Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format: National Examination 07-Mec-A6-1 Fluid Machinery, December 2016 — closed book, 3 hours. Section A (Calculative, Q1–Q5) and Section B (Descriptive, Q6–Q8); candidates do four of A and two of B for 60 marks. All eight questions are solved as a study resource. Values marked “from the figure” are read from the examination attachment drawings.
Reference texts: Dixon & Hall, Fluid Mechanics and Thermodynamics of Turbomachinery (7th ed.); Cohen, Rogers & Saravanamuttoo, Gas Turbine Theory (6th ed.); Turton, Principles of Turbomachinery; Çengel & Boles, Thermodynamics (9th ed.); Fox & McDonald, Introduction to Fluid Mechanics (9th ed.). Constants from the paper: $g=9.81\ \text{m/s}^2$, $c_p=1.005\ \text{kJ/kg\,K}$, $k=1.4$, $R=0.287\ \text{kJ/kg\,K}$, $\rho_{water}=1000\ \text{kg/m}^3$, $p_{atm}=100\ \text{kPa}$, $p_{vap}=2.34\ \text{kPa}$.
Given. A two-spool axial compressor with $T_1=80\,{}^\circ\text{C}=353.15\ \text{K}$, $p_1=80\ \text{kPa}$, inlet Mach $M=0.49$, inlet area $A=0.80\ \text{m}^2$, LP pressure ratio $r_{LP}=3$, HP pressure ratio $r_{HP}=5$ (7 stages each), and $\eta_c=0.90$ for both spools.
Given data
Quantity
Symbol
Value
Inlet temperature
$T_1$
353.15 K
Inlet pressure
$p_1$
80 kPa
Inlet Mach number
$M$
0.49
Inlet area
$A$
0.80 m²
LP / HP pressure ratio
$r_{LP},r_{HP}$
3 , 5
Isentropic efficiency
$\eta_c$
0.90
Find. LP and HP exit temperatures, the fifth-HP-stage bleed temperature, the inlet velocity, the mass flow, and the compressor drive power.
Fig. 1.1 — T–s diagram of the two-spool compression. Actual (irreversible) paths 1→2 (LP) and 2→3 (HP) lie to the right of the isentropic paths 1→2s, 2→3s; the vertical rise gives the temperature rise.
Approach. Treat each spool with the isentropic relation $T_s=T\,r^{(k-1)/k}$ corrected by $\eta_c$; the inlet velocity follows from the Mach number and sonic speed, the mass flow from continuity, and the power from a steady-flow energy balance $\dot P=\dot m c_p\Delta T$.
HP exit temperature. The HP spool takes air at $T_2$: $$T_{3s}=T_2 r_{HP}^{(k-1)/k}=497.8\times5^{0.2857}=788.5\ \text{K},\qquad T_3=T_2+\frac{T_{3s}-T_2}{\eta_c}=\boxed{820.8\ \text{K}\ (547.7\,{}^\circ\text{C})}$$
Bleed after the 5th of 7 HP stages. With equal pressure ratio per stage, the pressure ratio to the fifth stage is $r_{HP}^{5/7}=5^{0.714}=3.157$, so $$T_{5s}=T_2\,(3.157)^{0.2857}=691.4\ \text{K},\qquad T_5=T_2+\frac{T_{5s}-T_2}{\eta_c}=\boxed{712.9\ \text{K}\ (439.7\,{}^\circ\text{C})}$$
Inlet velocity. Sonic speed $a=\sqrt{kRT_1}=\sqrt{1.4\times287\times353.15}=376.7\ \text{m/s}$, hence $$V_1=M a=0.49\times376.7=\boxed{184.6\ \text{m/s}}$$ (this agrees with the 184 m/s stated in Question 2).
Mass flow. Inlet density $\rho_1=\dfrac{p_1}{RT_1}=\dfrac{80000}{287\times353.15}=0.789\ \text{kg/m}^3$, so by continuity $$\dot m=\rho_1 A V_1=0.789\times0.80\times184.6=\boxed{116.6\ \text{kg/s}}$$ (agrees with the 116 kg/s of Question 2).
Compressor drive power. Steady-flow energy balance across both spools: $$\dot P=\dot m c_p(T_3-T_1)=116.6\times1.005\times(820.8-353.15)=\boxed{54.8\ \text{MW}}$$
