Compressed Air Engineering Formula Register

Compressed Air EngineeringFormula Register

Choose the equation by its physical meaning, operating boundary and assumptions—not by the number it produces.

1. Engineering answer: an equation needs an operating boundary

It is not a wall of formulas detached from pressure, temperature, equipment state or measurement quality. A correct equation with the wrong airflow basis can produce a precisely wrong selection.

This edition contains 40 identified formula records. It brings together the supplied airflow, storage, electrical, pneumatic, thermal, sampling and diagnostic foundations, then adds explicit notation, dimensional explanations and reusable data fields. The register does not claim to contain every formula in every RC page or to validate all existing website tools.

Use it before entering a calculator, before comparing supplier figures and when reviewing a result that looks surprising. The practical question is not simply “Which equation produces a number?” It is “Does this equation answer this question with these data?” When a required basis is unknown, preserve the uncertainty or obtain the missing evidence instead of adding an arbitrary factor.

2. Where this register fits in the engineering centre

Those functions connect, but they are not interchangeable. A unit conversion can be dimensionally correct while a chosen physical model remains inappropriate. A reference case can pass while a live website adapter maps the wrong input column. A supplier sheet can be faithfully transcribed without proving the installed machine meets its duty.

It does not replace manufacturer selections, a pressure-vessel design, an electrical safety procedure, a laboratory method or a signed engineering decision. The downloadable register and examples are deliberately usable without making equipment-control changes or uploading private customer data.

3. The minimum record behind every formula

It applies the supplied data-dictionary and validation principles without rewriting older records. A formula record should be readable by an engineer and structured enough for a developer to reuse. A comment containing “assume normal conditions” is not a reference-state definition; a test report containing “passed” is not a method version.

4. Read dimensions and meaning as separate checks

Dimensional checking asks whether the operations produce the stated type of quantity. Pressure plus pressure produces pressure; power multiplied by time produces energy; actual volume flow divided by area produces velocity. It catches errors such as using millimetres as metres before squaring, mixing seconds and minutes, or reporting thermal duty in an electrical-energy field.

Semantic checking asks whether the quantities belong together. Package electrical power and motor shaft power share a dimension but have different boundaries. Reference airflow and actual pipe airflow share volume-flow units but differ in gas state. A temperature interval and a temperature value require different conversion rules. Dimensional consistency is necessary for these calculations; it is not sufficient to establish engineering validity.

Keep absolute pressure and kelvin temperature explicit in gas-state ratios. Keep gauge pressure and Celsius values available for readable field reports. Do not strip useful source terminology: store it alongside the normalized field. A calculation should preserve how a result was obtained, not erase the original input because it now has a standardized representation.

5. Choose the calculation family before choosing a number

Reuse them in future tool specifications and change records. Each formula card contains a numerical example for the stated assumptions; the examples are fictional reference inputs, not customer measurements, supplier guarantees or recommended settings.

6. Pressure and temperature reference formulas

F001 — Gauge to absolute pressure

p_abs = p_g + p_atm

Inputs and output: p_g: gauge pressure, bar; p_atm: matching atmospheric absolute pressure, bar. Output: p_abs, bar absolute

Dimensional check: Pressure + pressure = pressure; the reference changes, not the unit.

Use: Use when the gauge is referenced to the stated atmosphere at the relevant location. Keep the measurement point and atmospheric source with the result. Negative gauge readings can be meaningful; an atmosphere must not be added twice to an already absolute reading.

Limits: Require a known pressure reference and p_abs > 0 for downstream gas-state calculations. A sealed-gauge reference needs its documented relationship. A standard atmosphere is not automatically site atmosphere.

Reference example: p_g = 7.0 bar and p_atm = 0.98 bar give p_abs = 7.98 bar.

Basis: S1.

F002 — Compression pressure ratio

r_p = p_out_abs / p_in_abs

Inputs and output: p_out_abs and p_in_abs: compatible absolute pressures in the same unit. Output: r_p, dimensionless

Dimensional check: Pressure / pressure = 1.

Use: Use the actual inlet and outlet of the stated stage or package and a compatible operating state. The inlet after an intake restriction can differ from ambient pressure. Preserve whether the value describes a stage or the complete machine.

Limits: Require both pressures positive; flag a ratio below one in compression mode. A pressure ratio does not determine FAD, temperature rise or package power without another applicable model.

Reference example: 8.0 bar absolute / 1.0 bar absolute = 8.0; 7 bar gauge / 0 bar gauge is undefined, not equivalent.

Basis: S1.

F003 — Celsius value to kelvin

T_K = T_C + 273.15

Inputs and output: T_C: Celsius temperature value, not a temperature difference. Output: T_K, K

Dimensional check: Affine temperature-scale transformation; do not treat the offset as a measured heat input.

Use: Use an absolute temperature for gas-state ratios. Retain whether the original value is a process temperature, reference temperature or dew point. The numerical conversion does not change the physical measurement location or moisture basis.

Limits: For the gas models in this register require T_K > 0. Do not add 273.15 to a temperature interval. Converting a dew-point scale does not convert atmospheric dew point to pressure dew point.

Reference example: 20 °C = 293.15 K. A 20 K temperature difference is not 293.15 K.

Basis: S1, S6.

F004 — Fahrenheit temperature interval

delta_T_K = delta_T_F × 5/9

Inputs and output: delta_T_F: signed difference between compatible Fahrenheit readings. Output: delta_T_K, K

Dimensional check: A temperature interval scales by 5/9; no offset remains after subtraction.

Use: Use for a rise, approach or before/after difference expressed in Fahrenheit degrees. Preserve the sign and named points. A rise of 18 Fahrenheit degrees represents the same interval as 10 kelvin or 10 Celsius degrees.

Limits: For an absolute reading use T_C = (T_F − 32) × 5/9 instead. Never subtract 32 from a difference. A unit conversion is not a correction for different sensor locations or operating conditions.

Reference example: 18 °F difference × 5/9 = 10 K.

Basis: S1, S6.

7. Flow, density and gas-state formulas

F005 — CFM to cubic metres per minute

Q_m3_min = Q_ft3_min × 0.028316846592

Inputs and output: Q_ft3_min: volume flow on a declared, unchanged reporting basis. Output: Q_m3_min, m³/min on the same basis

Dimensional check: L³/time multiplied by an exact volume-scale factor remains L³/time.

Use: Use the international-foot definition for a pure unit conversion. Keep FAD, actual or reference-flow meaning as separate metadata. This conversion can be exact as a unit operation even when the original measured or brochure value is uncertain.

Limits: Do not use it to change pressure, temperature, gas composition or humidity. An unqualified CFM input may be converted numerically while still remaining unsuitable for a capacity or velocity comparison.

Reference example: 100 cfm = 2.8316846592 m³/min on the original basis.

Basis: S1, S2.

F006 — Gas density at a stated condition

rho = p_abs / (Z × R_specific × T_K)

Inputs and output: p_abs: Pa; Z: dimensionless; R_specific: J/(kg·K), matching gas; T_K: K. Output: rho, kg/m³

Dimensional check: Pa / [J/(kg·K) × K] = kg/m³.

Use: This is an explicitly added rearrangement of the gas-state model. For ideal-gas screening declare Z = 1; otherwise obtain applicable Z and gas-property information. Use gas identity and dry or humid composition consistently rather than adopting a hidden air-density constant.

Limits: Require positive pressure, temperature, R and Z. The equation does not calculate moisture composition, phase change or an unknown gas constant. It is not a compressor site-derating model.

Reference example: p = 100000 Pa, T = 300 K, illustrative R = 287 J/(kg·K), Z = 1 give rho = 1.161440186 kg/m³.

Basis: R1, S2. New explicit rearrangement; gas-state foundation cited.

F007 — Mass flow from actual volume flow

m_dot = rho × Q_actual

Inputs and output: rho: kg/m³ at the actual-flow condition; Q_actual: m³/s. Output: m_dot, kg/s

Dimensional check: kg/m³ × m³/s = kg/s.

Use: Match density to the same location, gas state and composition as the actual volume flow. A reference-volume flow requires reference density instead. This identity is useful for reconciling gas-state calculations without treating FAD as volume moving at pipe pressure.

Limits: Do not combine reference flow with discharge density. Convert minutes to seconds before this SI form. The present single-direction mode rejects negative flow; bidirectional measurements need a separately defined signed convention.

Reference example: rho = 1.2 kg/m³ and Q = 0.25 m³/s give m_dot = 0.30 kg/s.

Basis: S1, R1. New explicit mass-conservation identity.

F008 — Dry-gas volume-flow state conversion

Q2 = Q1 × (p1_abs/p2_abs) × (T2/T1) × (Z2/Z1)

Inputs and output: Q1: m³/min; p1,p2: same absolute-pressure unit; T1,T2: K; Z1,Z2: dimensionless. Output: Q2, m³/min at state 2

Dimensional check: Volume flow × dimensionless state ratios = volume flow.

Use: Use for the same dry-gas mass flow and composition, with no intervening loss or accumulation. Equal Z values cancel algebraically; equality is an assumption or documented fact, not a default universal property.

Limits: No zero or negative absolute inputs. Do not apply a second conversion to an already normalized output. Condensation, humid-air mass changes and compressor capacity at a different discharge pressure require their own methods.

Reference example: 10 m³/min at 1 bar absolute and 293.15 K becomes 1.25 m³/min at 8 bar absolute and the same temperature, with equal Z.

Basis: S2, R1.

8. Geometry and pneumatic-consumption formulas

F009 — Circular internal flow area

A = pi × D_internal² / 4

Inputs and output: D_internal: actual internal diameter, m. Output: A, m²

Dimensional check: Length squared = area.

Use: Use the internal bore of a circular pipe, cylinder or passage, with the geometry explicitly named. Nominal pipe size, outside diameter and hose marketing size are not interchangeable with internal diameter. Keep the dimensional source and any measurement uncertainty.

Limits: Require D > 0. Millimetres must become metres before squaring. Do not use circular area for a noncircular passage without a different geometry model; hydraulic diameter is not automatically the diameter of an equal-area circle.

Reference example: D = 0.050 m gives A = 0.001963495408 m².

Basis: S5.

F010 — Annular area

A_ann = pi × (D_outer² − D_inner²) / 4

Inputs and output: D_outer: outer flow boundary, m; D_inner: inner obstruction or rod diameter, m. Output: A_ann, m²

Dimensional check: Difference of squared lengths = area.

Use: For a cylinder this is the rod-side effective area. For a concentric drilling annulus it describes gross open area, not a full cuttings-transport model. Preserve which physical surfaces define both diameters rather than assuming a drill-rod catalogue size defines the complete annulus.

Limits: Require D_outer > D_inner >= 0. Eccentricity, joints, cuttings and local restrictions can affect real flow. Positive area alone does not establish adequate flushing or actuator performance.

Reference example: D_outer = 0.100 m and D_inner = 0.032 m give A_ann = 0.007049733914 m².

Basis: S5.

F011 — Mean actual-flow velocity

v_mean = Q_actual / A

Inputs and output: Q_actual: m³/s at the passage state; A: open flow area, m². Output: v_mean, m/s

Dimensional check: (m³/s)/m² = m/s.

Use: Use area-averaged velocity at the stated section after converting flow to actual conditions there. This is a continuity identity, not a recommendation for maximum pipe velocity. It does not describe local jets, boundary-layer velocity or a transient acoustic-wave speed.

Limits: Require A > 0 and an actual-flow basis. Using compressor FAD directly can greatly overstate pressurized-pipe velocity. A valid velocity calculation does not prove acceptable pressure loss, noise or drilling cuttings removal.

Reference example: Q_actual = 0.020 m³/s through A = 0.002 m² gives v_mean = 10 m/s.

Basis: S1, S2. New explicit continuity identity.

F026 — Powered chamber swept volume

V_swept = A_effective × stroke

Inputs and output: A_effective: m²; stroke: m, for the specific powered chamber. Output: V_swept, m³

Dimensional check: m² × m = m³.

Use: Use full-bore area for extension and annular area for a single-rod retraction chamber. Keep the two chamber volumes separate before building a cycle. A geometric volume describes displacement, not yet reference air consumed from the supply.

Limits: Require positive area and stroke. Dead volume, valve passages, tubing, cushion arrangements and leakage need explicit treatment where relevant. Do not infer delivered force or valve peak demand from swept volume alone.

Reference example: A = 0.002 m² and stroke = 0.5 m give 0.001 m³, or 1 litre.

Basis: S5.

F027 — Ideal geometric cylinder cycle air

V_cycle_ref = sum[V_chamber × (p_chamber_abs/p_ref_abs) × (T_ref/T_chamber) × (Z_ref/Z_chamber)]

Inputs and output: One term per powered filling stroke; volumes in m³; absolute pressures; temperatures in K. Output: V_cycle_ref, reference m³/cycle

Dimensional check: Chamber volume × state ratios = reference volume; cycle defines the event count.

A normal double-acting cycle includes one extension and one retraction; other machine sequences need their actual stroke counts. Average reference flow is V_cycle_ref multiplied by cycles per minute.

Limits: This convention is not a complete transient filling model. Initial trapped gas, residual/dead volume, pressure histories, valve losses and leakage may require refinements. Average cycle consumption cannot replace peak valve-flow selection.

Reference example: Two powered swept volumes totalling 0.002 m³ at 7 bar absolute versus 1 bar reference, equal T/Z, give 0.014 reference m³/cycle.

Basis: S5.

9. Time, electrical performance and delivery formulas

F012 — Reference volume from interval flow

V_ref = sum(Q_i × delta_t_i)

Inputs and output: Q_i: interval-mean reference flow, m³/min; delta_t_i: accepted non-overlapping interval, min. Output: V_ref, reference m³

Dimensional check: Volume/time × time = volume.

Use: Integrate only intervals with a documented meaning and compatible reference state. An interval-average record may be multiplied by its duration. Instantaneous samples require an explicitly selected integration method; the worksheet must not quietly treat every point as a held value.

Limits: Reject negative duration, overlaps and inconsistent reference bases. Missing intervals remain missing; report the covered-period total separately from an estimated full-period total. Do not count an imported duplicate twice.

Reference example: 20 m³/min for 10 min plus 18 m³/min for 20 min gives 560 reference m³.

Basis: S2, S8.

F013 — Time-weighted mean

x_mean = sum(x_i × delta_t_i) / sum(delta_t_i)

Inputs and output: x_i: compatible interval means; delta_t_i: positive supported durations in one time unit. Output: Same unit as x

Dimensional check: [Quantity × time]/time = quantity.

Use: Use for pressure, temperature, flow or power over a defined valid window. Each interval must represent the same quantity and measurement boundary. Equal weighting is appropriate only when the durations and representation justify it. Preserve the valid denominator with the output.

Limits: Require a positive total duration. Do not interpolate a missing peak or bridge a change in configuration without a documented model. A ratio of means and a mean of ratios are generally different operations.

Reference example: 80 °C for 10 min, 88 °C for 5 min and 84 °C for 20 min give 83.428571 °C over 35 valid min.

Basis: S6, S8.

F014 — Specific electrical power

SP = P_elec / Q_ref

Inputs and output: P_elec: kW; Q_ref: m³/min on the matching delivery basis. Output: kW per (m³/min)

Dimensional check: Power divided by volume flow; retained reporting units differ from kWh/m³.

Use: Use a matched operating point or a compatible ratio of time-weighted means. Preserve pressure, power boundary and airflow definition. A package ratio and a station-to-useful-delivery ratio answer different questions even if both outputs carry identical units.

Limits: Require Q_ref > 0. Zero-flow unloaded power remains reportable but has no finite P/Q ratio. A motor nameplate rating is not a substitute for measured package input. A smaller unmatched ratio is not proof of better performance.

Reference example: 123 kW / 20.4 m³/min = 6.029411765 kW per (m³/min).

Basis: S2, S8.

F015 — Electrical energy integration

E_elec = sum(P_i × h_i)

Inputs and output: P_i: accepted mean electrical input, kW; h_i: compatible interval duration, h. Output: E_elec, kWh electrical

Dimensional check: kW × h = kWh.

Use: Use measured interval energy directly where its boundary and validity are established; otherwise integrate appropriate mean power. Include loaded, unloaded, stopped auxiliary and transition states only as actually represented. Keep any estimated periods separate from measured integration.

Limits: Calendar hours cannot be assigned one observed load without a supported duty assumption. Do not add a package fan again if already included. Nonlinear or unbalanced electrical loads require suitable true-power measurement rather than a current shortcut.

Reference example: 120 kW for 1 h plus 30 kW for 0.5 h gives 135 kWh electrical.

Basis: S2, S4.

F016 — Specific electrical energy

e_elec = E_elec / V_ref; steady case: e_elec = P_elec / (60 × Q_ref)

Inputs and output: E_elec: kWh; V_ref: m³; steady P: kW and Q: m³/min. Output: e_elec, kWh per reference m³

Dimensional check: Electrical energy / reported gas volume; 60 reconciles minutes with hours.

Use: For varying operation use the ratio of integrated energy and delivered reference volume over matching intervals. The steady expression follows only when the selected power and flow genuinely represent the same condition. Keep upstream package and downstream useful-air boundaries distinct.

Limits: Require positive volume. Energy and volume measured on different time windows cannot be repaired by division. Include treatment losses and auxiliaries only once, according to the stated boundary.

Reference example: 123/(60 × 20.4) = 0.100490196 kWh/m³ for the stated steady point.

Basis: S2, S8.

F017 — Balanced three-phase electrical screening

P_screen = sqrt(3) × V_LL × I_L × PF / 1000

Inputs and output: V_LL: line-to-line RMS volts; I_L: line RMS amperes; PF: applicable true power factor. Output: P_screen, kW

Dimensional check: V × A gives apparent power; PF and 1/1000 convert to kW.

Use: Use only when the balanced-load and measurement assumptions are suitable. All inputs must describe the same operating condition and electrical boundary. This is a screening expression; direct true-power measurement remains preferable for audit and package performance.

Limits: Require non-negative current, positive voltage and 0 <= PF <= 1 in this consumption model. Distorted or unbalanced conditions are not resolved by entering an average current or displacement factor in place of the necessary power measurement.

Reference example: 400 V, 100 A and PF 0.9 give 62.353829072 kW under the declared balanced assumptions.

Basis: S4.

F018 — Two-state loaded/unloaded mean

P_mean = f_loaded × P_loaded + (1 − f_loaded) × P_unloaded

Inputs and output: f_loaded: share of defined running time; P_loaded,P_unloaded: kW at those states. Output: P_mean, kW

Dimensional check: Dimensionless time shares × power = power.

It preserves nonzero unloaded consumption instead of assuming that flow fraction equals power fraction. A separate stopped or purge state requires a multi-state extension.

Limits: Require 0 <= f_loaded <= 1 and supported state powers. Do not extrapolate to a variable-speed curve or combine enabled time with motor-running time. Transition duration and varying operating pressure may invalidate the simple representation.

Reference example: f_loaded = 0.6, P_loaded = 90 kW and P_unloaded = 30 kW give 66 kW.

Basis: S2.

F019 — Net useful delivery at steady inventory

Q_useful = Q_package − Q_purge − Q_drains − Q_other

Inputs and output: All flows: same reference m³/min and time/state; Q_other: identified non-overlapping boundary outflow. Output: Q_useful, reference m³/min

Dimensional check: Same-basis volume flows subtract to a volume flow.

Use: Use the stated stationary-inventory balance where receiver accumulation is negligible or separately accounted for. Each deduction must lie between the package and useful-delivery boundaries. A supplier net output that already includes a dryer cannot have the same purge deducted again.

Limits: An unknown deduction is not zero. Negative computed delivery signals an inconsistent boundary or input set for this mode. Short events with receiver charging or discharging need an inventory term, not this simplified balance.

Reference example: 20.4 − 1.2 − 0.1 − 0.2 = 18.9 reference m³/min.

Basis: S2, S8.

10. Storage and pressure-path formulas

F020 — Transient storage deficit

Q_deficit = max(Q_event − Q_supply, 0)

Inputs and output: Q_event,Q_supply: concurrent flow reaching the protected zone, same reference m³/min. Output: Q_deficit, reference m³/min

Dimensional check: Difference of same-basis volume flows; max with zero clips only the sizing deficit.

Use: Use the supply that actually reaches the zone during the event, not the compressor nameplate capacity elsewhere. The unclipped difference may be retained as a diagnostic field. The clipped result represents only the positive shortage to be covered by storage.

Limits: Inputs must be non-negative and have matching reference states. Zero deficit means this defined event requires no storage contribution; it does not prove that controls, piping, reserve or other events need no receiver.

Reference example: 18 m³/min demand minus 10 m³/min reaching the zone gives an 8 m³/min deficit.

Basis: S3.

F021 — Usable stored reference volume: general endpoints

V_release_ref = V_receiver × (Z_ref × T_ref / p_ref) × [p1/(Z1 × T1) − p2/(Z2 × T2)]

Inputs and output: V_receiver: m³; p1,p2,p_ref: common absolute unit; T1,T2,T_ref: K; Z values: dimensionless. Output: V_release_ref, reference m³

Dimensional check: m³ × (K/pressure) × (pressure/K) = m³.

It accounts for defined initial and final storage states rather than assuming pressure change alone establishes released mass. Equal reference and receiver temperatures and Z values recover the familiar pressure-band expression.

Limits: Require valid endpoint states and a positive mass-inventory difference for discharge mode. Endpoint assumptions do not predict the thermal transient; the flow path must still deliver the requested rate. This is not vessel mechanical design.

Reference example: At equal 300 K temperatures and Z = 1, V = 2 m³, p1 = 9 bar, p2 = 7 bar and p_ref = 1 bar give 4 reference m³.

Basis: S3. Source equation rearranged; derivation and scope explicit.

F022 — Receiver size for a net transient deficit

V_receiver = [Q_deficit × t × p_ref/(Z_ref × T_ref)] / [p1/(Z1 × T1) − p2/(Z2 × T2)]

Inputs and output: Q_deficit: reference m³/min; t: min; absolute pressures and kelvin temperatures as F021. Output: V_receiver, geometric m³

Dimensional check: Reference volume × pressure/K divided by pressure/K = geometric volume.

Use: Use a positive, defined net shortage and defensible endpoint states. With equal T and equal Z throughout, V = Q_deficit × t × p_ref/(p1 − p2). Isothermal receiver behaviour alone is not enough to cancel a different reference temperature; retain the receiver/reference temperature factor.

Limits: Require positive denominator and duration; zero deficit returns zero for this event. The selection still needs connection losses, recharge, control response and equipment review. No hidden margin or universal litres-per-kW allowance is applied.

Reference example: 8 m³/min for 20 s with p_ref = 1.01325 bar and a 1.5 bar band, equal T/Z, gives 1.801333333 m³.

Basis: S3.

F023 — Storage hold-up time

t_hold = V_release_ref / Q_deficit

Inputs and output: V_release_ref: reference m³ from F021; Q_deficit: positive reference m³/min. Output: t_hold, min

Dimensional check: Reference volume / reference volume per minute = minutes.

Use: Use only when the net deficit is represented as constant over the interval. For varying demand or supply, integrate the cumulative deficit against stored inventory instead. Retain minimum receiver pressure and the downstream service boundary.

Limits: Zero deficit makes this depletion-time question not applicable; do not return an infinite reliability guarantee. Delivery restriction may cause process failure before the inventory estimate is exhausted. Recharge and repeat-event spacing remain separate checks.

Reference example: 4 reference m³ / 6 reference m³/min = 0.666666667 min, or 40 s.

Basis: S3.

F024 — Paired segment pressure difference

delta_p = p_upstream − p_downstream

Inputs and output: Paired pressures: same unit, consistent references, identified points and operating time. Output: delta_p, signed pressure difference

Dimensional check: Pressure − pressure = pressure.

Use: Use synchronized or demonstrably compatible measurements along the relevant path. The signed difference is an observation; it does not by itself prove fouling or pipe undersizing. An intentional regulator step needs a separate interpretation from frictional loss.

Limits: Do not subtract measurements taken in different demand states. Do not add atmosphere to the difference. Negative values should trigger topology, backflow, timing and uncertainty review, not automatic conversion to a positive loss.

Reference example: 7.4 bar gauge upstream and 6.8 bar gauge downstream give 0.6 bar difference.

Basis: S1, S9.

F025 — Defined-path pressure budget

p_source_required = p_user_required + sum(delta_p_path) + provision

Inputs and output: All terms: same pressure unit; path losses: compatible simultaneous duty; provision: justified control/design allowance. Output: p_source_required on the user pressure-reference basis

Dimensional check: Pressure + pressures + pressure = pressure.

Use: Use the verified end-use requirement and the actual connected path. Keep intentional regulation and any allowance already present in the equipment requirement identifiable.

Limits: Do not sum maxima from unrelated times or duplicate common segments. An unexplained margin cannot replace loss-location evidence. A required subtotal above equipment limits calls for redesign or review, not permission to exceed a rating.

Reference example: 6.2 bar requirement + 0.35 bar distribution + 0.20 bar treatment + 0.10 bar stated provision = 6.85 bar.

Basis: S9.

11. Thermal-performance formulas

F028 — Named cooler approach

A_temperature = T_hot_outlet − T_cooling_inlet

Inputs and output: Temperatures: compatible Celsius or kelvin readings at named exchanger boundaries. Output: A_temperature, K interval

Dimensional check: Temperature difference has interval dimension, not an absolute-temperature ratio.

Use: Use the same defined points, duty and cooling configuration for comparisons. A warmer hot outlet can occur because cooling inlet temperature rose, even when the approach stayed unchanged. Keep the raw pair, not only the derived difference.

Limits: An approach value is not a universal exchanger-efficiency metric. Unknown cooling inlet, mismatched sensor response or changed load can invalidate the diagnostic comparison. No generic permissible approach is assigned.

Reference example: 54 °C hot outlet minus 38 °C cooling inlet = 16 K approach.

Basis: S6.

F029 — Single-phase liquid heat duty

H_dot = (rho_liquid × q_liquid/3600) × c_p × (T_out − T_in)

Inputs and output: rho: kg/m³; q_liquid: m³/h; c_p: kJ/(kg·K); temperatures define a K interval. Output: H_dot, kW thermal

Dimensional check: kg/s × kJ/(kg·K) × K = kJ/s = kW.

Use: Use a defined, approximately steady single-phase liquid boundary with applicable fluid properties. The sign follows the chosen inlet/outlet convention. A coolant warming as it passes the cooler carries positive heat under this expression.

Limits: No unknown density or specific heat may default silently to water. Glycol mixtures, phase change, changing stored heat and unmeasured bypass flow require other treatment. A higher temperature rise with lower flow can still mean less heat carried away.

Reference example: 6 m³/h, rho = 1000 kg/m³, c_p = 4.18 kJ/(kg·K) and rise 6 K give 41.8 kW thermal.

Basis: S6.

F030 — Thermal energy over an interval

E_thermal = sum(H_dot_i × h_i)

Inputs and output: H_dot_i: mean transferred thermal kW; h_i: accepted duration, hours. Output: E_thermal, kWh thermal

Dimensional check: Thermal kW × h = thermal kWh.

Use: Keep transferred heat separate from purchased electricity. An exchanger can move heat originating in several loads, and only a verified useful heat-recovery arrangement can support a displaced-energy calculation. This entry intentionally differs semantically from F015 despite equal dimensions.

Limits: Do not label heat transfer as an electricity saving. A constant-duty multiplication requires evidence that the duty represents the period. Transient heat storage and changes in fluid properties may limit the underlying duty estimate.

Reference example: 41.8 kW thermal for 2 h gives 83.6 kWh thermal.

Basis: S6.

12. Measurement-uncertainty formulas

F031 — Uncertainty of a sum with correlation

u_sum = sqrt(u_a² + u_b² + 2 × r_ab × u_a × u_b)

Inputs and output: u_a,u_b: standard uncertainties in the output quantity unit; r_ab: correlation coefficient. Output: u_sum, same quantity unit

Dimensional check: Square root of summed squared-unit terms returns the quantity unit.

Use: This is the two-input, unit-sensitivity form of first-order propagation. State why independence or a correlation value is appropriate. A shared calibration or common reference can make a zero-correlation assumption unsuitable.

Limits: Require non-negative standard uncertainties and −1 <= r_ab <= 1. Do not insert expanded uncertainties without converting to compatible standard uncertainties. This propagation does not repair biased, invalid or missing measurements.

Reference example: u_a = 0.0006, u_b = 0.0008, r_ab = 0 give u_sum = 0.001 in the same unit.

Basis: S7, R2.

F032 — Uncertainty of a difference with correlation

u_difference = sqrt(u_a² + u_b² − 2 × r_ab × u_a × u_b)

Inputs and output: u_a,u_b: standard uncertainties of the two compared values; r_ab: supported correlation. Output: u_difference, same quantity unit

Dimensional check: Signed sensitivity coefficients change the covariance term; units remain squared inside the root.

Use: Use for pressure, temperature or before/after differences when the model and uncertainty inputs are applicable. Positive correlation can reduce uncertainty in a difference while increasing it in a sum; the minus sign therefore matters.

Limits: A common error does not always cancel completely. Verify the measurement model and correlated components rather than setting r = 1 because the sensors share a brand. Unknown covariance may need a sensitivity study or qualified result.

Reference example: u_a = u_b = 0.4 K and r_ab = 0.5 give u_difference = 0.4 K.

Basis: S6, R2.

F033 — Uncertainty of a power-to-flow ratio

u_SP = abs(P/Q) × sqrt[(u_P/P)² + (u_Q/Q)² − 2 × r_PQ × (u_P/P) × (u_Q/Q)]

Inputs and output: P,Q: positive matched values; u_P,u_Q: standard uncertainties; r_PQ: supported correlation. Output: u_SP, same unit as P/Q

Dimensional check: Relative uncertainty terms are dimensionless; multiplied by the output quantity.

Use: Use this relative form where P and Q are positive and first-order propagation is adequate. Near zero flow or with strongly nonlinear uncertainty, a more suitable method may be needed. Keep pressure, timing, calibration and boundary contributions in the underlying measurement model.

Limits: Zero P makes this relative expression singular even when another derivative form exists. Zero Q makes the performance ratio invalid. Correlation and measurement uncertainty are not numerical rounding tolerance or supplier allowance.

Reference example: P = 100 kW, Q = 20 m³/min, u_P = 1 kW, u_Q = 0.2 m³/min and r = 0 give u_SP = 0.070710678.

Basis: S8, R2.

F034 — Expanded uncertainty

U = k_coverage × u_combined

Inputs and output: u_combined: standard uncertainty; k_coverage: declared positive coverage factor. Output: U, same unit as measurand

Dimensional check: Dimensionless coverage factor × quantity uncertainty = quantity uncertainty.

Use: Use the factor and reporting interpretation agreed for the uncertainty assessment. Retain u, k and the resulting U as separate fields. The supplied examples often use k = 2 as an illustrative choice; that choice alone does not establish a particular confidence level.

Limits: Do not confuse k_coverage with the gas heat-capacity ratio or a pipe-loss coefficient. Missing uncertainty is not zero. Expanded uncertainty is not a universal rule for accepting or rejecting a measurement near a limit.

Reference example: u = 0.001 mg/m³ and k = 2 give U = 0.002 mg/m³.

Basis: S7, S8.

The covariance signs in F031–F033 follow the sensitivities of the selected measurement model. Their first-order form is not a substitute for reviewing missing contributors, nonlinear behaviour or unsuitable input distributions. The register records the arithmetic separately from the measurement justification. [R2]

13. Sampling, coverage and event formulas

F035 — Accepted sample volume

V_sample = q_sample × t_active / 1000

Inputs and output: q_sample: L/min on the accepted basis; t_active: valid collection minutes. Output: V_sample, m³ on that basis

Dimensional check: L/min × min × 0.001 m³/L = m³.

Use: Use only the active, accepted collection interval. Separate flushing, stabilization, pauses and rejected periods. Where an instrument totaliser already measures accepted sample volume, do not deduct an interruption again.

Limits: A sample flow may be actual or reference volume; preserve that basis. A flow conversion does not correct sampling losses, unsuitable tubing or collection-medium breakthrough. A missing active duration prevents a quantified sample-volume result.

Reference example: 2 L/min for 20 accepted min gives 0.040 m³.

Basis: S7.

F036 — Particle number concentration

C_number = N_accepted / V_sample

Inputs and output: N_accepted: count for the named particle-size band; V_sample: accepted m³. Output: particles/m³ for that band and basis

Dimensional check: Dimensionless event count / volume = number concentration.

Use: Use the method-defined count and volume. Retain cumulative versus disjoint band definitions, any validated instrument correction and the sample identity. The calculation supplies concentration, not particle composition or a universal cleanliness classification.

Limits: Require V_sample > 0. Do not sum overlapping cumulative channels or treat an invalid blank as automatic subtraction. Zero recovered counts is a valid observation but not proof that the whole system contains no particles.

Reference example: 12 accepted counts / 0.040 m³ = 300 particles/m³.

Basis: S7.

F037 — Compatible oil-fraction total

C_total = sum(C_fraction_j)

Inputs and output: Each fraction: quantified mg/m³ with compatible point, state, method basis and no overlapping fraction. Output: C_total, mg/m³ for the complete specified fraction set

Dimensional check: Concentrations with identical basis sum to concentration.

Use: Use only after establishing which fractions the requirement includes and whether each result represents a distinct part. Keep completeness and result validity as separate fields.

Limits: A missing fraction or a less-than qualifier is not zero. The numerical runner accepts already-qualified numeric inputs; a production adapter must block an incomplete or censored total before calling this arithmetic. Summation cannot authorize utility or product release.

Reference example: Compatible complete quantified fractions 0.004 and 0.003 mg/m³ total 0.007 mg/m³.

Basis: S7.

F038 — Valid observation coverage

coverage_percent = 100 × t_valid / t_scheduled

Inputs and output: Durations: one unit and same assessed window, with non-overlapping valid support. Output: coverage_percent, %

Dimensional check: Time/time × 100 = percent.

Use: Use to communicate how much of the reporting window has usable evidence for a specified calculation. Coverage is purpose-specific: a pressure series can have better coverage than a paired pressure-and-flow comparison.

Limits: Require t_scheduled > 0 and 0 <= t_valid <= t_scheduled. A high percentage cannot substitute for the missing critical event. Coverage is not a confidence level, quality score or automatic acceptance measure.

Reference example: 45 valid min / 60 scheduled min × 100 = 75%.

Basis: S6, S7.

F039 — Observed threshold duration

t_exceed = sum[delta_t_i × I(x_i > limit)]

Inputs and output: x_i: classified interval values; delta_t_i: supported durations; limit and quantity basis explicit. Output: t_exceed, chosen time unit

Dimensional check: Time × dimensionless indicator summed over accepted intervals = time.

Use: This register uses strict greater-than comparison for its example. Declare another inequality where required. For interval averages, the output is duration assigned by that classification convention, not a reconstruction of every instantaneous threshold crossing.

Limits: No bridging unknown gaps. Equality is not exceedance under the strict rule. Missing periods make whole-window duration uncertain; an observed 5 minutes and a possible additional 5 minutes must not be reported as an exact 10.

Reference example: 80 °C for 10 min, 88 °C for 5 min and 84 °C for 20 min with strict limit 85 °C give 5 classified exceedance min.

Basis: S6, S7.

F040 — Duration of an interval union

t_union = sum(end_k − start_k) after merging overlapping intervals

Inputs and output: Intervals: start/end times on one timeline for the same entity or physical assessment scope. Output: t_union, chosen time unit

Dimensional check: Non-overlapping elapsed-time differences sum without double counting.

Use: Use to combine overlapping investigation, downtime or exposure-assessment windows for the same process. Preserve the original intervals and the merged result. Different affected processes must not be merged merely because their timestamps overlap.

Limits: Reject reversed intervals and unresolved timezones. Touching endpoints add no extra elapsed duration. A merged investigation window defines the time assessed; it does not prove continuous contamination or authorize a production disposition.

Reference example: [10,50) and [30,60) minutes merge to [10,60), giving 50 min rather than 70 min.

Basis: S7.

14. Worked example A: from reported air to pipe velocity

Fictional case: a verified dry reference flow is 12.0 m³/min at 1.0 bar absolute and 293.15 K. The corresponding pipe state is 7.0 bar gauge with a stated local atmosphere of 1.0 bar, and pipe gas temperature is 35 °C. Assume equal compressibility factors and unchanged dry-gas mass. The actual internal diameter is 50 mm. None of these inputs is a proposed universal design condition.

F001 gives 8.0 bar absolute in the pipe. F003 gives 308.15 K. F008 gives Q_pipe = 12 × (1/8) × (308.15/293.15) = 1.576752516 m³/min. Dividing by 60 gives 0.026279209 m³/s. F009 gives A = pi × 0.050²/4 = 0.001963495408 m². F011 therefore gives a mean velocity of approximately 13.383891 m/s.

Using 12 m³/min directly in the velocity calculation would give about 101.859164 m/s because the reference-to-pipe state conversion was omitted. That discrepancy is not a small design margin; it is a different physical quantity. The corrected velocity still does not approve the pipe diameter. Pressure loss, fittings, route, operating peaks and end-use pressure require their own assessment. The dependency chain is F001 + F003 → F008 → F009 + F011.

15. Worked example B: receiver gas temperature changes the result

Fictional event: net deficit is 8 reference m³/min for 20 seconds. Reference pressure is 1.01325 bar absolute and reference temperature is 293.15 K. The receiver is assumed isothermal at 313.15 K during the event, with an initial pressure of 9.01325 bar absolute and final pressure of 7.51325 bar absolute. All Z values are set to one explicitly.

The simpler equal-temperature calculation gives 8 × (20/60) × 1.01325/1.5 = 1.801333333 m³. Because receiver and reference temperatures differ, F022 gives 1.801333333 × (313.15/293.15) = 1.924228325 m³. The additional factor is 1.068224459, so the geometric volume rises by about 6.822446%. This is a physical state correction, not a discretionary safety margin.

For an existing 2.0 m³ receiver at the same assumed endpoints, F021 gives about 2.771673853 reference m³ usable release. F023 then gives about 20.787554 s of theoretical support at the stated constant deficit. The calculation does not establish that the connection can deliver the instantaneous rate or that the receiver returns to its starting condition before the next event.

No source file is overwritten. The generalized formula is preserved and the narrower reading is documented as a new applicability clarification.

16. Worked example C: integrate energy and air before dividing

Fictional operating record: a package delivers 20 m³/min at 120 kW for 10 minutes, then 18 m³/min at 112 kW for 20 minutes. Airflow shares one reference basis; power has the same package boundary. Assume these interval means adequately represent the full accepted 30-minute period.

F012 gives air volume 20 × 10 + 18 × 20 = 560 reference m³. F015 gives energy 120 × (10/60) + 112 × (20/60) = 57.333333333 kWh. F016 gives 57.333333333/560 = 0.102380952 kWh/m³. The time-weighted means are 18.666666667 m³/min and 114.666666667 kW; their F014 ratio is 6.142857143 kW per (m³/min), consistent with the energy result after division by 60.

A plain average of the two instantaneous specific-power values gives (6 + 6.222222222)/2 = 6.111111111, which does not match the defined period. Neither the shorter interval nor the smaller delivered volume should receive an arbitrary equal influence. Retain the raw interval definitions and use the integrated boundary. A missing five-minute interval would require qualified coverage, not insertion of the more favourable operating point.

17. Worked example D: valid arithmetic does not establish complete oil evidence

Fictional incident review: two required, non-overlapping oil fractions are quantified at 0.004 and 0.003 mg/m³ for compatible points and conditions. Their independent standard uncertainties are 0.0006 and 0.0008 mg/m³. The illustrative upper requirement is 0.010 mg/m³, with an explicitly selected interval rule.

F037 gives 0.007 mg/m³. F031 with r = 0 gives u = 0.001 mg/m³, and F034 with k = 2 gives U = 0.002 mg/m³. The comparison interval is 0.005–0.009 mg/m³. Under the selected upper-bound rule x + U <= L, this numerical comparison supports acceptance for that evidence set. It does not release production or certify untested points.

Now replace the second quantified fraction with a report of less than 0.008 mg/m³. A numeric sum using zero would be unsupported. Preserve the qualifier and stop the quantified total until the applicable method and decision treatment are resolved. The valid first fraction cannot fill the missing knowledge. Similarly, a passing repeat sample cannot erase the original incident or decide what happened during an unobserved period.

18. Build formula chains without hidden assumptions

A formula chain should pass a quantity record, not only a floating-point number. The record should contain value, unit, quantity kind, pressure or reference state, measurement boundary, timestamp or interval, quality status, source and correction history. F008 should not receive a bare number labelled “flow” when it needs a stated gas condition. F014 should not receive a motor rating when its numerator is package electrical input.

Explicit dependencies help both speed and quality. Once a reference-state conversion is reviewed, downstream calculations can reuse that version rather than reconstruct it independently. When an upstream unit mapping or measurement correction changes, the affected outputs should be marked stale until recalculated. Preserve earlier issued results instead of silently replacing their inputs beneath an unchanged conclusion.

Do not make a circular chain. A pipe flow estimated from compressor capacity cannot independently prove that capacity. A consumption estimate made from an assumed operating profile cannot validate the same profile. Identify the independent observation or supplier information that anchors the calculation before using its outputs as evidence.

19. Separate model limits, evidence gaps and numerical defects

A missing unit is different from an out-of-range value. An unmeasured physical state is different from a failed software test. Keep separate reason codes so the next person knows whether to repair the data import, obtain a measurement, revise the model or investigate the equipment. An all-purpose error badge wastes time by concealing the action needed.

20. Formula families intentionally not universalized

This edition does not invent one empirical pressure-drop formula for every compressed-air pipe, fitting and operating regime. It records paired pressure differences and pressure-budget arithmetic, while leaving full network friction, compressibility and local-loss modelling to an applicable selected method. A generic conversion coefficient without its unit convention and calibration range is not promoted into the register.

It also does not supply universal dryer correction factors, humidity-to-dew-point correlations, sonic-flow coefficients, compressor site derating, oil-purity limits, receiver control-cycle limits or useful heat-recovery percentages. These require additional method, gas-property, model or supplier evidence. Choked flow and transient cylinder filling are different questions from average consumption. Isentropic or polytropic compressor models do not become package-rating procedures merely by inclusion in a formula list.

Future additions should be appended as new identified records with sources and scope, not inserted by changing the meaning of an existing ID. Finance, carbon accounting and specialist borewell interpretations remain linked to their own pages where their boundaries are defined.

21. Editable worksheet A: formula applicability review

Duplicate this table for each material calculation or a clearly defined chain. The Word table is manually editable. Its empty fields are deliberate input spaces, not missing website copy. A calculation with outstanding applicability questions may remain useful as a labelled planning scenario without becoming a verified selection.

22. Editable worksheet B: input and transformation ledger

Preserve the original even when the conversion is straightforward. This makes it possible to detect a doubled pressure-reference change, the wrong time divisor, or a temperature-value conversion applied to a difference. The ledger should not be used to sanitize evidence by hiding an inconvenient raw observation. Estimates, corrections and measurements remain distinguishable.

23. Editable worksheet C: reference test and change record

A passing reference script demonstrates the selected local arithmetic, not that the production calculator imported the right data. Keep the target identity explicit. An updated formula may require a new reference case, a revised parser and a repeat of affected display/export checks. Unrelated pages and source files should not be regenerated as part of that change.

24. Frequently asked questions

Is this a list of universal compressor design formulas?

No. It is a scoped register of 40 relationships with their meanings and restrictions. Several are accounting identities or declared screening models. The register explains where supplier data, a different physical model or further measurement remains necessary.

Why can two correct calculators return different airflow values?

They may use different reference temperatures, absolute pressures, humidity conventions or delivery boundaries. Compare the quantity definitions before comparing the digits. A unit conversion alone cannot remove a difference in physical meaning.

Can I use gauge pressure in a ratio?

Use absolute pressure where the gas-state model requires it. A difference between compatible gauge readings may be valid because the shared atmosphere cancels; a compression ratio needs the individual absolute values.

Does isothermal storage mean the simple receiver formula always applies?

No. Isothermal means the receiver temperature is constant in the selected model. The reference-flow temperature may still differ. Use the general endpoint formula or retain the appropriate temperature and compressibility factors.

Why not use FAD directly for pipe velocity?

FAD describes delivered air on its declared inlet/reference basis. Velocity requires actual volume moving through the passage at its local state. Convert the gas state before dividing by area.

What happens when unloaded flow is zero?

Retain the valid unloaded electrical input and operating duration. Do not calculate a finite P/Q at zero flow. The energy consumed can still be included in the correct period and boundary.

Are measurement uncertainty and software tolerance the same?

No. Measurement uncertainty describes incomplete knowledge of the measurand under its model. Software comparison tolerance accommodates defined numerical representation or rounding differences. A supplier allowance and a design provision are separate again. [S8, R2]

Can a blank laboratory fraction be entered as zero?

No. Missing, qualified and quantified results have different meanings. The full requirement, fraction completeness and method must be resolved before a numerical total supports an acceptance conclusion.

Does passing every supplied numeric case certify the website?

No. The local reference cases test selected arithmetic. Production field mapping, semantic validation, browser behaviour, imports, exports and project applicability remain distinct tasks. The execution target and untested scope must remain visible.

Can additional equations be added later without restarting the register?

Yes. Add a stable new ID, sources, input definitions, restrictions, dimension explanation and reference cases. Version an existing record when its meaning or method changes; preserve the old issue and identify dependent tools.

25. Use the register in an actual engineering enquiry

Start with the question, then provide the source figures in their original form. Useful inputs include the compressor or instrument identity, duty point, relevant pressure and temperature, timeframe, diagram boundary and the equation or tool output being challenged. A screenshot of a final number is less useful than its source data and assumptions.

Primary action: Review My Compressed-Air Calculation. Secondary action: Open the Formula Applicability Register. The enquiry should request the formula ID or topic, original input units, expected result and observed result. Customer diagrams, supplier quotations and measurement records should remain private. A review request does not imply manufacturer authorization, laboratory accreditation or a guaranteed response time.