CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 76, 2019 

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

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Copyright © 2019, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-73-0; ISSN 2283-9216

Effect of Supersonic Nozzle Structure on Vapor Spontaneous 
Condensation 

Zhenya Duana,*, Longhui Lianga, Yushen Xieb, Zhiwei Maa, Chunliang Lic 
a
Qingdao University of Science and Technology, College of Mechanical and Electrical Engineering, Qingdao 266061, 

 China  
b
China University of Petroleum (Beijing), College of Chemical Engineering and Environment, Beijing 102200, China 

c
CNPC East China Design Institute Co., Ltd., Equipment Dept, Qingdao 266071, China 

 zyduan88@163.com 

To study the effect of the supersonic nozzle structure on the spontaneous condensation of water vapor, the 
wet steam model used to study the wet steam liquefaction of steam turbine was introduced into the numerical 
study of supersonic nozzle in supersonic cyclone separator, and the numerical results was in good agreement 
with the experimental data of the related literature. The supersonic nozzles, the contraction section with the 
Witozinsky curve and the bicubic curve, and the expansion section with conical curve and Foelsch curves, 
were designed. The influence of supersonic nozzle structure on water vapour liquefaction ability was 
investigated. And the following research results were obtained. When the structure of the supersonic nozzle 
was the same, the Witozinsky curve was selected for contraction section of the supersonic nozzle, and the 
nucleation rate of the droplet in the nozzle increased by 21.97 % and the number of droplets increased by 
32.16 %, which was helpful to the gas liquefaction in the nozzle. When the contraction curve of the supersonic 
nozzle was consistent, and the expansion curve was Foelsch curve, the gas nucleation rate and the number of 
droplets in the nozzle were greatly improved. The distribution of nozzle humidity was related to the structure of 
expansion section, but not to the structure of contraction section. The maximum humidity in nozzle depended 
on the outlet diameter of nozzle. When the other structures of the nozzle were consistent, the cone angle of 
divergent section increased from 2° to 8°, the peak value of the nucleation rate increased by 493.15 %, the 
maximum of the droplet number increased by 246.12 %, the outlet humidity of the nozzle increased by 73.37 
%, which greatly improved the gas liquefaction rate. In the case of gas liquefaction, increasing the cone angle 
of expansion section was conducive to liquefaction of water vapor.  

Introduction 
Supersonic cyclone separator is a new type of natural gas dehydration and heavy hydrocarbon removal 
device based on adiabatic expansion of rotating airflow, which integrates condensation and separation 
functions (Duan et al., 2018). As one of the core components of supersonic cyclone separator, supersonic 
nozzle can make natural gas expand adiabatically from subsonic state to supersonic state, and provide 
sufficient condition of low temperature and low pressure for spontaneous condensation of water vapor in 
natural gas (Wen et al., 2012). The spontaneous condensation process of water vapor in natural gas is an 
indispensable condition for natural gas purification process in supersonic cyclone separator. The design of 
supersonic nozzle can directly affect the separation efficiency of supersonic cyclone separator (Wang and Hu, 
2018). Therefore, the influence of supersonic nozzle structure on steam liquefaction should be considered in 
the design of supersonic nozzle for supersonic liquefaction. Han et al. (2016) studied the regularity of 
distribution of natural gas pressure, temperature, Mach number and other characteristic parameters on the 
axis line of the unit without considering the effect of condensation, and evaluated the performance of 
supersonic cyclone separator according to dew point drop and separation efficiency. Yang et al. (2017) 
applied nucleation and droplet growth theory to study the condensation process of water vapor in supersonic 
nozzle. The results showed that latent heat was released into vapor phase during spontaneous condensation, 
which leaded to jump of condensation parameters. Shooshtari and Shahsavand (2013) provided a new 

  DOI: 10.3303/CET1976087 

Paper Received: 11/03/2019; Revised: 01/04/2019; Accepted: 01/04/2019 
Please cite  this article as: Duan Z., Liang L., Xie Y., Ma Z., Li C., 2019, Effect of Supersonic Nozzle Structure on Vapor Spontaneous 
Condensation, Chemical Engineering Transactions, 76, 517-522  DOI:10.3303/CET1976087 

517



method for predicting droplet growth in binary mixtures in supersonic nozzles based on mass transfer rate 
calculation, which overcame the disadvantage that the estimation of droplet growth in pure fluids (such as 
steam) cannot be extended to binary or multicomponent systems. Jiang et al. (2010) established a numerical 
model for spontaneous condensation flow of two-component mixtures, and investigated the effect of entrance 
parameters on spontaneous condensation flow. Cao and Yang (2015) simulated the condensation 
characteristics of binary gas in supersonic nozzle by user defined function (UDF) in Fluent and analysed the 
effects of carrier gas, inlet pressure and inlet temperature on the distribution of condensation parameters. 
At present, the research on supersonic cyclone separator mainly focuses on structural optimization and 
internal condensation process, and seldom considers the influence of supersonic nozzle structure on 
condensation and liquefaction of water vapor. To accurately and simply study the influence of nozzle structure 
on water vapor condensation process, the wet steam model was used to carry out relevant research. The 
influence of supersonic nozzle structure on vapor liquefaction was systematically studied by changing the 
contraction section, expansion section line and expansion section cone angle of supersonic nozzle.  

Supersonic nozzle design 
At present, the design of supersonic nozzle for contraction section is mainly based on Witozinsky curve and 
Bicubic curve, and the design of expansion section is based on Foelsch curve method and conical curve 
method. 

1.1 Contraction section design 

Witozinsky curve: 

22

22
1

cr cr
32

1

1

1

1 1
1

1
3

x
LR R

R R x
L

  
−           = − −   

         
+  

     

(1) 

Bicubic curve: 

( )

3

cr
m2

1 cr m 1 1

3

cr
m2

1 cr 1 1m

1
1

1
1

1

R R x x
X

R R X L L

R R x x
X

R R L LX

    −
 = −    −    


   −
= −    − −    

， ≤

， ≥

 

(2) 

Where x is the axial distance of the nozzle, R is the radius of the cross section at x, and Xm is the relative 
position of the intersection of the front and back curves (Xm=0.5). 

1.2 Expansion section design 

Foelsch curve is designed based on method of characteristic curve, including throat transition section, straight 
line section and eliminate wave section. Because the design process of this method is rather complicated, it is 
no longer discussed here, and detailed design steps can be referred to literature (Liu et al., 2000). To simplify 
the design procedure and calculation of the expansion section of supersonic nozzle, the conical curve method 
can be employed to design the expansion section curve.  
Based on the above design method of supersonic nozzle, the main structural parameters of the designed 
supersonic nozzle are as follows: the length of contraction section L1 = 40 mm, the length of expansion section 
L2 = 110 mm, inlet diameter R1 = 40 mm, throat diameter Rcr = 10 mm. 

Numerical method 
The wet steam model in ANSYS Fluent 16.0 was employed to calculate the spontaneous condensation 
liquefaction process of water vapor in supersonic nozzle. The density-based implicit solver was utilized to 
solve the numerical calculation. The RNG k-ε model was employed for turbulence model and the second-order 
upwind scheme was employed for discretization of governing equations. In order to ensure the accuracy of 
solution, the standard residual of convergence was set to 10-6. The inlet boundary of supersonic nozzle was 
pressure inlet, and the total inlet pressure, static pressure, total temperature, turbulence intensity and 
hydraulic radius were set. The outlet boundary was pressure outlet, and the wall boundary of supersonic 
nozzle was non-slip, stationary and adiabatic wall. 

518



Results and analysis 
1.3 Model validation 

To validate the rationality and accuracy of the wet steam model used in this paper, the experiment of Moses 
and Stein (Moses and Stein, 1977) was employed to the model verification. In the experiment of Moses and 
Stein, the condensation process of water vapor in supersonic nozzle was studied. The throat of the nozzle is 
located at x=82.2 mm, and the throat cross section size is 10 mm×10 mm. The geometry of the supersonic 
nozzle used in the experiment is described in Figure 1. The subsonic part is composed of an arc with a radius 
of 53 mm, while the transonic and supersonic part is consisted of an arc with a radius of 686 mm. Under the 
conditions of inlet pressure 70,727.32 Pa and inlet temperature 377.15 K, the pressure value obtained by the 
experiment decreases gradually along the central axis of the nozzle, and when the water vapor condenses 
spontaneously, the pressure rises slightly and then decreases gradually, as shown in Figure 2. This is due to 
the condensation wave generated by latent heat of condensation released from the condensation process of 
water vapor, which leads to the increase of pressure. The wet steam model employed in this paper accurately 
predicts the spontaneous nucleation process of water vapor in supersonic nozzle, and the simulated pressure 
distribution is in good agreement with the experimental date of literature, which proves that the application of 
the wet steam model is completely credible in the study of water vapor liquefaction in supersonic nozzle. 
 

 

Figure 1: Geometry and dimensions of supersonic nozzles in the literature 

 

Figure 2: Comparison of numerical results in this paper with experimental data in literature 

1.4 Effect of nozzle contraction curve on spontaneous condensation  

Under the same structure of expansion section of supersonic nozzle, the geometric models of supersonic 
nozzle with the contraction section of Witozinsky curve and bicubic curve (hereinafter referred to as nozzle A 
and nozzle B) were established respectively to study the influence of the contraction section curve of 
supersonic nozzle on the variation of vapor liquefaction parameters. The inlet undercooling of nozzle A and B 
were -11.6 K and -12.6 K respectively, and with the expansion of water vapor in the nozzle, the undercooling 
increased continuously. The undercooling of nozzle A reached a peak value of 36.9 K at x= 46.5 mm, and 
nozzle B reached a peak value of 36.6 K at x= 45.5 mm. Water vapor began to condense, and the latent heat 
of condensation was generated to heat the water vapor in the nozzle. The increase of temperature led to the 
decrease of undercooling until the final stable value of 2 K was reached, as indicated in Figure 3(a). The 
undercooling of nozzle A was always higher than that of nozzle B in the contraction section (x=0~40 mm), 
which indicated that nozzle A could provide low temperature environment and power for droplet nucleation 
and growth for condensation of water in natural gas at the earliest stage. As can be seen from Figure 3(b), the 
maximum nucleation rates of nozzle A and B were 3.72×10

22 and 3.05×1022 m-3·s-1, respectively. Compared 

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0.00

0.02

0.04

0.06

y/
m

x/m

0.00

0.02

0.04

0.06

0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

p/
p 0

x/m

 Experimental data 
 Numerical results  

519



with nozzle B, the nucleation rate of nozzle A increased by 21.97 %. Figure 3(c) indicated that the maximum 
droplet number of nozzle A and B was 2.63×1022 and 1.99×1022 m-3, respectively. Compared with nozzle B, 
the droplet number of nozzle A increases by 32.16 %. Figure 3(d) described the distribution of humidity, and 
the nozzle contraction curve had little effect on humidity. Under the same structure, the humidity of steam in 
nozzle was related to the number of droplets and the diameter of droplets. The imbalance of flow was 
gradually restored by the combination of the number of droplets and the radius of the droplet, whether it was a 
large number of small droplets or a smaller number of droplets (Han et al., 2011). The above results showed 
that when the contraction curve was Witozinsky curve, the nucleation rate and droplet number in the nozzle 
were better than those of the bicubic curve, and the contraction curve of the nozzle had little effect on the 
humidity. 

 

Figure 3: Effect of contraction section curve on condensation liquefaction parameters 

1.5 Effect of nozzle expansion curve on spontaneous condensation 

Under the condition that the contraction section of supersonic nozzle was consistent and the outlet diameter 
was the same, the geometrical model of supersonic nozzle with expansion section curve as conical curve and 
Foelsch curve (hereinafter referred to as nozzle C and nozzle D) were established respectively to study the 
effect of the curve of supersonic nozzle expansion section on the process of water vapor liquefaction. 

 

Figure 4: Effect of contraction section curve on condensation liquefaction parameters 

520



In the region of droplet nucleation (x = 42.5~47 mm), the undercooling of nozzle C was lower than that of 
nozzle D. Correspondingly, in terms of droplet nucleation rate and droplet number distribution, nozzle D was 
better than nozzle C, as shown in Figure 4. The maximum nucleation rates of nozzle C and D were 
14.82×1022 and 22.09×1022 m-3·s-1 respectively, and the droplet numbers were 6.13×1022 and 9.85×1022 m-3 
respectively. Compared with nozzle C, the maximum nucleation rate and droplet number of nozzle D 
increased by 49.06 % and 60.69 %, respectively. Figure 4(d) showed the influence of nozzle contraction 
section curve on humidity distribution, and combining with Figure 3(d), it can be seen that the distribution of 
humidity in nozzle was only related to the structure of nozzle contraction section. The contraction section 
curve of nozzle affects the distribution of humidity, and the nozzle outlet diameter determined the value of the 
outlet humidity. The above analyses indicated that when the curve of expansion section was designed by the 
Foelsch curve, the nucleation rate and droplet number of water vapor in the nozzle were greatly improved, and 
the distribution of nozzle humidity was related to the structure of the expansion section, and the humidity at 
the nozzle outlet depended on the diameter of the nozzle outlet. 

4.4 Effect of cone angle of nozzle expansion section on spontaneous condensation 

The condensation process of water vapor in natural gas in supersonic cyclone separator mainly occurred in 
the expansion section of supersonic nozzle. When the conical curve was adopted in the nozzle expansion 
section, the effect of the conical angle of the nozzle expansion section on the condensation liquefaction 
process of water vapor was studied by changing the conical angle of the expansion section with keeping the 
other structures consistent. 
 

 

Figure 5: Effects of cone angle of expansion section on condensation liquefaction parameters 

Figure 5 showed the distribution of undercooling, nucleation rate, droplet number and humidity at the central 
axis of the nozzle in the cone angle range of 2°~8°, respectively. According to the theory of aerodynamics, 
when the inlet parameters of supersonic nozzle were the same and there was no shock wave in the nozzle, 
the larger the cone angle of the expansion section was, the greater the gas velocity was, and the greater the 
change intensity of pressure and temperature was at the same position. The analyses of the numerical results 
showed that the rapid change of gas temperature and pressure would make the undercooling increase faster 
when the cone angle was enlarged, so that the water vapor reached the Wilson point earlier, as shown in 
Figure 5(b)~Figure 5(d). When the peak value of undercooling of water vapor increased, the maximum 
condensation nucleation rate, the droplet number and outlet humidity of water vapor increased. When the 
cone angle increased from 2° to 8°, the peak value of nucleation rate increased 493.15 %, the maximum 
droplet number increased 246.12 %, the outlet humidity of nozzle increased 73.37 %, which greatly increased 
the liquefaction rate of water vapor. The above analyses indicated that the liquefaction rate of steam could be 
increased by increasing the cone angle of expansion section. With the increase of cone angle of nozzle 
expansion section, the temperature drop and pressure drop in supersonic nozzle with the same length of 
expansion section also increased. The excessive temperature drop and pressure drop could make the 

521



temperature and pressure of water vapor lower than the three-phase point and be in a state of non-
liquefaction, which would limit the condensation liquefaction process of water vapor in supersonic nozzle (Wen 
et al., 2015). In the design of supersonic nozzle for supersonic cyclone separator, it was necessary to choose 
the appropriate cone angle to improve the liquefaction rate of water vapor under the condition of ensuring 
water vapor liquefaction. 

5. Conclusions 
The wet steam model was introduced into the numerical calculation of wet steam liquefaction in supersonic 
nozzle, and the numerical model was verified and analyzed on the basis of relevant literature. The numerical 
result was in good agreement with the experimental data of the relevant literature, which proved the feasibility 
and accuracy of the wet steam model applied to the liquefaction of supersonic nozzle. 
In the case of the same structure of the expansion section of the supersonic nozzle, the Witozinsky curve was 
selected for the contraction section of the supersonic nozzle, comparing with the nozzle employed by bicubic 
curve, the nucleation rate of droplets in the nozzle increased by 21.97 % and the droplet number increased by 
32.16 %, which was helpful to the liquefaction of water vapor in the supersonic nozzle. When the contraction 
curve of the supersonic nozzle was the same and the expansion curve of the supersonic nozzle was Foelsch 
curve, the nucleation rate and droplet number of water vapor in the nozzle were greatly increased. The 
distribution of nozzle humidity had nothing to do with the line type of contraction section, but with the line type 
of expansion section. The humidity at nozzle outlet depended on the diameter of nozzle outlet. 
With the same nozzle structure, the cone angle of the expansion section increased from 2° to 8°, the peak 
nucleation rate increased by 493.15 %, the maximum droplet number increased by 246.12 %, and the outlet 
humidity of the nozzle increased by 73.37 %, which greatly improved the liquefaction rate of water vapor. In 
industrial applications, under the condition of ensuring water vapor liquefaction, increasing the cone angle of 
expansion section was beneficial to water vapor liquefaction. 

Acknowledgments 

This study was supported by the key research and development program (public welfare special) project of 
Shandong Province (2017GGX40113), the science & technology development project of Weihai 
(2016GGX022), the plan project of Qingdao applied basic research (17-1-1-93-jch). 

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