on regular and retrograde condensation in multiphase compressible flows

Accepted Manuscript

On regular and retrograde condensation in multiphase compressible flows

Lu Qiu, Yue Wang, Rolf D. Reitz

PII: S0301-9322(14)00084-6DOI: http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.05.004Reference: IJMF 2040

To appear in: International Journal of Multiphase Flow

Received Date: 12 January 2014Revised Date: 28 April 2014Accepted Date: 6 May 2014

Please cite this article as: Qiu, L., Wang, Y., Reitz, R.D., On regular and retrograde condensation in multiphasecompressible flows, International Journal of Multiphase Flow (2014), doi: http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.05.004

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Body Page 1 of 34 Body

On regular and retrograde condensation in multiphase compressible flows

Lu Qiu1,a), Yue Wang1,b) and Rolf D. Reitz1,c) 1Engine Research Center, Department of Mechanical Engineering,

University of Wisconsin-Madison

1500 Engineering Drive, Madison, WI 53706, USA

a): The corresponding author, email address: [email protected] b): Email address: [email protected] c): Email address: [email protected]

Abstract

Two distinctive condensation mechanisms of pure species, regular and retrograde

condensation, are investigated using a one-fluid model and a homogeneous phase equilibrium

model based on entropy maximization. Fluid dynamics simulations are performed to model

the regular condensation process of ethylene in a converging nozzle, and the retrograde

condensation process of a fluorinated compound in a shock tube. To our knowledge, the

present simulations are the first CFD simulations of retrograde condensation processes. The

simulations show reasonably good agreement with available experimental data both

quantitatively and qualitatively, which confirms the consistency between the theory-guided

simulations and experiments. For the supercritical injection problem, condensation is found

to occur after continuous expansion when ethylene is brought into the two-phase region from

a supercritical state. For the shock tube problem, both simulations and experiments show that

condensation occurs after the high pressure reflected shock is formed from the end wall.

Increase in the initial pressure ratio increases the incident shock strength and reinforces

condensation by elevating the liquid volume fraction. A complete liquefaction shock is found

at high incident shock Mach numbers when the compression is strong enough to send the

fluid from the pure vapor to the pure liquid state by crossing the two-phase mixture region.

After condensation, the condensed liquid phase is fully depleted as the pressure wave

expands and the fluid is brought back to the vapor state though a continuous evaporation


process.

Key words: liquefaction shock; retrograde condensation; phase transition; shock waves;

two-phase flows; equation of state; phase equilibrium

1. Introduction

Condensation and evaporation are common phase transition phenomena. Condensation

usually occurs for species experiencing an expansion process. Species exhibiting such

characteristics are referred to as regular or simple species, such as water, air and carbon

dioxide, etc. During the expansion process, a supersaturation condition is achieved, and when

the supersaturation exceeds a critical limit (i.e., the critical supersaturation ratio),

spontaneous condensation occurs. This kind of condensation has been experimentally

observed in supersonic nozzles for water [1-3] and in hypersonic wind channels for air [4, 5].

However, condensation due to compression was not studied until the early work of Thompson,

who investigated compression condensation both analytically, using thermodynamic analysis

[6], and experimentally in a shock tube [7] in the 1970s.

According to Thompson and Sullivan [6], compression-induced retrograde condensation

is seen for retrograde species, like benzene and octane, etc. Compression-induced

condensation occurs because these species have a large number of molecular degrees of

freedom (i.e., large heat capacity), and as a result, compression work can be stored as internal

energy with only a small temperature rise [6]. The distinctive and different behavior of

regular and retrograde fluids is illustrated in Figure 1, which presents the temperature-entropy

phase diagram of pure ethylene (C2H4), a regular species, and perfluoroheptane (C7F16), a

retrograde species that is used in the simulations to be presented later. The thermodynamic

properties used for the two species are given in Table 1. In Fig. 1, the upwards arrow

represents an isentropic compression process (through which temperature and pressure


increase) while the downwards arrow stands for an isentropic expansion process (through

which temperature and pressure decrease). More specifically, for C2H4, whether it is initially

in the pure liquid or vapor phase, only through an expansion process will the fluid enter the

two-phase region with condensation. In this paper, the two-phase region stands for

vapor-liquid equilibrium and the solid state is not considered. For C7F16, however, there are

two separate scenarios. If the fluid is initially in the vapor phase, only compression can bring

the fluid into the two-phase region with condensation. In contrast, if the fluid is initially in the

liquid phase, only an expansion can bring the fluid into the two-phase region with

evaporation. Understanding these basic condensation mechanisms is critical to comprehend

and explain the different physical problems associated with phase transitions of pure species,

which then serves as the basis for further examinations of the phase transition mechanisms of

mixtures.

Lin et al. [8] examined the injection of a supercritical binary fuel composed of methane

and ethylene. An opaque background in shadowgraph images appeared in the near-nozzle

region, which was thought to be due to the condensation of the fuel. The immediate

occurrence of condensation at the nozzle exit indicates that condensation may have already

occurred inside the nozzle. Indeed, experimental visualization performed using a transparent

injector for pure ethylene showed that the transparent passage became opaque “abruptly” at

the end of the convergent section. Lin et al. [8] attributed the near-spontaneous condensation

to homogeneous nucleation and calculated the nucleation rate and critical nucleus size using

the classical nucleation theory of Frenkel [9]. This non-equilibrium nucleation dynamics

analysis was also later adopted by Star et al. [10]. Based on the phase diagram in Fig. 1(a),

since ethylene is a regular species, its condensation is naturally connected to classical

homogeneous nucleation theory, for which supersaturation due to expansion is the physical

cause.


Dettleff et al. [7] experimentally studied retrograde condensation of a fluorinated species

in a shock tube. It was found that condensation could occur, and when it occurred, a clear

liquid phase was seen at the closed end of the tube driven section from photographs. Their

experimental work was a direct proof of the condensation mechanism for retrograde species

as the fluid only undergoes phase change upon compression by the reflected shock.

This paper investigates the condensation and evaporation characteristics of pure species

using a one-fluid model accompanied with a homogeneous phase equilibrium model. In

Section 2 the computational models for the fluid dynamics and thermodynamics are

introduced. Model validations are presented in Section 3, where the simulation results are

compared with available experimental data for both the regular and retrograde condensation

processes. Summary and conclusions are made at the end.

2. Computational Model

2.1 The Fluid Solver

The open-source computational fluid dynamics (CFD) code KIVA-3V Release 2 [11] was

used as the basic fluid solver. This code been widely applied for multi-dimensional diesel

spray combustion simulations (e.g., [12, 13]). Briefly stated, the solver uses an Arbitrary

Lagrangian Eulerian (ALE) approach to deal with moving boundaries such as the piston

motion. The standard KIVA code solves the conservation equations at each time step in three

consecutive steps. In Phase A, the source terms due to spray and combustion are solved (not

relevant to the current simulations). In Phase B the cell is moved with the local fluid velocity,

and the physical quantities (e.g., pressure, temperature and velocity), except for convection

effects, are solved. Finally, in Phase C (rezoning stage) the cell boundaries are mapped back

to the location where they should be from the specified mesh motion.

The current simulations consider real gas effects, as opposed to the ideal gas relation in

the standard KIVA code. The compressible flow solver in the KIVA code solves the partial


differential equation for the internal energy as [14]:

(1)

is the mixture density, is the velocity vector, is the pressure, and is the

turbulence dissipation rate. is the heat flux due to heat conduction and diffusion:

(2)

is the specific enthalpy of species i. and are the heat conduction coefficient and

diffusion coefficient, respectively. In the simulations to be presented below, and are

calculated through the dimensionless Prandtl and Schmidt numbers, and the default values

0.9 and 0.56 are used.

The energy equation is solved in three steps in Phase B, and each step is either a constant

volume or constant pressure process [14-16]. The focus here is the last step occurring at

constant pressure that is used to determine the B-state internal energy from a previous t-state

through:

(3)

with the diffusion term calculated using

(4)

is a parameter used to ensure numerical stability and hence it depends on local flow

situations. is the cell mass updated in Phase B. is the normal area vector of the cell

face a [14]. is the temperature determined from the t-state. After some manipulations as

detailed on Pages 10 and 11 in Ref. [15], the temperature in the B-state is updated through:


(5)

where superscript “n” refers to a previous time level. When ideal gas is assumed for the

mixture, as in the standard KIVA code, and in above equation can be easily

calculated. To consider the real gas effects, they must be calculated according to the selected

equation of state (EOS) model. When phase transition occurs, as will be discussion below

shortly, the two-phase mixture properties are determined using algebraic averages. For

instance, the mixture molar enthalpy is calculated as the molar average of the two

phases:

(6)

where is the mole fraction of the liquid phase. In this treatment, both the liquid and

vapor phases are treated as a homogeneous mixture at equilibrium.

Since only one set of conservation equations is solved for a two-phase mixture, this is

essentially a one-fluid model [17-19]. In a two-phase model, such as the volume of fluid (VoF)

method, the liquid/vapor interface is tracked by solving an extra transport equation of, for

instance, the liquid volume fraction. In the present one-fluid model, interfaces are not

resolved between the two phases in a cell. This means that capillary pressure effects are not

considered. However, the existence of phase interfaces can still be represented through the

liquid volume fraction. As the fluid solver updates the conditions of each cell, its

thermodynamic state (e.g., liquid volume fraction) also changes. As a result, a cell’s state can

change from a single-phase to a two-phase mixture, depending on the local conditions, and

hence the phase transition process is captured. In the present homogenous equilibrium model,

the two phases are assumed to have the same velocity, temperature and pressure, no subgrid


scale thermal and pressure gradients or relative motion exists. On the contrary, in two-fluid

model treatments, calculations of interfacial heat, mass and momentum transfer rates requires

the introduction of additional models.

2.2 The Thermodynamic Solver

In the standard KIVA code, the state of a cell is determined after the mass, momentum

and energy equations are solved. By assuming that thermodynamic equilibrium is attained

instantaneously, the cell temperature is determined directly from the updated internal energy

because it is only a function of temperature for ideal gases. Pressure is then calculated using

the ideal gas law. From the perspective of classical thermodynamics, because no finite time

scale is involved, the mixture in each computation cell immediately reaches a global

thermodynamically stable state. To consider real gas effects, the thermodynamic solver must

be modified accordingly, as now the internal energy is a function of both temperature and

pressure.

It should be noted that phase transition is predicted if the mixture potential energy can be

further lowered through phase separation. As a result, the phase equilibrium model is soundly

based on fundamental thermodynamic principles (i.e., the second law of thermodynamics).

For a specified thermodynamic system, its global stable state is the one with minimal

thermodynamic potential. For instance, when temperature, pressure and mole numbers are

specified, the global stable state has minimum Gibbs free energy [20]. Of course, once the

equilibrium state is determined, phase transition dynamics analysis requiring non-equilibrium

thermodynamics can be formulated. This is because any kinetic process should evolve toward

the equilibrium state and hence the equilibrium solution serves as a reference solution. It

should also be emphasized that the present equilibrium solution includes not only saturation

densities, temperature and pressure, etc., but also the equilibrium phase compositions.

The global relationship between the fluid and thermodynamic solvers can be seen from


either Fig. 3 in Ref. [21] or Fig. 1 in Ref. [22]. This one-fluid phase equilibrium model

assumes velocity, temperature and pressure equilibrium, so the rates of interface heat and

mass transfer are infinitely fast. After the fluid solver advances the specific internal energy

and species density for each computational cell in time, the thermodynamics solver is applied

to determine the state of each cell (e.g., temperature and pressure). For a mixture with

specified internal energy, volume and mole number of each component, the thermodynamic

equilibrium state has the maximum entropy [20]. Applying the entropy maximization

principle to solve practical isochoric-isoenergetic flash problems was proposed by Castier [23]

and its applications can be found in Refs. [22, 24, 25].

Entropy maximum oriented equilibrium calculations for a mixture are very complex as

partial derivatives of thermodynamic properties are needed (e.g., ,

and . , where is the fugacity coefficient). As a result, the more complex the

equation of state model, the more intensive the actual computation in the thermodynamic

solver. For pure species, the entropy maximization principle reduces to the constraints of

thermal, mechanical and phase equilibria. Along with mass and energy conservation, the

following algebraic equations are solved simultaneously [10, 26].

(7)

Superscript “spec” stands for specified inputs to the thermodynamic solver, is the volume

fraction of the vapor phase; is the saturation pressure of the pure species. is the

fugacity calculated for phase equilibrium. In the current simulations, the saturation pressure

line of a pure species is calculated by enforcing the equality of fugacity for vapor-liquid

equilibrium. The results are then saved in a tabulated table for future calls in solving Eq. (7)


iteratively. The specific internal energy of the mixture is calculated using the departure

function

(8)

where is the specific internal energy of the ideal gas, and it is only a function of

temperature. The internal energy departure is determined from the equation of state

(EOS) model. The Peng-Robinson EOS (PR EOS) [27] is adopted here due to its relative

simplicity and accuracy in calculating liquid density, and successes in investigating

multiphase flow problems [10, 16, 28, 29]. The PR EOS has the form

(9)

is the molar volume and is the universal gas constant. and are determined

using:

(10)

with

(11a)

An expanded formula [30] was used when the acentric factor is greater than 0.5:

(11b)

The compressibility factor, , an indication of departure from ideal gas behavior, is

calculated using

(12)

is determined using the following relationship:


(13)

The Mach number (Ma) is calculated using:

(14)

where is the isentropic sound speed, while is the fluid velocity magnitude. All other

thermodynamic properties, such as the fugacity coefficient, entropy departure, etc., are

calculated using the formulae found in Refs. [31] and [32].

3. Results

Simulation results were compared with experimental data for both regular and retrograde

condensation. The ideal gas properties, including the specific heat capacity and entropy, were

calculated using the NASA 7-coefficient polynomials [33]. Properties (e.g., the critical

properties) of pure species were obtained from the DIPPR database [34]. The reference state

was taken at 298.15 K and 1 bar for both the enthalpy and entropy.

3.1 Regular condensation

Regular condensation of ethylene (C2H4) is first considered. A rectangular nozzle with

exit cross section of 1 mm2 was used in the experimental work of Lin et al. [8]. All four

experimental conditions found in Ref. [10] were considered here, as listed in Table 2. Only

the first condition did not lead to condensation. The downstream chamber condition is

atmospheric, but the injection conditions vary.

Computational domain is shown in Figure 2, and more details of the geometries can be

found in Ref. [10]. Injection is oriented towards the +x direction, and in Figs. 3 and 4,

injection is from left to right. Pressure inflow and pressure outflow boundary conditions were

enforced on the left and right sides, respectively. The standard k-ε turbulence model was used,


and adiabatic and no-slip boundary conditions were applied on all walls. A three-block mesh

structure was used, with 220 grids points in the nozzle axis direction (x direction), 20 grid

points in the y direction, and 40 grid points across the nozzle cross-section (z direction),

making the final mesh composed of 176000 cells. Simulations with finer meshes showed that

this resolution gave adequately grid-independent results.

In the experiments [10] pressures were measured along the centerline (see red solid line in

Fig. 2) and a comparison with the predicted pressure distributions is presented in Figure 3. To

better illustrate the effects of phase change, simulation results using the ideal gas law without

phase change are also presented. Both ideal and real gas simulations match the experimental

pressure data well at the entrance of the convergent induction section. The real gas simulation

significantly improves the predictions in the straight nozzle section. It is seen that, at the end

of the convergent section, the ideal gas simulation shows a large pressure drop while the

pressure relaxation is not so strong when real gas effects are considered. At the same time,

with phase change included, the predicted pressure is more close to the experiments in the

near-nozzle area. This effect is especially noteworthy for Cases 2-4 for which a relatively

stronger condensation event occurs, compared to Case 1. It is hence concluded that both real

gas and phase change effects are critical in evaluating supercritical condensation processes

and they must be taken into account in simulations to capture phase transition processes. It is

encouraging to see that the current one-fluid approach gives reasonably good agreement with

the experimental data, given its relative low computational effort.

Figure 4 presents the liquid volume fraction and compressibility factors for the four cases.

For Cases 1 and 2, outside the condensing region the liquid volume fraction is zero, as they

are in a supercritical state with vapor-like density. For Case 1, condensation occurs, but it is

limited to the nozzle exit region. Note that the maximum liquid volume fraction is about

0.5% for Case 1, while the experiments show no visible condensation inside the nozzle. Star


et al. [10] showed similar results but the disparity was left unexplained. However, the

appearance of liquid phase is consistent with the experimental observation that condensation

did occur downstream of the nozzle (i.e., in the underexpanded jet outside the nozzle).

Therefore, it is believed that the outlet boundary condition used in the simulations may

contribute to the discrepancy, and such discrepancy would be mitigated if both the internal

and external flows were simulated together to avoid the uncertainty in the boundary

conditions. However, the complexity of expanding flow with imbedded shock waves and

phase change outside the nozzle would make this a more difficult simulation. For Case 2, the

two-phase region starts at the end of the convergent section. The liquid volume fraction keeps

increasing until it reaches a maximum at the entrance of the straight section, and it then

decreases all the way to the nozzle exit.

For Cases 3 and 4, the initial liquid volume fraction is unity since the fluid is in a

supercritical state with liquid-like density. In fact, evaporation occurs, and it is evidenced by

the decrease of the liquid volume fraction in the straight section. For these two cases, the

profiles of compressibility factor and liquid volume fraction are similar. Compared to Case 3,

the evaporation in Case 4 is less intensive, so the liquid volume fraction drops more slowly.

The liquid volume fraction decreases to 10% and 15% for Cases 3 and 4, respectively.

It is helpful to track the change of state of a fluid element as it travels through the nozzle

passage, such as those shown in Figure 5 for the state evolution along the nozzle centerline in

a phase diagram. Figure 5 (a) shows that for Cases 1 and 2 the fluid at the nozzle entrance is

initially in a vapor-like supercritical state. Because of the pressure gradient across the nozzle,

the fluid continuously expands during the trip to the exit. In the later stages of the expansion

process, when the supersaturation state is large enough, the fluid enters the vapor-liquid

two-phase region and condensation occurs. Since only a small portion penetrates into the

two-phase mixture region with a very feeble condensation, the state is close to the dew point


line. The process for Case 2 penetrates even further into the vapor dome, so the condensation

is stronger. For Cases 3 and 4, it is obvious that the evaporation occurs as the fluid crosses the

bubble point line, but because of the different initial state, the ending states are on different

constant-quality lines. Similarly, because the end state for Case 4 is closer to the bubble point

line, the liquid phase volume fraction is larger than that for Case 3.

The present phase-equilibrium approach does not give information about droplet sizes or

number densities, which requires a non-equilibrium approach, such as the study in [10] using

homogeneous nucleation theory. Non-equilibrium models also require that the equilibrium

solutions are available, so the present approach can also be used as a starting point for further

model development. However, it should be noticed that the simulations using classical

nucleation theory in Ref. [10] yield similar results to the present equilibrium solutions.

Specifically, the equilibrium and non-equilibrium simulations yield similar pressure profile

and the liquid volume fraction differs only less than 7% absolute. This information indicates

that the thermodynamic equilibrium assumption is an appropriate first estimation of the

non-equilibrium problem.

Indeed, there is still uncertainty surrounding classical nucleation theory models. For

example, experimental nucleation rates of nitrogen in a supersonic nozzle had been shown to

be underestimated by 13-16 orders of magnitude for the temperature range between (1.6 kPa,

39 K) and (7 kPa, 46 K) [35]. On the other hand, mean field kinetic nucleation theory [36, 37]

was found to do better (with an underestimation of only 2-3 orders of magnitude). In contrast,

classical nucleation theory was found to agree with experiment nucleation rates of methanol

in a supersonic nozzle to within only one order of magnitude [38]. Therefore, the

applicability of classical nucleation theory, such as that used in Ref. [10], needs to be further

validated for use in CFD simulations.


3.2 Retrograde condensation

A complete liquefaction shock is a compression discontinuity for which the upstream

state is vapor and the downstream state is pure liquid [7]. For the partial liquefaction shock,

the downstream state is in a two-phase mixture state. Both partial and complete condensation

observed in shock tube experiments are investigated here. To our knowledge, this is the first

time these experiments have been modeled using CFD. The shock tube is composed of a

driver section of 250 cm and a driven section of 242.6 cm. A 1-D domain was selected using

1500 grid points for each section. In the following discussion, “complete condensation” is

used interchangeably with “complete liquefaction shock” and “partial condensation” is used

interchangeably with “partial liquefaction shock.” Together three different initial conditions

are considered for the driven section: 0.51, 0.67 and 0.99 bar. In the shock tube experiments

[7], the driver gas section was filled with nitrogen and the driven section used a fluorinated

compound, perfluoro-dimethycyclohexane (C8F16). Noticing that available thermodynamic

properties of this species are limited, perfluoroheptane (C7F16) was used instead in the present

simulations. Dettleff et al.[7] also argued that the results would not be significantly different

if other heavy hydrocarbons were used, and C7F16 also experiences retrograde behavior (as

seen in Fig. 1 (b)). Its constant volume specific heat at the critical point is calculated to

be 48.5 using the Peng-Robinson equation of state. This is in accord with the conclusion

of Thompson and Sullivan [6] that it is necessary that to permit the possibility

of a complete liquefaction shock. On the other hand, for a partial liquefaction shock, only that

is required.

The shock tube experiment in Ref. [7] was designed to produce liquefaction as the

incident shock reflects from the closed end of the tube. A schematic plot is shown in Fig. 6 to

illustrate the shock tube configuration. The driver section is at State 4 while the driven

section is at State 1. Upon releasing the diaphragm, incident compression shock and


expansion waves form. The state behind the incident shock front is State 2. The section

between contact surface and the leading expansion wave is at State 3. Finally, the region

between the reflected shock wave and the contact surface is at State 5.

The effects of the incident shock Mach number (Ms) are shown in Fig. 7. Figure 7 (a)

shows that pressure P2 downstream of the incident shock does not show a strong non-linear

behavior, as would be expected for ideal gases from the normal shock relations. In contrast,

real gas effects lead to an almost-linear relationship. The pressure P5 and temperature T5 are

shown in Figs. 7 (b) and (c). As the shock strength increases, P5 increases too. It is noted that

the profile of P5 does not show liquefaction effects, as there are no obvious changes in its

slope, as also noticed in Ref. [7]. On the contrary, the plot of T5 is a good indication of the

onset of condensation, which shows a “distorted, angular S-form” [7]. The two

discontinuities on the slope of T5 represent the start of partial and complete condensation,

respectively [7].

The distorted “S” shape can be explained as follows. During the transition from the vapor

to the two-phase state at low Ms, partial condensation occurs, so T5 has a sudden increase, as

seen in Fig. 7 (c). Similarly, during the transition from a two-phase state to a single liquid

state or to a supercritical state at higher Ms, heating effects are lost and the slope of T5

decreases, as also seen in Fig. 7 (c). The shock Mach number required for partial and

complete condensation is specifically marked in all the plots of Fig. 7. Figure 7 (d) plots P5

versus T5 with the saturation line. It is clearly seen that the fluid in State 5 experiences a state

change from superheated vapor to a two-phase mixture, and then to a compressed liquid and

finally to the supercritical state.

Some discrepancies between the simulation and experiments are observed in Fig. 7. This

could be due to two reasons. First, as mentioned, a slightly different species (i.e., C7F16) was

used in the simulations while C8F16 was used in the experiments. However, the general trends


of the experiments are well captured, as seen from the close-to-linear relationship between P2

and Ms, the non-linear relationship between P5 and Ms, and the “S” shape relationship

between T5 and Ms on the occurrence of partial and complete condensation. Second, it should

be noted that flows in a shock tube are intrinsically three dimensional and involve complex

shock wave turbulence boundary layer interactions and flow separations [39-41], and phase

change further complicates the flow. However, the current 1D simplification provides a

reasonable match on the global parameters (e.g., P2, P5 and T5 in Fig. 7).

Additional simulations were performed to characterize the effects of incident shock

strength on the onset of condensation. Figure 8 presents the start of condensation (i.e., partial

liquefaction) at different Ms for the three P1’s. From P1=0.99 to P1=0.51 bar, as the pressure is

reduced, the Ms required for condensation increases. In other words, condensation is

postponed when P1 is reduced. This is expected, as the reflected shock, which depends on the

initial pressure ratio P4/P1, must be strong enough to bring the vapor phase into the two-phase

mixture region. Similarly, for each specified P1, the onset of condensation is advanced when

Ms or P4/P1 becomes larger.

Two special cases were chosen for further analyses. Case 1 has partial condensation

occurring at P1=0.67 bar and P4= 4.0 bar. Case 2 is at P1=0.99 bar and P4= 8.5 bar, under

which strong (actually complete) condensation occurs. For Case 1, the predicted temperature

and liquid volume fraction are plotted in Fig. 9 at t=16.0 ms (a) and t=18.0 ms (b). These two

time instants are after the reflected shock has formed. The liquid volume fraction has already

reached 2% by t=16.0 ms. It is noticed that there is a plateau of temperature in the

condensation region. Because of the phase change, a large temperature gradient appears.

From t=16.0 to t=18.0 ms, the two-phase region is transported along with the pressure wave,

and the previous temperature gradient is disrupted. Compared with the results at t=16.0 ms,

the decrease liquid volume fraction indicates that evaporation continues to occur while the


local fluid is experiencing expansion.

Results for Case 2 are presented in Figure 10 at two instants as well. Condensation has

already occurred at t=13.28 ms. But at this time the condensation is so strong that full

condensation, i.e., a complete liquefaction shock, occurs. The shock strength is so high that

100% of the original gaseous fluid is converted to liquid phase. Therefore, a larger local

gradient is found for both the pressure and temperature when compared with Case 1. At

t=15.0 ms, the liquid phase amount has dropped as evaporation occurs, similar to Case 1.

However, since the vapor-liquid interface is distinctive due to the strong liquid volume

fraction gradient, the evaporative cooling leads to sharp change of temperature.

The two numerical simulations above indicate that partial or complete condensation

occurs when the reflected shock forms as the incident shock is reflected from the closed end

of the tube. Because pressure is enhanced through the reflected shock (also seen through the

color scale), the appearance of the liquid phase can only be due to the compression, and not

through expansion (such as the condensation seen in nozzles and wind tunnels, as mentioned

above). Therefore, the present CFD simulation results are consistent with the condensation

mechanism for this retrograde fluid (refer to Fig. 1 (b)). In addition, it is remarked that the

condensation region is limited to lie between the contact surface and the right-hand end of the

tube, where the pressure is high enough to be able to trigger the phase change.

Since P5 is highest upon pressure wave reflection, it is more intuitive to monitor the

dynamic evolution of the properties at the right-hand tube end. The histories of pressure and

liquid volume fraction of two phases are presented in Fig. 11 for the two cases above. For

both cases, the abrupt increase in the pressure profile (e.g., Fig. 11 (a) and (b)) is a clear

indication of the arrival of the incident shock, which is closely followed by the condensation,

given that the shock strength is strong enough. The following pressure plateau indicates the

state behind the reflected liquefaction shock (i.e., P5). The relative length of the plateau is


approximately the residence time of the shock, which is in agreement with the experimental

measurements in Ref. [7]. For Case 1, the partial condensation event takes place at t=13.86

ms. After an additional 0.8 ms, the liquid phase occupies 2% of the volume. There is a period

of a further 2.5 ms, during which the state (pressure and liquid phase amount) does not vary

too much, indicating that a quasi-steady state is obtained. This quasi-steady state is the

preferred state in shock tube to provide relatively constant high temperature and pressure

conditions. After t=17.0 ms, evaporation starts and the liquid phase is continuously depleted.

Compared with the feeble condensation above with only 2% liquid volume conversion,

the situation changes dramatically when strong condensation takes place. Firstly,

condensation occurs earlier and it uses 75% less time than for Case 2 to convert all the

gaseous fluid to the pure liquid phase. Secondly, the oscillations seen in the pressure profile

are stronger due to the stronger phase transition events and the quasi-steady state duration is

reduced (Fig. 11(b)). Thirdly, after the evaporation occurs, within about 75% of the time used

in Case 1, the liquid volume fraction decreases from 100% to about 3%. It is also evident that

the following liquid volume fraction increases due to repeated condensation. However,

evaporation still dominates as the remaining liquid phase is converted to vapor promptly, so it

is not specifically marked in Fig. 11 (b).

It is helpful to analyze these dynamic processes at the right-hand end of the tube in a

phase diagram, which is illustrated in Fig. 12 on temperature-entropy diagrams for the two

cases. The numbers in the brackets are the starting and ending times for the state. For both

cases, it is noticed that the fluid evolves between the single phase and the two-phase state.

For Case 1, spontaneous condensation occurs as the fluid is compressed to cross the dew

point line and enter the two-phase state (e.g., Fig. 12 (a)). The fluid stays in the two-phase

mixture state until evaporation occurs. After the evaporation is completed, the fluid enters the

vapor phase again. During the evaporation in the two-phase state, the temperature and


pressure decrease along the saturation line. It is also found that the evolution of the fluid in

the two-phase mixture state and the vapor state after t=19.05 ms is close to an isentropic

process.

Compared to Case 1, because the shock strength in Case 2 is higher, the fluid is

continuously compressed into the two-phase mixture state, and further into the pure liquid

state in a much shorter period (see Fig. 12 (b)). In fact, only about 0.15 ms is needed for the

fluid to move from the dew point line to reach the bubble point line. In addition, it is noticed

that the compression process wanders its way from the pure vapor to the pure liquid state.

The pressure in the liquid state is higher than the critical pressure and the fluid is considered

to be at the supercritical state with liquid-like density. After t=14.55 ms, the expansion leads

to continuous evaporation, bringing the fluid back into the two-phase mixture region.

Different from before, from t=14.55 to 17.93 ms, the evaporation process that converts all the

liquid to the pure vapor state is more close to isentropic than the previous condensation

process. After t=17.93 ms, the fluid finally returns to the original vapor state and further

expansion is close to isentropic.

Finally, three more cases are selected to illustrate the effects of P4 (equivalently Ms) on

the condensation intensity. Figures 13 (a), (c) and (e) present the global x-t wave diagrams at

P4=4.5, 5.0 and 6.0 bar with constant P1=0.51 bar, respectively. The colors are based on

pressure values but to increase the color contrast, the logarithm of pressure is used. It is seen

that P5 increases with P4. At the same time, condensation becomes more intensive, as seen

from the blow-up plots of liquid volume fraction contours shown in Figs. 13 (b), (d) and (f)

(note the scales are different). More specifically, the maximum liquid volume fraction is

found to be 8%, 25% and 100% for the three conditions considered, corresponding to a

partial, intermediate and strong condensation, respectively.


4. Summary and Conclusions

Computational fluid dynamic simulations were performed for two-phase flows in a

converging nozzle and a shock tube to investigate phase transition behavior for both regular

and retrograde condensation of pure species. The fluid solver is based on the KIVA 3v release

2 open source code. But, to consider the real gas effects, the thermodynamic relationship was

updated using the Peng-Robinson equation of state. Accordingly, the thermodynamic solver

was also updated to account for the fact that the internal energy is not only a function of

temperature. A homogeneous phase equilibrium model based on entropy maximization was

used as the phase transition model.

The simulation results show reasonable agreement with available experimental data using

the present one-fluid approach with a realistic equation of state model. They also conform to

the expected behavior from thermodynamic analyses based in temperature-entropy phase

diagrams, and the deduced condensation mechanisms for the different fluids. Therefore, the

present theory-led numerical simulations are consistent with experiments. For ethylene,

regular condensation can only occur due to an expansion process, which brings the fluid to a

supersaturation state until spontaneous condensation occurs. The simulation results were

verified against measured pressure distributions for four conditions. On the other hand,

condensation can occur due to compression with a retrograde fluid. The present shock tube

simulations with a fluorocarbon fluid were found to predict well both qualitatively and

quantitatively when compared with available experimental data. Therefore, the present

approach can be applied to study phase transition behavior for both regular and retrograde

condensation in multiphase compressible flows.

Specifically for the retrograde fluid condensation, the following conclusions can be drawn

from the present simulations and the comparisons with the classical experiments of Dettleff et

al. [7]:


(1) Consistent with the experiments, condensation occurs in the reflected shock region at

the right-hand end of the shock tube, where only the retrograde fluid exists.

(2) Partial condensation occurs when the incident shock is strong enough that the fluid

can be compressed from the vapor state to the two-phase mixture state.

(3) Complete condensation occurs when the incident shock is strong enough that the fluid

can be further compressed from the vapor state to the two-phase mixture state, and to

the compressed liquid state or supercritical state with liquid-like properties.

(4) The temperature at State 5 (behind the reflected shock) is a better indicator of

condensation than pressure.

(5) The vapor-to-liquid conversion (i.e., liquid volume fraction) is enhanced when the

incident shock Mach number or P4 increases.

(6) Later after the condensation, the expanding wave leads to continuous evaporation

until the liquid phase is fully depleted.

Acknowledgements

The research work was conducted at the Engine Research Center and sponsored by

Department of Energy (DOE) and Sandia National Laboratories through the Advanced

Engine Combustion Program (MOU 04-S-383) under the management of Dr. Dennis L.

Siebers.

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(a)

(b)

Fig. 1. Temperature-entropy phase diagram of pure species. (a): Regular fluid ethylene (C2H4). (b)

Retrograde fluid perfluoroheptane (C7F16). The downwards arrow shows an isentropic expansion process

while the upwards arrow is an isentropic compression process. Also shown are the vapor dome, critical

point and isobaric lines at four pressures (i.e., 0.2, 0.4 and 0.8 Pc,). The supercritical region where both

temperature and pressure are above the critical point is specifically highlighted.

Fig. 2. 3D view of the computational mesh used for pure ethylene injection. Red solid line shows where

pressure transducers are located. Dashed line shows injection direction.


(a)

(b)

(c)

(d)

Fig. 3. Comparison of pressure profiles with experimental data along the centerline of the rectangular

nozzle for the four conditions listed in Table 2. Injection is from left to right. Refer to Fig. 2 for the

centerline location. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. Also shown are simulation results using

the ideal gas law without phase change.


Fig. 4. Calculated liquid volume fraction and compressibility factors of the two phases along the centerline

of the rectangular nozzle for the four conditions listed in Table 2. Injection is from left to right. Refer to Fig.

2 for the centerline location. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

(a)

(b)

(c)

(d)


(a)

(b)

Fig. 5. Evolution of thermodynamic state along centerline of rectangular nozzle on temperature-entropy

phase diagrams for the four cases listed in Table 2. Refer to Fig. 2 for the centerline location. (a) Cases 1

and 2. (b) Cases 3 and 4. Arrow indicates the direction of the state change. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Position-time wave diagram for the shock tube configuration. Test section is near the right end. The

driver section ○4 and driven section ○1 are initially separated by the diaphragm. Contact surface is

shown by the solid (blue) line. ○2 and ○3 represent the state behind the incident shock and contact

surface, respectively.


(a)

(b)

(c)

(d)

Fig. 7. Comparison with experimental data as a function of incident shock Mach number. (a) Pressure P2.

(b) Pressure P5. (c) Temperature T5. The two discontinuities of slope represent onset of partial

condensation and complete condensation, respectively. (d) Pressure P5 versus temperature T5.

Fig. 8. Effects of P1 and incident shock Mach number on the onset of condensation.


(a)

(b)

Fig. 9. Predicted contact discontinuity, temperature, and liquid volume fraction for partial liquefaction

shock at P1=0.67 bar and P4=4.0 bar. (a) At 16.0 ms. (b) At 18.0 ms. Note that the liquid volume fraction is

enlarged 10 times. Vertical dashed line indicates the right-hand end wall location.


(a)

(b)

Fig. 10. Predicted contact discontinuity, temperature, and liquid volume fraction for partial liquefaction

shock at P1=0.99 bar and P4=9.5 bar. (a) At 13.28 ms. (b) At 15.0 ms. Vertical dashed line indicates the

right-hand end wall location.


(a)

(b)

Fig. 11. Evolution of liquid volume fraction and pressure at the right-hand end of the shock tube. (a)

Partial liquefaction shock at P1=0.67 bar and P4=4.0 bar. (b) Complete liquefaction shock P1=0.99 bar and

P4=9.5 bar.

(a)

(b)

Fig. 12. Evolution of state at the right-hand end of the shock tube in temperature-entropy phase diagram.

(a) Partial liquefaction shock at P1=0.67 bar and P4=4.0 bar. (b) Complete liquefaction shock P1=0.99 bar

and P4=9.5 bar. Numbers in legend show the time period.


x (cm)

Tim

e (

ms)

P1=0.51 bar

P4=4.5 bar

log(P)

-250 -200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

10

12

14

16

18

20

22

-0.2

0

0.2

0.4

0.6

0.8

1

Contact surface

(a)

x (cm)

Tim

e (

ms)

P1=0.51 bar

P4=4.5 bar

liquid volumefraction

220 225 230 235 24010

12

14

16

18

20

22

0.0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Contact surface

(b)

x (cm)

Tim

e (

ms)

P1=0.51 bar

P4=5.0 bar

log(P)

-250 -200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

10

12

14

16

18

20

22

-0.2

0

0.2

0.4

0.6

0.8

1

Contact surface

(c)

x (cm)

Tim

e (

ms)

P1=0.51 bar

P4=5.0 bar


220 225 230 235 24010

12

14

16

18

20

22

0

0.05

0.1

0.15

0.2

Contact surface

(d)

x (cm)

Tim

e (

ms)

P1=0.51 bar

P4=6.0 bar

log(P)

-250 -200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

10

12

14

16

18

20

22

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Contact surface

(e)

x (cm)

Tim

e (

ms)

P

1=0.51 bar

P4=6.0 bar


220 225 230 235 24010

12

14

16

18

20

22

0

0.2

0.4

0.6

0.8

Contact surface

(f)

Fig. 13. Global x-t wave diagram of logarithm of pressure and liquid volume fraction at P1=0.51 bar. (a) and

(b) Partial condensation at P4=4.5. (c) and (d) Intermediate condensation at P4=5.0. (e) and (f) Strong

condensation at P4=6.0.


Table 1. Thermodynamic properties of C2H4 and C7F16.

species [K]

[bar]

[-]

C2H4 282.34 50.41 0.0862

C7F16 475.65 16.1 0.5429

Table 2: Simulation conditions for supercritical injection of ethylene.

Case

Injection conditions Chamber conditions

Condensation? Pressure

(bar)

Temperature

(K)

Pressure

(bar)

Temperature

(K)

1 55.18 309.2 1.37 298.0 No

2 55.42 289.0 1.37 298.0 Yes

3 55.64 283.9 1.37 298.0 Yes

4 54.94 281.5 1.37 298.0 Yes

Highlights

Highlights Page 1 of 1 Highlights

Highlights:

Regular and retrograde condensation is simulated using phase equilibrium model.

Supercritical nozzle flow of C2H4 with condensation is simulated.

Shock tube experiments with compression-induced condensation are simulated.

Theory-guided simulation is found to be consistent with experiments.

on regular and retrograde condensation in multiphase compressible flows

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