Explain velocity triangles in turbomachinery and derive Euler's turbomachinery equation.

Velocity triangles relate absolute velocity (C), relative velocity (W), and blade velocity (U) at inlet and outlet of turbomachine rotors: C = W + U (vector addition). Components: C_u (tangential/whirl) and C_m (meridional/flow). Euler's turbomachinery equation from angular momentum: specific work w = U₂C_u2 - U₁C_u1, positive for compressors/pumps (energy added), negative for turbines (energy extracted). For pumps, head H = (U₂C_u2 - U₁C_u1)/g. Design parameters: flow coefficient φ = C_m/U, head coefficient ψ = gH/U², degree of reaction R = (W₂²-W₁²)/(C₂²-C₁²+W₂²-W₁²). Blade angles determined from triangles: tan β = C_m/(U-C_u). Slip factor σ = C_u2_actual/C_u2_ideal accounts for imperfect guidance. Understanding triangles essential for blade design, off-design analysis, and performance prediction.