Formula
Losses per kg fuel, each then divided by the gross calorific value:
\(L_{dg} = m_{g,dry}\left[h_{g}(T_g) - h_{g}(T_{ref})\right] \qquad L_{H} = m_{H_2O,H}\left[h_v(T_g) - h_f(T_{ref})\right] \qquad L_{M} = M\left[h_v(T_g) - h_f(T_{ref})\right]\)
\(L_{\omega} = \omega\,m_a\left[h_v(T_g) - h_v(T_{ref})\right] \qquad L_{CO} = y_{CO}\,\frac{m_{g,dry}}{M_{g,dry}}\,M_{CO}\,\Delta h_{CO} \qquad L_{ub} = C_{ub}\,\Delta h_{C} \qquad L_{r} = (A + C_{ub})\,c_{p,r}\,(T_r - T_{ref})\)
Credits for heat entering above the reference temperature:
\(B_{a} = m_a\left[h_{air}(T_a) - h_{air}(T_{ref})\right] \qquad B_{\omega} = \omega\,m_a\left[h_v(T_a) - h_v(T_{ref})\right]\)
\(\eta_{GCV} = 1 - \frac{\sum L}{GCV} + \frac{\sum B}{GCV} \qquad \eta_{NCV} = \eta_{GCV}\,\frac{GCV}{NCV}\)
Variables
| Symbol | Meaning | SI unit |
|---|---|---|
| \(C\), \(H\), \(O\), \(N\), \(S\), \(A\), \(M\) | ultimate analysis as fired, mass fractions | – |
| \(GCV_{in}\) | gross calorific value entered (0 = estimate by Channiwala–Parikh) | J/kg |
| \(EA\) | excess air fraction | – |
| \(\omega\) | air humidity, kg water per kg dry air | kg/kg |
| \(T_a\) | combustion air temperature entering the boiler boundary | K |
| \(T_g\) | flue gas temperature leaving the boundary (after the air heater) | K |
| \(T_{ref}\) | reference temperature, 298.15 K (25 °C) | K |
| \(y_{CO}\) | CO in dry flue gas, volume fraction | – |
| \(C_{ub}\) | unburned carbon in residue per kg fuel | kg/kg |
| \(T_r\) | temperature at which residue (ash) leaves | K |
| \(m_a\), \(m_{g,dry}\) | actual dry air and dry flue gas per kg fuel (from the combustion balance) | kg/kg |
| \(m_{H_2O,H}\) | water formed from hydrogen per kg fuel | kg/kg |
| \(M_{g,dry}\), \(M_{CO}\) | molar mass of dry flue gas and of CO | kg/kmol |
| \(h_g\), \(h_{air}\) | enthalpy of dry flue gas and of air (Shomate integration) | J/kg |
| \(h_v\) | enthalpy of water vapour at low pressure (IF97, 1 psia) | J/kg |
| \(h_f\) | enthalpy of liquid water at 25 °C (IF97) | J/kg |
| \(\Delta h_{CO}\) | heat of burning CO to CO₂, 10 160 Btu/lb = 23.63 MJ/kg | J/kg |
| \(\Delta h_{C}\) | heat of burning carbon to CO₂, 14 500 Btu/lb = 33.73 MJ/kg | J/kg |
| \(c_{p,r}\) | mean specific heat of residue, 0.16 Btu/(lb·°F) = 670 J/(kg·K) | J/(kg·K) |
| \(GCV\), \(NCV\) | gross and net calorific value used | J/kg |
| \(L_{dg}\), \(L_{H}\), \(L_{M}\), \(L_{\omega}\), \(L_{CO}\), \(L_{ub}\), \(L_{r}\) | losses: dry gas, water from hydrogen, fuel moisture, air moisture, CO, unburned carbon, residue | – |
| \(L_{rc}\), \(L_{un}\) | radiation and convection, unaccounted (entered as fractions of input) | – |
| \(\sum L\), \(\sum B\) | total losses and total credits, fractions of GCV | – |
| \(B_{a}\), \(B_{\omega}\) | credits for entering dry air and its moisture | – |
| \(\eta_{GCV}\), \(\eta_{NCV}\) | efficiency on the gross and on the net basis | – |
Method and standards
The energy balance (indirect) method of ASME PTC 4-2013 (owner decision 2026-11-09): losses and credits referred to 25 °C on the gross calorific value. The combustion balance is the same as in the combustion calculator (exact molar ratios, PTC 4 air). Gas enthalpies are integrated from NIST Shomate heat capacities of the actual flue gas composition rather than taken from a fixed 1.005 kJ/(kg·K); water vapour enthalpy comes from IF97 at 1 psia as PTC 4 §5.19.5 specifies. Radiation and unaccounted losses are entered, as the standard measures or agrees them.
Assumptions and limits
- Steady state, single fuel, no flue gas recirculation or sorbent; fuel sensible heat credit is not included.
- The flue gas temperature must be above about 39 °C (saturation at 1 psia) and above the entering air temperature.
- Residue sensible heat uses a mean specific heat (PROPOSED in the constants table) instead of the PTC 4 polynomial; the difference is below 0.02 % of input.
References
- ASME PTC 4-2013 Fired Steam Generators §5.14 (losses) and §5.15 (credits)
- Channiwala & Parikh (2002) for GCV when not entered
Questions and answers
Why are losses and credits referred to 25 °C instead of the air temperature?
PTC 4 fixes the reference at 25 °C (77 °F) so that results are comparable between tests. Air entering above 25 °C brings heat in, which appears as a credit; the two together give the same answer as referencing to the air temperature when the air is at ambient.
What is the difference between the gross and net efficiency?
The gross (HHV) basis counts the latent heat of the water in the flue gas as a loss; the net (LHV) basis leaves it out of both input and loss. The net figure is 4–6 points higher for coal and 9–11 points higher for natural gas. State the basis whenever you quote an efficiency.
Where do I get the unburned carbon?
From the ash analysis: carbon in bottom ash and fly ash times the ash flows, divided by the fuel flow. 0.3–1 % of fuel is typical for stokers and fluidised beds, below 0.1 % for pulverised coal and nil for oil.
How does this differ from BS 845 / IS 8753?
The loss list is the same in substance. BS 845 references losses to the ambient air temperature and has no separate credits; PTC 4 references to 25 °C and adds credits. For air at 25 °C the two give identical results.