2482
J. Phys. Chem. 1987, 91, 2482-2486
Homogeneous Condensation in N2, Ar, and H20 Free Jets Georg Koppenwallner* and Carl Dankert DFVLR Institute for Experimental Fluid Mechanics, 0-3400 Gottingen, Federal Republic of Germany (Received: June 17, 1986)
Condensation in free jet expansions from sonic orifices is studied with nitrogen, water vapor, and argon as test gases. The stagnation temperature To is varied from 80 to 450 K by a liquid nitrogen cooler or a hot water bath, and in the case of water vapor by an electric driven steam generator. The stagnation pressure powas changed between 0.2 to 3 bar. The nozzle throat diameters are 5 and 10 mm. Static pressures are measured along the center line of nitrogen jets and the static temperatures are calculated by isentropic theory up to condensation onset. The condensation onset points on the center line of all free jets are detected by laser light scattering. The nitrogen laser data from free jets fit perfectly to static pressure measurements which are also compared to the classical homogeneous nucleation theory; the argon laser data are compared to experiments in free jets, nozzles, and shock tubes.
Introduction Measurements of condensation in supersonic flows can be conducted with static pressure probesZ or static temperature or with mass spectrometers detecting cluster size population in the flow field.I2-I5 This report deals with experimental studies of large free jets (nozzle throat diameters of 5 and 10 mm) with stagnation conditions near the vapor pressure line of pure gases Ar). The homogeneous condensation on the flow field (Nz, HzO, axis of supersonic free jets] can be described by the stream tube theory for flows with heat a d d i t i ~ n l ~as, ’long ~ as the mass reduction of the gas flow due to condensation is small. A static probe device2 is used to detect the static pressure rise due to condensation, and these experiments can be compared with calculations based on classical nuclection theory.3 Optical methods like light scattering are used in supersonic flow since 195118to detect condensate. The various laser light scattering experim e n t ~ in, which ~ ~ ~G. ~D.~Stein ~ ~was markedly involved, show
that the condensate radii can be assumed to range from 10 to 100 A. Therefore the scattered light can be described by Rayleigh scattering theory.9x21-23
Theory Free Jet Theory. The flow field of an isentropic supersonic free jet is shown in Figure 1 and can be calculated on the axis by the stream tube theory” with three basic equations: continuity: -1 -dP + - - 1+ d-Up dx U dx
1 dA A dx
(1)
momentum:
en ergy : dP + - 1- =dP - -1 o
( I ) Ashkenas, H.; Sherman, F. S. Rarefied Gas Dyn. 1966, 2, 84. (2) Dankert, C.; Koppenwallner, G. Rarefied Gas Dyn. 1979, 2, 1107. (3) Diiker, M.; Koppenwallner, G. Rarefied Gas Dyn. Pt. 2 1981, 1190. (4) Matthew, M. W.; Steinwandel, J. J. Aerosol Sci. 1983, 14, 755. (5) de Boer, B. G.; Kim, S. S.; Stein, G. D. Rarefied Gas Dyn. 1979, 2, 1151. (6) Pierce, T.; Sherman, P. M.; McBride, D. D. Astronaut. Acta, 1971, 16, 1. (7) Lewis, J. W. L.; Williams, W. D. “Argon Condensation in Free Jet Expansion”, AEDC-TR-74-32, 1974. (8) Stein, G. D. “Argon Nucleation in a Supersonic Nozzle”, Report of Office of Naval Research No. AD-A 007357/7GI, 1974. (9) Williams, W. D.; Lewis, J. W. L. ‘Experimental Study of Condensation Scaling Laws for Reservoir and Nozzle Parameters and Gas Species”, AIAA 14th Aerospace Sciences Meeting, Washington DC, 1976. (IO) Wu, B. J. C.; Wegener, P. P.; Stein, G. D. J. Chem. Phys. 1978,69 p 1776. (1 1) Kimura, A.; Tanaka, K.; Nishida, M. Rarefied Gas Dyn. 1980, 2, 1155. (12) Hagena, 0. F.; Obert, W. J. Chem. Phys. 1972, 56, 1793. (13) Golomb, D.; Good, R. E.; Bailey, A. B.; Busby, M. R.; Dawburn, R. J . Chem. Phys. 1972, 57, 3844. (1 4) Knof, H. Massenspektrometrie von Kondensarionskeimen in der Gasphase; Physik Verlag GmbH; Weinheim/Bergstrasse, 1974. (15) Obert, W. Rarefied Gas Dyn. 1979, 2, 1181. (16) Wegener, P. P. Noneguilibrium Flows; Marcel Dekker: New York, 1965. (17) Zierep, J. AGARDograph 1974, No. 191. (18) Durbin, E. J. ‘Optical Methods Involving Light Scattering for Measuring Size and Concentration of Condensation Particles in Supercooled Hypersonic Flows”, NASA-TN-2441, 195 1. (19) Daum, F. L.; Farrell, C. A. “Light Scattering Instrumentation for Detecting Air Condensation in a Hypersonic Wind Tunnel”, 4th International Congress on Instrumentation in Aerospace Simulation Facilities, Brussels, June, 1971, p 209.
=o
p dx
KP dx
(3)
The description of a free jet in the subsonic and transsonic region is based on experiment^.',^*^^ The hypersonic expansion (x/D > 4) can be calculated by the free jet theory of Ashkenas and Sherman. I The Mach number distribution and the normalized temperature gradient along the free jet axis are given in Figure 2 with nitrogen (K = 1.40) as a test gas. The sub- and transsonic distributions s ~ ~ a static pressure probe; in the are based on m e a s ~ r e m e n t with supersonic part a Pitot pressure probe was used in addition. The cooling rates in such free jets are shown in Figure 3, normalized to the orifice diameter and the stagnation temperature. The maximum cooling rates are observed near the orifice at M = I . The Condensing Free Jet. Figure 4 shows the vapor pressure curve of nitrogen on a p-T graph and sonic isentrops. Possible stagnation conditions ( M = 0) for free jet experiments are also plotted. During the isentropic expansion the vapor pressure curve is crossed from the gas phase to the liquid at MI and from the liquid to the solid phase (or from gas to solid) at M,. The gas expands further on isentropically till at a corresponding supersaturation the condensation process starts. Flows with heat addition and mass reduction due to condensation were studied by Zierep.17 With the assumption that the (20) Kim, S. S.; Shi, D. C.; Stein, G. D. Rarefied Gas Dyn. 1980,2, 1211. (21) Williams, W. D.; Lewis, J. W. L. AIAA J. 1975, 13, 709. (22) Kerker, M. The Scattering of Light; Academic: New York, 1969. (23) Born, M.; Wolf, E. Principles ofopfics;Pergamon: New York 1983. (24),Dankert, C.; Koppenwallner, G. “Homogene Gaskondensation in Expansionsstromungen - Aufbau des Analysesystems und vorbereitende Messungen”, DFVLR-AVA Bericht IB 252-75 H 06, Gottingen, 1975.
0022-3654187 12091-2482$01.50/0 0 1987 American Chemical Societv
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2483
Homogeneous Condensation in Free Jets free jet boundary
D T
-1
0
2
-
4
;-
6
8
Figure 3. Temperature gradient in a sub- and supersonic free jets.
I
I
gas
,
I
1
I
I
x
D 3
(M)k=,o
nitrogen
1.40
0.40
8.7
argon
1.67
0.075
(p)x-,o
15
D-
6.10-5
2.10-~
Figure 1. The isentropic free jet.
Figure 4. Vapor pressure curve of nitrogen and some isentropics.
The classical theory is based on (1) the nucleation process; (2) the growth of individual droplets; and (3) integration of droplet formation and droplet growth along the free jet to obtain the total number density of droplets and the condensed mass fraction. We used the classical nucleation theory of Vollmer as described by Wegener16 with the following relation^:^ (1) The size r* and the surface area o* for a critical droplet are
Figure 2. Sub- and transsonic free jet measurements: supersonic theory,’ K
(2) The equilibrium number density n,* of critical droplets is n,* = n e x p ( -AG* F) (7)
= 1.40.
size of the free jet is not influenced by condensation the three basic equations (eq 1-3) can be setup for condensing flows: conti nuit y :
(3) The nucleation rate (rate of formation of supercritical droplets of radius r > r*, i.e., nuclei) is J JK = J = nnO* exp(
-%);
momentum stays unchanged (2) energy:
with -2
c ! E + U d U = - -Hcond dmcond
(5) dx dx m dx This system (eq 4 , 2 , and 5 ) of coupled differential equations can be s01ved3J7if A(x) is given. In hypersonic flows (@ >> 1) and small condensed mass fraction the heat release, the condensed mass, the static pressure jump, and the static temperature can be calculated.2 To define condensation onset we used the change of static pressure p ( x ) , where a 10% increase is defined to be the condensation onset point, called Wilson point. The Classical Condensation Theory. This theory must give for the above equations the condensation rate input dmmnd/dx.
kT
3
Ps( T )
Our preexponential factor nnQ* equals Wegener’s K2 factor (ref 16, p 176): JK = J / n is the rate of nuclei formation per unit time and molecule. For droplet growth the “Hertz-Knudsen” formula or the Biihler-Oswatitsch relation can be
’
(25) Dankert, C. Rarefied Gus Dyn. 19$4,2,983-990. (26) Oswatitsch, K. Z . Angew. Mechanik 1942, 22, 1-14. (27) Wagner, B.; Duker, M. “Prediction of Condensation Onset and Growth in the European Transonic Wind Tunnel ETW”, AGARD No. 348, 1984, p 13-1.
2484
The Journal of Physical Chemistry, Vol. 91. No. 10, 1987
Koppenwallner and Dankert
I
xi0
To
I
I
I
I
I
4
5
6
/
-
F w r e 5. Nucleation rate JK and number density jet. po
I
I
nozzle
nd of nuclei in the free windtunnel
-
,
1
0
1
2
3
8-
free jet
Figure 7. Measurements of static pressure in nitrogen free jets with nucleation and condensation. laser*
/-I
/-
"
/ /
'.(
Mach disk
liquid N,
Figure 6. Experimental setup.
Our integration scheme along the free jet is explained in ref 3 and 27. In Figure 5 we show an example calculation of the nucleation rate JK along a free jet expansion and the integrated number density of nuclei formed during this expansion. Nucleation rate JK shows a maximum shortly before the condensation onset. Further example calculation showed that the value of the JK maximum strongly depends on the physical data of surface tension used in the nucleation rate equation. The position of the JK maximum along the expansion depend however only weekly on these input data.
Experimental Methods h e r Arrangements and Wind Tunnel. The measurements are conducted in the low density wind tunnel of the DFVLR-AVA Gottingen. A schematical view of the experimental setup is given in Figure 6 . The wind tunnel diameter is 400 mm. Test gases are nitrogen ( K = 1.40), water vapor ( K = 1.33), and argon ( K = 1.67). In case of water vapor as the test gas all pipes outside and (28) Vas, F. L.; Koppenwallner, G . "The Princeton University HighPressure Hypersonic Nitrogen Tunnel N-3", Department of Aerospace Mechanical Science, Princeton University, 1964, Report 690. (29) Hefer, G. "Die zweite Mepstrecke des hypersonischen Vakuumwindkanals der AVA - Baubeschreibung und Betriebsverhalten", DLR-FB 70-42, 1970. (30) Diiker, M. "Nitrogen Condensation in Stream Tubes Duplicating the Airfoil Flow in a Cryogenic Transsonic Wind Tunnel", DFVLR-AVA Bericht IB 222-83 A 08, GBttingen, 1983.
14
0
e xperimentsZ
- - theory 26,27
1
2
3
8-
4
5
6
7
8
Figure 8. Condensed mass fraction along free jet axis compared to classical theory.
inside the wind tunnel are kept at 380 K to avoid condensation in the pipes. The stagnation pressure can be varied from po = 0.2 to 3 bar, and the stagnation temperature from To= 80 to 500 K. The sonic orifices2s diameters are D = 5 and 10 mm. The light source is an argon ion laser (4 W) with a wavelength of A = 5145 A and 0.2-W output adjusted. The laser is located outside the wind tunnel and aimed through a window kept at 300 K and through the stagnation chamber, the nozzle throat, and the expansion field downstream on-axis of the free jet. The scattered light is observed under 90' through a second window a t 300 K by a simple observation device with the eye piece on a traversing mechanism and a millimeter position readout. When condensation occurs the droplets scatter enough laser light t o see a lightened blue tube on the free jet axis from condensation onet point to the Mach disk downstream. Pressure Measurements. To measure the static pressure, a 1-mm central probe with pressure holes is installed on the center line of the free jet. It can be moved along the center line, thus enabling static pressure measurements from the stagnation chamber through the throat into the hypersonic part of the free jet. Pitot pressure surveys are also conducted along the nozzle axis, but no condensation effects on Pitot pressure could be found.2 Results A sample of the static pressure results for the 5-mm throat diameter and nitrogen is given in Figure 7. The stagnation
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2485
Homogeneous Condensation in Free Jets
t
-
-.
1.106
Y
0 .I. 1061 50
100
150
TO
IO
050
10
025
200 K 250
Figure 11. Temperature gradients at condensation onset in free jets.
35t
\ mLond=30%
\
\
‘
25%
20%
15%
Equilibrium condensed
s-so
18
I
fractions I
16
I
Q
po=0.5 bar
I
20
22
R
Figure 9. Specific entropy during condensation in a N2 free jet: mm, po = 1 bar, To = 150 K.
D = 10
100 K
I.
suDersonic nozzles’
I
i
f
60
I
100
)
To
150
I K
200
Figure 12. Condensation onset in N2 free jets; comparison of laser light
scatteringZ5and static pressure probe.2
Torr
-
100
Ts Figure 10. Condensation onset in N, flows: Tw-T8graph.
pressure is 3 bar, and the stagnation temperature is varied from 280 to 110 K. At condensation onset, the static pressure starts to increase above the corresponding isentropic value. The calculated beginning of nucleation is also given in this plot. The condensed mass fraction can be derived from eq 2,4, and 5 and the hypersonic assumption A@ >> l S 2A sample of the obtained results is given in Figure 8 for a 5-mm nitrogen free jet. The solid curve represents the measurements, the pointed line is a calculation using the Hertz-Knudsen formula,16and the dashed curve represents the Oswatitsch-Buhler t h e ~ r y ~with ~ , ~a’ surface tension depending on temperature. In Figure 9 the expansion along the axis of a condensing nitrogen free jet is plotted on a T-S diagram. Beginning with the Wilson point the condensation proceeds until nearly saturation conditions are reached, and the expansion then continues isentropically without further condensation on a new isentrop. Near the Wilson point, the equilibrium condensed mass fraction would be 27%. Such a high rate is, however, never reached in the expanding nitrogen free jet, see Figure 8. Figure 10 shows the experimental Wilson point in nitrogen flows of supersonic nozzlesg and free jetsZ in a T,-T, diagram. All experimental data show an adiabatic supercooling of T,-Tw = 35 K. In addition the location of the maximum nucleation rate
I 0.001
5 10
I
8/ P-
20
30
TW
I 40
50
K 60
Figure 13. Onset of condensation in supersonic nitrogen flows: p-T
diagram. calculated for nitrogen free jets3 is shown, given slightly lower values of T,-T,. The temperature gradients at the experimental Wilson points are shown in Figure 11, plotted in K/s as function of the stagnation temperature. All free jet values are between lo5 and 2 X lo6 K/s. In our nitrogen experiments the relaxation time, defined as the time from saturation to condensation, is about 20 ~ s .
2486 The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 200
" o o
I
A
0 0
3.
,/'
~
91
To :LOOK po ~0.5...0.9 bar n =1.33 1 7
'
1
1
I
Koppenwallner and Dankert All measured data are on a curve that diverges from the vaDor pressure line at higher expansion rates. The laser data in H 2 0 free jets are plotted in Figure 14. Again a large condensation onset delay was observed. A series of experiments on condensation onset in argon flows is shown in Figure 15: laser light scattering in free jet, static probes in free jets and nozzles, Ar-He mixtures in nozzles, and shock tube data. The supercooling rates of the argon free jets, having extremely high pressure and temperature gradients, are highest,'^^.^^ followed by the supersonic nozzles,6v'0 and in shock tubes the condensation onset is found near the vapor pressure line.4 I
Conclusion The free jet expansion is a useful tool to study condensation onset phenomena. In order to detect condensation static pressure probes and laser light scattering were used. Both methods give comparable results. In the p T phase diagram the condensation onset data of nitrogen merge into one curve, which indicates that condensation onset is quite independent from jet size however strongly dependent on the expansion isentrope. Calculations with classical condensation theory could only reproduce the measurements after empirical adjustment of the surface tension law for solid nitrogen. Acknowledgment. This contribution is dedicated to our scientific friend Gil Stein. Gil Stein was one of the outstanding contributors for the condensation sessions at the biannual International Symposia on Rarefield Gas Dynamics. We and this scientific community lost a good friend and an excellent scientific collegue. List of Symbols A
D HWnd mCond
incan: M nc
n
n n*
PW
01 10
50
30 Tw
70
O* P PK Pw Po p,(T) Pstat
90 K 100
Figure 15. Onset of condensation in supersonic argon flows: p-T diagram.
r* (S -
T Ts
SO)/R
Tstat
TW
A sample of the light scattering results in nitrogen free jetsZ5 is given in Figure 12 for nozzle throat diameters of 10 mm and a stagnation pressure of 0.5 bar. The varied stagnation temperature is plotted vs. the Wilson point on the free jet axis. In addition to the laser light scattering measurements the data of the on-axis2 static probe are given. Both methods fit perfectly within the error bars. The pressure ratio of stagnation to chamber pressure is po/pKL 3000. All Wilson points of supersonic nitrogen flows (free jets and supersonic nozzles) are shown in Figure 13 on a p-T diagram.
TO U
vm X
XO
(x/D),
AG* K
x 0
P
area of stream tube, m2 throat diameter, mm condensation enthalpy, J/kg condensed mass, kg condensed mass fraction Mach number number density of droplets number density of molecules flux of molecules to unit surface area number of molecules in critical droplet surface area of critical droplet pressure, bar, Torr chamber pressure, bar pressure a t Wilson point, Torr stagnation pressure, bar vapor pressure, Torr static pressure, Torr radius of critical droplet specific entropy, (S - S,)/R = 3.5 In T [K] - In p [bar] temperature, K temperature at saturation point, K static temperature, K temperature at Wilson point, K stagnation temperature, K flow velocity, m/s specific volume per molecule in the condensed state axial distance from nozzle throat, mm virtual source point, mm condensation onset point (Wilson point) Gibbs free formation enthalpy of a critical droplet ratio of specific heats wavelength, 8, surface tension, surface energy per unit area density, kg/m2
