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4.3 · Calculate combined-cycle thermal efficiency

Learn to calculate combined-cycle thermal efficiency through clear examples and targeted practice.

University of Alberta MEC E 340: Applied Thermodynamics

Combined Power and Cogeneration

MEC E 340 Applied Thermodynamics · Study topic 4.3

A combined-cycle plant couples a gas-turbine power cycle to a steam power cycle. The gas branch produces work, and some energy in its exhaust is transferred to the steam branch, which produces additional work. For the overall efficiency, treat both branches as one plant: count the external fuel heat once, and add the net work outputs of both cycles. Use states 1–4 to identify the gas-cycle branch and states 5–8 to identify the steam-cycle branch; these numbers are labels only, not supplied property values. Assume steady operation and negligible changes in kinetic and potential energy unless a problem says otherwise. The examples give all required energy rates and efficiencies directly, so no property-table values are needed.

What you will learn

  • Identify the external heat input and net work outputs for a combined gas–steam power plant.
  • Use supplied heat-recovery and cycle-efficiency data to calculate combined-cycle thermal efficiency.
  • Keep branch and overall energy balances consistent, including energy that leaves without being recovered.

1. Define the plant boundary and performance measure

Draw the overall boundary around the gas and steam cycles together. The external input in the examples is fuel energy supplied per unit time, denoted by Q˙fuel\dot Q_{\mathrm{fuel}}. Useful output is the sum of the two cycles’ net work rates. A net work rate already accounts for internal work demands, such as compressor or pump work, when the problem defines it that way.
The combined-cycle thermal efficiency compares useful work output with external heat input. Use the same units and time basis for every rate, such as megawatts. The result is dimensionless and is commonly reported as a percentage.
The heat transferred from gas exhaust to the steam cycle is internal to the combined-plant boundary. It is not another fuel input. If the problem gives only the gas-cycle efficiency, convert it to gas-cycle net work using the gas cycle’s stated heat-input basis before combining outputs.
ηCC=W˙GT,net+W˙ST,netQ˙fuel\eta_{\mathrm{CC}}=\frac{\dot W_{\mathrm{GT,net}}+\dot W_{\mathrm{ST,net}}}{\dot Q_{\mathrm{fuel}}}
  • Set the overall boundary around both cycles.
  • Count external fuel heat once and add net work rates.
  • Check whether supplied work values are net or gross.

2. Balance each branch without losing energy

At steady state, a branch energy balance accounts for energy entering, energy leaving, and work. For the gas branch in the simplified examples, divide energy leaving the branch into recoverable exhaust energy and other energy leaving without recovery. The latter can include exhaust energy not designated as available to the recovery system and any other stated losses. This distinction prevents an incomplete balance from implying that all fuel input becomes work or recoverable exhaust heat.
For the steam branch, the heat actually delivered to it is the relevant heat input for its stated thermal efficiency. If only a fraction of available exhaust heat reaches the steam cycle, apply that recovery fraction first. Then use the steam-cycle efficiency to find its net work.
For an overall plant calculation, heat passed from one branch to the other cancels as an internal transfer. The overall balance therefore compares external fuel input with total plant work output and energy ultimately rejected or otherwise leaving the overall boundary. Use supplied branch data to calculate work; do not invent missing property values.
Q˙fuel=W˙GT,net+Q˙exh,avail+Q˙other\dot Q_{\mathrm{fuel}}=\dot W_{\mathrm{GT,net}}+\dot Q_{\mathrm{exh,avail}}+\dot Q_{\mathrm{other}}
  • Keep recoverable exhaust heat distinct from other energy leaving the gas branch.
  • Apply the steam-cycle efficiency to heat delivered to that cycle.
  • Internal heat transfer is not an additional overall-plant input.

3. Calculate and check the combined result

A dependable calculation follows the energy path. First identify gas-cycle net work, or calculate it from the gas-cycle efficiency and its heat input. Next determine the heat delivered to the steam cycle from the available exhaust heat and any recovery fraction. Then calculate steam-cycle net work, add the branch work rates, and divide by the external fuel heat rate.
The gas-branch balance is a useful consistency check. If supplied values do not account for all input energy, the remaining amount must be acknowledged as other energy leaving the branch or as another explicitly stated term. Do not force a balance by treating an unreported amount as recoverable steam-cycle heat.
Check that all rates use the same units, that the result is dimensionless, and that the total net work is less than the external fuel input for the supplied positive-input examples. When the steam branch produces positive work, combined efficiency should exceed the gas-cycle efficiency calculated on the same fuel-input basis.
W˙ST,net=ηSTQ˙recovered\dot W_{\mathrm{ST,net}}=\eta_{\mathrm{ST}}\dot Q_{\mathrm{recovered}}
  • Apply recovery fraction before steam-cycle efficiency.
  • Add work rates, not cycle efficiencies.
  • Check both the branch energy accounting and overall efficiency.

Worked example

Combine supplied net work rates

A plant receives fuel energy at 100 MW. The gas cycle produces 34 MW net work. Of the energy leaving the gas branch, 50 MW is available for exhaust-heat recovery; the remaining branch energy leaves without recovery. The steam cycle converts 30% of the heat delivered to it into net work, and all 50 MW reaches it. Find the combined-cycle thermal efficiency. Treat the rates as supplied steady-state data.
  1. Account for energy leaving the gas branch
    The gas-branch input not converted to work or identified as recoverable exhaust leaves by other routes. This term closes the simplified branch balance; it is not steam-cycle input.
    Q˙other=100−34−50=16 MW\dot Q_{\mathrm{other}}=100-34-50=16\ \mathrm{MW}
  2. Calculate steam-cycle work
    All 50 MW of available exhaust heat reaches the steam cycle, so apply the supplied steam efficiency to 50 MW.
    W˙ST,net=0.30(50 MW)=15 MW\dot W_{\mathrm{ST,net}}=0.30(50\ \mathrm{MW})=15\ \mathrm{MW}
  3. Find overall efficiency
    Add the two net work rates and divide by external fuel input. The recovered heat is an internal transfer, not an added input.
    ηCC=34+15100=0.49=49%\eta_{\mathrm{CC}}=\frac{34+15}{100}=0.49=49\%
Answer: The combined-cycle thermal efficiency is 49%.
Check: The plant delivers 49 MW of net work from 100 MW of fuel input. The gas-branch balance also accounts for 16 MW leaving without recovery, so the 50 MW recovery figure is not incorrectly treated as all of the branch’s rejected energy.

Worked example

Include incomplete exhaust-heat recovery

A combined plant receives fuel heat at 240 MW. Its gas cycle produces 84 MW net work, and 120 MW of energy leaving the gas branch is available to the recovery system. The balance of gas-branch energy leaves without recovery. Only 90% of the available 120 MW reaches the steam cycle. The steam cycle’s net thermal efficiency, based on heat reaching it, is 32%. Calculate the combined-cycle thermal efficiency.
  1. Close the gas-branch energy accounting
    The fuel input not appearing as gas-cycle work or available exhaust energy is energy leaving by other routes. The given available exhaust heat is therefore not assumed to be all energy rejected by the gas branch.
    Q˙other=240−84−120=36 MW\dot Q_{\mathrm{other}}=240-84-120=36\ \mathrm{MW}
  2. Find heat delivered to the steam cycle
    The recovery fraction applies to the 120 MW available to the recovery system.
    Q˙recovered=0.90(120 MW)=108 MW\dot Q_{\mathrm{recovered}}=0.90(120\ \mathrm{MW})=108\ \mathrm{MW}
  3. Calculate steam work and overall efficiency
    Use the steam efficiency on the 108 MW delivered, then add steam and gas net work before dividing by the external fuel input.
    W˙ST,net=0.32(108)=34.56 MW,ηCC=84+34.56240=0.494=49.4%\dot W_{\mathrm{ST,net}}=0.32(108)=34.56\ \mathrm{MW},\quad \eta_{\mathrm{CC}}=\frac{84+34.56}{240}=0.494=49.4\%
Answer: The combined-cycle thermal efficiency is 49.4%.
Check: The total net work is 118.56 MW, less than the 240 MW external input. The gas-branch remainder is 36 MW, while only 108 MW of the 120 MW available exhaust heat reaches the steam cycle.

Worked example

Use gas-cycle efficiency as an intermediate step

A plant receives 500 MW of fuel heat. The gas-cycle thermal efficiency is 38%, and 180 MW of energy leaving the gas branch is available before heat recovery. Any remaining gas-branch energy leaves without recovery. A recovery system transfers 60% of the available exhaust heat to the steam cycle, which converts 28% of its received heat to net work. Find the combined-cycle efficiency.
  1. Calculate gas-cycle net work
    The gas-cycle efficiency is defined on the supplied 500 MW fuel input, so it gives the gas-cycle net work rate directly.
    W˙GT,net=0.38(500 MW)=190 MW\dot W_{\mathrm{GT,net}}=0.38(500\ \mathrm{MW})=190\ \mathrm{MW}
  2. Check the gas-branch balance and find recovered heat
    The difference between fuel input, gas work, and available exhaust energy leaves without recovery. The recovery fraction then determines the heat delivered to the steam cycle.
    Q˙other=500−190−180=130 MW,Q˙recovered=0.60(180)=108 MW\dot Q_{\mathrm{other}}=500-190-180=130\ \mathrm{MW},\quad \dot Q_{\mathrm{recovered}}=0.60(180)=108\ \mathrm{MW}
  3. Calculate steam work and combined efficiency
    Apply the steam efficiency to 108 MW, add the resulting steam work to gas-cycle net work, and divide by the original fuel input.
    W˙ST,net=0.28(108)=30.24 MW,ηCC=190+30.24500=0.44048≈44.0%\dot W_{\mathrm{ST,net}}=0.28(108)=30.24\ \mathrm{MW},\quad \eta_{\mathrm{CC}}=\frac{190+30.24}{500}=0.44048\approx44.0\%
Answer: The combined-cycle thermal efficiency is approximately 44.0%.
Check: The result is above the gas-cycle efficiency of 38% because the steam branch adds positive work from recovered heat. Total net work is 220.24 MW, below the 500 MW fuel input.

Common mistakes and how to avoid them

Adding gas-cycle and steam-cycle efficiencies.
Correction: The efficiencies have different heat-input bases. Find each branch’s net work, add the work rates, and divide by the combined plant’s external fuel heat input.
Counting recovered exhaust heat as a second fuel input.
Correction: Recovered heat is transferred within the overall plant boundary. Count external fuel heat once.
Treating available exhaust heat as all energy leaving the gas branch.
Correction: If the branch input does not balance using work and the stated available exhaust energy, identify the remaining energy as other energy leaving, or include any additional stated balance terms.
Applying steam-cycle efficiency to all available exhaust heat when only part is recovered.
Correction: First calculate the heat delivered using the recovery fraction, then apply the steam-cycle efficiency.
Subtracting compressor or pump work again from a value already labelled net work.
Correction: Follow the problem’s definition. A supplied net work value already includes internal work requirements.

Lesson summary

  • Use one boundary around the gas and steam cycles for overall efficiency.
  • Add gas- and steam-cycle net work, then divide by external fuel heat input.
  • Keep available exhaust energy distinct from other energy leaving the gas branch.
  • Apply heat-recovery fractions before the steam-cycle efficiency and check the balances.

Check your understanding

Question 1

A gas cycle produces 40 MW net work from 100 MW fuel input. A steam cycle produces 10 MW net work from recovered exhaust heat. What is the combined-cycle efficiency?
  1. 40%
  2. 50%
  3. 20%
  4. 80%
Show answer and explanation
50%
The total net work is 50 MW. Dividing by the 100 MW external fuel input gives 50%.

Question 2

A recovery system delivers 75 MW to a steam cycle with 24% net thermal efficiency. What steam-cycle net work rate follows from these supplied values?
  1. 18 MW
  2. 24 MW
  3. 51 MW
  4. 99 MW
Show answer and explanation
18 MW
The steam-cycle net work is 0.24 times 75 MW, which equals 18 MW.

Key terms

Combined-cycle thermal efficiency
Total net work output from the gas and steam cycles divided by external heat supplied to the combined plant.
Available exhaust energy
The stated portion of energy leaving the gas branch that is available to the heat-recovery system.
Recovered heat
The portion of available exhaust energy that actually reaches the steam cycle.
Net work
Useful work output after accounting for work consumed within a cycle, when the supplied value is defined as net.

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