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Modeling of CO2 Homogeneous and Heterogeneous Condensation Plumes† Zheng Li, Jiaqiang Zhong, and Deborah A. Levin* Department of Aerospace Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: May 01, 2009; ReVised Manuscript ReceiVed: July 24, 2009
A pure CO2 homogeneous condensation flow is simulated with a new direct simulation Monte Carlo (DSMC)based condensation model, and comparisons of the measured [Ramos, A.; Ferna´ndez, J. M.; Tejeda, G.; Montero, S. Phys. ReV. A 2005, 72, 3204] and simulated CO2 cluster size and number density distributions were found to agree well for low stagnation pressures. The same carbon dioxide homogeneous condensation model was then applied to the study of an expanding heterogeneous condensation flow of a 5% CO2 and 95% mixture [Williams, W. D.; Lewis, J. W. L. DTIC Paper No. AEDC-TR-80-16, 1980]. To simulate this case, a new kinetic-based model of N2 molecules condensing on CO2 nuclei was developed using molecular dynamics techniques and was implemented in the DSMC simulation of the expanding mixture. Pure nitrogen flow for the same expansion conditions was observed to not produce any clusters. It was found that incorporation of the heterogeneous condensation process of N2 molecules condensing on CO2 nuclei causes the average N2 cluster size to increase from 10 (the homogeneous condensation result) to about 2000. The simulated Rayleigh scattering intensity results were found to agree well with the experimental data. I. Introduction Condensation processes can occur when rocket plume temperatures decrease during the expansion process, causing the plume gas number density to decrease and the cluster particle size to increase. Due to the existence of condensed cluster particles, condensation impacts not only the plume radiation behavior but also the possibility of plume interactions with satellite surfaces. For example, it was found that the existence of particles in the Bow Shock Ultraviolet 2 plume causes the ultraviolet radiation intensity at 230 nm to increase by a factor of 300 at an altitude of 118 km.1 Rochelle et al.2 considered the potential problems of plume impingement on the International Space Station components such as heating, unbalanced surface pressure distributions, and contamination. For reasons such as these, it is therefore important to study condensation phenomenon in freely expanding, high-pressure plumes to the space near-vacuum condition. Whether the nature of the condensation is homo- or heterogeneous depends on the properties of the initial nuclei particles in the plume. In homogeneous condensation, initial nuclei are created in the supersaturated condensable gas environment. Therefore, a large degree of supersaturation is required to induce a significant amount of homogeneous condensation. In heterogeneous condensation, the source of nuclei can be existing foreign particles, such as environmental dust, incomplete combustion soot particles, or surface sputtering particles. However, even in the absence of particulates, heterogeneous condensation caused by the presence of multiple gaseous species can occur, particularly in operational rocket plume exhausts that typically consist of mixtures of simple gaseous species. Initial nuclei can be created out of the more easily condensable trace species, such as water and carbon dioxide, followed by condensation of the less condensable species, such as oxygen and nitrogen, on the water or carbon dioxide cluster nuclei, even though the oxygen and nitrogen gas pressure is less than their saturation pressure. †
Part of the “Barbara J. Garrison Festschrift”. * Corresponding author. E-mail:
[email protected].
A homogeneous condensation model has been developed in previous work to simulate a free expanding argon (a LennardJones system similar to CO2) condensation plume using the direct simulation Monte Carlo (DSMC) method.3-6 Our previous work can be briefly summarized as follows. On the basis of the mass, momentum, and energy conservation relationship, a comprehensive DSMC condensation model, including nucleation, cluster-monomer sticking and nonsticking collision, and cluster evaporation model, was developed.3 Using the molecular dynamics technique, the cluster-monomer sticking collision model was verified in ref 4. A hybrid MD-DSMC approach was developed to construct a kinetic nucleation model for generating initial dimer clusters from a triple collision process.5 Finally, in ref 6, a kinetics-based evaporation model was obtained from unimolecular dissociation theory and directly verified by molecular dynamics (MD). The purpose of this work is to develop a heterogeneous condensation model based on the presence of initial nuclei derived from homogeneous condensation, and the model will be compared with free expanding heterogeneous condensation plumes measured at Arnold Engineering Development Center (AEDC).7 In these experiments, homogeneous condensation from a pure nitrogen plume was not observed in the Rayleigh scattering intensity measurements. However, when 5% of the N2 molecules in the noncondensible plume were replaced by CO2 molecules, a sudden increase in the Rayleigh scattering intensity was observed, indicating the presence of a large amount of clusters in the mixture-expanding plume. Therefore, the measured Rayleigh scattering intensity in the mixture of 5% CO2 and 95% N2 is not due to homogeneous condensation, but most likely is due to the heterogeneous condensation of N2 gas molecules condensing on CO2 nuclei. The simulations conducted here provide the first test of the heterogeneous condensation hypothesis of this data. The prediction of cluster size and number density in a CO2 homogeneous condensation plume provides a fundamental basis for further modeling of a heterogeneous condensation plume. The classical nucleation theory (CNT)8 is chosen to create initial
10.1021/jp9040698 2010 American Chemical Society Published on Web 10/16/2009
Modeling of CO2 Condensation Plumes nuclei for modeling homogeneous condensation, although the validity of CNT theory can be questioned9-11 for various species and plume conditions. In the CNT theory, the nucleation process is modeled using the properties of the bulk fluid, such as its density and surface tension. The theory can be parametrized to predict the dependence of the rate on supersaturation, but not over a broad temperature range. For example, on the basis of an extended set of experimental data, Wolk et al.12 found that the classical nucleation theory (CNT) water nucleation rate over a temperature from 220 to 260 K had to be corrected as a function of vapor temperature, but for temperatures under 220 K, no correct rate is currently available. Since the critical cluster size can be as small as tens of molecules, one may expect that the bulk parameter for surface tension, required for evaluation of the nucleation rate, is not appropriate.13,10 Therefore, before undertaking the modeling of the heterogeneous mixture case, we validate our CO2 homogeneous condensation models to be used in the DSMC simulations. To that end, the paper is organized as follows: A brief description of the DSMC method of modeling transitional flows is given, and the homo- and heterogeneous condensation models that are used in the simulations are discussed in Section II. In Section III, the fidelity of the CO2 homogeneous condensation model is assessed by comparing the simulated cluster number density, average cluster size, and gas mole fraction to the Rayleigh and Raman scattering experimental data of Ramos et al.14 in a free expanding plume. This data is unique because it provides not just the usual terminal cluster size and concentration15 but also both radial and axial profiles of the cluster and gas number densities, cluster size, and gas rotational temperature throughout the expansion. Comparisons between simulations and experiment are provided for a range of stagnation conditions, which corresponds to different degrees of condensation. The simulations are shown to predict the terminal condensate flow properties; however, for higher stagnation pressures, the predicted near-nozzle exit characteristics deviate from the experiment. Reasons for this deviation are explained. Since the CO2-N2 mixture condensation plume data is at lower stagnation conditions, the same CO2 homogeneous condensation model is used to model the CO2-CO2 collisions. In addition, a molecular-dynamics-based CO2-N2 heterogeneous condensation model is developed, and dimer formation probability through termolecular collisions and cluster-monomer/cluster-cluster interaction models are integrated into DSMC simulations. The simulated Rayleigh scattering intensity along the plume center line is then compared to the AEDC measured data in Section IV. Conclusions are summarized in Section V. II. Direct Simulation Monte Carlo Condensation Models In the DSMC16 procedure, the entire flow volume is divided into cells and filled with simulation particles that represent real molecules and clusters. The cell size is defined by the local mean free path, and the number of the simulation particles representing a given species (molecular species or clusters) in the cell is defined as the ratio of the number of real particles to a preset value, typically referred to as FN (the number of real particles represented by a simulated particle). Each simulated molecule is characterized by spatial coordinates, velocity, internal energy, and species-related mass. The simulated clusters, in addition, are characterized by the number of molecules they comprise. The basic principle of DSMC is that the continuous process of particle movement and interaction is uncoupled. First, at each time step, every particle is moved according to its velocity. Particles that leave the computational domain are no longer
J. Phys. Chem. C, Vol. 114, No. 12, 2010 5277 tracked, and molecules that collide with a solid boundary (wall) change their velocities according to the appropriate boundary conditions. Next, the interaction between the particles is modeled by appropriate coarse-grained models; for example, by collision models that include the simulation of pair collisions and the change of molecular velocities according to momentum and energy conservation of the pair. Moreover, the collision step includes the redistribution of translational and internal energies and possible chemical reaction processes. The DSMC computational domain is divided into cells, with cell size on the order of the local mean free path, and the time step is chosen to be on the order of the local mean collision time such that the particles move only a fraction of the cell size during the time step. The collision process of each simulated particle is locally calculated within a cell. The number of potential collision pairs is chosen by the majorant collision frequency scheme,17 (1/2)N(N - 1)FN(σcr)max∆t/Vc, where N is the number of simulated molecules in a cell, FN is the number of real molecules represented by a simulated DSMC molecule, σ is the collision cross section, cr is the collision relative velocity, and Vc is the cell volume. Note that in the majorant collision frequency scheme, the computing time is linear with N. The collision probability of the majorant collision frequency scheme is σcr/(σcr)max. To characterize the molecular collisions in DSMC, the variable hard sphere (VHS) model16 is used. It assumes that the total collision cross section, σ, depends on the relative collision velocity, cr, as σ ) σref((cref)/(cr))2ν, where the subscript ref refers to reference value, and ν is a constant related to the species viscosity. Note that in the VHS model, molecular scattering is isotropic in the center of mass reference frame. The Larsen-Borgnakke model16 is employed to characterize the redistribution of the total collision energy among the translational and internal modes of the colliding molecules according to local equilibrium distributions on the basis of the number of degrees of freedom. For rotational and vibrational relaxation numbers, the variable temperature model of Parker18 was used with values of T* and Zr,∞ as 18.1 and 91.5 K for both CO2 and N2 and values of θV as 959.66 and 3371 K for CO2 and N2, respectively. The computational cost of the DSMC approach can be high due to small grid cell sizes, tiny time steps, and large numbers of simulated molecules. To decrease the computational cost, a two-dimensional axisymmetric scheme is used to simulate condensation flow in a freely expanding plume instead of a 3D simulation. Since the number density of various species in the flow may be different by several orders of magnitude, species weighting factors are used to ensure a reasonable number of simulated molecules represent each species. Since the DSMC approach is used in this work to simulate supersonic flow, the simulation can start from a starting surface where the flow is supersonic and the gas density is sufficiently low that the DSMC computational cost is affordable. The flow upstream of the starting surface is simulated by a continuum (Navier-Stokes) approach. In the DSMC condensation model, three types of interactions exist:(a)molecule-molecule(M-M)collisions,(b)molecule-cluster (M-C) collisions, and (c) cluster-cluster (C-C) collisions. Moreover, nucleation and evaporation are two key models in our DSMC simulation of condensation coupled flow. Whenever an event of one of these processes occurs, mass, momentum, and energy conservation equations are used to determine the new velocities and temperatures of the gas and cluster particles involved. Details of these models are reviewed below and can
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also be found in refs 3, 19, and 20, The DSMC-based SMILE code21 is modified in this work to simulate the two-phase expanding condensation plumes. The kinetic and internal energies of a monomer and a cluster are given as
1 Ekinm ) mV2 2
(1)
ξ Eintm ) kT 2
(2)
gas constant, T is the temperature, and S is the degree of supersaturation defined as
S)
(3)
Eintc ) jCpmT
(4)
where m is the mass of one molecule of the given species, j is the number of monomers in a cluster, V is the velocity, k is the Boltzmann constant, T is the temperature, ξ is the number of internal degrees of freedom, and Cp is the specific heat capacity for the cluster. In this work, we used 1880 and 2042 kJ/kg K as heat capacity constants for CO2 and N2, respectively. A cluster is assumed to have three degrees of freedom for translational motion, three for rotational motion, and an additional internal energy proportional to its heat capacity and temperature. A. Monomer-Monomer Collisions. Collisions between gas monomers are modeled using the variable hard sphere (VHS) model,16 a standard DSMC approach. Monomer-monomer collisions are treated in the same way whether the two colliding monomers are of the same or different species. Momentum and energy transfer by collisions can also occur, causing changes in the kinetic and internal energies of both molecules. The Larsen-Borgnakke model16 is used to distribute the energy between the translational and internal modes and to determine the postcollisional states of the two monomers. B. Nucleation. To incorporate homogeneous condensation into DSMC, models are needed for the formation and growth of condensed particles or clusters. The formation of clusters, called nucleation, occurs when a vapor exists in a supersaturated state, where the vapor pressure is higher than the saturation pressure. The supersaturated state is a metastable state that can exist in regions where there are no foreign particles, surfaces, or nuclei. Once a nucleus forms in the region, the condensation process will be set in motion. Several theories have been developed to model this process, such as CNT,8 the dynamic nucleation theory,13 and the kinetic nucleation models.5 In this work, we will use CNT,8,22,23 a commonly used approach, because it may be readily evaluated and, in some cases, has shown good agreement with experimental data.3 In a supersaturated vapor environment with no nuclei, density fluctuations in the vapor can lead to the formation of a nucleus. The nucleus formation is associated with an increase in the free energy of the system. In CNT, the increase in free energy, ∆F, is described by the bulk properties of the condensable species according to
4 ∆F ) 4πr2σ - πr3FlRT ln S 3
(5)
where r is the radius of the newly formed cluster, σ is the surface tension, Fl is the liquid (cluster) density, R is the species-specific
(6)
where P is the condensable gas pressure and Ps is the saturation pressure. For CO2, we use the saturation pressure of Span et al.,24
() [( ) ( )
ln 1 Ekinc ) mjV2 2
P Ps
Tt Ps T T ) a 1+ a2 1 Pt T 1 Tt Tt
1.9
(
+ a3 1 -
T Tt
)] 2.9
(7)
with Tt ) 216.592 K, Pt ) 0.517 95 MPa, a1 ) -14.740 846, a2 ) 2.432 701 5, and a3 ) -5.306 177 8. The maximum value of ∆F for a particular value of S is associated with a cluster radius known as the critical radius, r*. The value of r* is given as
r/ )
2σ FlRT ln S
(8)
In CNT, nuclei with radii larger than the critical radius are stable and will grow in size by consuming vapor molecules. Nuclei with radii smaller than the critical radius will decay to monomers. Therefore, in the simulations presented in this work, clusters are introduced into the flow through nucleation at the critical radius. The rate at which initial nuclei are introduced into the flow, according to CNT, is given as
J)
(
2 -4πr/2σ 2σ FV exp 3kT πm3 Fl
)
(9)
Equation 9 assumes that the nucleation process does not significantly impact the vapor number density. In the DSMC simulation, the nucleation process is implemented by introducing clusters at the critical size into the computational domain. The number of DSMC simulated cluster particles to be formed, Nc, is then calculated on the basis of the nucleation rate and given as
Nc )
J∆tV FNWc
(10)
where J is the nucleation rate, ∆t is the time step, V is the collision cell volume, FN is the number of real particles represented by one DSMC simulated particle, and Wc is the cluster weighting factor. Each newly formed cluster is placed at a random location in the cell. From CNT, the cluster is assumed to be in thermal equilibrium with the surrounding gas. Therefore, the cluster temperature and velocity are determined from the local gas conditions, and the conditions of the gas in the cell are adjusted to account for the presence of the new cluster. If a cluster of size j is formed, j monomers are removed from the cell to conserve mass. To satisfy conservation of energy, the latent heat released by the cluster formation is distributed uniformly to each monomer in the cell. The energy released, L, is assumed to have the following form,
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L ) Hvmj
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(11)
where Hv is the latent heat of vaporization for the condensable species. In this work, we used 574 and 199 kJ/kg as the latent heat constants for CO2 and N2 clusters, respectively. C. Cluster-Monomer Nonsticking Collisions. When a collision occurs between a cluster and a monomer, it can result in either a sticking or nonsticking process on the basis of a sticking probability, q, which is a function of collision relative velocity, cluster size, and temperature. The modeling of sticking collisions will be discussed in the next subsection, and the determination of the sticking probability will be discussed in the next section. In addition, if a noncondensible gas is present, then any collision between a monomer of noncondensible gas and a cluster is considered nonsticking. In a nonsticking collision, momentum and energy can be transferred but mass cannot; that is, the cluster size does not change. The momentum and energy conservation for a nonsticking collision, using the equations for cluster and monomer energy given in eqs 1-4, are
mV bm1 + mjV bc1 ) mV bm2 + mjV bc2
(12)
ξ1 1 j 1 mVm12 + kTm1 + mVc12 + jCpmTc1 ) mVm22 + 2 2 2 2 ξ2 j 2 kT + mVc2 + jCpmTc2 (13) 2 m2 2 In eqs 12 and 13 and subsequent equations, the subscript m represents a monomer; c represents a cluster; and the subscripts 1 and 2 represent the values before and after the collision, respectively. To solve for the postcollisional states of the cluster and monomer, it is assumed that the temperature of the monomer after the collision is equal to the temperature of the cluster after the collision. Since the gas vibrational temperature is low, the number of degrees of freedom of the monomer is the same before and after the collision. Using simple collision theory, the mean kinetic energy in the center of mass system is 2kTc1.16 Using the known values of the cluster and monomer precollisional velocities and temperatures, as well as the cluster size, and eqs 12 and 13, the velocities and temperatures of the collision pair after the collision can be calculated. D. Condensation. A sticking collision between a cluster and a monomer is referred to as condensation. When a condensation event occurs, the monomer effectively becomes a part of the cluster, and the cluster size, j, increases by 1, and a gas particle is removed from the computational domain. In this model, the monomer transfers all of its momentum and energy to the cluster. The momentum and energy conservation for a condensation event are given as
mV bm1 + mjV bc1 ) m(j + 1)V bc2 ξ1 1 1 mVm12 + kTm1 + mjVc12 + jCpmTc1 + Hvm ) 2 2 2 1 m(j + 1)Vc22+(j + 1)CpmTc2 2
(14)
(15)
The velocities and temperatures of the cluster and monomer before the collision are known as well as the initial cluster size,
j, and the cluster size after the collision, j + 1. Therefore, the postcollisional cluster velocity and temperature can be calculated using eqs 14 and 15. The rate of condensation can be approximated by
C)
4πr2qP
√2πmkTv
(16)
where it has been assumed that the monomers are in an equilibrium Maxwellian velocity distribution at the gas temperature, and the sticking probability, q, is usually within 0.1-1. E. Cluster-Cluster Collisions. A cluster-cluster collision is similar to a cluster-monomer collision in that the outcome can be either sticking or nonsticking. A sticking collision between two clusters is referred to as coalescence, whereas a nonsticking collision can result in either reflexive or stretching separation. The outcome of a cluster-cluster collision is determined using the Ashgriz-Poo model,25 which has been used to model coalescence in water droplet flows and validated using MD simulations.26 In the Ashgriz-Poo model, three factors affect the outcome of the collision: cluster size ratio, impact parameter, and Weber number. The cluster size ratio is the ratio of the diameter of the smaller cluster to that of the larger cluster. The impact parameter is the normalized distance between the centers of the undisturbed clusters at the point of closest approach, perpendicular to the relative velocity vector. The Weber number, We, is given by
We )
FdV2 σ
(17)
where F is the liquid density of the cluster, d is the diameter of the smaller cluster, V is the relative velocity of the two clusters, and σ is the liquid surface tension. Coalescence is more likely to occur for smaller values of We.26 If coalescence occurs, the event is treated similarly to condensation. The two clusters combine to form one large cluster, and the momentum and energy conservation equations for coalescence are given as
mjAb V cA1 + mjBb V cB1 ) m(jA + jB)b V c2 1 1 mj V 2 + jACpmTcA1 + mjBVcB12 + jBCpmTcB1 ) 2 A cA1 2 2 (jA + jB)Vc2 + (jA + jB)CpmTc2
(18) 1 m 2 (19)
where the subscripts A and B are used to differentiate the two clusters. The velocities, temperatures, and sizes prior to collision are known, and the cluster size after the collision is given as the sum of the sizes of the two original clusters. Therefore, the postcollisional velocity and temperature of the new cluster can be calculated using eqs 18 and 19, respectively. If the collision is determined to be nonsticking, the collision is modeled by the standard DSMC collision model, taking into account the masses of each cluster in the collision. F. Evaporation. In the evaporation process, monomers are removed from clusters and are placed in the computational domain as separate entities. The evaporation rate, given by
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E)
4πr2Ps
√2πmkTc
( )
exp
2σ FRTcr
Li et al.
(20)
can be determined by assuming that the cluster of critical size is in a metastable state where its condensation and evaporation rates are equal. It is also assumed that molecules evaporate from the cluster one at a time so that each evaporation event involves the removal of one monomer from a cluster. When an evaporation event occurs, the cluster size is reduced by one, and one new monomer is randomly placed in the DSMC computational cell. The momentum and energy conservation equations are given as
mjV bc1 ) mV bm2 + m(j - 1)b V c2
Figure 1. System sketch of the CO2 expanding jet. The domain was divided into six computational zones, where the left, middle, and right two zones represent the feeding pipe, the converging part, and the pipe nozzle, respectively.
(21)
1 1 mjVc12 + jCpmTc1 - Hvm ) m(j - 1)Vc22 + (j 2 2 ξ2 1 1)CpmTc2 + mVm22 + kTm2 (22) 2 2 Again, the velocity, temperature, and size of the cluster prior to evaporation are known, and the size of the cluster after evaporation is one less than the initial size. To solve for the velocities and temperatures of the monomer and cluster after evaporation, two assumptions must be made. First, it is assumed that the temperature of the monomer after evaporation is equal to the temperature of the cluster before evaporation. Second, as in Section IIC, it is assumed that the mean postcollisional kinetic energy in the center of mass system is 2kTc1.16 This allows us to calculate the magnitude of the relative velocity after the collision, and the direction of the relative velocity vector is chosen randomly. Then using eqs 21 and 22, the postcollisional states of the cluster and monomer can be determined. III. DSMC Simulations of CO2 Homogeneous Condensation In addition to Rayleigh scattering intensities, Ramos et al.14 recently used linear Raman spectroscopy to study cluster formation in CO2 condensation plumes. Five supersonic free expansion plumes of CO2 at stagnation pressures of 1-5 bar and a stagnation temperature of 294 K were expanded through a sonic pipe nozzle with a throat diameter of 0.313 mm. The experimental data along the plume center line provided by this work is used to validate our DSMC CO2 homogeneous condensation simulations as is discussed in this section. A. Near-Field Fluid Field Calculation. The simulations under consideration are challenging due to the large variation in number density from the chamber to the plume far field. The Knudsen number, defined as the ratio of mean free path to the characteristic length (the nozzle diameter), is on the order of 10-5 at the feeding pipe. This low Knudsen number indicates that the flow is continuum in nature and a Navier-Stokes solution is appropriate. Therefore, to model the flow inside the pipe-shaped nozzle, we employ a Navier-Stokes calculation followed by DSMC.27 Since the flow is expanding, the Knudsen number increases very quickly, and the continuum assumption breaks down outside the nozzle. The first Navier-Stokes calculation (designated as “near-field”) is started inside the feeding pipe and carried out to a small region beyond the pipe nozzle. The converged near-field macroparameters of gas temperatures, velocities, and number densities are then used to
Figure 2. Density contours from Navier-Stokes calculation (3-bar case). The nozzle exit is located at X ) 0.0.
create a starting surface for the DSMC, “far-field”, calculation. The starting surface is located just downstream of the pipe nozzle on which the flow is supersonic. Detailed modeling of the reservoir nozzle flow was undertaken because the experimental setup did not employ a standard convergent-divergent nozzle shape or an orifice geometry. Therefore, it could not be assumed that the flow at the nozzle throat or the end of the pipe nozzle was sonic. Figure 1 shows our Navier-Stokes computational domain (commercial software GASP28 was used). The CO2 gas with a stagnation pressure from 1 to 5 bar and a temperature of 294 K enters the domain from a 2-mm-wide feed pipe that converges to a pipe nozzle through a cone with a half angle of 30°. The gas accelerates as it moves through the converged cone and the pipe nozzle, and then it expands freely to the vacuum environment, where we define two supersonic boundaries. The left and lower boundaries were defined with fixed pressure and temperature and axisymmetric conditions, respectively. Three adiabatic wall boundaries were defined at the feed pipe, and the convergent cone and pipe nozzle and second-order extrapolation boundary conditions were used at the two outlets. The outside vacuum domain was chosen to be sufficiently large to allow the flow to become supersonic, but not so large that Navier-Stokes equations would be invalid. The domain was divided into several zones to achieve a better mesh resolved solution. The computational domain was divided into about 100 000 mesh cells, and it took 30 h of computational time on 40 2.0-GHz processors for the relative residue to be 10-6. Figure 2 shows the number density contours of the Navier-Stokes calculation for the 3 bar case. The gas density and temperature are constant inside the feed pipe (where the velocity is very low). They decrease slightly in the cone and pipe nozzle and then decrease rapidly outside the nozzle. On the other hand, the Mach number increases as the gas accelerates and increases to 1.7 at the nozzle outlet. In the near-field region, the density drops by about a factor of 2 from the nozzle exit to vacuum; the temperature drops from 220 to 190 K; and in the corresponding zoomed plots, shown in Figure 3, the Mach number increases from 1.3 to 1.7. Note that the flow at the exit is not sonic, but is already supersonic. Since the Navier-Stokes equations are valid only in the continuum region (inside nozzle and in the near nozzle region), we then construct a starting surface at the nozzle exit, as shown in Figure 3, to initiate the DSMC calculation.
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Figure 3. Enlargement of Mach number contours from the Navier-Stokes calculation close to the pipe exit (3-bar case).
Figure 4. Number density contour from DSMC calculation (3-bar case). Here, D is the nozzle diameter of the CO2 expanding jet.
B. DSMC Non-Condensation Calculation. In the DSMC calculation, the outside computational domain was extended to 12-by-6 nozzle diameters, from the original Navier-Stokes computational domain of 0.5 by 1 diameters. The 2D axisymmetric DSMC code, SMILE,21 was used to simulate the gas expansion from the starting surface, located at the nozzle exit. The domain was divided into 400 × 200 cells with up to 200 × 200 subcells for each cell so that the cell size is comparable to mean free path. We used a time step of 1 × 10-9 s, which is smaller than the local mean time between collisions and 1 million simulated particles, each of which represented 2 × 107 CO2 molecules. The simulation was carried out for 400 000 steps, or 70 computation hours, on 32 2-Ghz processors. Figure 4 shows the DSMC number density contours, which demonstrate the typical behavior of free gas expansions to vacuum with the density dropping by 3 orders of magnitude. Note that the axial distance starting from the nozzle exit is normalized to the nozzle diameter, D. Similarly, as can be seen in Figure 5, the Mach number increases from 1.3 to 11, and the number density drops by 3 orders of magnitude along the plume center line. Figure 5 also shows that the translational and rotational temperatures are close and decrease rapidly from 230 to 10 K along the nozzle center line. Note that the vibration temperature remains constant, since the vibrational relaxation number is large compared to the number of possible collisions in the rapidly expanding flow. The simulation results are shown for a stagnation pressure of 3 bar and are similar for the lower and higher stagnation pressures. Figure 6 shows the pressure, saturation pressure, and saturation ratio along the nozzle center line. The saturation pressure deceases rapidly as the gas temperature decreases, but the gas pressure decreases slowly compared to the saturation pressure. The saturation ratio, defined as the ratio of the gas to saturation pressure, therefore increases from
