## 2 Experimental Set-Up of the Calibration  
In this work, the experiment under study is the calibration of a flow meter against a standard proving tank. Next, we will show how the classic GUM method is applied. The result of the calibration is the ratio between the volume measured by the meter and the reference volume \(V_{\text{REFERENCE}}\), in this case, the proving tank volume. This ratio is referred to as the Meter Factor (\(MF\)) [9], so  
\[MF_{i}=\frac{V_{\text{REFERENCE},i}}{V_{\text{METER},i}} \tag{1}\]  
The subscript \(i\) refers to the fact that the \(MF\) is calculated three times because the experiment comprises three measurements or runs. The reference volume needs to be corrected because the calibration normally takes place at a temperature \(\overline{t}_{i}\) different from the reference temperature \(t_{\text{REF}}\) (15 \({}^{\circ}\)C), so Equation (1) becomes  
\[MF_{i}=\frac{V_{\text{TANK},i}\big{\{}1+k_{v}\big{[}\overline{t}_{i}-t_{\text{ REF}}\big{]}\big{\}}}{V_{\text{METER},i}} \tag{2}\]  
With \(k_{v}\) being the cubic expansion coefficient of the tank material, \(V_{\text{TANK},i}\) is the volume measured by the tank at temperature \(\overline{t}_{i}\), that is the average of the three temperature indications (\(t_{i1,2,3}\)) measured at the tank:  
\[\overline{t}_{i}=\frac{t_{i1}+t_{2}+t_{i3}}{3} \tag{3}\]  
Since three runs are measured, \(MF\) is  
\[\overline{M}\ \overline{F}=\sum_{i=1}^{3}\frac{MF_{i}}{n} \tag{4}\]  
There is also a tolerance repeatability limit above which the calibration would be withdrawn.  
The schematic calibration set-up can be seen in Figure 1, as it appears in the control room of the calibration facility. This scheme includes all the components of the calibration facility and not only the ones used for this work. The calibration lab consists of two pipe provers (prover 1 and prover 2, seen in Figure 1) which receive the real products that are previously filtered by the filtering system (see real products for calibrations in Figure 1 and the filtering system in such figure). The location of the meters under calibration are also indicated (meters under calibration in the diagram). There is a dashed line that connects the 5000 L standard proving tank (at the bottom of Figure 1 and appearing as a 5000 L container) with the meter under calibration. So, the meters can be calibrated either by the tanks or by the provers. A Coriolis meter can be calibrated in the same way as a positive displacement meter if it is fitted with a system that translates its measurement signal into electrical pulses.  
Figure 2 presents a photograph of the 5000-litre tank and a meter under calibration (in the figure, the meter is a Coriolis mass flow meter).  
The general technical specifications of Coriolis meters are as follows.  
* Accuracy of 0.1% to 2%. When operating at low flow rates, accuracy increases as the flow rate increases. As the flow rate reaches higher values, accuracy stabilizes. Therefore, in general, accuracy may depend on the flow rate, although not specified by the manufacturer. For example, in [4], flow rate coefficients in the order of 0.001% h/m\({}^{3}\) are determined in the range of 60 m\({}^{3}\)/h to 165 m\({}^{3}\)/h.
* Typical turndown ratio of 100:1. If the meter has a curved tube, it can increase up to 200:1, while if it has a straight tube (like the one in the Figure 1) the turndown ratio is limited to 30:1 to 50:1 and the accuracy is also worse.
* Zero stability. Due to mechanical tolerances, process effects, and even installation, the Coriolis meter may give a measurement signal in zero flow conditions. This value is small in absolute terms, but it has an impact on low flow rates, where the Coriolis effect is smaller. Coriolis meters must be \"zeroed\" in zero flow conditions to correct for this systematic effect.