Reheat, regenerative and supercritical Rankine cycles

Lesson 5 ended with a puzzle. Raising the boiler pressure raises efficiency but wets the turbine exhaust, and even at high pressure the average temperature of heat addition stays far below the turbine inlet temperature because so much heat goes into warming cold feedwater. Large steam plants solve both problems with two modifications, reheat and regenerative feedwater heating, and the most advanced ones go further by operating above the critical pressure of water. To see what these changes are worth, Tomás builds a model of a utility-scale plant at 8 MPa and 500 °C with a 10 kPa condenser, and then adds the modifications one at a time. He also discovers that Northgate already has one of them: its deaerator is an open feedwater heater.

The ideal reheat Rankine cycle

In a reheat cycle, steam expands in a high-pressure turbine to an intermediate pressure, returns to the boiler to be reheated at roughly that pressure, and then expands in a low-pressure turbine to the condenser. Reheating moves the low-pressure expansion line to the right on the T-s diagram, so the exhaust is much drier. Total heat input and total turbine work are

Reheat is mainly a way to keep the exhaust dry so that higher boiler pressures can be used. Its effect on efficiency depends on the reheat pressure. If the reheat pressure is high, the added heat goes in at a high average temperature but the gain in exit quality is small; if it is low, the exhaust is very dry but the added heat goes in at a low average temperature. The best reheat pressure is usually about a quarter of the boiler pressure. Very large plants use double reheat, though the extra boiler and piping costs limit it to the largest, highest-pressure units.

The regenerative cycle and feedwater heaters

The second weakness of the simple cycle is the cold feedwater. In Tomás's base plant, the pump delivers water at about 46 °C to a boiler that boils at 295 °C, and all the heat needed to warm the water is added at low temperature. The regenerative cycle warms the feedwater internally with steam bled, or extracted, from the turbine. The bled steam has already done some work; it gives up its remaining latent heat to the feedwater instead of to the condenser. The average temperature of heat addition rises, and so does the efficiency.

In an open feedwater heater, also called a direct-contact heater, the bled steam mixes with the feedwater. Both streams enter at the same pressure, and the mixture leaves as saturated liquid at that pressure, which then needs its own pump to reach boiler pressure. With a fraction \(y\) of the steam bled at state 7, the heater's energy balance gives

In a closed feedwater heater, the bled steam condenses on the outside of tubes through which the feedwater flows, so the two streams can be at different pressures. The condensed steam, called drips, is either pumped forward into the feedwater line or cascaded backward through a trap to a lower-pressure heater or the condenser. Closed heaters avoid the need for a pump after each heater, but they are more expensive and cannot heat the feedwater quite to the saturation temperature of the bled steam. Most plants combine several closed heaters with one open heater that also serves as a deaerator.

That last point is Northgate's. Its deaerator at 120 kPa is an open feedwater heater: steam from the district main, or from the turbine exhaust, mixes with condensate returning from the buildings, heats it to saturation and drives out dissolved oxygen and carbon dioxide that would corrode the boilers. Graham has always thought of the deaerator as water treatment. Tomás points out that it also raises the feedwater temperature to about 105 °C, which is regeneration.

Modeling the improvements

The program runs three ideal cycles: (a) the simple cycle at 8 MPa and 500 °C, (b) the same with reheat at 1 MPa back to 500 °C, and (c) reheat plus one open feedwater heater at 0.5 MPa fed with steam bled from the low-pressure turbine. Superheated-steam properties at intermediate states are found by linear interpolation in the steam tables, which the program does explicitly.

# Reheat and regeneration for a utility-scale plant: 8 MPa, 500 C, 10 kPa condenser (ideal cycles)
def interp(x, x1, y1, x2, y2): return y1 + (x - x1) * (y2 - y1) / (x2 - x1)

# Saturation (kPa): hf, hfg, sf, sfg, vf; superheat (kPa, C): (h, s), rounded from standard steam tables
sat = {10: (191.81, 2392.1, 0.6492, 7.4996, 0.001010), 500: (640.09, 2108.0, 1.8604, 4.9603, 0.001093)}
sup = {(8000, 500): (3399.5, 6.7266), (1000, 200): (2828.3, 6.6956), (1000, 250): (2943.1, 6.9265),
       (1000, 500): (3479.1, 7.7642), (500, 300): (3064.6, 7.4614), (500, 400): (3272.4, 7.7956)}

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

h1, vf10 = sat[10][0], sat[10][4]
h3, s3 = sup[(8000, 500)]

# (a) simple cycle
wp = vf10 * (8000 - 10); h2 = h1 + wp
h4, x4 = wet(10, s3)
eta_a = ((h3 - h4) - wp) / (h3 - h2)
print(f"(a) simple:   eta = {eta_a:.3f}, exit quality = {x4:.3f}, wnet = {h3 - h4 - wp:.1f} kJ/kg")

# (b) reheat at 1 MPa back to 500 C
(hA, sA), (hB, sB) = sup[(1000, 200)], sup[(1000, 250)]
h4r = interp(s3, sA, hA, sB, hB); T4r = interp(s3, sA, 200, sB, 250)
h5, s5 = sup[(1000, 500)]
h6, x6 = wet(10, s5)
wt = (h3 - h4r) + (h5 - h6); qin = (h3 - h2) + (h5 - h4r)
eta_b = (wt - wp) / qin
print(f"(b) reheat:   HP exit {T4r:.1f} C, h = {h4r:.1f}; eta = {eta_b:.3f}, exit quality = {x6:.3f}, wnet = {wt - wp:.1f} kJ/kg")

# (c) reheat plus one open feedwater heater at 0.5 MPa, fed by steam bled from the LP turbine
(hC, sC), (hD, sD) = sup[(500, 300)], sup[(500, 400)]
h7 = interp(s5, sC, hC, sD, hD); T7 = interp(s5, sC, 300, sD, 400)
wp1 = vf10 * (500 - 10); h8 = h1 + wp1                       # condensate pump to the heater
h9, vf5 = sat[500][0], sat[500][4]                           # saturated liquid leaves the heater
wp2 = vf5 * (8000 - 500); h10 = h9 + wp2                      # feed pump to the boiler
y = (h9 - h8) / (h7 - h8)                                    # fraction bled, from the heater energy balance
wt = (h3 - h4r) + (h5 - h7) + (1 - y) * (h7 - h6)
wpump = (1 - y) * wp1 + wp2
qin = (h3 - h10) + (h5 - h4r)
eta_c = (wt - wpump) / qin
print(f"(c) + FWH:    bleed at 0.5 MPa, {T7:.1f} C, h = {h7:.1f}; y = {y:.4f}; eta = {eta_c:.3f}, wnet = {wt - wpump:.1f} kJ/kg")

# Average heat-addition temperatures and steam flow for 500 MW net
Tl = 45.81 + 273.15
for name, eta in (("simple", eta_a), ("reheat", eta_b), ("reheat + FWH", eta_c)):
    print(f"   {name:13s} average heat-addition temperature = {Tl / (1 - eta) - 273.15:5.1f} C")
print(f"steam to the HP turbine for 500 MW net, case (c): {500e3 / (wt - wpump):.0f} kg/s")

(a) simple:   eta = 0.394, exit quality = 0.810, wnet = 1261.2 kJ/kg
(b) reheat:   HP exit 206.7 C, h = 2843.7; eta = 0.408, exit quality = 0.949, wnet = 1565.6 kJ/kg
(c) + FWH:    bleed at 0.5 MPa, 390.6 C, h = 3252.9; y = 0.1463; eta = 0.428, wnet = 1449.2 kJ/kg
   simple        average heat-addition temperature = 253.3 C
   reheat        average heat-addition temperature = 265.8 C
   reheat + FWH  average heat-addition temperature = 284.4 C
steam to the HP turbine for 500 MW net, case (c): 345 kg/s

The last lines of output convert each efficiency into an equivalent average temperature of heat addition with \(\bar T_H = T_L/(1 - \eta)\): 253 °C for the simple cycle, 266 °C with reheat and 284 °C with reheat and regeneration. Every improvement works by lifting \(\bar T_H\), exactly as the Carnot-like form of the efficiency predicts.

Choosing extraction pressures

With one heater, the best extraction pressure gives a feedwater temperature roughly halfway between the condensing and boiling temperatures. With several heaters, a good first estimate divides the temperature rise from condenser to boiler into equal steps. Each added heater raises efficiency by a smaller amount, and the number in a plant is an economic choice: typical large units have six to eight. The deaerator usually sits near the middle of the train at a pressure of several hundred kilopascals to a megapascal, so that it operates above atmospheric pressure and air leaks out rather than in.

Supercritical and ultra-supercritical plants

Above the critical point of water, 22.06 MPa and 374 °C, there is no boiling: the water warms continuously from a dense liquid-like fluid to a light vapor-like one without a two-phase region. A supercritical boiler is therefore a once-through design without a steam drum, and the heat-addition line on the T-s diagram climbs above the dome rather than crossing it. Removing the constant-temperature boiling plateau raises the average temperature of heat addition substantially.

Supercritical units typically operate at about 24 MPa with main and reheat temperatures of 540 to 570 °C; ultra-supercritical units go to 25 to 30 MPa and about 600 to 620 °C. Their net efficiencies reach roughly 42 to 47 percent on the lower heating value of the fuel, against 35 to 38 percent for older subcritical units. Research programs on advanced ultra-supercritical steam aim at 700 °C and above, which requires nickel-based alloys in the hottest parts. Carbon dioxide can also be used as the working fluid in a supercritical Brayton-like cycle; supercritical CO2 cycles are compact and promising for nuclear, solar and waste-heat applications.

Many US coal units have retired over the past fifteen years and few new ones are planned, but the same steam-cycle principles apply to the remaining supercritical coal units, to the steam side of combined-cycle plants and to biomass, nuclear and concentrated solar plants. Nuclear plants are a special case: their steam is saturated or only slightly superheated at about 6 to 7 MPa (roughly 900 to 1000 psia), so their efficiency is near 33 to 35 percent, and moisture separators and reheaters between turbine stages keep the low-pressure turbines dry.

Back at Northgate

The utility-plant model was a detour, but it pays off at Northgate. The deaerator, as an open feedwater heater, sets the feedwater temperature that the boilers see. Raising the deaerator pressure from 120 kPa to 300 kPa would raise the feedwater temperature from about 105 °C to 134 °C and reduce the fuel used to make steam, at the cost of using more 1 MPa district steam in the deaerator. Whether that pays depends on how the steam would otherwise be used, a question that Lesson 7's cogeneration analysis answers. Tomás also notices that the district condensate returns from the buildings at 60 to 80 °C, already warm: a district system is, in effect, a giant regenerative loop in which the buildings are the feedwater heaters.

Kofi adds a design point. If Northgate adds a heat-recovery steam generator behind a gas turbine, its economizer section will do the job that feedwater heaters do in a utility plant, using exhaust heat that would otherwise go up the stack. In combined-cycle plants, bleeding steam for feedwater heating is usually avoided, because the exhaust gas is a better source of low-temperature heat.

For US customary readers, the same analysis works with steam tables in Btu/lbm, psia and °F: 8 MPa and 500 °C is about 1160 psia and 932 °F, with \(h = 1462\) Btu/lbm, and the reheat pressure of 1 MPa is 145 psia. Utility heat rates are quoted in Btu/kWh; 42.8 percent corresponds to 3412/0.428 = 7970 Btu/kWh for the ideal cycle alone, and real plants with boiler and auxiliary losses report net heat rates of roughly 8,700 to 10,500 Btu/kWh.