Evaluate how effectively a boiler or industrial furnace turns fuel into useful heat, using the heat-loss method and the Siegert formula. Adjust the fuel, excess air and flue-gas temperature to see the stack loss, thermal efficiency and the room left for energy savings in real time.
Parameters
Fuel
Sets the Siegert factor f and the stoichiometric max CO₂
Excess-air ratio λ
Ratio of actual air supplied to the theoretical air
Flue-gas temperature
℃
Exhaust temperature at the chimney inlet
Inlet (ambient) air temperature
℃
Temperature of the combustion air drawn into the burner
Fuel heat input
kW
Heat input on a heating-value basis (burner firing rate)
Other losses (radiation, unburnt)
%
Sum of casing radiation, unburnt loss, blowdown loss, etc.
Results
—
Stack loss (%)
—
Flue-gas CO₂ (%)
—
Other losses (%)
—
Thermal efficiency η (%)
—
Useful heat output (kW)
—
Heat lost up the stack (kW)
—
Furnace / boiler energy flow
Fuel and air burn at the burner, useful heat flows to the load, and the hot flue gas rises up the chimney. Arrow and plume sizes scale with each share of the fuel energy.
Dry flue-gas loss q_stack (Siegert formula) and the flue-gas CO₂ concentration. t_flue: flue-gas temperature, t_air: combustion-air temperature, λ: excess-air ratio.
$$\eta=100-q_{stack}-q_{other}$$
Thermal efficiency η (heat-loss method). q_other: other losses such as radiation and unburnt fuel. f is the fuel-specific Siegert factor and λ the excess-air ratio.
What is Furnace and Boiler Efficiency?
🙋
"Thermal efficiency" of a factory boiler means how much of the fuel I burned actually did useful work, right? How is it measured?
🎓
Exactly. There are two ways. One is the "input-output method" — measure the fuel heat going in and the heat carried out by the steam, then divide. But measuring steam flow accurately is surprisingly hard. So the method used most in the field is the "heat-loss method". You flip the logic: start at 100% and subtract the heat that escaped. η = 100% − the sum of the losses. This tool uses that heat-loss method.
🙋
I see — counting what escaped is easier. So where does the biggest loss escape?
🎓
By far the "stack loss" — the heat in the flue gas going up the chimney. Think about it: after combustion the exhaust leaves the stack at 200 or 300 °C. All the sensible heat carried away by that hot gas is loss. Raise the "flue-gas temperature" on the left and you will see the stack loss climb and the efficiency drop. With the default natural gas, the stack loss alone is about 7%.
🙋
7%! So how do I reduce that stack loss?
🎓
There are two levers. One is "lower the excess air". If you supply more air than combustion needs, you heat up that surplus air — mostly non-burning nitrogen — and throw it away. Lowering the excess-air ratio λ both reduces the volume of exhaust to be heated and raises the CO₂ concentration in the flue gas, which enlarges the denominator of the Siegert formula. Both effects cut the loss. The other lever is "lower the flue-gas temperature": recover heat with an economizer or air pre-heater so the temperature you throw away itself drops.
🙋
Then should I keep cutting the excess air all the way down to exactly 1.0?
🎓
That is the trap. An excess-air ratio of 1.0 is "exactly the theoretical air", the stoichiometric point. But real combustion never mixes perfectly, so right at the stoichiometric point some regions run short of oxygen and leave unburnt fuel. Soot and carbon monoxide form, and that becomes an "unburnt-fuel loss". It is not just waste — CO is a poisoning hazard. So real plants keep λ around 1.1 to 1.3 with a little margin. Working that knife-edge between the "efficiency peak" and the "incomplete-combustion boundary" is where combustion management shows its skill.
Frequently Asked Questions
The standard method in practice is the heat-loss method. The idea is simple: efficiency η = 100% − the sum of the losses (%). By far the largest loss is the dry flue-gas (stack) loss — the sensible heat carried up the chimney by the hot exhaust. This tool computes the stack loss with the Siegert formula q_stack = f·(t_flue − t_air)/CO₂%, adds the other losses (radiation, unburnt fuel), and subtracts the total from 100% to give the thermal efficiency.
f is the fuel-specific Siegert factor (fuel constant), derived from the carbon-to-hydrogen ratio and the heating value of the fuel. This tool uses f=0.38 for natural gas, f=0.50 for fuel oil and f=0.65 for coal. A fuel with a larger f produces a larger stack loss at the same flue-gas temperature and CO₂ concentration. The Siegert formula is a long-standing practical equation that estimates the flue-gas loss from just three quantities: flue-gas temperature, combustion-air temperature and flue-gas CO₂ concentration.
Lowering the excess-air ratio λ reduces the surplus air — most of which is inert nitrogen that gets heated for nothing — so the very volume of exhaust to be heated drops. At the same time the flue-gas CO₂ concentration rises as CO₂% = CO₂max/λ, which enlarges the denominator of the Siegert formula and cuts the stack loss. But it cannot be cut too far: below the stoichiometric point combustion becomes incomplete, producing soot and carbon monoxide. That is an unburnt-fuel loss — wasteful and dangerous at the same time.
The stack loss is proportional to the difference (t_flue − t_air), so lowering the flue-gas temperature reduces the loss linearly. In real plants an economizer (feed-water pre-heater) or air pre-heater is placed in the exhaust path to recover the otherwise wasted sensible heat. As a rule of thumb, lowering the flue-gas temperature by about 20 °C raises the efficiency by about 1%. However, going too low lets the water vapour in the exhaust condense and causes low-temperature corrosion from sulphuric acid, so the temperature must stay above the acid dew point.
Real-World Applications
Energy audits of industrial boilers: For steam boilers in food, chemical, textile and paper plants, just measuring the flue-gas temperature and CO₂ concentration (or residual O₂) lets you estimate the thermal efficiency immediately with the same Siegert formula this tool uses. In an energy audit, the engineer first pins down "what efficiency are we at now" with this quick calculation, then estimates how many points can be gained by optimising excess air or adding an economizer.
Combustion management of industrial furnaces: In heating furnaces, heat-treatment furnaces and glass-melting furnaces, the stack loss is also the largest loss item. Continuous reheating furnaces use regenerative burners that return the exhaust heat to the combustion air, sharply lowering the flue-gas temperature and lifting efficiency. Raise the inlet-air temperature in this tool and the effect of air pre-heating shows directly as an efficiency gain.
Combustion control and O₂ trim: Modern boilers carry zirconia O₂ sensors that continuously measure the oxygen in the flue gas and an "O₂ trim control" that throttles the airflow with dampers or an inverter. Residual O₂ maps one-to-one to the excess-air ratio λ, so lowering O₂ means lowering λ means cutting the stack loss. The excess-air slider in this tool is a hands-on way to feel the region this trim control aims at.
Fuel switching and CO₂ emissions: Switching from coal to fuel oil, or from fuel oil to natural gas, lowers the Siegert factor f, so the stack loss falls and the efficiency rises even at the same operating conditions. Switch the fuel preset in this tool to compare quantitatively how a fuel change affects efficiency and CO₂ emissions. It works as a starting point for the preliminary study of decarbonisation and fuel conversion.
Common Misconceptions and Pitfalls
The most common one is "just lower the flue-gas temperature and efficiency rises without limit". The stack loss is indeed proportional to (t_flue − t_air), but if you cool the flue gas too far, the water vapour in it reaches its dew point and condenses. For sulphur-bearing fuels such as fuel oil and coal, the condensate reacts with sulphur trioxide to form sulphuric acid and severely corrodes the economizer and the flue. This is "low-temperature corrosion", and the flue-gas temperature has a lower limit set by the acid dew point. As a guide, keep it around 100 °C even for natural gas and above 150 °C for high-sulphur fuels. Chasing efficiency alone by cooling the flue gas shortens the equipment life.
Next, confusing whether efficiency is on a higher-heating-value (HHV) or lower-heating-value (LHV) basis. The Siegert formula in this tool counts the sensible heat carried by the water vapour in the exhaust, but assumes the latent heat released when that vapour condenses (the heat of vaporisation) is not recovered. This corresponds to an efficiency on the LHV basis. A condensing boiler that recovers the latent heat too can show an "efficiency" above 100% on the LHV basis. When comparing catalogue figures, always check whether they are HHV- or LHV-based. A different basis can shift the number by 5 to 10% for the very same equipment.
Finally, assuming the rated efficiency is always achieved. A boiler's efficiency is highest at the rated catalogue point and gets worse at low load. As the load drops, the share of radiation loss from the casing relatively increases, and at low firing it is harder to throttle the excess air, so the stack loss tends to rise too. On top of that, soot and scale (water deposits) on the heat-transfer surfaces block the heat path, push up the flue-gas temperature and lower efficiency. The calculation in this tool is an instantaneous estimate; the annual average efficiency of a real plant, once load variation, start-stop cycles and fouled surfaces are folded in, is normally a few points below the rated value.
How to Use
Enter excess air percentage (typically 3–15% for natural gas, 15–25% for coal) using the slider or numeric input.
Set flue-gas temperature in °C (stack temperatures range 150–250°C for modern boilers, 200–350°C for older units).
Input ambient air temperature and fuel energy input in kW, then execute the calculation to obtain efficiency η and stack heat losses.
Worked Example
Natural gas boiler: fuel input 500 kW, excess air 8%, flue-gas temperature 180°C, ambient 25°C. Stack loss calculation yields ~8.2% heat lost up the stack (≈41 kW), CO₂ concentration ~9.5%, and thermal efficiency η ≈ 89.3%, leaving useful heat output of 446.5 kW. Reducing flue-gas temperature to 160°C increases efficiency to ~91.1% by lowering convective and radiative losses.
Practical Notes
Excess air above 12% in gas furnaces triggers disproportionate stack loss; coal and oil systems require higher margins (20–25%) to ensure complete combustion.
Install economizers or air-preheaters to recover stack heat when flue-gas exceeds 200°C; each 10°C reduction typically gains 1–2% efficiency.
Monitor CO₂ levels (should be 9–12% for natural gas) to detect air-leakage or burner maladjustment that masquerades as efficiency loss.
Industrial furnaces with refractory degradation show unmeasured losses (scaling, radiation) not captured in stack loss alone; cross-validate with fuel consumption audits.