In a multi-stage axial flow compressor, the work done factor $\Omega$ (or $\lambda$) is introduced in design calculations to account for:

(A) The growth of annulus wall boundary layers that creates non-uniform spanwise axial velocity profiles across downstream stages

(B) Mechanical friction losses in the shaft bearings and drive gears

(C) Combustion pressure loss across the flame holder

(D) Supersonic shock wave detachment on transonic rotor blade tips

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In axial compressors, 1D mean-line design assumes a uniform axial velocity $C_a$ across the blade height. However, in reality, viscous boundary layers develop along both the inner hub and outer casing walls as flow progresses through successive stages. This boundary layer displacement forces fluid through the central annulus core at a higher actual axial velocity than predicted by 1D continuity, reducing the aerodynamic blade incidence and decreasing the stage temperature rise. ### 2. Analytical Definition The work done factor $\Omega$ is defined as: $$\Omega = \frac{\text{Actual Work Input}}{\text{Theoretical Work Input based on mean-line } C_a} < 1$$ Typically, $\Omega \approx 0.98$ for Stage 1 and decreases steadily to $\approx 0.85$ for Stage 8+. ### 3. Distractor Forensics - **Option A:** Correct fluid-mechanical reason. - **Option B:** Bearing friction is accounted for by mechanical transmission efficiency $\eta_m$, not work done factor. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Work done factor is strictly a fluid dynamic blockage effect, NOT mechanical friction. - **Standard Reference:** Saravanamuttoo, H.I.H., *Gas Turbine Theory*, Chapter 5.

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