# Traceability of the Micro Scale Pipe Viscometer for Traceable Calibration of Dynamic Viscosity
## 3 Results and Discussion
### Calibration of the Inner Diameter of the Microtube
#### 3.1.3 Glass, Stainless Steel and PEEK Microtube Using the Flow Calibration Method  
The flow calibration method shows laminar flow behavior over the full range of the generated flow rates as shown in Figure 6 for the glass, stainless steel and PEEK beige microtube. The data were fitted with a linear fit with forced zero intercept to obtain the  
Figure 4: Zoomed-in cross section of the microtube shown in Figure 3 for the determination of inner diameter using the \(\upmu\)-CT method. The average inner diameter of the soda glass microtube measured at nine different positions is (0.1269 \(\pm\) 0.0010) mm.  
Figure 3: Cross-section of the glass microtube determined by means of \(\upmu\)-CT method.  
slope \(\Delta\)P/Q. As mentioned earlier, the inner diameter is then calculated according to Equation (2) by taking into account the slope of the fit \(\Delta\)P/Q, the dynamic viscosity of pure water obtained by calculation with the measured temperature and the length of the microtube. The results are listed in Table 6.  
It is important to mention here that the two pressure drops of the connectors between the ends of the microtube and the pressure sensors have to be taken into account if they are not negligible compared to the main pressure drop over the microtube. Otherwise, the inner diameter is underestimated. This correction has been applied in these measurements, although the nominal inner diameter of the micro tubes are of an order of 0.13-0.15 mm and much smaller than the inner diameter of the connectors (more than 2 mm for this setup). Thus, the corrections due to the pressure drop in the connectors are negligible compared to the main pressure drop over the micro tube.  
The results of the inner diameter of the soda glass micro tube obtained with both methods are in perfect agreement. For the case of the stainless steel micro tube, the structure of the cross-sections lead to the assumption that the hydrodynamic diameter has to be  
\begin{table}
\begin{tabular}{c c c} \hline
**Material** & **Flow Calibration Method (mm)** & \(\mu\)**-CT Method (mm)** \\ \hline Soda glass & 0.1267 \(\pm\) 0.0003 & 0.1269 \(\pm\) 0.0010 \\ Stainless steel & 0.1257 \(\pm\) 0.0003 & 0.144 \(\pm\) 0.010 \\ PEEK beige & 0.1515 \(\pm\) 0.0004 & Not applied \\ \hline \end{tabular}
\end{table}
Table 6: Determination of the inner diameter by means of the flow calibration and \(\mu\)-CT methods. The results include the expanded uncertainty (\(k\) = 2).  
Figure 5: Cross-section of the stainless steel microtube determined by means of the \(\mu\)-CT method. Due to the noncircular shape of the cross sections, the volumes were determined to calculate an equivalent inner diameter as (0.144 \(\pm\) 0.010) mm.  
Figure 6: Pressure drop as a function of flow rate for the laminar flow of water through the soda glass micro tube (light blue circle), stainless steel microtube (black triangles) and PEEK Beige microtube (beige diamonds).  
smaller, as the structure represents a more random surface with a high surface roughness instead of a circular cross-section with smooth surface roughness. Thus, both methods show different results, as expected. Therefore, the stainless steel microtube is less feasible for the pipe viscometer.