Formula
\(Q_{dg} = m_{g,dry}\left[h_g(T_g) - h_g(T_a)\right] \qquad Q_{H} = \frac{M_{H_2O}}{M_{H_2}}\,H\left[h_v(T_g) - h_f(T_a)\right] \qquad Q_{M} = M\left[h_v(T_g) - h_f(T_a)\right] \qquad Q_{\omega} = \omega\,m_a\left[h_v(T_g) - h_v(T_a)\right]\)
\(Q_{st} = Q_{dg} + Q_{H} + Q_{M} + Q_{\omega} \qquad L_i = \frac{Q_i}{GCV} \qquad \bar c_{p,g} = \frac{h_g(T_g) - h_g(T_a)}{T_g - T_a}\)
Variables
| Symbol | Meaning | SI unit |
|---|---|---|
| \(GCV\) | gross calorific value of the fuel as fired | J/kg |
| \(T_g\) | flue gas temperature at the stack | K |
| \(T_a\) | combustion air temperature (reference for the losses) | K |
| \(m_{g,dry}\) | dry flue gas per kg fuel | kg/kg |
| \(m_{a}\) | dry combustion air per kg fuel | kg/kg |
| \(H\) | hydrogen in the fuel as fired, mass fraction | – |
| \(M\) | moisture in the fuel as fired, mass fraction | – |
| \(\omega\) | air humidity, kg water per kg dry air | kg/kg |
| \(y_{CO_2}\), \(y_{O_2}\) | CO₂ and O₂ in the dry flue gas, volume fractions (N₂ + Ar by difference) | – |
| \(h_g\) | enthalpy of the dry flue gas (Shomate integration over the entered composition) | J/kg |
| \(h_v\) | enthalpy of water vapour at 1 psia (IF97) | J/kg |
| \(h_f\) | enthalpy of saturated liquid water at \(T_a\) (IF97) | J/kg |
| \(M_{H_2O}/M_{H_2}\) | water formed per kg hydrogen, 8.936 | kg/kg |
| \(Q_{dg}\), \(Q_{H}\), \(Q_{M}\), \(Q_{\omega}\), \(Q_{st}\) | losses per kg fuel: dry gas, water from hydrogen, fuel moisture, air moisture, total | J/kg |
| \(L_{dg}\), \(L_{H}\), \(L_{M}\), \(L_{\omega}\), \(L_{st}\) | the same as fractions of GCV | – |
| \(\bar c_{p,g}\) | mean specific heat of the dry flue gas between \(T_a\) and \(T_g\) | J/(kg·K) |
Method and standards
The loss definitions of ASME PTC 4 §5.14 applied with the combustion air temperature as the reference, which is the form used by the BS 845 / IS 8753 audit methods. Dry gas enthalpy is integrated from NIST Shomate heat capacities of the entered CO₂/O₂/N₂ composition instead of a constant 1.005 kJ/(kg·K); the water terms use IF97 enthalpies of vapour at 1 psia and of liquid at the air temperature, which is what the latent-heat-plus-superheat approximation (2 442 + 1.88 ΔT) stands in for.
Assumptions and limits
- Argon and SO₂ in the dry gas are lumped into N₂ (error below 0.2 % of the dry gas loss).
- Complete combustion; CO, unburned carbon, radiation and residue losses are not included here; use the boiler efficiency calculator for the full balance.
- Stack temperature above about 39 °C (water vapour at 1 psia).
References
- ASME PTC 4-2013 §5.14.2–5.14.4 (loss definitions)
- NIST-JANAF Shomate heat capacities
- IAPWS-IF97
Questions and answers
What is the rule of thumb for the dry gas loss?
About 1 % of input for every 20 K of stack temperature above the air temperature at normal excess air, which this calculator reproduces: 12.5 kg of dry gas at a mean cp of 1.02 kJ/(kg·K) over 120 K is 1.53 MJ, or 7.7 % of a 20 MJ/kg coal.
Why is the reference here the air temperature and not 25 °C?
This is the quick stack-loss form, which many audit methods (BS 845, IS 8753) use: losses counted from the air temperature with no separate credits. The full PTC 4 calculator references to 25 °C and adds credits; for air at 25 °C the two agree.
Which dry gas quantity do I enter?
The dry flue gas mass per kg fuel from the combustion calculator at the measured excess air, or a measured flue gas flow divided by the fuel flow and corrected for moisture.