22-Mec-A1 Applied Thermodynamics and Heat Transfer · December 2013
Question 2 of 8: Reheat Rankine Power Plant
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Reference texts: Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed., McGraw-Hill) — ideal-gas processes, gas power cycles, 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) — natural-convection and internal-flow correlations, lumped-capacitance transients, radiation exchange, and the LMTD/ε–NTU heat-exchanger method. Freon-12 property data are read from the appendix supplied with the exam; steam properties from standard tables. This is an open-book, equal-value paper; candidates answer any five of the eight questions (three from one Part and two from the other) — all eight are worked here as a study resource.
Question 2: Reheat Rankine Power Plant (equal value)
Given. A single mass flow $\dot m$ circulates. HP inlet 3.5 MPa / 350 °C; HP exhaust and reheat pressure 0.5 MPa; reheat to 350 °C; LP exhaust (condenser) 7.5 kPa; condensate leaves at 30 °C and is pumped to 3.5 MPa. $\eta_{T}=0.85$ (each turbine), $\eta_{P}=0.80$, and the combined turbine output is $\dot W_T=1000$ kW.
State
Condition
h (kJ/kg)
s (kJ/kg·K)
1 — HP inlet
3.5 MPa, 350 °C
3104.7
6.665
2 — HP exhaust
0.5 MPa (wet, actual)
2745.1
—
3 — reheat exit
0.5 MPa, 350 °C
3168.1
7.634
4 — LP exhaust
7.5 kPa (wet, actual)
2498.9
—
Find. (a) the circulating steam rate $\dot m$; (b) the pump power $\dot W_P$; (c) the cycle thermal efficiency $\eta_\text{th}$.
Figure 2 — T–s schematic: 1→2 high-pressure expansion to 0.5 MPa, 2→3 reheat at 0.5 MPa to 350 °C, 3→4 low-pressure expansion to 7.5 kPa. Actual (irreversible) exhaust states lie to the right of the isentropic end points. States 1 and 3 are both at 350 °C; state 2 sits essentially on the saturated-vapour line ($x_2\approx0.998$) and state 4 inside the dome ($x_4\approx0.97$).
Approach. Fix each turbine exhaust from its isentropic end state and the 85 % efficiency, sum the stage works, and use the 1000 kW output to size $\dot m$. The pump is handled with $w_P=v_f\,\Delta P/\eta_P$, and the thermal efficiency compares net work to the boiler-plus-reheat heat input.
HP stage (3.5 MPa → 0.5 MPa). Isentropic $s_{2s}=s_1=6.665$; at 0.5 MPa ($s_f=1.8607$, $s_{fg}=4.9606$) the quality is $x_{2s}=(6.665-1.8607)/4.9606=0.968$, so $h_{2s}=640.2+0.968(2108)=2681.7$ kJ/kg. Applying the efficiency,$$h_2=h_1-\eta_T(h_1-h_{2s})=3104.7-0.85(423.0)=2745.1,\quad \boxed{w_{HP}=359.6\ \text{kJ/kg}}.$$
LP stage (reheat 0.5 MPa, 350 °C → 7.5 kPa). After reheat $h_3=3168.1$, $s_3=7.634$. Isentropic to 7.5 kPa ($s_f=0.5764$, $s_{fg}=7.6750$): $x_{4s}=(7.634-0.5764)/7.6750=0.920$, $h_{4s}=168.8+0.920(2405.3)=2380.9$ kJ/kg. Then$$h_4=h_3-\eta_T(h_3-h_{4s})=3168.1-0.85(787.3)=2498.9,\quad \boxed{w_{LP}=669.2\ \text{kJ/kg}}.$$
(a) Steam mass flow. The same $\dot m$ passes both turbines, so the combined specific work is $w_T=w_{HP}+w_{LP}=1028.7$ kJ/kg and$$\dot m=\frac{\dot W_T}{w_T}=\frac{1000}{1028.7}=\boxed{0.972\ \text{kg/s}}.$$
(b) Pump power. Pumping subcooled liquid ($v_f\approx0.001004\ \text{m}^3/\text{kg}$) from 7.5 kPa to 3500 kPa, $w_{P,s}=v_f\Delta P=0.001004(3492.5)=3.51$ kJ/kg; with $\eta_P=0.80$, $w_P=3.51/0.80=4.38$ kJ/kg, so$$\dot W_P=\dot m\,w_P=0.972(4.38)=\boxed{4.26\ \text{kW}}.$$
(c) Thermal efficiency. Heat is added in the boiler ($h_1-h_{\text{pump out}}$, with $h_{\text{pump out}}=h_f(30^\circ\text{C})+w_P=125.7+4.4=130.1$) and in the reheater ($h_3-h_2$): $q_\text{in}=(3104.7-130.1)+(3168.1-2745.1)=3397.6$ kJ/kg. With net work $w_\text{net}=w_T-w_P=1024.3$,$$\eta_\text{th}=\frac{w_\text{net}}{q_\text{in}}=\frac{1024.3}{3397.6}=\boxed{0.302\;(30.2\%)}.$$
Check
Steam properties are read from standard tables with linear interpolation at 3.5 MPa / 350 °C (between the 3.0 and 4.0 MPa columns); a ±0.1 % shift in the tabulated enthalpies moves $\dot m$ by only a few thousandths of a kg/s. The pump work is a <0.5 % correction to the cycle output, as expected for a liquid.
