**3. Discussion**

The multiphase flow profile independent liquid flow rate measurement accuracy presented in Section 2 is remarkable, considering the highly complex liquid holdup and liquid flow velocity correlations that are observed in these multiphase flows (see Figure 6), especially when the flow was disturbed by the downstream valve. Therefore, these results prove that a low-field MR-based flow metering apparatus can be applied to ensure representative sampling of the multiphase flow in a wide range of multiphase flow profiles and a wide dynamic range in terms of the average gas and liquid flow rates. In addition, this proves that flow regime identification is possible using MR measurement methods, which can be an important asset in the industry. For instance, in oil production and process optimization, where slugs may induce excessive structural vibration in piping systems causing component failures due to fatigue or resonance [2,28,29]. Research into the multiphase flow can benefit from measurement equipment that does not disturb the flow. Finally, the frequency encoding-based liquid holdup determination method applied in this study is shown to be robust enough to be applied to multiphase flows, opening up MR-based imaging opportunities in industrial multiphase flow applications.

#### **4. Materials and Methods**

The flow experiments presented in this work were performed using the M-PHASE 5000 multiphase flow meter developed by KROHNE [30] and shown in Figure 8. The 3.5 m long instrument is designed around a horizontal glass fiber reinforced epoxy (GRE) flow tube that is available in 2, 3, and 4 pipe sizes. A 3 pipe was used in this work, which has an 80 mm internal diameter. The main magnet section was constructed using a two-ring, 90 cm long, 0.2 T Halbach magnet with a length-to-radius ratio of 6. It contains a cylindrical region-of-interest (ROI) of 10 cm length and 10 cm diameter that was passively shimmed to a homogeneity of about 1000 ppm. A 12.5 cm long solenoid-shaped volume coil with an inner diameter of 12 cm, and a 40 cm long *z*-gradient coil with an inner diameter of 15 cm are centered on the ROI. The RF coil was used for both transmission of RF pulses and reception of NMR signals and was driven at 8.5 MHz using an RF power of 1.3 kW. The gradient coil was operated using a continuous current that generated a gradient field strength of 23.5 mT/m (equivalent to 10 kHz/cm). All electronics required for the NMR measurements and data transfer to a control computer are integrated into two flame-proof boxes that are mounted directly onto the flow meter. The instrument is additionally equipped with a pre-magnetization section consisting of 3 identical, two-ring, 30 cm long, unshimmed, 0.2 T Halbach magnets. The pre-magnetization length can be varied by selectively activating pre-magnetization sections by rotating the inner ring with respect to the outer ring in the Halbach section by 180 degrees.

**Figure 8.** Annotated photograph of the KROHNE MPHASE 5000 MR-based multiphase flow meter used for the flow experiments presented in this work. Image courtesy of KROHNE [30].

The flow experiments presented in this work were performed on water-air mixtures for a wide range of flow rate combinations using the maximum pre-magnetization length. A schematic representation of the flow loop used is presented in Figure 9. Water flow was controlled by using 3 commercially available submersible garden water pumps that could be powered on independently. These pumps were placed in a 1 m3 Industrial Bulk Container (IBC) tank and yielded a combined maximum water flow rate of 48 m3/h. The flow loop was kept at atmospheric pressure via a vent in the IBC. The water injection point

in the flow tubing for each pump was fitted with a ball valve that allowed for fine-tuning of the water injection for each individual pump. Whenever a pump was inactive, the ball valve allowed this pump to act as a controlled bypass for lowering flow rates through the magnet. This way, the superficial water flow velocity could be varied from about 0.5 cm/s up to 3 m/s. Air injection from a central laboratory compressed air supply was controlled using a needle valve and the superficial gas flow velocity could be varied from about 5 cm/s up to 3.5 m/s, corresponding to a maximum gas flow rate of 60 m3/h through the multiphase flow meter. The reference volumetric flow rate for injected water was measured using a commercial electromagnetic flow meter (EMF in Figure 9) that has an accuracy better than 0.2% [31], while a commercial Coriolis flow meter with accuracy better than 0.5% [32] was used for air mass flow measurement. The air mass flow rate was converted to a volumetric air flow rate using dry air PVT calculations [33] based on the temperature and pressure measurements that are integrated into the M-PHASE 5000 multiphase flow meter. The flow loop piping layout was U-shaped and had a total straight flow length of 2 m (25D) applied upstream and downstream of the multiphase flow meter for flow conditioning. A ball valve was added just before the flow return connection to the IBC tank, which allowed the effect of flow disturbances on multiphase flow profiles and multiphase flow measurement accuracy to be studied by partially closing this valve. Multiphase flow profiles during the tests could be observed through the 1.5 m long transparent pipe section placed in front of the multiphase flow meter. Some example flow profile videos captures are provided in the supplementary information. Based on the observations during the tests, a flow map could be created to help predict the flow profile in the flow loop as a function of the gas and liquid flow rates. This flow map was presented as Figure 4 in Section 2.

**Figure 9.** Schematic representation of the two-phase water-air flow loop used for the flow experiments presented in this work. Note that the air injection point, flow meter, and downstream ball valve of the flow loop were all placed at the same elevation above the IBC, making the piping horizontal over the entire length of the two-phase flow path.

Flow experiments were performed using broadband excitation constant-gradient LFA-CPMG pulse sequences [33,34] using 45◦ flip angle pulses of duration *t*pulse = 10 μs, and 2*τ* = 800 μs echo spacing. This pulse sequence is shown schematically in Figure 10. To ensure the maximum initial signal amplitude and uniform spectral width of both excitation and refocusing pulses, pulse duration was kept the same for both excitation and refocusing pulses. Low flip angle pulse sequences can be used to determine the frequency spectrum of the sample in the ROI even in situations where limited SNR is available by combining the data from several echoes [35]. In addition, the amplitude decay of the LFA-CPMG signals with time due to the convective outflow of spins from the ROI, as obtained from the envelope of the spin-echo maximum amplitudes, can be used to derive average flow velocity

information [9]. The number of acquired echoes and the wait time between consecutive pulse sequence executions were optimized in each flow experiment using the integrated flow measurement optimization feature of the KROHNE M-PHASE 5000. This algorithm actively tunes the number of echoes in real-time to match the lowest flow velocity component that occurs in the multiphase flow during the flow experiment. The wait time is set to 2 times the echo train length to ensure the sample is fully refreshed between consecutive pulse sequence executions. The liquid holdup was obtained by integrating the liquid distribution image obtained from the first 20 echoes and taking the ratio of this integral with the integral of a full pipe water reference measurement.

**Figure 10.** Schematic representation of the broadband excitation constant-gradient LFA-CPMG pulse sequence used in flow experiments. RF pulses with on-resonance flip angle α◦ are indicated by black rectangles. Digital acquisition (DAQ) of spin-echoes is represented in blue. The field gradient *G*z is represented in dark red.

Prior to the two-phase flow experiments, velocity determination was calibrated, and liquid distribution image-based liquid holdup determination was validated.

Pure water flow experiments were performed on a dedicated single phase flow loop at KROHNE to calibrate the slope of the flow velocity determination via the convective decay of the LFA-CPMG signals. Figure 11 shows the relation between the reference flow velocity and the convective decay rate, *Rv*, for 17 different flow velocities up to 11 m/s. This convective decay rate was determined by fitting an exponential decay to *T*2,eff-corrected LFA-CPMG signals. The *T*2,eff used for correction was determined as the effective *T*<sup>2</sup> decay obtained from a static LFA-CPMG experiment performed prior to each flow experiment. The use of an exponential convective decay model is based on the work by Petrova et al. [34] that showed the asymptotic form of the *T*2,eff-corrected signal in low flip angle CPMGs to be exponential. The exponential fit was validated to be a better fit to our data than the linear fit method that is applied in flow measurement using bulk CPMGs.

The liquid holdup determination was validated by filling the multiphase flow loop shown in Figure 9 completely with water and by draining the piping in a controlled way via the ball valve that is placed downstream of the flow meter. A total of 19 different liquid levels were created this way, ranging from 100% down to 0.5% liquid holdup. The combined 1D liquid distribution images of these liquid level steps are shown in Figure 12, where the full semi-circle at the top left indicates the full pipe liquid experiment. When stepping from this 100% liquid holdup experiment down to lower liquid holdups, a progressively bigger portion of the semi-circle is cut-off due to the absence of liquid. As reference liquid holdup, the signal amplitude of a bulk spin-echo (*A*BSE) was used. The reference liquid holdup for experiment *i* can be derived from the bulk spin-echo amplitude using the relation

**Figure 11.** The reference liquid flow velocity, *u*liq, as a function of the convective decay rate, *Rv*, that was obtained by fitting an exponential decay to *T*2,eff-corrected LFA-CPMG signals. The solid line indicates the calibration function used in the two-phase flow experiments.

$$h\_{\rm liq,i} = \frac{A\_{\rm BSE,i}}{A\_{\rm BSE,100\%}} \, \tag{6}$$

in which *A*BSE,100% is the bulk spin-echo amplitude obtained for a full pipe of liquid. The relation between the liquid holdup obtained using the bulk spin-echo and obtained from the liquid distribution images as acquired using the LFA-CPMG frequency-encoded spinechoes is shown in Figure 13. A one-to-one correspondence between both methods exists over the entire range, indicating the robustness of the liquid distribution image-based liquid holdup determination method, even at liquid holdups down to a few percent.

**Figure 12.** The set of 1D liquid distribution images acquired in the static verification experiments represented as a surface plot in which liquid distribution images are sorted from the highest to lowest measured holdup. The axis labeled as *z* indicates the height along the flow tube with the pipe axis located at *z* = 0 mm. The separate experiments are well recognized as steps in the surface as a progressively bigger portion of the semi-circle is cut-off due to the absence of liquid.

**Figure 13.** The liquid holdup as determined from a bulk spin-echo experiment as a function of the liquid holdup as determined from the 1D liquid distribution images obtained using a broad band constant-gradient LFA-CPMG pulse sequence. The measurements for 19 different liquid holdups show a 1-to-1 correspondence between the two methods.

#### **5. Patents**

Patent pending, provisional application number is DE 10 2021 111 162.5.

**Supplementary Materials:** The following videos of example flow experiments are available online. Video S1: Slug\_slow\_Qgas11.5\_Qliq7.3.mp4 (Qgas = 11.5 m3/h and Qliq = 7.3 m3/h), Video S2: Slug\_fast\_Qgas43.2\_Qliq21.2.mp4 (Qgas = 43.2 m3/h and Qliq = 21.1 m3/h), Video S3: Wavy\_Qgas46.1 \_Qliq2.6.mp4 (Qgas = 46.1 m3/h and Qliq = 2.6 m3/h), Video S4: Stratified\_Qgas46.6\_Qliq1.1.mp4 (Qgas = 46.6 m3/h and Qliq = 1.1 m3/h).

**Author Contributions:** Conceptualization, R.R.T. and L.M.C.C.; data curation, L.M.C.C.; methodology, R.R.T. and L.M.C.C.; software, L.M.C.C.; validation, R.R.T. and L.M.C.C.; formal analysis, R.R.T. and L.M.C.C.; investigation, R.R.T. and L.M.C.C.; visualization, R.R.T. and L.M.C.C.; writing original draft, R.R.T.; writing—review and editing, R.R.T. and L.M.C.C. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to KROHNE Messtechnik GmbH proprietary rights.

**Acknowledgments:** The authors would like to acknowledge Jankees Hogendoorn for his organizational support to the publication of this work and would like to thank the development team at KROHNE for their support in the development of the experimental facilities used for the flow measurements presented in this work. In this regard, the authors are especially indebted to Coert Kriger and Juan Pablo Nicoloff for their instrumental support during the experiments.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Sample Availability:** Not available.
