## 3 Experimental Setup and Uncertainty Analysis
### Uncertainty Estimation Model  
In the performance evaluation of the microbubble pump by experimental measurements, the uncertainty and error rate were analyzed to evaluate the reliability of the measured data. Error is simply the difference between the true value (xtrue) and the measured value (xmeas), whereas uncertainty is related to the probability that the true value will fall within a specified range, determined by a confidence interval. Whenever a physical quantity is measured, its true value is always unknown, as the measured value is surely affected by errors. As the number of readings (\(n\)) is increased at the same operating and external conditions, the true value tends to be closer to the measured value, i.e., errors are minimized (as n\(\rightarrow\)infinity, xtrue= xmeas). So, the value of \(n\) should be as large as possible for more accurate results.  
The possible types of error that can impact measured values include zero errors, calibration errors, instrument sensitivity, data transport errors, statistical errors during data reduction, conceptual errors such as the accuracy of the applied theory and instrument full scale (FS) reading errors. It is impossible to determine all types of error and how each error can individually impact the overall uncertainty. However, the following equation  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Flow State** & **Flow Coefficient** & **Pressure Coefficient** \\ \hline Underflow & 0.24 & 4.4 \\ Underflow & 0.31 & 3.3 \\ Design & 0.39 & 2.0 \\ Overflow & 0.44 & 1.4 \\ Overflow & 0.47 & 1.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experiment conditions.  
Figure 5: Picture of the pump experimental setup.  
can be used to determine the overall symmetric uncertainty limits in the case of the pump experiments [18]:  
\[\mu\ =0.5*\left(\frac{\sigma}{\sqrt{n}}\right)*t_{CI}(v) \tag{6}\]  
where \(\mu\) is the overall uncertainty, \(\sigma\) is the standard deviation, \(t_{CI}(v)\) is the distribution factor at a degree of freedom \(v\), and \(n\) is the number of measured data. \(\frac{\sigma}{\sqrt{n}}\) is the standard error (\(\epsilon\)), so the above equation becomes:  
\[\mu\ =0.5*(\epsilon)*t_{CI}(v) \tag{7}\]  
The degree of freedom, in this case, can be defined as:  
\[v=n-1 \tag{8}\]  
\[\sigma=\mp\sqrt{\frac{1}{v}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}} \tag{9}\]  
where \(X_{i}\) is an arbitrary value of a measured variable and \(\overline{X}\) is the mean value of the measured variable:  
\[\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i} \tag{10}\]  
\(CI\) in \(t_{CI}(v)\) is the confidence interval, which is the probability that the population mean of the measured variable will fall within the estimated range by using the sample mean. If we consider a 95% confidence interval (assuming a Gaussian (normal) distribution), then \(CI\) can be mathematically defined as:  
\[CI=Prob\left(X_{i}-1.96\sigma\leq\overline{X_{pop}}\leq X_{i}+1.96\sigma \right)=0.95 \tag{11}\]  
Once the values of \(CI\) and \(v\) are determined, a distribution table can be used to estimate the value of \(t_{CI}(v)\). So, Equation (6) can be used to calculate the uncertainty of independently measured variables, which are the inlet and the outlet pressure in the present study.