# Noninvasive Magnetic-Marking-Based Flow Metering with Optically Pumped Magnetometers
## 3 Results  
### Volumetric Flow Measurement  
Pump rates were varied from 150 mL/min to 275 mL/min in steps of 25 mL/min. According to Equation (2) these pump rates correspond to flow velocities from 12.7 cm/s to 23.3 cm/s. For this velocity range the maximum Reynolds Number \(Re\) is  
\[Re=\frac{vd\rho}{\eta}=1165, \tag{6}\]  
where \(d\) is the tube diameter, \(\rho\) is the density of water at 20 \({}^{\circ}\)C and \(\eta\) is the dynamic viscosity of water at 20 \({}^{\circ}\)C. Thus, \(Re\) being smaller than \(Re_{crit}=2300\), we assume that we have laminar flow. Hence, the apparatus measures the laminar flow of water inside the tube.  
For each pump rate the syringe was emptied completely. The batch volume and the set pump rate determine the measuring time. In Figure 4 streamed data from the 200 mL/min measurement are shown as an example. The OPM data were periodically modulated by the RF pulses and yielded a signal amplitude of about 60 pT. Please note that the Signal amplitude off-set was corrected in such a way that it was set to 0 before the start of the measurement.  
The flow rate and volumetric flow calculation was done as explained in Equations (1) and (2). For each pump rate, data were recorded until the syringe was completely emptied. Depending on the flow rate, every measurement yielded 5 to 14 data points. The flow velocity for a given pump rate was determined by the average of the individual values \(t_{D}-t_{E}\) in one recording. The error on the average flow velocity was calculated from the scatter of the individual measured values for each pump rate. The results are listed in  
Figure 4: Data stream showing all relevant parameters for a flow measurement. The water velocity is 200 mL/min. The pump is turned on at 5 s. Then, the polarized water flows past the OPM. This increases the magnetic field measured to a plateau region of 300 pT. At 25 s the RF coil is switched on, which periodically changes the water pre-polarization amplitude by about 60 pT. As explained in Figure 1, the OPM measures the vector component of the water polarization in z direction. Thus, it detects how much of the water pre-polarization is destroyed by the RF pulse. Residual fields in the shielded environment are on the order of nT. For this reason, the data are shifted to start at zero. Please note: The reference voltage plotted in the upper channel was recorded using the “trigger input” of the FieldLine OPM interface. The resulting line shape does not exactly reproduce the applied sinusoidal envelope of the RF pulses. It is only used to determine \(t_{E}\).  
Table 1 and shown in Figure 5. On average, the relative error on the volumetric flow \(Q\) and thus on the detected flow velocity \(v\) was 3%. In addition, the predicted pump rate of the syringe pump was mostly within one standard deviation of the detected volumetric flow. For 275 mL/min the predicted pump rate lays within two standard deviations.  
For any practical application of the presented scheme, the dynamic range of the procedure was limited by two factors. The first limitation derived from the decay of the medium polarization. For too-small flow velocities, the signal would decay before it arrived in front of the sensor. The second limitation was caused by bandwidth of the OPM sensor (currently 100 Hz). This limited the accuracy of the determination of \(t_{D}\) and, hence, of \(\Delta t\) and the flow velocity.