## 2 Literature Review
### Turbocharging Map Scaling  
Matching a turbocharger map to any given engine model is a difficult task to accomplish; several different methods throughout literature have been attempted. However, all methods start with the same fundamental principle of scaling the map in terms of dimensionless coefficients. By using this method, a map is obtained that is representative of a family of turbochargers of similar performances but of different operating conditions and sizes. For turbomachinery in the gas turbine industry, the following scaling law relationship Equation (1) is used to define the pressure ratio as a function of rotor tip Mach number, the mass flow coefficient, the Reynolds number, and the Mach number based on inlet conditions [11]. \[\frac{P_{d}}{P_{u}}=f\bigg{[}\frac{N_{TC}\,D}{\sqrt{RT_{u}}},\ \frac{\dot{m} \sqrt{RT_{u}}}{P_{u}D^{2}},R_{e},\ M_{n}\bigg{]} \tag{1}\]  
Equation (1) shows the pressure ratio for a compression process. For a turbine, it would be for the expansion process which would be the inverse of the left-hand side of Equation (1), but the right-hand side stays the same. The above equation allows for the ability to treat a tested turbocharger performance as a black box system, and similar pressure ratios can be obtained for compressors and turbines of the same family by keeping these parameters constant at each point. Thus, this assumption leads to the first stage used in most literature to scale maps to their dimensionless coefficients, giving a generic map that will govern the family. This is extremely useful when designing new equipment,  
Figure 1: Comparison of turbine total to static efficiency vs. blade-to-gas-speed ratio between a radial and axial turbine. as it is far easier and quicker to simply scale a map theoretically to test turbomachinery, at various diameters, than to manufacture and test. The laws can yield comparable results for differently sized turbomachinery; however, an exact match between the similar turbomachinery cannot be achieved [12]. Therefore, to simplify the scaling of turbomachinery, it can be assumed for small scaling factors of diameter that the Reynolds number can be free to vary and that the gas-specific properties of the flow are constant [13]. This assumption has also been used in an alternative method for scaling using the polytropic infinitesimal stage [14]. Therefore, when scaling, Equation (1) can be further reduced to Equation (2). \[\frac{P_{d}}{P_{u}}=f\bigg{[}\frac{N_{TC}\,D}{\sqrt{T_{u}}},\,\,\frac{in\, \sqrt{T_{u}}}{P_{u}D^{2}}\bigg{]} \tag{2}\]  
Equation (2) along with the black-box treatment of the turbomachinery can be better put into context by looking at a performance map shown below in Figure 2.  
Even with the scaling laws simplified, matching a turbocharger to an engine is a far more difficult task in itself, as the turbomachinery diameter is not chosen but needs to be obtained to provide the desired performance, and this task needs to be accomplished simultaneously for both compressor and turbine. [PERSON] et al. [15] showed good results for scaling process by first identifying the compressor diameter and the torque required from the turbine to reach the necessary boost, yielding good turbocharger performance results at various engine sizes. However, the author does not show whether the results were matched to specific engine performances. Nevertheless, the Ricardo WAVE balances the turbocharger shaft speed using a torque balance equation, meaning this method is viable [16]. When scaling for pressure ratios, the results shown have a good correlation between differently sized turbomachinery. However, most methods investigated have not considered scaling the efficiency, which leads to the questions, should the efficiency be scaled, or is it acceptable to assume the change as negligible to the final results? The efficiency can be established as a function of the corrected shaft speed and the pressure ratio as shown in Equation (3) [15].