Steam System and Boiler Efficiency

Steam System and Boiler Efficiency

Steam System and Boiler Efficiency: the Heat-Loss Method, the Steam-System Balance and the Loss Estimates

What the tool does

The tool takes one boiler and its steam system as entered and works through five linked estimates: the boiler efficiency by the heat-loss (indirect) method, with the direct input-output method beside it when the fuel input is metered; the steam, feedwater, blowdown, flash, condensate, makeup and deaerator balance on IAPWS-IF97 properties, from which the fuel input, the annual fuel cost and the fuel cost of steam follow; the water and chemical arithmetic of the blowdown; the steam lost through failed traps and leaks and the heat lost from bare pipe; and the power a backpressure turbine would recover from steam that a pressure-reducing valve throttles. Every method is a closed form or a textbook balance written out on the page with its numbers. The public sources are the DOE Steam Tip Sheets and the Improving Steam System Performance sourcebook, Army TM 5-650 Central Boiler Plants (fuel analyses and the blowdown appendix) and the IAPWS and NASA property formulations; ASME PTC 4 and the ABMA radiation chart are cited for the method and the radiation loss, not reproduced.

Conventions

Pressures are entered as gauge values and the tool adds the standard atmosphere (14.696 psi, 101.325 kPa). Temperatures are plain numbers (°F in USCS mode, °C in SI mode) and are not converted by the unit switch, nor are the heating value, the fuel price, the insulation conductivity and the other labelled plain quantities; flows, pressures and lengths go through the unit dictionary. Efficiencies are on the higher-heating-value basis unless the line says LHV. Money is printed with two decimals and every annual figure uses the entered hours per year.

Fuel analysis and combustion stoichiometry

The fuel is described by its ultimate analysis as fired (carbon, hydrogen, oxygen, nitrogen, sulfur, ash and moisture in per cent by mass) and its higher heating value. The five selectable fuels are the analyses printed in the footnotes of Army TM 5-650 Tables 1-4 to 1-8 (natural gas, No. 2 and No. 6 oil, and two bituminous coals); the natural-gas footnote lists 0.8 % carbon dioxide beside the elements, and the tool folds it into the carbon and oxygen columns (0.8 % CO₂ = 0.218 % C + 0.582 % O), which is the same mass balance because that carbon needs exactly the oxygen it already carries. The lower heating value is HHV − m_w h_fg(25 °C), with m_w the water from the hydrogen (18.015/2.016 = 8.94 kg per kg of H) plus the fuel moisture.

Per kilogram of fuel the oxygen demand is n_O2 = C/12.011 + H/4.032 + S/32.065 − O/31.999 mol (carbon to CO₂, hydrogen to H₂O, sulfur to SO₂, less the oxygen the fuel brings), and the theoretical dry air is n_O2/0.20948 mol at the U.S. Standard Atmosphere 1976 composition (N₂ 78.084 %, O₂ 20.948 %, Ar 0.934 %, CO₂ 0.031 %, 28.964 kg/kmol). With excess air e the dry products are CO₂ (fuel carbon burned plus the air's own), SO₂, N₂ (fuel plus air), Ar and the surplus oxygen 0.20948 e n_air,th; the water is the fuel water plus ω times the dry-air mass, with the humidity ratio ω = 0.622 p_v/(p − p_v) from the entered air temperature and relative humidity. Unburned carbon (a fraction of the fuel carbon left in the refuse) is removed from the carbon burned, and carbon monoxide in the dry gas replaces carbon dioxide mole for mole and frees half a mole of oxygen per mole.

The excess air is recovered from a measured dry-basis flue-gas oxygen x by the dry-gas balance, e = x (D_fuel + a n_air,th) / [n_air,th (0.20948 − x)], where D_fuel is the dry products of the fuel alone and a the non-oxygen fraction of air; from a carbon-dioxide reading by the corresponding closed form; with carbon monoxide present the closed form is refined by a secant iteration on the full balance. The CO₂ route cannot tell excess air from a fuel-rich flame and is bounded by the stoichiometric CO₂ of the fuel.

The heat-loss method

Each loss is the heat that leaves the boiler envelope per unit mass of fuel, divided by the higher heating value, referred to the combustion-air temperature entering the boiler (ASME PTC 4 uses the same reference; its full loss list and its correction terms are not reproduced):

Dry flue gas: L_dg = Σ_i n_i [h_i(T_stack) − h_i(T_air)] over CO₂, SO₂, N₂, Ar, O₂ and CO, the sensible enthalpies from the NASA Glenn seven-coefficient fits (McBride, Zehe and Gordon 2002) through the shared property library; SO₂ is taken with the molar enthalpy of CO₂ (the two molar heat capacities agree within a few per cent and SO₂ is below 0.1 % of the gas) and CO with that of N₂.

Water from the hydrogen and the fuel moisture: L_fw = m_w [h_v(T_stack, p_w) − h_f(T_air)]. The higher heating value counts this water condensed at the reference temperature, so the loss is the whole difference between vapour at the stack temperature and its partial pressure p_w = y_H2O p_atm (IAPWS-IF97 region 2) and saturated liquid at the reference: the latent heat and the vapour's sensible heat together. The air moisture entered as vapour, so it carries only its sensible rise, L_aw = ω m_air [h_v(T_stack) − h_v(T_air)].

Unburned carbon: the carbon in the refuse times its heat of combustion, 393,510 J/mol / 12.011 g/mol = 32.76 MJ/kg (14,087 Btu/lb), from the NASA Glenn enthalpy of formation of CO₂. Carbon monoxide: the moles of CO times 282.98 kJ/mol, the heat that CO → CO₂ would still release. Radiation and convection: entered as a percentage of the input, read from the ABMA radiation-loss chart for the boiler's rated output and load (about 0.5 to 1.5 % for a packaged boiler at full load, more at part load because the surface loss is nearly constant while the input falls). Unaccounted: an entered margin for the sensible heat of the refuse, sootblowing steam, fuel sensible heat and the manufacturer's margin.

η_HHV = 1 − Σ L. The combustion efficiency printed beside it is 100 % less the three stack losses alone (dry gas, fuel water, air moisture), which is the quantity tabulated by TM 5-650 Tables 1-4 to 1-8 and by Steam Tip Sheet #4 against the flue-gas oxygen and the stack-minus-air temperature; the tool reproduces those tables within 0.4 points for gas and oil (the manual's own calculation is not stated; its coal tables run 0.7 to 1.4 points below the tool, consistent with a moisture or refuse allowance). The LHV efficiency is η_HHV × HHV/LHV. Figure 1 plots η_HHV against stack temperature for several oxygen levels and Figure 2 the losses against excess air at the entered stack temperature; the slopes are the rule of thumb of Tip Sheet #4, about 1 % per 40 °F of stack temperature or 15 % of excess air.

Direct method: when the fuel input is metered, η_direct = [ṁ_gen (h_s − h_fw) + ṁ_bd (h_bw − h_fw)] / Q_fuel, the heat absorbed by the steam and the blowdown over the measured input, with the generation and blowdown of the balance below. ASME PTC 4 prefers the heat-loss method because the measured quantities enter the result with a smaller weight; a direct figure well below the heat-loss one points to a loss the accounting missed or to meter error.

Steam properties and the system balance

All water and steam properties come from IAPWS-IF97 through the shared fluid-property library: the header steam dry saturated at the drum pressure or superheated at the entered temperature; the feedwater saturated liquid at the deaerator pressure (the deaerator holds the feedwater at its saturation temperature, 227 °F at 5 psig; pump work neglected); the blowdown saturated liquid at the drum pressure; the condensate and the makeup as liquid at their temperatures; the flash steam and the vent as saturated vapour at their pressures.

The boiler generates ṁ_gen = ṁ_s + ṁ_da, the steam delivered to the plant plus the deaerator's own heating steam, and takes feedwater ṁ_fw = ṁ_gen + ṁ_bd. The blowdown follows one of four rules: from the solids balance ṁ_bd TDS_bw = ṁ_mu TDS_mu (the condensate carries no solids), so that the cycles of concentration on the makeup are C = TDS_bw/TDS_mu and ṁ_bd = ṁ_mu/C (Steam Tip Sheet #9); from an entered C on the makeup; from the cycles on the feedwater of TM 5-650 Appendix D, B = F/C, which gives ṁ_bd = ṁ_gen/(C − 1); or as an entered fraction b of the feedwater, ṁ_bd = b ṁ_gen/(1 − b). The deaerator obeys ṁ_fw + ṁ_vent = ṁ_c + ṁ_mu + ṁ_fl + ṁ_da and ṁ_fw h_fw + ṁ_vent h_g,da = ṁ_c h_c + ṁ_mu h_mu + ṁ_fl h_g,fl + ṁ_da h_s, with the condensate ṁ_c = r ṁ_s, the vent a fraction of the feedwater and the flash steam ṁ_fl = x ṁ_bd. Because the deaerator steam is itself generated by the boiler, the equations are solved together by a fixed-point iteration on ṁ_da, which converges in a few steps; the test script checks the result against a direct linear solve.

Flash steam: when saturated liquid at the drum pressure is let down to the flash-tank pressure, the fraction x = (h_f,drum − h_f,flash)/h_fg,flash flashes to vapour (14.9 % from 150 psig to 5 psig); the tool sends it to the deaerator, where it displaces heating steam, and the drain leaves as saturated liquid at the flash pressure. A makeup heat exchanger on the drain recovers an entered fraction of the drain's heat above the raw makeup temperature (Steam Tip Sheet #10 uses 90 %) and warms the makeup before the deaerator. The blowdown heat reported is ṁ_bd (h_bw − h_mu), the enthalpy of the drum water above the raw makeup that replaces it; the flash steam and the exchanger recover part of it and the rest is lost to the drain, costed as fuel over the boiler efficiency. The value of one percentage point of blowdown is printed for the same reason Steam Tip Sheet #9 gives: every pound blown down leaves the drum carrying the drum enthalpy.

Fuel: the boiler duty Q_out = ṁ_gen (h_s − h_fw) + ṁ_bd (h_bw − h_fw) over the heat-loss efficiency is the fuel input on the HHV basis, and the fuel flow is that input over the heating value. The fuel cost of steam in the form of Steam Tip Sheet #15 is the fuel price times (h_s − h_fw)/η per unit of steam; the effective cost divides the whole annual fuel bill by the steam actually delivered to the plant, so it carries the deaerator steam and the blowdown.

Economizer and air preheater: an exchanger of effectiveness ε on the stack gas recovers Q = ε C_min (T_stack − T_cold) per unit mass of fuel, the gas-side capacity rate taken as the mean slope of the flue-gas enthalpy between the two temperatures and the cold side the feedwater (ṁ_fw c_p per unit of fuel) or the combustion air (A_wet c_p,air). The new stack temperature is the one at which the gas holds the enthalpy left, and the losses are recomputed with it, still referred to the ambient air (the preheated air returns its heat to the furnace inside the envelope). The fuel saving is 1 − η_old/η_new for the same steam duty. The page warns when the economizer water would approach the drum saturation temperature and when the recovered stack temperature comes within 10 K of the water dew point of the gas, below which a condensing economizer would recover latent heat the sensible model does not credit and a sulfur fuel would corrode.

Blowdown water and chemicals

The makeup for the year is the makeup flow times the hours, priced per 1,000 gallons or per cubic metre. The chemical arithmetic is that of TM 5-650 Appendix D-4: the chemical held at a residual in the boiler water leaves with the blowdown and must be replaced, feed = ṁ_bd × residual / active fraction of the product (the manual's example: 2,105 lb/h of blowdown at 60 ppm of phosphate is 3.03 lb/day of phosphate, 7.58 lb/day of a 40 % product). An oxygen scavenger consumed by the feedwater oxygen is fed on the feedwater instead; the library carries that form, the page prints the residual form.

Steam trap and leak losses

A trap that has failed open discharges steam to the condensate system, in effect to atmosphere, through its orifice. For dry saturated steam the discharge is choked whenever the line pressure exceeds about 1.74 atmospheres, and Napier's formula gives W = p A/70 lb/s with p in psia and A in square inches, times a discharge coefficient C_d; the tool uses it in SI (W = C_d A p × 1.4566 × 10⁻³ s/m) with C_d = 0.72 for a re-entrant orifice, the coefficient behind the table of Steam Tip Sheet #1 (Boiler Efficiency Institute). The table's cells are 1.3 % above Napier's constant; the ideal-gas choked-flow equation with k = 1.3 and the saturation temperature agrees with Napier within 1 % at 100 to 150 psig. The lost steam is costed at the boiler's fuel cost of steam for the entered hours (the sheet's example uses 8,760 h and $10 per 1,000 lb).

Pipe heat loss and insulation

A horizontal pipe at the line saturation temperature loses q′ = (h_c + h_r) π D (T_s − T_a) per unit length in still air. The convection coefficient is the Churchill and Chu (1975) correlation for the horizontal cylinder, Nu = {0.60 + 0.387 Ra^1/6/[1 + (0.559/Pr)^9/16]^8/27}², with the air properties at the film temperature from the convection library (U.S. Standard Atmosphere 1976 viscosity and conductivity); the radiation coefficient is ε σ (T_s⁴ − T_a⁴)/(T_s − T_a). With insulation of thickness t and conductivity k the heat conducted, 2π k (T_p − T_s)/ln(D_o/D_p), must equal the surface loss of the jacket at its own emissivity, and the jacket temperature is found by bisection; this is the calculation of the 3E Plus program (ASTM C680), without the pipe-wall resistance, wind or the temperature dependence of k, which is entered at its mean temperature. The bare loss per 100 ft and 8,760 h is printed on the basis of the Steam Tip Sheet #2 table: the tool's values run 4 to 15 % above the table's (the table's emissivity and correlation are not stated), so the table is reproduced in trend, not cell by cell. Figure 3 plots the loss against thickness; the first inch does most of the work and the economic thickness is where the fuel saved per added inch no longer pays for it.

Backpressure turbine versus pressure-reducing valve

Steam let down through a PRV keeps its enthalpy (isenthalpic throttling) and arrives superheated at the low pressure. Through a turbine the isentropic end state has s_2s = s₁, the actual exit h₂ = h₁ − η_s (h₁ − h_2s), and the shaft work is h₁ − h₂, times the generator efficiency for the electrical output; the tool prints the output per 1,000 lb/h (or per tonne per hour), the figure of the chart in Steam Tip Sheet #20, which assumes 50 % isentropic and 96 % generator efficiency and dry saturated inlet steam. The exhaust carries less heat than the PRV steam, so a process that condenses it to saturated liquid at the low pressure needs more steam in the ratio (h₁ − h_f,2)/(h₂ − h_f,2); the extra steam is raised at the boiler efficiency and its fuel, over the kilowatt-hours generated, is the cost of the generated power. Tip Sheet #20's shortcut, fuel price × 0.003412 MMBtu/kWh / (η_boiler η_generator), charges the whole turbine work as heat lost and is printed beside the exact figure; Tip Sheet #22 walks the same comparison for a 600 psig boiler and reads $0.029/kWh. The annual saving is the output times the hours times the difference between the purchased and the generated price; equipment and installation costs are outside the tool. Figure 4 plots the output per unit of steam against the exhaust pressure for the entered inlet state.

Checks

The test script runs 524 checks: the five TM 5-650 fuel analyses and the stoichiometry of each against an independent numpy script (theoretical air, dry-gas analysis, the closed-form inversions against a root search); the default boiler's losses against the same script; cells of the TM 5-650 combustion-efficiency tables and of the Steam Tip Sheet #4 table; twenty seeded boilers with perturbed analyses, humidity, unburned carbon and carbon monoxide; the IF97 states against the NIST WebBook (IAPWS-95) at the tool's pressures; the flash fractions, the deaerator balance against a linear solve, the TM 5-650 Appendix D examples D-1 to D-4 to the pound; the Tip Sheet #9, #10, #15, #20, #22 and #23 arithmetic to the printed dollar; Napier against the Tip Sheet #1 table and the choked-flow equation; the pipe loss against the numpy model and the Tip Sheet #2 table; the turbine states against the WebBook and the Tip Sheet #22 exhaust temperature (527 °F printed, 526.3 °F computed); and the tool itself through the CLI harness in both unit systems.

Limits

One boiler, one header, one deaerator. No condensing economizer (latent recovery below the dew point), no part-load efficiency curve (enter the stack temperature and oxygen of the load in question), no fuel sensible heat or sootblowing steam beyond the unaccounted margin, no pipe-wall resistance or wind in the insulation estimate, no condensing turbine or extraction, and no equipment or installation costs. The ASME consensus water-chemistry limits, the ABMA radiation chart and the ASME PTC 4 loss corrections are entered values or cited only.