12.4 Vapour Power Cycles: Ideal, Reheat & Regenerative Rankine Cycles

11.4 Vapour Power Cycles: Ideal, Reheat & Regenerative Rankine Cycles

Coal-fired thermal power generation produces the overwhelming majority of baseload electricity across India, consuming over $75%$ of Coal India Limited's annual coal production. The Rankine cycle and its advanced modifications (reheat, regeneration, supercritical steam parameters) form the theoretical and thermodynamic foundation of utility power engineering.

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1. Carnot Vapour Cycle & Practical Limitations

While the Carnot cycle achieves maximum theoretical efficiency between $T_H$ and $T_L$, it is impractical for condensable vapours because:

1. Two-Phase Compression: Compression of wet steam (state 4 to 1) requires a compressor handling liquid droplets and vapour simultaneously, causing blade cavitation, droplet erosion, and severe mechanical inefficiency.
2. Superheating Incompatibility: Superheating steam at constant temperature requires continuous pressure reduction during heat addition, which is impossible in conventional boiler tube headers.
3. Low Work Ratio: Work consumed by two-phase compression is huge ($w_c \approx 40%-50% \text{ of } w_t$), making the cycle highly sensitive to component irreversibilities.

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2. The Simple Ideal Rankine Cycle

The Rankine cycle resolves these limitations by completely condensing steam to saturated liquid in the condenser and pumping liquid water into the boiler.

       RANKINE CYCLE: T-s DIAGRAM              RANKINE CYCLE FLOW DIAGRAM
    T ^                                      +--------------------------+
      |           1 (Superheated)            |          BOILER          |
      |          /|                          +--------------------------+
      |   Boiler/ |                              ^                  |
      |  4-----*  | Turbine                      | q_in             | High P Steam
      |  |     |  | Expansion                    |                  v
      |  |     |  |                           [PUMP]            [TURBINE] --> W_t
      |  3-----*--2 (Condenser)                  ^                  |
      |        |                             w_p |                  | Low P Steam
      +--------+-------------------> s           |    +-----------+ v
                                                 +----| CONDENSER |
                                                      +-----------+

Four Steady-Flow Processes

1. Process 1-2: Reversible adiabatic (isentropic) expansion in turbine ($s_1 = s_2$).
2. Process 2-3: Constant pressure heat rejection in condenser ($P_2 = P_3 = P_{cond}$). Steam condensed to saturated liquid ($x_3 = 0$).
3. Process 3-4: Reversible adiabatic (isentropic) pumping in feed pump ($s_3 = s_4$).
4. Process 4-1: Constant pressure heat addition in boiler/superheater ($P_4 = P_1 = P_{boiler}$).

Energy Analysis per kg of Steam

• Turbine Work Output ($w_t$): $w_t = h_1 - h_2$
• Pump Work Input ($w_p$): $w_p = h_4 - h_3 = \int_{P_{cond}}^{P_{boiler}} v_f \, dP \approx v_{f3} (P_{boiler} - P_{cond}) \times 10^3 \quad (\text{kJ/kg})$ (Where $v_{f3} \approx 0.0010\text{ m}^3/\text{kg}$, pressures in $\text{MPa}$).
• Net Work Output ($w_{net}$): $w_{net} = w_t - w_p = (h_1 - h_2) - (h_4 - h_3)$
• Boiler Heat Input ($q_{in}$): $q_{in} = h_1 - h_4 = (h_1 - h_3) - w_p$
• Thermal Efficiency ($\eta_{Rankine}$): $\eta_{Rankine} = \frac{w_{net}}{q_{in}} = \frac{(h_1 - h_2) - w_p}{h_1 - h_4}$

Thermal Performance Indices

• Specific Steam Consumption (Steam Rate, SSC): $\text{SSC} = \frac{3600}{w_{net}} \quad \left(\frac{\text{kg}}{\text{kWh}}\right)$
• Heat Rate (HR): Heat input required per unit of electrical output: $\text{HR} = \frac{3600 \cdot q_{in}}{w_{net}} = \frac{3600}{\eta_{th}} \quad \left(\frac{\text{kJ}}{\text{kWh}}\right)$
• Work Ratio ($r_w$): $r_w = \frac{w_{net}}{w_t} = \frac{w_t - w_p}{w_t} \approx 0.985 - 0.995 \quad (w_p \ll w_t)$

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3. Parametric Effects on Rankine Cycle Performance

+-------------------------------------------------------------------------+
|                 PARAMETRIC SENSITIVITY IN RANKINE CYCLE                 |
+-----------------------+---------------------+---------------------------+
| Parameter Change      | Effect on $\eta$ & $W_{net}$ | Practical Constraints     |
+-----------------------+---------------------+---------------------------+
| **Increase Boiler     | $\eta_{th} \uparrow$| Exhaust steam moisture    |
| Pressure ($P_b$)**    | $T_{mean,in} \uparrow$| increases ($x_2 \downarrow$), leading to  |
|                       |                     | heavy LP blade erosion.   |
+-----------------------+---------------------+---------------------------+
| **Increase Superheat  | $\eta_{th} \uparrow$| Limited by metallurgical  |
| Temp ($T_{sup}$)**    | $w_{net} \uparrow$  | limits of boiler/turbine  |
|                       | $x_2 \uparrow$ (Drier) | alloy steels ($565-600^{\circ}\text{C}$). |
+-----------------------+---------------------+---------------------------+
| **Lower Condenser     | $\eta_{th} \uparrow$| Limited by cooling water  |
| Pressure ($P_c$)**    | $w_{net} \uparrow$  | ambient temperature       |
|                       | $T_{sink} \downarrow$| ($30^{\circ}\text{C} \implies P_c \approx 5-10\text{ kPa}$).|
+-----------------------+---------------------+---------------------------+

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4. Reheat Rankine Cycle

To exploit the thermodynamic benefits of ultra-high boiler pressures without suffering excessive moisture at turbine exhaust, steam is expanded in a High-Pressure (HP) turbine, returned to the boiler reheater, and then expanded in a Low-Pressure (LP) turbine.

        REHEAT RANKINE CYCLE: T-s DIAGRAM
    T ^
      |           1 (HP Inlet)      3 (Reheat Inlet)
      |          /|                /|
      |   Boiler/ |  HP Exp       / |  LP Exp
      |  6-----*  2 (Reheat) ----*  |
      |  |     |                    |
      |  5-----*--------------------4 (Condenser Inlet)
      +---------------------------------------> s

Thermodynamic Energy Formulation

• Total Turbine Work ($w_t$): $w_t = w_{t,HP} + w_{t,LP} = (h_1 - h_2) + (h_3 - h_4)$
• Total Heat Added ($q_{in}$): $q_{in} = q_{boiler} + q_{reheater} = (h_1 - h_6) + (h_3 - h_2)$
• Reheat Efficiency ($\eta_{Reheat}$): $\eta_{Reheat} = \frac{(h_1 - h_2) + (h_3 - h_4) - w_p}{(h_1 - h_6) + (h_3 - h_2)}$

Optimum Reheat Pressure

• Practical empirical optimum: $P_{reheat} \approx (0.20 - 0.25) \times P_{boiler}$.
• Primary Goal: Ensures LP turbine exhaust dryness fraction $x_4 > 0.88 - 0.90$, avoiding water droplet erosion on turbine blade tips.

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5. Regenerative Rankine Cycle with Feedwater Heaters (FWH)

In regeneration, steam is bled from intermediate turbine stages to preheat compressed feedwater prior to boiler entry. This raises the mean temperature of heat addition ($T_{m1}$), driving cycle efficiency toward the Carnot limit.

                   FEEDWATER HEATER (FWH) ARCHITECTURES
+------------------------------------+------------------------------------+
| Open (Direct-Contact) FWH          | Closed (Shell-and-Tube) FWH        |
+------------------------------------+------------------------------------+
| Bled steam and feedwater mix       | Bled steam condenses on shell;     |
| directly at common pressure.       | feedwater flows through tubes.     |
|                                    |                                    |
| Requires separate feed pump for    | Does not require separate pump per |
| each open heater stage.            | heater; uses steam trap drains.    |
|                                    |                                    |
| Acts as a **Deaerator** to strip   | Operates at high pressures         |
| dissolved $O_2$ and $CO_2$ gases.  | downstream of main boiler feed pump|
+------------------------------------+------------------------------------+

   OPEN FEEDWATER HEATER (DEAERATOR) ENERGY BALANCE
          Bled Steam: m kg @ h_2
                   |
                   v
   (1 - m) kg      +---------------+      1.0 kg Saturated Liquid
   --------------> |   OPEN FWH    | ---------------------------->
   Feedwater @ h_6 +---------------+      @ P_2 (h_7)

Mass & Energy Balance for Single Open FWH

For $1\text{ kg}$ of steam entering turbine, $m\text{ kg}$ is bled at state 2, leaving $(1 - m)\text{ kg}$ to expand to condenser state 3:

$m \cdot h_2 + (1 - m) \cdot h_6 = 1 \cdot h_7$ $m = \frac{h_7 - h_6}{h_2 - h_6}$

• Turbine Work: $w_t = (h_1 - h_2) + (1 - m)(h_2 - h_3)$
• Heat Supplied: $q_{in} = h_1 - h_8$ (where $h_8$ is feed pump exit enthalpy to boiler)

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6. Binary Vapour Cycles (Mercury-Steam)

No single fluid possesses ideal thermodynamic properties across the entire temperature range of $30^{\circ}\text{C}$ to $600^{\circ}\text{C}$:

• Water has high critical temperature ($373.95^{\circ}\text{C}$) but requires excessive pressure ($22.06\text{ MPa}$).
• Mercury (Hg) has high critical temperature ($898^{\circ}\text{C}$) and low saturation pressures at elevated temperatures ($P_{sat} = 1.0\text{ bar}$ at $356^{\circ}\text{C}$), but extremely low vapour pressure at ambient temperatures.

In a Binary Vapour Cycle, Mercury serves as the topping fluid and Steam as the bottoming fluid. The Mercury condenser acts as the Steam boiler.

$\eta_{overall} = \eta_{Hg} + \eta_{Steam} - \eta_{Hg} \cdot \eta_{Steam}$

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7. Worked Numerical Examples

Example 1: Simple Ideal Rankine Cycle Analysis

Problem: A steam power plant operates on a simple ideal Rankine cycle between boiler pressure $P_1 = 3.0\text{ MPa}$ and condenser pressure $P_2 = 10\text{ kPa}$. Steam enters the turbine as superheated vapour at $T_1 = 400^{\circ}\text{C}$. Steam properties:

• State 1 ($3\text{ MPa}, 400^{\circ}\text{C}$): $h_1 = 3231.7\text{ kJ/kg}, s_1 = 6.9235\text{ kJ/kg}\cdot\text{K}$
• State 2 ($10\text{ kPa}$ sat): $h_f = 191.8\text{ kJ/kg}, h_{fg} = 2392.1\text{ kJ/kg}, s_f = 0.6492\text{ kJ/kg}\cdot\text{K}, s_{fg} = 7.4996\text{ kJ/kg}\cdot\text{K}, v_f = 0.00101\text{ m}^3/\text{kg}$

Calculate:

1. Turbine work output ($w_t$).
2. Pump work input ($w_p$).
3. Cycle thermal efficiency ($\eta_{th}$).
4. Specific Steam Consumption ($\text{SSC}$).

Solution:

1. Isentropic expansion ($s_2 = s_1 = 6.9235\text{ kJ/kg}\cdot\text{K}$): $s_2 = s_f + x_2 s_{fg} \implies 6.9235 = 0.6492 + x_2 (7.4996)$ $x_2 = \frac{6.9235 - 0.6492}{7.4996} = \frac{6.2743}{7.4996} = 0.8366$ $h_2 = h_f + x_2 h_{fg} = 191.8 + (0.8366 \times 2392.1) = 191.8 + 2001.23 = 2193.03\text{ kJ/kg}$ $w_t = h_1 - h_2 = 3231.7 - 2193.03 = 1038.67\text{ kJ/kg}$

2. Pump work ($w_p$): $w_p = v_f (P_b - P_c) = 0.00101\text{ m}^3/\text{kg} \times (3000 - 10)\text{ kPa} = 0.00101 \times 2990 = 3.02\text{ kJ/kg}$ $h_4 = h_3 + w_p = 191.8 + 3.02 = 194.82\text{ kJ/kg}$

3. Thermal efficiency: $w_{net} = w_t - w_p = 1038.67 - 3.02 = 1035.65\text{ kJ/kg}$ $q_{in} = h_1 - h_4 = 3231.7 - 194.82 = 3036.88\text{ kJ/kg}$ $\eta_{th} = \frac{w_{net}}{q_{in}} = \frac{1035.65}{3036.88} = 0.3410 \text{ (34.10\%)}$

4. Specific Steam Consumption: $\text{SSC} = \frac{3600}{w_{net}} = \frac{3600}{1035.65} = 3.476\text{ kg/kWh}$

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Example 2: Open Feedwater Heater Mass Fraction

Problem: In a regenerative steam cycle, steam enters the turbine at $h_1 = 3400\text{ kJ/kg}$. Steam is bled at $P_2 = 0.6\text{ MPa}$ ($h_2 = 2800\text{ kJ/kg}$) into an open feedwater heater. Saturated liquid leaves the heater at $h_7 = 670\text{ kJ/kg}$. Feedwater enters the heater from the condensate pump at $h_6 = 180\text{ kJ/kg}$. Find the mass fraction $m$ of steam bled per kg of boiler steam flow.

Solution: Applying steady-state energy balance to the open FWH: $m \cdot h_2 + (1 - m) \cdot h_6 = h_7$ $m (h_2 - h_6) = h_7 - h_6$ $m = \frac{h_7 - h_6}{h_2 - h_6} = \frac{670 - 180}{2800 - 180} = \frac{490}{2620} = 0.1870\text{ kg of bled steam per kg boiler steam}$