Steam and Gas Turbines, Rankine Cycle

What to remember

• The Rankine cycle is the standard cycle of a steam power plant: boiler (constant pressure heat addition), turbine (isentropic expansion), condenser (constant pressure heat rejection) and pump (isentropic compression). Efficiency = net work / heat supplied.
• Superheat, reheat, regeneration (feed water heating), higher boiler pressure and lower condenser pressure raise cycle efficiency. Reheat mainly keeps the exit steam dry; regeneration raises the mean temperature of heat addition.
• The Brayton cycle is the ideal gas turbine cycle: efficiency = 1 - 1/r^((γ-1)/γ) depends only on pressure ratio r. Intercooling, reheating and regeneration improve work output or efficiency.

1. The ideal Rankine cycle

A steam power plant works on a closed loop of water and steam. The four processes are:

• 1-2: pump raises the pressure of saturated liquid (isentropic).
• 2-3: boiler and superheater add heat at constant pressure.
• 3-4: steam expands in the turbine (isentropic, ideal).
• 4-1: condenser removes heat at constant pressure and temperature.

Formulas (per kg of steam, enthalpy h in kJ/kg):

• Turbine work WT = h3 - h4.
• Pump work WP = v × (P2 - P1), where v is the specific volume of water (about 0.001 m³/kg). Pump work is very small compared with turbine work.
• Heat supplied Qin = h3 - h2 (h2 = h1 + WP).
• Net work = WT - WP.
• Rankine efficiency = (WT - WP) / Qin.
• Back work ratio = WP / WT (very small for steam, about 1% or less).
• Specific steam consumption = 3600 / net work (kg/kWh).
• Heat rate = 3600 / efficiency (kJ/kWh).

The Carnot cycle is not used with steam because pumping a wet mixture is difficult and the cycle gives very low work per kg. The Rankine cycle replaces it by condensing completely to liquid, so only liquid is pumped.

Worked example. Turbine inlet enthalpy 3200 kJ/kg, exit enthalpy 2200 kJ/kg, condensate enthalpy 200 kJ/kg, pump work 10 kJ/kg. Turbine work = 1000. Net work = 990. Heat supplied = 3200 - 210 = 2990. Efficiency = 990/2990 = 33.1%.

Pump work example. v = 0.001 m³/kg, pressure rise 10 MPa = 10,000 kPa. WP = 0.001 × 10,000 = 10 kJ/kg.

2. How to improve the Rankine cycle

• Reheat cycle: steam is taken out after partial expansion, reheated to nearly the original temperature in the boiler, and expanded in the low pressure turbine. Exit dryness is improved, so blade erosion is reduced. Dryness at exit is kept above about 0.88.
• Regenerative cycle: small amounts of steam are bled from the turbine to heat the feed water. Open feed water heaters (direct contact) mix steam and water and need an extra pump. Closed heaters (shell and tube) do not mix; drains are cascaded back or pumped forward. The ideal limit of infinite heaters approaches Carnot-like behaviour for heat addition.
• Supercritical units operate above the critical point of water (about 22.1 MPa and 374 °C), where there is no boiling and no separate liquid and vapour phase.
• APGENCO runs coal-based thermal stations in Andhra Pradesh, such as Dr. Narla Tata Rao thermal station at Ibrahimpatnam, Rayalaseema thermal station at Kalamalla and the Krishnapatnam station. They use Rankine-based units with superheat, reheat and regenerative heating. Check the latest official release for unit ratings.

3. Steam turbines: types and compounding

Steam turbines convert the energy of steam into rotation. Steam is first expanded in nozzles (or fixed blades) to gain velocity, then acts on moving blades.

• Degree of reaction = enthalpy drop in moving blades / enthalpy drop in the whole stage. It is 0 for pure impulse and 0.5 for Parsons.
• Compounding reduces rotor speed. A single-stage De Laval turbine has a very high rotor speed, which is not practical for large power.
• Critical pressure ratio for a nozzle: about 0.546 for superheated steam, about 0.577 for dry saturated steam and 0.528 for air. At this ratio the throat is choked and flow is at maximum discharge. Convergent nozzle is used when exit pressure is above critical; convergent-divergent nozzle when below.
• Governing: throttle governing, nozzle governing and bypass governing control the steam flow with the load.
• Other parts: labyrinth gland seals, diaphragms, casing, rotor and reduction gearing.

4. Velocity diagram, blade efficiency and work

Let V1 = absolute steam velocity at nozzle exit, α = nozzle angle, Vb = blade (mean) speed, Vw = whirl component.

• Work per kg = Vb × (Vw1 + Vw2) for blade with Vw directions added (ΔVw).
• Blade efficiency = work done per kg / (V1²/2) = 2 Vb ΔVw / V1².
• Blade speed ratio ρ = Vb / V1.
• For a single-stage impulse turbine with symmetrical blades and no friction, efficiency = 4ρ(cos α - ρ). It is maximum at ρ = cos α / 2, and the maximum is cos² α.
• For a Parsons (50% reaction) turbine, maximum efficiency = 2cos² α / (1 + cos² α) at ρ = cos α.
• For a Curtis wheel with n rows of moving blades, the optimum speed ratio is cos α / (2n). In a two-row Curtis wheel the work ratio of first to second row is about 3 : 1.
• Axial thrust arises in reaction turbines because of pressure drop across moving blades; it is balanced by a dummy piston.
• Stage efficiency = actual enthalpy drop / isentropic enthalpy drop. Reheat factor (about 1.03 to 1.04) is the ratio of the sum of stage isentropic drops to the total isentropic drop; it is greater than 1 in multistage turbines.

5. Condensers

A condenser lowers the exhaust pressure and recovers condensate.

• Jet (direct contact) condenser: cooling water mixes with steam; used where condensate is not needed.
• Surface condenser: shell and tube; cooling water in tubes; condensate reused in the boiler.
• Air must be removed continuously by an air ejector or vacuum pump, since air spoils vacuum. By Dalton's law, the condenser pressure is the sum of steam partial pressure and air partial pressure.
• Vacuum efficiency = actual vacuum / ideal vacuum possible. Condenser efficiency = rise in cooling water temperature / (saturation temperature - inlet water temperature).
• Cooling towers (natural draught, induced draught) cool the circulating water.

6. Gas turbine and the Brayton cycle

Processes of the ideal air-standard Brayton cycle: 1-2 isentropic compression, 2-3 constant pressure heat addition (combustion), 3-4 isentropic expansion, 4-1 constant pressure heat rejection.

• Efficiency = 1 - 1/r^((γ-1)/γ), where r = P2/P1. For air, γ = 1.4 and (γ-1)/γ = 0.2857.
• Equivalent form: efficiency = 1 - T1/T2. If T1 = 300 K and T2 = 600 K, efficiency = 50%.
• Back work ratio = compressor work / turbine work. It is large (about 40-60%) in a gas turbine, unlike steam.
• Pressure ratio for maximum specific net work: r = (T3/T1)^(γ / (2(γ-1))). Here T3 is maximum cycle temperature and T1 is minimum.
• Open cycle: air drawn from the atmosphere, exhaust released. Closed cycle: working fluid recirculated with heat exchangers.
• Intercooling between compressor stages reduces compressor work. Reheating between turbine stages increases turbine work. Used alone, both lower cycle efficiency because more heat is added, but they raise net work.
• Regeneration uses the hot exhaust to preheat compressed air; it improves efficiency and works best at low pressure ratios. Effectiveness of regenerator = (actual air temperature rise) / (maximum possible rise).
• Isentropic efficiencies: compressor = ideal work / actual work; turbine = actual work / ideal work.
• Combined cycle: gas turbine exhaust runs a heat recovery steam generator (HRSG) that drives a steam turbine. This gives higher overall efficiency than either cycle alone.

Exam traps

• Pump work is tiny; it is not equal to turbine work. Do not ignore it unless told so.
• Reheat is mainly to improve dryness; efficiency gain is small. Regeneration is for efficiency.
• Raising boiler pressure raises efficiency but lowers exit dryness.
• Impulse: pressure drop only in nozzles. Reaction: drop in both fixed and moving blades.
• De Laval is simple impulse; Curtis is velocity compounding; Rateau is pressure compounding.
• Maximum blade efficiency of single-stage impulse is cos² α at ρ = cos α / 2, not at ρ = cos α (that is for Parsons).
• Brayton efficiency depends only on pressure ratio (and γ), not on maximum temperature.
• Intercooling alone and reheating alone reduce efficiency but increase work.

One-liners

• 1. Rankine cycle has four processes: two isobaric and two isentropic.
• 2. Rankine pump work = v ΔP.
• 3. Critical pressure ratio for superheated steam nozzle is about 0.546.
• 4. For air the critical pressure ratio is 0.528.
• 5. Parsons turbine has degree of reaction 0.5.
• 6. Curtis turbine is velocity compounded.
• 7. Rateau turbine is pressure compounded.
• 8. Brayton efficiency = 1 - 1/r^((γ-1)/γ).
• 9. Regeneration in gas turbines is most useful at low pressure ratio.
• 10. Air in a condenser is removed by an air ejector.
• 11. Heat rate = 3600 / efficiency in kJ/kWh.
• 12. In a combined cycle the HRSG links gas turbine exhaust to the steam cycle.

Practice questions

1. In the ideal Rankine cycle, heat is rejected at constant pressure in which component?

   1. Turbine
   2. Condenser
   3. Boiler
   4. Feed pump
   Answer

   B. Condenser

   Steam condenses at constant pressure and temperature in the condenser.

2. The work required by the feed pump in a Rankine cycle is given by

   1. P2 / P1
   2. h3 - h4
   3. v (P2 - P1)
   4. T ΔS
   Answer

   C. v (P2 - P1)

   Water is nearly incompressible, so pump work = specific volume × pressure rise.

3. The critical pressure ratio for a nozzle expanding superheated steam is about

   1. 0.546
   2. 0.528
   3. 0.577
   4. 0.333
   Answer

   A. 0.546

   Superheated steam (n = 1.3) gives about 0.546; 0.528 is for air and 0.577 for dry saturated steam.

4. The degree of reaction of a Parsons turbine is

   1. 25%
   2. 100%
   3. 0%
   4. 50%
   Answer

   D. 50%

   Parsons turbine has equal enthalpy drop in fixed and moving blades.

5. A Curtis turbine is an example of

   1. Velocity compounded impulse turbine
   2. Single-stage reaction turbine
   3. Reaction turbine
   4. Pressure compounded impulse turbine
   Answer

   A. Velocity compounded impulse turbine

   Curtis wheel absorbs the nozzle velocity in several moving rows after one pressure drop.

6. A Rateau turbine uses

   1. Pressure compounding
   2. Velocity compounding
   3. Pressure-velocity reaction only
   4. No nozzles
   Answer

   A. Pressure compounding

   Pressure is dropped in several stages, each with its own nozzles and a single moving row.

7. A Rankine cycle has turbine work 800 kJ/kg, pump work 8 kJ/kg and heat supplied 2400 kJ/kg. The efficiency is

   1. 36%
   2. 30%
   3. 40%
   4. 33%
   Answer

   D. 33%

   Net work = 800 - 8 = 792; 792/2400 = 0.33.

8. Water (v = 0.001 m³/kg) is pumped through a pressure rise of 5 MPa. The pump work in kJ/kg is

   1. 50
   2. 5
   3. 0.5
   4. 500
   Answer

   B. 5

   WP = 0.001 × 5000 kPa = 5 kJ/kg.

9. The main purpose of reheating steam in a Rankine cycle is to

   1. Eliminate the condenser
   2. Reduce boiler pressure
   3. Improve dryness of steam at turbine exit
   4. Reduce pump work
   Answer

   C. Improve dryness of steam at turbine exit

   Reheat keeps the final steam drier and reduces blade erosion; efficiency gain is small.

10. An ideal Brayton cycle has compressor inlet temperature 300 K and compressor exit temperature 450 K. Its efficiency is

    1. 50%
    2. 25%
    3. 33.3%
    4. 66.7%
    Answer

    C. 33.3%

    Efficiency = 1 - T1/T2 = 1 - 300/450 = 0.333.

11. Consider: 1. The back work ratio of a gas turbine plant is much larger than that of a steam plant. 2. Pump work in a steam plant is a large fraction of turbine work. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    1 is correct because compression of gas takes a large share of turbine work; 2 is wrong since liquid pumping needs little work.

12. A plant has thermal efficiency 36%. The heat rate in kJ/kWh is

    1. 12,960
    2. 10,000
    3. 7,200
    4. 3,600
    Answer

    B. 10,000

    Heat rate = 3600 / 0.36 = 10,000 kJ/kWh.

13. A Rankine cycle has a net work of 900 kJ per kg of steam. The specific steam consumption in kg/kWh is

    1. 2
    2. 3
    3. 5
    4. 4
    Answer

    D. 4

    SSC = 3600 / 900 = 4 kg/kWh.

14. A single-stage frictionless impulse turbine has nozzle angle 60°. The maximum blade efficiency (symmetrical blades) is

    1. 75%
    2. 25%
    3. 50%
    4. 12.5%
    Answer

    B. 25%

    Maximum efficiency = cos² α = cos² 60° = 0.25.

15. For a single-stage impulse turbine with symmetrical blades, maximum blade efficiency occurs when blade speed ratio equals

    1. cos α
    2. 2 cos α
    3. sin α / 2
    4. cos α / 2
    Answer

    D. cos α / 2

    Efficiency 4ρ(cos α - ρ) is maximum at ρ = cos α / 2.

16. The optimum blade speed ratio of a two-row Curtis wheel is

    1. cos α / 8
    2. cos α
    3. cos α / 4
    4. cos α / 2
    Answer

    C. cos α / 4

    For n rows the optimum ratio is cos α / (2n); n = 2 gives cos α / 4.

17. Consider: 1. Raising boiler pressure raises Rankine efficiency. 2. Raising boiler pressure makes the exit steam drier. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Higher pressure raises efficiency but the exit steam becomes wetter, so 2 is wrong.

18. When boiler pressure is increased at constant maximum temperature, the dryness fraction at turbine exit

    1. Increases
    2. Remains the same
    3. Becomes exactly 1
    4. Decreases
    Answer

    D. Decreases

    Expansion starts from a lower-enthalpy state on the Mollier chart, giving wetter exhaust.

19. Lowering condenser pressure in a Rankine cycle is limited mainly by

    1. Boiler pressure
    2. Temperature of the cooling water
    3. Pump efficiency
    4. Superheater size
    Answer

    B. Temperature of the cooling water

    Saturation temperature in the condenser cannot fall below the cooling water temperature.

20. Intercooling between compressor stages in a gas turbine

    1. Increases compressor work
    2. Eliminates the combustor
    3. Reduces compressor work
    4. Reduces turbine work
    Answer

    C. Reduces compressor work

    Cooler gas has lower specific volume, so less compression work is needed.

21. A regenerator in a gas turbine plant is most effective when the pressure ratio is

    1. High
    2. Low
    3. Very high
    4. Equal to the optimum for maximum work only
    Answer

    B. Low

    Turbine exhaust must be hotter than compressor exit; this holds at low pressure ratios.

22. An ideal Brayton cycle has r^((γ-1)/γ) = 2. Its efficiency is

    1. 75%
    2. 25%
    3. 50%
    4. 33%
    Answer

    C. 50%

    η = 1 - 1/2 = 0.5.

23. For maximum specific net work, the compressor exit temperature is √(T1 T3). With T1 = 300 K and T3 = 1200 K, it is

    1. 450 K
    2. 750 K
    3. 900 K
    4. 600 K
    Answer

    D. 600 K

    √(300 × 1200) = √360000 = 600 K.

24. In a combined cycle plant, the exhaust of the gas turbine is used in a

    1. Heat recovery steam generator
    2. Open feed water heater
    3. Cooling tower
    4. Reheat condenser
    Answer

    A. Heat recovery steam generator

    The HRSG raises steam for a steam turbine, improving overall efficiency.

25. In an open (direct contact) feed water heater

    1. Steam and water are kept apart by tubes
    2. Only air is heated
    3. Bled steam mixes directly with feed water
    4. No extra pump is needed
    Answer

    C. Bled steam mixes directly with feed water

    Mixing gives saturated liquid at heater pressure; a pump is needed after it.

26. The reheat factor in a multistage steam turbine is generally

    1. Greater than 1
    2. Zero
    3. Exactly 1
    4. Less than 1
    Answer

    A. Greater than 1

    Heat from stage losses is partly recovered in later stages; the factor is about 1.03-1.04.

27. Dryness fraction at the exit of a steam turbine is usually kept above about

    1. 0.30
    2. 0.88
    3. 0.60
    4. 0.50
    Answer

    B. 0.88

    Wetter steam causes blade erosion.

28. The Carnot cycle is not used as the ideal steam cycle mainly because

    1. It has too high efficiency
    2. Water cannot boil
    3. Turbines cannot expand steam
    4. Pumping a wet mixture is difficult
    Answer

    D. Pumping a wet mixture is difficult

    Rankine condenses fully so only liquid is pumped.

29. The critical point of water is at about

    1. 1 MPa and 100 °C
    2. 10 MPa and 311 °C
    3. 0.6 kPa and 0.01 °C
    4. 22.1 MPa and 374 °C
    Answer

    D. 22.1 MPa and 374 °C

    Above this point there is no distinct liquid-vapour boundary.

30. Consider: 1. Brayton cycle efficiency depends on pressure ratio. 2. Brayton cycle efficiency (ideal) is independent of the maximum temperature. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    η = 1 - 1/r^((γ-1)/γ) contains only r and γ, so both statements are correct.

31. A Rankine cycle has turbine inlet enthalpy 3000 kJ/kg, condensate enthalpy 200 kJ/kg and pump work 10 kJ/kg. Heat supplied per kg is

    1. 2790 kJ
    2. 2810 kJ
    3. 2990 kJ
    4. 2800 kJ
    Answer

    A. 2790 kJ

    Feed water enters the boiler at 200 + 10 = 210; Qin = 3000 - 210 = 2790.

32. Air leakage into a surface condenser

    1. Spoils the vacuum and must be removed by an air ejector
    2. Improves the vacuum
    3. Has no effect
    4. Increases condensate temperature only
    Answer

    A. Spoils the vacuum and must be removed by an air ejector

    Air adds partial pressure (Dalton's law) and acts as a poor heat conductor.

33. A surface condenser is preferred to a jet condenser in power plants because

    1. It is cheaper in all cases
    2. Condensate can be reused as boiler feed
    3. It needs no cooling water
    4. It has no tubes
    Answer

    B. Condensate can be reused as boiler feed

    In a jet condenser, cooling water mixes with condensate, contaminating it.

34. The degree of reaction of a pure impulse turbine stage is

    1. 0.5
    2. 1
    3. Greater than 1
    4. Zero
    Answer

    D. Zero

    No enthalpy drop occurs in the moving blades.

35. Velocity compounding is used mainly to

    1. Remove the nozzle
    2. Increase blade length
    3. Reduce the rotor speed
    4. Increase steam pressure
    Answer

    C. Reduce the rotor speed

    Velocity is absorbed over several moving rows, so a lower blade speed is acceptable.

36. For a Parsons (50% reaction) turbine, maximum blade efficiency occurs at speed ratio

    1. cos² α
    2. cos α
    3. 2 cos α
    4. cos α / 2
    Answer

    B. cos α

    Maximum efficiency = 2cos²α / (1 + cos²α) at ρ = cos α.

37. Blade speed is 200 m/s and the change in whirl velocity is 600 m/s. Work done per kg of steam in kJ is

    1. 240
    2. 120
    3. 60
    4. 30
    Answer

    B. 120

    Work = Vb × ΔVw = 200 × 600 = 120,000 J/kg = 120 kJ/kg.

38. Nozzle exit velocity is 1000 m/s, blade speed 250 m/s and change in whirl velocity 1200 m/s. The blade efficiency is

    1. 60%
    2. 48%
    3. 75%
    4. 30%
    Answer

    A. 60%

    η = 2 Vb ΔVw / V1² = 2 × 250 × 1200 / 10⁶ = 0.6.

39. An ideal gas turbine has T1 = 300 K, T2 = 600 K, T3 = 1200 K. The turbine exit temperature T4 is

    1. 300 K
    2. 1200 K
    3. 600 K
    4. 900 K
    Answer

    C. 600 K

    T3/T4 = T2/T1 (same pressure ratio), so T4 = 1200 × 300/600 = 600 K.

40. Consider: 1. A reaction turbine develops axial thrust on the rotor. 2. In an impulse turbine the pressure drops across moving blades. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Pressure drop across moving blades causes axial thrust in reaction turbines; impulse blades have no pressure drop, so 2 is wrong.

41. Actual enthalpy drop in a turbine is 800 kJ/kg and the isentropic drop is 1000 kJ/kg. The isentropic efficiency is

    1. 125%
    2. 20%
    3. 90%
    4. 80%
    Answer

    D. 80%

    Efficiency = actual / ideal = 800/1000.

42. Which of the following is a method of governing a steam turbine?

    1. Rheostat governing
    2. Nozzle governing
    3. Gear governing
    4. Blade polishing
    Answer

    B. Nozzle governing

    Throttle, nozzle and bypass governing control steam flow with load.

43. Compared with a steam power plant, a gas turbine plant generally has

    1. Slower start-up
    2. Smaller back work ratio
    3. Larger condenser
    4. Quicker start-up and lower cooling water need
    Answer

    D. Quicker start-up and lower cooling water need

    No boiler or condenser is needed in an open gas turbine cycle.

44. Match the turbine with its feature: 1 De Laval, 2 Curtis, 3 Rateau, 4 Parsons. Features: P Pressure compounded, Q Single stage impulse, R 50% reaction, S Velocity compounded.

    1. 1-R, 2-P, 3-S, 4-Q
    2. 1-S, 2-Q, 3-R, 4-P
    3. 1-Q, 2-S, 3-P, 4-R
    4. 1-P, 2-R, 3-Q, 4-S
    Answer

    C. 1-Q, 2-S, 3-P, 4-R

    De Laval is single-stage impulse, Curtis velocity compounded, Rateau pressure compounded, Parsons 50% reaction.

45. When the exit pressure of a nozzle is higher than the critical pressure, the suitable nozzle is

    1. Convergent-divergent
    2. Divergent only
    3. Any shape gives choking
    4. Convergent
    Answer

    D. Convergent

    Without choking, the flow expands fully in a convergent nozzle; the divergent part is needed only below critical pressure.