## 3 Reduced Mass Flow Fitting
### Coefficient Analysis and Calibration  
As mentioned above, the fitting is performed as two individual turbines with their corresponding turbine entry map data and, each _MFR_ map has been fitted separately to study the behavior of the coefficients, as shown in Figure 11. It is worth highlighting that, according to the _MFR_ definition in Equation (1), for an increasing _MFR_, the turbine mass flow parameter of the shroud/long volute entry increases and hub/short volute decreases. * Coefficient \(a\): It represents the rotor discharge coefficient as shown in Equations (8) and (19). The rotor effective flow area is variable with _MFR_, considering the amount of mass flow entering into each branch, i.e., growing as the mass flow grows, and this is true according to the definition of the discharge coefficient. Therefore, it is regarded as some quadratic behavior with _MFR_ and the magnitude of this coefficient should be between 0 and 1. It is observed that the coefficient \"\(a\)\" trends are exactly as expected. \[A_{4,i}^{j}=A_{4,i}^{j\,geom}\cdot a\] (19)
* Coefficient \(b\): The significance of this parameter is from velocity triangle speed ratio, and its order of magnitude can be estimated using Equation (20) as described in [30]. This coefficient is dependent on the mass flow passing through the turbine. More mass flow in a turbine branch gives higher \"\(b\)\" values than lower mass flow. This is due to higher radial velocities at rotor inlet, for the same peripheral speed and expansion ratio values. This trend is clearly shown in Figure 11c,d for both branches of every turbine type. \[ \begin{split} b&=\left(\frac{v_{0}}{C_{ss}} \right)^{2}+\left(\frac{w_{3}}{C_{ss}}\right)^{2}=\frac{A_{\text{eff}}}{A_{0}} \cdot\left(\frac{1}{\Pi_{T}}\right)^{\left(\frac{1}{7}\right)}+O\left[10^{-1 }\right]\\ &=O\left[10^{-1}\right]+O\left[10^{-1}\right]\to O\left[10^{-1} \right]\leq b\leq O\left[10^{0}\right]\end{split} \] (20)
* Coefficient \(c\): It is the ratio between rotor and stator discharge coefficients (\"\(a\)\" and \"\(C_{Ds}\)\"), as shown in Equation (21). As the effective flow area of the stator is variable with the mass flow ratio in the turbine, the value of \"\(c\)\" is expected to increase. Besides, \"\(c\)\" must always justify Equation (21), i.e., as an effective section introduced through the discharge coefficient (\(C_{Ds}\)) should not be higher than unity [30]. \[C_{Ds}=\left(\frac{a}{c}\right)\leq 1\] (21)
* Coefficient \(d\): It is related to the pressure ratio in the rotor with respect to the total turbine pressure ratio [30]. Its value decreases with more flow into the branch and tends to 0.5. This implies that, with high mass flow in the branch, the pressure ratio in the rotor tends to be half of the total pressure ratio in the turbine. With low mass flow, most of the expansion is occurring in the rotor. Both are the expected logical trends. Summarizing, all the coefficients values and trends with _MFR_ have physical meaning, as they are derived from theoretical considerations [30]. These coefficients showed the exact expected trends with the various _MFR_s in both turbines, asymmetrical dual-volute and symmetrical twin-entry. By reviewing the fitting coefficients of each turbine entry of the two different radial in-flow types and for several _MFR_s in Figure 11, it is possible to impose those physical trends with _MFR_ for each turbine entry: a linear trend for coefficients \"\(b\)\", \"\(c\)\", and \"\(d\)\" and a quadratic behavior for \"\(a\)\". In this approach, seven coefficients of each turbine entry can be adjusted using a non-linear fitting method for all \(MFR\)s of a given branch at the same time. This implies that one single fitting procedure and one set of parameters will be needed for each entry of the turbine to predict the reduced mass flow in all flow admission conditions. Further elaborating this conclusion, only measuring three \(MFR\)s is enough for fitting the whole range of turbine mass flow parameter: any other non-measured \(MFR\)s can be predicted.