# Traceability of the Micro Scale Pipe Viscometer for Traceable Calibration of Dynamic Viscosity
## 2 Materials and Methods
### Flow Calibration Method  
Another method for the calibration of the inner diameter of the micro tubes is the flow calibration method in the laminar regime. A reference liquid with known dynamic viscosity is used to measure the pressure drop as a function of the flowrate and to calculate the inner diameter of the microtube according to Equation (2):  
\[r=\sqrt[4]{\frac{8\cdot\eta\cdot Q\cdot L}{\pi\cdot\Delta P}}, \tag{2}\]  
where \(\eta\) is the dynamic viscosity, \(\Delta P\) is the pressure drop over the microtube, \(Q\) is the volumetric flowrate, \(L\) is the length of the micro tube, and \(r\) is the inner radius of the micro tube.  
The reference liquid chosen is pure water as the viscosity and the density formulas as a function of temperature are well established and these quantities can be determined by measuring the temperature of the pure water. The dynamic viscosity and the density of the pure water are calculated according to equations from the NIST database [19] by measuring the temperature in the tubing before and after the microtube. Measurements are performed in the corresponding flowrate range to obtain pressure drops in the range from 0.5 bar to 8.0 bar. Thus, the pressure drop over the micro tube as a function of flowrate can be fitted linearly in the laminar regime to obtain the slope \(\Delta\)P/Q and then calculate the inner diameter of the micro tube.  
For the determination of the inner diameter of the micro tubes with the flow calibration method, the following values of the dynamic viscosity and density of pure water were taken from the NIST database [19] (listed in Table 3), which is based on the IAPWS R12-08 [20].  
The values at 1 bar absolute pressure are applied, which corresponds roughly to the ambient pressure of the laboratory conditions. The change rate of the dynamic viscosity as a function of the pressure is \(-4\cdot 10^{-5}\) mPa\(\cdot\)s/bar. As the measurements are performed for pressure drops in the range from 0.5 bar to 8.0 bar, the maximum average pressure in the liquid over the full capillary is half of the maximum pressure drop, 4.0 bar. This implies a maximum change in the dynamic viscosity of \(1.6\cdot 10^{-4}\) mPa\(\cdot\)s, which corresponds to 0.016% of the dynamic viscosity at ambient conditions. This contribution is negligible as we assume the uncertainty contribution from the calculation of the dynamic viscosity of pure water with the measured temperature as 0.5% (see Table 4).  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Soda Glass Micro Tube** & **Stainless Steel Micro Tube** \\ \hline Positions & 9 & 2 \\ X-ray tube voltage & 100 kV & 160 kV \\ Target power & 10 W & 5 W \\ X-ray filter & 0.03 mm Al & 0.1 mm Cu \\ Exposure time & 1.3 s & 5.5 s \\ Projections & 800 & 800 \\ Voxel size & 2.8 \(\mu\)m & 1.0 \(\mu\)m \\ \hline \end{tabular}
\end{table}
Table 2: \(\mu\)-CT measurement conditions.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Reference Liquid** & \
\begin{tabular}{c} **Absolute Pressure** \\ **(bara)** \\ \end{tabular} &
\begin{tabular}{c} **Dynamic Viscosity** \\ **(mPa\(\cdot\)s)** \\ \end{tabular} & **Density (kg/m\({}^{3}\))** \\ \hline H\({}_{2}\)O & 1.0 & 0.97052 & 997.93 \\ H\({}_{2}\)O & 2.0 & 0.97048 & 997.97 \\ H\({}_{2}\)O & 4.0 & 0.97041 & 998.07 \\ H\({}_{2}\)O & 8.0 & 0.97026 & 998.25 \\ \hline \end{tabular}
\end{table}
Table 3: Density and dynamic viscosity of water according to the NIST database [19] at a temperature of 21.3 \({}^{\circ}\)C and several values of absolute pressure.