# Investigation of the Effects of Different Working Fluids on Compressor Cascade Performance
## 2 Theoretical Analysis  
The physical properties of working fluid are significant factors affecting the compressor performance. For subsonic compressor cascades, coefficient of total pressure loss and static pressure ratio are considered as important indexes in order to judge compressor cascades performance. To have further understanding of the influence of physical properties of working fluid on performances of the subsonic compressor cascade, the theoretical analysis method will be used in this section to establish the relationship of physical properties with the coefficient of total pressure loss and static pressure ratio in compressor cascade. The flow in the compressor cascade may be considered as one-dimensional (1D) steady flow. For one-dimensional steady flow, the continuity equation can be written as follows:  
\[\frac{d\rho}{\rho}+\frac{dV}{V}+\frac{dA}{A}\;=\;0 \tag{1}\]  
The flow in compressor cascade can be almost adiabatic, so the Bernoulli equation becomes:  
\[\frac{dp}{p}+\gamma Ma^{2}(\frac{dV}{V}+2f\frac{dx}{D})\;=\;0 \tag{2}\]The energy equation becomes:  
\[\frac{dT}{T}+(\gamma-1)Ma^{2}\frac{dV}{V}\ =\ 0 \tag{3}\]  
The logarithmic differential of the equation of state can be obtained as follows:  
\[\frac{dp}{p}-\frac{d\rho}{\rho}-\frac{dT}{T}\ =\ 0 \tag{4}\]  
According to the definition of Mach number:  
\[Ma\ =\ \frac{V^{2}}{\gamma R_{g}T} \tag{5}\]  
Equation (5) can also be written as:  
\[\frac{dV}{V}\ =\ \frac{dMa}{Ma}+\frac{dT}{2T} \tag{6}\]  
And:  
\[\frac{dp^{*}}{p^{*}}\ =\ \frac{dp}{p}+\frac{\gamma Ma^{2}}{(1+\frac{\gamma-1}{2 }Ma^{2})}\frac{dMa}{Ma} \tag{7}\]  
From Equations (1)-(4), (6) and (7), the Equations can be obtained as:  
\[\frac{dp^{*}}{p^{*}}\ =\ -2\gamma Ma^{2}f\frac{dx}{D} \tag{8}\]  
\[\frac{dp}{p}\ =\ \frac{\gamma Ma^{2}}{1-Ma^{2}}\frac{dA}{A}+\frac{1+(\gamma-1 )Ma^{2}}{1-Ma^{2}}\frac{dp^{*}}{p^{*}} \tag{9}\]  
Static pressure ratio and coefficient of total pressure loss can be written as:  
\[\omega\ =\ \frac{p_{1}^{*}-p_{2}^{*}}{p_{1}^{*}} \tag{10}\]  
\[\frac{dp}{p}+\gamma Ma^{2}(\frac{dV}{V}+2f\frac{dx}{D})\;=\;0 \tag{2}\]0  
With the use of different working fluids in same cascade, the relationship of total pressure loss coefficient becomes:  
\[\frac{dp}{p}+\gamma Ma^{2}(\frac{dV}{V}+2f\frac{dx}{D})\;=\;0 \tag{2}\]1  
The friction factor f is related to the Reynolds number and the roughness of the compressor cascade. Once Reynolds numbers of two working fluids are equal, Equation (12) can be written as:  
\[\frac{dp}{p}+\gamma Ma^{2}(\frac{dV}{V}+2f\frac{dx}{D})\;=\;0 \tag{2}\]2  
From Equations (9)-(11), Equation (9) can be obtained as:  
\[\frac{dp}{p}+\gamma Ma^{2}(\frac{dV}{V}+2f\frac{dx}{D})\;=\;0 \tag{2}\]3When different working fluids used in the same compressor cascade, the relationship between static pressure ratio becomes:  
\[\frac{dp}{p}+\gamma Ma^{2}(\frac{dV}{V}+2f\frac{dx}{D})\;=\;0 \tag{2}\]4  
As shown in Equation (13), compressor cascade coefficient of total pressure loss with working fluid \"A\" can be acquired when the coefficient of total pressure loss in the compressor cascade with working fluid \"B\", heat capacity ratio and Mach numbers of both fluids are already known. Working fluid \"A\" coefficient of total pressure loss will increase with the increased heat capacity ratio of working fluid \"A\". Static pressure ratio operating with working fluid \"A\" can be acquired when static pressure ratio and the coefficient of total pressure loss of working fluid \"B\", heat capacity ratio of both working fluids and Mach number of both working fluids are already known, as per Equation (15). Static pressure ratio of a compressor operating on working fluid \"A\" increase with the increase of the heat capacity ratio of working fluid \"A\" and decrease with the increase in total pressure loss coefficient.