Euler equation rewritten with fluid angles and mass flows
v 1 = u 1 tan β r 1 {\displaystyle v_{1}=u_{1}\tan \beta _{r1}}
v 2 = u 2 tan β 2 = ω r 2 − u 2 tan β r 2 {\displaystyle v_{2}=u_{2}\tan \beta _{2}=\omega r_{2}-u_{2}\tan \beta _{r2}}
T t 2 T t 1 = 1 + ( ω r 2 ) 2 c p T t 1 [ 1 − u 2 ω r 2 ( tan β r 2 + u 1 r 1 u 2 r 2 tan β 1 ) ] {\displaystyle {\frac {T_{t2}}{T_{t1}}}=1+{\frac {(\omega r_{2})^{2}}{c_{p}T_{t1}}}[1-{\frac {u_{2}}{\omega r_{2}}}(\tan \beta _{r2}+{\frac {u_{1}r_{1}}{u_{2}r_{2}}}\tan \beta _{1})]}
The blade speed dimensionless group:
( ω r 2 ) 2 c p T t 1 {\displaystyle {\frac {(\omega r_{2})^{2}}{c_{p}T_{t1}}}}
![{\displaystyle {\frac {T_{t2}}{T_{t1}}}=1+{\frac {(\omega r_{2})^{2}}{c_{p}T_{t1}}}[1-{\frac {u_{2}}{\omega r_{2}}}(\tan \beta _{r2}+{\frac {u_{1}r_{1}}{u_{2}r_{2}}}\tan \beta _{1})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae9dbdd166d5397bc3f0fce739279cde8a879d7)
