Formula solvers starting with F — 66 solvers

F-Value (Equivalent Time at Reference Temperature)

F = t \times 10^{\,(T - T_{ref})/z}

Water TreatmentThe F-value: time spent at one temperature expressed as the equivalent time at a reference temperature. F₀ is the special case with the reference at 121.1 °C and z at 10 C°, the convention for spore-forming bacteria in thermal food processing. It is a bookkeeping conversion, and it lets the come-up, hold and cool-down of a whole cycle be added into one number.

Fabric Cover Factor — Warp and Weft Combined

K = K_1 + K_2 - \frac{K_1 K_2}{28}

Textiles & FabricThe two thread systems of a woven cloth do not cover independently: where a warp thread crosses a weft thread, the same square millimetre is covered twice and must only be counted once. Subtracting the overlap gives a single figure for how much of the cloth is thread and how much is hole — the number that decides whether a fabric is a sheer, a shirting or a sailcloth.

Fabric Weight from Construction — GSM

G = n_1 \rho_1 (1 + c_1) + n_2 \rho_2 (1 + c_2)

Textiles & FabricWhat a square metre of woven cloth weighs, built from the specification rather than a scale: for each thread system, threads per unit length times mass per unit length, raised by the crimp because the yarn in the cloth is longer than the cloth it sits in. Threads per metre multiplied by kilograms per metre is kilograms per square metre — the arithmetic is dimensionally exact, with no conversion factor to lose.

Factor of Safety

FS = \frac{\sigma_{u}}{\sigma_{allow}}

Strength of MaterialsMechanicsPhysicsFactor of safety as ultimate or yield strength divided by the allowable working stress, the engineer's declared margin of ignorance.

Factor of Safety Against Sliding

FS = \frac{W\tan\delta}{P_h}

Soil MechanicsMechanicsFactor of safety of a retaining structure against base sliding, comparing frictional resistance under its weight with the driving horizontal thrust.

Factorial

n! = n \times (n-1) \times \cdots \times 2 \times 1

ProbabilityAlgebraThe product of every whole number from n down to 1, counting the ways that n distinct objects can be arranged in order.

Fan Affinity Law — Airflow vs Speed

\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsPhysicsFan airflow in CFM changes in direct proportion to wheel speed, the first law used when re-sheaving a belt-driven air handler.

Fan Affinity Law — Power vs Speed

\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsPhysicsFan brake power varies with the cube of wheel speed — the law behind variable-air-volume energy savings and behind burnt-out re-sheaved motors.

Fan Affinity Law — Static Pressure vs Speed

\frac{SP_2}{SP_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsPhysicsFan static pressure rises with the square of wheel speed, the reason a modest re-sheave can overpressurise ductwork and blow out flex connections.

Fan Brake Horsepower

BHP = \frac{Q \cdot SP}{6356 \, \eta}

HVAC & HydronicsFluid MechanicsPhysicsShaft power a fan absorbs; the 6356 divisor assumes cubic feet per minute, inches of water gauge and horsepower at the given efficiency.

Faraday's Law of Electrolysis (m = QM/nF)

m = \frac{Q M}{n F}

ChemistryFaraday's law of electrolysis: the mass plated out at an electrode from the charge passed, the molar mass, and the electrons transferred per ion.

Faraday's Law of Induction

\varepsilon = N \frac{\Delta\Phi}{\Delta t}

Electricity & MagnetismPhysicsThe EMF induced in a coil of N turns by flux changing at ΔΦ/Δt (magnitude form).

Federal Bridge Gross Weight Formula

W = 500 \left( \frac{L\,N}{N-1} + 12\,N + 36 \right)

Axle Loads & PavementThe federal limit on how much any group of consecutive axles may weigh, set by how far apart the outer axles of the group are spread. Written into 23 U.S.C. § 127(a)(2), it is the reason a five-axle semi needs 51 feet from steer to rear trailer axle before it can gross 80,000 pounds.

Feed Conversion Ratio

FCR = \frac{F}{G}

Livestock & FeedFeed consumed per unit of liveweight gained — the headline efficiency figure for any growing animal, and the one that decides whether a feeding period paid for itself.

Feed Cost per Unit of Gain

c_g = \frac{C_f}{G}

Livestock & FeedWhat a kilogram of liveweight cost in feed — the feed bill divided by the weight it produced. The figure that decides whether to sell an animal now or feed it further.

Feed Line (q-Line)

y = \frac{q}{q - 1}\,x - \frac{z_F}{q - 1}

Chemical EngineeringThe line the feed's thermal condition draws on a McCabe–Thiele diagram. Both operating lines must cross on it, and where they cross is the optimum feed stage — so q decides where the feed nozzle goes.

Fenske Equation (Minimum Stages)

N_{min} = \frac{\ln\!\left[\frac{x_D}{1 - x_D}\cdot\frac{1 - x_B}{x_B}\right]}{\ln \alpha}

ChemistryThe number of theoretical stages a distillation column would need at total reflux — the absolute floor, before any reflux ratio is chosen. A real column needs roughly twice this many trays.

Fibre V-Number (Normalised Frequency)

V = \frac{2\pi a \, \mathrm{NA}}{\lambda}

OpticsOne dimensionless number that decides everything about how a step-index fibre guides. Below V = 2.405 — the first zero of the Bessel function J₀ — only the fundamental mode propagates and the fibre is single-mode; above it, more and more modes appear. Gloge published the weakly-guiding analysis this rests on in 1971.

Fibre Volume Fraction from Weight Fraction

V_f = \frac{W_f / \rho_f}{W_f / \rho_f + (1 - W_f) / \rho_m}

Strength of MaterialsThe conversion between what a burn-off or acid-digestion test measures and what every micromechanics equation wants. They are not the same number and they are not close: carbon fibre is half again as dense as cured epoxy, so a 70% fibre laminate by weight is only about 61% by volume. Getting this backwards is the most common large error in composite calculations, and it always runs in the unsafe direction.

Fick Principle for Oxygen Consumption

\dot{V}\mathrm{O_2} = \mathrm{CO} \cdot (C_a\mathrm{O_2} - C_v\mathrm{O_2}) \cdot 10

Biomedical & ClinicalAdolf Fick's 1870 principle: oxygen consumption equals blood flow times the oxygen the blood gives up crossing the tissues.

Fick's First Law of Diffusion

J = D\,\frac{C_1 - C_2}{L}

ChemistryPhysicsSteady-state diffusive flux down a concentration gradient. Written here for a slab of thickness L with a fixed concentration on each face, which is the form used for membranes, coatings and stagnant films. The flux J is in mol per square metre per second.

Field Efficiency

e = \frac{C_e}{C_t}

Farm OperationsThe share of a machine's theoretical capacity it actually delivers, once turning, filling, unloading, adjusting and overlap are counted. Measured, it is the honest input to every capacity estimate.

Field Emergence

E = \frac{P}{S}

Crop ProductionThe share of sown seed that became a counted plant. Measured after the fact, it is the single most useful number a grower can carry into next spring's seeding rate calculation.

File Transfer Time

t = \frac{S}{R}

Computer ScienceTime to move a file at a sustained throughput: size divided by rate. Enter the size in megabytes and the rate in megabytes per second, and the answer comes back in seconds.

Fillet Weld Capacity from Throat Area

F = \tau \, A_t

Welding & JoiningThe load a fillet weld can carry: the allowable stress on the weld metal multiplied by the area of the throat resisting it. A fillet weld is checked in shear on its throat whichever direction the load pulls — that is the convention every structural code shares.

Fillet Weld Effective Throat

a = 0.707 \, z

Welding & JoiningThe shortest distance from the root of an equal-leg fillet weld to its face — the plane the weld actually fails on. It is the leg size multiplied by 0.707, and using the leg where the throat belongs overstates the weld by 41 per cent.

Fillet Weld Size for a Load per Unit Length

w = \frac{f}{0.707 \, \tau_{a}}

Machine DesignThe leg size an equal-leg fillet weld needs to carry a stated load per unit length at a stated allowable stress. The form weld design is actually done in, because a continuous weld's length is set by the joint, not by the designer.

Filtration Rate (Filter Loading Rate)

v_f = \frac{Q}{A}

Water & WastewaterWater TreatmentFluid MechanicsFiltration rate through a granular media filter: flow divided by filter bed area, the approach velocity reported in metres per day.

Fin Efficiency (Straight Fin)

\eta_f = \frac{\tanh(mL)}{mL}

Heat TransferThermodynamicsEfficiency of a straight fin with an adiabatic tip, comparing its real duty with the duty it would give if it were all at base temperature.

Fin Heat Transfer Rate

\dot{Q}_f = \eta_f \, h A_f \, \Delta T_b

Heat TransferThermodynamicsHVAC & HydronicsDuty of a fin or finned surface: the ideal convective rate over the whole fin area, derated by the fin efficiency.

Fin Parameter mL (Straight Fin)

mL = L \sqrt{\frac{2h}{k t}}

Heat TransferThermodynamicsThe dimensionless group governing straight-fin performance, combining fin length, thickness, material conductivity and the surface film coefficient.

Final Velocity (Uniform Acceleration)

v = v_0 + a t

MechanicsPhysicsFinal velocity after accelerating uniformly from an initial velocity for a given time.

Finnie Erosion of a Ductile Metal

W = \frac{c \, m \, V^{2}}{8 \, p} \, f(\alpha)

Erosion & SedimentIain Finnie's 1960 model in Wear treats an impacting hard particle as a tiny cutting tool: it ploughs a groove out of a ductile surface, and the volume it removes goes as the square of the impact velocity, inversely with the target's flow stress, and with a strongly angle-dependent shape factor that peaks at a SHALLOW angle. That shallow-angle peak is the signature of ductile erosion, and it is the opposite of how brittle materials behave.

First-Order Chlorine Decay

C = C_0 \, e^{-k t}

Water TreatmentFree chlorine decays exponentially as it reacts with organics, biofilm and pipe wall. The bulk decay coefficient k is measured in a bottle test on your own water; the residual falls by the same fraction in every equal interval of time.

First-Order Integrated Rate Law

[\mathrm{A}] = [\mathrm{A}]_0\,e^{-kt}

ChemistryGives the concentration remaining in a first-order reaction, the exponential decay that governs radioactive decay and most drug clearance.

Fixed-End Moment — Fixed-Fixed Beam, Uniform Load

M_F = \frac{w L^{2}}{12}

Strength of MaterialsCivil & SurveyingMechanicsMoment at each built-in end of a fixed-fixed beam under a uniformly distributed load, wL²/12. Midspan carries only wL²/24, half as much again the other way.

Flash Steam Mass Rate

\dot{m}_f = \frac{\%F}{100} \, \dot{m}_c

ThermodynamicsWater TreatmentFlash steam a condensate stream releases when it is let down to a lower pressure, from the flash percentage and the condensate rate through the trap.

Flash Steam Percentage

\%F = \frac{h_{f1} - h_{f2}}{h_{fg2}} \times 100

Water TreatmentThermodynamicsPercentage of hot condensate that flashes to steam when let down to a lower pressure, from the saturated liquid and latent enthalpies.

Flash-to-Bang Distance to a Lightning Strike

d = \left(331.3 + 0.606\,T_C\right) t

Storm & SkyLight arrives effectively instantly and sound does not, so the gap between the flash and the thunder is a distance divided by the speed of sound. Because that speed follows air temperature, the temperature is an input here rather than a constant hidden inside a folk rule — and the answer is a measurement, never a safety margin.

Flat Roof Snow Load (ASCE 7)

p_f = 0.7\,C_e\,C_t\,I_s\,p_g

Storm & SkyThe ASCE 7 flat-roof snow load: the ground snow load from the code's map, reduced by 0.7 because a roof sheds and drifts differently from open ground, then adjusted for wind exposure, for how warm the roof is underneath, and for what the building is for. A teaching aid, not a design authority — the governing code is.

Flight Line Spacing from Sidelap

S = W_c \, (1 - p_{side})

Photogrammetry & Remote SensingHow far apart the flight lines go: the across-track ground width of the frame, less the fraction that must overlap the neighbouring line. Sidelap is the overlap between ADJACENT LINES — not between consecutive frames, which is endlap and a different number on a different sensor edge.

Flywheel Energy Fluctuation

E = I \omega^{2} C_{s}

Machine DesignThe energy a flywheel exchanges as its speed swings across one cycle, set by its inertia, its mean speed and the coefficient of speed fluctuation allowed. The relation that sizes every flywheel on a press, engine or compressor.

Focal Length of a Spherical Mirror

f = \frac{R}{2}

OpticsPhysicsA spherical mirror focuses parallel rays at half its radius of curvature.

Focused Spot Diameter of a Gaussian Beam

d_f = \frac{4 M^{2} \lambda f}{\pi D}

OpticsThe 1/e² diameter a collimated Gaussian beam of diameter D is brought to by a lens of focal length f. It is the working spot-size equation of laser cutting, welding and marking, and it says the three ways to a smaller spot are a shorter lens, a bigger input beam, and a better beam.

Food-to-Microorganism (F/M) Ratio

\frac{F}{M} = \frac{Q \, S_0}{V \, X}

Water & WastewaterWater TreatmentFood-to-microorganism ratio: BOD applied per day divided by the mass of mixed liquor solids holding it, in kg BOD per kg MLVSS per day.

Forage Demand in Animal Unit Months

F = n \, i \, t

Livestock & FeedThe forage a herd will consume over a grazing period, from the animal units, the intake per animal unit and the length of the season. One animal unit month is conventionally 780 lb — about 354 kg — of dry matter.

Force Between Parallel Wires

F = \frac{\mu_0 I_1 I_2 \ell}{2\pi d}

Electricity & MagnetismPhysicsMagnetic force between two parallel current-carrying wires separated by distance d.

Force Transmissibility

TR = \sqrt{\frac{1 + (2\zeta r)^{2}}{(1 - r^{2})^{2} + (2\zeta r)^{2}}}

Machine DesignThe fraction of a machine's shaking force that reaches the floor through its mounts, as a function of the frequency ratio and the damping. Below a ratio of √2 it is greater than 1 — the isolator amplifies.

Fouled Overall Coefficient

\frac{1}{U_f} = \frac{1}{U_c} + \frac{1}{h_f}

Heat TransferThermodynamicsHVAC & HydronicsAdds a fouling deposit as one more resistance in series, reducing the clean overall coefficient to the fouled value used for design margin.

Fouling Factor on an Overall Coefficient

\frac{1}{U_f} = \frac{1}{U_c} + R_f

Heat TransferThermodynamicsHVAC & HydronicsService-condition U of a heat exchanger, adding the TEMA fouling factor as an extra area-specific resistance on top of the clean coefficient.

Fourier Number

\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}

Heat TransferThermodynamicsDimensionless time for transient conduction, written from conductivity, density and specific heat so no diffusivity input is needed.

Fractional Canopy Cover from LAI

f_c = 1 - e^{-k L}

Forestry & Forest EcologyThe share of the ground a canopy actually shades, as the complement of what its Beer's law lets through. It approaches 100 % but never reaches it, which is the honest reason a real canopy always has a few sunflecks on the floor.

Fractional Conversion from Concentration

X = \frac{C_{A0} - C_A}{C_{A0}}

Chemical EngineeringThe fraction of the limiting reactant that has been consumed, read straight off an inlet and an outlet analysis. Every reactor design equation on this site is written in terms of this one number, so it is worth getting the basis right before anything else.

Francis Formula: Rectangular Weir

Q = 3.33 \, L \, H^{3/2}

Water & WastewaterFluid MechanicsFlow over a suppressed rectangular weir by the Francis formula, with the trade constant 3.33 for crest length and head in feet.

Freezing-Point Depression

\Delta T_f = K_f \, b

ChemistryGives how far a dissolved solute lowers a solvent's freezing point from the molality and the solvent's cryoscopic constant.

Freight Cost per Unit

c_u = \frac{R \, W_c}{n}

Freight & ShippingWhat the freight adds to one unit of product: a rate per hundredweight, applied to the chargeable weight, spread over the pieces on the bill. The last step that turns a shipping decision into a number the business recognises.

Friction Power Loss at a Sliding Contact

P = \mu N v

Machine DesignThe friction force multiplied by the speed it is dragged through: the power a sliding contact turns into heat. Every watt of it appears at the interface, and where that heat goes decides whether the contact survives — most sliding parts are limited by temperature long before they are limited by strength.

Froude Number (Open Channel)

Fr = \frac{V}{\sqrt{g \, D}}

Fluid MechanicsCivil & SurveyingWater & WastewaterFroude number of an open channel: the ratio of flow velocity to the speed of a shallow-water wave, with D the hydraulic depth (flow area divided by top width).

Fuel Consumption (L/100 km)

C = \frac{100\,V}{d}

Everyday & HealthFuel consumption in litres per hundred kilometres from the fuel burned and the distance covered.

Fuel Use per Unit Area

F = \frac{q}{C}

Farm OperationsFuel burned per hectare, from the burn rate per hour and the effective field capacity. The form the number has to be in before it can be compared to anything or budgeted against a crop.

Fundamental Frequency of a String

f = \frac{v}{2L}

Waves & OscillationsPhysicsLowest standing-wave frequency of a string fixed at both ends.

Fundamental of a Closed Pipe

f = \frac{v}{4L}

Waves & OscillationsPhysicsA pipe closed at one end resonates with a quarter wavelength inside, giving a fundamental of v/4L.

Fundamental Period of a Soil Site

T = \dfrac{4H}{V_s}

SeismologyThe period at which a soft soil layer over stiff rock resonates: four times its thickness divided by its shear wave velocity. The number that explains why identical buildings a block apart can fare completely differently.

Fundamental Traffic Flow Relation

q = k\,v

Traffic Flow & SignalsFlow equals density times space mean speed. Three quantities describe a stream of traffic and this identity ties them together, so measuring any two fixes the third — which is how a pair of loop detectors reports a speed it never measured.

Fused Filament Extrusion Flow Balance

\frac{\pi}{4} \, d_f^{2} \, v_f = w \, h \, v_p

Additive ManufacturingWhat goes into the hot end must come out of the nozzle: the cross-section of the filament times how fast it is pushed equals the cross-section of the deposited bead times how fast the head moves. The mass balance behind every extrusion multiplier, every E-step calibration and every under-extruded print.

Future Value of an Annuity (Regular Deposits)

\mathit{FV} = D\,\frac{(1+i)^n - 1}{i}

Money & BusinessWhat an equal deposit made at the end of every period grows to after n periods at rate i — the replacement-fund and savings-plan formula.