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Two-phase expansion is the process where a fluid undergoes a pressure drop through or in the liquid–vapor dome. Due to the volume increase in volumetric expanders, a pressure drop occurs in the fluid resulting in flashing phenomena occurring. These phenomena have been studied before in other processes such as two-phase flows or static flash. However, this has not been extensively studied in volumetric expanders and is mostly neglected.

two-phase expansion
volumetric expander

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Volumetric expanders are preferred for the low-temperature heat-to-power application as they have low rotational speeds with low flow rates. This matches well with the current requirements of ORCs. Furthermore, they can cope with relatively high-pressure ratios compared to single-stage turbo expanders and can also inherently handle liquid–vapor mixtures. Both of these traits are beneficial to the TLC and PEORC.

Each type of volumetric expander has some benefits and drawbacks, these are discussed by ^{[1]}^{[2]}. Currently, only two types are primarily considered for the use of two-phase expanders. These are the screw or Lysholm expander and the piston or reciprocating expander. The most promising type for two-phase expansion is the Lysolm expander. The main benefit of this type is that both phases can be considered to be mixed well. Giving acceptable results when thermodynamic equilibrium is assumed ^{[3]}^{[4]}^{[5]}. This implies that a simple homogeneous mixture model in thermodynamic equilibrium could be considered. Other types of screw expanders, such as the single screw, were deemed too mechanically complex ^{[6]}. However, even these are appearing as a possible alternative ^{[7]}^{[8]}.

The reciprocating expander is interesting for applications with higher pressure ratios and lower velocities. The main benefit of this expander type is the built-in volume ratio (BVR). The reciprocating expander is the type that has the largest BVR of all volumetric expander types. This trait is in accordance with the requirements for two-phase expanders which require large volume ratios to accompany the flashing process. A drawback of using this type of expander is the modeling of the expansion process. Assuming thermodynamic equilibrium does not give acceptable results with actual experimental results ^{[9]} as it did for Lysholm expanders. It requires a better understanding of the flashing process taking place during expansion.

In volumetric expanders, this sudden depressurization is a result of the increasing working volume. Due to the shape of a reciprocating expander, the flashing process could be compared to static flash experiments ^{[10]}. Static flash stands for the case where the fluid remains static in the horizontal direction during the flashing process ^{[11]}. These processes have been researched before. For example, as occurring in the desalination process ^{[12]} of drinking water production from seawater or as a process taking place in safety valves ^{[13]}. The phenomena and data found in literature about static flashing could ^{[14]} prove to also be useful in describing the phenomena taking place in volumetric expanders, and most notably reciprocating expanders.

One of the first analytical models for the two-phase Lysholm expanders is described by ^{[3]}. The authors mostly assume thermodynamic equilibrium and thus apply the HEM. The authors compared their analytical model with their performed experiments. Both methods showed an increase in internal efficiency with the rotor speed. Importantly, the authors also looked into a flashing delay. Included by adding a certain time delay between the start of the intake stroke and instantaneous flashing to equilibrium. The authors conclude that this can be omitted as the results for no delay time correspond just as well to their experiments. A similar methodology to ^{[3]} was applied by ^{[4]}. However, compared to a broader experimental dataset. The authors showed that the working fluid choice and correct rotor profiles are important in the design of TLC with a twin screw expander. Ref. ^{[15]} created a chamber model for a twin-screw expander within commercial software with the capability of integration in an entire TLC cycle. This methodology is specific to the machine as it requires geometrical data regarding cell volume evolution, suction and discharge ports as well as the multiple leakage paths in Lysholm expanders ^{[16]}. This model also uses thermodynamic equilibrium properties, and thus applies the HEM. The authors found a significant impact of the intake manifold expansion on the overall machine performance. Therefore, the authors designed a Lysholm expander with a variable built-in volume ratio ^{[17]}. The simulations show higher total power outputs for smaller BVR due to the higher mass flow rates, while the specific power decreases. Lower BVR also results in higher volumetric efficiencies but lower isentropic efficiencies due to under-expansion. In general, the simulation is capable to optimize the total power output in different operation conditions by varying the BVR. Lastly, Ref. ^{[5]} also assumed thermodynamic equilibrium in their initial simulation procedure. Additionally, they also assume that the work is performed by the vapor phase. Later, Ref. ^{[18]} modeled the chambers with flash vaporization based on an equation found by experimental spray flash evaporation. This model assumes a minimum superheat of 1K before evaporation occurs. The amount of evaporation is a fraction of the evaporation needed to achieve equilibrium which was experimentally determined. Thus a boiling delay model was used. The model with thermal disequilibrium predicted slightly lower internal power and isentropic efficiency, but only in the order of 3%. This is in line with the general finding that the equilibrium assumption can be used for Lysholm expanders.

After experimental determination of a temperature difference between the liquid and vapor phase, Ref. ^{[19]} constructed a five equation flashing model consisting of three energy continuity equations (of the liquid, vapor, and housing) combined with the mass continuity and the interfacial exchange model to predict the experimental data. Ref. ^{[20]} compared the model proposed by ^{[19]} with the HEM for different expander frequencies and initial superheats. The authors find discrepancies of up to 8% in the isentropic efficiency and expansion work between the two models. These are attributed to the disequilibrium losses by the authors. Ref. ^{[21]} proposed a design for a two-phase reciprocating expander consisting of a cyclone separator and the piston itself. The applied modeling for this design was also based on equilibrium assumptions. The authors found isentropic efficiencies in the range of 65 to 85% depending on the working fluid and operational regime. They also noted lower efficiencies for higher engine speeds. Ref. ^{[22]} used this design with the model of ^{[19]} to estimate the intake losses. For this design, the authors found a linear relation between the intake losses and the intake ratio, defined as the intake time to the expansion time. Ref. ^{[10]} also modeled a two-phase expander with a cyclone separator. The authors applied an evaporation model based on static flash pool evaporation and separated the phases in the cyclone and piston parts. Due to the better predicted performance at higher evaporation rates, the authors consider rotary expanders a better match for the two-phase expansion process.

Screw expanders have seen the most modeling of their use with two-phase expansion. Most of these assumed thermal equilibrium throughout expansion as these conform well with the available experimental data. The impact of the inlet port was found to have a non-negligible impact on the operation of the machine. More research about the design of the inlet manifold for two-phase expanders would thus be advised instead of basing the design on the machines compressor operation. When flash evaporation within the chamber was assumed, a discrepancy of only 3% was found with the equilibrium model. Reciprocating expanders on the other hand do require some modeling technique that takes into account the metastable conditions. Only one model of this type was available which takes the metastable condition in the working chamber into account via the mixture model. Other models split up the working volume in the expansion chamber and a phase separator cyclone. This process will have to be studied further, and described with more methods, to better understand the phenomena. These insights could make it possible to better design two-phase volumetric expanders.

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Heule, X.V.; Paepe, M.D.; Lecompte, S. Modeling of Two-Phase Volumetric Expanders. Encyclopedia. Available online: https://encyclopedia.pub/entry/25602 (accessed on 07 February 2023).

Heule XV, Paepe MD, Lecompte S. Modeling of Two-Phase Volumetric Expanders. Encyclopedia. Available at: https://encyclopedia.pub/entry/25602. Accessed February 07, 2023.

Heule, Xander Van, Michel De Paepe, Steven Lecompte. "Modeling of Two-Phase Volumetric Expanders," *Encyclopedia*, https://encyclopedia.pub/entry/25602 (accessed February 07, 2023).

Heule, X.V., Paepe, M.D., & Lecompte, S. (2022, July 28). Modeling of Two-Phase Volumetric Expanders. In *Encyclopedia*. https://encyclopedia.pub/entry/25602

Heule, Xander Van, et al. ''Modeling of Two-Phase Volumetric Expanders.'' *Encyclopedia*. Web. 28 July, 2022.

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