22-Mec-A1 Applied Thermodynamics and Heat Transfer · May 2013
Question 2 of 8: Two-Stage Reheat Steam Turbine with Extraction
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper: National Examinations — 07-Mec-A1 Applied Thermodynamics and Heat Transfer, May 2013. Open-book, 3-hour paper. Part A (Thermodynamics, Q1–Q4) and Part B (Heat Transfer, Q5–Q8); a complete paper is any five questions — three from one part and two from the other. Full worked solutions to all eight questions are given below.
Reference texts: Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed., McGraw-Hill) — gas cycles, steam tables, reheat Rankine and vapour-compression refrigeration; Çengel & Ghajar, Heat and Mass Transfer (6th ed.) and Incropera, DeWitt, Bergman & Lavine, Fundamentals of Heat and Mass Transfer (8th ed., Wiley) — conduction with critical radius, natural-convection and internal-flow correlations, and the ε–NTU heat-exchanger method. Freon-12 and steam property data are taken from the appendix supplied with the exam and standard tables.
Question 2: Two-Stage Reheat Steam Turbine with Extraction (equal value)
Given. Throttle steam 4.50 MPa / 350 °C; HP exhaust and reheat pressure 150 kPa; process extraction 10,900 kg/hr = 3.028 kg/s at 150 kPa; reheat to 300 °C; condenser 7.5 kPa; $\dot W=3730$ kW; $\eta_{HP}=0.84$, $\eta_{LP}=0.81$.
State
Condition
h (kJ/kg)
s (kJ/kg·K)
1 — HP inlet
4.5 MPa, 350 °C
3081.3
6.518
2 — HP exhaust
150 kPa (wet, actual)
2527.7
—
3 — reheat exit
150 kPa, 300 °C
3073.1
8.028
4 — LP exhaust
7.5 kPa (wet, actual)
2612.8
—
Find. The boiler steam-generation rate $\dot m_1$ (kg/s) that delivers 3730 kW of turbine output while bleeding 3.028 kg/s at 150 kPa.
Figure 2 — T–s schematic: 1→2 high-pressure expansion to 150 kPa, extraction of 10,900 kg/hr, 2→3 reheat at 150 kPa to 300 °C, 3→4 low-pressure expansion to 7.5 kPa. Actual (irreversible) states lie to the right of the isentropic end points.
Approach. Fix each stage exit with an isentropic end-state and the given stage efficiency, compute the specific work of each stage, then write a power balance with the full flow through the HP stage and the reduced flow (after extraction) through the LP stage.
HP stage exit (150 kPa). Isentropic: $s_{2s}=s_1=6.518$; with $s_f=1.4337$, $s_{fg}=5.7894$ the quality is $x_{2s}=(6.518-1.4337)/5.7894=0.878$, so $h_{2s}=467.13+0.878(2226.0)=2422.2$ kJ/kg. Applying the efficiency,$$h_2=h_1-\eta_{HP}(h_1-h_{2s})=3081.3-0.84(659.1)=2527.7\ \text{kJ/kg},\quad \boxed{w_{HP}=553.6\ \text{kJ/kg}}.$$
Reheat and LP stage exit (7.5 kPa). After reheat, $h_3=3073.1$, $s_3=8.028$. Isentropic to 7.5 kPa ($s_f=0.5764$, $s_{fg}=7.6750$): $x_{4s}=(8.028-0.5764)/7.6750=0.971$, $h_{4s}=168.79+0.971(2406.0)=2504.8$ kJ/kg. Then$$h_4=h_3-\eta_{LP}(h_3-h_{4s})=3073.1-0.81(568.3)=2612.8\ \text{kJ/kg},\quad \boxed{w_{LP}=460.3\ \text{kJ/kg}}.$$
Power balance and boiler capacity. The full flow $\dot m_1$ passes the HP stage; only $(\dot m_1-\dot m_\text{ext})$ reaches the LP stage after the 3.028 kg/s bleed. With $\dot W=3730$ kW,$$\dot m_1 w_{HP}+(\dot m_1-\dot m_\text{ext})w_{LP}=\dot W\;\Rightarrow\;\dot m_1=\frac{3730+3.028(460.3)}{553.6+460.3}=\boxed{5.05\ \text{kg/s}}\;(\approx 18{,}200\ \text{kg/hr}).$$
Check
The state properties are read from steam tables with linear interpolation at 4.5 MPa/350 °C and logarithmic-in-pressure interpolation for the 150 kPa, 300 °C reheat point; a ±0.1 % shift in the tabulated enthalpies moves $\dot m_1$ by only a few hundredths of a kg/s. "Boiler capacity" is taken as the main steam generated at 4.5 MPa (the reheater is a separate duty).
