Power cycles: the Rankine cycle and combined heat and power

Kingfisher's boilers raise about 60 t/h of steam, and nearly all of it is used at 400 kPa to heat the distillation columns, evaporators and dryers. At the moment the boilers make steam at that low pressure directly. Ingrid's study asks whether the boilers should instead make high-pressure steam at 4 MPa and 400 °C, pass it through a back-pressure turbine to generate electricity, and send the turbine exhaust to the existing 400 kPa header. To answer, the team needs the thermodynamics of steam power cycles: what a conventional power plant achieves, why its efficiency is limited, and why combining power generation with process heating changes the economics completely.

Why not a Carnot cycle?

Lesson 7 showed that a Carnot cycle sets the upper limit for any engine between two temperatures. A Carnot cycle using steam would take in heat at constant temperature by boiling water, expand the vapour isentropically, condense it at constant temperature, and compress the wet mixture isentropically back to saturated liquid. Two steps are impractical. Compressing a two-phase mixture is very difficult for real pumps and compressors, and expanding wet steam to low quality erodes turbine blades. Also, holding all heat input at the boiling temperature wastes the high flame temperature available in a boiler. The practical alternative is the Rankine cycle.

The ideal Rankine cycle

The Rankine cycle has four steps. Feedwater is pumped as a liquid from condenser pressure to boiler pressure, which takes little work because liquids have small volumes. The boiler heats it, boils it and superheats it at constant pressure. The turbine expands the steam to condenser pressure, ideally isentropically. The condenser turns the exhaust back to saturated liquid at constant pressure, rejecting heat to cooling water. Each step is a steady-flow process, so the energy balances of Lesson 10 apply to each one.

Improving the Rankine cycle

Three changes raise the average temperature at which heat is added or lower the temperature at which it is rejected, and each improves efficiency. Raising the boiler pressure raises the boiling temperature, though at fixed turbine inlet temperature it also makes the exhaust wetter. Superheating to a higher temperature adds heat at higher temperatures and dries the exhaust. Lowering the condenser pressure lowers the rejection temperature, down to the limit set by available cooling water. Large utility plants combine all three, with supercritical boiler pressures of 25 MPa or more and steam temperatures near 600 °C, reaching efficiencies of about 40 to 45 percent.

Reheating, in which partly expanded steam returns to the boiler for further heating before a second turbine stage, raises efficiency modestly and, more importantly, keeps the turbine exhaust dry enough to protect the blades. Modern turbines tolerate exhaust qualities down to about 0.88 to 0.90; the ideal cycle above, with an exhaust quality of 0.816, would be too wet in practice.

Regeneration with feedwater heaters

In the simple cycle, cold feedwater from the condenser enters the boiler and must be heated from about 46 °C using high-temperature heat. Regeneration heats the feedwater with steam extracted from the turbine part way through its expansion. That steam has already done some work, and its remaining latent heat preheats the feedwater, so the boiler adds heat at a higher average temperature. An open feedwater heater mixes the extracted steam directly with the feedwater; a closed heater transfers heat through tubes. Large plants use six to eight heaters in series.

Combined heat and power

A condensing power plant throws away about two thirds of its fuel energy in the condenser because the heat is rejected near room temperature, where it has little use. Many process plants, Kingfisher included, need large amounts of low-pressure steam anyway. Combined heat and power, or cogeneration, puts the turbine between the high-pressure boiler and the process: the turbine exhausts at the process steam pressure, and the process, not a condenser, uses the exhaust heat. Almost all the extra fuel burned to make high-pressure steam turns into electricity, because the heat that would have been rejected is already needed.

# Rankine cycles and combined heat and power for Kingfisher Bioenergy (approximate steam-table values)
h1, s1 = 3213.6, 6.7690                         # 4 MPa, 400 C superheated steam
sat = {10: (191.83, 2392.8, 0.6493, 7.5009, 0.001010),     # P (kPa): hf, hfg, sf, sfg, vf
       400: (604.74, 2133.8, 1.7766, 5.1193, 0.001084)}
P_boiler = 4000.0                               # kPa

def expand(P, s=s1):
    hf, hfg, sf, sfg, vf = sat[P]
    x = (s - sf) / sfg
    return hf + x * hfg, x

def simple_rankine(eta_t=1.0, eta_p=1.0, P_cond=10):
    hf, hfg, sf, sfg, vf = sat[P_cond]
    h2s, x2s = expand(P_cond)
    wt = eta_t * (h1 - h2s)
    wp = vf * (P_boiler - P_cond) / eta_p
    q_in = h1 - (hf + wp)
    return wt, wp, q_in, (wt - wp) / q_in, x2s

wt, wp, q, eta, x = simple_rankine()
print(f"Ideal Rankine 4 MPa/400 C to 10 kPa: x2 = {x:.4f}, turbine {wt:.1f}, pump {wp:.2f}, heat in {q:.1f} kJ/kg, efficiency {eta:.3f}")
wt, wp, q, eta, x = simple_rankine(0.85, 0.80)
print(f"Real Rankine (turbine 0.85, pump 0.80): net work {wt - wp:.1f} kJ/kg, efficiency {eta:.3f}")

# Regenerative cycle with one open feedwater heater at 400 kPa (ideal turbine and pumps)
h6, _ = expand(400)                              # extraction steam
h2s, _ = expand(10)
hf10, _, _, _, vf10 = sat[10]; hf400, _, _, _, vf400 = sat[400]
h_p1 = hf10 + vf10 * (400 - 10)                  # condensate pump outlet
y = (hf400 - h_p1) / (h6 - h_p1)                 # fraction extracted, from the heater energy balance
h_p2 = hf400 + vf400 * (P_boiler - 400)          # feed pump outlet
wt = (h1 - h6) + (1 - y) * (h6 - h2s)
wp = (1 - y) * vf10 * (400 - 10) + vf400 * (P_boiler - 400)
q_in = h1 - h_p2
print(f"Regenerative cycle: extraction fraction y = {y:.4f}, net work {wt - wp:.1f} kJ/kg, heat in {q_in:.1f} kJ/kg, efficiency {(wt - wp) / q_in:.3f}")

# Combined heat and power: back-pressure turbine exhausting to the 400 kPa process header
m = 60.0 / 3.6                                  # kg/s of process steam (60 t/h)
eta_t, eta_boiler = 0.80, 0.84
w = eta_t * (h1 - h6)
power = m * w
extra_fuel = power / eta_boiler                 # extra fuel energy compared with raising the steam at low pressure
print(f"CHP: {m:.2f} kg/s through the turbine gives {power/1000:.2f} MW of shaft power")
print(f"Extra boiler fuel about {extra_fuel/1000:.2f} MW, so fuel-to-power efficiency is about {power / extra_fuel:.2f}")
print(f"A condensing power plant at 35 % efficiency would burn {power / 0.35 / 1000:.1f} MW of fuel for the same power")

Ideal Rankine 4 MPa/400 C to 10 kPa: x2 = 0.8159, turbine 1069.6, pump 4.03, heat in 3017.7 kJ/kg, efficiency 0.353
Real Rankine (turbine 0.85, pump 0.80): net work 904.1 kJ/kg, efficiency 0.300
Regenerative cycle: extraction fraction y = 0.1654, net work 975.7 kJ/kg, heat in 2605.0 kJ/kg, efficiency 0.375
CHP: 16.67 kg/s through the turbine gives 7.04 MW of shaft power
Extra boiler fuel about 8.38 MW, so fuel-to-power efficiency is about 0.84
A condensing power plant at 35 % efficiency would burn 20.1 MW of fuel for the same power

The program confirms each result. It also shows that a real condensing cycle with turbine and pump efficiencies of 0.85 and 0.80 reaches only about 30 percent, which puts the combined heat and power figure of 84 percent in perspective.