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A two-phase approach for simulation of water-flooded twin-screw machines validated for expander applications

A two-phase approach for simulation of water-flooded twin-screw machines validated for expander... International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 A two-phase approach for simulation of water-flooded twin-screw machines validated for expander applications A Nikolov, A Brümmer TU Dortmund University, Faculty of Mechanical Engineering, Chair of Fluidics, Leonhard-Euler-Str. 5, 44227 Dortmund, Germany E-mail: [email protected], [email protected] Abstract. In the lower and medium power range, twin-screw machines offer a high potential for energy conversion in various compressor applications with liquid injection or as expanders with respect to electrical power generation from regenerative and exhaust heat sources in trilateral or wet Rankine cycle systems, for instance. Aiming high efficiencies and reliability, the design of liquid-flooded twin-screw machines as a critical system component presents particular challenges for the engineers. Hence, reliably representative simulations to guide design are mandatory. In this context, this study presents a two-phase approach for simulation of the operational behaviour of water-flooded twin-screw machines. The thermodynamic fluid state is calculated using the humid air model that considers the two-phase mixture in thermal equilibrium. Additionally, dissipative two-phase mass flow rates are predicted regarding a slip- flow model and two-phase discharge coefficients. The proposed two-phase approach including liquid distribution in the working chamber is validated for expander applications considering available experimental data of the test twin-screw expander SE 51.2 in terms of indicator diagrams, indicated power, mass flow rate, and outlet temperature. 1. Introduction Liquid-flooded twin-screw machines are widely applied as compressors providing benefits in terms of two-phase operation, such as high pressure ratios in a single stage together with low thermal stress at the same time, clearance sealing effects or lubrication of the moving machine parts. While water- flooded process-gas compressors have long been in operation, water-injection in air or hydrogen twin- screw compressors gain more importance in recent years. Recovering exhaust heat from low-grade heat sources, for example, in trilateral or wet organic Rankine cycle systems, twin-screw expanders capable of dealing with large amounts of liquid can be applied as an alternative to turbines avoiding wet expansion. Moreover, injecting an auxiliary liquid during expansion could provide benefits in terms of utilisation of available pressure potentials, such as in (natural) gas reducing stations, where only a small temperature drop during expansion might be allowed. In this context, injected liquid, e.g. water, in combination with an exhaust or other regenerative heat source acts also as heat carrier. In order to theoretically determine the operational behaviour of water-flooded twin-screw machines with respect to system design, appropriate two-phase models, e.g. for chamber model simulations, considering the thermodynamic fluid properties of the air-water mixture and adequately reflecting the mass flow rates are required. With regard to the thermodynamic modelling of water injection into the working chamber of twin-screw compressors, a study presented in [1] considers a water evaporation model in order to determine the influence of very low amount of liquid injection on the operation of a twin-screw compressor. In terms of two-phase flows, only a few contributions in the literature deal with non-homogeneous two-phase mass flow rate models in rotary displacement machines. Generally, Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 homogeneous approaches are applied as in [2]. In the same manner, in [3] and [4], Sangfors simulates the operation of liquid-flooded twin-screw machines based on homogeneous two-phase clearance flow considerations. In [5], Vasuthevan and Brümmer present thermodynamic modelling of a twin-screw expander in a trilateral flash cycle regarding homogeneous two-phase inlet, outlet, and clearance flows. In contrast, a separated flow model with liquid entrainment as defined by Smith in [6] is applied by Bell in [7] and by Lemort in [8] to determine inlet and outlet flows in scroll expanders. Moreover in [7], Bell implements a constant two-phase discharge coefficient of 0.77 in order to predict the flow losses. In [9], Huagen et al. present a two-phase clearance flow model based on the slip model with liquid entrainment according to [6] and use a discharge flow model as proposed by Lin in [10] to calculate indicator diagrams of an oil-injected twin-screw compressor. In [11], a comprehensive flow path modelling in a water-lubricated air twin-screw compressor is conducted applying different two-phase models to each clearance and opening. The following study deals with an approach for simulations of water-flooded twin-screw machines. In this context, a two-phase model according to humid air [12] proposed for the calculation of the thermodynamic properties of the working fluid is presented. Moreover for calculation of pressure- driven two-phase mass flow rates according to [13], assumptions as with the modelling of the discharge behaviour of clearances and openings as well as liquid distribution in the working chamber during expander operation are made. To verify the presented two-phase approach, multi-chamber simulations of the water-flooded twin- screw expander prototype SE 51.2 [14–16] are performed by means of the simulation tool KaSim [17]. The simulation results are compared to experimental data according to [15, 16] by means of expander (two-phase) mass flow rate, internal (simulated) and indicated (measured) power, pressure depending on the working chamber volume, as well as outlet temperature. 2. Thermodynamic properties of air-water two-phase mixtures The following section addresses the fundamentals for the calculation of the thermodynamic state of the working fluid in water-flooded twin-screw machines. In this context, the humid air concept [12] is proposed considering a mixture of ideal fluids in thermodynamic (thermal) equilibrium. The two-phase mixture consists of the incompressible fluid water in liquid or solid state of matter, its corresponding gaseous phase steam, and dry air. The thermodynamic fluid properties with respect to the humid air model are listed in table A1 (Appendix). Density of the incompressible phase—water or ice—is considered constant. Moreover, within the relevant pressure and temperature range of expander operation, the specific heat capacity of each fluid is assumed constant, too. The composition of a water-air mixture is specified by water load 𝑋 —as commonly used in the context of humid air—representing the ratio of water mass 𝑚 to dry-air mass 𝑚 : 𝑤 𝑑𝑎 𝑋 ∶= . (1) 𝑑𝑎 Since in this work a steady-state process is considered, dry air mass 𝑚 and water mass 𝑚 in 𝑑𝑎 𝑤 equation (1) can be substituted by the corresponding mass flow rates 𝑚 ̇ and 𝑚 ̇ with regard to two- 𝑑𝑎 𝑤 phase flows. Hence, water load 𝑋 can be expressed as follows: 𝑚 ̇ 𝑋 ∶= . (2) 𝑚 ̇ 𝑑𝑎 By definition, water mass 𝑚 and mass flow rate 𝑚 ̇ consider the entire amount of water molecules in 𝑤 𝑤 liquid, solid, and gaseous state of matter in the two-phase mixture. As schematically illustrated in figure 1, the volume 𝑉 of the air-water mixture is composed of the sub-volumes of dry air 𝑉 , water 𝑑𝑎 steam 𝑉 , and the incompressible phase of water 𝑉 (or ice for temperatures below the triple-point 𝑤𝑠 𝑙𝑖𝑞 temperature of water 𝑇 = 273.16 K): 𝑡𝑟 2 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑉 = 𝑉 + 𝑉 = 𝑉 + 𝑉 + 𝑉 . (3) 𝑔 𝑙𝑖𝑞 𝑑𝑎 𝑤𝑠 𝑙𝑖𝑞 A phase boundary between the volumes of the incompressible and the gaseous phase is present, whereas dry air and steam share the same volume. Considering the gaseous phase in the humid air mixture consisting of dry air and water steam, the overall static pressure 𝑝 of the mixture equals the sum of the partial pressures 𝑝 of the single gases in the mixture as expressed by Dalton’s law [12]: 𝑝 = ∑ 𝑝 . (4) 𝑗 =1 Partial pressure 𝑝 of any gaseous component is related to the gaseous volume 𝑉 . Taking the example 𝑗 𝑔 of humid air, the sum of the partial pressure of dry air 𝑝 and water steam 𝑝 provides the overall 𝑑𝑎 𝑤𝑠 static pressure 𝑝 . At the same time, the overall static pressure 𝑝 imposed on the surface of the incompressible phase represents the pressure of the two-phase mixture. As for ideal gases, the partial pressure of each gaseous fluid can be expressed as follows: 𝑚 ∙ 𝑅 ∙ 𝑇 𝑗 𝑗 𝑝 ∶= . (5) A mass-specific gas constant 𝑅 is assigned to each gaseous fluid in the mixture. The maximum mass of water steam 𝑚 in the mixture is related to saturated humid air and can be calculated from the 𝑤𝑠 ,𝑡𝑠𝑎 ideal gas law: 𝑝 ∙ 𝑉 𝑤𝑠 ,𝑡𝑠𝑎 𝑔 𝑚 = . (6) 𝑤𝑠 ,𝑡𝑠𝑎 𝑅 ∙ 𝑇 𝑤𝑠 Here, the saturation pressure of water steam 𝑝 is a function of the temperature and can be 𝑤𝑠 ,𝑡𝑠𝑎 calculated, for instance, by means of the Antoine equation [12] or as proposed in [18]. Additionally, the saturation pressure 𝑝 of water steam depends on the overall static pressure 𝑝 , if solubility of steam 𝑤𝑠 ,𝑡𝑠𝑎 molecules in the liquid phase are taken into account [18]. In the following considerations, this effect is neglected. Figure 1. Volume 𝑉 of the two-phase mixture containing the sub-volumes of dry air 𝑉 , water steam 𝑉 , and liquid water 𝑉 . 𝑑𝑎 𝑤𝑠 𝑙𝑖𝑞 ∆ International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 If less water mass 𝑚 than the maximum mass of water steam 𝑚 at given temperature 𝑤 ,𝑠𝑎𝑡 is available in the humid air mixture, partial pressure 𝑝 is lower than the saturation pressure 𝑝 of water steam. This case can be quantified by means of relative humidity 𝜑 dividing ,𝑠𝑎𝑡 the partial pressure 𝑝 by the saturation pressure of water steam 𝑝 : ,𝑠𝑎𝑡 𝑤𝑠 𝜑 ∶= . (7) 𝑤𝑠 ,𝑡𝑠𝑎 For a relative humidity in the range of 0 < 𝜑 ≤ 1, all water molecules in the mixture exist as steam. For a relative humidity of 100 % (𝜑 = 1), saturation water load 𝑋 is calculated as follows: 𝑡𝑠𝑎 𝑚 𝑅 𝑝 𝑤𝑠 ,𝑡𝑠𝑎 𝑑𝑎 𝑤𝑠 ,𝑡𝑠𝑎 𝑋 = = ∙ . (8) 𝑡𝑠𝑎 𝑚 𝑅 𝑝 − 𝑝 𝑑𝑎 𝑤𝑠 𝑤 𝑠 ,𝑡𝑠𝑎 By analogy with water load 𝑋 , the composition of the two-phase water-air mixture can be expressed by means of mass dryness fraction 𝑥 relating the mass 𝑚 of the gaseous humid air phase consisting of dry air and water steam to the mass 𝑚 of the mixture as follows: 𝑚 + 𝑚 𝑑𝑎 𝑤𝑠 𝑥 ∶= = . (9) 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 As with two-phase mass flow rate calculations discussed later in section 3.3, mass dryness fraction 𝑥 is referred to the ratio of the gaseous mass flow rate 𝑚 ̇ to the mass flow rate 𝑚 ̇ representing the sum of the single mixture components dry air and water: 𝑚 ̇ 𝑚 ̇ + 𝑚 ̇ 𝑔 𝑑𝑎 𝑤𝑠 (10) 𝑥 ∶= = . 𝑚 ̇ 𝑚 ̇ + 𝑚 ̇ 𝑑𝑎 𝑤 From equation (1), equation (8), and equation (9), mass dryness fraction can be expressed as follows: 1 + 𝑋 𝑡𝑠𝑎 (11) 𝑥 = . 1 + 𝑋 In contrast to the conventional consideration of the triple point of water as a zero energy level, the energy conservation equations regarding (specific) enthalpy and internal energy of humid air are derived with respect to the absolute zero from thermodynamic point of view (0 K). This approach provides throughout positive values for the extensive state variables internal energy and enthalpy as required by the simulation environment in this study. In contrast to internal energy and enthalpy, entropy of the mixture is related to the triple-point temperature of water, since entropy supports the calculations of isentropic change in state as an auxiliary state variable allowing negative values. For better understanding of the two-phase approach using the absolute zero temperature of 0 K as a zero energy level, specific aspects considering change in aggregate state of water are explained. In figure 2, the correlation between temperature and pressure as well as the different states of matter— steam, liquid, and ice—in terms of specific enthalpy are schematically illustrated. This can be transferred to specific internal energy too. In the region of two aggregate states—liquid water and steam (𝑇 > 𝑇 ) or ice and steam (𝑇 < 𝑇 )— 𝑡𝑟 𝑡𝑟 pressure is a function of temperature. The boundaries of the two-phase region correspond to a saturated state of the gaseous phase. With regard to the triple point, all three aggregate states are present. Here, 𝑡𝑟 𝑡𝑟 𝑡𝑟 the difference of the specific enthalpies (∆ℎ , ∆ℎ , and ∆ℎ ), in the literature also referred to as 𝑠𝑢 𝑒𝑣 𝑚𝑒 𝑤𝑠 𝑤𝑠 𝑤𝑠 𝑤𝑠 𝑤𝑠 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 latent heats, of two saturated aggregate states define the corresponding specific enthalpy required for a complete change in aggregate state. Outside the two-phase region, only one fluid state of matter exists. The reference condition providing enthalpies for change in state of matter is randomly set in the triple point of water. In general, these energies could be determined at any valid temperature for the transition between two aggregate states. overheated steam saturated steam liquid + steam ice + steam ice liquid temperature [K], pressure [Pa] Figure 2. Specific enthalpy ℎ of water as a function of pressure 𝑝 and temperature 𝑇 . As illustrated in figure 2, specific enthalpy can be calculated by addition of different specific enthalpy levels beginning from the absolute zero temperature. For instance, specific enthalpy of liquid 𝑡𝑟 water ℎ equals the sum of specific enthalpy ℎ of ice, specific melting enthalpy ∆ℎ , and the 𝑙𝑖𝑞 ,1 𝑖𝑐𝑒 ,𝑡𝑟 difference of specific enthalpy of water at temperature 𝑇 and triple-point temperature 𝑇 . Specific 1 𝑡𝑟 𝑡𝑟 melting enthalpy ∆ℎ represents the energy released during freezing of liquid water or required to bring water from solid to liquid aggregate state in the triple point. In the same manner with reference to the thermodynamic absolute zero point, specific enthalpy ℎ of steam can be calculated considering 𝑤𝑠 ,1 the specific enthalpy of ice ℎ at triple-point temperature 𝑇 , the specific sublimation enthalpy 𝑖𝑐𝑒 ,𝑡𝑟 𝑡𝑟 𝑡𝑟 ∆ℎ , and the difference of specific enthalpy ℎ and ℎ of steam at temperature 𝑇 and triple- 𝑤𝑠 ,1 𝑤𝑠 ,𝑡𝑟 1 point temperature 𝑇 . 𝑡𝑟 In the following, the calculation of specific volume, internal energy, enthalpy, and entropy of the two-phase air-water mixture are presented. All fluid specific parameters and enthalpies in terms of change in aggregate state of water can be obtained from table A1 (Appendix). 2.1. Specific volume Specific volume 𝑣 of the two-phase mixture is defined as the ratio of the volume 𝑉 occupied by the 1+𝑥 two-phase mixture divided by the mass of dry air 𝑚 : 𝑑𝑎 -1 specific enthalpy [J∙kg ] 𝑠𝑢 𝑚𝑒 𝑚𝑒 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑣 ∶= . (12) 1+𝑥 𝑑𝑎 The specific volume 𝑣 referred to the entire mass 𝑚 of the two-phase mixture equals: 𝑉 𝑉 −1 𝑣 ∶= = = 𝑣 ∙ (1 + 𝑋 ) . (13) 1+𝑥 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 As illustrated in figure 1, the entire fluid volume 𝑉 as in equation (3) consists of the partial volumes of the incompressible phase of water as well as the gaseous phase including dry air and steam. Combining equation (3) and equation (12), for saturated humid air (𝑋 > 𝑋 ) and temperatures above the triple 𝑡𝑠𝑎 point of water (𝑇 > 𝑇 ), specific volume 𝑣 of the two-phase fluid equals on a pro rata basis the sum 𝑡𝑟 1+𝑥 of the single specific volumes of each mixture component: 𝑇 𝑇 ( ) ( ) 𝑣 = 𝑣 + 𝑋 ∙ 𝑣 + 𝑋 − 𝑋 ∙ 𝑣 = 𝑅 ∙ + 𝑋 ∙ 𝑅 ∙ + 𝑋 − 𝑋 ∙ 𝑣 . (14) 1+𝑥 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑡𝑠𝑎 𝑙𝑖𝑞 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑡𝑠𝑎 𝑙𝑖𝑞 𝑝 𝑝 Here for the incompressible phase, specific volume 𝑣 of liquid water is considered. In contrast, at 𝑙𝑖𝑞 temperatures below the triple point (𝑇 < 𝑇 ), specific volume 𝑣 of the two-phase fluid is regarded 𝑡𝑟 1+𝑥 to ice specific volume 𝑣 : 𝑖𝑐𝑒 𝑇 𝑇 ( ) 𝑣 = 𝑅 ∙ + 𝑋 ∙ 𝑅 ∙ + 𝑋 − 𝑋 ∙ 𝑣 . (15) 1+𝑥 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑝 𝑝 For single-phase humid air (𝑋 ≤ 𝑋 ), no incompressible phase is present and the expression for 𝑡𝑠𝑎 specific volume 𝑣 reduces to: 1+𝑥 𝑇 𝑇 𝑣 = 𝑅 ∙ + 𝑋 ∙ 𝑅 ∙ . (16) 1+𝑥 𝑑𝑎 𝑤𝑠 𝑝 𝑝 2.2. Internal energy Internal energy 𝑈 of the two-phase fluid is composed of the internal energies of dry air and water: 𝑈 = 𝑈 + 𝑈 = 𝑚 ∙ 𝑢 + 𝑚 ∙ 𝑢 . (17) 𝑑𝑎 𝑤 𝑑𝑎 𝑑𝑎 𝑤 𝑤 Here, 𝑢 and 𝑢 are the specific internal energies of dry air and water respectively. On the one hand, 𝑑𝑎 𝑤 referring internal energy 𝑈 of the mixture to the mass of dry air 𝑚 only, specific internal energy 𝑢 , 𝑑𝑎 1+𝑥 as commonly used in the literature, is specified: 𝑢 ∶= = 𝑢 + 𝑋 ∙ 𝑢 . (18) 1+𝑥 𝑑𝑎 𝑤 𝑑𝑎 On the other hand, relating internal energy 𝑈 to the entire mass 𝑚 of the mixture, specific internal energy 𝑢 of the two-phase mixture defined as 𝑈 𝑈 𝑢 ∶= = (19) 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 can be calculated from equation (18): 6 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 −1 (20) 𝑢 = 𝑢 ∙ (1 + 𝑋 ) . 1+𝑥 With regard to the thermodynamic absolute zero (𝑇 = 0 K), specific internal energy 𝑢 of saturated 1+𝑥 and unsaturated humid air (𝑋 ≤ 𝑋 ) can be obtained from the specific internal energies of dry air and 𝑡𝑠𝑎 water steam: 𝑡𝑟 𝑡𝑟 𝑢 = 𝑐 ∙ 𝑇 + 𝑋 ∙ {𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ (𝑣 − 𝑣 ) + 𝑐 ∙ (𝑇 − 𝑇 )}. (21) 1+𝑥 𝑣 ,𝑑𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑤𝑠 𝑖𝑐𝑒 𝑣 ,𝑤𝑠 𝑡𝑟 For temperatures above the triple-point temperature of water (𝑇 > 𝑇 ), specific internal energy 𝑢 of 𝑡𝑟 1+𝑥 the two-phase fluid can be calculated with reference to saturated humid air and liquid water in the mixture (𝑋 > 𝑋 ): 𝑡𝑠𝑎 𝑡𝑟 𝑡𝑟 ( ) ( ) 𝑢 = 𝑐 ∙ 𝑇 + 𝑋 ∙ {𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ 𝑣 − 𝑣 + 𝑐 ∙ 𝑇 − 𝑇 } + 1+𝑥 𝑣 ,𝑑𝑎 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑤𝑠 𝑖𝑐𝑒 𝑣 ,𝑤𝑠 𝑡𝑟 (22) 𝑡𝑟 ( ) { ( )} 𝑋 − 𝑋 ∙ 𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ (𝑣 − 𝑣 ) + 𝑐 ∙ 𝑇 − 𝑇 . 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑙𝑖𝑞 𝑖𝑐𝑒 𝑙𝑖𝑞 𝑡𝑟 𝑡𝑟 In equation (22), 𝑣 is referred to temperature and pressure in the triple point of water. For 𝑤𝑠 temperatures below the triple point (𝑇 < 𝑇 ), specific internal energy 𝑢 is determined as follows: 𝑡𝑟 1+𝑥 𝑡𝑟 𝑡𝑟 ( ) ( ) 𝑢 = 𝑐 ∙ 𝑇 + 𝑋 ∙ {𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ 𝑣 − 𝑣 + 𝑐 ∙ 𝑇 − 𝑇 } + 1+𝑥 𝑣 ,𝑑𝑎 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑤𝑠 𝑖𝑐𝑒 𝑣 ,𝑤𝑠 𝑡𝑟 (23) ( ) 𝑋 − 𝑋 ∙ 𝑐 ∙ T. 𝑡𝑠𝑎 𝑖𝑐𝑒 2.3. Enthalpy Enthalpy 𝐻 of the two-phase mixture results from internal energy 𝑈 and the product of pressure 𝑝 and volume 𝑉 : 𝐻 ∶= 𝑈 + 𝑝 ∙ 𝑉 . (24) Relating enthalpy 𝐻 to the mass of dry air 𝑚 , specific enthalpy ℎ of the two-phase mixture can be 𝑑𝑎 1+𝑥 obtained from equation (24) as follows: ℎ ∶= = 𝑢 + 𝑝 ∙ 𝑣 . (25) 1+𝑥 1+𝑥 1+𝑥 𝑑𝑎 Specific enthalpy ℎ considering the entire mass 𝑚 of the two-phase mixture can be determined as follows: 𝐻 𝐻 −1 ℎ ∶= = = ℎ ∙ (1 + 𝑋 ) . (26) 1+𝑥 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 2.4. Entropy Entropy 𝑆 of the two-phase fluid is the sum of the entropies of each component in the mixture also including entropy of mixing of the gaseous fluids. Dividing entropy 𝑆 by the mass 𝑚 of dry air, 𝑑𝑎 specific entropy 𝑠 is defined as follows: 1+𝑥 𝑠 ∶= . (27) 1+𝑥 𝑑𝑎 Specific entropy 𝑠 related to the entire mass 𝑚 of the two-phase mixture is introduced as: 𝑠𝑢 𝑚𝑒 𝑠𝑢 𝑠𝑢 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑆 𝑆 −1 𝑠 ∶= = = 𝑠 ∙ (1 + 𝑋 ) . (28) 1+𝑥 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 In contrast to specific internal energy 𝑢 and specific enthalpy ℎ , specific entropy 𝑠 is 1+𝑥 1+𝑥 1+𝑥 calculated with regard to the triple-point temperature of water 𝑇 . When calculating the specific entropy 𝑡𝑟 𝑠 of the two-phase fluid, a distinction must again be made between unsaturated and saturated humid 1+𝑥 air as well as for liquid water or ice as incompressible fluid [12]. Considering saturated and unsaturated humid air (𝑋 ≤ 𝑋 ) only, following expression is applicable: 𝑡𝑠𝑎 𝑇 𝑝 ( ) 𝑠 = (𝑐 + 𝑋 ∙ 𝑐 ) ∙ 𝑙𝑛 ( ) − 𝑅 + 𝑋 ∙ 𝑅 ∙ 𝑙𝑛 ( ) + 1+𝑥 𝑝 ,𝑑𝑎 𝑝 ,𝑤𝑠 𝑑𝑎 𝑤𝑠 𝑇 𝑝 𝑡𝑟 𝑡𝑟 (29) 𝑡𝑟 ∆ℎ 𝑋 ∙ + ∆ 𝑠 (𝑋 ). 𝑡𝑟 Here, ∆ 𝑠 (𝑋 ) is referred to as specific entropy of mixing of ideal gases with respect to dry air and steam and can be calculated as follows: 𝑅 𝑅 𝑅 𝑅 𝑑𝑎 𝑑𝑎 𝑑𝑎 𝑑𝑎 ∆ 𝑠 (𝑋 ) = 𝑅 ∙ {( + 𝑋 ) ∙ 𝑙𝑛 ( + 𝑋 ) − 𝑋 ∙ 𝑙𝑛 (𝑋 ) − ∙ 𝑙𝑛 ( ) } . (30) 𝑤𝑠 𝑅 𝑅 𝑅 𝑅 𝑤𝑠 𝑤𝑠 𝑤𝑠 𝑤𝑠 In terms of saturated humid air (𝑋 > 𝑋 ) at temperatures above the triple point (𝑇 > 𝑇 ), specific 𝑡𝑠𝑎 𝑡𝑟 entropy 𝑠 of the two-phase fluid can be determined from: 1+𝑥 𝑇 𝑝 𝑠 = (𝑐 + 𝑋 ∙ 𝑐 ) ∙ 𝑙𝑛 ( ) − (𝑅 + 𝑋 ∙ 𝑅 ) ∙ 𝑙𝑛 ( ) + 1+𝑥 𝑝 ,𝑑𝑎 𝑡𝑠𝑎 𝑝 ,𝑤𝑠 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑇 𝑝 𝑡𝑟 𝑡𝑟 (31) 𝑡𝑟 ∆ℎ 𝑇 ( ) ( ) 𝑋 ∙ + ∆ 𝑠 𝑋 + 𝑋 − 𝑋 ∙ 𝑐 ∙ 𝑙𝑛 ( ) . 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑙𝑖𝑞 𝑇 𝑇 𝑡𝑟 𝑡𝑟 At temperatures below the triple point (𝑇 < 𝑇 ), specific entropy 𝑠 is calculated using the 𝑡𝑟 1+𝑥 expression: 𝑇 𝑝 ( ) 𝑠 = (𝑐 + 𝑋 ∙ 𝑐 ) ∙ 𝑙𝑛 ( ) − 𝑅 + 𝑋 ∙ 𝑅 ∙ 𝑙𝑛 ( ) + 1+𝑥 𝑝 ,𝑑𝑎 𝑡𝑠𝑎 𝑝 ,𝑤𝑠 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑇 𝑝 𝑡𝑟 𝑡𝑟 (32) 𝑡𝑟 𝑡𝑟 ∆ℎ ∆ℎ 𝑇 𝑋 ∙ + ∆ 𝑠 (𝑋 ) − (𝑋 − 𝑋 ) ∙ { − 𝑐 ∙ 𝑙𝑛 ( ) }. 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑇 𝑇 𝑇 𝑡𝑟 𝑡𝑟 𝑡𝑟 3. Two-phase chamber model simulation Chamber model simulations are commonly applied in the theoretical analysis of positive displacement machines considering one or more cyclically changing working chambers [19]. Including the law of conservation of mass and energy, a numerical solver based on the time-step method—KaSim—solves the thermodynamic and fluid-mechanical equations of the multi-chamber model [17]. For the following theoretical analyses in this work, KaSim is adapted to the requirements of two-phase simulations. 3.1. Two-phase initialisation in KaSim For purposes of simplicity with respect to chamber model simulations, it is assumed that spatial gradients in the intensive state variables within a fluid capacity are insignificant. The two-phase fluid state is defined by means of the time-dependent extensive state variables internal energy 𝑈 (𝑡 ) neglecting kinetic and potential energy, volume 𝑉 (𝑡 ) of the fluid capacity, and mass 𝑚 (𝑡 ) together with mass dryness fraction 𝑥 (𝑡 ), figure 3. Additionally, enthalpy 𝐻 (𝑡 ) of the two-phase fluid is specified in order to reduce 𝑚𝑒 𝑒𝑣 𝑒𝑣 𝑒𝑣 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 the amount of time-consuming iterations during calculation of the intensive state variables pressure 𝑝 and temperature 𝑇 . In KaSim, fluid capacities (chambers, ports, pipes etc.) are initialised by means of pressure 𝑝 , temperature 𝑇 , capacity volume 𝑉 , and mass dryness fraction 𝑥 . rotor-tip (housing) mechanical thermal two-phase clearance (hc) energy energy fluid connection front clearance (fc) two-phase , , , fluid capacity inlet opening (io) , , outlet port , x , chamber 1 chamber 2 … chamber n mass flow rate inlet port Figure 3. Geometry abstraction of an exemplary twin-screw expander geometry with regard to a chamber model for the simulation tool KaSim [17] including selected two-phase fluid capacities and connections as well as parameters for their specification. The thermodynamic change in state of the fluid in a capacity results from transfer of, e.g. mechanical or thermal energy and mass. For example, mass exchange is performed using connections representing clearances or openings in the machine. Currently for connections affiliated to moving boundaries, only pressure-driven (Poiseuille) mass flow rates are considered neglecting the effect of Couette flow. Two- phase flow connections (clearances, inlets and outlets, leakage paths etc.) are specified by a cross- sectional area 𝐴 (𝑡 ), 𝑖𝑓𝑖𝑐𝑒𝑜𝑟 and 𝑧𝑧𝑙𝑒𝑛𝑜 discharge coefficients 𝐶 (𝑡 ), flow coefficients 𝛼 (𝑡 ), a connection-specific mass dryness fraction 𝜒 (𝑡 ), and a two-phase flow regime (ℎ𝑜𝑔𝑒𝑛𝑒𝑜𝑜𝑚𝑢𝑠 or with phase 𝑠𝑙𝑖𝑝 ), see figure 3. Each connection requires information input with regard to the two-phase flow regime. A ℎ𝑜𝑔𝑒𝑛𝑒𝑜𝑜𝑚𝑢𝑠 flow regime implies two-phase flows at equal phase flow velocities. The selection of a 𝑠𝑙𝑖𝑝 -flow allows for calculation of theoretical mass flow rates as for separated two-phase flows with different phase flow velocities represented by a so-called velocity slip ratio. Considering the discharge behaviour of the connections for two-phase flows, a distinction between 𝑧𝑛𝑜 𝑧 𝑙𝑒 and 𝑖𝑓𝑖𝑐𝑒𝑜𝑟 flows is 9 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 made. Two-phase flow losses in nozzle or orifice-like connections represented by the discharge coefficient 𝐶 (𝑡 ) are calculated depending on the fluid state of the connected capacities as proposed in [13]. According to equation (40), the flow coefficient 𝛼 (𝑡 ) provides a user-specific constant input reflecting wall friction in addition to the interphase friction loss as calculated by the two-phase nozzle discharge coefficient approach presented in [13]. During simulation, the discharge behaviour of a fluid connection remains fixed and no transition between the nozzle and orifice approach is allowed. To model liquid distribution in the fluid capacity, a connection-specific mass dryness fraction 𝜒 can be explicitly considered in terms of mass flow rate calculation: 𝜒 ∶= 𝑥 . (33) 𝑐𝑜𝑛 Connection mass dryness fraction 𝜒 can be specified constant or equal to mass dryness fraction 𝑥 of the two-phase fluid in the source capacity (capacity of higher pressure considering two connected fluid capacities). With respect to a constant 𝜒 , mass dryness fraction 𝑥 of the source capacity fluid must exhibit values between 0.01 and 0.9. Outside this limits, 𝜒 is dynamically initialised during simulation with mass dryness fraction 𝑥 of the source capacity fluid rather than using the initially specified constant value preventing capacity mass dryness fractions 𝑥 greater than unity or less than zero. 3.2. Change in state of the two-phase fluid In this study, adiabatic fluid capacities without heat flows over the control volume boundaries are considered. Hence, change in fluid state results either from change in capacity volume or mass and, correspondingly, enthalpy transfer during a simulation time step. The new internal energy 𝑈 of the (two-phase) fluid can be calculated from the initial state according to internal energy 𝑈 as follows: 𝑈 = 𝑈 + 𝑊 + ∑ 𝐻 . 1 0 𝑘 (34) Here, 𝑊 indicates the work performed in terms of volume change and 𝐻 represents enthalpy of each exchanged fluid element over the control volume boundaries. Assuming isentropic volume change, work is calculated as follows: 𝑊 = − ∫ = 𝑈 − 𝑈 . (35) 0 1,𝑠 At constant initial entropy 𝑆 and water load 𝑋 during volume change, internal energy 𝑈 of humid 0 0 1,𝑠 air is iteratively calculated from the new intensive state variables pressure 𝑝 and temperature 𝑇 : 1 1 𝑈 = 𝑈 (𝑉 , 𝑝 , 𝑇 , 𝑋 , 𝑆 ). (36) 1,𝑠 1 1 1 0 0 If the thermodynamic state of water in the two-phase mixture after change in volume corresponds to the triple-point state, internal energy of the two-phase fluid is calculated by a linear interpolation between internal energy according to liquid state of water as in equation (22) and solid state according to equation (23). For a constant fluid capacity volume, the fluid state alters as a result of mass and corresponding enthalpy transfer. For two-phase flows, enthalpy 𝐻 of each exchanged fluid element is calculated from specific enthalpy ℎ according to equation (26) factored by the two-phase mass 𝑚 : 𝑘 𝑘 −1 𝐻 = 𝑚 ∙ ℎ = 𝑚 ∙ ℎ ∙ (1 + 𝑋 ) . (37) 𝑘 𝑘 𝑘 𝑘 1+𝑥 ,𝑘 𝑘 𝑝𝑑𝑉 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 Enthalpy 𝐻 of the fluid elements exchanged between two adjacent fluid capacities is related to the fluid capacity at higher pressure. Further, during balancing internal energy within a specific fluid capacity, elements flowing in are considered positive and such leaving the control volume negative. 3.3. Calculation of two-phase mass flow rate For the calculation of the two-phase mass flow rate, a two-phase separated flow (slip-flow) method as proposed in [13] is applied. Here, the mass flow rate 𝑚 ̇ results from a theoretical two-phase mass flow rate 𝑚 ̇ factored by a discharge coefficient 𝐶 including the entire flow losses: 𝑡 ℎ 𝑑 𝑚 ̇ = 𝐶 ∙ 𝑚 ̇ . (38) 𝑑 𝑡 ℎ The theoretical two-phase mass flow rate 𝑚 ̇ through restrictions is derived from the conservation laws 𝑡 ℎ of momentum and mass as presented amongst others by Chisholm in [20] and Morris in [21]. The discharge coefficient 𝐶 allows for the entire momentum loss and considers, by definition, a contraction coefficient 𝐶 depending on Mach number effects factored by the so-called coefficient of velocity 𝐶 𝑐 𝑣 including friction effects [21–24]: 𝐶 = 𝐶 ∙ 𝐶 . (39) 𝑑 𝑐 𝑣 For orifice flows, the discharge coefficient 𝐶 includes momentum losses only due to contraction effects (vena contracta), and the coefficient of velocity 𝐶 equals unity. The orifice approach (𝐶 = 𝐶 ) is 𝑣 𝑑 𝑐 preferred with respect to inlet and outlet connections in twin-screw expanders. For narrow flow sections or nozzle flows, no flow contraction is considered (𝐶 = 1), and the discharge coefficient primarily allows for momentum losses due to interphase and wall friction (𝐶 = 𝐶 ). This method for flow loss 𝑑 𝑣 calculation is applied to any clearance in the twin-screw-expander or leakage paths where no contraction of the two-phase flow jet is expected. Since the approach regarding two-phase discharge coefficients for nozzle flows presented in [13] reflects only interphase friction effects, in the current study the coefficient of velocity is factored by an empirical flow coefficient 𝑎 as follows: 𝐶 = 𝐶 ∙ 𝑎 . (40) 𝑣 𝑣 The coefficient of velocity 𝐶 is addressed to interphase friction as reported in [13] depending mainly on mass dryness fraction while the flow coefficient 𝑎 reflects wall friction effects. 4. Geometry of SE 51.2 The two-phase approach for the simulation of water-flooded twin-screw expanders proposed in this work is verified by means of existing experimental data of the twin-screw expander SE 51.2 including integral operational parameters as well as indicator diagrams presented in [14–16]. In figure 4, a 3D model of the test twin-screw expander prototype SE 51.2 including positions of the high-resolution pressure indication transmitters used to record indicator diagrams is illustrated. SE 51.2 is a twin-screw expander prototype without timing gears. The transmission of torque occurs directly via contact between the rotor flanks. The screw rotors are hardened and have a tough tungsten carbide/carbon (WC/C) wear-protection coating, so seizure is avoided in dry-running or water-flooded operation. Both fixed and loose bearing sets are grease lubricated, so no oil supply is necessary. Details about the expander geometry parameters are listed in table 1. 11 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 inlet port 0 2 3 4 axial outlet, inlet area curve volume curve outlet area curve floating bearing casing 100 10 inlet plate 90 9 (control edge) 80 8 0 1 2 3 4 5 6 70 7 coupling 60 6 casing 50 5 40 4 30 3 rotor casing 20 2 10 1 male rotor 0 0 100 200 300 400 500 600 700 800 female rotor male rotor rotational angle [ ] fixed bearing casing Figure 4. Positions of pressure indication transmitters (0…6), inlet and outlet area, as well as volume curve of the test twin-screw expander SE 51.2 including ranges of male rotor rotational angle corresponding to each pressure transmitter [15, 16]. Table 1. Geometry parameters of the test twin-screw expander SE 51.2. designation unit male rotor (mr) female rotor (fr) number of lobes 𝑧 [-] 3 5 diameter [mm] 71.8 67.5 rotor lead [mm] 181.8 303 wrap angle [°] 200 -120 rotor length [mm] 101 rotor profile [-] modified asymmetric SRM axis-centre distance [mm] 51.2 internal volume ratio [-] 2.5 displaced volume per male rotor rotation [cm ] 286 front clearance height ℎ (high pressure) [mm] 0.1 𝑓𝑐 ,ℎ𝑝 front clearance height ℎ (low pressure) [mm] 0.17 𝑓𝑐 ,𝑙𝑝 rotor-tip (housing) clearance height ℎ [mm] 0.08 ℎ𝑐 5. Simulation setup In the following section, the multi-chamber model of the twin-screw expander SE 51.2 including specific input with respect to the modelling of liquid distribution and the boundary conditions for the simulations are presented. 5.1. Multi-chamber model of SE 51.2 The multi-chamber model used for the simulation of the twin-screw expander SE 51.2 is illustrated in figure 5. The working chambers are numbered consecutively beginning from the low pressure domain. Each working chamber is divided into two fluid sub-capacities belonging to the male and female rotor. Pressure balance connections provide equal pressures in two corresponding male and female sub- capacities during each calculation time step. Nevertheless, different temperatures and mass dryness fractions of these sub-capacities result after pressure balancing. chamber volume [cm ] inlet and outlet area [cm ] International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 high pressure (hp) domain fluid connections inlet (hp) F 8 outlet (lp) pressure balance connection (lp) M 6 F 7 pressure balance connection (hp) M 5 F 6 blow hole (lp) blow hole (hp) M 4 F 5 housing clearance M 3 F 4 front and interlobe clearances M 2 F 3 are not illustrated M 1 F 2 M .. male rotor chamber F .. secondary rotor chamber F 1 subsequent chamber indication low pressure (lp) domain Figure 5. Multi-chamber model (adiabatic, front and interlobe clearances not illustrated) of SE 51.2 containing fluid capacities and connections. In terms of flow loss calculation, as presented in table 2, all clearances of SE 51.2 are initialised with respect to nozzle discharge coefficient including the coefficient of velocity factored by a user-specified flow coefficient according to equation (40). Inlets and outlets are modelled as orifice flow restrictions according to equation (39) considering only contraction effects. Table 2. Specific parameters for the initialisation of the chamber model of SE 51.2 with respect to two- phase flow connections. flow discharge dryness fraction flow coefficient connection regime behaviour 𝝌 𝒂 inlet opening (hp) slip orifice 𝜒 = 𝑥 1.0 𝑖𝑜 outlet opening (lp) slip orifice 1.0 𝜒 = 𝑥 𝜒 = 𝑥 rotor-tip clearance (mr) slip nozzle 1.0 ℎ𝑐 , rotor-tip clearance (fr) slip nozzle 𝜒 = 𝑥 0.8 ℎ𝑐 ,𝑓𝑟 blow hole slip nozzle 𝜒 = 𝑥 1.0 𝑏 ℎ interlobe clearance slip nozzle 1.0 𝜒 = 0.9 𝑖𝑐 𝜒 = 0.9 front clearance slip nozzle 0.8 𝑓𝑐 Due to the complexity of predicting the liquid distribution in the working chamber and thus at the clearances of twin-screw machines, no holistic models are available in the literature. Therefore, assumptions have to be made. On the one hand, the liquid flows in radial direction as a result of 𝑚𝑟 𝑜𝑜 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 centrifugal forces where it spreads over the housing surface. On the other hand, Couette flows additionally influence the amount of liquid directly at the clearance. Due to centrifugal forces, it could be assumed that all clearances except blow holes and rotor-tip clearances remain relatively dry and are not significantly sealed by the liquid. Therefore, high mass dryness fractions are expected there. For the calculation of the pressure-driven two-phase clearance mass flow rate, rotor-tip clearance and blow hole mass dryness fraction is recommended to be initialised by values equal to those of the chamber fluid or less. In this work, the liquid distribution in the working chamber is modelled initialising mass dryness fraction of each clearance as proposed in table 2. For example, housing clearances and blow holes are initialised with a mass dryness fraction related to the source chamber value. In contrast, interlobe and front (axial) clearances are considered relatively “dry” according to a dryness faction of 0.9. These assumptions are supported, in particular, by video recordings of the working cycle (not presented here). 5.2. Simulation boundary conditions The simulations of the operational behaviour of SE 51.2 are performed according to the experimental investigations as presented in [16]. The range of pressure, temperature, inlet and outlet mass dryness fraction, and male rotor tip-speed respectively for simulation and comparison with experimental data are listed in table 3. Table 3. Boundary conditions for the simulation of SE 51.2. parameter value -1 maximum male rotor tip-speed 𝑢 67.7 m∙s 5 5 5 3∙10 Pa, 4∙10 Pa, 5∙10 Pa inlet pressure 𝑝 (absolute) outlet pressure 𝑝 (absolute) 1∙10 Pa inlet temperature 𝜗 55 °C (𝑥 = 0.5), 90 °C (𝑥 = 1.0) 𝑖 𝑖 𝑖 0.5 (water-flooded), 1.0 (dry-running) inlet/outlet mass dryness fraction 𝑥 /𝑥 𝑖 𝑜 6. Results In the following section, the two-phase approach for simulation of water-flooded twin-screw machines is validated comparing chamber-model simulation results to experimental data for the expander SE 51.2. In this context, indicator diagrams, expander mass flow rates, experimentally determined indicated and simulated internal powers, as well as outlet temperatures are considered. Indicated power is calculated by means of the experimentally recorded chamber pressure during a working cycle and the corresponding chamber volume as follows: 𝑃 = − 𝑛 · 𝑧 · ∮ 𝑝 · . (41) 𝑖𝑛𝑑 First, calculation and experimental results for dry-running operation (𝑥 = 1) are presented in figure 6 in order to analyse the simulation model regardless of liquid effects. Mass flow rate as well as indicated and internal power are examined as a function of male rotor tip-speed 𝑢 at different inlet pressures 𝑝 . As expected, simulated and experimental mass flow rate and power increase at rising rotor speed and inlet pressure due to increasing number of working cycles and inlet density respectively. With regard to inlet throttling losses at increasing rotor tip-speed, both expander parameters exhibit a degressive progression due to declining chamber pressure during the filling process. Moreover, for a combination of high rotor speeds and low inlet pressures, overexpansion as observed in the indicator 5 -1 diagram for 𝑝 = 3∙10 Pa and 𝑢 = 60 m∙s negatively impacts the expander performance. 𝑚𝑟 𝑚𝑟 𝑚𝑟 𝑚𝑟 𝑑𝑉 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 0.16 -16 0.14 -14 0.12 -12 0.1 -10 0.08 -8 0.06 -6 0.04 -4 0.02 -2 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 -1 -1 male rotor tip-speed [m∙s ] male rotor tip-speed [m∙s ] [10 Pa] 3 4 5 = 90 °C experiment = 1 simulation 5.5 5.5 inlet control edge inlet control edge 5.0 5.0 4.5 4.5 4.0 4.0 -1 -1 = 3.8 m∙s = 52.8 m∙s 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 -6 3 -6 3 volume [10 m ] volume [10 m ] Figure 6. Expander mass flow rate, simulated internal and experimental indicated power, as well as indicator diagrams at different inlet pressures and rotor speeds for dry-running operation (𝑥 = 1). Deviations between simulation and experimental results are mainly attributed to either modelling of clearances and openings or dynamic effects during the experiments and uncertainties of the measurement. In figure 6, the greatest disagreement between simulation and experiment is observed for -1 5 5 increasing rotor tip-speeds 𝑢 > 30 m∙s at inlet pressure 𝑝 = 4∙10 Pa and 𝑝 = 5∙10 Pa. Here, the 𝑖 𝑖 simulated mass flow rate and internal power are lower than the experimental values. This difference can be traced back to the orifice discharge coefficients applied to the expander inlet during the simulation (see table 2). For single-phase flows, in contrast to nozzles approaching the theoretical mass flow rate, orifices are associated with flow losses due to contraction effects. As reported in [13], the inlet geometry -1 mass flow rate [kg∙s ] pressure [10 Pa] 5 internal/indicated power [kW] pressure [10 Pa] 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 based on the twin-screw expander SE 51.2 exhibits a transition from an orifice to nozzle discharge behaviour as the opening becomes slip-shaped with the end of the chamber filling. Therefore, as with inlet discharge coefficients related to orifices, the flow loss during chamber filling might be overestimated, since, in this study, the inlet discharge behaviour is fixed and cannot be changed during simulation progress. In the same context, the measured indicator diagrams illustrated for -1 𝑢 = 52.8 m∙s show higher working chamber pressure during volume increase compared to the simulation results. Additionally, as observed during chamber filling in the indicator diagrams and reported in [15], the pressure oscillation in the inlet port directly influence the chamber pressure. Hence in contrast to the chamber model simulation not considering dynamic fluid inertia effects, the experiments reveal higher fluid density in the working chamber during chamber filling. With regard to low rotor tip-speed in figure 6, in particular at 𝑝 = 3∙10 Pa, simulated expander mass -1 flow rate is slightly overpredicted. In the indicator diagrams for 𝑢 = 3.8 m∙s , the simulated chamber pressure during expansion is lower than the experimentally recorded. Along with a higher simulated than measured mass flow rate, this indicates overestimated internal leakages that could be affiliated with the interlobe clearance directly connecting high-pressure working chambers with chambers discharging into the low-pressure port of the expander. Due to, for example, thermal deformation or bearing clearance, the interlobe clearance could experience significant relative change in height compared to its initial design dimension. The interlobe clearance leakage permanently induces loss flows that reduce the chamber pressure and cannot be utilised at lower pressure in terms of chamber refilling as with housing and front clearances or blow holes. In figure 7, simulation and experimental results under water-flooded operating conditions (𝑥 = 0.5) are presented. Generally as a function of rotor speed and inlet pressure, two-phase mass flow rate as well as internal and indicated power exhibit similar characteristics as for dry-running operation in figure 6. The analysis of the water-flooded expander at increasing rotor tip-speed reveals simulated internal power slightly lower than the experimentally determined indicated power. Here again as with dry-running operation, this could be attributed to dynamic effects during the chamber filling or to the inlet area modelling with respect to flow losses. At the same time, for low rotor speeds and increasing inlet pressure, the simulation provides expander two-phase mass flow rates slightly less than the experimentally recorded. The maximum deviation is in the range of 12 % at the lowest rotor tip-speed and highest inlet pressure investigated. In connection with the proposed liquid distribution in this study, this gives a rise to the assumption that a relevant internal leakage is underestimated. As the indicator diagrams show, during the internal expansion, an internal leakage path associated with chamber refilling might be the reason for this difference due to the initially higher and afterwards lower pressure levels compared to the measurements. In this context, the liquid sealing effect with respect to blow holes and the rotor-tip clearances might be overestimated proposing their initialisation as with the chamber mass dryness fraction. As recorded images of the working cycle (not presented here) indicate, liquid distribution at the rotor-tip clearances and blow holes dynamically changes during the expansion progress providing an evidence for the relevance of its appropriate modelling. Hence, the assumption as made within this study has to be considered only as an initial approach. However with the aim of reliable simulations of liquid-flooded twin-screw machines, a high potential for developing appropriate liquid-distribution models in future works exists. 𝑚𝑟 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 0.32 -16 0.28 -14 0.24 -12 0.2 -10 0.16 -8 0.12 -6 0.08 -4 0.04 -2 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 -1 -1 male rotor tip-speed [m∙s ] male rotor tip-speed [m∙s ] [10 Pa] 3 4 5 = 55 °C experiment = 0.5 simulation 5.5 5.5 inlet control edge inlet control edge 5.0 5.0 4.5 4.5 4.0 4.0 -1 -1 = 3.8 m∙s = 52.8 m∙s 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 -6 3 -6 3 volume [10 m ] volume [10 m ] Figure 7. Expander mass flow rate, simulated internal and experimental indicated power, as well as indicator diagrams at different inlet pressures and rotor speeds for dry-running operation (𝑥 = 0.5). Figure 8 illustrates the measured and simulated expander outlet temperature depending on rotor speed and inlet pressure. At this point it must be noted that, in contrast to the experimental data, the simulated temperatures do not include any losses as with hydraulic and mechanical friction. Basically, expander outlet temperature decreases due to declining impact of internal leakages at rising rotor speed, on the one hand, and higher utilisable pressure ratio related to increasing inlet pressure, on the other hand. In terms of dry-running operation (𝑥 = 1), the simulated temperatures deviate from the experimental values by maximum 20 K. As one reason for this result, the greater throttling loss as mentioned above is identified and, thus, lower pressure ratios are available for expansion associated -1 mass flow rate [kg∙s ] pressure [10 Pa] 5 internal/indicated power [kW] pressure [10 Pa] International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 with lower temperature drops. Applying a nozzle discharge coefficients to the inlet opening of SE 51.2, lower expander outlet temperatures are simulated attributed to higher chamber pressures after disconnecting the working chamber from the high-pressure port (not presented here). Hence, the maximum temperature difference declines to 12 K. Additionally, in the simulation, overestimated gaseous flows through the interlobe clearance bypass the working chamber and the relatively high temperatures of the leakage flow resulting from the corresponding isenthalpic change in state contribute to a further temperature increase downstream of the expander. With regard to water-flooded operation (𝑥 = 0.5), expander outlet temperatures are relatively accurately predicted by the simulation. Significant deviations are observed at an inlet pressure of 𝑝 = 5∙10 Pa and low rotor speeds -1 𝑢 < 30 m∙s . For two-phase flows, these can be explained by measurement uncertainties at different phase temperatures unlike the assumptions of thermal equilibrium as with the humid air model. [10 Pa] 3 4 5 = 90 °C = 55 °C experiment = 1 = 0.5 simulation 370 370 = 363.15 K 350 350 330 330 = 328.15 K 310 310 290 290 270 270 250 250 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 -1 -1 male rotor tip-speed [m∙s ] male rotor tip-speed [m∙s ] Figure 8. Expander outlet temperature as a function of male rotor tip-speed at different inlet pressures for dry-running (𝑥 = 1) and water-flooded (𝑥 = 0.5) operation. 𝑖 𝑖 7. Conclusion and outlook Within the framework of this study, a two-phase approach for simulation of water-flooded twin-screw machines is validated for expander applications. The thermodynamic properties of the two-phase mixture of dry air and water are represented by the humid air model considering both phases in thermal equilibrium. Applying a two-phase slip-flow flow model, mass flow rates are calculated factoring a theoretical mass flow rate by geometry-specific discharge coefficients reflecting flow losses as in nozzle and orifice restrictions. Liquid distribution in the expander working chamber directly influencing the two-phase clearance flows is modelled in a simple way using clearance-specific mass dryness fractions. Due to centrifugal forces, it is assumed that most of the liquid distributes towards the radial clearances, while the axial and interlobe clearances are associated with relatively low amount of liquid. In this context, radial rotor-tip clearances and blow holes are initialised with mass dryness fraction of the source fluid capacity and the remaining clearances are referred to a constant mass dryness fraction of 0.9. outlet temperature [K] outlet temperature [K] 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 The single- and two-phase simulations reveal a relatively high degree of agreement with experimental data considering expander mass flow rate, internal power, outlet temperature, and chamber pressure depending on volume. At low rotor tip-speeds related to significant clearance flows, for both dry-running and liquid-flooded operation, simulated internal power matches the experimentally determined indicated power within a small range of deviation. Here for dry-running operation, the simulations exhibit a slight overprediction of clearance flows resulting in higher expander mass flow rates. Nevertheless, the comparison of simulated and measured chamber pressure reveals an appropriate prediction of internal leakages contributing to chamber refilling. Regarding the two-phase simulation, the assumed liquid distribution provides greater sealing effect indicated by underestimation of expander mass flow rates and lower chamber pressures during chamber refilling compared to the experimental data. At higher rotor speeds for both dry-running and liquid-flooded operation, simulated internal power and mass flow rate are underpredicted, in particular, due to dynamic effects during chamber filling not reflected by the simulation. Accordingly, higher outlet temperatures are simulated under dry-running conditions. In contrast, for water-flooded operation, deviations between simulated and experimental expander outlet temperatures are mainly traced back to uncertainties linked to two-phase temperature measurements. In the present work, the liquid distribution in the working chamber is identified as a major challenge for further development of the two-phase simulation approach. In this context, more accurate and detailed models for the initialisation of mass dryness fraction of the clearances of twin-screw machines for expander and compressor applications are required. Since at high rotor speeds the effect of two- phase Couette flow increases, it has to be considered too. Moreover in terms of inlet throttling, flow loss calculation could be improved with respect to transition between an orifice and nozzle discharge coefficient during simulation depending on the shape of the inlet geometry. Nomenclature symbol unit designation area 𝐴 𝑚 −1 ( ) specific heat capacity of the incompressible phase 𝑐 𝐽 ∙ ∙ 𝐾 𝐶 - (two-phase) contraction coefficient 𝐶 - calculated (two-phase) discharge coefficient −1 ( ) isobaric specific heat capacity 𝑐 𝐽 ∙ ∙ 𝐾 −1 𝑐 𝐽 ∙ ( ∙ 𝐾 ) isochoric specific heat capacity 𝐶 - (two-phase) coefficient of velocity −1 ℎ 𝐽 ∙ specific enthalpy 𝐻 𝐽 enthalpy 𝑚 mass of the two-phase mixture 𝑚 mass of dry air 𝑑𝑎 mass of water steam 𝑤𝑠 mass of liquid water 𝑙𝑖𝑞 −1 𝑚 ̇ ∙ 𝑠 mass flow rate of the two-phase mixture −1 𝑚 ̇ ∙ 𝑠 mass flow rate of dry air 𝑑𝑎 −1 𝑚 ̇ ∙ 𝑠 mass flow rate of water steam 𝑤𝑠 −1 𝑚 ̇ ∙ 𝑠 mass flow rate of liquid water 𝑙𝑖𝑞 −1 𝑛 rotational speed pressure 𝑃 𝑊 power −1 𝑅 𝐽 ∙ ( ∙ 𝐾 ) specific gas constant −1 𝑠 ( ) specific entropy 𝐽 ∙ ∙ 𝐾 −1 entropy 𝑆 𝐽 ∙ 𝐾 −1 time 𝑡 𝑠 𝑘𝑔 𝑘𝑔 𝑃𝑎 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑇 𝐾 temperature −1 𝑢 rotor tip-speed 𝑚 ∙ 𝑠 −1 𝑢 specific internal energy 𝐽 ∙ 𝑈 𝐽 internal energy 3 −1 𝑣 specific volume 𝑚 ∙ volume 𝑉 𝑚 work 𝑊 𝐽 - mass dryness fraction 𝑋 - water load 𝑧 - number of rotor lobes α - flow coefficient 𝜗 °𝐶 temperature 𝜑 - relative humidity 𝜒 - connection mass dryness fraction 𝜒 ≔ 𝑥 𝑐𝑜𝑛 abbreviation/index designation 0 initial fluid state 1 changed fluid state 1+x related to dry air mass bh blow hole ch chamber con connection da dry-air ev evaporation ex exchanged fc front clearance fr female rotor g gaseous hc rotor-tip (housing) clearance hp high pressure i inlet ic interlobe clearance ice ice ind indicated io inlet opening j index for a gas component in a gaseous fluid mixture k index for exchanged fluid element liq liquid (water) lp low pressure me melting mr male rotor o outlet oo outlet opening s isentropic sat saturation su sublimation th theoretical tr triple w water (liquid and steam) ws water steam 𝑘𝑔 𝑘𝑔 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 Appendix A Table A1. Constant fluid properties for the thermodynamic state calculation of the two-phase fluid mixture in thermal equilibrium with regard to humid air [12]. designation unit value -1 -1 [J∙kg ∙K ] 287.05 specific gas constant of dry air 𝑅 𝑑𝑎 -1 -1 [J∙kg ∙K ] 461.5 specific gas constant of water steam 𝑅 𝑤𝑠 -1 -1 isobaric specific heat capacity of dry air 𝑐 [J∙kg ∙K ] 1004.6 𝑝 ,𝑑𝑎 -1 -1 isobaric specific heat capacity of water steam 𝑐 [J∙kg ∙K ] 1863 𝑝 ,𝑤𝑠 -1 -1 isochoric specific heat capacity of dry air 𝑐 [J∙kg ∙K ] 717.55 𝑣 ,𝑑𝑎 -1 -1 isochoric specific heat capacity of water steam 𝑐 [J∙kg ∙K ] 1401.5 𝑣 ,𝑤𝑠 -1 -1 liquid water specific heat capacity 𝑐 [J∙kg ∙K ] 4191 𝑙𝑖𝑞 -1 -1 ice specific heat capacity 𝑐 [J∙kg ∙K ] 2070 𝑖𝑐𝑒 3 -1 specific volume of liquid water 𝑣 [m ∙kg ] 0.001002 𝑙𝑖𝑞 3 -1 [m ∙kg ] 0.001087 specific volume of ice 𝑣 𝑖𝑐𝑒 [K] 273.16 triple temperature 𝑇 𝑡𝑟 [Pa] 611.657 triple pressure 𝑝 𝑡𝑟 -1 𝑡𝑟 [kJ∙kg ] 333.4 specific melting enthalpy ∆ℎ at 𝑇 𝑡𝑟 -1 𝑡𝑟 specific vaporisation enthalpy ∆ℎ at 𝑇 [kJ∙kg ] 2500.9 𝑡𝑟 -1 𝑡𝑟 specific sublimation enthalpy ∆ℎ at 𝑇 [kJ∙kg ] 2834.3 𝑡𝑟 References [1] Rane S, Kovačević A, Stošić N and Stupple G 2019 On Numerical Investigation of Water Injection to Screw Compressors Proceedings of the ASME International Mechanical Engineering Congress and Exposition 2018 ASME 2018 International Mechanical Engineering Congress and Exposition (Pittsburgh, Pennsylvania, USA, 09.11.2018 - 15.11.2018) (New York, N.Y.: The American Society of Mechanical Engineers) [2] Räbiger K, Maksoud T, Ward J and Hausmann G 2008 Theoretical and experimental analysis of a multiphase screw pump, handling gas–liquid mixtures with very high gas volume fractions Experimental Thermal and Fluid Science 32 1694–701 [3] Sangfors B 1984 Computer Simulation of the Oil Injected Twin Screw Compressor International Compressor Engineering Conference International Compressor Engineering Conference (Purdue, Indiana) (International Compressor Engineering Conference) ed Purdue University Libraries (Purdue, Indiana) [4] Sangfors B 1998 Computer Simulation of Effects From Injection of Different Liquids in Screw Compressors International Compressor Engineering Conference International Compressor Engineering Conference (Purdue) (International Compressor Engineering Conference) ed Purdue University Libraries [5] Vasuthevan H and Brümmer A 2016 Thermodynamic Modeling of Screw Expander in a Trilateral Flash Cycle 23rd International Compressor Engineering Conference at Purdue: West Lafayette, Indiana, USA, 11-14 July 2016 (Purdue) ed Purdue University Libraries [6] Smith S L 1969 Void Fractions in Two-Phase Flow: A Correlation Based upon an Equal Velocity Head Model Proceedings of the Institution of Mechanical Engineers 184 647–64 [7] Bell I 2011 Theoretical and Experimental Analysis of Liquid Flooded Compression in Scroll Compressors PhD thesis Purdue University [8] Lemort V December 19th, 2008 Contribution to the Characterization of Scroll Machines in Compressor and Expander Modes (Liège: Université de Liège, Belgique) [9] Huagen W, Ziwen X and Pengcheng S 2004 Theoretical and experimental study on indicator diagram of twin screw refrigeration compressor International Journal of Refrigeration 27 331–8 𝑠𝑢 𝑒𝑣 𝑚𝑒 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 [10] Lin Z H 1982 Two-phase flow measurements with sharp-edged orifices International Journal of Multiphase Flow 8 683–93 [11] Wang C, Xing Z, Chen W, Sun S and He Z 2019 Analysis of the leakage in a water-lubricated twin-screw air compressor Applied Thermal Engineering 155 217–25 [12] Baehr H D and Kabelac S 2012 Thermodynamik: Grundlagen und technische Anwendungen (Springer-Lehrbuch) 15th edn (Berlin u.a.: Springer Vieweg) [13] Nikolov A and Brümmer A 2022 Two-phase mass flow rate through restrictions in liquid-flooded twin-screw compressors or expanders International Journal of Refrigeration (submitted) [14] Nikolov A and Brümmer A 2014 Influence of water injection on the operating behaviour of screw expanders: Experimental investigation International Conference on Screw Machines 2014 (TU Dortmund University, Germany, 23rd-24th September) (VDI-Berichte vol 2228) ed VDI (Düsseldorf: VDI-Verl.) pp 43–60 [15] Nikolov A and Brümmer A 2016 Analysis of Indicator Diagrams of a Water Injected Twin-shaft Screw-type Expander 23rd International Compressor Engineering Conference at Purdue: West Lafayette, Indiana, USA, 11-14 July 2016 (Purdue) ed Purdue University Libraries [16] Nikolov A and Brümmer A 2018 Impact of different clearance heights on the operation of a water-flooded twin-screw expander—experimental investigations based on indicator diagrams International Conference on Screw Machines (TU Dortmund University, Germany, 18th-19th September) ed IoP Conf. Series: Material Science and Engineering [17] Janicki M 2007 Modellierung und Simulation von Rotationsverdrängermaschinen PhD thesis Fakultät Maschinenbau, TU Dortmund University [18] IAPWS - International Association for the Properties of Water and Steam 2011 Revised Release on the Pressure along the Melting and Sublimation Curves of Ordinary Water Substance (IAPWS R14-08(2011)) (Plzeň, Czech Republic: IAPWS) http://www.iapws.org/ (accessed 8 Aug 2017 ) [19] Janicki M and Kauder K 2003 Adiabatic Modelling and Thermodynamic Simulation of Rotary Displacement Machines International Conference on Compressors and Their Systems (Cass Business School, City University, London, UK, 7-10 September) (IMechE conference transactions) ed City University London (Bury St Edmunds: Professional Engineering) pp 511–9 [20] Chisholm D 1983 Two-phase flow in pipelines and heat exchangers (London: Godwin) [21] Morris S D 1991 Compressible gas-liquid flow through pipeline restrictions Chemical Engineering and Processing: Process Intensification 30 39–44 [22] Morris S D 1990 Discharge coefficients for choked gas—liquid flow through nozzles and orifices and applications to safety devices Journal of Loss Prevention in the Process Industries 3 303–10 [23] Trutnovsky K and Komotori K 1981 Berührungsfreie Dichtungen (Düsseldorf: VDI-Verlag) [24] Jobson D A 1955 On the Flow of a Compressible Fluid through Orifices Proceedings of the Institution of Mechanical Engineers 767–76 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IOP Conference Series: Materials Science and Engineering IOP Publishing

A two-phase approach for simulation of water-flooded twin-screw machines validated for expander applications

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Abstract

International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 A two-phase approach for simulation of water-flooded twin-screw machines validated for expander applications A Nikolov, A Brümmer TU Dortmund University, Faculty of Mechanical Engineering, Chair of Fluidics, Leonhard-Euler-Str. 5, 44227 Dortmund, Germany E-mail: [email protected], [email protected] Abstract. In the lower and medium power range, twin-screw machines offer a high potential for energy conversion in various compressor applications with liquid injection or as expanders with respect to electrical power generation from regenerative and exhaust heat sources in trilateral or wet Rankine cycle systems, for instance. Aiming high efficiencies and reliability, the design of liquid-flooded twin-screw machines as a critical system component presents particular challenges for the engineers. Hence, reliably representative simulations to guide design are mandatory. In this context, this study presents a two-phase approach for simulation of the operational behaviour of water-flooded twin-screw machines. The thermodynamic fluid state is calculated using the humid air model that considers the two-phase mixture in thermal equilibrium. Additionally, dissipative two-phase mass flow rates are predicted regarding a slip- flow model and two-phase discharge coefficients. The proposed two-phase approach including liquid distribution in the working chamber is validated for expander applications considering available experimental data of the test twin-screw expander SE 51.2 in terms of indicator diagrams, indicated power, mass flow rate, and outlet temperature. 1. Introduction Liquid-flooded twin-screw machines are widely applied as compressors providing benefits in terms of two-phase operation, such as high pressure ratios in a single stage together with low thermal stress at the same time, clearance sealing effects or lubrication of the moving machine parts. While water- flooded process-gas compressors have long been in operation, water-injection in air or hydrogen twin- screw compressors gain more importance in recent years. Recovering exhaust heat from low-grade heat sources, for example, in trilateral or wet organic Rankine cycle systems, twin-screw expanders capable of dealing with large amounts of liquid can be applied as an alternative to turbines avoiding wet expansion. Moreover, injecting an auxiliary liquid during expansion could provide benefits in terms of utilisation of available pressure potentials, such as in (natural) gas reducing stations, where only a small temperature drop during expansion might be allowed. In this context, injected liquid, e.g. water, in combination with an exhaust or other regenerative heat source acts also as heat carrier. In order to theoretically determine the operational behaviour of water-flooded twin-screw machines with respect to system design, appropriate two-phase models, e.g. for chamber model simulations, considering the thermodynamic fluid properties of the air-water mixture and adequately reflecting the mass flow rates are required. With regard to the thermodynamic modelling of water injection into the working chamber of twin-screw compressors, a study presented in [1] considers a water evaporation model in order to determine the influence of very low amount of liquid injection on the operation of a twin-screw compressor. In terms of two-phase flows, only a few contributions in the literature deal with non-homogeneous two-phase mass flow rate models in rotary displacement machines. Generally, Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 homogeneous approaches are applied as in [2]. In the same manner, in [3] and [4], Sangfors simulates the operation of liquid-flooded twin-screw machines based on homogeneous two-phase clearance flow considerations. In [5], Vasuthevan and Brümmer present thermodynamic modelling of a twin-screw expander in a trilateral flash cycle regarding homogeneous two-phase inlet, outlet, and clearance flows. In contrast, a separated flow model with liquid entrainment as defined by Smith in [6] is applied by Bell in [7] and by Lemort in [8] to determine inlet and outlet flows in scroll expanders. Moreover in [7], Bell implements a constant two-phase discharge coefficient of 0.77 in order to predict the flow losses. In [9], Huagen et al. present a two-phase clearance flow model based on the slip model with liquid entrainment according to [6] and use a discharge flow model as proposed by Lin in [10] to calculate indicator diagrams of an oil-injected twin-screw compressor. In [11], a comprehensive flow path modelling in a water-lubricated air twin-screw compressor is conducted applying different two-phase models to each clearance and opening. The following study deals with an approach for simulations of water-flooded twin-screw machines. In this context, a two-phase model according to humid air [12] proposed for the calculation of the thermodynamic properties of the working fluid is presented. Moreover for calculation of pressure- driven two-phase mass flow rates according to [13], assumptions as with the modelling of the discharge behaviour of clearances and openings as well as liquid distribution in the working chamber during expander operation are made. To verify the presented two-phase approach, multi-chamber simulations of the water-flooded twin- screw expander prototype SE 51.2 [14–16] are performed by means of the simulation tool KaSim [17]. The simulation results are compared to experimental data according to [15, 16] by means of expander (two-phase) mass flow rate, internal (simulated) and indicated (measured) power, pressure depending on the working chamber volume, as well as outlet temperature. 2. Thermodynamic properties of air-water two-phase mixtures The following section addresses the fundamentals for the calculation of the thermodynamic state of the working fluid in water-flooded twin-screw machines. In this context, the humid air concept [12] is proposed considering a mixture of ideal fluids in thermodynamic (thermal) equilibrium. The two-phase mixture consists of the incompressible fluid water in liquid or solid state of matter, its corresponding gaseous phase steam, and dry air. The thermodynamic fluid properties with respect to the humid air model are listed in table A1 (Appendix). Density of the incompressible phase—water or ice—is considered constant. Moreover, within the relevant pressure and temperature range of expander operation, the specific heat capacity of each fluid is assumed constant, too. The composition of a water-air mixture is specified by water load 𝑋 —as commonly used in the context of humid air—representing the ratio of water mass 𝑚 to dry-air mass 𝑚 : 𝑤 𝑑𝑎 𝑋 ∶= . (1) 𝑑𝑎 Since in this work a steady-state process is considered, dry air mass 𝑚 and water mass 𝑚 in 𝑑𝑎 𝑤 equation (1) can be substituted by the corresponding mass flow rates 𝑚 ̇ and 𝑚 ̇ with regard to two- 𝑑𝑎 𝑤 phase flows. Hence, water load 𝑋 can be expressed as follows: 𝑚 ̇ 𝑋 ∶= . (2) 𝑚 ̇ 𝑑𝑎 By definition, water mass 𝑚 and mass flow rate 𝑚 ̇ consider the entire amount of water molecules in 𝑤 𝑤 liquid, solid, and gaseous state of matter in the two-phase mixture. As schematically illustrated in figure 1, the volume 𝑉 of the air-water mixture is composed of the sub-volumes of dry air 𝑉 , water 𝑑𝑎 steam 𝑉 , and the incompressible phase of water 𝑉 (or ice for temperatures below the triple-point 𝑤𝑠 𝑙𝑖𝑞 temperature of water 𝑇 = 273.16 K): 𝑡𝑟 2 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑉 = 𝑉 + 𝑉 = 𝑉 + 𝑉 + 𝑉 . (3) 𝑔 𝑙𝑖𝑞 𝑑𝑎 𝑤𝑠 𝑙𝑖𝑞 A phase boundary between the volumes of the incompressible and the gaseous phase is present, whereas dry air and steam share the same volume. Considering the gaseous phase in the humid air mixture consisting of dry air and water steam, the overall static pressure 𝑝 of the mixture equals the sum of the partial pressures 𝑝 of the single gases in the mixture as expressed by Dalton’s law [12]: 𝑝 = ∑ 𝑝 . (4) 𝑗 =1 Partial pressure 𝑝 of any gaseous component is related to the gaseous volume 𝑉 . Taking the example 𝑗 𝑔 of humid air, the sum of the partial pressure of dry air 𝑝 and water steam 𝑝 provides the overall 𝑑𝑎 𝑤𝑠 static pressure 𝑝 . At the same time, the overall static pressure 𝑝 imposed on the surface of the incompressible phase represents the pressure of the two-phase mixture. As for ideal gases, the partial pressure of each gaseous fluid can be expressed as follows: 𝑚 ∙ 𝑅 ∙ 𝑇 𝑗 𝑗 𝑝 ∶= . (5) A mass-specific gas constant 𝑅 is assigned to each gaseous fluid in the mixture. The maximum mass of water steam 𝑚 in the mixture is related to saturated humid air and can be calculated from the 𝑤𝑠 ,𝑡𝑠𝑎 ideal gas law: 𝑝 ∙ 𝑉 𝑤𝑠 ,𝑡𝑠𝑎 𝑔 𝑚 = . (6) 𝑤𝑠 ,𝑡𝑠𝑎 𝑅 ∙ 𝑇 𝑤𝑠 Here, the saturation pressure of water steam 𝑝 is a function of the temperature and can be 𝑤𝑠 ,𝑡𝑠𝑎 calculated, for instance, by means of the Antoine equation [12] or as proposed in [18]. Additionally, the saturation pressure 𝑝 of water steam depends on the overall static pressure 𝑝 , if solubility of steam 𝑤𝑠 ,𝑡𝑠𝑎 molecules in the liquid phase are taken into account [18]. In the following considerations, this effect is neglected. Figure 1. Volume 𝑉 of the two-phase mixture containing the sub-volumes of dry air 𝑉 , water steam 𝑉 , and liquid water 𝑉 . 𝑑𝑎 𝑤𝑠 𝑙𝑖𝑞 ∆ International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 If less water mass 𝑚 than the maximum mass of water steam 𝑚 at given temperature 𝑤 ,𝑠𝑎𝑡 is available in the humid air mixture, partial pressure 𝑝 is lower than the saturation pressure 𝑝 of water steam. This case can be quantified by means of relative humidity 𝜑 dividing ,𝑠𝑎𝑡 the partial pressure 𝑝 by the saturation pressure of water steam 𝑝 : ,𝑠𝑎𝑡 𝑤𝑠 𝜑 ∶= . (7) 𝑤𝑠 ,𝑡𝑠𝑎 For a relative humidity in the range of 0 < 𝜑 ≤ 1, all water molecules in the mixture exist as steam. For a relative humidity of 100 % (𝜑 = 1), saturation water load 𝑋 is calculated as follows: 𝑡𝑠𝑎 𝑚 𝑅 𝑝 𝑤𝑠 ,𝑡𝑠𝑎 𝑑𝑎 𝑤𝑠 ,𝑡𝑠𝑎 𝑋 = = ∙ . (8) 𝑡𝑠𝑎 𝑚 𝑅 𝑝 − 𝑝 𝑑𝑎 𝑤𝑠 𝑤 𝑠 ,𝑡𝑠𝑎 By analogy with water load 𝑋 , the composition of the two-phase water-air mixture can be expressed by means of mass dryness fraction 𝑥 relating the mass 𝑚 of the gaseous humid air phase consisting of dry air and water steam to the mass 𝑚 of the mixture as follows: 𝑚 + 𝑚 𝑑𝑎 𝑤𝑠 𝑥 ∶= = . (9) 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 As with two-phase mass flow rate calculations discussed later in section 3.3, mass dryness fraction 𝑥 is referred to the ratio of the gaseous mass flow rate 𝑚 ̇ to the mass flow rate 𝑚 ̇ representing the sum of the single mixture components dry air and water: 𝑚 ̇ 𝑚 ̇ + 𝑚 ̇ 𝑔 𝑑𝑎 𝑤𝑠 (10) 𝑥 ∶= = . 𝑚 ̇ 𝑚 ̇ + 𝑚 ̇ 𝑑𝑎 𝑤 From equation (1), equation (8), and equation (9), mass dryness fraction can be expressed as follows: 1 + 𝑋 𝑡𝑠𝑎 (11) 𝑥 = . 1 + 𝑋 In contrast to the conventional consideration of the triple point of water as a zero energy level, the energy conservation equations regarding (specific) enthalpy and internal energy of humid air are derived with respect to the absolute zero from thermodynamic point of view (0 K). This approach provides throughout positive values for the extensive state variables internal energy and enthalpy as required by the simulation environment in this study. In contrast to internal energy and enthalpy, entropy of the mixture is related to the triple-point temperature of water, since entropy supports the calculations of isentropic change in state as an auxiliary state variable allowing negative values. For better understanding of the two-phase approach using the absolute zero temperature of 0 K as a zero energy level, specific aspects considering change in aggregate state of water are explained. In figure 2, the correlation between temperature and pressure as well as the different states of matter— steam, liquid, and ice—in terms of specific enthalpy are schematically illustrated. This can be transferred to specific internal energy too. In the region of two aggregate states—liquid water and steam (𝑇 > 𝑇 ) or ice and steam (𝑇 < 𝑇 )— 𝑡𝑟 𝑡𝑟 pressure is a function of temperature. The boundaries of the two-phase region correspond to a saturated state of the gaseous phase. With regard to the triple point, all three aggregate states are present. Here, 𝑡𝑟 𝑡𝑟 𝑡𝑟 the difference of the specific enthalpies (∆ℎ , ∆ℎ , and ∆ℎ ), in the literature also referred to as 𝑠𝑢 𝑒𝑣 𝑚𝑒 𝑤𝑠 𝑤𝑠 𝑤𝑠 𝑤𝑠 𝑤𝑠 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 latent heats, of two saturated aggregate states define the corresponding specific enthalpy required for a complete change in aggregate state. Outside the two-phase region, only one fluid state of matter exists. The reference condition providing enthalpies for change in state of matter is randomly set in the triple point of water. In general, these energies could be determined at any valid temperature for the transition between two aggregate states. overheated steam saturated steam liquid + steam ice + steam ice liquid temperature [K], pressure [Pa] Figure 2. Specific enthalpy ℎ of water as a function of pressure 𝑝 and temperature 𝑇 . As illustrated in figure 2, specific enthalpy can be calculated by addition of different specific enthalpy levels beginning from the absolute zero temperature. For instance, specific enthalpy of liquid 𝑡𝑟 water ℎ equals the sum of specific enthalpy ℎ of ice, specific melting enthalpy ∆ℎ , and the 𝑙𝑖𝑞 ,1 𝑖𝑐𝑒 ,𝑡𝑟 difference of specific enthalpy of water at temperature 𝑇 and triple-point temperature 𝑇 . Specific 1 𝑡𝑟 𝑡𝑟 melting enthalpy ∆ℎ represents the energy released during freezing of liquid water or required to bring water from solid to liquid aggregate state in the triple point. In the same manner with reference to the thermodynamic absolute zero point, specific enthalpy ℎ of steam can be calculated considering 𝑤𝑠 ,1 the specific enthalpy of ice ℎ at triple-point temperature 𝑇 , the specific sublimation enthalpy 𝑖𝑐𝑒 ,𝑡𝑟 𝑡𝑟 𝑡𝑟 ∆ℎ , and the difference of specific enthalpy ℎ and ℎ of steam at temperature 𝑇 and triple- 𝑤𝑠 ,1 𝑤𝑠 ,𝑡𝑟 1 point temperature 𝑇 . 𝑡𝑟 In the following, the calculation of specific volume, internal energy, enthalpy, and entropy of the two-phase air-water mixture are presented. All fluid specific parameters and enthalpies in terms of change in aggregate state of water can be obtained from table A1 (Appendix). 2.1. Specific volume Specific volume 𝑣 of the two-phase mixture is defined as the ratio of the volume 𝑉 occupied by the 1+𝑥 two-phase mixture divided by the mass of dry air 𝑚 : 𝑑𝑎 -1 specific enthalpy [J∙kg ] 𝑠𝑢 𝑚𝑒 𝑚𝑒 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑣 ∶= . (12) 1+𝑥 𝑑𝑎 The specific volume 𝑣 referred to the entire mass 𝑚 of the two-phase mixture equals: 𝑉 𝑉 −1 𝑣 ∶= = = 𝑣 ∙ (1 + 𝑋 ) . (13) 1+𝑥 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 As illustrated in figure 1, the entire fluid volume 𝑉 as in equation (3) consists of the partial volumes of the incompressible phase of water as well as the gaseous phase including dry air and steam. Combining equation (3) and equation (12), for saturated humid air (𝑋 > 𝑋 ) and temperatures above the triple 𝑡𝑠𝑎 point of water (𝑇 > 𝑇 ), specific volume 𝑣 of the two-phase fluid equals on a pro rata basis the sum 𝑡𝑟 1+𝑥 of the single specific volumes of each mixture component: 𝑇 𝑇 ( ) ( ) 𝑣 = 𝑣 + 𝑋 ∙ 𝑣 + 𝑋 − 𝑋 ∙ 𝑣 = 𝑅 ∙ + 𝑋 ∙ 𝑅 ∙ + 𝑋 − 𝑋 ∙ 𝑣 . (14) 1+𝑥 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑡𝑠𝑎 𝑙𝑖𝑞 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑡𝑠𝑎 𝑙𝑖𝑞 𝑝 𝑝 Here for the incompressible phase, specific volume 𝑣 of liquid water is considered. In contrast, at 𝑙𝑖𝑞 temperatures below the triple point (𝑇 < 𝑇 ), specific volume 𝑣 of the two-phase fluid is regarded 𝑡𝑟 1+𝑥 to ice specific volume 𝑣 : 𝑖𝑐𝑒 𝑇 𝑇 ( ) 𝑣 = 𝑅 ∙ + 𝑋 ∙ 𝑅 ∙ + 𝑋 − 𝑋 ∙ 𝑣 . (15) 1+𝑥 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑝 𝑝 For single-phase humid air (𝑋 ≤ 𝑋 ), no incompressible phase is present and the expression for 𝑡𝑠𝑎 specific volume 𝑣 reduces to: 1+𝑥 𝑇 𝑇 𝑣 = 𝑅 ∙ + 𝑋 ∙ 𝑅 ∙ . (16) 1+𝑥 𝑑𝑎 𝑤𝑠 𝑝 𝑝 2.2. Internal energy Internal energy 𝑈 of the two-phase fluid is composed of the internal energies of dry air and water: 𝑈 = 𝑈 + 𝑈 = 𝑚 ∙ 𝑢 + 𝑚 ∙ 𝑢 . (17) 𝑑𝑎 𝑤 𝑑𝑎 𝑑𝑎 𝑤 𝑤 Here, 𝑢 and 𝑢 are the specific internal energies of dry air and water respectively. On the one hand, 𝑑𝑎 𝑤 referring internal energy 𝑈 of the mixture to the mass of dry air 𝑚 only, specific internal energy 𝑢 , 𝑑𝑎 1+𝑥 as commonly used in the literature, is specified: 𝑢 ∶= = 𝑢 + 𝑋 ∙ 𝑢 . (18) 1+𝑥 𝑑𝑎 𝑤 𝑑𝑎 On the other hand, relating internal energy 𝑈 to the entire mass 𝑚 of the mixture, specific internal energy 𝑢 of the two-phase mixture defined as 𝑈 𝑈 𝑢 ∶= = (19) 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 can be calculated from equation (18): 6 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 −1 (20) 𝑢 = 𝑢 ∙ (1 + 𝑋 ) . 1+𝑥 With regard to the thermodynamic absolute zero (𝑇 = 0 K), specific internal energy 𝑢 of saturated 1+𝑥 and unsaturated humid air (𝑋 ≤ 𝑋 ) can be obtained from the specific internal energies of dry air and 𝑡𝑠𝑎 water steam: 𝑡𝑟 𝑡𝑟 𝑢 = 𝑐 ∙ 𝑇 + 𝑋 ∙ {𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ (𝑣 − 𝑣 ) + 𝑐 ∙ (𝑇 − 𝑇 )}. (21) 1+𝑥 𝑣 ,𝑑𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑤𝑠 𝑖𝑐𝑒 𝑣 ,𝑤𝑠 𝑡𝑟 For temperatures above the triple-point temperature of water (𝑇 > 𝑇 ), specific internal energy 𝑢 of 𝑡𝑟 1+𝑥 the two-phase fluid can be calculated with reference to saturated humid air and liquid water in the mixture (𝑋 > 𝑋 ): 𝑡𝑠𝑎 𝑡𝑟 𝑡𝑟 ( ) ( ) 𝑢 = 𝑐 ∙ 𝑇 + 𝑋 ∙ {𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ 𝑣 − 𝑣 + 𝑐 ∙ 𝑇 − 𝑇 } + 1+𝑥 𝑣 ,𝑑𝑎 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑤𝑠 𝑖𝑐𝑒 𝑣 ,𝑤𝑠 𝑡𝑟 (22) 𝑡𝑟 ( ) { ( )} 𝑋 − 𝑋 ∙ 𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ (𝑣 − 𝑣 ) + 𝑐 ∙ 𝑇 − 𝑇 . 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑙𝑖𝑞 𝑖𝑐𝑒 𝑙𝑖𝑞 𝑡𝑟 𝑡𝑟 In equation (22), 𝑣 is referred to temperature and pressure in the triple point of water. For 𝑤𝑠 temperatures below the triple point (𝑇 < 𝑇 ), specific internal energy 𝑢 is determined as follows: 𝑡𝑟 1+𝑥 𝑡𝑟 𝑡𝑟 ( ) ( ) 𝑢 = 𝑐 ∙ 𝑇 + 𝑋 ∙ {𝑐 ∙ 𝑇 + ∆ℎ − 𝑝 ∙ 𝑣 − 𝑣 + 𝑐 ∙ 𝑇 − 𝑇 } + 1+𝑥 𝑣 ,𝑑𝑎 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑡𝑟 𝑡𝑟 𝑤𝑠 𝑖𝑐𝑒 𝑣 ,𝑤𝑠 𝑡𝑟 (23) ( ) 𝑋 − 𝑋 ∙ 𝑐 ∙ T. 𝑡𝑠𝑎 𝑖𝑐𝑒 2.3. Enthalpy Enthalpy 𝐻 of the two-phase mixture results from internal energy 𝑈 and the product of pressure 𝑝 and volume 𝑉 : 𝐻 ∶= 𝑈 + 𝑝 ∙ 𝑉 . (24) Relating enthalpy 𝐻 to the mass of dry air 𝑚 , specific enthalpy ℎ of the two-phase mixture can be 𝑑𝑎 1+𝑥 obtained from equation (24) as follows: ℎ ∶= = 𝑢 + 𝑝 ∙ 𝑣 . (25) 1+𝑥 1+𝑥 1+𝑥 𝑑𝑎 Specific enthalpy ℎ considering the entire mass 𝑚 of the two-phase mixture can be determined as follows: 𝐻 𝐻 −1 ℎ ∶= = = ℎ ∙ (1 + 𝑋 ) . (26) 1+𝑥 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 2.4. Entropy Entropy 𝑆 of the two-phase fluid is the sum of the entropies of each component in the mixture also including entropy of mixing of the gaseous fluids. Dividing entropy 𝑆 by the mass 𝑚 of dry air, 𝑑𝑎 specific entropy 𝑠 is defined as follows: 1+𝑥 𝑠 ∶= . (27) 1+𝑥 𝑑𝑎 Specific entropy 𝑠 related to the entire mass 𝑚 of the two-phase mixture is introduced as: 𝑠𝑢 𝑚𝑒 𝑠𝑢 𝑠𝑢 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑆 𝑆 −1 𝑠 ∶= = = 𝑠 ∙ (1 + 𝑋 ) . (28) 1+𝑥 𝑚 𝑚 + 𝑚 𝑑𝑎 𝑤 In contrast to specific internal energy 𝑢 and specific enthalpy ℎ , specific entropy 𝑠 is 1+𝑥 1+𝑥 1+𝑥 calculated with regard to the triple-point temperature of water 𝑇 . When calculating the specific entropy 𝑡𝑟 𝑠 of the two-phase fluid, a distinction must again be made between unsaturated and saturated humid 1+𝑥 air as well as for liquid water or ice as incompressible fluid [12]. Considering saturated and unsaturated humid air (𝑋 ≤ 𝑋 ) only, following expression is applicable: 𝑡𝑠𝑎 𝑇 𝑝 ( ) 𝑠 = (𝑐 + 𝑋 ∙ 𝑐 ) ∙ 𝑙𝑛 ( ) − 𝑅 + 𝑋 ∙ 𝑅 ∙ 𝑙𝑛 ( ) + 1+𝑥 𝑝 ,𝑑𝑎 𝑝 ,𝑤𝑠 𝑑𝑎 𝑤𝑠 𝑇 𝑝 𝑡𝑟 𝑡𝑟 (29) 𝑡𝑟 ∆ℎ 𝑋 ∙ + ∆ 𝑠 (𝑋 ). 𝑡𝑟 Here, ∆ 𝑠 (𝑋 ) is referred to as specific entropy of mixing of ideal gases with respect to dry air and steam and can be calculated as follows: 𝑅 𝑅 𝑅 𝑅 𝑑𝑎 𝑑𝑎 𝑑𝑎 𝑑𝑎 ∆ 𝑠 (𝑋 ) = 𝑅 ∙ {( + 𝑋 ) ∙ 𝑙𝑛 ( + 𝑋 ) − 𝑋 ∙ 𝑙𝑛 (𝑋 ) − ∙ 𝑙𝑛 ( ) } . (30) 𝑤𝑠 𝑅 𝑅 𝑅 𝑅 𝑤𝑠 𝑤𝑠 𝑤𝑠 𝑤𝑠 In terms of saturated humid air (𝑋 > 𝑋 ) at temperatures above the triple point (𝑇 > 𝑇 ), specific 𝑡𝑠𝑎 𝑡𝑟 entropy 𝑠 of the two-phase fluid can be determined from: 1+𝑥 𝑇 𝑝 𝑠 = (𝑐 + 𝑋 ∙ 𝑐 ) ∙ 𝑙𝑛 ( ) − (𝑅 + 𝑋 ∙ 𝑅 ) ∙ 𝑙𝑛 ( ) + 1+𝑥 𝑝 ,𝑑𝑎 𝑡𝑠𝑎 𝑝 ,𝑤𝑠 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑇 𝑝 𝑡𝑟 𝑡𝑟 (31) 𝑡𝑟 ∆ℎ 𝑇 ( ) ( ) 𝑋 ∙ + ∆ 𝑠 𝑋 + 𝑋 − 𝑋 ∙ 𝑐 ∙ 𝑙𝑛 ( ) . 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑙𝑖𝑞 𝑇 𝑇 𝑡𝑟 𝑡𝑟 At temperatures below the triple point (𝑇 < 𝑇 ), specific entropy 𝑠 is calculated using the 𝑡𝑟 1+𝑥 expression: 𝑇 𝑝 ( ) 𝑠 = (𝑐 + 𝑋 ∙ 𝑐 ) ∙ 𝑙𝑛 ( ) − 𝑅 + 𝑋 ∙ 𝑅 ∙ 𝑙𝑛 ( ) + 1+𝑥 𝑝 ,𝑑𝑎 𝑡𝑠𝑎 𝑝 ,𝑤𝑠 𝑑𝑎 𝑡𝑠𝑎 𝑤𝑠 𝑇 𝑝 𝑡𝑟 𝑡𝑟 (32) 𝑡𝑟 𝑡𝑟 ∆ℎ ∆ℎ 𝑇 𝑋 ∙ + ∆ 𝑠 (𝑋 ) − (𝑋 − 𝑋 ) ∙ { − 𝑐 ∙ 𝑙𝑛 ( ) }. 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑡𝑠𝑎 𝑖𝑐𝑒 𝑇 𝑇 𝑇 𝑡𝑟 𝑡𝑟 𝑡𝑟 3. Two-phase chamber model simulation Chamber model simulations are commonly applied in the theoretical analysis of positive displacement machines considering one or more cyclically changing working chambers [19]. Including the law of conservation of mass and energy, a numerical solver based on the time-step method—KaSim—solves the thermodynamic and fluid-mechanical equations of the multi-chamber model [17]. For the following theoretical analyses in this work, KaSim is adapted to the requirements of two-phase simulations. 3.1. Two-phase initialisation in KaSim For purposes of simplicity with respect to chamber model simulations, it is assumed that spatial gradients in the intensive state variables within a fluid capacity are insignificant. The two-phase fluid state is defined by means of the time-dependent extensive state variables internal energy 𝑈 (𝑡 ) neglecting kinetic and potential energy, volume 𝑉 (𝑡 ) of the fluid capacity, and mass 𝑚 (𝑡 ) together with mass dryness fraction 𝑥 (𝑡 ), figure 3. Additionally, enthalpy 𝐻 (𝑡 ) of the two-phase fluid is specified in order to reduce 𝑚𝑒 𝑒𝑣 𝑒𝑣 𝑒𝑣 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 the amount of time-consuming iterations during calculation of the intensive state variables pressure 𝑝 and temperature 𝑇 . In KaSim, fluid capacities (chambers, ports, pipes etc.) are initialised by means of pressure 𝑝 , temperature 𝑇 , capacity volume 𝑉 , and mass dryness fraction 𝑥 . rotor-tip (housing) mechanical thermal two-phase clearance (hc) energy energy fluid connection front clearance (fc) two-phase , , , fluid capacity inlet opening (io) , , outlet port , x , chamber 1 chamber 2 … chamber n mass flow rate inlet port Figure 3. Geometry abstraction of an exemplary twin-screw expander geometry with regard to a chamber model for the simulation tool KaSim [17] including selected two-phase fluid capacities and connections as well as parameters for their specification. The thermodynamic change in state of the fluid in a capacity results from transfer of, e.g. mechanical or thermal energy and mass. For example, mass exchange is performed using connections representing clearances or openings in the machine. Currently for connections affiliated to moving boundaries, only pressure-driven (Poiseuille) mass flow rates are considered neglecting the effect of Couette flow. Two- phase flow connections (clearances, inlets and outlets, leakage paths etc.) are specified by a cross- sectional area 𝐴 (𝑡 ), 𝑖𝑓𝑖𝑐𝑒𝑜𝑟 and 𝑧𝑧𝑙𝑒𝑛𝑜 discharge coefficients 𝐶 (𝑡 ), flow coefficients 𝛼 (𝑡 ), a connection-specific mass dryness fraction 𝜒 (𝑡 ), and a two-phase flow regime (ℎ𝑜𝑔𝑒𝑛𝑒𝑜𝑜𝑚𝑢𝑠 or with phase 𝑠𝑙𝑖𝑝 ), see figure 3. Each connection requires information input with regard to the two-phase flow regime. A ℎ𝑜𝑔𝑒𝑛𝑒𝑜𝑜𝑚𝑢𝑠 flow regime implies two-phase flows at equal phase flow velocities. The selection of a 𝑠𝑙𝑖𝑝 -flow allows for calculation of theoretical mass flow rates as for separated two-phase flows with different phase flow velocities represented by a so-called velocity slip ratio. Considering the discharge behaviour of the connections for two-phase flows, a distinction between 𝑧𝑛𝑜 𝑧 𝑙𝑒 and 𝑖𝑓𝑖𝑐𝑒𝑜𝑟 flows is 9 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 made. Two-phase flow losses in nozzle or orifice-like connections represented by the discharge coefficient 𝐶 (𝑡 ) are calculated depending on the fluid state of the connected capacities as proposed in [13]. According to equation (40), the flow coefficient 𝛼 (𝑡 ) provides a user-specific constant input reflecting wall friction in addition to the interphase friction loss as calculated by the two-phase nozzle discharge coefficient approach presented in [13]. During simulation, the discharge behaviour of a fluid connection remains fixed and no transition between the nozzle and orifice approach is allowed. To model liquid distribution in the fluid capacity, a connection-specific mass dryness fraction 𝜒 can be explicitly considered in terms of mass flow rate calculation: 𝜒 ∶= 𝑥 . (33) 𝑐𝑜𝑛 Connection mass dryness fraction 𝜒 can be specified constant or equal to mass dryness fraction 𝑥 of the two-phase fluid in the source capacity (capacity of higher pressure considering two connected fluid capacities). With respect to a constant 𝜒 , mass dryness fraction 𝑥 of the source capacity fluid must exhibit values between 0.01 and 0.9. Outside this limits, 𝜒 is dynamically initialised during simulation with mass dryness fraction 𝑥 of the source capacity fluid rather than using the initially specified constant value preventing capacity mass dryness fractions 𝑥 greater than unity or less than zero. 3.2. Change in state of the two-phase fluid In this study, adiabatic fluid capacities without heat flows over the control volume boundaries are considered. Hence, change in fluid state results either from change in capacity volume or mass and, correspondingly, enthalpy transfer during a simulation time step. The new internal energy 𝑈 of the (two-phase) fluid can be calculated from the initial state according to internal energy 𝑈 as follows: 𝑈 = 𝑈 + 𝑊 + ∑ 𝐻 . 1 0 𝑘 (34) Here, 𝑊 indicates the work performed in terms of volume change and 𝐻 represents enthalpy of each exchanged fluid element over the control volume boundaries. Assuming isentropic volume change, work is calculated as follows: 𝑊 = − ∫ = 𝑈 − 𝑈 . (35) 0 1,𝑠 At constant initial entropy 𝑆 and water load 𝑋 during volume change, internal energy 𝑈 of humid 0 0 1,𝑠 air is iteratively calculated from the new intensive state variables pressure 𝑝 and temperature 𝑇 : 1 1 𝑈 = 𝑈 (𝑉 , 𝑝 , 𝑇 , 𝑋 , 𝑆 ). (36) 1,𝑠 1 1 1 0 0 If the thermodynamic state of water in the two-phase mixture after change in volume corresponds to the triple-point state, internal energy of the two-phase fluid is calculated by a linear interpolation between internal energy according to liquid state of water as in equation (22) and solid state according to equation (23). For a constant fluid capacity volume, the fluid state alters as a result of mass and corresponding enthalpy transfer. For two-phase flows, enthalpy 𝐻 of each exchanged fluid element is calculated from specific enthalpy ℎ according to equation (26) factored by the two-phase mass 𝑚 : 𝑘 𝑘 −1 𝐻 = 𝑚 ∙ ℎ = 𝑚 ∙ ℎ ∙ (1 + 𝑋 ) . (37) 𝑘 𝑘 𝑘 𝑘 1+𝑥 ,𝑘 𝑘 𝑝𝑑𝑉 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 Enthalpy 𝐻 of the fluid elements exchanged between two adjacent fluid capacities is related to the fluid capacity at higher pressure. Further, during balancing internal energy within a specific fluid capacity, elements flowing in are considered positive and such leaving the control volume negative. 3.3. Calculation of two-phase mass flow rate For the calculation of the two-phase mass flow rate, a two-phase separated flow (slip-flow) method as proposed in [13] is applied. Here, the mass flow rate 𝑚 ̇ results from a theoretical two-phase mass flow rate 𝑚 ̇ factored by a discharge coefficient 𝐶 including the entire flow losses: 𝑡 ℎ 𝑑 𝑚 ̇ = 𝐶 ∙ 𝑚 ̇ . (38) 𝑑 𝑡 ℎ The theoretical two-phase mass flow rate 𝑚 ̇ through restrictions is derived from the conservation laws 𝑡 ℎ of momentum and mass as presented amongst others by Chisholm in [20] and Morris in [21]. The discharge coefficient 𝐶 allows for the entire momentum loss and considers, by definition, a contraction coefficient 𝐶 depending on Mach number effects factored by the so-called coefficient of velocity 𝐶 𝑐 𝑣 including friction effects [21–24]: 𝐶 = 𝐶 ∙ 𝐶 . (39) 𝑑 𝑐 𝑣 For orifice flows, the discharge coefficient 𝐶 includes momentum losses only due to contraction effects (vena contracta), and the coefficient of velocity 𝐶 equals unity. The orifice approach (𝐶 = 𝐶 ) is 𝑣 𝑑 𝑐 preferred with respect to inlet and outlet connections in twin-screw expanders. For narrow flow sections or nozzle flows, no flow contraction is considered (𝐶 = 1), and the discharge coefficient primarily allows for momentum losses due to interphase and wall friction (𝐶 = 𝐶 ). This method for flow loss 𝑑 𝑣 calculation is applied to any clearance in the twin-screw-expander or leakage paths where no contraction of the two-phase flow jet is expected. Since the approach regarding two-phase discharge coefficients for nozzle flows presented in [13] reflects only interphase friction effects, in the current study the coefficient of velocity is factored by an empirical flow coefficient 𝑎 as follows: 𝐶 = 𝐶 ∙ 𝑎 . (40) 𝑣 𝑣 The coefficient of velocity 𝐶 is addressed to interphase friction as reported in [13] depending mainly on mass dryness fraction while the flow coefficient 𝑎 reflects wall friction effects. 4. Geometry of SE 51.2 The two-phase approach for the simulation of water-flooded twin-screw expanders proposed in this work is verified by means of existing experimental data of the twin-screw expander SE 51.2 including integral operational parameters as well as indicator diagrams presented in [14–16]. In figure 4, a 3D model of the test twin-screw expander prototype SE 51.2 including positions of the high-resolution pressure indication transmitters used to record indicator diagrams is illustrated. SE 51.2 is a twin-screw expander prototype without timing gears. The transmission of torque occurs directly via contact between the rotor flanks. The screw rotors are hardened and have a tough tungsten carbide/carbon (WC/C) wear-protection coating, so seizure is avoided in dry-running or water-flooded operation. Both fixed and loose bearing sets are grease lubricated, so no oil supply is necessary. Details about the expander geometry parameters are listed in table 1. 11 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 inlet port 0 2 3 4 axial outlet, inlet area curve volume curve outlet area curve floating bearing casing 100 10 inlet plate 90 9 (control edge) 80 8 0 1 2 3 4 5 6 70 7 coupling 60 6 casing 50 5 40 4 30 3 rotor casing 20 2 10 1 male rotor 0 0 100 200 300 400 500 600 700 800 female rotor male rotor rotational angle [ ] fixed bearing casing Figure 4. Positions of pressure indication transmitters (0…6), inlet and outlet area, as well as volume curve of the test twin-screw expander SE 51.2 including ranges of male rotor rotational angle corresponding to each pressure transmitter [15, 16]. Table 1. Geometry parameters of the test twin-screw expander SE 51.2. designation unit male rotor (mr) female rotor (fr) number of lobes 𝑧 [-] 3 5 diameter [mm] 71.8 67.5 rotor lead [mm] 181.8 303 wrap angle [°] 200 -120 rotor length [mm] 101 rotor profile [-] modified asymmetric SRM axis-centre distance [mm] 51.2 internal volume ratio [-] 2.5 displaced volume per male rotor rotation [cm ] 286 front clearance height ℎ (high pressure) [mm] 0.1 𝑓𝑐 ,ℎ𝑝 front clearance height ℎ (low pressure) [mm] 0.17 𝑓𝑐 ,𝑙𝑝 rotor-tip (housing) clearance height ℎ [mm] 0.08 ℎ𝑐 5. Simulation setup In the following section, the multi-chamber model of the twin-screw expander SE 51.2 including specific input with respect to the modelling of liquid distribution and the boundary conditions for the simulations are presented. 5.1. Multi-chamber model of SE 51.2 The multi-chamber model used for the simulation of the twin-screw expander SE 51.2 is illustrated in figure 5. The working chambers are numbered consecutively beginning from the low pressure domain. Each working chamber is divided into two fluid sub-capacities belonging to the male and female rotor. Pressure balance connections provide equal pressures in two corresponding male and female sub- capacities during each calculation time step. Nevertheless, different temperatures and mass dryness fractions of these sub-capacities result after pressure balancing. chamber volume [cm ] inlet and outlet area [cm ] International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 high pressure (hp) domain fluid connections inlet (hp) F 8 outlet (lp) pressure balance connection (lp) M 6 F 7 pressure balance connection (hp) M 5 F 6 blow hole (lp) blow hole (hp) M 4 F 5 housing clearance M 3 F 4 front and interlobe clearances M 2 F 3 are not illustrated M 1 F 2 M .. male rotor chamber F .. secondary rotor chamber F 1 subsequent chamber indication low pressure (lp) domain Figure 5. Multi-chamber model (adiabatic, front and interlobe clearances not illustrated) of SE 51.2 containing fluid capacities and connections. In terms of flow loss calculation, as presented in table 2, all clearances of SE 51.2 are initialised with respect to nozzle discharge coefficient including the coefficient of velocity factored by a user-specified flow coefficient according to equation (40). Inlets and outlets are modelled as orifice flow restrictions according to equation (39) considering only contraction effects. Table 2. Specific parameters for the initialisation of the chamber model of SE 51.2 with respect to two- phase flow connections. flow discharge dryness fraction flow coefficient connection regime behaviour 𝝌 𝒂 inlet opening (hp) slip orifice 𝜒 = 𝑥 1.0 𝑖𝑜 outlet opening (lp) slip orifice 1.0 𝜒 = 𝑥 𝜒 = 𝑥 rotor-tip clearance (mr) slip nozzle 1.0 ℎ𝑐 , rotor-tip clearance (fr) slip nozzle 𝜒 = 𝑥 0.8 ℎ𝑐 ,𝑓𝑟 blow hole slip nozzle 𝜒 = 𝑥 1.0 𝑏 ℎ interlobe clearance slip nozzle 1.0 𝜒 = 0.9 𝑖𝑐 𝜒 = 0.9 front clearance slip nozzle 0.8 𝑓𝑐 Due to the complexity of predicting the liquid distribution in the working chamber and thus at the clearances of twin-screw machines, no holistic models are available in the literature. Therefore, assumptions have to be made. On the one hand, the liquid flows in radial direction as a result of 𝑚𝑟 𝑜𝑜 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 centrifugal forces where it spreads over the housing surface. On the other hand, Couette flows additionally influence the amount of liquid directly at the clearance. Due to centrifugal forces, it could be assumed that all clearances except blow holes and rotor-tip clearances remain relatively dry and are not significantly sealed by the liquid. Therefore, high mass dryness fractions are expected there. For the calculation of the pressure-driven two-phase clearance mass flow rate, rotor-tip clearance and blow hole mass dryness fraction is recommended to be initialised by values equal to those of the chamber fluid or less. In this work, the liquid distribution in the working chamber is modelled initialising mass dryness fraction of each clearance as proposed in table 2. For example, housing clearances and blow holes are initialised with a mass dryness fraction related to the source chamber value. In contrast, interlobe and front (axial) clearances are considered relatively “dry” according to a dryness faction of 0.9. These assumptions are supported, in particular, by video recordings of the working cycle (not presented here). 5.2. Simulation boundary conditions The simulations of the operational behaviour of SE 51.2 are performed according to the experimental investigations as presented in [16]. The range of pressure, temperature, inlet and outlet mass dryness fraction, and male rotor tip-speed respectively for simulation and comparison with experimental data are listed in table 3. Table 3. Boundary conditions for the simulation of SE 51.2. parameter value -1 maximum male rotor tip-speed 𝑢 67.7 m∙s 5 5 5 3∙10 Pa, 4∙10 Pa, 5∙10 Pa inlet pressure 𝑝 (absolute) outlet pressure 𝑝 (absolute) 1∙10 Pa inlet temperature 𝜗 55 °C (𝑥 = 0.5), 90 °C (𝑥 = 1.0) 𝑖 𝑖 𝑖 0.5 (water-flooded), 1.0 (dry-running) inlet/outlet mass dryness fraction 𝑥 /𝑥 𝑖 𝑜 6. Results In the following section, the two-phase approach for simulation of water-flooded twin-screw machines is validated comparing chamber-model simulation results to experimental data for the expander SE 51.2. In this context, indicator diagrams, expander mass flow rates, experimentally determined indicated and simulated internal powers, as well as outlet temperatures are considered. Indicated power is calculated by means of the experimentally recorded chamber pressure during a working cycle and the corresponding chamber volume as follows: 𝑃 = − 𝑛 · 𝑧 · ∮ 𝑝 · . (41) 𝑖𝑛𝑑 First, calculation and experimental results for dry-running operation (𝑥 = 1) are presented in figure 6 in order to analyse the simulation model regardless of liquid effects. Mass flow rate as well as indicated and internal power are examined as a function of male rotor tip-speed 𝑢 at different inlet pressures 𝑝 . As expected, simulated and experimental mass flow rate and power increase at rising rotor speed and inlet pressure due to increasing number of working cycles and inlet density respectively. With regard to inlet throttling losses at increasing rotor tip-speed, both expander parameters exhibit a degressive progression due to declining chamber pressure during the filling process. Moreover, for a combination of high rotor speeds and low inlet pressures, overexpansion as observed in the indicator 5 -1 diagram for 𝑝 = 3∙10 Pa and 𝑢 = 60 m∙s negatively impacts the expander performance. 𝑚𝑟 𝑚𝑟 𝑚𝑟 𝑚𝑟 𝑑𝑉 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 0.16 -16 0.14 -14 0.12 -12 0.1 -10 0.08 -8 0.06 -6 0.04 -4 0.02 -2 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 -1 -1 male rotor tip-speed [m∙s ] male rotor tip-speed [m∙s ] [10 Pa] 3 4 5 = 90 °C experiment = 1 simulation 5.5 5.5 inlet control edge inlet control edge 5.0 5.0 4.5 4.5 4.0 4.0 -1 -1 = 3.8 m∙s = 52.8 m∙s 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 -6 3 -6 3 volume [10 m ] volume [10 m ] Figure 6. Expander mass flow rate, simulated internal and experimental indicated power, as well as indicator diagrams at different inlet pressures and rotor speeds for dry-running operation (𝑥 = 1). Deviations between simulation and experimental results are mainly attributed to either modelling of clearances and openings or dynamic effects during the experiments and uncertainties of the measurement. In figure 6, the greatest disagreement between simulation and experiment is observed for -1 5 5 increasing rotor tip-speeds 𝑢 > 30 m∙s at inlet pressure 𝑝 = 4∙10 Pa and 𝑝 = 5∙10 Pa. Here, the 𝑖 𝑖 simulated mass flow rate and internal power are lower than the experimental values. This difference can be traced back to the orifice discharge coefficients applied to the expander inlet during the simulation (see table 2). For single-phase flows, in contrast to nozzles approaching the theoretical mass flow rate, orifices are associated with flow losses due to contraction effects. As reported in [13], the inlet geometry -1 mass flow rate [kg∙s ] pressure [10 Pa] 5 internal/indicated power [kW] pressure [10 Pa] 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 based on the twin-screw expander SE 51.2 exhibits a transition from an orifice to nozzle discharge behaviour as the opening becomes slip-shaped with the end of the chamber filling. Therefore, as with inlet discharge coefficients related to orifices, the flow loss during chamber filling might be overestimated, since, in this study, the inlet discharge behaviour is fixed and cannot be changed during simulation progress. In the same context, the measured indicator diagrams illustrated for -1 𝑢 = 52.8 m∙s show higher working chamber pressure during volume increase compared to the simulation results. Additionally, as observed during chamber filling in the indicator diagrams and reported in [15], the pressure oscillation in the inlet port directly influence the chamber pressure. Hence in contrast to the chamber model simulation not considering dynamic fluid inertia effects, the experiments reveal higher fluid density in the working chamber during chamber filling. With regard to low rotor tip-speed in figure 6, in particular at 𝑝 = 3∙10 Pa, simulated expander mass -1 flow rate is slightly overpredicted. In the indicator diagrams for 𝑢 = 3.8 m∙s , the simulated chamber pressure during expansion is lower than the experimentally recorded. Along with a higher simulated than measured mass flow rate, this indicates overestimated internal leakages that could be affiliated with the interlobe clearance directly connecting high-pressure working chambers with chambers discharging into the low-pressure port of the expander. Due to, for example, thermal deformation or bearing clearance, the interlobe clearance could experience significant relative change in height compared to its initial design dimension. The interlobe clearance leakage permanently induces loss flows that reduce the chamber pressure and cannot be utilised at lower pressure in terms of chamber refilling as with housing and front clearances or blow holes. In figure 7, simulation and experimental results under water-flooded operating conditions (𝑥 = 0.5) are presented. Generally as a function of rotor speed and inlet pressure, two-phase mass flow rate as well as internal and indicated power exhibit similar characteristics as for dry-running operation in figure 6. The analysis of the water-flooded expander at increasing rotor tip-speed reveals simulated internal power slightly lower than the experimentally determined indicated power. Here again as with dry-running operation, this could be attributed to dynamic effects during the chamber filling or to the inlet area modelling with respect to flow losses. At the same time, for low rotor speeds and increasing inlet pressure, the simulation provides expander two-phase mass flow rates slightly less than the experimentally recorded. The maximum deviation is in the range of 12 % at the lowest rotor tip-speed and highest inlet pressure investigated. In connection with the proposed liquid distribution in this study, this gives a rise to the assumption that a relevant internal leakage is underestimated. As the indicator diagrams show, during the internal expansion, an internal leakage path associated with chamber refilling might be the reason for this difference due to the initially higher and afterwards lower pressure levels compared to the measurements. In this context, the liquid sealing effect with respect to blow holes and the rotor-tip clearances might be overestimated proposing their initialisation as with the chamber mass dryness fraction. As recorded images of the working cycle (not presented here) indicate, liquid distribution at the rotor-tip clearances and blow holes dynamically changes during the expansion progress providing an evidence for the relevance of its appropriate modelling. Hence, the assumption as made within this study has to be considered only as an initial approach. However with the aim of reliable simulations of liquid-flooded twin-screw machines, a high potential for developing appropriate liquid-distribution models in future works exists. 𝑚𝑟 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 0.32 -16 0.28 -14 0.24 -12 0.2 -10 0.16 -8 0.12 -6 0.08 -4 0.04 -2 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 -1 -1 male rotor tip-speed [m∙s ] male rotor tip-speed [m∙s ] [10 Pa] 3 4 5 = 55 °C experiment = 0.5 simulation 5.5 5.5 inlet control edge inlet control edge 5.0 5.0 4.5 4.5 4.0 4.0 -1 -1 = 3.8 m∙s = 52.8 m∙s 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 -6 3 -6 3 volume [10 m ] volume [10 m ] Figure 7. Expander mass flow rate, simulated internal and experimental indicated power, as well as indicator diagrams at different inlet pressures and rotor speeds for dry-running operation (𝑥 = 0.5). Figure 8 illustrates the measured and simulated expander outlet temperature depending on rotor speed and inlet pressure. At this point it must be noted that, in contrast to the experimental data, the simulated temperatures do not include any losses as with hydraulic and mechanical friction. Basically, expander outlet temperature decreases due to declining impact of internal leakages at rising rotor speed, on the one hand, and higher utilisable pressure ratio related to increasing inlet pressure, on the other hand. In terms of dry-running operation (𝑥 = 1), the simulated temperatures deviate from the experimental values by maximum 20 K. As one reason for this result, the greater throttling loss as mentioned above is identified and, thus, lower pressure ratios are available for expansion associated -1 mass flow rate [kg∙s ] pressure [10 Pa] 5 internal/indicated power [kW] pressure [10 Pa] International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 with lower temperature drops. Applying a nozzle discharge coefficients to the inlet opening of SE 51.2, lower expander outlet temperatures are simulated attributed to higher chamber pressures after disconnecting the working chamber from the high-pressure port (not presented here). Hence, the maximum temperature difference declines to 12 K. Additionally, in the simulation, overestimated gaseous flows through the interlobe clearance bypass the working chamber and the relatively high temperatures of the leakage flow resulting from the corresponding isenthalpic change in state contribute to a further temperature increase downstream of the expander. With regard to water-flooded operation (𝑥 = 0.5), expander outlet temperatures are relatively accurately predicted by the simulation. Significant deviations are observed at an inlet pressure of 𝑝 = 5∙10 Pa and low rotor speeds -1 𝑢 < 30 m∙s . For two-phase flows, these can be explained by measurement uncertainties at different phase temperatures unlike the assumptions of thermal equilibrium as with the humid air model. [10 Pa] 3 4 5 = 90 °C = 55 °C experiment = 1 = 0.5 simulation 370 370 = 363.15 K 350 350 330 330 = 328.15 K 310 310 290 290 270 270 250 250 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 -1 -1 male rotor tip-speed [m∙s ] male rotor tip-speed [m∙s ] Figure 8. Expander outlet temperature as a function of male rotor tip-speed at different inlet pressures for dry-running (𝑥 = 1) and water-flooded (𝑥 = 0.5) operation. 𝑖 𝑖 7. Conclusion and outlook Within the framework of this study, a two-phase approach for simulation of water-flooded twin-screw machines is validated for expander applications. The thermodynamic properties of the two-phase mixture of dry air and water are represented by the humid air model considering both phases in thermal equilibrium. Applying a two-phase slip-flow flow model, mass flow rates are calculated factoring a theoretical mass flow rate by geometry-specific discharge coefficients reflecting flow losses as in nozzle and orifice restrictions. Liquid distribution in the expander working chamber directly influencing the two-phase clearance flows is modelled in a simple way using clearance-specific mass dryness fractions. Due to centrifugal forces, it is assumed that most of the liquid distributes towards the radial clearances, while the axial and interlobe clearances are associated with relatively low amount of liquid. In this context, radial rotor-tip clearances and blow holes are initialised with mass dryness fraction of the source fluid capacity and the remaining clearances are referred to a constant mass dryness fraction of 0.9. outlet temperature [K] outlet temperature [K] 𝑚𝑟 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 The single- and two-phase simulations reveal a relatively high degree of agreement with experimental data considering expander mass flow rate, internal power, outlet temperature, and chamber pressure depending on volume. At low rotor tip-speeds related to significant clearance flows, for both dry-running and liquid-flooded operation, simulated internal power matches the experimentally determined indicated power within a small range of deviation. Here for dry-running operation, the simulations exhibit a slight overprediction of clearance flows resulting in higher expander mass flow rates. Nevertheless, the comparison of simulated and measured chamber pressure reveals an appropriate prediction of internal leakages contributing to chamber refilling. Regarding the two-phase simulation, the assumed liquid distribution provides greater sealing effect indicated by underestimation of expander mass flow rates and lower chamber pressures during chamber refilling compared to the experimental data. At higher rotor speeds for both dry-running and liquid-flooded operation, simulated internal power and mass flow rate are underpredicted, in particular, due to dynamic effects during chamber filling not reflected by the simulation. Accordingly, higher outlet temperatures are simulated under dry-running conditions. In contrast, for water-flooded operation, deviations between simulated and experimental expander outlet temperatures are mainly traced back to uncertainties linked to two-phase temperature measurements. In the present work, the liquid distribution in the working chamber is identified as a major challenge for further development of the two-phase simulation approach. In this context, more accurate and detailed models for the initialisation of mass dryness fraction of the clearances of twin-screw machines for expander and compressor applications are required. Since at high rotor speeds the effect of two- phase Couette flow increases, it has to be considered too. Moreover in terms of inlet throttling, flow loss calculation could be improved with respect to transition between an orifice and nozzle discharge coefficient during simulation depending on the shape of the inlet geometry. Nomenclature symbol unit designation area 𝐴 𝑚 −1 ( ) specific heat capacity of the incompressible phase 𝑐 𝐽 ∙ ∙ 𝐾 𝐶 - (two-phase) contraction coefficient 𝐶 - calculated (two-phase) discharge coefficient −1 ( ) isobaric specific heat capacity 𝑐 𝐽 ∙ ∙ 𝐾 −1 𝑐 𝐽 ∙ ( ∙ 𝐾 ) isochoric specific heat capacity 𝐶 - (two-phase) coefficient of velocity −1 ℎ 𝐽 ∙ specific enthalpy 𝐻 𝐽 enthalpy 𝑚 mass of the two-phase mixture 𝑚 mass of dry air 𝑑𝑎 mass of water steam 𝑤𝑠 mass of liquid water 𝑙𝑖𝑞 −1 𝑚 ̇ ∙ 𝑠 mass flow rate of the two-phase mixture −1 𝑚 ̇ ∙ 𝑠 mass flow rate of dry air 𝑑𝑎 −1 𝑚 ̇ ∙ 𝑠 mass flow rate of water steam 𝑤𝑠 −1 𝑚 ̇ ∙ 𝑠 mass flow rate of liquid water 𝑙𝑖𝑞 −1 𝑛 rotational speed pressure 𝑃 𝑊 power −1 𝑅 𝐽 ∙ ( ∙ 𝐾 ) specific gas constant −1 𝑠 ( ) specific entropy 𝐽 ∙ ∙ 𝐾 −1 entropy 𝑆 𝐽 ∙ 𝐾 −1 time 𝑡 𝑠 𝑘𝑔 𝑘𝑔 𝑃𝑎 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 𝑘𝑔 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 𝑇 𝐾 temperature −1 𝑢 rotor tip-speed 𝑚 ∙ 𝑠 −1 𝑢 specific internal energy 𝐽 ∙ 𝑈 𝐽 internal energy 3 −1 𝑣 specific volume 𝑚 ∙ volume 𝑉 𝑚 work 𝑊 𝐽 - mass dryness fraction 𝑋 - water load 𝑧 - number of rotor lobes α - flow coefficient 𝜗 °𝐶 temperature 𝜑 - relative humidity 𝜒 - connection mass dryness fraction 𝜒 ≔ 𝑥 𝑐𝑜𝑛 abbreviation/index designation 0 initial fluid state 1 changed fluid state 1+x related to dry air mass bh blow hole ch chamber con connection da dry-air ev evaporation ex exchanged fc front clearance fr female rotor g gaseous hc rotor-tip (housing) clearance hp high pressure i inlet ic interlobe clearance ice ice ind indicated io inlet opening j index for a gas component in a gaseous fluid mixture k index for exchanged fluid element liq liquid (water) lp low pressure me melting mr male rotor o outlet oo outlet opening s isentropic sat saturation su sublimation th theoretical tr triple w water (liquid and steam) ws water steam 𝑘𝑔 𝑘𝑔 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. Series: Materials Science and Engineering 1267 (2022) 012020 doi:10.1088/1757-899X/1267/1/012020 Appendix A Table A1. Constant fluid properties for the thermodynamic state calculation of the two-phase fluid mixture in thermal equilibrium with regard to humid air [12]. designation unit value -1 -1 [J∙kg ∙K ] 287.05 specific gas constant of dry air 𝑅 𝑑𝑎 -1 -1 [J∙kg ∙K ] 461.5 specific gas constant of water steam 𝑅 𝑤𝑠 -1 -1 isobaric specific heat capacity of dry air 𝑐 [J∙kg ∙K ] 1004.6 𝑝 ,𝑑𝑎 -1 -1 isobaric specific heat capacity of water steam 𝑐 [J∙kg ∙K ] 1863 𝑝 ,𝑤𝑠 -1 -1 isochoric specific heat capacity of dry air 𝑐 [J∙kg ∙K ] 717.55 𝑣 ,𝑑𝑎 -1 -1 isochoric specific heat capacity of water steam 𝑐 [J∙kg ∙K ] 1401.5 𝑣 ,𝑤𝑠 -1 -1 liquid water specific heat capacity 𝑐 [J∙kg ∙K ] 4191 𝑙𝑖𝑞 -1 -1 ice specific heat capacity 𝑐 [J∙kg ∙K ] 2070 𝑖𝑐𝑒 3 -1 specific volume of liquid water 𝑣 [m ∙kg ] 0.001002 𝑙𝑖𝑞 3 -1 [m ∙kg ] 0.001087 specific volume of ice 𝑣 𝑖𝑐𝑒 [K] 273.16 triple temperature 𝑇 𝑡𝑟 [Pa] 611.657 triple pressure 𝑝 𝑡𝑟 -1 𝑡𝑟 [kJ∙kg ] 333.4 specific melting enthalpy ∆ℎ at 𝑇 𝑡𝑟 -1 𝑡𝑟 specific vaporisation enthalpy ∆ℎ at 𝑇 [kJ∙kg ] 2500.9 𝑡𝑟 -1 𝑡𝑟 specific sublimation enthalpy ∆ℎ at 𝑇 [kJ∙kg ] 2834.3 𝑡𝑟 References [1] Rane S, Kovačević A, Stošić N and Stupple G 2019 On Numerical Investigation of Water Injection to Screw Compressors Proceedings of the ASME International Mechanical Engineering Congress and Exposition 2018 ASME 2018 International Mechanical Engineering Congress and Exposition (Pittsburgh, Pennsylvania, USA, 09.11.2018 - 15.11.2018) (New York, N.Y.: The American Society of Mechanical Engineers) [2] Räbiger K, Maksoud T, Ward J and Hausmann G 2008 Theoretical and experimental analysis of a multiphase screw pump, handling gas–liquid mixtures with very high gas volume fractions Experimental Thermal and Fluid Science 32 1694–701 [3] Sangfors B 1984 Computer Simulation of the Oil Injected Twin Screw Compressor International Compressor Engineering Conference International Compressor Engineering Conference (Purdue, Indiana) (International Compressor Engineering Conference) ed Purdue University Libraries (Purdue, Indiana) [4] Sangfors B 1998 Computer Simulation of Effects From Injection of Different Liquids in Screw Compressors International Compressor Engineering Conference International Compressor Engineering Conference (Purdue) (International Compressor Engineering Conference) ed Purdue University Libraries [5] Vasuthevan H and Brümmer A 2016 Thermodynamic Modeling of Screw Expander in a Trilateral Flash Cycle 23rd International Compressor Engineering Conference at Purdue: West Lafayette, Indiana, USA, 11-14 July 2016 (Purdue) ed Purdue University Libraries [6] Smith S L 1969 Void Fractions in Two-Phase Flow: A Correlation Based upon an Equal Velocity Head Model Proceedings of the Institution of Mechanical Engineers 184 647–64 [7] Bell I 2011 Theoretical and Experimental Analysis of Liquid Flooded Compression in Scroll Compressors PhD thesis Purdue University [8] Lemort V December 19th, 2008 Contribution to the Characterization of Scroll Machines in Compressor and Expander Modes (Liège: Université de Liège, Belgique) [9] Huagen W, Ziwen X and Pengcheng S 2004 Theoretical and experimental study on indicator diagram of twin screw refrigeration compressor International Journal of Refrigeration 27 331–8 𝑠𝑢 𝑒𝑣 𝑚𝑒 International Conference on Screw Machines 2022 IOP Publishing IOP Conf. 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