In 1D steady adiabatic compressible flow of an ideal gas through a constant-area duct with wall friction (Fanno flow), as an initially subsonic stream ($M_1 < 1$) flows along the duct in the downstream direction:

(A) The Mach number increases toward unity ($M \to 1$)

(B) The total (stagnation) pressure decreases continuously ($p_0 \downarrow$)

(C) The total (stagnation) temperature remains strictly constant ($T_0 = \text{constant}$)

(D) The static pressure increases continuously along the duct ($p \uparrow$)

Answer Key:

A, B, C

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Energy Conservation:** Because the duct is adiabatic ($q = 0$) and does no shaft work ($w = 0$), $h_0 = c_p T_0 = \text{constant}$ throughout (Statement C is TRUE). 2. **Stage 2: Second Law of Thermodynamics:** Wall shear friction is an irreversible dissipative process. Therefore, entropy increases ($ds > 0$). 3. **Stage 3: Stagnation Pressure Loss:** $\Delta s = -R \ln(p_{02}/p_{01}) > 0 \implies p_{02} < p_{01}$ (Statement B is TRUE). 4. **Stage 4: Mach Number and Pressure Trend:** Fanno differential relation: $\frac{dM^2}{M^2} = \frac{\gamma M^2 (1 + \frac{\gamma-1}{2} M^2)}{1 - M^2} \frac{4f dx}{D}$. For $M < 1$, $dM^2/dx > 0$ (Mach number rises toward 1, Statement A is TRUE). Consequently, static pressure drops ($dp/dx < 0$, Statement D is FALSE). 5. **Stage 5: Official Reference:** Ascher H. Shapiro, *Compressible Fluid Flow*, Vol. 1, Chapter 6.

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