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The spheroidal state of a waterdrop
Kistemakei , 1963
J.
Physica 29 96-104
THE SPHEROIDAL STATE OF A vVATERDROP THE LEIDENFROST PHENOMENON *)
J. KISTEMAKER FOM-Laborn.torium voor Massascheidiug, Amsterdam, Nederland.
Synopsis After a historical introduction of the Leidenfrost phenomenon, some experiments are described to determine heatflow, drop to wall distance and evaporation rate for a water drop above a highly polished brass block in a temperature range between 100° and 500 aC, at atmospheric pressure, in air. Distances vary between 30 and 60 microns. At IS5 aC and lower temperatures we found nucleate boiling. A formula for the distance based on aerodynamical flow calculations, is presented.
1. Introduction. The study of the behaviour ot liquid drops on hot surfaces belongs to the eldest subjects of study in physics. In 1732 already the famous Dutch scientist Bo er h a av e-) gave a beautiful description of a drop of alcohol floating on hot iron, without catching fire. After another short description by El d er-), the German scientist Leid enfr ost") (1756) did extensive research on the subject. His name was since then associated with the phenomenon. Characteristic for the phenomenon is that liquid drops, if not too large, try to take a spheroidal form instead of wettening the underlying surface. It was the Frenchman Bou t.i g ny«) who described this shape first, which caused the second indicative name for the Leidenfrost phenomenon. In those early days the physical background of the floating phenomenon was seriously discussed. Possibilities considered then were: a. floating on its own vapour; b. a repulsive 'heat force' ; c. differential adhesive and cohesive forces. The points of view under band c were abandoned, when physical insight in the molecular character of matter grew (Maxwell, Boltzmann). Star k 5) described in 1898 the influence of changes in surface tension *) Most of the experimental work was done in the Kamerlingh Ormes Laboratory of the Leyden University, in 1944/46, together with Mr. C. ]. van del' I-I am. Conclusive measurements and studies were done in the FOM-Laboratory for Mass Separation/Amsterdam, with the assistance of Mr. A. Ha r i n g.
-
96 -
THE SPHEROIDAL STATE OF A WATERDROP
97
along the drop surface. This causes the observed turbulence and the fact that the drop liquid is undercooled. He also discussed the observed rapid oscillations of the lower surface of the drop, causing destruction if the temperature of the metal falls below a certain minimum. In recent times (1937) Schmid and Sp ei d e l.") considered the distance of the drop to the metal surface as a function of the heat transport through the gas layer, whereas in 1947 Pleteneva and Reh b i n d.er") studied the rate of evaporation of floating drops. A strange aspect has always been that drops of paraffin wax and other oils with extreme low vapour pressures still seem to show the Leidenfrost phenomenon. This made Holst Web ers) think that thermo-molecular pressure effects (radio-meter effects) playa role of importance here, and this led us to starting some research in the Kamerlingh annes Laboratory in Leyden, in the difficult years 1944/46, in doing direct measurements on the temperature distribution in the metal block underneath the drop, as well as direct determination of the distance of a drop to the metal surface. This work was not finished however, because of external circumstances. Recent developments created new interest, although these studies are all dealing with large volumes of liquid. The study of heat transfer to a 'cooling' liquid drew attention since nuclear reactors and high pressure steam plants have asked for a solution of the problem how to make this transfer as large as possible. Extensive research was done by Van Wijk and Van St r a Ien"). Increasing the temperature difference LIT between the wall and the liquid, one passes first through the convection regIon, then through the so called region of nucleate boiling, where locally on the wall vapour bubbles are generated. Raising the temperature of the wall still more, we pass through an unstable transition zone to the film boiling zone, where the heating surface is covered with a coherent layer of vapour (Leidenfrost phenomenon). This results in a partial thermal insulation. The heat flux decreases severely causing an increase of the wall temperature which can be catastrophic in case of steam plants 10)11). In the FOM-laboratory for Mass Separation in Amsterdam we added some more work in 1962, so that now a more or less complete picture of this interesting phenomenon can be given for water drops, under atmospheric conditions. Certainly, this work can be repeated on a much broader basis in laboratories more dedicated to this subject than our laboratories are. We present these data here, however, because of the new experimental approach of this classical phenomenon.
2. The experiment. The drop was floating above a highly polished brass block which was electrically heated up to 500°C. To keep the drop in equilibrium the horizontal position could be adjusted with screws. A thin glass
98
J.
KISTE lIL-\KER
pipet conti nu ou sly injected a flow of water (of abou t 15°C) into th e top of the w ate rdrop , so that its size did not change. We t ook advantage of the pipet to ke ep the d rop just sym metrically around the sno ut oft he pipet ina well d efined , quiet position, becau se of the surface t ension a ction (seefigure I).
F ig . 1. A wat er drop ab ove a hi ghl y po lished (500°C) brass block, in the spheroidal state. The size of the drop was kept const a nt by feed ing water continuously t hroug h t he visible glas s tube. This tube fixes the position of the drop b y s ur fac e tension.
The height of the drop floating above the brass sur face was photographed with X-rays . It was necessary to work with the focus F of the X-ray tube at the same height as the upp er rim of the brass block, which can be controlled very accurately (angle of 1 arc minute) from th e presence of total reflection of the X-rays on the metal sur face, visible on the photographic plate. Good contrast was obtained by resolving some barium ac etate in the waterdrop. X-ray photography was a n ecessity as normal optic al methods proved unreliable, because of false reflections and refra ction. At different heights the te mperat ure in t he drop was measured by using thermocouple Th. S. Th e t emperature gradient in th e sur face layer of th e brass block was measured by using thermocouples Th . I, 2, 3 and 4 (vide figure 2). The dista nce between the thermocouples was 3 mm . A row of four narrow holes (of 1 mm diam et er) had been bored in the brass block, which was approximat ely 2 em t hick. Through each bore hole an insulated const antan wire ran to the t op of the block, where it s upp er 0.5 mm was brazed by means of silver.
THE SPHEROIDAL STATE OF A WATERDROl'
99
For the determination of temperature differences between the brazing spots of 1, 2, 3 or 4, the relative potential difference between two spots was measured.
Th.5
Ph.
Fig. 2. Arrangement to investigate the behaviour of a water drop above a hot brass surface. Thickness of the block; about 2 ern. Th. 1, 2, 3, 4 and 5 are thermocouples. B are the bore holes of 1 mm diameter. P is a pipet filled with water. F is an adjustable focus of an X-ray tube. Ph is a photographic plate. Some bariurnacetate was resolved in the water of the drop.
3. Observations. The temperature underneath the waterdrop was approximately 4°C lower than the block temperature in the range from 200°C to 500°C, and it was independent of the temperature in the room. The temperature fall took place over a distance of 1 em outside of the edge of the drop, whereas there was hardly any gradient underneath the drop (see figure 3).
edge
I--_-I--"'-----v 3
/
cent re
21_r rnrn
s'c
Fig. 3. The differences in temperature underneath and close to a waterdrop having a diameter (dl of 12 nun, at a block temperature of about 500°C, being independent of the room temperature and measured at atmospheric pressure, in air.
100
J.
KISTEMAKER
Some results - for a drop diameter of 12 mm - are in tables I and II. TABLE I Block temperature (in 0c)
Smallest distance between drop and hot surface (measured)
185
violent oscillations
??
200
30 microns
82
500
60 microns
92
I
Average temperature of the water in the drop (in "C)
Inside the drop one can observe heavy turbulences and temperature differences of a few degrees, with the lowest temperature in top of the drop. At 18SoC block temperature suddenly violent oscillations start. Vapour bubbles escape through the middle of the liquid. This state shows an explosive evaporation of the waterclrop. At temperatures lower than 185°C and at a pressure of one atmosphere the stable state in the Leidenfrost phenomenon does not exist, if water is used in atmospheric air. TABLE II Block temperature in "C 243
I
Drop diameter in mrn 12
Evaporation of water in mmv/sec
330
12
2.8
470
12
4.2
I
2.1
The measured thickness of the waterdrop was in all cases 3.7 mm in the middle of the drop, and approximately 2.5 mm at the edge. The rate of evaporation of the water drop is visible from table II and in figure 4. 4. Discuss-ion. a. The radiometer effect. The temperature gradient underneath the waterdrop is far too small to give a radiometer effect 8) of any importance. Moreover, it would cause a negative pressure excess. Whether the effect is also negligible under a paraffin drop, is an. open question, however. Temperatures and distances should be measured in that case.
b. The hea.t flux through the gas-film. The temperature gradient inside the brass block can be used to calculate the heat flux via the gas-film to the drop. That this can be clone is because of the relatively small heat transfer of a brass block at temperatures up to 500°C, to air of room temperature, which is 5.4 X 10- 4 cal sec-1cm 2 °C-112), whereas the heat transfer via the gas-film to water is: 1 X 10-2 cal sec-1 em- 2 °C- 1 10) 12). Radiation
THE SPHEROIDAL STATE OF A W:\TERDROP
101
losses are still a factor 4 smaller than the heat transfer to air, so that we can neglect these. Taking the temperature gradient inside the brass block the same as the gradient measured along the surface of this block (figure 3), we find a total heat flux:
Q with 0
=
=
Oli. dT/dR
=
3.3 cal/sec,
5 ern- ((estimated flux area inside the brass), Ii. = 0.25 cal cm-1 seer! degreer! (for brass),
clTldR
=
3°e em-I.
ci
E
~
500
400
JOO
200
'--~---~------'-------'
Fig. 4. The speed of evaporation of a waterdrop above a heated, highly polished brass block. Diameter of the waterdrop: 12 rnrn. Thickness of the waterdrop: 3.7 mm in the middle; 2.5 mm at the edge. Pressure: 1 atmosphere (air).
The diameter of the drop has always been kept at 12 mm. This corresponds with a heat flux per degree temperature difference of 0.8' 10-2 cal sec-1 cm-2 °C-l with the brass at 500 0 e and the drop at about 90°C, which is in good agreement with literature. The thermal conductivity of water vapour at 300°C is approximately 1 X 10- 4 cal sec- 1 cm-1 degree-I. In case of our drop above a metal plate of 500°C, one finds that the heat transport would be about 6.6 cal sec-I, which is a factor 2 too large. This can be explained from a temperature jump at the liquid surface, caused by an apparent heat resistance, making dTjdR at the metal surface smaller than one would expect. c. The gas prod uetion. The observed heat transport leads to an evaporation of water at the lower siele of the drop. The warming up of the
J.
102
KISTEMAKER
- - - - - -_ . _ - - - - -
continuous flux of qIiq mm 3/sec of cold water flowing into the drop must be taken into account.
Q = O· (rT
+ [100 -
tl'oomJ c) 'qli'! rT = 540 cal (heat of evaporation of water) ; qIiq = volume of water evaporated per sec. ; c = specific heat of water; o = area of lower side of the drop. With Q = 3.3 cal sec"! we find for our drop with a wall temperature of 50QoC a qlll] = 5 rnmf sec- 1 - in good agreement with the observed quantity read from figure 4. If we take the average temperature of the gas in the film 300°C, this corresponds with a volume of gas of qgas = 11 cm 3 sec 1 (for total drop area, lower side). cm- 3
This gas quantity has to escape via the gas film and is the apparent reason of the elevation of the waterclrop. During this calculation we have supposecl that the diffusive evaporation from the unclercoolecl top surface of the drop can be neglected (in one atmosphere air). d. The thickness of the gas film. To discuss the observed flow of heat, the evaporation rate and the thickness of the gas film at about 500°C wall temperature, we need the equations of hydrodynamics. Considering a stationary state and an incompressible fluid, the main equation is:
1tP grad v2
-
p[v rot
vJ
=
-
grad p
+ 1](Llv + grad div v).
(a)
Here, v
= speed of the gas in em secv- ;
p
= density of the gas in grams;
p = pressure in dynes cm- 2 ; = viscosity of the gas in poises. grad div v = a (incompressible fluid).
1]
In cylindrical co-ordinates related to our drop problem. with the z-axis parallel to the drop-axis, this equation reduces to: 2
},p ov ~
or
=
..». + or
1]
(~ !'!- + r or
1 02v -
or 2
!.-) . r 2
(b)
We have supposed two things: I. All parts of the lower surface of the drop evaporate with the same speed of qgas em sec1 gas of atmospheric pressure. The temperature of this gas is supposed to be (T wall + T dl'OP) /2. This means, that the continuity equation (mass balance) in our case gives for the space between drop and metal wall:
f
surface
u- df
=
fvolumo
div v elt = :n;r 2 qgas'
(c)
THE SPHEROIOAL STATE OF A WAT ER DROP
103
Taking v as a mean speed value over the height of the gap h, this leads t o:
2nrhv = nr 2qg3>j'
_
(d)
or :
rqgas
(e )
V=--.
2ft
= 6 X 10-3 ern, and = 0.6 ern, this mean s that for qg,," ~ 11 cm 3 sec- 1 for total area
For a gap height It r givin g
v
~
500 cm sec-I, which is a relatively low value.
I I. All rotational t erms are small. We consider only the main stream in the medium plane between the hot wall and the lower side of the drop. This seems to be correct as viscosity terms are negligible with speeds of the order of 500 em sec l . Integration of equ ation (b) gives:
~pv2 = Po - P + 'YJ J(2.r !~ - ~) dr or + 1 02~ 81'2 1'2 from which we deduce by substitution of
v from
(e)
q
1'2 2
tp -4k 2 = po - p.
(I )
Substitution of: P H , O.go.><
at 100°C, 1 atm = 6 X 10- 4 gram ; at 300°C, 1 atm = 4 X 10- 4 gram ;
r = R = 0.6 em, as used in our exp eriments, 2.5 mm water ~ 100 dynes (pressure difference in H 2 0 -gas between middle of ga p, center of drop, and the edge of the drop), gives simply :
po - p ~ 3 .7 -
rr:':
It
=
~Rqga" V2~P
(g)
and the h valeus in the third column of table III. One might compare these results with Schmid and Speidel's empirical formula 7):
where : Ava pOlll' ~
1 X 10-4 cal sec-1 crrr ! degree" (at 300°C) ; T m et al - T b oili ng = temperature differ enc e between metal and boiling point of water; T b oll tng - TU'lUId. = t emperature differen ce between b oiling point and real t emperature of the wat erdrop ; p = pressure durin g the experiment.
104
THE SPHEROIDAL STATE OF A WATERDROP
The results are in table III. TABLE III Values of 11 in microns 1310ck temperature in °C
I
200 500
lz measured
I
Our equation (g)
I
60
I
20
30
I
46
Schmid and Speidel 12
I
100
We used the experimental numbers from tables I and II. A value of iJp of 200 dynes, instead of the 100 which we took, would give perfect agreement of our experiments with equation (g). A higher value of iJp seems reasonable indeed, as the gas at the edge of the gap - underneath the drop - starts to expand and the difference of height in the waterlevel in the drop gives, therefore, too small pressure differences. I thank Ir. L. H. J. Wachters for a useful discussion on equation (b). This work was partially financed by the Stichting vo or Fundamenteel Onderzoek der Materie, and the Stichting voor Zuiver Wetenschappelijk Onderzoek. Received 30·8·62
REFERENCES
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
For a small review, sec: Partington, ]. R, An advanced treatise on physical chemistry, II (1955) 282. Boerhaavc, H., Elemcnta chemiae Lugdunum Batavorum (Leyden) 4 tI (1732) 258. Eller,]. T'h. Mem, Acad. Berlin (1746) 43. Leidenfrost,]. G., De aquae communis nonnullis qualltatlbus tractatus, Duisburgi (1756) 30. 13outigny, P. M., Ann. Chirn. I) (1843) 350. Stark, J., Ann. d.Phys. 301 (1898) 306. Sc h ml d, G. and Speidel, H., Z. f. Elektrochemie 43 (1937) 187. Pleteneva and Rehbinder,]. phys, Chern, USSR 21 (1946) 961; J. phys, Chern. USSR 22 (1947) L Holst Weber, S. Th., Kg!. Danske Vld , Selsk., Mat. Fys. Medel. 14 (1937), 13. Van Wijk, W. Rand VanStralen, S. J. D., Physica zn (1962) 150. Jakob, M., Heat transfer, Vol. I, Wiley, New York, 1950. Mc Ad a m s, W. H., Heat transmission, 3rd Edit., McGrawHill, New York, 1945. Van Stralen, S.]. D., Ned. Tijdschr. Natk, 28 (1962) 10.
J.
Physica 29 96-104
THE SPHEROIDAL STATE OF A vVATERDROP THE LEIDENFROST PHENOMENON *)
J. KISTEMAKER FOM-Laborn.torium voor Massascheidiug, Amsterdam, Nederland.
Synopsis After a historical introduction of the Leidenfrost phenomenon, some experiments are described to determine heatflow, drop to wall distance and evaporation rate for a water drop above a highly polished brass block in a temperature range between 100° and 500 aC, at atmospheric pressure, in air. Distances vary between 30 and 60 microns. At IS5 aC and lower temperatures we found nucleate boiling. A formula for the distance based on aerodynamical flow calculations, is presented.
1. Introduction. The study of the behaviour ot liquid drops on hot surfaces belongs to the eldest subjects of study in physics. In 1732 already the famous Dutch scientist Bo er h a av e-) gave a beautiful description of a drop of alcohol floating on hot iron, without catching fire. After another short description by El d er-), the German scientist Leid enfr ost") (1756) did extensive research on the subject. His name was since then associated with the phenomenon. Characteristic for the phenomenon is that liquid drops, if not too large, try to take a spheroidal form instead of wettening the underlying surface. It was the Frenchman Bou t.i g ny«) who described this shape first, which caused the second indicative name for the Leidenfrost phenomenon. In those early days the physical background of the floating phenomenon was seriously discussed. Possibilities considered then were: a. floating on its own vapour; b. a repulsive 'heat force' ; c. differential adhesive and cohesive forces. The points of view under band c were abandoned, when physical insight in the molecular character of matter grew (Maxwell, Boltzmann). Star k 5) described in 1898 the influence of changes in surface tension *) Most of the experimental work was done in the Kamerlingh Ormes Laboratory of the Leyden University, in 1944/46, together with Mr. C. ]. van del' I-I am. Conclusive measurements and studies were done in the FOM-Laboratory for Mass Separation/Amsterdam, with the assistance of Mr. A. Ha r i n g.
-
96 -
THE SPHEROIDAL STATE OF A WATERDROP
97
along the drop surface. This causes the observed turbulence and the fact that the drop liquid is undercooled. He also discussed the observed rapid oscillations of the lower surface of the drop, causing destruction if the temperature of the metal falls below a certain minimum. In recent times (1937) Schmid and Sp ei d e l.") considered the distance of the drop to the metal surface as a function of the heat transport through the gas layer, whereas in 1947 Pleteneva and Reh b i n d.er") studied the rate of evaporation of floating drops. A strange aspect has always been that drops of paraffin wax and other oils with extreme low vapour pressures still seem to show the Leidenfrost phenomenon. This made Holst Web ers) think that thermo-molecular pressure effects (radio-meter effects) playa role of importance here, and this led us to starting some research in the Kamerlingh annes Laboratory in Leyden, in the difficult years 1944/46, in doing direct measurements on the temperature distribution in the metal block underneath the drop, as well as direct determination of the distance of a drop to the metal surface. This work was not finished however, because of external circumstances. Recent developments created new interest, although these studies are all dealing with large volumes of liquid. The study of heat transfer to a 'cooling' liquid drew attention since nuclear reactors and high pressure steam plants have asked for a solution of the problem how to make this transfer as large as possible. Extensive research was done by Van Wijk and Van St r a Ien"). Increasing the temperature difference LIT between the wall and the liquid, one passes first through the convection regIon, then through the so called region of nucleate boiling, where locally on the wall vapour bubbles are generated. Raising the temperature of the wall still more, we pass through an unstable transition zone to the film boiling zone, where the heating surface is covered with a coherent layer of vapour (Leidenfrost phenomenon). This results in a partial thermal insulation. The heat flux decreases severely causing an increase of the wall temperature which can be catastrophic in case of steam plants 10)11). In the FOM-laboratory for Mass Separation in Amsterdam we added some more work in 1962, so that now a more or less complete picture of this interesting phenomenon can be given for water drops, under atmospheric conditions. Certainly, this work can be repeated on a much broader basis in laboratories more dedicated to this subject than our laboratories are. We present these data here, however, because of the new experimental approach of this classical phenomenon.
2. The experiment. The drop was floating above a highly polished brass block which was electrically heated up to 500°C. To keep the drop in equilibrium the horizontal position could be adjusted with screws. A thin glass
98
J.
KISTE lIL-\KER
pipet conti nu ou sly injected a flow of water (of abou t 15°C) into th e top of the w ate rdrop , so that its size did not change. We t ook advantage of the pipet to ke ep the d rop just sym metrically around the sno ut oft he pipet ina well d efined , quiet position, becau se of the surface t ension a ction (seefigure I).
F ig . 1. A wat er drop ab ove a hi ghl y po lished (500°C) brass block, in the spheroidal state. The size of the drop was kept const a nt by feed ing water continuously t hroug h t he visible glas s tube. This tube fixes the position of the drop b y s ur fac e tension.
The height of the drop floating above the brass sur face was photographed with X-rays . It was necessary to work with the focus F of the X-ray tube at the same height as the upp er rim of the brass block, which can be controlled very accurately (angle of 1 arc minute) from th e presence of total reflection of the X-rays on the metal sur face, visible on the photographic plate. Good contrast was obtained by resolving some barium ac etate in the waterdrop. X-ray photography was a n ecessity as normal optic al methods proved unreliable, because of false reflections and refra ction. At different heights the te mperat ure in t he drop was measured by using thermocouple Th. S. Th e t emperature gradient in th e sur face layer of th e brass block was measured by using thermocouples Th . I, 2, 3 and 4 (vide figure 2). The dista nce between the thermocouples was 3 mm . A row of four narrow holes (of 1 mm diam et er) had been bored in the brass block, which was approximat ely 2 em t hick. Through each bore hole an insulated const antan wire ran to the t op of the block, where it s upp er 0.5 mm was brazed by means of silver.
THE SPHEROIDAL STATE OF A WATERDROl'
99
For the determination of temperature differences between the brazing spots of 1, 2, 3 or 4, the relative potential difference between two spots was measured.
Th.5
Ph.
Fig. 2. Arrangement to investigate the behaviour of a water drop above a hot brass surface. Thickness of the block; about 2 ern. Th. 1, 2, 3, 4 and 5 are thermocouples. B are the bore holes of 1 mm diameter. P is a pipet filled with water. F is an adjustable focus of an X-ray tube. Ph is a photographic plate. Some bariurnacetate was resolved in the water of the drop.
3. Observations. The temperature underneath the waterdrop was approximately 4°C lower than the block temperature in the range from 200°C to 500°C, and it was independent of the temperature in the room. The temperature fall took place over a distance of 1 em outside of the edge of the drop, whereas there was hardly any gradient underneath the drop (see figure 3).
edge
I--_-I--"'-----v 3
/
cent re
21_r rnrn
s'c
Fig. 3. The differences in temperature underneath and close to a waterdrop having a diameter (dl of 12 nun, at a block temperature of about 500°C, being independent of the room temperature and measured at atmospheric pressure, in air.
100
J.
KISTEMAKER
Some results - for a drop diameter of 12 mm - are in tables I and II. TABLE I Block temperature (in 0c)
Smallest distance between drop and hot surface (measured)
185
violent oscillations
??
200
30 microns
82
500
60 microns
92
I
Average temperature of the water in the drop (in "C)
Inside the drop one can observe heavy turbulences and temperature differences of a few degrees, with the lowest temperature in top of the drop. At 18SoC block temperature suddenly violent oscillations start. Vapour bubbles escape through the middle of the liquid. This state shows an explosive evaporation of the waterclrop. At temperatures lower than 185°C and at a pressure of one atmosphere the stable state in the Leidenfrost phenomenon does not exist, if water is used in atmospheric air. TABLE II Block temperature in "C 243
I
Drop diameter in mrn 12
Evaporation of water in mmv/sec
330
12
2.8
470
12
4.2
I
2.1
The measured thickness of the waterdrop was in all cases 3.7 mm in the middle of the drop, and approximately 2.5 mm at the edge. The rate of evaporation of the water drop is visible from table II and in figure 4. 4. Discuss-ion. a. The radiometer effect. The temperature gradient underneath the waterdrop is far too small to give a radiometer effect 8) of any importance. Moreover, it would cause a negative pressure excess. Whether the effect is also negligible under a paraffin drop, is an. open question, however. Temperatures and distances should be measured in that case.
b. The hea.t flux through the gas-film. The temperature gradient inside the brass block can be used to calculate the heat flux via the gas-film to the drop. That this can be clone is because of the relatively small heat transfer of a brass block at temperatures up to 500°C, to air of room temperature, which is 5.4 X 10- 4 cal sec-1cm 2 °C-112), whereas the heat transfer via the gas-film to water is: 1 X 10-2 cal sec-1 em- 2 °C- 1 10) 12). Radiation
THE SPHEROIDAL STATE OF A W:\TERDROP
101
losses are still a factor 4 smaller than the heat transfer to air, so that we can neglect these. Taking the temperature gradient inside the brass block the same as the gradient measured along the surface of this block (figure 3), we find a total heat flux:
Q with 0
=
=
Oli. dT/dR
=
3.3 cal/sec,
5 ern- ((estimated flux area inside the brass), Ii. = 0.25 cal cm-1 seer! degreer! (for brass),
clTldR
=
3°e em-I.
ci
E
~
500
400
JOO
200
'--~---~------'-------'
Fig. 4. The speed of evaporation of a waterdrop above a heated, highly polished brass block. Diameter of the waterdrop: 12 rnrn. Thickness of the waterdrop: 3.7 mm in the middle; 2.5 mm at the edge. Pressure: 1 atmosphere (air).
The diameter of the drop has always been kept at 12 mm. This corresponds with a heat flux per degree temperature difference of 0.8' 10-2 cal sec-1 cm-2 °C-l with the brass at 500 0 e and the drop at about 90°C, which is in good agreement with literature. The thermal conductivity of water vapour at 300°C is approximately 1 X 10- 4 cal sec- 1 cm-1 degree-I. In case of our drop above a metal plate of 500°C, one finds that the heat transport would be about 6.6 cal sec-I, which is a factor 2 too large. This can be explained from a temperature jump at the liquid surface, caused by an apparent heat resistance, making dTjdR at the metal surface smaller than one would expect. c. The gas prod uetion. The observed heat transport leads to an evaporation of water at the lower siele of the drop. The warming up of the
J.
102
KISTEMAKER
- - - - - -_ . _ - - - - -
continuous flux of qIiq mm 3/sec of cold water flowing into the drop must be taken into account.
Q = O· (rT
+ [100 -
tl'oomJ c) 'qli'! rT = 540 cal (heat of evaporation of water) ; qIiq = volume of water evaporated per sec. ; c = specific heat of water; o = area of lower side of the drop. With Q = 3.3 cal sec"! we find for our drop with a wall temperature of 50QoC a qlll] = 5 rnmf sec- 1 - in good agreement with the observed quantity read from figure 4. If we take the average temperature of the gas in the film 300°C, this corresponds with a volume of gas of qgas = 11 cm 3 sec 1 (for total drop area, lower side). cm- 3
This gas quantity has to escape via the gas film and is the apparent reason of the elevation of the waterclrop. During this calculation we have supposecl that the diffusive evaporation from the unclercoolecl top surface of the drop can be neglected (in one atmosphere air). d. The thickness of the gas film. To discuss the observed flow of heat, the evaporation rate and the thickness of the gas film at about 500°C wall temperature, we need the equations of hydrodynamics. Considering a stationary state and an incompressible fluid, the main equation is:
1tP grad v2
-
p[v rot
vJ
=
-
grad p
+ 1](Llv + grad div v).
(a)
Here, v
= speed of the gas in em secv- ;
p
= density of the gas in grams;
p = pressure in dynes cm- 2 ; = viscosity of the gas in poises. grad div v = a (incompressible fluid).
1]
In cylindrical co-ordinates related to our drop problem. with the z-axis parallel to the drop-axis, this equation reduces to: 2
},p ov ~
or
=
..». + or
1]
(~ !'!- + r or
1 02v -
or 2
!.-) . r 2
(b)
We have supposed two things: I. All parts of the lower surface of the drop evaporate with the same speed of qgas em sec1 gas of atmospheric pressure. The temperature of this gas is supposed to be (T wall + T dl'OP) /2. This means, that the continuity equation (mass balance) in our case gives for the space between drop and metal wall:
f
surface
u- df
=
fvolumo
div v elt = :n;r 2 qgas'
(c)
THE SPHEROIOAL STATE OF A WAT ER DROP
103
Taking v as a mean speed value over the height of the gap h, this leads t o:
2nrhv = nr 2qg3>j'
_
(d)
or :
rqgas
(e )
V=--.
2ft
= 6 X 10-3 ern, and = 0.6 ern, this mean s that for qg,," ~ 11 cm 3 sec- 1 for total area
For a gap height It r givin g
v
~
500 cm sec-I, which is a relatively low value.
I I. All rotational t erms are small. We consider only the main stream in the medium plane between the hot wall and the lower side of the drop. This seems to be correct as viscosity terms are negligible with speeds of the order of 500 em sec l . Integration of equ ation (b) gives:
~pv2 = Po - P + 'YJ J(2.r !~ - ~) dr or + 1 02~ 81'2 1'2 from which we deduce by substitution of
v from
(e)
q
1'2 2
tp -4k 2 = po - p.
(I )
Substitution of: P H , O.go.><
at 100°C, 1 atm = 6 X 10- 4 gram ; at 300°C, 1 atm = 4 X 10- 4 gram ;
r = R = 0.6 em, as used in our exp eriments, 2.5 mm water ~ 100 dynes (pressure difference in H 2 0 -gas between middle of ga p, center of drop, and the edge of the drop), gives simply :
po - p ~ 3 .7 -
rr:':
It
=
~Rqga" V2~P
(g)
and the h valeus in the third column of table III. One might compare these results with Schmid and Speidel's empirical formula 7):
where : Ava pOlll' ~
1 X 10-4 cal sec-1 crrr ! degree" (at 300°C) ; T m et al - T b oili ng = temperature differ enc e between metal and boiling point of water; T b oll tng - TU'lUId. = t emperature differen ce between b oiling point and real t emperature of the wat erdrop ; p = pressure durin g the experiment.
104
THE SPHEROIDAL STATE OF A WATERDROP
The results are in table III. TABLE III Values of 11 in microns 1310ck temperature in °C
I
200 500
lz measured
I
Our equation (g)
I
60
I
20
30
I
46
Schmid and Speidel 12
I
100
We used the experimental numbers from tables I and II. A value of iJp of 200 dynes, instead of the 100 which we took, would give perfect agreement of our experiments with equation (g). A higher value of iJp seems reasonable indeed, as the gas at the edge of the gap - underneath the drop - starts to expand and the difference of height in the waterlevel in the drop gives, therefore, too small pressure differences. I thank Ir. L. H. J. Wachters for a useful discussion on equation (b). This work was partially financed by the Stichting vo or Fundamenteel Onderzoek der Materie, and the Stichting voor Zuiver Wetenschappelijk Onderzoek. Received 30·8·62
REFERENCES
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
For a small review, sec: Partington, ]. R, An advanced treatise on physical chemistry, II (1955) 282. Boerhaavc, H., Elemcnta chemiae Lugdunum Batavorum (Leyden) 4 tI (1732) 258. Eller,]. T'h. Mem, Acad. Berlin (1746) 43. Leidenfrost,]. G., De aquae communis nonnullis qualltatlbus tractatus, Duisburgi (1756) 30. 13outigny, P. M., Ann. Chirn. I) (1843) 350. Stark, J., Ann. d.Phys. 301 (1898) 306. Sc h ml d, G. and Speidel, H., Z. f. Elektrochemie 43 (1937) 187. Pleteneva and Rehbinder,]. phys, Chern, USSR 21 (1946) 961; J. phys, Chern. USSR 22 (1947) L Holst Weber, S. Th., Kg!. Danske Vld , Selsk., Mat. Fys. Medel. 14 (1937), 13. Van Wijk, W. Rand VanStralen, S. J. D., Physica zn (1962) 150. Jakob, M., Heat transfer, Vol. I, Wiley, New York, 1950. Mc Ad a m s, W. H., Heat transmission, 3rd Edit., McGrawHill, New York, 1945. Van Stralen, S.]. D., Ned. Tijdschr. Natk, 28 (1962) 10.