## 2 Experimental Study
### Performance Parameters  
The experiments provided the necessary data to calculate turbine map parameters: reduced mass flow, reduced turbine speed, expansion ratio, adiabatic efficiency, and blade to jet speed ratio. As described by [PERSON] et al. [12], representing the double entry turbine performance parameters with the average values between the two entries (e.g., expansion ratio and turbine scroll temperature) showed an impact on resulting maps and made it difficult to analyze the parameters. Moreover, representing both branches with one turbine flow parameter, the mass flow distribution between respective entries under equal and unequal admission conditions is not known. Thus, as detailed in [12], the two entries are treated separately for calculating the turbine map parameters as follows. In the following equations, the term \(i\) represents the generic code for the turbine inlet branch. The reduced turbine speed \(n_{\text{red}}\) is computed for each branch using appropriate turbine scroll inlet temperature, as shown Equation (2). \[n_{red,i}=\frac{n}{\sqrt{T_{0t,i}}} \tag{2}\]  
where \(n\) is the turbocharger speed and \(T_{0t,i}\) is the turbine total inlet temperature at the corresponding turbine entry. The turbine reduced mass flow rate \(\dot{m}_{\text{red}}\) is computed for each entry, as shown in Equation (3)  
\[\dot{m}_{red,i}=\frac{\dot{m}_{i}\cdot\sqrt{T_{0t,i}}}{p_{0t,i}} \tag{3}\]  
Figure 4: Dual-volute turbocharger setup in the gas stand. where \(\dot{m}_{i}\) and \(p_{0t,i}\) are the turbine mass flow rate and total inlet pressure for a corresponding entry. The turbine total to static pressure ratio \(\Pi_{i(0t,A)}\) is estimated by Equation (4). \(p_{4}\) is the common turbine outlet pressure and therefore it does not have an index. \[\Pi_{i_{i}(0t,A)}=\frac{p_{0t,i}}{p_{4}} \tag{4}\]  
When the twin-entry/dual-volute turbine is working in an engine, the inlet conditions are changed in both branches due to the staggered firing orders in the cylinders [40] and corresponding interval opening of the cylinders valves [41]. Accordingly, it was assumed that the power produced by each turbine branch is different [12]. Therefore, the turbine total to static efficiency was computed as two individual turbines, as shown in Equation (5), on the basis of the enthalpy-entropy adiabatic expansion of the turbine shown in Figure 5.  
Hereafter, it is referred as apparent efficiency. \[\eta_{MFR_{x}(t/s)}^{i}=\frac{T_{0t,i}-T_{4t}^{MFR_{x}}}{T_{0t,i}-T_{4s,i}} \tag{5}\]  
where \(T_{4t}^{MFR_{x}}\) is the mass averaged mixed turbine outlet total temperature that is measured in the gas stand. Since the temperature that is coming from the individual turbine branches is mixed at the turbine outlet station (see Figure 5), it is a convenient method to assume the outlet stations are at equal temperatures [12]. \(T_{4s,i}\) is the isentropic turbine temperature and it is estimated independently for both turbine entries as shown in Equation (6). \[T_{4s,i}=T_{0t,i}\cdot\left(\frac{1}{\Pi_{i_{i}(0t,A)}}\right)^{\frac{7-1}{7}} \tag{6}\]  
Figure 5: Enthalpy–entropy expansion process in twin-entry/dual-volute radial turbines. Lastly, Equation (7), shows the definition of a blade to jet speed ratio, which is estimated for each branch with their corresponding inlet temperature and expansion ratio. \[\sigma_{i}=\frac{2\cdot\pi\cdot n\cdot r_{3}}{\sqrt{2\cdot c_{p}\cdot T_{i,0t} \cdot\left[1-\left(\frac{1}{\Pi_{i,(0,4)}}\right)^{\frac{\gamma-1}{\gamma}} \right]}} \tag{7}\]  
By treating both turbine inlet branches separately, it is possible to obtain two different efficiency maps linked to the mass flow parameter maps, as shown in Figure 6 (here, the experimental results of dual-volute turbocharger are shown, since the followed procedure is the same as for the twin-entry turbine that was determined in [12]). The mass flow parameter and efficiency data are normalized by the experimental maximum values of every scroll.