet temperatures. Different pressures of vapor space b were used in the determination, ranging from 25 to 75% of the presssure of vapor space a. The variation of the pressure in b within this range caused no significant difference in the pressure reading of space a. This shows that the leakage through the annular gap between the liquid jet stream and receiving orifice I was small enough not to cause significant error in
Values determined using jet tensimeter of this work
0 Data b y Linder ( 1 9 3 1 ) h Value used b y Voitkevich (1 9 6 3 ) X Calculated value (Dreirbach, 1 9 5 5 )
+ Jet assembly
Vapor pressure of p-cymene
Calculated value (McDonald et al., 1 9 5 9 )
the pressure reading of space a. This leakage was also large enough to bleed off any uncondensable gas generated in the system, because the pressure of space a stayed constant. The measured vapor pressure data of p-cymene were plotted in Figure 3 in comparison with some data found in the literature. A total of 181 vapor pressure readings was taken within the temperature range of 5" to 16.5"C. The deviation between these data points and the best line drawn through them never exceeded 0.001 mm of Hg. Only this line is shown in Figure 3 for reasons of clarity. The saturation vapor pressures of distilled water were also determined by a jet tensimeter. The deviations from the handbook values were random and within the uncertainty of the manometer and thermometer readings (less than 0.5%). This shows that the jet tensimeter can be used as a convenient tool for determining vapor pressures of liquids. If the ambient temperature is much different from that of the jet liquid, a thermal jacket is desirable to keep the wall of the jet housing a t the same or slightly higher temperature than that of the jet liquid. Experimental Results. The experimentally determined evaporation rates a t moderate p,/po ratio are believed to be more reliable, because the relative error of the measured pn by the inclined manometer is larger when pm is small; and when p,/pb is large, the net rate of evaporation is low and therefore the relative error of the measured rate of evaporation is higher. The experimental results a t two different p,/pb values were plotted in Figure 1 as G/wb us. the time of exposure, where W is the average rate of evaporation over the exposure time of the liquid jet, and wb is defined as pb
d2&;
These experimental results were compared
with the theoretical curves ( A ) based on the Schrage method of correction for the effect of vapor return and the theoretical curves ( B ) assuming a correction factor of unity (Maa, 1967, 1969). The agreement between the experimental data and curves A suggests that Schrage's method of computing the rate of vapor return to an evaporating liquid surface is satisfactory. Refined Thermal Gradient Calculation
I n previous work (Maa, 1967, 1969), the liquid surface temperature during evaporation or condensation was calcuInd. Eng. Chem. Fundam., Vol. 9, No. 2, 1 9 7 0
285
I
I
I
1
I
I
I
I
@ = tA l w d t
c1
's! X
was used to compare with the experimentally measured values. The agreement was satisfactory, but there was some discrepancy which grew larger a t higher mass transfer rates (Figure 4). I n the process of evaporation, the vapor-liquid interface is moving toward the liquid side because of the loss of surface liquid. This causes higher surface temperature and therefore higher rate of evaporation. The reverse is true for condensation. This velocity of the phase boundary is related to the rate of mass transfer as
t
N
€. 0 d
IS tl
w
h
P
phlr
u = - = -
(Tt? - T,)
(9)
When this factor is taken into consideration, Equation 3 becomes
bT - + u - = =bT dt ax
A T , "C Figure 4.
Condensation and evaporation rates Qf water as functions of AT
sec, Ta (for condensation) = 0.78'C t = 0.95 X Experimental data 0 Condensation h Evaporation Theoretical curves A. Evaporation, without correction for loss of surface liquid 6. Evaporation, with correction for loss of surface liquid C. Condensation, without correction for gain of surface liquid D. Condensation, with correction for gain of surface liquid
at
b2T 6x2
a-
=
-
2)
(-I-) ax bT b2T
Equation 10 becomes
(3)
and the boundary conditions are
k
a' = a
bX2
ax2
lated from the assumption that the latent heat was supplied or removed through the liquid by conduction. The Fourier's conduction equation is
bT -=
If one defines
b2T
This looks similar to Equation 3, except that a' is not a constant but a function of 2: and t. However, since the effect of the movement of the phase boundary on the rate of mass transfer is not large (less than 5% in the present instance), it is perhaps satisfactory to consider a' as a constant. The value of a' is estimated by first assuming
-1- t$)Jg)&
bT b2T ax bx2
=
(13)
and then taking its averaged value over the time of exposure. Combining Equations 6, 9, 11, and 13, the estimated a' becomes
h(T1, - T,) = h,w
(4)
The solution of Equation 3 for the case of constant h, the interfacial heat transfer coefficient, was shown to be and the adjusted thermal conductivity is
k ' = k . -a' a
and
Since w and T,,are variables, as shown by Equations 1 and 5, h is not a constant. Hence, its averaged value,
h
=
+l
h dt
(7)
was used for the calculation of liquid surface temperature and mass transfer rates at various times of exposure. The time averaged mass transfer rate, 286 Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
The values of a' and k' were first computed for the time of exposure using a step-by-step method; they were then used instead of a and k for the calculation of surface temperatures and mass transfer rates. These calculated rates of evaporation and condensation for water were plotted against A T , the absolute value of Tb - T,, in Figure 4, in comparison with the experimental data and theoretical curves of the previous work (Maa, 1969). In this figure, the p , and T , values for evaporation were always made the same as those for condensation of the same AT. These curves calculated using adjusted thermal diffusivity and conductivity agree better with the experimental
t
data than those computed previously. This demonstrates that using constants a’ and k’ and the method of estimating their values reported here are satisfactory.
v
Conclusions
a
w
The Schrage method of correction for the mass vapor movement is verified experimentally a t moderate p,/po ratio. The experimental rates of evaporation and condensation coincide with the theoretical values corrected for the movement of the phase boundary due to the loss and gain, respectively, of surface liquid. These not only confirm the assumptions for the heat transfer calculation but also establish with new certainty that the evaporation and condensation coefficients are unity for common liquids. Nomenclature
h
=
p, p,, R T Tb T, Tt, T, AT
= = = = = = =
interfacial heat transfer coefficient, cal sec-I cm-2 deg-I h = time average of h h, = latent heat of vaporization, cal g-I k = thermal conductivity of the liquid, cal sec-I cm-I deg-l M = molecular weight, g mole-‘ pb = saturation vapor pressure corresponding to Tal dyne
ern+
pressure of vapor region saturation vapor pressure corresponding to Ti, gas constant, 08.314 x lo7 erg deg.-’ mole-’ temperature, K bulk liquid temperature saturation temperature corresponding to p , liquid surface temperature = temperature of vapor region = the absolute value of Tb - T ,
= exposure time, sec
u = average flow rate
x a
r P
of gas molecules from phase interface, cm sec-’ = velocity of liquid, cm sec-I = mass rate of evaporation or condensation, g ern+ sec-1 = time average of w = distance, directing from phase interface to liquid region, cm = thermal diffusivity of the liquid, cm2 sec-1 = correction factor, dimensionless = density of liquid, g cm-a
literature Cited
Bonacci, J. C., Eagleton, L. C., OSW Symposium, Rochester, N. Y., NOV.13-15,1966. Dreisbach, R. R., Advan. Chem. Sw.,No. 15 (1955). Hickman, K., Ind. Eng. Chem. 46, 1442-6 (1954). Hickman, K., Proceedings of First International Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965. Jamieson, D. T., 3rd Symposium on Thermophysical Properties, Purdue University, Lafayette, Ind., March 1965. Linder, E. G . , J . Phys. Chem. 35,531 (1931). Maa, J. R., IND. ENG.CHEM.FUNDAM. 6, 504-18 (1967). Maa, J. R., IND. ENG.CHEM.FUNDAM. 8, 564-70 (1969). McDonald, R. A., Shrader, S. A., Stull, D. R., J. Chem. Eng. Datu 4, 311-13 (1959). Mills, A. F., Technical Report on NSF GP-2520, Ser. 6, Issue 39, University of California, 1965. Nabavian, K., Bromley, L. A., Chem. Eng. Sci. 18, 651-60 (1963). Schrage, R. W., “Theoretical Study of Interphase Mass Transfer,” Columbia University Press, New York, 1953. Standart, G., Cihla, Z., Collection Czechoslov. Chem. Commun. 23, 1608-18 (1958). Voitkevich, S. A., Zh. Fiz. Khim. 37 ( 5 ) , 1029-36 (1963). RECEIVED for review May 28, 1969 ACCEPTED February 25, 1970 Work aided by a grant from the Office of Saline Water, United States Department of the Interior.
An Improved Equation of State Otto Redlich and Victoria B. T. Ngol Inorganic Materials Research Division, Lawrence Radiation Laboratory, and Department of Chemical Engineering, University of California, Berkeley, Calif. 94790
Repeated attempts have been made in the last 20 years to improve the equation of state of Redlich and Kwong by introduction of a third parameter. A new improvement systematically attacks the principal shortcoming of the old equation, its failure near the critical point.
Tm
GENERAL PROBLEM of an equation of state suitable for practical applications has been repeatedly discussed. Various authors (Chueh and Prausnitz, 1967, 1968; Joffe and Zudkevich, 1961; Wilson, 1966; and others) applied and modified the equation of Redlich and Kwong (1949). The two main problems have always been to make the equation more flexible by introduction of a third parameter, and to obtain an improved representation of the neighborhood of the critical point. At the same time, the good behavior of the old equation 1
Present address, Shell Development Co., Emeryville, Calif.
at high pressures and a t high temperatures was to be conserved. Any new equation should also be cubic in the volume, a t least in its principal terms. I n two previous attempts (Redlich and Dunlop, 1963; Redlich et al., 1965), the improvement was achieved by adding a function of temperature and pressure to the compressibility factor ZK computed by the old equation. Stringent conditions for this function could be formulated and the simplest suitable function resulted from a rather cumbersome comparison with observed data. The disadvantage was the complicated nature of the additional functions, although they Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970 287
