Reference
Formula Library
Searchable engineering formulas with variables, units, worked examples, and assumptions.
All
Instrumentation
Fluid Mechanics
Control Systems
Power Plant
Process
Mechanical
Civil
Control
4-20 mA Signal Scaling
PV = LRV + [(I − 4)/(20 − 4)] × (URV − LRV)
Converts a linear 4-20mA transmitter signal to an engineering value given the lower and upper range values.
VariablesPV = process value, I = current signal (mA), LRV/URV = lower/upper range values
UnitsmA, engineering units
ExampleRange 0-100 bar, I = 14.5 mA → PV = 0 + [(14.5-4)/16]×100 = 65.625 bar
ApplicationAny linear 4-20mA transmitter or I/O card scaling
AssumptionsLinear transmitter response, no damping/offset error
StandardISA/IEC 60381
Orifice Plate Flow
Q = Cd · E · (π/4)·d² · √(2·ΔP/ρ), E = 1/√(1−β⁴)
Volumetric flow rate through a square-edged concentric orifice plate.
VariablesQ = flow, Cd = discharge coefficient, E = velocity of approach factor, d = bore, ΔP = differential pressure, ρ = density, β = d/D
Unitsm3/s, Pa, kg/m3, m
Exampled=0.05m, D=0.1m, ΔP=5000Pa, ρ=1000kg/m3, Cd=0.6 → Q ≈ 0.0112 m3/s
ApplicationCustody metering, process flow measurement
AssumptionsFully developed turbulent flow, square-edged plate, incompressible fluid
StandardISO 5167 (simplified constant-Cd form)
Liquid Valve Sizing (Cv)
Cv = Q(gpm) / √(ΔP(psi)/SG)
Required flow coefficient for a control valve in non-flashing, non-cavitating liquid service.
VariablesCv = valve flow coefficient, Q = flow rate, ΔP = pressure drop, SG = specific gravity
Unitsgpm, psi
ExampleQ=100gpm, ΔP=25psi, SG=1.0 → Cv = 100/√25 = 20
ApplicationControl valve preliminary sizing
AssumptionsNo flashing/cavitation, negligible piping geometry factor (Fp≈1)
StandardISA-75.01 (simplified)
Plant Heat Rate
Heat Rate (kcal/kWh) = 860.42 / Overall Efficiency
Relates thermal efficiency to the fuel energy required per unit of electrical output.
Variables860.42 = kcal equivalent of 1 kWh
Unitskcal/kWh, fraction
ExampleEfficiency = 38% → Heat rate = 860.42/0.38 ≈ 2264 kcal/kWh
ApplicationThermal power plant performance and fuel cost estimation
AssumptionsSteady-state operation, efficiency includes boiler×turbine×generator
StandardIEEE/ASME PTC-6 concepts (indicative use)
RTD Callendar-Van Dusen
R(t) = R0·(1 + A·t + B·t²) [t ≥ 0°C]
Standard resistance-temperature relationship for platinum RTDs.
VariablesR0 = resistance at 0°C, A = 3.9083e-3, B = -5.775e-7
UnitsΩ, °C
ExamplePt100 at 100°C → R ≈ 138.5 Ω
ApplicationTemperature measurement with Pt100/Pt1000 elements
AssumptionsIEC 60751 class curve, negligible self-heating and lead-wire error
StandardIEC 60751
DP-Flow Square Root Relationship
Flow% = √(ΔP%)
Differential-pressure flow transmitters produce a signal proportional to the square of flow; extraction linearizes it.
VariablesFlow% = percent of max flow, ΔP% = percent of max differential pressure
Units% span
ExampleΔP at 25% of span → Flow = √0.25 = 50% of max flow
ApplicationOrifice/venturi/flow-nozzle DP flow measurement
AssumptionsBeta ratio and fluid density constant across the range
StandardISO 5167 / general DP flow theory
Hydrostatic Pressure / Level
P = ρ · g · h
Pressure exerted by a column of liquid, used for level measurement via pressure/DP transmitters.
VariablesP = pressure, ρ = fluid density, g = gravitational acceleration, h = liquid column height
UnitsPa, kg/m3, m/s2, m
Exampleρ=1000kg/m3, h=2m → P = 1000×9.807×2 = 19,614 Pa
ApplicationOpen/closed tank level measurement
AssumptionsConstant density, static (no acceleration) column
StandardFundamental fluid statics
PID Controller (parallel form)
Output = Bias + Kp·e + Ki·∫e dt + Kd·de/dt
Standard parallel-form proportional-integral-derivative control law.
Variablese = SP−PV error, Kp/Ki/Kd = gains, Bias = output at zero error
Unitscontroller engineering units, seconds
ExampleSP=50, PV=45, Kp=2 → P-term = 2×5 = 10
ApplicationProcess control loop tuning and output prediction
AssumptionsLinear control law, no output clamping/anti-windup modeled
StandardISA-5.1 / IEC 61131-3 concepts
Reynolds Number (pipe flow)
Re = 4·ṁ / (π·μ·D)
Dimensionless number indicating laminar vs turbulent flow regime.
Variablesṁ = mass flow rate, μ = dynamic viscosity, D = pipe internal diameter
Unitskg/s, Pa·s, m
Exampleṁ=5kg/s, μ=0.001Pa·s, D=0.1m → Re ≈ 63,700 (turbulent)
ApplicationFlow meter selection, discharge coefficient validity checks
AssumptionsFully developed, Newtonian fluid
StandardFundamental fluid mechanics
Pump Head & Hydraulic Power
P(kW) = ρ·g·Q·H / (1000·η)
Shaft power required to deliver a given flow against a given head at a stated pump efficiency.
Variablesρ = density, Q = flow (m3/s), H = head (m), η = pump efficiency (fraction)
Unitskg/m3, m3/s, m, kW
Exampleρ=1000, Q=0.05m3/s, H=30m, η=0.75 → P ≈ 19.6 kW
ApplicationPump selection and motor sizing
AssumptionsConstant density, steady flow, efficiency includes hydraulic+mechanical losses
StandardFundamental hydraulics
Simply Supported Beam Deflection (Point Load)
δ_max = PL³ / (48EI)
Maximum deflection of a simply supported beam under a center point load, classical Euler-Bernoulli beam theory.
Variablesδ_max = max deflection, P = point load, L = span, E = modulus of elasticity, I = moment of inertia
UnitsN, m, Pa, m⁴ → result in m
ExampleP=10000 N, L=4 m, E=200 GPa, I=6.6667×10⁻⁵ m⁴ → δ = (10000×4³)/(48×200e9×6.6667e-5) = 0.001 m = 1 mm
ApplicationSimply supported beam design and deflection serviceability check
AssumptionsLinear elastic material, small deflections, uniform cross-section
StandardClassical mechanics of materials (Euler-Bernoulli beam theory)
Euler Column Buckling
P_cr = π²EI / (KL)²
Elastic critical buckling load for a slender column, for any standard end-support condition via the effective length factor K.
VariablesP_cr = critical load, E = modulus of elasticity, I = moment of inertia, K = effective length factor, L = unsupported length
UnitsPa, m⁴, m → result in N
ExamplePinned-pinned (K=1), L=4 m, E=200 GPa, I=6.6667×10⁻⁵ m⁴ → P_cr = π²×200e9×6.6667e-5 / 4² ≈ 8,224,700 N
ApplicationPreliminary column buckling check
AssumptionsPerfectly elastic, initially straight, centrally loaded column — no safety factor included
StandardClassical mechanics of materials (Euler 1757)
Concrete Nominal Mix Quantities
Material volume = Dry volume × (ratio part / sum of ratio parts)
Cement, sand and aggregate quantities for an IS 456:2000 nominal concrete mix, using the standard 1.54 dry volume factor.
VariablesDry volume = wet volume × 1.54 (typical), ratio = fixed cement:sand:aggregate proportion by grade
Unitsm³
ExampleM20 (1:1.5:3), 1 m³ wet volume → dry volume 1.54 m³ → cement = 1.54×(1/5.5) = 0.28 m³ → ×1440 kg/m³ / 50 kg/bag ≈ 8.06 bags
ApplicationMaterial quantity estimation for ordinary grade concrete (M5–M20)
AssumptionsIS 456:2000 nominal mix method; M25 and above require a design mix, not covered by this method
StandardIS 456:2000, Table 9 (nominal mixes)
Rebar (Reinforcement Bar) Weight
W = ρ × (πd²/4) × L (site rule of thumb: W/L ≈ d²/162 kg/m)
Weight of a rebar/reinforcement bar from its diameter and length, using steel density; the familiar d²/162 site approximation gives the same result within about 0.1%.
VariablesW = weight, ρ = steel density (7850 kg/m³), d = bar diameter (mm), L = length (m)
Unitsmm, m → result in kg
Example16 mm bar, 1 m long → W = 7850 × (π×0.016²/4) × 1 = 1.578 kg (rule of thumb 16²/162 = 1.580 kg)
ApplicationReinforcement material takeoff and procurement
AssumptionsStandard steel density 7850 kg/m³
StandardStandard construction industry practice
Rankine Active Earth Pressure
Kₐ = tan²(45° − φ/2) σₐ = Kₐσv − 2c√Kₐ
Active earth pressure coefficient and pressure on a retaining wall from Rankine theory, for a cohesionless or cohesive soil.
VariablesKₐ = active pressure coefficient, φ = soil friction angle, σv = vertical (overburden) stress, c = cohesion
Unitsdegrees, kPa → result in kPa
Exampleφ=30°, depth=3 m, γ=18 kN/m³, c=0 → Kₐ = tan²(30°) = 0.333 → σₐ = 0.333×(18×3) = 18 kPa
ApplicationRetaining wall design, preliminary lateral earth pressure estimate
AssumptionsVertical wall face, horizontal backfill, no wall friction (real design should also check Coulomb theory)
StandardRankine earth pressure theory (1857)
Manning's Equation (Open Channel Flow)
V = (1/n) × R^(2/3) × S^(1/2)
Mean flow velocity in an open channel under uniform flow conditions, from the channel's roughness, hydraulic radius, and slope.
VariablesV = velocity, n = Manning's roughness coefficient, R = hydraulic radius (A/P), S = longitudinal channel slope
UnitsSI units (k=1); m/s result
ExampleRectangular channel, b=2 m, depth=1 m, n=0.013, S=0.001 → A=2 m², P=4 m, R=0.5 m → V=1.53 m/s, Q=3.06 m³/s
ApplicationOpen channel and storm drain flow capacity calculations
AssumptionsUniform (steady, normal) flow, constant cross-section and slope
StandardManning's equation (Robert Manning, 1889)
Effective Stress (Terzaghi)
σ' = σ − u
Effective stress carried by the soil skeleton, equal to total stress minus pore water pressure — the fundamental principle underlying settlement, bearing capacity, and slope stability analysis.
Variablesσ' = effective stress, σ = total stress (from soil self-weight), u = pore water pressure
UnitskPa
ExampleDepth 5 m, water table at 2 m, γ(above)=17, γ(below)=19 kN/m³ → σ=17×2+19×3=91 kPa, u=9.81×3=29.43 kPa → σ'=61.57 kPa
ApplicationFoundation settlement and bearing capacity analysis
AssumptionsHydrostatic pore pressure (no seepage flow)
StandardTerzaghi's principle of effective stress (1925)
Pump Specific Speed
Ns = N√Q / H^0.75
Dimensional specific speed used to indicate centrifugal pump impeller geometry family (radial, mixed, or axial flow) at a given duty point.
VariablesNs = specific speed, N = pump speed (rpm), Q = flow at best efficiency point (US gpm), H = head per stage (ft)
UnitsUS customary (rpm, gpm, ft)
ExampleN=1770 rpm, Q=500 gpm, H=150 ft → Ns = 1770√500/150^0.75 ≈ 923 (radial flow range)
ApplicationEarly-stage pump type / impeller family selection
AssumptionsUS customary unit form; numerically different in other unit systems for the same physical pump
StandardStandard turbomachinery similarity theory
HVAC Sensible Heat Load
Q(BTU/hr) = 1.08 × CFM × ΔT(°F) Q(kW) = 1.2 × (L/s) × ΔT(°C)
Standard ASHRAE air-side sensible heat formula, relating airflow and temperature change to cooling or heating load.
VariablesQ = sensible heat load, CFM/L⁄s = airflow, ΔT = temperature difference across the coil or space
UnitsImperial (CFM, °F) or metric (L/s, °C)
Example2000 CFM, 20°F ΔT → Q = 1.08×2000×20 = 43,200 BTU/hr = 3.6 tons
ApplicationHVAC sensible cooling/heating load estimation
AssumptionsSensible heat only — excludes latent (moisture) load, solar gain, occupancy and infiltration
StandardASHRAE Fundamentals
Bolt Torque-Tension Relationship
T = K × D × F
Standard fastener torque-tension relationship relating tightening torque to target clamp force via the nut factor K.
VariablesT = tightening torque, K = nut factor (lubrication/thread condition dependent, typically 0.10–0.20), D = bolt diameter, F = target clamp force
UnitsN, m → result in N·m
ExampleM12 bolt (D=0.012 m), K=0.2, F=20,000 N → T = 0.2×0.012×20000 = 48.0 N·m
ApplicationBolted joint tightening specification
AssumptionsK is not universal — varies with lubrication and plating; use the bolt's own datasheet value where available
StandardShigley's Mechanical Engineering Design; standard fastener engineering practice
Refrigeration Ton Conversion
1 ton = 12,000 BTU/hr = 3.5168 kW
Standard definition of a ton of refrigeration, used throughout HVAC and refrigeration capacity ratings.
VariablesDefined from the latent heat to freeze one short ton of 0°C water to ice in 24 hours
Unitstons, kW, BTU/hr
Example5 tons × 3.5168 = 17.58 kW
ApplicationConverting between chiller/AC capacity units
AssumptionsExact definitional constant, not an approximation
StandardStandard refrigeration industry definition
Duct Sizing (Velocity Method)
Area = Flow ÷ Velocity
Required duct cross-sectional area and equivalent round-duct diameter from airflow and a chosen design velocity.
VariablesArea = required duct cross-section, Flow = airflow (CFM or L/s), Velocity = target design velocity
UnitsImperial (CFM, fpm) or metric (L/s, m/s)
Example2000 CFM at 1000 fpm → Area = 2 ft² → round duct diameter ≈ 19.1 in
ApplicationSizing a single HVAC duct section for a target velocity
AssumptionsTarget velocity is a design choice (noise/pressure-drop trade-off), not a fixed physical constant
StandardASHRAE Fundamentals duct design guidance
Gear Ratio (Speed and Torque)
N₁/N₂ = D₂/D₁ = T₂/T₁
Ideal kinematic relationship between gear tooth counts, rotational speed, and torque across a gear mesh.
VariablesN = rotational speed, D = pitch diameter (or tooth count), T = torque; subscripts 1=driving gear, 2=driven gear
Unitsrpm, N·m
Example1800 rpm input, 20 teeth driving 60 teeth → ratio 3:1 → output speed 600 rpm
ApplicationGearbox/gear train speed and torque calculation
AssumptionsIdeal kinematic relationship — real torque output is reduced by mesh efficiency losses
StandardStandard mechanical power transmission theory
Belt and Pulley Speed Ratio
N₁ × D₁ = N₂ × D₂
Exact kinematic relationship between belt-driven pulley diameters and rotational speeds, assuming no belt slip.
VariablesN = rotational speed, D = pulley diameter; subscripts 1=driving pulley, 2=driven pulley
Unitsrpm, mm
Example1450 rpm motor with 100 mm pulley driving a 250 mm pulley → output speed = 1450×100/250 = 580 rpm
ApplicationBelt drive speed reduction/increase calculation
AssumptionsNo belt slip — real belts under heavy load can slip slightly, reducing actual output speed below this ideal value
StandardStandard mechanical power transmission theory
SAMA Logic Blocks (Summer, Select, Limit, AND/OR)
Summer: out = Σ(gain_i × input_i) + bias High/Low Select: out = max/min(inputs)
Standard SAMA (Scientific Apparatus Makers Association) function blocks used throughout combustion and boiler control interlock/permissive logic diagrams.
VariablesSummer: weighted sum plus bias; High/Low Select: outputs the highest/lowest input; Limiters clamp output to a bound; AND/OR/NOT are standard boolean logic (>=0.5 treated as TRUE)
UnitsEngineering units of the signal being summed/selected/limited; dimensionless for logic blocks
ExampleHigh Select of fuel demand (70%) and air feedforward (65%) → output = max(70,65) = 70%, the standard cross-limiting air floor
ApplicationBuilding and simulating custom interlock/permissive control logic from standard symbols
AssumptionsStatic/algebraic blocks only — no dynamic (lag, integrator, timer) blocks, which need time-stepped simulation instead
StandardSAMA (Scientific Apparatus Makers Association) PMC 22.1 standard symbols
Cable Overall Diameter & Gland Size
OD = [(d_cond + 2t_ins) × k_layup + 2t_inner + 2t_armour] + 2t_outer, then match OD to a standard gland's clamping range
Builds up a multicore cable's overall diameter layer by layer -- conductor, insulation, the geometric lay-up factor for the core/pair count, inner sheath, optional armour, and outer sheath -- then selects the standard cable gland whose clamping range covers that OD.
Variablesd_cond = conductor diameter, t_ins = insulation thickness, k_layup = lay-up (cabling) factor for the element count, t_inner/t_outer = sheath thickness, t_armour = armour wire diameter
Unitsmm
Example4 pair × 1.5 sqmm instrumentation cable, unarmoured → OD ≈ 14.1 mm → M25 gland (10-17 mm range)
ApplicationSelecting cable glands for junction boxes, marshalling cabinets, and field instrument enclosures
AssumptionsStandard IEC 60228 stranding allowance, IS 5831/IS 1554-based insulation thickness, geometric circle-packing lay-up factors (exact for 1/2/3/4/7 cores, interpolated for other standard counts). An estimate, not a specific manufacturer's datasheet -- actual cable OD varies between manufacturers even at the same nominal size.
StandardIEC 60228, IS 1554, IS 5831, IEC 62444
Boiler Efficiency (Indirect / Heat Loss Method)
η = 100 − (L1+L2+L3+L4+L5+L6+L7+L8), L1 = m×Cp×(Tf−Ta)/GCV×100, L2 = 9H₂×[584+Cp(Tf−Ta)]/GCV×100
Computes boiler efficiency from individually calculated heat losses (dry flue gas, H2 in fuel, moisture in fuel/air, CO, surface radiation, unburnt carbon) rather than an assumed efficiency percentage. The standard method used for real boiler performance testing.
Variablesm = dry flue gas mass (kg/kg fuel), Cp = specific heat, Tf/Ta = flue gas/ambient temperature, H2 = hydrogen % in fuel, GCV = gross calorific value
Unitskcal/kg, %, °C
ExampleCoal (C=55%, H2=3.5%, moisture=8%), GCV=4200 kcal/kg, flue gas O2=5%, Tf=160°C, Ta=30°C → efficiency ≈ 85%
ApplicationReal boiler performance testing and energy audits (ASME PTC 4.1 / BEE India standard procedure) — available as Mode 5 of the Thermal Plant Estimator
AssumptionsL1-L5 computed from fuel ultimate analysis and flue gas data. L6 (surface loss) and L7/L8 (unburnt carbon in ash) require direct measurement and are user-supplied, not derived. Result is GCV-basis — differs from NCV-basis figures often quoted for high-hydrogen fuels like natural gas.
StandardASME PTC 4.1, BEE India standard procedure, BS 845 / DIN 1942 equivalent