# Friction Factor Correlation for Regenerator Working
## 3 Experimental Setup
### Data Collection and Reduction  
The phase angle between pressure and velocity is controlled via the RLC network as shown in Figure 2. The valves are adjusted accordingly until the travelling-wave condition is achieved. The phase is observed by monitoring the phase angle, \(\varphi\), of impedance at location 5. The impedance at location 5, \(Z_{5}\), is given as:  
\[Z_{5}=\frac{P_{5}}{u_{5}}=|Z_{5}|(\cos\varphi+i\sin\varphi). \tag{8}\]  
The terms \(P_{5}\), \(u_{5}\), and \(\varphi\) represent the oscillating pressure, oscillating velocity and phase angle of impedance at location P5, respectively. The phase angle of impedance represents the phase difference between pressure and velocity at that location. When the phase angle of impedance is zero, the imaginary part of impedance becomes zero. Hence oscillating pressure and oscillating velocity are both contributing only to the real value of impedance. As a result, pressure and velocity are in phase [22]. Pressures are measured in the experiment. The velocity at location 5, \(u_{5}\), is determined using a Transfer Matrix Method. The Transfer Matrix Method has originally been used to measure the acoustical properties of components such as the porous media in the study by [PERSON] and [PERSON] [23] and [PERSON] et al. [24]. [PERSON] et al. [24] applied a linear thermoacoustic theory to estimate acoustic characteristics of a regenerator. In the current study, the method is used to theoretically provide information on variables not measured in the experiment. Lossless thermoacoustic model is used to relate pressure, \(P\), and velocity, \(u\), at two locations. This is shown in a matrix form as follows:  
\[\left[ \begin{array}{c}\frac{\partial P}{\partial x}\\ \frac{\partial u}{\partial x}\end{array} \right]=\left[ \begin{array}{cc}0&-i \omega\rho_{m}\\ \frac{i\omega}{\gamma p_{m}}&0\end{array} \right]\left[ \begin{array}{c}P\\ u\end{array} \right]. \tag{9}\]  
These lossless equations are based on assumptions that the flow is adiabatic and inviscid. The assumptions are considered valid between points 3 and 5 due to the short distance between them and the absence of the influence of temperature within the area (as indicated in Figure 2). The solutions of Equation (9) may be shown in a matrix form as:  
\[\left[ \begin{array}{c}P_{5}\\ u_{5}\end{array} \right]=\left[ \begin{array}{cc}m_{11}&m_{12}\\ m_{21}&m_{22}\end{array} \right]\left[ \begin{array}{c}P_{3}\\ u_{3}\end{array} \right]. \tag{10}\]  
Four equations are needed to solve for the four unknowns. Two equations are obtained from the lossless thermoacoustic model with analytical solutions (as given by [PERSON] [25]) cast into the matrix form of Equation (10) to represent \(m_{11}\) and \(m_{21}\) and are shown as follows:  
\[m_{11}=\cos(kx), \tag{11}\]  
\[m_{21}=\frac{i}{\rho a}\sin(kx). \tag{12}\]  
Two more equations are obtained through the wave characteristics following the analysis of [PERSON] and [PERSON] [23]. The flow is reciprocal and the area of interest is symmetrical. For symmetrical system, \(m_{11}\) equals \(m_{22}\). The reciprocity of the flow requires that the determinant of the transfer matrix be unity. These two characteristics are shown as follows:  
\[m_{11}=m_{22}, \tag{13}\]  
\[m_{11}m_{22}-m_{12}m_{21}=1. \tag{14}\]The transfer matrix method used in the experiment is finally shown as:  
\[\left[ \begin{array}{c}P_{5}\\ u_{5}\end{array} \right]=\left[ \begin{array}{cc}\cos(kx)&i\rho a\sin(kx)\\ \frac{i}{\rho a}\sin(kx)&\cos(kx)\end{array} \right]\left[ \begin{array}{c}P_{3 }\\ u_{3}\end{array} \right]. \tag{15}\]  
The terms \(\omega\), \(\rho\), and \(a\) are the angular velocity, density and sound speed, respectively. The velocity at location \(P_{3}\), \(u_{3}\), is calculated using the lumped element method with the equation given by [PERSON] [25]:  
\[u_{3}=\frac{i\omega V_{c}P_{4}}{S\gamma p_{m}}. \tag{16}\]  
In the lumped element method, the velocity is calculated using information gained from the compliance.