Two-Stage Reheat Turbine — Boiler Capacity: 17-Phys-B6 Applied Thermodynamics and Heat Transfer, May 2013, Q2

Question 2: Two-Stage Reheat Turbine — Boiler Capacity (Part A, 20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Steam enters the HP stage at $4.50$ MPa, $350^{\circ}\text{C}$; HP exhaust/extraction at $150$ kPa with $\dot m_{extract}=10{,}900$ kg/hr; remaining steam reheated to $300^{\circ}\text{C}$ at $150$ kPa, expanded through the LP stage to $7.5$ kPa; $\dot W_{turbine}=3730$ kW; $\eta_{HP}=84\%$, $\eta_{LP}=81\%$.

Find. The boiler (total steam) capacity $\dot m_{tot}$ in kg/s.

Figure — T–s diagram: HP expansion 1→2 (dashed 2s = isentropic, solid 2 = actual), constant-pressure reheat 2→3 at 150 kPa, LP expansion 3→4 (dashed 4s = isentropic, solid 4 = actual, exiting into the superheated region at 7.5 kPa).

Approach. Get steam-table properties at each state (open-book — any standard steam table), apply each stage's isentropic efficiency to get the actual enthalpy drop, then close a power balance across both stages with the extraction removed between them.

State 1 — HP inlet. At $4.50$ MPa, $350^{\circ}\text{C}$ (superheated): $h_1=3081.5$ kJ/kg, $s_1=6.515$ kJ/kg·K.
State 2s/2 — HP exhaust. Isentropic expansion to $150$ kPa ($s_{2s}=s_1$) lands in the two-phase region at $T_{sat}=111.4^{\circ}\text{C}$, giving $h_{2s}=2421.0$ kJ/kg. Applying the stage efficiency, $$h_2=h_1-\eta_{HP}(h_1-h_{2s})=3081.5-0.84(3081.5-2421.0)=\boxed{2526.7\ \text{kJ/kg}},$$ so $w_{HP}=h_1-h_2=554.8$ kJ/kg.
State 3 — after reheat. At $150$ kPa, $300^{\circ}\text{C}$ (superheated): $h_3=3073.3$ kJ/kg, $s_3=8.028$ kJ/kg·K.
State 4s/4 — LP exhaust. Isentropic expansion to $7.5$ kPa ($s_{4s}=s_3$) gives a wet state $h_{4s}=2504.6$ kJ/kg ($x_{4s}=0.971$). Applying the LP stage efficiency, $$h_4=h_3-\eta_{LP}(h_3-h_{4s})=3073.3-0.81(3073.3-2504.6)=\boxed{2612.6\ \text{kJ/kg}}$$ (this actually lands just INTO the superheated region, $T_4\approx60.5^{\circ}\text{C}$, since the 81%-efficient expansion generates enough irreversibility to keep the steam dry — a useful, if slightly counter-intuitive, check on the arithmetic), so $w_{LP}=h_3-h_4=460.7$ kJ/kg.
Boiler capacity from the power balance. The full flow $\dot m_{tot}$ does HP work; only the remaining flow $\dot m_{tot}-\dot m_{extract}$ (with $\dot m_{extract}=10{,}900/3600=3.028$ kg/s) does LP work: $$\dot W_{turbine}=\dot m_{tot}\,w_{HP}+(\dot m_{tot}-\dot m_{extract})\,w_{LP}.$$ Solving for $\dot m_{tot}$, $$\dot m_{tot}=\frac{\dot W_{turbine}+\dot m_{extract}\,w_{LP}}{w_{HP}+w_{LP}} =\frac{3730+3.028\times460.7}{554.8+460.7}=\boxed{5.05\ \text{kg/s}}.$$

Final results — Question 2
Quantity	Value
$h_2$ (HP actual exit)	$2526.7$ kJ/kg
$h_4$ (LP actual exit)	$2612.6$ kJ/kg ($T_4\approx60.5^{\circ}\text{C}$, superheated)
$w_{HP}$, $w_{LP}$	$554.8$, $460.7$ kJ/kg
Boiler capacity $\dot m_{tot}$	$\boxed{5.05\ \text{kg/s}}$ ($\approx18{,}170$ kg/hr)

Topic: Multi-stage steam-turbine power balance with a process extraction and reheat between stages.

Key relations: Isentropic stage efficiency $\eta_s=(h_{in}-h_{out})/(h_{in}-h_{out,s})$; steady-flow power balance $\dot W=\sum\dot m\,\Delta h$ per stage, with the mass flow reduced by any extraction taken between stages.

Why this works: the HP stage processes the FULL boiler flow (extraction happens at its exit), while the LP stage only sees what remains after the process steam is pulled off — so the two stages' work terms are weighted by two different mass flow rates in the same power balance, not one common $\dot m$.

Common pitfall: applying $\dot m_{tot}$ to both stages (ignoring the extraction) or applying the stage efficiency to the wrong enthalpy drop direction (efficiency always REDUCES the actual work recovered relative to the isentropic ideal, so actual $h_{out}$ is HIGHER than $h_{out,s}$ for an expansion).

Source: Çengel & Boles, Thermodynamics: An Engineering Approach, Ch. 10 (Rankine cycle with reheat and process extraction), Ch. 7 (isentropic turbine efficiency).

Question 2 of 8: Two-Stage Reheat Turbine — Boiler Capacity

Question 2: Two-Stage Reheat Turbine — Boiler Capacity (Part A, 20 marks)