Shock loss measurements in non-ideal supersonic flows of organic vapors

Non-ideal compressible fluid dynamics (NICFD) is a branch of gas dynamics that studies flows of dense vapors occurring in the close proximity of the vapor–liquid equilibrium and the critical point, so in conditions in which the ideal gas law does not properly describe the thermodynamics involved (Thompson 1971). Compressibility factor Z is defined in Eq. (1) where P is pressure, T is temperature, \(\rho\) is density and R is the gas constant.It represents the departure of the fluid volumetric behavior from that of an ideal gas. It is therefore identically equal to 1 in case of ideal gas behavior and possibly differs from 1 otherwise: It is thus an indication of the level of non-ideality. The term non-ideal in NICFD indeed refers to this latter aspect, which has a number of direct gas dynamic “side effects” distinctly separating such flows from those of ideal gases. For example, isentropic expansions show a non-ideal dependence on total conditions, analogously to shock waves, which no longer depend on the pre-shock Mach number only.

Non-ideal flows occur in a vast range of engineering processes, from rocket propulsion to industrial and chemical activities. Examples are the oil and gas, heat pumps and refrigeration fields, and even pharmaceuticals production with the use of the rapid expansion of supercritical solutions (Helfgen et al. 2003). In the power generation field, non-ideal flows occur in supercritical carbon dioxide (\(\text {sCO}_2\)) power cycles and are particularly important in organic Rankine cycles (ORCs), one of the most relevant application fields for NICFD. ORCs are preferred to conventional steam cycles when low to medium source temperature and low to medium power output are considered, thanks to their low cost, plant simplicity and thermodynamic efficiency. Fluids usually employed in ORCs feature high complexity and molecular weight, and turbine expansion occurs in the dense gas region near the saturation curve and the critical point. As a result, turbine flows are highly supersonic and show strong non-ideal flow effects (Romei et al. 2020), requiring accurate design tools accounting for these aspects in order to achieve high turbine efficiency, which in turn strongly impacts cycle efficiency (Colonna et al. 2015; Macchi and Astolfi 2016).

Comparison of numerical tools for design and analysis, ranging from preliminary loss correlations to full computational fluid dynamics (CFD) simulations, with experiment is relatively rare. This is because detailed experimental data characterizing non-ideal flows for ORC applications are currently not widely available in the open literature due to the intrinsic difficulties in running dedicated experimental facilities. ORC working fluids, like many others of interest in the NICFD field, are liquids at standard room temperature and pressure. Typical ORC inlet turbine flows are instead at saturated, superheated or supercritical conditions, with temperatures and pressures  in the range 100–\(400~^\circ \hbox {C}\) and 10–\(50~\text {bar}\) (Macchi and Astolfi 2016). Thus, in order to reproduce realistic conditions in a wind tunnel, a closed gas cycle or a phase transition thermodynamic cycle must be put in place (Spinelli et al. 2020). These are noticeably more complicated and expensive with respect to operation with incondensable gases such as air, where compressed air storage tanks or continuous loops are often sufficient to carry out an experimental campaign. Moreover, measurement procedures are also more complex due to the high fluid temperature involved and condensation issues in pneumatic lines. Also, the non-ideal flow field dependence on stagnation conditions significantly increases the number of flows that have to be experimentally reproduced for a complete characterization. Indeed, a polytropic ideal gas flow in a fixed geometry features trends in Mach number, pressure and temperature ratios that are independent of the actual total conditions and only depend on molecular complexity through the specific heat ratio \(\gamma\). In case of non-ideal flows instead, testing needs to cover the whole range of possible operating total temperature and pressure conditions.

Despite these difficulties, several active plants exist and are starting to provide valuable experimental data, mostly on relatively simple yet extremely useful geometries such as converging–diverging nozzles. These allow to reproduce elementary flows important for fundamental NICFD studies and are also the simplest geometry representative of blade passages in ORC turbines. Among these, so-called nozzle-fitted facilities is the Test Rig for Organic VApors (TROVA) (Spinelli et al. 2013) at the Laboratory of Compressible fluid dynamics for Renewable Energy Applications (CREA Lab) of Politecnico di Milano, where all the experimental campaigns concerning the present work were carried out. Other plants of this kind are the ORCHID (Head et al. 2016) at TU Delft, the CLOWT (Reinker et al. 2017) at Muenster University of Applied Sciences and the dense gas blowdown facility at Imperial College London (Robertson et al. 2020).

Several turbine-fitted facilities instead including all typical components of an organic Rankine cycle also exist, such as the LUT micro-ORC test rig at Lappeenranta - Lahti University of Technology (Turunen-Saaresti et al. 2017). The ORCHID at TU Delft is designed to also operate with a turbine instead of a nozzle, but testing to date was performed with the latter only. These test rigs are mainly devoted to performance measurement of the different components and of the overall thermodynamic cycle and are less suited to provide detailed flow information than nozzle-fitted facilities.

Due to the peculiarity of non-ideal vapor flows in ORCs, measurements such as velocity magnitude and direction, mass flow rate and turbine performance, which are routinely carried out in more standard cycles and turbomachinery (e.g., gas turbines operating with air and combustion gases), are not often performed yet. This is mainly because no appropriately calibrated instrumentation for non-ideal conditions is currently available. Indeed, none of the previously mentioned wind tunnels for non-ideal flows is routinely employed as a dedicated calibration facility for pressure probes.

Research efforts are now starting to move toward this direction. Results on the performance of a rotatable cylinder Pitot probe in high-subsonic flows with fluid \(\hbox {Novec}^{\text {TM}}\) 649 at the CLOWT plant were very recently presented (Reinker et al. 2020) as part of a preliminary study in order to establish measurement techniques for determination of Mach numbers in high-subsonic and transonic organic vapor flow fields.

The present authors also developed a pneumatic system for pressure probe measurements in flows of organic vapors in non-ideal conditions that sets the foundation for future directional pressure probes calibration and use in the characterization of such flows (Conti et al. 2022), and which is indeed the basis of the work here documented.

Considering the above, even blade cascade testing, quite common in the design process of gas and steam turbines, is instead significantly more complex in the case of non-ideal flows due to the previously mentioned difficulties in running dedicated wind tunnels and the lack of established probe measurement methodologies. Cascade testing is recently starting to take place for such flows. To the authors’ knowledge, the first experimental campaign of this kind was carried out at Whittle Laboratories of Cambridge University in a newly modified transient wind tunnel of Ludwieg tube type, where annular turbine cascade flows of R134a were characterized with pressure measurements (Baumgärtner et al. 2019). Wake measurements of R134a flows in the same cascade were taken with a wedge total pressure probe with substantial complementary use of CFD calculation. The latter was necessary to overcome the unavailability of a Mach number measurement upstream of the probe allowing to calculate shock losses at the probe tip in order to retrieve the pre-shock total pressure and evaluate cascade losses (Baumgärtner et al. 2020).

Linear blade cascade testing was recently carried out in the CLOWT facility at Muenster University in order to verify theoretical predictions of critical and choking Mach numbers of organic vapor flows (aus der Wiesche et al. 2021).

With the objective of measuring cascade losses to validate low- and high-fidelity turbine design tools, a linear cascade experiment representative of the stator operation of axial/radial turboexpanders for ORC systems will be concluded in the very near future at the TROVA wind tunnel at CREA Lab of Politecnico di Milano with siloxane MM (Manfredi et al. 2021).

Given the context above, the objective of the present research is twofold. A probe is inserted in non-ideal supersonic flow of siloxane MM (commonly employed in medium-/high-temperature ORCs) causing a detached bow shock locally normal to its tip. The first tests of this kind are here carried out to verify the correct functioning of the complete measurement system for direct shock loss measurements, with a view of establishing reliable experimental procedures. These will be fundamental for accurate measurements in non-ideal flows of particular interest for organic Rankine cycle applications, such as in blade cascades testing (Manfredi et al. 2021).

The experimental campaigns here reported also allow to directly characterize non-ideal shock losses at varying levels of non-ideality in operating conditions representative of the first-stage stator of ORC turbines, thus contributing to filling the current literature gap in available experimental data.

The present paper is structured as follows.

Section 2 illustrates the facility, instrumentation, nozzle and pneumatic system employed in the experimental campaigns here considered.

Section 3 reports results on preliminary tests with nitrogen as operating fluid.

Section 4 analyzes experimental campaigns with siloxane MM, with particular emphasis on highlighting the effects of flow non-ideality.

Section 5 draws the conclusions of this work and suggests future outlooks.

Preliminary tests with nitrogen were carried out prior to the experimental campaign with MM in order to assess the measurement system performance. Reliability of experimental results was verified through repeated testing, although only one exemplary test (TROVA-N\(_2\)) is reported here for brevity. The schlieren image in Fig. 5 shows that the wind tunnel operating regime is indeed the expected one, with supersonic flow impinging on the probe tip resulting in a bow shock that is locally normal to it.

Figure 6 illustrates measured pressures during test TROVA-N\(_2\). It was performed with line flushing to test the full procedure for MM experimental runs, although it was unnecessary here due to the impossibility of condensation. Pressure \(P_{\text {t,line}}\) is in perfect agreement with the reference total pressure in the plenum \(P_{\text {t,ref}}\) after line flushing has ended, indicating no issues on the upstream total pressure line to the differential transducer. The two pressures are within error bars of one another from test time \(t=5.9~\hbox {s}\), so less than \(0.2~\hbox {s}\) after nitrogen flushing was stopped at pressure peak. Flushing pressure was chosen as representative of expected peak pressure with MM and is slightly high for the present nitrogen testing conditions.

Pressure \(P_\text {a}\) is larger with respect to \(P_\text {b}\) and \(P_{\text {fs}}\), consistently with tap position along the nozzle, as per Figs. 2 and  3.

The plot also reports the total pressure loss across the shock \(\Delta P_\text {t}\) directly measured with the differential transducer. It is compared to the theoretical shock loss calculated from total conditions, measured \(P_{\text {fs}}\) and assuming a normal shock at the probe tip. \(\Delta P_{\text {t}}\) calc. from \(P_{\text {fs}}\) was determined by numerically solving mass, momentum and energy conservation equations across the shock coupled with the Span–Wagner thermodynamic model through the FluidProp library (Span and Wagner 2003). This is a state-of-the-art multiparameter model that provides accurate thermodynamic properties even close to the critical point. A functional form in terms of the reduced Helmoltz free energy, as a function of the inverse reduced temperature and reduced density, is provided for the fundamental relation linking all thermodynamic properties of a simple system in a stable equilibrium state. For siloxane MM, implemented model parameters were first reported in Colonna et al. (2006, 2008) and were more recently improved by Thol et al. (2016, 2017).

Analytical Rankine–Hugoniot equations for a polytropic ideal gas with specific heat ratio \(\gamma =1.4\) were used to verify the above shock loss calculation method, which is generalized to non-ideal flows, for following use with MM.

The uncertainty associated with calculated shock total pressure loss was determined by propagating measurement uncertainty in total temperature, total pressure and free-stream pressure. Given the very large number of calls to the thermodynamic model, Monte Carlo-like methods are too computationally expensive, and a procedure based on polynomial chaos expansions was thus implemented using the UqLab framework (Marelli and Sudret 2021) .

The top two plots in Fig. 7 illustrate trends in the compressibility factor evaluated at total conditions \(Z_\text {T}\) and in the free-stream Mach number \(M_{\text {fs}}\). The latter is calculated from total pressure, total temperature and measured free-stream pressure using the Span–Wagner thermodynamic model and assuming an isentropic flow. Both quantities confirm the polytropic ideal gas behavior of the flow, with constant values of \(Z_\text {T}=1\) and of \(M_{\text {fs}}\simeq 1.7\) that are independent of the actual total conditions.

To evaluate loss measurements quality, the loss coefficient Y was defined as:where the numerator represents shock total pressure loss and the denominator is pre-shock kinetic head. The loss coefficient was calculated using both the measured total pressure difference and the one calculated from \(P_{\text {fs}}\), as reported in the third plot from the top in Fig. 7, and referred to in the legend as \(Y_{\text {meas}}\) and \(Y_{\text {calc}}\), respectively. Both losses show a constant value during the test, highlighting the ideal gas behavior of nitrogen at the considered mild thermodynamic conditions, given the independence of shock losses from the varying total quantities.

The difference \(\Delta Y = Y_{\text {meas}} - Y_{\text {calc}}\) (bottom plot in Fig. 7) is also fairly constant and below \(1\%\), with the agreement between measured and theoretical shock losses indicating a good measurement system performance.

The schlieren image in Fig. 8 shows that the wind tunnel operates as intended during experimental campaigns with siloxane MM too. Analogously to test TROVA-N\(_2\) with nitrogen, supersonic flow impinges on the probe tip with a bow shock that is locally normal to it.

As evidenced in Fig. 9, the higher shock slope compared to nitrogen indicates a lower pre-shock Mach number, linked to the higher siloxane MM molecular complexity. Indeed, since nozzle geometry is fixed, nitrogen flow reaches a pre-shock Mach number of \(M_{\text {fs}}\simeq 1.7\), while MM vapor is in the range \(M_{\text {fs}}=1.47-1.57\), depending on total conditions.

The stronger density gradients compared to test TROVA-N\(_2\) allow to visualize the Mach lines due to surface roughness of nozzle profiles.

Reliability of experimental results was verified through repeated testing (not reported for brevity), and only exemplary test TROVA-MM is analyzed here, with measured pressures reported in Fig. 10. Pressures \(P_{\text {t,line}}\) and \(P_{\text {t,ref}}\) are always within error bars of one another throughout the test after line flushing has ended. Pressure \(P_\text {a}\) is larger with respect to \(P_\text {b}\) and \(P_{\text {fs}}\), consistently with tap position along the nozzle. Figure 10 also shows the total pressure loss across the shock \(\Delta P_\text {t}\) directly measured with the differential transducer, and the one calculated from total conditions, \(P_{\text {fs}}\) and balance equations across the shock (together with its associated propagated uncertainty), as previously explained.

The two top plots in Fig. 11 report trends in the compressibility factor at total conditions \(Z_\text {T}\) and in the free-stream Mach number \(M_{\text {fs}}\). Contrarily to nitrogen, both quantities vary significantly. \(Z_\text {T}\) increases from values just below 0.7 toward 1 during the test, indicating more dilute conditions as the test proceeds. The free-stream Mach number has a value of \(M_{\text {fs}}=1.47\) at test start and increases to \(M_{\text {fs}}=1.57\), showing a marked non-ideal dependence on total pressure and temperature. The loss coefficient Y, defined in Eq. 2, is also reported in Fig. 11 for measured and calculated losses, together with their difference \(\Delta Y\). Contrarily to test TROVA-N\(_2\) with nitrogen, losses are not constant during testing with siloxane MM, again highlighting the non-ideal dependence of the flow field on total conditions. The agreement between experimental and theoretical loss coefficient is satisfactory, with a difference always below \(2 \%\) throughout the test, with a slight decrease toward more ideal conditions to values consistent with nitrogen testing.

The present work documents the first ever direct measurements of total pressure losses across shocks in non-ideal supersonic flows representative of first-stage turbine stators in organic Rankine cycles.

Preliminary tests with nitrogen at free-stream Mach number \(M_{\text {fs}}\simeq 1.7\) were carried out to verify the measurement procedure in ideal gas flows.

Experimental campaigns were then carried out with siloxane MM vapor at free-stream Mach number \(M_{\text {fs}} \sim 1.5\), with varying total conditions and level of non-ideality. Direct total pressure measurements were compared with losses calculated using pre-shock quantities and by solving conservation equations across a normal shock employing a state-of-the-art thermodynamic model. Observed differences were always below \(2 \%\), attesting the satisfactory reliability of the implemented experimental procedure.

Test results were then analyzed by separating the impact of pre-shock Mach number variation due to the blowdown nature of the plant, and direct non-ideality effects on shock intensity. As a general finding, the campaign here reported allowed to verify that non-ideality has a direct significant impact on shock intensity. Even at the considered mildly non-ideal conditions with \(Z_\text {T}\gtrsim 0.70\), it was responsible for a stronger shock compared to the ideal gas limit at same free-stream Mach number, with differences as large as \(6\%\).

The present research contributes to establishing reliable experimental procedures for shock loss measurements in non-ideal flows. This sets the foundations for future pressure probes calibration and use for characterization of non-ideal supersonic flows of organic vapors, such as in the testing of ORC turbine blade cascades. Moreover, the results here reported contribute to filling the literature gap concerning experimental data on shock losses in non-ideal flows and are thus useful for the validation of related numerical design and analysis tools.