Regenerative cycle: 4 MPa, 400°C, open heater

Assumptions: ideal regenerative Rankine cycle with one open feed water heater; turbine and pumps isentropic; exit of heater is saturated liquid at 400 kPa; pump work included. Property values are taken from the steam tables (IAPWS-IF97 values, as in Cengel's tables).

              1
  +--------+------>+---------------+   +-----+
  | Boiler |       |    Turbine    |===| Gen |
  |  + S/H |       +---------------+   +-----+
  +--------+          | m kg (2)  | (1-m) kg (3)
      ^ 7             v           v
      |         +-----------+ +-----------+
  +--------+  6 |  Open FWH | | Condenser |
  | Pump 2 |<---| (mixing)  | +-----------+
  +--------+    +-----------+       | 4
                      ^ 5           |
                 +--------+         |
                 | Pump 1 |<--------+
                 +--------+

Steam table data

At 4 MPa, 400 °C (superheated): $h_1 = 3214.4$ kJ/kg, $s_1 = 6.7712$ kJ/kg K.

State 2 (bled steam, 400 kPa, $s_2 = s_1$)

$$\begin{aligned} x_2 &= \frac{6.7712 - 1.7766}{5.1188} = 0.9757 \\ h_2 &= 604.72 + 0.9757 \times 2133.33 = 2686.3\ \text{kJ/kg} \end{aligned}$$

State 3 (exhaust, 10 kPa, $s_3 = s_1$)

$$\begin{aligned} x_3 &= \frac{6.7712 - 0.6492}{7.4997} = 0.8163 \\ h_3 &= 191.81 + 0.8163 \times 2392.07 = 2144.5\ \text{kJ/kg} \end{aligned}$$

Pumps

$$\begin{aligned} W_{p1} &= v_f (p_5 - p_4) = 0.001010(400 - 10) = 0.39\ \text{kJ/kg} \\ h_5 &= 191.81 + 0.39 = 192.20\ \text{kJ/kg} \\ W_{p2} &= 0.001084(4000 - 400) = 3.90\ \text{kJ/kg} \\ h_7 &= h_6 + W_{p2} = 604.72 + 3.90 = 608.62\ \text{kJ/kg} \end{aligned}$$

Bled steam fraction (energy balance on open FWH)

$m h_2 + (1-m) h_5 = h_6$:

$$m = \frac{h_6 - h_5}{h_2 - h_5} = \frac{604.72 - 192.20}{2686.3 - 192.20} = 0.1654\ \text{kg/kg}$$

Work and heat (per kg of steam at boiler exit)

$$\begin{aligned} W_T &= (h_1 - h_2) + (1-m)(h_2 - h_3) \\ &= 528.1 + 0.8346 \times 541.8 = 980.3\ \text{kJ/kg} \\ W_P &= (1-m) W_{p1} + W_{p2} = 0.33 + 3.90 = 4.23\ \text{kJ/kg} \\ W_{net} &= 980.3 - 4.23 = 976.1\ \text{kJ/kg} \\ Q_s &= h_1 - h_7 = 3214.4 - 608.62 = 2605.7\ \text{kJ/kg} \end{aligned}$$

Cycle efficiency

$$\eta = \frac{W_{net}}{Q_s} = \frac{976.1}{2605.7} = 0.3746$$

Answer: cycle efficiency $\approx 37.5\%$ (bled fraction $m = 0.165$ kg per kg of steam).

  T
  ^                    1
  |                   /|
  |      f----------g  |
  |     /              |
  |    7               |
  |    6 . . . . . . . 2  bleed (m kg)
  |   /                |
  |  5                 |
  |  4-----------------3  (1-m) kg
  +-------------------------> s
  4-5 pump 1, 5 and 2 mix in FWH -> 6,
  6-7 pump 2, 7-f-g-1 boiler,
  1-2-3 turbine, 3-4 condenser

For comparison, a simple Rankine cycle between the same limits gives about 35.3%, so one open heater raises the efficiency by about 2 percentage points.