Optimal operation of refrigeration oriented supersonic separators for natural gas dehydration via heterogeneous condensation

Accepted Manuscript Optimal Operation of Refrigeration Oriented Supersonic Separators for Natural Gas Dehydration via Heterogeneous Condensation S.H. Rajaee Shooshtari, A. Shahsavand PII: DOI: Reference:

S1359-4311(17)36945-4 https://doi.org/10.1016/j.applthermaleng.2018.04.109 ATE 12106

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

30 October 2017 10 February 2018 22 April 2018

Please cite this article as: S.H.R. Shooshtari, A. Shahsavand, Optimal Operation of Refrigeration Oriented Supersonic Separators for Natural Gas Dehydration via Heterogeneous Condensation, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/j.applthermaleng.2018.04.109

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Optimal Operation of Refrigeration Oriented Supersonic Separators for Natural Gas Dehydration via Heterogeneous Condensation S.H. Rajaee Shooshtari, A. Shahsavand1 Department of Chemical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract Refrigerative supersonic separators (3S's) have found extensive applications in natural gas industry, especially for dew point corrections of water vapor and heavier hydrocarbons. Previous studies indicated that even at heavy refrigeration duties and staggering radial acceleration values (>500,000g) generated inside 3S units, the condensed water (or hydrocarbon) droplets should have a minimum diameter of 2 micrometers to provide sufficiently large dehydration efficiencies. To promote the overall condensation rate and facilitate the separation of condensed phase from natural gas streams, certain rates of edible salt particles are assumed in the present article to be injected into the gas flow at the 3S unit entrance. The plenum chamber static vanes are also positioned after the throat location to enjoy the full swirling effect. Our simulation results indicate that by using the optimal structure of the 3S unit for a typical case study, the minimum solid particle injection rate is around 2.4 weight percent to achieve almost complete overall separation efficiency of the condensed water droplets. Moreover, the overall pressure recovery of the entire 3S unit can be boosted up to 83% for such optimal structure.

Keywords:

Refrigeration,

Supersonic

Separator,

Heterogeneous condensation, Solid particle injection

1

Corresponding author,

Email: [email protected]

1

Swirl,

Natural

gas

dehydration,

1. Introduction World energy consumption is forecasted to be monotonically increased from 524 quadrillion Btu in 2010 to 820 quadrillion Btu in 2040 [1]. Around 25% of this gigantic energy consumption should be provided by natural gas resources. Natural gas is a more environmentally attractive fuel compared with other hydrocarbons and continues to be favored due to abundant resources and robust production. Natural gas processing facilities are often far from final consumers and pipelines are traditionally used to transport natural gases to their ultimate destinations [2]. Dehydration of these methane rich gases is necessary to prevent many difficulties such as corrosion and hydrate formation in the corresponding trunk-lines [3]. Conventional processes such as absorption, adsorption and cryogenic dehumidification methods are energy demanding, expensive with adverse environmental impacts. The refrigeration oriented supersonic separators provides a high efficiency, more reliable and cost effective replacement for conventional processes such as Joule-Thompson valves and cryogenic hydraulic turbines [3]. In a pioneering research, Jassim et al. [4,5] employed computational fluid dynamic (assuming real gas behavior) for investigating the effects of both nozzle geometries and swirl flow on high-pressure natural gas dehydration in a Laval nozzle. Karimi and Abdi [6] investigated one dimensional flow of a natural gas mixture containing methane, ethane, propane and water vapor inside Laval nozzle in the absence of both swirl and condensation processes. Malyshkina [7,8] theoretically studied the gas dynamics behavior inside a supersonic separator. Two-dimensional Euler model was used to investigate the swirling flow of natural gas across Laval nozzle. In 2011 and in a series of research articles, the swirling flow of natural gas inside 3S unit was investigated by Wen et al. [9-11]. The first article [9],emphasized that the natural gas pressure, temperature and its tangential velocity had non-uniform distributions in the presence of swirl, which may affect separation of water vapor and heavy hydrocarbons. In the subsequent one

2

[10], three different swirl strengths were investigated inside supersonic separator. It was concluded that the moderate swirl provided the best performance by creating both lowest temperature and highest centrifugal acceleration during the separation of water and hydrocarbons from a natural gas stream inside 3S unit. Later one, the effects of nozzle geometry on swirl efficiency and corresponding separation performances were discussed in sufficient details [11]. Wen et al. [12] predicted particles separation characteristics due to the swirl motion using Discrete Particle Method by assuming typical size of the liquid particle in the range of 0.1– 4µm. Yang et al. [13] investigated the swirling flow inside the supersonic separators while the swirl generator was located in the supersonic region. Artificial neural networks were also used to design both pilot and industrial scales 3S units for natural gas dehumidification processes [14]. Effects of various diffusers geometries were numerically studied by Wen et al [15] on shock wave and pressure recovery performances for the single phase gas flow inside various 3S units. They concluded that the conical diffuser positioned after the straight tube provided the best pressure recovery performance for the dehydration of saturated natural gas. In a similar research, Yang et al. [16] successfully compared the pressure recovery inside 3S unit via both numerical and theoretical (analytical) methods predictions. The FLUENT software was used in the numerical section. Vaziri and shahsavand [17] performed maximization of swirl strength and centrifugal acceleration for single phase flow inside 3S unit via considering various inlet flow velocity components. Castier [18,19] employed thermodynamic models to study various distributions of natural gases intensive properties across the entire converging-diverging nozzles. Secchi et al. [20] also proposed a one dimensional mathematical model for preliminary design of 3S units to extract heavier hydrocarbon fractions from natural gases.

3

Bian e al. [21] used three dimensional simulations to investigate the single phase flow of natural gas inside 3S units. The optimal combinations of inlet and exit pressures were computed to provide both minimum pressure loss and adequate temperature profile across the entire device. Surprisingly, they reported that the inlet pressure of 600 kPa and pressure loss ratio of 47.5% were best choices for their case study. Effects of three different delta wings (small, medium and large) were analyzed on the swirling flow of the dry natural gas by Wen et al. [22] in the absence of any condensation. They concluded that the large delta wing configuration can lead to collection efficiency of around 75% for 4μm hypothetically assumed droplets. In a similar research, Wen et al. [23] investigated the effects of swirl (generated via various blades with different angles) on the flow characteristics inside 3S units in the absence of condensation. They reported that the strong swirl can cause non-uniform distributions for gas pressure, temperature and corresponding velocity, which had different effects on the performance of 3S unit. The optimal swirling device was consisted of 4–8 blades with 40–70 degrees, angles. In 2017, Liu and Liu [24] numerically investigated the effects of various turbulence models on the flow characteristics of natural gas inside 3S units. They showed that the Reynolds stress model provided better predictions for the entire flow parameters. In the same year, the optimization of static vanes in the plenum chamber of 3S unit was studied by Yang et al. [25]. Interestingly, they concluded that the expansion characteristic and swirl motion were opposed to each other. More recently, Niknam et al. [26] used the well-known Aspen HYSYS software to mimic the 3S unit performance by considering a combination of conventional unit operations such as expander unit, phase separator and compressor unit. An algorithm was presented to find a control function to compensate the backpressure of the nozzle within the feed pressure variations.

4

Yang and Wen [27] predicted the particle behavior and collection efficiency inside 3S unit with strong swirl without considering the actual condensation process. Their results indicated that the collection efficiency reaches over 80% for hypothetical droplets with assumed diameters of 1.5μm. Hu et al. [28] numerically investigated the 3S unit performance when equipped with reflow channel exhibiting axial or tangential outlet velocity components. The simulation results indicated that the use of reflow channels provide better refrigeration performance. As in previous studies, no actual condensation was considered. In addition to water vapor and hydrocarbon separations inside 3S units, removal of other natural gas impurities such as CO2 and H2S was recently investigated. Sun et al. [29, 30] investigated the effects of inlet conditions on CO2 and H2S condensation from CH4-CO2 and CH4-H2S binary gas mixtures via supersonic nozzles. Arinelli et al. [31] compared the use of 3S units for treating of CO2 rich humid natural gas with conventional Water Dew Point Adjustment (WDPA) and Hydrocarbon Dew Point Adjustment (HCDPA) process. They reported that supersonic separator performs more adequately for WDPA and HCDPA compared to solvent absorption and Joule-Thomson Expansion (JTE) processes. But it should be combined with Membrane Permeation (MP) process to provide best performance. Jiang et al. [32] studied CO2 separation from natural gas via 3S unit using Discrete Particle Method, assuming no real condensation. Their results showed that the hypothetically assumed CO2 droplets (with diameter > 1.5μm) are expected to be completely separated from natural gas stream. Our latest works [33-35], addressed other capabilities of 3S units such as separation of water vapor, hydrogen sulfide and hydrocarbons from natural gases along with maximization of energy recovery inside Laval nozzles. Haghighi et al. [36] reviewed most of the relevant researches in 2015 and emphasized that the size of particulate phase (liquid droplets) are usually in the range of 1 to 1000 nanometers and "requires some attention in both modeling and experimental studies".

5

It is evident that for a given swirl number, separation efficiencies of the condensed phase strongly depend on the size of liquid droplets. Successful enlargement of these droplets have crucial effect on the separation performances of the 3S units. This issue becomes more significant, when only water vapor is condensed and separated from methane or a lean natural gas, in the absence of heavier hydrocarbons condensation [33]. The heterogeneous condensations may be used to enlarge the condensed droplets. Enlargement of excessively small droplets via heterogeneous condensation over solid particles was used in some industrial processes. For example, enlargement of extremely small particles (from a few nanometers to some microns) via employment of heterogeneous condensation lead to efficient removal of such fine particles [37]. Heidenreich and Ebert [38] reported the successful employment of heterogeneous condensation for enlargement and separation of water vapor droplets from gas streams. At the start of millennium, Heidenreich et al. [39] investigated efficient separation of fine particles inside packed columns by enlargement of particles via heterogeneous condensation. Successful separation via heterogeneous condensation of water vapor over fine particles were also reported for various flue gas desulfurization processes [40-42]. Most recently, Xu et al. [43] investigated heterogeneous condensation of water vapor in moderated growth tube. Their results indicated that using moderated growth tube provided larger particles compared to conventional growth tubes. They also studied the particles enlargement at high concentrations using heterogeneous condensation of water vapor [44]. The effects of super-saturation, particle size, residence time and particle wettability were investigated. In the present study, heterogeneous condensation of natural gas humidity on the injected sodium chloride (NaCl) particles is investigated as a mean to encourage the dehydration process inside 3S unit. The swirl generator will be positioned at the end of first diffuser to maximize the radial

6

acceleration. The combination of straight tube and second diffuser is used for efficient liquid collection and proper pressure recovery. The optimal rate of injected solid particles will also receive proper attention for effective dehumidification of a certain lean natural gas via a 3S unit. 2. Mathematical model As shown in Figure 1, any industrial scale 3S unit is usually comprised of 4 distinct sections recognized as the plenum chamber, the Laval nozzle, a straight tube with liquid collection point and finally, the pressure recovery section. The static vanes of the plenum chamber may be positioned at either ends of the Laval nozzle. In this article, the plenum chamber is assumed to be positioned at the end of Laval nozzle prior to straight tube section. The following paragraphs describe the mathematical models for each of the above segments.

Figure 1: Schematic representation of a 3S unit equipped with straight tube [36]

7

2.1 Laval nozzle Mass, energy and momentum balance governing equations should be solved simultaneously with proper equations of state to find the entire required distributions of pressure, temperature, velocity, Mach number and the gas density across Laval nozzle. Table 1 presents the most basic forms of the above governing equations and their corresponding final differential form. Derivation of the entire equations had been presented elsewhere in our previous articles [33-35].

Table1: Gas flow equations in Laval nozzle NO

Gov. Eqn.

Basic form

Final Differential form d G

1

G

t  m L m G m S m

Continuity



2

3

4

5

Momentum Equation

PengRobinson (PR) Equation of state (EOS)

Energy equation

Mach number

dA dU G  A UG

dmL 0 mt  mL  mS

f  U 2 dx dP  G G P 2P de

d  mGU G  mLU L  mSU S    AdP 



f A GU G2 dx 2d e



mtU G dU G AP U G

d dT dP  X G Y G  0 P G TG

P  RTG G ( B1  B2 G )

Parameters , , mt mL mG and mS : Total, liquid, gas and solid mass flow rates G : Gas density UG : Gas velocity A: Cross sectional area P: Fluid pressure f: Friction factor x: Axial direction de: Hydraulic diameter B1 and B2: PR parameters TG: Gas temperature   P  X G  P  G T G

Y  UG2 U L2  m ( h  )  m ( h  ) L L  G G 2 2   d 0 US2     mS (hS  2 ) 

Z  M2 

dTG P  Y  dP  1   TG G c pTG  X  P 

U G2 G  P

8

TG P

 P     TG  G

h dmL U G2 dU G  fg 0 c pTG U G C pTG mt

h: Enthalpy Cp: specific heat at constant pressure hfg: latent heat of condensation

dU d dZ dP 2 G  G  Z UG G P

Z: square of Mach number γ: Specific heat ratio

When the entering gas contains solid particles, the liquid phase is created due to homogenous and heterogeneous condensation processes across the entire lengths of both Laval nozzle and the straight tube. While, heterogeneous condensation on the surface of NaCl crystals can occur very early and prior to throat location, but homogenous nucleation happens much later when the super-saturation ratio exceeds unity. As before, Table 2 depicts the entire equation sets required to compute the total condensation rate inside Laval nozzle [36,45-47]. The overall computational algorithm for prediction of various gas and liquid distributions across the Laval nozzle has been received proper attention in our previous articles [33,35]. A few modifications are required here to address simultaneous occurrence of both homogeneous and heterogeneous condensations.

2.2 Straight tube equations As previously mentioned, the 3S unit swirl generator is assumed to be connected between Laval nozzle and straight tube to maximize the radial acceleration. Evidently, all solid particles are previously covered with condensed layer of liquid phase (water) before entering the straight tube. Afterwards, exceedingly large radial accelerations induced via the generated swirl will throw away all particles towards the straight tube wall. While the sufficiently large particles will reach the walls, other smaller particles will be carried away with the dried gas stream. The straight tube length should be greater than a certain critical value to prevent subsonic flow occurrence before liquid collection point [20]. Table 3 provides necessary equations for prediction of solid particles radial displacements inside the straight tube, required for successful liquid collection before pressure recovery section [20,4849]. Evidently, all gas distributions (e.g. Mach number, pressure and temperature) at the straight tube inlet are previously computed via equations of Table 1. The effect of plenum chamber on the velocity components of the entire fluid flow is assumed separately.

9

Table 2: Condensation equations in Laval nozzle NO

Equation

6

Critical radius in homogeneous condensation

7

Algebraic equations

Nucleation rate

9

Wolk and Strey correction

10

heat transfer coefficient

11

Droplet temperature

12

Growth rate

2 w  Lw RTG ln Sw

rc 

 Lw : Liquid water density S w : Super-saturation ratio

aw : water activity

Psat ,solution  aw .Psat

Surface tension and vapor pressure correction

8

Parameters  w : water surface tension

M L : Molality B : constant qc: Condensation coefficient  4 rc 2 w  2 w exp    Gw : Water vapor density  mw3 3 kT G   mw : mass of a single water molecule k: Boltzmann constant

 w,solution   w  B.M L

J c  qc

2 Gw  Lw

J H 2O  J c exp(27.56 



 r

6.5 103 ) TG

1 1

2 8  Kn 1.5Pr 1  

TL  T sat ( Pw )  T sat ( Pw )  TG 

dr  TL  TG  dt  L h fg

rc r

n

j 1

13

 j 1  4 / 3  L   J iVsegi    i 1 

 r

3  rhom, j   rhom 3

hom



n

14

Heterogeneous condensation rate

mL,h et   4 / 3  L Nin  j 1

 r

h et

15

Total liquid mass flow rate

 : Thermal conductivity r: Droplet radius Pr: Prandtl number Kn: Knudsen number Tsat: Saturation temperature Pw: Water partial pressure hfg: Latent heat of condensation

mL ,hom    4 / 3 rc3  L J j Vsegj  

Homogenous condensation rate

-

 rh et , j   rh3et 3

mL  mL,hom  mL,h et



j: Segment counter n: Number of segments Vsegj: Volume of segment j rhom: Droplet radius in homogeneous condensation Nin: Number of solid particles per second at the nozzle inlet rhet: Particle radius (with added water layer) in heterogeneous condensation -

Equations 16 and 17 are used to find the required length of the straight tube which should be larger than the previously mentioned critical value computable via equation 20. The combination of equations 18 and 19 predict the required friction factor corrected for the swirl effect. The entire computational algorithm for estimation of all of the required parameters at the end of the straight tube via the solution of Table 3 equations, is presented in Figure 2.

10

Table 3: Straight tube equations NO

Equation

16

Particles residence time

17

Straight tube length

18

Friction factor

19

Friction factor in the presence of swirl

20

21

22

23

Final model

t

Parameters G : Viscosity of gas phase Dd: Droplet diameter ρL: Liquid density ρG: Gas density ω: Angular velocity R: Pipe radius r: Droplet radial position Uaxial: Gas axial velocity component K: 0.1 mm D: pipe diameter Re: Reynolds number y: Tangent of swirl angle

18G R ln( ) Dd2 (  L  G ) 2 r

Lt  t U axial

 K 65  f  0.11    D Re 

0.25

1.5 fs  1  2 y 0.4 Fr 0.1  f

Fr: Froude number

Critical straight tube length (Lmax)

 1 M (  1) M  D  1 Lmax    ln  2 2 2  (  1) M 2  4 f s  M

M: Mach number at any point of the straight tube.

Temperature ratio in straight tube

 1 T 2  *   1 2 T 1 M 2

T*: Temperature in critical point

Pressure ratio in straight tube

 1    P 1  2    * P M  1  1 M 2   2 

Density ratio in straight tube

  1 2  *   2 M      1   1 M 2   2 

2

2

11

0.5

P*: Pressure in critical point

0.5

ρ*: Density in critical point

START

Input: TG, P, UG, Mach number and ρG at the straight tube (ST) inlet Assume static vanes angles and corresponding angular velocity (ω) Calculate ST length (Lt) from equations 16 & 17 Compute maximum length of ST (Lmax) from equations 18-20 No

Lt < Lmax Yes

Assume Mach number at the end of ST (M2)

Calculate Lmax2 based on M2 from equation (20) Ltnew=Lmax-Lmax2

No │Ltnew-Lt│< Tolerance ≈ 0

Yes Calculate T *, P* and ρ* from equations (21, 22, 23) using M, T, P and ρ at ST inlet

Calculate T, P and ρ from equations (21, 22, 23) for ST outlet using outlet * * * Mach number and T , P and ρ computed in the previous step Calculate UG at the ST outlet from equation (5)

STOP

Figure 2: Flow chart for computation of straight tube length and its required parameters.

12

2.3 Pressure recovery model After collection of the condensed phase at the end of the straight tube, natural gas enters a second diffuser for pressure recovery purpose. Assuming 100 percent liquid collection, the rate of liquid mass flow rate ( m L ) should be set to zero for the next segment. Once again, Table 1 equations should be used to compute all of the required operational parameters (T, P, Ma, G) for both sides of the normal shock occurring in the diffuser (except normal shock faces). Table 4 tabulates the required equations for prediction of the normal shock parameters at its both faces for a given shock location [16, 35]. The 3S exit pressure computed so forth should be checked to be greater than the minimum required pipeline inlet pressure. Table 4: Normal shock wave equations NO 24

Equation

Final model

2 Z u    1  1

Pressure in downstream of shock location

Pd  Pu

25

Square of Mach number in downstream of shock location

(  1) Z u  2 Zd  2 Z u  (  1)

26

Temperature in downstream of shock location

Parameters u: upstream of shock location Z: square of Mach number γ: specific heat ratio

 (2 Zu  (  1))  ((  1) Zu  2)  TGd  TGu   (  1)2 Zu  

3. Grid size verification and Model validation In order to ensure that the simulation results are independent of the recruited grid sizes, Figure 3 illustrates both computed pressure distributions along 3S unit and the corresponding homogeneous nucleation rates for three different step sizes of 0.006, 0.0001 and 0.00001 meters. For all cases, the entering saturated natural gas pressure and the corresponding temperature were assumed to be 7 MPa and 310 K, respectively and 6×1013 NaCl particles per second were assumed to be injected at the inlet to provide simultaneous homogeneous and

13

heterogeneous condensations inside 3S unit. As can be seen, there is no practical difference between the results obtained via moderate and fine step sizes. The former step size of h=0.0001 m is used in the entire simulations of the present work.

a)

b)

Figure 3: Verification of the step-size (h) for a typical one dimensional grid structure. a) Pressure distribution and b) homogeneous nucleation profile.

Similar to Ding et al [46], simulation results of Dykas [45] are used to validate the above model predictions, due to the lack of experimental data for heterogeneous condensation inside Laval nozzle. Dykas [45] solved 3D Euler equation set to predict both homogeneous and heterogeneous condensations of steam inside supersonic nozzle. Ding et al. [46] investigations via computational fluid dynamic for hetero-homogeneous nucleation inside nozzle-diffuser lead to very similar results as Dykas and therefore omitted in our validation diagrams of Figures 4-6. In both studies, steam flow inside two dimensional Barschdorff nozzle (with 60 mm throat height and 584 mm wall curvature radius) was investigated. Inlet stagnation pressure and the corresponding temperature were 78500 Pa and 380.55 K, respectively. Figure 4 compares our

14

presented model results with Dykas’s predictions [45]. As can be seen, both pressure ratio and wetness fraction (y) distributions are practically same for the case of homogeneous condensation and in the absence of solid particles (nP=0 kg-1).

Figure 4: Successful validation of our present model predictions with the Dykas [45] simulation results for pressure and wetness fraction (y) distributions during homogeneous condensation.

Two different situations may be recognized when the inlet steam contains appreciable amounts of sufficiently large solid particles, as shown in Figures 5 & 6. Figure 5 depicts our simulation results for simultaneous homogeneous and heterogeneous condensations when moderate number of inlet NaCl solid particles (nP=1015 kg-1 and dp=10-8 m) are used. As can be seen, fairly close agreements exist between previously computed pressure distribution and wetness fraction results obtained by Dykas with our predicted values [45].

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Figure 5: Proper validation of our model predictions with Dykas simulation results [45] for pressure ratios (P/P0) and various wetness fractions (y) across Laval nozzle. (np=1015 kg-1 & dp=10-8 m)

As can be seen, two wetness fraction distributions across the entire length of Laval nozzle correspond to heterogeneous condensation and total (combination of homogeneous and heterogeneous) condensation. While acceptable validations are achieved, relatively small deviations (which occur at the lateral section of the Laval nozzle) are mainly due to completely different models employed in two independent studies of ours and Dykas. In the second case, extremely large concentrations of the solid particles are assumed to be injected into the entering fluid. Evidently, homogenous nucleation will not happen and the condensation will be solely due to heterogeneous growth phenomenon over the surface of solid particles. Figure 6 shows very close agreements between our model predictions with the simulation results presented by Dykas when the inlet steam contains large amounts of NaCl particles (r = 10-8 m) injection rates with mass concentration of nP=1016 kg-1.

16

Due to the positive values of Gibb's free energy for the homogeneous nucleation process and in the presence of such high concentration of NaCl particles, the condensation phenomenon is entirely heterogeneous and therefore, sudden jumps in all distributions will not be anticipated. Incremental release of latent heat during the heterogeneous condensation leads to small differences between the so called real condensing system distributions and those of isentropic situation (without any condensation).

Figure 6: Successful validation of our model predictions with Dykas simulation results [45] for isentropic & condensing pressure ratios (P/P0) and wetness fraction (y) across Laval nozzle. (np=1016 kg-1 & dp=10-8 m)

4. Simulation results The mathematical model described in section 2 is used to investigate the effect of heterogeneous condensation on the natural gas dehydration efficiency when it is flowing across a 3S unit. The entire set of equations (depicted in Tables 1-4) are solved using a combination of 4th order RungeKutta method and fixed point iteration technique as denoted in Figure 2. 17

For successful dehydration of a natural gas, two distinct specifications should be satisfied. First, the water content of natural gas should be reduced to its permissible value of less than 7 lbm/MMSCF. In this work, this criterion is assumed to be 2 lbm/MMSCF, to ensure sufficient dehydration, because some of the condensed water is evaporated after the collection point and returns back to the dried natural gas stream. Furthermore, the condensed droplets should be sufficiently large to reach the straight tube walls under the induced radial acceleration generated via swirl effect and provide adequate water separation efficiency. Heterogeneous condensation of water vapor on large injection rates of solid particles will bypass the nucleation process and encourages the dehydration via growth phenomenon inside the straight tube of the 3S unit.

4.1. Case study description Figure 7 illustrates the two dimensional schematic representation of the 3S unit employed in the current work for investigating the performance of our presented model. Table 5 provides the required 3S geometries. The wet fluid entering 3S unit is assumed to contain methane rich natural gas saturated with water vapor with the stagnation pressure and temperature of 7 MPa and 310 K, respectively.

4.2. Effect of injection rates on the homogeneous nucleation process To ensure that no homogeneous nucleation will occur inside 3S unit, sufficient number of NaCl particles should be injected at its entrance. A uniform particle size distribution (PSD) of 2µm is used to establish sufficient dehydration inside the straight tube. Figure 8 illustrates the presented model simulation results for variations of homogeneous nucleation phenomenon with the particles injection rate.

18

Figure 7: Two dimensional schematic diagram of the 3S unit used in this research.

Table 5: The 3S unit geometrical parameters used for simulations.

Parameter

Symbol Unit

Value

Nozzle inlet radius

Ri

m

0.2

Nozzle throat radius

Rth

m

0.0566

Converging length

Lc

m

0.5

Converging angle

α

degree 16

Diverging angle

β

degree 8

Diverging length

Ld

m

0.0445

Straight tube length

Lt

m

0.15*

Diffuser angle

γ

degree 8

Diffuser length

Ldif

m

0.3

* The optimal value of the straight tube length will be computed later via Figure 11. As can be seen, homogeneous nucleation reduces and the corresponding peak diminishes while shifting away from the throat location as the solid particles injection rate increases. The reason behind such interesting phenomenon lies in the signs of Gibbs free energies (GFEs) for the corresponding nucleation and growth processes.

19

Figure 8: Our simulation results for the effects of solid particles injection rate (number per time) on the homogeneous nucleation process.

It is previously well established that the nucleation process has positive GFE while the growth process has negative GFE. To overcome this positive free energy barrier, the super-saturation ratio (P/P*) in the fluid should exceed unity to force the occurrence of homogeneous nucleation process. Increasing the number of injected particles will encourages the spontaneous rate of heterogeneous condensation of water vapor over the solid particles surfaces and reduces the water content of natural gas. Evidently, at lower water contents the nucleation process will be considerably harder to happen and less probable, especially when sufficient number of particles are available in the gas stream which can practically act as some pseudo-nuclei to promote the growth process.

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In other words, the super-saturation ratio tends towards unity as the growth process proceeds due to the condensation of water vapor over the heterogeneous solid particles. As Figure 8 clearly shows, the heterogeneous growth process prevails for particle injection rates of greater than 9×1013 s-1. At such extreme cases, the homogeneous nucleation is self-inhibited due to its positive GFE value and the total condensation rate is almost equal to the heterogeneous condensation rate. Figure 9 provides other similar diagrams for some particle injection rates which depicts variations of cumulative heterogonous condensation rate across the Laval nozzle until the overall water content of the gas phase reduces down to 2 lbm/MMSCF.

Figure 9: Effects of particles different flow rates (numbers per time) on the cumulative heterogeneous condensation for fixed overall condensation rate to reduce the water content of the dry gas to permissible value.

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For example, 45% of the overall water vapor liquefaction occurs via the heterogonous condensation at the end of diffuser, when the injected rate is around 4×1013 numbers/second. As Figure 9 implies, the total condensation is entirely due to heterogeneous condensation when the solid particle injected rate is greater than 9×1013 numbers/second. It is interesting to note that for all cases, the heterogeneous condensation initially starts near the entrance and quite ahead of the throat location. For very small number of injected particles, the homogeneous nucleation after the throat location creates a huge shower of droplets and the homogeneous condensation practically prevails. Therefore, the water content of dry natural gas reaches to its permissible value of 2 lbm/MMSCF closer to the throat location compared to heterogeneous case. On the other hand, extremely small droplets have practically extremely small chances to reach the straight tube walls. Figures 8 and 9 imply that the number of injected particles should be larger than 9×1013 numbers/s to achieve a successful separation of the condensed phase.

4.3. Performance of the 3S unit at the optimal rate of particle injection Figure 10 illustrates the three key distributions of: a) water vapor mole fraction inside gas stream (yH2O), b) the corresponding condensed liquid mass flow rate (mL) and c) the particles radiuses (Rd) along the entire length of Laval nozzle for the above minimum required rate of injected particles. As mentioned earlier, Figure 10 clearly demonstrates that for sufficiently large number of injected particles, the heterogeneous condensation starts much earlier than the throat and considerable amount of water vapor condenses before the gas arrives the throat location. The heterogeneously formed condensed liquid mass flow rate reaches around 0.12 kg/s at the Laval nozzle outlet which reduces the mole fraction of water vapor inside the natural gas from its initial value of 0.00088 down to around 0.00004 at the entrance of the straight tube. Similarly, the size of heterogeneous droplets formed by the condensation of

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water vapor over the solid particles slightly increase from its initial value of 2 µm to its final value of around 2.03 µm. Such thin layer of condensed water over the injected solid particles is sufficient to reduce the water content of the working fluid below its permissible value of 2 lbm/MMSCF.

Figure 10: Distributions of water vapor mole fraction, liquid mass flow rate and particles sizes along the Laval nozzle when N=91013 s-1

To assist the separation of the heterogeneous droplets, the gas enters the straight tube in order to provide adequate residence time for the particles to settle and reach the tube walls. According to the Fanno line concept, the velocity of the fluid decreases inside a straight tube at supersonic adiabatic condition and a critical length (denoted by Lmax) exists which at that point the fluid Mach number becomes unity. Evidently, the corresponding tube length should be smaller than the above critical value of Lmax to prevent water evaporation inside the straight tube.

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Assuming that the static vanes are positioned at the entrance of the straight tube, Figure 11 illustrates the distribution of the heterogeneous particles separation efficiencies along the straight tube length for different static vanes angles. The swirl (vane) angle (θ) is defined as the ration of arc-tangent of the velocity ratio defined as the ratio of tangential velocity over the axial velocity.

Figure 11: Separation efficiency distributions along straight tube for various swirl angles.

The filled markers of Figure 11 show the corresponding maximum admissible length of the straight tube (Lmax) for each swirl angle. By increasing the value of static vanes angle from 15 to 35 degrees, the required straight tube length decreases due to rapid increase in radial acceleration value. As can be seen, for vanes angles smaller than 30 degrees, the required straight tube length exceeds Lmax and may lead to severe operational problems such as reevaporation of the condensed phase. For the selected vanes angle of 35 degrees, the value of

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straight tube length of 0.15m (as reported in Table 5) is required so that 100% of the heterogeneous droplets can reach the wall. As Figure 11 clearly shows, the required straight tube length is larger than its corresponding L max value, for smaller angles. Figure 12 shows the distributions of pressure ratio and Mach number along the 3S unit. Initially, the gas enters the Laval nozzle at subsonic condition (M<<1) and expands to supersonic velocity with the aid of a nozzle diffuser with the geometry previously described in Table 5 and schematically depicted in Figure 7. Due to transformation of potential energy into kinetic energy in an adiabatic expansion process, the gas velocity and the corresponding Mach number increase at the expense of severe static pressure reduction. After first diffuser, the gas enters the plenum chamber which is positioned at the straight tube entrance. Due to the extremely large swirl generated by the static vanes (with θ=35 degrees), the entire heterogeneous particles can reach the straight tube walls before arriving at the collection point. Regarding Fanno line concept for supersonic flow inside a tube and as Figure 12 clearly depicts, the fluid pressure rises inside the straight tube due to the frictional losses while its Mach number decreases more rapidly. After passing the collection point and leaving almost all the condensed liquid behind, the dry gas enters the second diffuser for pressure recovery. By resorting to a carefully adjusted normal shock occurrence inside second diffuser, sufficiently large pressure recovery can be accomplished. To ensure maximum separation at the collection point, the normal shock should happen adequately far from the collection point. Evidently, the Mach number increases in this gap and the fluid pressure decreases accordingly. A sudden drop in Mach number from supersonic to subsonic condition will lead to a large pressure jump at the second face of the normal shock. The pressure recovery continues as the fluid travels towards the end of the second diffuser and leaves the 3S unit. For the present case study the pressure recovery was around 79% which is quite acceptable for most practical situations.

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First Diffuser

Nozzle

Straight Second diffuser Tube (Pressure recovery section)

Figure 12: Pressure ratio and Mach number distributions across the entire length of 3S unit.

The actual position of the normal shock wave is a very delicate matter. If it is adjusted too far from the collection point, the pressure recovery may suffer drastically. On the other hand, if the normal shock is adjusted too close to the collection point, any small deviation in the exit pressure can push the normal shock earlier than the collection point and complete vaporization of the condensed phase may occur. For this reason, a small but reasonable distance should be left between the collection point and the normal shock to ensure stable and smooth operation.

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4.4. Search for the optimal value of first diffuser angle As mentioned in our previous articles [14, 35], the 3S diffuser angle has significant effect on its overall performance during the homogeneous nucleation process. To investigate such effect in the presence of heterogeneous condensation, similar simulations are carried out for smaller diffuser angles and the corresponding performances of the 3S unit are depicted in Figures 13 to 15. The static vanes angles are kept fixed at previously found value of 35 degrees. As pointed out previously, larger diffuser angles may lead to severe operational difficulties inside Laval nozzle and is not desirable for practical applications [35].

Figure 13: Effect of diffuser angles on the minimum rate of injected solid particles required to eliminate the homogeneous nucleation process.

Figure 13 illustrates the variations of the minimum required amount of injected particles (with initial diameter of 2μm) for elimination of the homogeneous nucleation with the

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corresponding diffuser angle. As can be seen, less solid particles are required for lower diffuser angles. Because, as discussed in our previous work [35], both pressure and the corresponding temperature distributions drop much slower for smaller diffuser angles (β). Therefore, for larger diffuser angles the temperature drop is so fast that premature appearance of the homogeneous nucleation would be observed. On the other hand, for smaller diffuser angles, the peak of the homogeneous nucleation process moves far from the throat location and provides more favorable conditions for heterogeneous condensation. To avoid homogeneous nucleation, more solid particle should be injected into the gas stream at the entrance of the Laval nozzle, when larger diffuser angles are used. For the present case study, to avoid homogeneous nucleation at least 3 kg/s solid particles should be injected into 125.7 kg/s gas flow (around 2.4% of the total mass flow), when the diffuser angle is around 2 degrees. Figure 13 clearly shows that smaller diffuser angles are more preferable from solid particle injection rate point of view. To specify the exact optimal diffuser angle in the presence of heterogeneous condensation, other aspects of such complex process should be considered. Figure 14 illustrates that the diverging angle have a minor effect on the optimally required straight tube length (Lt) but it can severely affect the corresponding critical length (Lmax). All diagrams depicted in Figure 14 are computed for a fixed natural gas water vapor content of 2 lbm per one million cubic feet of gas at the standard condition, while entering the straight tube. Furthermore, minimum amount of solid particles obtained from Figure 13 is used for any diffuser angle. As shown, lower diffuser angles lead to smaller values of L max at relatively constant length of straight tube (Lt). This is evident, since at isentropic conditions and for supersonic flow, the fluid Mach number leaving the first diffuser will increase as the corresponding angle increases. According to equation 20, the value of L max also increases as the Mach number

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becomes larger. Figure 14 shows that for extremely small diffuser angles, the overall length of the 3S unit increases asymptotically. As illustrated in Figure 13, this shortcoming can be acceptable, because for smaller values of β (leading to longer diffusers) the particle injection rates drop drastically. On the hand, the diffuser angle could not be less than 1.3 degrees, because for lower β values the required straight tube length exceeds the corresponding critical length, which could not be afforded for practical applications. To be on the safe side, the previously found value of β = 2° would be optimum for the present case study with the particle injection rate of 3kg/s.

Figure 14: Effects of first diffuser angle on the diffuser length (Ld), fluid Mach number at the straight tube entrance (Main), required and maximum tube lengths (Lt & Lmax)

In our previous work [35], it was reported that higher diverging angles (up to 8 degrees) provide better pressure recoveries for homogeneous condensation inside 3S units. Figure 15 demonstrates that in the case of heterogeneous condensation inside 3S unit, the overall pressure

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recovery slightly increases as the first diffuser angle (β) decreases. The best pressure recovery situation happens when the diverging angle is around 2 degrees. In other words, by decreasing the 3S diffuser angle down to 2°, the overall length of the diffuser section increases but both the pressure recovery and solid injection rates will be sufficiently improved.

Figure 15: Effects of first diffuser angle on the pressure distribution and pressure recovery along 3S.

The combination of Figures 11 to 15 clearly illustrates that for the present case study the 3S unit with the first diffuser angle of 2 degrees provides complete recovery of the initial water content with minimum particle injection rate and maximum pressure recovery along the 3S unit. In other words, all particles will reach to the walls of the straight tube and successful dehydration (total separation) will be achieved when the entering natural gas contains 2.4 weight percent solid particles with the radius of 2µm. Furthermore, the optimal values obtained in this work for the supersonic section vane angles and the overall pressure recovery were around 35 degrees and 83%, respectively.

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5. Conclusion Previous studies concluded that for sufficiently large separation efficiency inside 3S units, the size of particulate phase should be greater than 2μm [27]. Others pointed out that such important issue has not been addressed properly and "requires some attention in both modeling and experimental studies" [36]. For the first time, this article introduces a detailed model for injection of NaCl particles to promote water separation efficiency inside supersonic separators via heterogeneous condensation. The present study clearly demonstrated that many parameters such as particle injection rate, first diffuser angle, supersonic plenum chamber vane angles and straight tube length have profound effect on the overall performance and separation efficiency of the 3S units, when heterogeneous condensation is used. Our simulation results indicated that injection of at least 2.4 weight percent solid particles with sufficiently large diameters into natural gas stream at the entrance of 3S unit with first diffuser angle of 2° and the plenum chamber vane angles of around 35 degrees leads to complete separation of water droplets without any homogeneous nucleation. With such optimal structure and geometry, the overall pressure recovery of the 3S unit can be increased as high as 83%.

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Research Highlights

 3S units can potentially eliminate the entire water content of natural gas streams.  The condensed droplets should be sufficiently large for successful dehydration.  Heterogeneous dehydration via injection of NaCl particles at 3S inlet is studied.  Static swirling vanes are positioned post throat location for full swirling effect.  Injection of 2.4 wt% solid particles (R=2μm) provided best separation performance.

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