Formula solvers starting with B — 98 solvers

Back Azimuth

\alpha_b = \alpha \pm 180^{\circ}

Civil & SurveyingTrigonometryReverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside the zero to three sixty range.

Backwash Water Volume

V_{bw} = v_b \, A \, t

Water & WastewaterWater TreatmentFluid MechanicsWater consumed by one filter backwash, from the backwash rise rate, the filter bed area and the duration of the wash.

Bandwidth from Q and Centre Frequency

BW = \frac{f_{0}}{Q}

Electricity & MagnetismElectrical TradeWidth of a resonant circuit's passband between its half-power points, set entirely by the resonant frequency and the quality factor.

Bandwidth-Delay Product

\mathit{BDP} = R\,\mathit{RTT}

Computer ScienceHow many bits are in flight on a link at any instant: the rate multiplied by the round-trip time. A sender whose window is smaller than this can never fill the pipe, however fast the link is.

Bank Shot Rail Contact Point

x_c = \frac{x_1 y_2 + e\,x_2 y_1}{y_2 + e\,y_1}

Billiards & Cue SportsMechanicsWhere on the rail to send the ball for a one-cushion bank. The schoolbook mirror method reflects the target through the rail and aims at the image — which is this formula with e = 1. Real cushions rebound wider than they receive, so the true contact point sits shifted, and this is the mirror method corrected for restitution.

Banked Curve Angle

\theta = \arctan\!\left(\frac{v^{2}}{r g}\right)

MechanicsPhysicsBank angle that lets a vehicle round a curve of a given radius at a given speed with no reliance on sideways friction.

Barlow's Formula (Pipe Pressure Rating)

P = \frac{2 S t}{D}

HVAC & HydronicsFluid MechanicsMechanicsInternal pressure a pipe can hold from wall stress, wall thickness and outside diameter — the thin-wall hoop-stress relation used by pipeline codes.

Barometric Pressure with Altitude

P = P_0 \, e^{-Mgz/RT}

Air Quality & DispersionAtmospheric pressure at a height above a reference level, for an isothermal layer. The exponential decay that gives the atmosphere a scale height of about 8.4 km.

Barton Q-System Rock Mass Quality

Q = \frac{RQD}{J_n} \cdot \frac{J_r}{J_a} \cdot \frac{J_w}{SRF}

Rock MechanicsBarton, Lien and Lunde's 1974 tunnelling quality index, fitted to some 200 case records at the Norwegian Geotechnical Institute. Six ratings multiplied in three pairs: how blocky the rock is, how well the blocks interlock, and what the water and the stress field are doing to them. It spans six orders of magnitude, from 0.001 for squeezing ground to 1000 for massive unjointed rock.

Barton Roof Support Pressure (simplified)

P_{roof} = \frac{2}{J_r}\,Q^{-1/3}

Rock MechanicsThe form of Barton's roof support pressure used when the rock mass has three or more joint sets, where the block size term has saturated and drops out. Two ratings carry the whole estimate: the tunnelling quality index Q and the joint roughness number Jr. Everything true of the fuller form is true of this one — it is a fit to case records, and the leading 2 is 2 kg/cm² rather than a pure number.

Barton Roof Support Pressure (with Jn)

P_{roof} = \frac{2\sqrt{J_n}}{3\,J_r}\,Q^{-1/3}

Rock MechanicsBarton, Lien and Lunde's 1974 estimate of the permanent support pressure a tunnel roof will need, from the rock mass quality Q, the joint set number and the joint roughness. This is the form for masses with FEWER than three joint sets, where the block size term still matters; at Jn = 9 it becomes the simplified form exactly. It is a curve fitted through some 200 case records, not a mechanics result.

Basal Metabolic Rate (Harris-Benedict, Men)

\mathrm{BMR} = 88.362 + 13.397\,m + 4.799\,h - 5.677\,a

Biomedical & ClinicalEveryday & HealthResting energy expenditure for men in kcal per day, by the 1984 Roza and Shizgal revision of the 1919 Harris-Benedict equations.

Basal Metabolic Rate (Harris-Benedict, Women)

\mathrm{BMR} = 447.593 + 9.247\,m + 3.098\,h - 4.330\,a

Biomedical & ClinicalEveryday & HealthResting energy expenditure for women in kcal per day, by the 1984 Roza and Shizgal revision of the 1919 Harris-Benedict equations.

Basal Metabolic Rate (Mifflin-St Jeor)

\mathrm{BMR} = 10\,m + 6.25\,h - 5\,a + s

Biomedical & ClinicalEveryday & HealthMifflin and St Jeor's 1990 fit for resting energy expenditure in kcal per day, from mass in kilograms, height in centimetres and age in years.

Base Saturation

BS = 100 \, \frac{B}{CEC}

Crop ProductionThe share of a soil's exchange sites occupied by the basic cations — calcium, magnesium, potassium and sodium — rather than by acidity. The number behind a lime recommendation.

Basquin S-N Relation

\sigma_a = \sigma_f' \, (2N_f)^{b}

Strength of MaterialsMechanicsThe straight line an S-N curve becomes on log-log paper: alternating stress against reversals to failure, with the fatigue strength coefficient and exponent from a materials database.

Bayes' Theorem (Two Hypotheses)

P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B \mid A) \, P(A) + P(B \mid A^{c}) \, \left(1 - P(A)\right)}

ProbabilityStatisticsUpdates a prior belief into a posterior after evidence arrives, weighing the true-positive rate against the false-positive rate.

Beam Deflection — Simply Supported, Centre Load

\delta = \frac{P L^{3}}{48 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with I entered in m⁴.

Beam Deflection — Simply Supported, Uniform Load

\delta = \frac{5 w L^{4}}{384 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I entered in m⁴.

Beam Quality Factor M² (θ = M²λ/πw₀)

\theta = \frac{M^{2} \lambda}{\pi w_0}

OpticsSiegman's correction that turns the ideal-Gaussian relation into one a real laser obeys. M² is the factor by which a beam's waist-divergence product exceeds the diffraction limit, it is measured rather than chosen, and it is never below 1.

Bearing Basic Rating Life (L₁₀)

L_{10} = \left( \frac{C}{P} \right)^{p}

Machine DesignHow long a rolling bearing lasts before one in ten of a large group of identical bearings has failed by fatigue, in millions of revolutions. The load ratio raised to a power — cubed for ball bearings, 10/3 for roller bearings — which is why halving the load multiplies life eightfold.

Bearing Capacity Factor Nc

N_c = (N_q - 1)\cot\phi

Soil MechanicsMechanicsPrandtl's cohesion bearing capacity factor Nc derived from Nq and the friction angle, tending to 5.14 as the friction angle goes to zero.

Bearing Capacity Factor Nq

N_q = e^{\pi\tan\phi}\,\tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsPrandtl–Reissner surcharge bearing capacity factor Nq from the friction angle, the value tabulated by Meyerhof, Hansen and Vesic.

Beat Frequency

f_{\text{beat}} = f_1 - f_2

Waves & OscillationsPhysicsAcoustics & NoiseTwo nearby tones interfere to produce a loudness pulse at their difference frequency, with f₁ the higher of the pair.

Bed Shear Stress in an Open Channel

\tau_0 = \rho \, g \, R \, S

Erosion & SedimentThe force the flow drags along the bed, per unit of bed area, from nothing but a weight balance: the downslope component of the water's own weight in a reach has nowhere to go except into the boundary. Hydraulic radius times energy slope times unit weight, and it is the number every sediment threshold is compared against.

Beer Colour (Morey's Equation)

SRM = 1.4922 \times MCU^{0.6859}

Brewing & FermentationTurns malt colour units into a predicted beer colour on the SRM scale. Morey's power law is the correlation most recipe software uses, and it exists because colour does not add up linearly.

Beer–Lambert Law (A = εbc)

A = \varepsilon\, b\, c

ChemistryRelates a sample's absorbance to its molar absorptivity, path length, and concentration — the working equation of UV-Vis spectrophotometry.

Belt Length — Open Drive

L = 2C + \frac{\pi}{2}(D + d) + \frac{(D - d)^2}{4C}

Machine DesignThe length of a belt running open — both pulleys turning the same way — from the two pulley diameters and the centre distance. Two straight spans, two arcs, and a small correction for the fact that the arcs are not half-circles when the pulleys differ.

Belt Power from Tight and Slack Tensions

P = (F_1 - F_2) \, v

Machine DesignThe power a belt actually delivers: the difference between the tight-side and slack-side tensions — the net pull — multiplied by the belt speed. The tensions themselves do no work; only their difference does.

Belt Speed

v = \pi \, d \, n

Machine DesignHow fast the belt itself is travelling: the pulley's circumference times its rotational speed. The figure that decides how much power a belt of a given section can carry, and the one printed in feet per minute on every North American drive chart.

Bend Allowance

BA = \theta \left( R + K t \right)

Machining & TurningThe length of material consumed by a bend: the arc the neutral axis travels through, from where the bend starts to where it ends. The K-factor says where inside the thickness that neutral axis actually lies, and it is the only part of this equation that is not pure geometry.

Bend Deduction

BD = 2 \left( R + t \right) \tan\!\left( \frac{\theta}{2} \right) - BA

Machining & TurningThe number a flat pattern is actually laid out from. Add the two flange lengths as if the part had a perfectly sharp corner, subtract this deduction, and you have the flat blank. It is twice the outside setback less the bend allowance, and it exists because drawings dimension flanges to the outside of the part while material follows the neutral axis.

Bending Stress (σ = Mc/I)

\sigma = \frac{M c}{I}

Strength of MaterialsMechanicsPhysicsBending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I entered in m⁴.

Bending Stress from Section Modulus (σ = M/S)

\sigma = \frac{M}{S}

Strength of MaterialsMechanicsPhysicsBending stress straight from the moment and a tabulated section modulus S in m³, the everyday form used with steel tables.

Bernoulli's Equation (Two Points)

P_1 + \tfrac{1}{2}\rho v_1^{2} + \rho g z_1 = P_2 + \tfrac{1}{2}\rho v_2^{2} + \rho g z_2

Fluid MechanicsPhysicsConservation of mechanical energy along a streamline: pressure, velocity and elevation trade against one another for an incompressible fluid with no friction and no pump between the two points.

Betz Limit — Maximum Extractable Wind Power

P_{max} = \frac{16}{27} \cdot \tfrac{1}{2} \rho A v^{3}

Solar & Wind PowerThe most power any device can take from an open stream of moving air: 16/27, or 59.26%, of the kinetic flux through its swept area. Betz proved it in 1920 from momentum and mass conservation alone, with no reference to blades.

Binomial Coefficient (n Choose k)

\binom{n}{k} = \frac{n!}{k! \, (n-k)!}

AlgebraThe number of ways to choose k items from n, which is also the coefficient of a single term in a binomial expansion.

Binomial Distribution Mean

\mu = n p

ProbabilityStatisticsExpected number of successes across n independent trials that each succeed with probability p — the mean of the binomial distribution.

Binomial Distribution Variance

\sigma^{2} = n p (1 - p)

ProbabilityStatisticsSpread of the number of successes across n independent trials, at its largest when the per-trial chance p sits at one half.

Binomial Probability

P(X = k) = \binom{n}{k} p^{k} (1 - p)^{\,n-k}

ProbabilityStatisticsChance of exactly k successes in n independent trials that each succeed with the same fixed probability p, as with coin tosses.

Biomass Yield Coefficient Y(X/S)

Y_{X/S} = \frac{X - X_0}{S_0 - S}

Chemical EngineeringGrams of cells made per gram of substrate eaten. It is the number that turns a fermentation plan into a shopping list — how much sugar buys how much biomass — and the number that tells you, when it comes out wrong, that something is being oxidised to carbon dioxide or excreted as a by-product instead of becoming cells.

Biot Number

\mathrm{Bi} = \frac{h L_c}{k}

Heat TransferThermodynamicsRatio of internal conduction resistance to surface convection resistance; below 0.1 a body may be treated as having one uniform temperature.

Birthday Collision Probability (Exponential Approximation)

p \approx 1 - e^{-n^{2}/(2N)}

Cryptography & Key SizingChance that n values drawn independently from a space of N contain at least one repeat. This is the exponential APPROXIMATION, and it exists because the exact all-distinct product cannot be evaluated at the sizes cryptography works in: N = 2^128 would need 2^64 factors. It is excellent when n is far below N and it overstates the collision chance when n is a large fraction of N.

Birthday Problem (All Distinct)

P = \frac{N!}{(N - n)! \; N^{\,n}}

ProbabilityStatisticsChance that n independent picks from N equally likely options are all different, the engine behind the birthday paradox.

Bit Rate from Baud Rate

R = B\,n

Computer ScienceBit rate from a symbol rate and the number of bits each symbol carries. Baud counts symbols per second; a 16-QAM symbol carries 4 bits, so 2400 baud is 9600 bits per second.

Blade Loading Coefficient

C_T/\sigma = \frac{C_T}{\sigma}

Aerospace & FlightThrust coefficient divided by solidity: how hard each unit of blade area is working, rather than each unit of disc. It is proportional to the mean blade lift coefficient, and it is the number that says how close the rotor is to stall.

Blank Diameter for a Cylindrical Cup

D = \sqrt{d^{2} + 4 d h}

Machining & TurningThe diameter of the flat disc that becomes a flat-bottomed cylindrical cup, taken from surface-area equivalence: the blank has the same area as the bottom disc plus the wall it unrolls into. A starting point for a trial blank, not a finished answer.

Blast Burden (Konya's Formula)

B = 3.15 \, D_e \left( \frac{SG_e}{SG_r} \right)^{1/3}

Rock MechanicsHow far back from the free face a blast hole should be drilled, from Konya's formula as published in FHWA-HI-92-001. The burden is the most important single dimension in a bench blast: get it right and the rock moves, get it too large and the shot is confined, too small and the energy goes into the air as noise, flyrock and airblast.

Blast Vibration: Scaled Distance and Peak Particle Velocity

PPV = K \left( \frac{D}{\sqrt{W}} \right)^{-\beta}

Rock MechanicsThe standard blast vibration prediction: peak particle velocity falls off as a power of the square-root scaled distance, which is the distance to the point of interest divided by the square root of the charge fired on one delay. The FORM comes from Siskind and colleagues at the US Bureau of Mines; the two CONSTANTS come from your own site and from nowhere else.

Blending Two Feeds to a Protein Target

x = \frac{a_t - a_2}{a_1 - a_2}

Livestock & FeedThe share of the higher-protein ingredient needed when blending two feeds to hit a target analysis — the Pearson square, written as the algebra it always was.

Blending Two Fertilisers to a Target

x = \frac{a_t - a_2}{a_1 - a_2}

Crop ProductionThe share of the higher-analysis product needed when blending two fertilisers to hit a target analysis — the Pearson square, written as the algebra it actually is.

Blood Alcohol Estimate (Widmark Equation)

\mathrm{BAC} = \frac{100\,A}{r\,m} - \beta t

Everyday & HealthChemistryWidmark's 1932 estimate of blood alcohol concentration from alcohol consumed, body mass, a distribution ratio and elapsed time. An estimate only, never a fitness-to-drive test.

Blow-Up Ratio and Blown Film Gauge

\mathrm{BUR} = \frac{D_b}{D_d} \qquad t = \frac{h_d}{\mathrm{BUR} \cdot \mathrm{DDR}} = \frac{h_d\, D_d}{D_b\, \mathrm{DDR}}

Polymers & Plastics ProcessingA blown-film bubble is a mass balance you can watch. The melt leaves an annular die of gap h_d and diameter D_d, the internal air inflates it to a bubble diameter D_b — that ratio is the blow-up ratio — and the nip pulls it up faster than it left, which is the draw-down ratio. Whatever the two stretches multiply to, the gauge divides by.

Blowdown Rate from Cycles

B = \frac{E}{\text{COC} - 1}

Water TreatmentFluid MechanicsBlowdown a cooling tower must bleed to hold a target cycles of concentration, given its evaporation rate.

Board Basis Weight (Grammage)

W = \frac{m}{A}

Packaging & Box StrengthMass divided by area — the way paper and combined board have always been sold, and the number that connects a blank size to a freight cost and a fibre bill. The whole difficulty is the units: grammage is grams per square metre almost everywhere, and North American board is quoted in pounds per thousand square feet, which is a different number for the same board.

Board Feet of Lumber

BF = \frac{t \, w \, L}{12}

Trades & ConstructionThe lumberyard's volume unit: thickness in inches times width in inches times length in feet, divided by twelve. One board foot is 144 cubic inches.

BOD Mass Loading

W = Q \, C

Water & WastewaterWater TreatmentChemistryMass of BOD, COD or solids arriving per unit time from a flow and its concentration, on the sanitary basis of 1 mg/L = 1 g/m³.

BOD Removal Efficiency

E = \frac{C_i - C_e}{C_i} \times 100

Water & WastewaterWater TreatmentPercent removal across a treatment unit or a whole plant, from the influent and effluent concentrations of BOD, TSS or any pollutant.

Body Mass Index (BMI)

\mathrm{BMI} = \frac{m}{h^2}

Everyday & HealthBody mass index: mass divided by the square of height, the population screening ratio defined by Quetelet in 1832.

Body Surface Area (Du Bois Formula)

\mathrm{BSA} = 0.007184 \cdot h_{cm}^{0.725} \cdot m_{kg}^{0.425}

Everyday & HealthBody surface area by the 1916 Du Bois power fit, still the reference against which newer formulas are compared.

Body Surface Area (Mosteller Formula)

\mathrm{BSA} = \sqrt{\frac{h_{cm} \cdot m_{kg}}{3600}}

Everyday & HealthBody surface area in square metres by Mosteller's 1987 square-root simplification of the older exponential fits.

Body Surface Area Dosing (mg per m²)

D = D_A \cdot \mathrm{BSA}

Biomedical & ClinicalThe dose for a person from a protocol quoted per square metre of body surface area, the convention used across oncology.

Bohr Model Energy Levels

E_n = -\frac{13.606\ \mathrm{eV}}{n^{2}}

Modern PhysicsChemistryPhysicsEnergy of the hydrogen atom's nth level: −13.606 eV divided by n².

Boiler Blowdown Percent

\%B = \frac{\text{TDS}_{fw}}{\text{TDS}_b} \times 100

Water TreatmentChemistryBoiler blowdown as a percentage of feedwater, set by the ratio of feedwater TDS to the maximum TDS allowed in the drum.

Boiler Blowdown Rate from Steam Rate

B = \frac{S}{\text{COC} - 1}

Water TreatmentThermodynamicsContinuous blowdown a steam boiler must carry, in pounds per hour, from its steam production rate and target cycles of concentration.

Boiler Cycles of Concentration

\text{COC} = \frac{\text{TDS}_b}{\text{TDS}_{fw}}

Water TreatmentChemistryCycles of concentration in a steam boiler: boiler water TDS divided by feedwater TDS, since steam leaves the dissolved solids behind.

Boiler Horsepower to Heat Output

Q = 33{,}475 \; \text{BHP}

Water TreatmentThermodynamicsConverts boiler horsepower to heat output using the ASME definition of 33,475 BTU per hour per boiler horsepower.

Boiler Horsepower to Steam Rate

S = 34.5 \; \text{BHP}

Water TreatmentThermodynamicsSteam output of a boiler from its horsepower rating, at the ASME definition of 34.5 lb/h of steam from and at 212 °F per BHP.

Boiler Makeup from Condensate Return

M = S\left(1 - \frac{\%CR}{100}\right)

Water TreatmentThermodynamicsFresh makeup water a steam plant must treat, from the steam production rate and the fraction of condensate that comes back.

Boiler or Furnace Output from Input

\dot{Q}_{out} = \dot{Q}_{in} \, \eta

HVAC & HydronicsThermodynamicsUsable heat delivered by a boiler or furnace from its fuel input rate and its efficiency rating, the nameplate arithmetic behind AFUE.

Boiling-Point Elevation

\Delta T_b = K_b \, b

ChemistryGives how far a dissolved solute raises a solvent's boiling point from the molality and the solvent's ebullioscopic constant.

Boilup Ratio

V_B = \frac{V}{B}

Chemical EngineeringThe vapour the reboiler raises divided by the bottoms drawn off beneath it. The stripping section's answer to the reflux ratio, and the term the steam bill is really written in.

Bolt Preload from Torque (T = KDF)

T = K D F

Strength of MaterialsMechanicsPhysicsBolt preload from tightening torque using the nut factor K, T = KDF, the field method behind every published torque spec.

Bond Number / Eötvös Number (Gravity against Surface Tension)

Bo = \frac{\Delta\rho \, g \, L^{2}}{\sigma}

Fluid MechanicsThe weight of a body of fluid set against the surface tension holding it together. Below about 1 capillarity rules and a drop is a sphere, a meniscus climbs a tube, a bubble stays round; above it gravity rules and the drop puddles, the meniscus flattens, the bubble goes cap-shaped. Chemical engineers call the same ratio the Eötvös number, and it is the same equation with a different name on it.

Bond's Law and the Work Index

E = 10 \, W_i \left( \frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}} \right)

Rock MechanicsFred Bond's 1952 relation for the energy it takes to grind ore: the specific energy depends on the difference in the reciprocal square roots of the product and feed sizes, scaled by a work index measured for that ore. It is the backbone of every grinding circuit sized in the last seventy years, and it is a correlation rather than a mechanism.

Bounce Height from Coefficient of Restitution

h_2 = e^{2} h_1

MechanicsPhysicsHeight a dropped ball rebounds to, from the drop height and the coefficient of restitution of the bounce.

Box Compression Service Derating

BCT_{svc} = BCT_{lab} \, f_h \, f_t \, f_o \, f_m

Packaging & Box StrengthThe chain of multipliers between a laboratory compression figure and the load a box will actually hold in a warehouse. Every factor on this page is a user input with NO default, on purpose. The circulating tables of humidity, storage-time, overhang and interlock numbers are unattributed restatements of a copyrighted handbook, and this site will not restate them either. What it will do is tell you what the open literature has measured, and warn you where the numbers everybody quotes have been shown to be wrong.

Box Perimeter

Z = 2 \left( L + W \right)

Packaging & Box StrengthThe distance around a rectangular box's footprint, and the Z that both McKee formulas want. It is deliberately not the girth used for shipping-size rules, and it takes no notice of the depth: what carries a top load is the four side panels standing on their edges, so the loaded length is the perimeter of the plan.

Boyle's Law

P_1 V_1 = P_2 V_2

ThermodynamicsChemistryPhysicsAt constant temperature, pressure times volume stays constant for a fixed amount of gas.

Brake Torque from Friction

T = \mu F r n

Machine DesignThe torque a friction brake develops: the coefficient of friction times the actuating force times the effective radius, once for each rubbing surface. The starting point for every brake and holding-clamp calculation.

Brayton Cycle Efficiency (Pressure Ratio)

\eta = 1 - \frac{1}{r_p^{\left(\gamma - 1\right)/\gamma}}

ThermodynamicsPhysicsAir-standard efficiency of a gas turbine from compressor pressure ratio alone. Turbine inlet temperature does not appear, which is why the equation says nothing about the specific work — the thing that actually limits real machines.

Break-Even Price

p_b = \frac{C}{Y}

Farm OperationsThe price the crop has to fetch, at the yield expected or already in the bin, before it has paid for itself. The number that belongs on the wall beside the futures screen during marketing season.

Break-Even Quantity

Q = \frac{F}{p - v}

Money & BusinessUnits that must be sold before fixed costs are covered: fixed cost divided by the contribution margin, price minus variable cost per unit.

Break-Even Yield

Y_b = \frac{C}{p}

Farm OperationsThe yield the crop has to make, at a price already contracted or assumed, before it has paid for itself. The most-used number on any grain budget, and the one that decides whether an acre gets seeded.

Breakpoint Chlorine-to-Ammonia Ratio

R = \frac{\mathrm{Cl_2}}{\mathrm{NH_3\text{-}N}}

Water TreatmentChemistryThe weight ratio of chlorine dose to ammonia nitrogen, which must reach about 7.6 to 1 to pass breakpoint and produce a free chlorine residual.

Breeder's Equation

R = h^2 S

Plant Breeding & GeneticsHow far a population moves in one generation of selection: the selection differential multiplied by narrow-sense heritability. The central equation of applied breeding.

Brewhouse Efficiency

E = \frac{m_a}{m_p}

Brewing & FermentationThe share of the extract locked in the grain that actually reached the kettle. The definitional form — extract recovered over extract available — from which every working efficiency figure descends.

Brewster's Angle

\tan\theta_B = \frac{n_2}{n_1}

OpticsPhysicsThe incidence angle at which reflected light is completely polarized.

Briggs Buoyancy Flux

F = \frac{g \, v_s \, d^2 (T_s - T_a)}{4 \, T_s}

Air Quality & DispersionThe buoyancy flux parameter of a hot stack plume, from exit velocity, stack diameter and the temperature difference against ambient. The input to every Briggs plume-rise equation.

Briggs Plume Rise (Neutral and Unstable)

\Delta h = \frac{1.6 \, F^{1/3} x^{2/3}}{u}

Air Quality & DispersionHow far a buoyant plume climbs above the stack at a given downwind distance, in neutral or unstable air — the Briggs two-thirds law that underlies most regulatory dispersion modelling.

Brinkman Number

\mathrm{Br} = \frac{\mu \, v^{2}}{k \, \Delta T}

Heat TransferHeat generated by shearing the fluid against heat carried away by conduction. It is why a polymer melt leaves a die hotter than it entered, why a journal bearing runs warm on its own oil, and why viscous fluids in small passages cannot be treated as isothermal.

Broad-Crested Weir Discharge

Q = C_d \left(\tfrac{2}{3}\right)^{3/2} \sqrt{g} \; b \, H^{3/2}

Fluid MechanicsCivil & SurveyingWater & WastewaterDischarge over a broad-crested weir or a level spillway crest, derived by forcing critical depth over the crest. In SI the constant group is 1.705, so Q = 1.705 C_d b H^1.5 with lengths in metres.

Broad-Sense Heritability

H^2 = \frac{V_G}{V_P}

Plant Breeding & GeneticsThe share of the variation you can see that is genetic rather than environmental. The starting question of any breeding programme: is this trait worth selecting on at all?

Build Time Estimate

t_b = n \, ( t_s + t_r )

Additive ManufacturingLayers multiplied by what each layer costs: the time the beam spends scanning it plus the time the recoater spends spreading the next one. The estimate every quotation starts from, and the one every scheduler learns to pad.

Bulk Modulus (K = ΔP·V₀/ΔV)

K = \frac{\Delta P \, V_0}{\Delta V}

Strength of MaterialsMechanicsPhysicsBulk modulus from the pressure rise and the volume change it produces; water sits near 2.2 GPa and hydraulic oil near 1.5 GPa.

Buoyant Force (Archimedes' Principle)

F_b = \rho V g

Fluid MechanicsPhysicsThe upward force on a submerged body equals the weight of the fluid it displaces, with g = 9.80665 m/s².

Burn Time from Propellant Load

t_b = \frac{m_p}{\dot m}

Aerospace & FlightHow long an engine can fire: propellant on board divided by the rate it leaves at. Trivial arithmetic, and the number a whole stage is designed around — nozzle cooling, tank size and structural loads all follow from it.

Bushels to Mass by Test Weight

m = n \, w_b

Crop ProductionThe mass of a lot of grain quoted in bushels, using the legal test weight of the crop — 56 lb for corn, 60 for wheat and soybeans, 48 for barley, 32 for oats. A bushel of grain is a weight, not a volume.