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Temperature determinations of the vapor—liquid interface surface temperature determinations
Temperature Determinations of the Vapor-Liquid Interface Surface Temperature Determinations K E N N E T H H I C K M A N AND W I L L I A M KAYSER t Distillation Research Laboratory, Rochester Institute of Technology, Rochester, New York 14614 Received February 28, 1975; accepted June 17, 1975
The difficulty in determining the temperature precisely at the interface of a liquid with a gas, usually air, is a common experience, brought to our notice again by Johansson's Letter-to-the-Editor (1) and the reply by Cini el al. (2). Both letters cover real situations confronting the experimenter; the liquid may be volatile or nonvolatile and in contact with air or other gas transporting varying quantities of vapor, often while temperature differences with the environment are accommodated. Physical techniques and expertise are debated which permit approach to measurement of true interfacial temperature. The limiting difficulty is that placing a sensor no matter how tiny in the liquid surface measures an average boundary layer temperature but not necessarily Ttr, the temperature of the instant top row 2 of molecules. If this could be done, it would interfere with and displace the top row, affording a classic example of the Schroedinger paradox that "to measure is to disturb." However, there is an indirect method of great precision that can be applied to volatile liquids if the operation can be conducted in a closed system with exclusion of air or other gas. Proof will be offered below that the surface temperature Ttr of a liquid in contact with its pure saturated vapor will be at Now at Exxon Nuclear Co., Inc., Richland, Washington ~A momentarily precise b u t statistically diffuse enfitity (3, 4).
substantially the mean equilibrium boiling point despite minor fluctuations in heat input to the surface. There are the added stipulations that the vapor pressure shall be above 10 Torr and the rate of evaporation or condensation shall not be unusually large. If the liquid is water and the measurements are made under ambient conditions, the surface temperature can be calculated from the barometric reading and the Steam Tables (5). In the absence of a barometer, or if, in the case of other substances, reliable liquid-vapor pressure curves are not available, the interfacial temperature will be the same as indicated by a thermometer (or equivalent) situated in a suitable (6) reference boiler maintained at the same pressure, e.g., included in the vapor manifold. In real systems, the temperature of the test liquid is likely to fluctuate. When the liquid cools, there is net condensation of vapor on the surface; when the temperature rises, there is net evaporation. Since traces of foreign gas can buffer condensation, it is preferred to keep the liquid slightly superheated (a real input of 3-8 W/500 ml, surface area ~ 1 0 0 cm 2) to ensure a net loss of vapor and a further scavenging from the system of any residual foreign gas. Experience shows that the average (inhomogeneous) temperature of the bulk liquid will then be 0.1-0.4 ° above true b.p. (7). I t will be less accurate to use averages of subsurface temperature in
578 Journal of Colloid and Interface Science, VoI. 52, No. 3, SeDtember 1975
Copyright © 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
SURFACE TEMPERATURE DETERMINATIONS calculations than the temperature corresponding to the measured equilibrium vapor pressure. If the supporting calculations offered below should suggest that our recommendations are purely theoretical, we point out that they are experimental, involving specified conditions and an operational change, e.g., from a thermocouple sensor to an absolute manometer. They are also theoretical in the sense that both depend on the theory or the accumulated data of the art. A user of the thermocouple assumes an absence of thermal gradient between the true interface and the sensor; the signal is electrical, to be interpreted by a meter, in turn calibrated from a derived relation between mV and T °. The manometer is characterized by the density of it filling and is actuated directly frora the interface in question. In the suggested operational range ~1333 pascals (10 Torr or ram Hg) to 10 ~ pascals (1 atm) the mass exchange between liquid and vapor is tremendous, at 100°C 973 g sac-~ per 100 cm ~ of water surface and at 15°C (12.8 Tort) 18.5 g sac-~ per 100 cm ~. The equivalent two-way self-compensating heat flux is 524 kg cal and 10.87 kg cal sac-1 per 100 cm 2, respectively. An extreme thermal leakage of 10 W postulated in our model (and skeletonized in Fig. 1) is equivalent to 0.00239 kg cal sac-~, comparatively a trifling quantity. For calculating the effect of this on the temperature of the surface, we select the Clausius-Clapeyron equation originally formulated for true equilibrium conditions and use the time-honored device of making a small disturbance of equilibrium, i.e., by adding a moiety of heat energy and withdrawing an equivalent quantity of extra vapor. The liquid surface is assumed to be free from obstructive adsorbate and offers a vapor-liquid exchange coefficient E ~ 1.0 (8, 9). The two-way rate of transport W1 of vapor across the surface can then be calculated from the H e r t z -
579
Oondenser O
(~Thermo- :o \ \ \ meter ~ ~
IManometer
or or Barometer
iv1
,-'::;:" 'i Reference Thermocouple FIG. 1.
Knudsen equation (10, 11): W1 = 0.0583 p~(M/T~r,)}
(g/cra 2 sac)
[1]
where pl is the vapor pressure (ram Hg) of the liquid at the surface temperature (°K), T~r, and M is the molecular weight of the fluid. At equilibrium, W1 equals the rate of evaporation as well as the rate of condensation from the surface. The heat flux Q of the vaporization process, equal to that for the condensation process, is given by:
Q = wlx
[2]
where X is the heat of vaporization of the fluid (cal/g). Now the addition of a small quantity of heat q to the surface will increase the rate of evaporation; the increase is given by: q W2 - W1 = - = 0.0583 (M)~
L(T~)~ where V/~ is the two-way rate of transport of vapor across the surface that occurs when the heat flux is Q 4- q; P~, the vapor pressure of the liquid at Ttr~, can be rewritten in terms of X, PI, Ttrl, and T~r~ using the
Journal of Colloid and Interface Science, VoI. 52, No. 3, September 1975
HICKMAN AND KAYSER
580
TABLE INCREASES IN SURFACE TEMPERATURES Tt,1 (°C)
t)1 (mm Hg)
100 75 50 25 16 8 4 0 --4 --8 --15
760.00 289.10 92.51 23.756 13.634 8.045 6.101 4.579 3.410 2.514 1.436
I
ACCOMPANYINGKNOWNHEAT INPUT TO SURFACE
W~ (g/sec cm 2)
X (cal/gm)
0 (cal/see cm~)
9.73 3.83 1.27 0.340 0.198 0.119 0.0906 0.0685 0.0514 0.0382 0.0221
538.7 554.0 568.4 582.3 587.0 591.2 593.3 595.4 597.5b 601.2b 603.2 b
5242 2123 721.9 198.0 116.2 70.35 53.75 40.78 30.71 22.97 13.33
q~/O
4.56 X 1.13 X 3.31 X 1.21 X 2.06 X 3.40 X 4.45 X 5.86 X 7.78 X 1.04 X 1.79 X
10-~ 10-5 10-5 10-~ 10-4 10-4 10-4 10-4 10-~ 10-a 10-3
nTtr (°C)
1.30 X 2.72 X 6.71 X 2.04 X 3.23 X 5.03 X 6.36 X 8.11 X 1.04 X 1.35 X 2.19 X
10-4 10-~ 10-~ 10-3 10-3 10-3 10-~ 10-3 I0-~ 10-2 10-2
p2 (mm I-It)
P~ p~ (ram Hg)
760.0035 289.1033 92.5131 23.7589 13.6368 8.0477 6.1037 4.5817 3.4127 2.5166 1.4386
0.0035 0.0033 0.0031 0.0029 0.0028 0.0027 0.0027 0.0027 0.0027 0.0026 0.0026
- -
q = 0.1 W/cm 2 = 0.0239 cal/sec cm~. b Extrapolated. Clausius-Clapeyron equation:
p~ = pl exp[
pl ~. +
XM ( 1
R
Tt,:IT,~2
1
.
[4]
Substituting expression [-4-] into [3-1 and simplifying, assuming Tt~Ttr2 ~ Ttr~~, the increase in surface temperature, ATtr, can be determined. AT~r
--
T ~ -- T m
=
[5]
0.0583M~X2pl Values of ATt~ for water are listed in Table I for the addition of heat, q = 10 W / 1 0 0 cm 2, to the surface. This represents at least twice the h e a t i n p u t required to sustain a small increase in superheat 2~t, of the bulk liquid. Increases in surface t e m p e r a t u r e in the usual range of l a b o r a t o r y temperatures, 16-25°C, listed in Column 7, are thus only a few thousandths of a degree even when h e a t is leaking into the equipment a t a greater rate t h a n is likely to obtain during a critical m e a s u r e m e n t of surface properties. This is not surprising when q is compared to the total h e a t equivalent Q of the vaporization process. Similarly, convection currents in the
liquid modify the surface properties only slightly. If a b u o y a n t riser strikes the interface with a local h e a t increment 10 times t h a t reaching the surface elsewhere, on the average, Ttr at the point of i m p a c t will be ~ 0 . 0 1 ° above surrounding areas and a descending striation cooled b y evaporation cannot leave the surface depressed below T t ~ The m a x i m u m difference between local areas of surface is thus unlikely to exceed 0.01 °, corresponding to a change in surface tension of water, at 20°C, for instance, of a scarcely measurable 0.0015 dynes. I n the equally unlikely circumstance of the exchange coefficient being depressed from unity, for instance b y a semipermeable layer to E "-~ 0.1, 2~Ttr would rise to 0.1 ° corresponding to A3, ~ 0.015 dynes in the example cited. T h e small values of A T ~ predicted here inevitably will be compared with the much larger values usual in jet t e n s i m e t r y (3, 8) where ATtr is often > 1°: I n such cases, p l is generally less t h a n 10 Torr, p2 is a small fraction of p~, and exposure of the surface to evaporation or condensation is ~ 0 . 0 0 1 sec or less. T h e p a r a m e t e r s of the two situations thus differ a thousandfold. A d o p t i o n of the v a p o r pressure m e t h o d of surface t e m p e r a t u r e d e t e r m i n a t i o n does n o t necessarily banish the t e m p e r a t u r e sensor b u t
Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975
SURFACE TEMPERATURE DETERMINATIONS r a t h e r defines its true function, to measure t e m p e r a t u r e s and t e m p e r a t u r e gradients in the vicinity of the surface. CONCLUSION I t is shown t h a t the true t e m p e r a t u r e of the surface of a clean volatile liquid, for example water, in contact with its s a t u r a t e d v a p o r can be inferred with a c e r t a i n t y of 0.002 ° from b a r o m e t r i c readings under a m b i e n t conditions even though h e a t energy m a y be inserted or removed equivalent of 0.1 W cm -2 of surface. Such a system and m e t h o d permits d e t e r m i n a t i o n of surface tension and other physical characteristics with improved accuracy. ACKNOWLEDGMENT This work was supported by the Office of Water Research and Technology under Grant No. 14-30-3236.
581
REFERENCES 1. JOHAI,ISSON, K., J. Colloid Interjace Sci. 48, 176 (1974). 2. CINI, R., LoGLIO, G., AND FICALBI,A., Y. Colloid Interface Sci., to appear. 3. HICKMAI~, K., "Proceedings First International Symposium on Water Desalination," Vol. I., pp. 180-223 (U.S. Dept of the Interior, Washing, D.C., 1965). 4. BuFf, F. P., LOVETT,R. A., AND STILLINGER, F. H., JE., Phys. Ray. Letters 15, 621 (1965). 5. BAIN, R. W. (Ed.), "Steam Tables 1964," Her Majesty's Stationary Office, Edinburgh, United Kingdom, 1964. 6. KAYSER,W., "Evaporation of Water from Aqueous Interfaces," OSW Report #808, PB 215-184, Fig. 39, p. 102, items A, K, N, and O, distributed by N.I.T.S., U.S. Dept. of Commerce, 5285 Port Royal Rd., Springfield, Va. 22151. 7. HICKMAN, K., AND WHITE, I., Science 172, 718 (1971). 8. MAA, J., Ind. Eng. Chem. Fundam. 9, 283 (1970). 9. HICI¢~tAN,K., Desalination 1, 13 (1966). t0. HERTZ, H., Ann. Phys. 17, 177 (1882). 1l. Kx:IqDs~!, M., Ann. Phys. 47, 697 (1915).
Journal of Colloid and Interface Science. Vol. 52, No. 3, September 1975
The difficulty in determining the temperature precisely at the interface of a liquid with a gas, usually air, is a common experience, brought to our notice again by Johansson's Letter-to-the-Editor (1) and the reply by Cini el al. (2). Both letters cover real situations confronting the experimenter; the liquid may be volatile or nonvolatile and in contact with air or other gas transporting varying quantities of vapor, often while temperature differences with the environment are accommodated. Physical techniques and expertise are debated which permit approach to measurement of true interfacial temperature. The limiting difficulty is that placing a sensor no matter how tiny in the liquid surface measures an average boundary layer temperature but not necessarily Ttr, the temperature of the instant top row 2 of molecules. If this could be done, it would interfere with and displace the top row, affording a classic example of the Schroedinger paradox that "to measure is to disturb." However, there is an indirect method of great precision that can be applied to volatile liquids if the operation can be conducted in a closed system with exclusion of air or other gas. Proof will be offered below that the surface temperature Ttr of a liquid in contact with its pure saturated vapor will be at Now at Exxon Nuclear Co., Inc., Richland, Washington ~A momentarily precise b u t statistically diffuse enfitity (3, 4).
substantially the mean equilibrium boiling point despite minor fluctuations in heat input to the surface. There are the added stipulations that the vapor pressure shall be above 10 Torr and the rate of evaporation or condensation shall not be unusually large. If the liquid is water and the measurements are made under ambient conditions, the surface temperature can be calculated from the barometric reading and the Steam Tables (5). In the absence of a barometer, or if, in the case of other substances, reliable liquid-vapor pressure curves are not available, the interfacial temperature will be the same as indicated by a thermometer (or equivalent) situated in a suitable (6) reference boiler maintained at the same pressure, e.g., included in the vapor manifold. In real systems, the temperature of the test liquid is likely to fluctuate. When the liquid cools, there is net condensation of vapor on the surface; when the temperature rises, there is net evaporation. Since traces of foreign gas can buffer condensation, it is preferred to keep the liquid slightly superheated (a real input of 3-8 W/500 ml, surface area ~ 1 0 0 cm 2) to ensure a net loss of vapor and a further scavenging from the system of any residual foreign gas. Experience shows that the average (inhomogeneous) temperature of the bulk liquid will then be 0.1-0.4 ° above true b.p. (7). I t will be less accurate to use averages of subsurface temperature in
578 Journal of Colloid and Interface Science, VoI. 52, No. 3, SeDtember 1975
Copyright © 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
SURFACE TEMPERATURE DETERMINATIONS calculations than the temperature corresponding to the measured equilibrium vapor pressure. If the supporting calculations offered below should suggest that our recommendations are purely theoretical, we point out that they are experimental, involving specified conditions and an operational change, e.g., from a thermocouple sensor to an absolute manometer. They are also theoretical in the sense that both depend on the theory or the accumulated data of the art. A user of the thermocouple assumes an absence of thermal gradient between the true interface and the sensor; the signal is electrical, to be interpreted by a meter, in turn calibrated from a derived relation between mV and T °. The manometer is characterized by the density of it filling and is actuated directly frora the interface in question. In the suggested operational range ~1333 pascals (10 Torr or ram Hg) to 10 ~ pascals (1 atm) the mass exchange between liquid and vapor is tremendous, at 100°C 973 g sac-~ per 100 cm ~ of water surface and at 15°C (12.8 Tort) 18.5 g sac-~ per 100 cm ~. The equivalent two-way self-compensating heat flux is 524 kg cal and 10.87 kg cal sac-1 per 100 cm 2, respectively. An extreme thermal leakage of 10 W postulated in our model (and skeletonized in Fig. 1) is equivalent to 0.00239 kg cal sac-~, comparatively a trifling quantity. For calculating the effect of this on the temperature of the surface, we select the Clausius-Clapeyron equation originally formulated for true equilibrium conditions and use the time-honored device of making a small disturbance of equilibrium, i.e., by adding a moiety of heat energy and withdrawing an equivalent quantity of extra vapor. The liquid surface is assumed to be free from obstructive adsorbate and offers a vapor-liquid exchange coefficient E ~ 1.0 (8, 9). The two-way rate of transport W1 of vapor across the surface can then be calculated from the H e r t z -
579
Oondenser O
(~Thermo- :o \ \ \ meter ~ ~
IManometer
or or Barometer
iv1
,-'::;:" 'i Reference Thermocouple FIG. 1.
Knudsen equation (10, 11): W1 = 0.0583 p~(M/T~r,)}
(g/cra 2 sac)
[1]
where pl is the vapor pressure (ram Hg) of the liquid at the surface temperature (°K), T~r, and M is the molecular weight of the fluid. At equilibrium, W1 equals the rate of evaporation as well as the rate of condensation from the surface. The heat flux Q of the vaporization process, equal to that for the condensation process, is given by:
Q = wlx
[2]
where X is the heat of vaporization of the fluid (cal/g). Now the addition of a small quantity of heat q to the surface will increase the rate of evaporation; the increase is given by: q W2 - W1 = - = 0.0583 (M)~
L(T~)~ where V/~ is the two-way rate of transport of vapor across the surface that occurs when the heat flux is Q 4- q; P~, the vapor pressure of the liquid at Ttr~, can be rewritten in terms of X, PI, Ttrl, and T~r~ using the
Journal of Colloid and Interface Science, VoI. 52, No. 3, September 1975
HICKMAN AND KAYSER
580
TABLE INCREASES IN SURFACE TEMPERATURES Tt,1 (°C)
t)1 (mm Hg)
100 75 50 25 16 8 4 0 --4 --8 --15
760.00 289.10 92.51 23.756 13.634 8.045 6.101 4.579 3.410 2.514 1.436
I
ACCOMPANYINGKNOWNHEAT INPUT TO SURFACE
W~ (g/sec cm 2)
X (cal/gm)
0 (cal/see cm~)
9.73 3.83 1.27 0.340 0.198 0.119 0.0906 0.0685 0.0514 0.0382 0.0221
538.7 554.0 568.4 582.3 587.0 591.2 593.3 595.4 597.5b 601.2b 603.2 b
5242 2123 721.9 198.0 116.2 70.35 53.75 40.78 30.71 22.97 13.33
q~/O
4.56 X 1.13 X 3.31 X 1.21 X 2.06 X 3.40 X 4.45 X 5.86 X 7.78 X 1.04 X 1.79 X
10-~ 10-5 10-5 10-~ 10-4 10-4 10-4 10-4 10-~ 10-a 10-3
nTtr (°C)
1.30 X 2.72 X 6.71 X 2.04 X 3.23 X 5.03 X 6.36 X 8.11 X 1.04 X 1.35 X 2.19 X
10-4 10-~ 10-~ 10-3 10-3 10-3 10-~ 10-3 I0-~ 10-2 10-2
p2 (mm I-It)
P~ p~ (ram Hg)
760.0035 289.1033 92.5131 23.7589 13.6368 8.0477 6.1037 4.5817 3.4127 2.5166 1.4386
0.0035 0.0033 0.0031 0.0029 0.0028 0.0027 0.0027 0.0027 0.0027 0.0026 0.0026
- -
q = 0.1 W/cm 2 = 0.0239 cal/sec cm~. b Extrapolated. Clausius-Clapeyron equation:
p~ = pl exp[
pl ~. +
XM ( 1
R
Tt,:IT,~2
1
.
[4]
Substituting expression [-4-] into [3-1 and simplifying, assuming Tt~Ttr2 ~ Ttr~~, the increase in surface temperature, ATtr, can be determined. AT~r
--
T ~ -- T m
=
[5]
0.0583M~X2pl Values of ATt~ for water are listed in Table I for the addition of heat, q = 10 W / 1 0 0 cm 2, to the surface. This represents at least twice the h e a t i n p u t required to sustain a small increase in superheat 2~t, of the bulk liquid. Increases in surface t e m p e r a t u r e in the usual range of l a b o r a t o r y temperatures, 16-25°C, listed in Column 7, are thus only a few thousandths of a degree even when h e a t is leaking into the equipment a t a greater rate t h a n is likely to obtain during a critical m e a s u r e m e n t of surface properties. This is not surprising when q is compared to the total h e a t equivalent Q of the vaporization process. Similarly, convection currents in the
liquid modify the surface properties only slightly. If a b u o y a n t riser strikes the interface with a local h e a t increment 10 times t h a t reaching the surface elsewhere, on the average, Ttr at the point of i m p a c t will be ~ 0 . 0 1 ° above surrounding areas and a descending striation cooled b y evaporation cannot leave the surface depressed below T t ~ The m a x i m u m difference between local areas of surface is thus unlikely to exceed 0.01 °, corresponding to a change in surface tension of water, at 20°C, for instance, of a scarcely measurable 0.0015 dynes. I n the equally unlikely circumstance of the exchange coefficient being depressed from unity, for instance b y a semipermeable layer to E "-~ 0.1, 2~Ttr would rise to 0.1 ° corresponding to A3, ~ 0.015 dynes in the example cited. T h e small values of A T ~ predicted here inevitably will be compared with the much larger values usual in jet t e n s i m e t r y (3, 8) where ATtr is often > 1°: I n such cases, p l is generally less t h a n 10 Torr, p2 is a small fraction of p~, and exposure of the surface to evaporation or condensation is ~ 0 . 0 0 1 sec or less. T h e p a r a m e t e r s of the two situations thus differ a thousandfold. A d o p t i o n of the v a p o r pressure m e t h o d of surface t e m p e r a t u r e d e t e r m i n a t i o n does n o t necessarily banish the t e m p e r a t u r e sensor b u t
Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975
SURFACE TEMPERATURE DETERMINATIONS r a t h e r defines its true function, to measure t e m p e r a t u r e s and t e m p e r a t u r e gradients in the vicinity of the surface. CONCLUSION I t is shown t h a t the true t e m p e r a t u r e of the surface of a clean volatile liquid, for example water, in contact with its s a t u r a t e d v a p o r can be inferred with a c e r t a i n t y of 0.002 ° from b a r o m e t r i c readings under a m b i e n t conditions even though h e a t energy m a y be inserted or removed equivalent of 0.1 W cm -2 of surface. Such a system and m e t h o d permits d e t e r m i n a t i o n of surface tension and other physical characteristics with improved accuracy. ACKNOWLEDGMENT This work was supported by the Office of Water Research and Technology under Grant No. 14-30-3236.
581
REFERENCES 1. JOHAI,ISSON, K., J. Colloid Interjace Sci. 48, 176 (1974). 2. CINI, R., LoGLIO, G., AND FICALBI,A., Y. Colloid Interface Sci., to appear. 3. HICKMAI~, K., "Proceedings First International Symposium on Water Desalination," Vol. I., pp. 180-223 (U.S. Dept of the Interior, Washing, D.C., 1965). 4. BuFf, F. P., LOVETT,R. A., AND STILLINGER, F. H., JE., Phys. Ray. Letters 15, 621 (1965). 5. BAIN, R. W. (Ed.), "Steam Tables 1964," Her Majesty's Stationary Office, Edinburgh, United Kingdom, 1964. 6. KAYSER,W., "Evaporation of Water from Aqueous Interfaces," OSW Report #808, PB 215-184, Fig. 39, p. 102, items A, K, N, and O, distributed by N.I.T.S., U.S. Dept. of Commerce, 5285 Port Royal Rd., Springfield, Va. 22151. 7. HICKMAN, K., AND WHITE, I., Science 172, 718 (1971). 8. MAA, J., Ind. Eng. Chem. Fundam. 9, 283 (1970). 9. HICI¢~tAN,K., Desalination 1, 13 (1966). t0. HERTZ, H., Ann. Phys. 17, 177 (1882). 1l. Kx:IqDs~!, M., Ann. Phys. 47, 697 (1915).
Journal of Colloid and Interface Science. Vol. 52, No. 3, September 1975