An Experimental Investigation of Supercritical Methane Injection Characteristics in a CO₂ Environment

Industries and governments are working assiduously to achieve the goal of clean and reliable high power generation. The supercritical CO₂ (sCO₂) power cycle is a new technology to accomplish that goal. The latest studies showed that sCO₂ turbines exhibit a higher power output [1], are cost-efficient [2], and require less space for operation than steam turbines. The Allam cycle (direct-fired sCO₂ cycle) works in the direction of capturing the CO₂ and attaining the high thermal efficiencies without losing the output power for compression while reaching pressure values beyond the critical pressure of CO₂ (73.7 bar). For instance, the Allam cycle generates higher efficiency than steam turbines with 95% CO₂ dilution by working at an optimum pressure of 300 bar. Recently, there have been numerous research and publications into various fundamental aspects of the Allam cycle and its components from our group [3–27].

Jet mixing [28] is a vital parameter that strongly affects the thermal efficiencies of cycles through its entrainment and thus mixing of fuel and oxidizer. Nevertheless, it has not been thoroughly investigated in high-pressure conditions. The fluid will be in a supercritical regime at high operating pressures, so jet mixing should be understood in supercritical conditions. Overall, as reported previously, a fluid changes its normal behavior under these supercritical conditions [29].

The literature explains the vital role of jet divergence in subcritical to supercritical conditions. The first experimental study on the injection of CO₂ into an N₂ environment at supercritical pressures was reported by Newman and Brzustowski [30]. They observed widening of jet profile and finely atomized spray at supercritical conditions. Chehroudi et al. presented their findings in Ref. [31] on round jet growth rates with an injection of N₂, O₂, and He into the chamber with several media including N₂, He, and mixtures of O₂ and N₂ at subcritical to supercritical conditions. Furthermore, they added that the fluid structure appeared like a classical liquid jet breakup at subcritical thermodynamic conditions. In contrast, atomization was inhibited above critical values due to the vanishing effect of heat vaporization. In the study [32], the authors injected the liquid oxygen in gas phase hydrogen at a maximum limiting pressure of 100 bar and temperature ranging from 100 K to 370 K. Their results explicitly mentioned that at supercritical conditions. The jet will not be regarded as spray formation instead it is fluid/fluid mixing.

A comprehensive experimental study by Oschwald et al. [33] elucidated drastic variation of jet visualization appearance using an injection of cryogenic N₂ into gaseous N₂ at subcritical to supercritical pressures. They observed that the injected fluid and surrounding media exhibited gas-gas mixing behavior at critical conditions. The reason was attributed to the disappearance of surface tension and vanishing of evaporation enthalpy. A similar experimental study conducted in Ref. [34] concluded domination of the diffusion process over atomization of droplets due to the absence of surface tension at supercritical conditions for jet disintegration. In contrast, atomization inundated the spray diffusion process at subcritical pressures. The relevant research conducted by experimental studies [35,36] substantiated previously mentioned jet behavior.

Most theories that address jet mixing and jet spreading identify the chamber to injectant density ratio as the main parameter. Abramovich et al.'s turbulent gas jet theory provided semi-empirical equations for linear growth or jet spreading angles corresponding to density variation. They implemented the mixing length concept to establish the correlations for turbulent submerged gas jets, where a moving fluid was injected into a stagnant fluid. Dimotakis in Ref. [38] investigated the disproportioned entrainment of the fluid amount in the mixing layer from each stream. In addition, based on this observation, he derived an equation for two-dimensional incompressible variable-density spatially growing mixing layers. Following Dimotakis' theory, scientists such as Brown and Roshko [39], Papamoschou and Roshko [40] proposed theoretical corelations for incompressible variable-density temporally growing gaseous mixing layers.

Furthering the discussion, Lamanna et al. [41] studied the disintegration regimes for injection of n-hexane jets into argon environment at subcritical to supercritical conditions. They concluded that the atomization process was a dominant disintegration phenomenon; the classical thermodynamic states of fluids account for the spreading rate. Their trends matched the spreading rates predicted by Reitz and Brocco's [42] equations. The experimental planar laser-induced fluorescence (PLIF) study by Roy [43] investigated the disintegration of supercritical jets. It stated that supercritical jets behaved as gas jets injected into gas surrounding, which agreed with the theoretical analysis by Abramovich's theory. A numerical approach was taken by Zong et al. [44] showed the inconsistencies in the numerical and experimental studies. The authors attributed the discrepancies to the shadowgraph technique used previously, as it saturated the images of low-density areas.

In addition, some studies [45–47] executed vital research on the injection of methane into the air, though hardly any literature explains the injection of methane into CO₂ media. Therefore, qualitative and quantitative approaches have been taken in this study that characterizes methane and CO₂ in the chamber filled with CO₂. Consequently, the authors executed experimental campaigns that include a pressure range of 10 bar to 200 bar for CO₂, 10 bar to 220 bar injection pressure for methane, and a temperature of 323 K and 353 K. Diagnostic techniques such as schlieren imaging and methane laser absorption spectroscopy were applied simultaneously to reveal the precise quantitative data for the spreading rate of jets at given conditions as a function of density variation.

Experiments were conducted in the facility mounted on the optical bench where simultaneous optical diagnostic setups such as the schlieren imaging technique for jet spreading visualization and absorption spectroscopy for methane detection are implemented. The arrangement of components such as different valves, devices, and instruments required for the experimental facility is depicted in Fig. 1.

The experiments were performed in two different constant volume chambers. Experiments conducted in the spherical chamber (more details about this facility in our prior work [48–52]) reached 140 bar, whereas, in the cylindrical chamber (components in Refs. [5,53]), experiments were performed at high pressure exceeding 200 bar. The pressure chambers are built with stainless steel and can operate at maximum pressures of 140 bar and 600 bar, respectively. Therm craft manufactured furnace (XSB-12-12-18-IC-Va P) was incorporated to ensure uniform heating of the spherical chamber. Similarly, for the cylindrical chamber, a custom-built heating jacket was employed. The K-type thermocouples, which measure the temperature of working fluids before entering the chamber within the accuracy of 0.7%, were attached to the surfaces of the chamber and manifold. As it is critical to build and perform experiments directly at high pressures, a series of low-pressure experiments were conducted known as the “ramp-up phase” to get the hands-on experience to operate the valves and read the instrument values and experimental procedure in the spherical chamber with a maximum pressure of 140 bar. Succeeding the ramp-up experiments, the authors moved on to the next high-pressure cylindrical chamber. The nitrogen drove gas booster/pump (DLE 30-1-NN-C-FEC-M) is manufactured by Maximator Technologies Inc. (Fairview, PA) has been employed to achieve the required high pressure of CO₂ and methane in the chamber and manifold, respectively. The spherical chamber has two optical accesses with 2-in. sapphire windows. Furthermore, the cylindrical chamber facility includes two optical accesses with 3-in. diameter sapphire windows.

The manifold built with stainless steel tube was installed to inject the working fluids CO₂ and methane into the chamber through separate injectors. The CO₂ and methane were injected from the tubing arrangement in the manifold at required thermophysical conditions. Two pressure transducers were mounted for the injector and the chamber pressure. The manifold system utilized a digital omega pressure transducer (DPG80001-5K, 0.1–5,000 PSI) with 0.25% of full-scale terminal point accuracy. The volume of the nominal manifold was increased by converting the straight tube into coiled tubing and implemented before the solenoid valve. An electronically actuated solenoid valve was mounted on the manifold, which remotely controlled the methane injection flow.

For the experimental campaign, the injector dimensions were selected to maintain the fully developed flow at the output of the injector tip. The spherical chamber has an injector tube with a diameter and length of 0.0275 in. and 23.62 in., respectively. Similarly, the injector tube has a diameter and length of 0.03 in. and 4.14 in. for the cylindrical chamber, respectively. Schlieren imaging and methane laser absorption optical diagnostic techniques are depicted in Fig. 1. Schlieren imaging develops the flow interaction visualization and extracts the density gradient data. This technique was accomplished using a LED light source. The diffused light from LED was focused and collimated through a concave mirror of a focal length of 500 mm. Mirrors were arranged to direct these collimated light rays to a high-pressure chamber's high-speed camera (SA-Z Photron). The high-speed camera has a high spatial resolution and respective frame rates. The range 512 × 512 resolution with a frame rate of 5000 Hz and 10,000 Hz was selected to improve visualization jets. A laser absorption optical technique was implemented to detect the start of methane flow at the output of the tip of the injector. As methane molecules have an absorption spectrum at 3.39 μm wavelength He-Ne laser source (REO, Model No. 32172), an infrared laser source at the required wavelength was directed through the iris and reflected by mirrors. In addition, a beam splitter was mounted to split the laser line and send the signal to the reference detector and others through the chamber and transmitted detector (VIGO Mfg., Model No. PVI-2TE-5/MP-DC-5M-F-M8). The reference detector measures the intensity of the reference range signal; hence, the leading sensor measures the intensity of the transmitted sample beam. The laser signal was collected through ni-daq and labview code.

The experiments were executed by implementing schlieren imaging and laser absorption optical diagnostic techniques concurrently. The chamber was filled with CO₂ at the required pressure and temperature. The digital pressure transducer was mounted near the chamber pressure. The settling down period of CO₂ in the chamber varies by temperature and pressure conditions and ranges from 10 min to 1 h. The settled CO₂ was identified through the schlieren image. The valve near the chamber was closed, and the entire manifold was evacuated and vacuumed to ensure no CO₂ in any manifold. Methane was introduced through an injector, and required inlet pressure and temperature conditions are measured with the pressure transducer and thermocouples mounted on the manifold. The methane was injected into a chamber filled with CO₂ by remotely actuating the solenoid valve. The opening time for the solenoid valve ranged from 50 ms to 500 ms, depending on the experimental conditions. The chamber was evacuated to the atmosphere through the exhaust land line also vacuumed until chamber pressure showed 0.01 psi.

Before injecting the methane, the high-speed camera, the solenoid valve, the LED light source, the 3.39 μm laser, and detectors were turned on. The camera was employed simultaneously to capture the visual images and the methane transmitted signal targeted by the transmitted detector. Experimental data were collected through labview code after every experiment was plotted to extract the results.

An in-house python code was implemented to analyze the jet spreading angle results. The steps involved in this method are represented in Figs. 2 and 3 and explained below.

Figure 2 shows the variation of transmitted laser signal intensity from the given elements such as methane and CO₂ in volts, along with the time. When CO₂ is filled in the chamber, some quantity remains inside for a length and diameter of 0.2 and 0.03 inches of the injector. Hence, when the solenoid valve is actuated, the CO₂ inside the injector will initially be pushed into the chamber, then methane will be injected into the chamber. Therefore, a straight horizontal (0.1 s–0.14 s) plot indicates that CO₂ at the tip of the injector is being pushed into a high-pressure chamber. Subsequently, the straight line is followed by a declining fluctuating curve that implies methane is injected inside the chamber. The plot helps choose the time window for estimating the time of the methane coming out of the injector inside the chamber. The time window for jet cone angle calculation was considered when the curve started to decline, and the jet hit the opposite wall of the chamber. Schlieren image video helped to observe the time of the methane jet hitting the wall.

Figure 3 shows the step-by-step procedure followed to estimate the jet cone angle. First, the footage captured was applied to convert the video into a series of images. In the second step, the region of interest (ROI) was selected, reducing the time for processing. In the subsequent step, the background was removed for the desired ROI. The surface of the jet was whited out to observe only the jet interaction. Due to the jet surface whitening out process, edge detection can easily detect the edge of the jet, shown by the dots. Linear fit with the dot is conducted to calculate the jet spreading angle.

The jet spreading angle is a vital parameter that quantifies jet development and mixing. Comprehensive and intense research on jet analysis has been a course of interest for many years. Literature studies [33,54] provided the knowledge of jet growth variation with density ratio. However, other factors such as pressure ratios and reduced pressures could significantly affect jet spreading angles. Hence, this study discusses the comprehensive effect of these parameters on jet divergence angles, also called jet spreading angles of methane injection into the CO₂ environment at subcritical to supercritical thermodynamic conditions.

Abramovich's theory introduced the semi-empirical model for jet divergence variation in submerged turbulent gas jets for density ratios of less than one. In this study, the flowing methane is injected into a stagnant CO₂ filled in the chamber. Hence, it is considered as a submerged turbulent gas jet. Both methane and CO₂ are in the gas phase in a subcritical regime. Furthermore, in the supercritical regime, each fluid appears like a gas as the diffusion process dominates the evaporation process but has a density like liquid. However, it has been observed from Fig. 4 that the experimental results do not agree with the curve estimated through Abramovich's model at higher density ratios.

At both temperatures, the jet spreading angle is increased nonlinearly. The tan(angle) grows from 0.1 to 0.5 quantitatively for a density ratio in the range of 1 to 4. From the previous study [55], it has been observed that at near supercritical thermodynamic conditions, the density of CO₂ changes drastically, whereas methane grows very slowly. Precisely, the higher density ratio cases in this literature correspond to the supercritical stage of both the injectant and chamber fluid. Hence at supercritical conditions, jet growth and mixing improve, and thus, the jet entrains more fluid along its path.

Figure 5 shows the variation of jet spreading angle with density ratios for injection of CO₂ into CO₂ filled in the chamber at temperatures of 323 K and 353 K and at subcritical to supercritical pressures. Theoretically, the density ratio of injection of CO₂ into CO₂ should be around one at a constant temperature. As mentioned previously, the CO₂ at the edge of the injector has been pushed into the chamber by pressurized methane in the manifold. Therefore, the density of chamber CO₂ differs slightly from injectant CO₂. Fluid density changes somewhat at a constant temperature, mainly for the experimental conditions. The pressure ratio of injectant fluid to the chamber fluid pressure is 1.5; nevertheless, the density ratio is around one. The jet growth decreases along with density ratios observed in each figure at both temperatures. The reason is attributed to the almost similar density due to the same fluid element. As the fluids are the same, there will be a hindrance to the penetration of the CO₂ jet into CO₂ jet. The curves plotted for empirical equations in Fig. 5 indicate that the jet divergence increases with the density ratio. But none of the given theories are applicable for predicting jet divergence angles for a similar chamber and injectant fluid. Therefore, the jet grows and mixes well in methane injection into CO₂, whereas, for the case of similar fluid injection, the jet penetration was restricted, leading to less jet development and mixing.

Reduced pressure is specifically defined as the ratio of chamber pressure to the critical pressure of the injectant fluid. It signifies the relative subcritical or supercritical stage of injectant fluid. In addition, it also indicates the structural changes in the behavior of jets. Specifically, for methane, the critical thermodynamic condition occurs at a pressure of 45.9 bar and temperature of 190.4 K, and for CO_2, it is at 73.7 bar and 304.1 K, respectively. The jet spreading angles were obtained for a range of reduced pressure from 0.2 to 5 for methane injection and injection of CO₂ separately in a chamber filled with CO₂ in a cylindrical chamber. An increase in the reduced pressures from 0.2 to 5 indicates that chamber pressure increases.

Figure 6 represents the jet growth variation with reduced pressures for a wide pressure range of 10 bar to 200 bar and temperatures of 323 K and 353 K for methane injected into the CO₂ environment. For all the cases, the pressure ratio, injectant to chamber pressure, was kept constant at 1.5. The reduced pressure below one indicates where subcritical methane is injected into subcritical CO₂. The jet spreading angle growth is almost constant for this case, as observed in Fig. 6. In a subcritical regime, gas-gas mixing is regarded as both fluids are in their gaseous phase. The jet disappears due to the diffusion process rather than vaporization [43,56].

The methane reaches its supercritical stage for a reduced pressure range between 1 and 1.6, whereas CO₂ remains in a subcritical phase. Therefore, the supercritical methane is injected into the subcritical CO₂. As the methane density increases substantially in the supercritical stage, it mixes well with subcritical CO₂. The jet development improvement is observed from an increase in jet spreading angles in Fig. 6. Compared to subcritical cases, the jet spreading angles increase as methane pressure increases when methane turns supercritical; similar observations were found in a study by Roy et al. [56].

For reduced pressures above 1.6, the chamber pressure becomes critical, and hence, the author observed the case of supercritical methane injection into supercritical CO₂. In a supercritical regime, CO₂ density increases tremendously compared to supercritical methane. From Fig. 6, the jet spreading angle increases with reduced pressures after 1.6. In supercritical regimes, the CO₂ structure appears like gas but has density as the liquid. Furthermore, the absence of latent heat indicates that the evaporation process is not valid. Therefore, jet fluid disappears due to the diffusion process. The features mentioned above for fluids in a supercritical regime are also mentioned in a study under cryogenic conditions [44]. At higher injectant and chamber pressures, the jet spreading angle increases further. From Fig. 6, the higher reduced pressures from 4 to 5 show higher chamber pressures of around 200 bar. As the jet spreading angle increases at high pressures, the jet encompasses more fluid in the chamber.

Figure 7 explains the behavior of jet spreading angles for a range of reduced pressure from 0.2 to 2 at different pressure ratios for CO₂ injection into CO₂ filled in the chamber. The dataset encompasses the pressure range of 10 bar to 150 bar and at temperatures of 323 K and 353 K. It is observed that, at a specific pressure ratio, the jet spreading angle increases as reduced pressure increases. For instance, at Pr of 1.5, the jet spreading increases from 0.1 to 0.3 for a reduced pressure range of 0.2 to 1. Above 0.7 of reduced pressures indicates the injection of subcritical CO₂ into supercritical CO₂. Hence, sudden jump in jet spreading angle was observed after 0.7 of reduced pressure. Considering the case at P_r of 1.2 at both the temperatures, the jet divergence is more at 353 K than divergence at 323 K. Hence, fluid mixing will be smoother at higher temperatures.

As mentioned previously, the jet spreading angle increases at higher chamber pressures, and the dataset at a temperature of 323 K and P_r of 1.3 same is observed. These data points are at a significantly higher pressure than critical thermodynamic values of CO₂. In addition, such cases show the behavior of supercritical fluid injection into the supercritical fluid environment. It interprets smooth injection and jet mixing processes at high operating pressures.

The pressure ratio is the ratio of the chamber to injectant pressures; it represents the relative magnitude of injection pressures compared to chamber pressures. For analysis of jet divergence with varied P_r, the case of methane injection into CO₂ was considered at temperatures of 323 K and 353 K. Experiments at a given temperature and pressure less than 50 bar were conducted in a spherical chamber. At the same time, other experiments were conducted in a cylindrical chamber. The following part explains the effects of pressure ratio on jet spreading angles.

Figure 8 presents the effect of pressure ratio on jet spreading angles at different chamber pressures of 20 bar, 50 bar, and 80 bar at a temperature of 323 K for methane injected into CO₂. The range of P_r varied from 1.4 to 2. At 20 bars, the subcritical methane was injected into subcritical CO₂. An increase in P_r interprets an increase in injection pressure; hence, as the injection pressure increases, the jet spreading angle increases at constant chamber pressure. At chamber pressure of 50 bars, the CO₂ is in a subcritical regime, whereas injected methane is in a supercritical phase. Hence, such a case is an instance of supercritical to subcritical injection. Again, there is an increase in jet spreading angle for a rise in P_r. At constant chamber pressure and temperature, there will be an elevation in the injection pressure of methane that raises its density. Hence jet mixing becomes effortless.

In the case of chamber pressure of 80 bars and temperature of 323 K in Fig. 8, a reverse trend was observed, where a reduction in the jet divergence angles was discerned with an increase in P_r. The chamber pressure of 80 bar suggests a supercritical environment and need higher pressure for injectant fluid, so ultimately, both the fluid will be supercritical regime. Unlike methane densities, CO₂ densities vary drastically in the supercritical regime. Hence even though the methane injection pressure rises, its density is not enough to penetrate through CO₂ in the chamber.

Figure 9 explicitly shows the jet spreading with a pressure ratio at a temperature of 353 K for methane injected into the CO₂ environment. Similar trends at a temperature of 323 K were observed. At subcritical chamber pressures, the subcritical injection pressures of methane penetrate and mix well with chamber gas. The P_r of 1.5 and chamber pressure of 30 bar show the injected methane is in the supercritical regime. Therefore, injected methane jet spreading angles increase with P_r. However, at supercritical CO₂ pressures of the chamber, the injected CO₂ will also be in the supercritical regime, and hence the jet divergence decreases as P_r increases.

When the methane in the injector comes out into the chamber filled with CO₂, density gradients are observed due to the density difference between fluids. The visualization method is adopted to study the jet interaction with chamber fluid. Figure 10 represents jet interface visualization captured through a schlieren imaging setup for three jet and chamber pressure combinations in a cylindrical chamber. The combinations include (i) injection of subcritical methane into subcritical CO₂, (ii) injection of supercritical methane into subcritical CO_2, and (iii) injection of supercritical methane into supercritical CO₂. Figure 10(a) shows the qualitative image for subcritical methane injection into the subcritical CO₂ environment at P_r of 1.45 and a chamber pressure of 25 bars. The subcritical phase occurs below the critical values of both fluids, so jet and chamber fluid appear as gas and have densities like gas. The mixing occurs as soon as jet fluid comes out of the injector. As there is gas-gas mixing, no evidence of droplet formation appears. The jet disappears classically due to the diffusion process. The term “jet disappears” indicates that the methane jet is well mixed with CO₂ in the chamber.

Figure 10(b) represents a case of injection of supercritical methane into a subcritical CO₂ environment at a chamber pressure of 50 bars, P_r of 1.5, and a density ratio of 2.15. Although methane is in the supercritical state, its density change is minimal than those variations for CO₂ in its subcritical phase. Still, methane structure appears like gas and liquid; it disappears without droplet formation phenomena. There are no ligament formations that are observed in a liquid jet breakup. Hence, diffusion process dominates over the evaporation process, and the mixing is considered gas-gas mixing.

Figure 10(c) indicates jet flow visualization where both fluids are at supercritical conditions. Above critical values, the CO₂ density enhances tremendously. Hence, the density ratio is 3.15. At these conditions, the enthalpy of evaporation vanishes, and gas–gas mixing is governed by a diffusion process. These features indicated that the mixing exhibits gas–gas behavior at and above critical pressure values. The visualization image confirms the elements, as there are no droplet and ligament formations. Unlike other studies [8,18] mentioned in the introduction, the present literature studies low-density fluid injection into high-density fluids. Nevertheless, methane penetrates through high-density supercritical CO₂, as its structure appears as gas. The study [57] substantiates the visualized jet mixing characterization observed with acquired images.

This study investigated methane and CO₂ jets into a chamber filled with CO₂ at different subcritical to supercritical pressure and temperature conditions in the well-designed constant volume spherical and cylindrical rigs. Schlieren imaging and methane absorption spectroscopy were applied to determine jet divergence angles. Unlike previous studies, this work observed jet mixing interaction to inject low dense methane into high-density CO₂. Furthermore, the effects of density ratios reduced pressure and pressure ratio on jet spreading angles were analyzed.

Density ratios enhance the jet mixing interaction by increasing the jet divergence angle. In addition, due to the vanishing effect of the vaporization process and little surface tension, it behaves like a gas and has a density like liquid. Hence, methane and CO₂ exhibit gas-gas mixing. For injection of CO₂ into CO₂, though at constant temperature density for the same fluid, it differs due to pressure difference. Moreover, a significant reduction in jet divergence angle was observed for a minor change in density ratios. Therefore, the jet injection process negatively affects it as it goes into a higher-density region.

Investigated reduced pressure is a vital parameter that relates the critical pressure of the injectant fluid to chamber pressure. For all the cases, jet divergence angles increased as reduced pressure increased. Higher pressure enhances the capabilities of jet mixing with chamber fluid.

In the case of injection of subcritical methane into subcritical CO₂ and supercritical methane into subcritical CO₂, as the pressure ratio increases, the jet divergence angle increases. Whereas, when both fluids transformed into the supercritical phase, it was not easy for supercritical methane to penetrate supercritical CO₂. Therefore, the jet spreading angle decreased. Similar conclusions were applicable for the injection of CO₂ into CO₂ at subcritical to supercritical pressure conditions.

The jet visualization method was adopted to see the jet mixing with the surrounding fluid. Methane and CO₂ are in their gas phase at subcritical conditions and have low densities. At the same time, no traces of droplet formation or ligament formation occurred under supercritical conditions. As there is no vaporization process, the emission of jet fluid takes place solely due to the diffusion process. Therefore, at supercritical conditions, both fluid exhibits a gas-gas mixing phenomenon.

This study provides the first insight into the supercritical mixing effects of methane in a CO₂ environment. The results will have applications in advanced power generation and propulsion engines where such environments are encountered. Specifically, the Allam cycle development and deployment employing direct-fired CO₂ combustion will benefit from this work.

• UCF and the Office of Naval Research (ONR) (Award No. N00014-18-1-2362; Funder ID: 10.13039/100000006).

• Department of Energy (DOE) (Award No. DE-FE0025260; Funder ID: 10.13039/100000015).

P_r =

pressure ratio

P_C =

chamber pressure

sCO₂ =

supercritical CO₂

P_M =

manifold pressure

θ =

jet divergence angle