formula.expert

Formula solvers — H

0–9 · 1A · 84B · 98C · 181D · 94E · 80F · 66G · 65H · 72I · 46J · 6K · 18L · 63M · 105N · 46O · 21P · 138Q · 4R · 113S · 194T · 104U · 8V · 31W · 51X · 3Y · 6Z · 3

72 of 1701 solvers. Browse by category instead →

Hailstone Terminal Velocity

vt=4 g ρi D3 Cd ρav_t = \sqrt{\dfrac{4\,g\,\rho_i\,D}{3\,C_d\,\rho_a}}

Storm & SkyBalance a hailstone's weight against the drag on a sphere and the fall speed drops out: proportional to the square root of the diameter, so a stone four times as wide falls twice as fast. A two-centimetre stone arrives near 20 m/s, a golf ball near 30, and a softball near 45.

Half-Life and Decay Constant

t1/2=ln⁡2λt_{1/2} = \frac{\ln 2}{\lambda}

ChemistryPhysicsConverts between a half-life and the exponential decay constant via the factor ln 2 = 0.6931471806.

Half-Life Decay

N=N0(12)t/t1/2N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}

ChemistryPhysicsGives the quantity remaining after repeated halvings over an elapsed time measured in half-lives.

Half-Life of a Second-Order Reaction

t1/2=1k [A]0t_{1/2} = \frac{1}{k\,[\mathrm{A}]_0}

ChemistryGives the half-life of a second-order reaction, which unlike a first-order half-life depends on the starting concentration.

Half-Life of a Second-Order Reaction

t1/2=1k CA0t_{1/2} = \frac{1}{k\,C_{A0}}

Chemical EngineeringHow long a second-order reaction takes to consume half its reactant. Unlike first-order decay, this time depends on the starting concentration — and each successive half-life is twice as long as the one before it.

Hall–Petch Relation

σy=σ0+ky d−1/2\sigma_y = \sigma_0 + k_y \, d^{-1/2}

Metallurgy & Heat TreatmentFiner grains make a stronger metal, and the strength climbs as the inverse square root of the grain diameter. Hall measured it on mild steel in 1951 and Petch on cleavage in 1953, and it is the one strengthening mechanism that costs nothing in toughness — every other route to strength trades ductility away.

Halpin–Tsai Transverse Modulus

E2=Em 1+ξηVf1−ηVf,η=Ef/Em−1Ef/Em+ξE_2 = E_m \, \frac{1 + \xi \eta V_f}{1 - \eta V_f}, \quad \eta = \frac{E_f/E_m - 1}{E_f/E_m + \xi}

Strength of MaterialsThe semi-empirical form that actually predicts the transverse stiffness of a lamina, sitting above the inverse rule of mixtures and far below the longitudinal one. One fitted parameter, the reinforcing factor ξ, carries everything the two bounds cannot see: the shape of the fibre cross-section and how the fibres are packed against one another.

Hamming Code Parity Bits

2r≥m+r+12^{r} \ge m + r + 1

Computer ScienceParity bits r needed to correct any single-bit error in m data bits. The r check bits must be able to name every bit position plus the no-error case, which is where the +1 comes from.

Hardness Load Removed per Regeneration

m=C Vm = C \, V

Water TreatmentChemistryGives the mass of hardness as CaCO₃ a softener must remove between regenerations from the raw hardness and the volume of water treated.

Hardness Removal Efficiency and Leakage

R=Cin−CoutCinR = \frac{C_{\text{in}} - C_{\text{out}}}{C_{\text{in}}}

Water TreatmentChemistryGives the fraction of hardness a softener or membrane removes from the inlet and outlet concentrations, with the remainder being leakage to service.

Hardy–Weinberg Heterozygote Frequency

H=2p(1−p)H = 2p(1-p)

Plant Breeding & GeneticsThe proportion of heterozygotes expected in a population at Hardy–Weinberg equilibrium, from one allele's frequency. The null model every real population is compared against.

Harmon Peaking Factor

PF=1+144+P/1000PF = 1 + \frac{14}{4 + \sqrt{P/1000}}

Water & WastewaterWater TreatmentFluid MechanicsHarmon peaking factor for sanitary sewer design: the ratio of peak hourly to average daily flow for a served population.

Harmonic Frequencies

fn=nf1f_n = n f_1

Waves & OscillationsPhysicsThe nth harmonic of a vibrating system is n times its fundamental frequency.

Harmonic Mean of Two Numbers

H=2aba+bH = \frac{2ab}{a + b}

AlgebraThe reciprocal-based average of two numbers, the right average for rates such as speeds covering equal distances.

Harvest Index

HI=YgBHI = \frac{Y_g}{B}

Crop ProductionThe share of a crop's above-ground biomass that ends up as harvested grain. The single number that describes what plant breeding did to cereals in the twentieth century.

Hash Collision Resistance (Birthday Bound)

n1/2≈2ln⁡2  ⋅  2 b/2n_{1/2} \approx \sqrt{2\ln 2}\;\cdot\;2^{\,b/2}

Cryptography & Key SizingNumber of messages that must be hashed before a collision becomes as likely as not, for a hash with a b-bit output. It is about 2^(b/2), and the constant sqrt(2 ln 2) = 1.1774 is the exact leading factor. NIST SP 800-107 Rev. 1 uses this to state that an approved hash function of b bits provides b/2 bits of collision resistance.

Hausner Ratio

HR=ρtρbHR = \frac{\rho_t}{\rho_b}

Additive ManufacturingTapped density divided by apparent (bulk) density: how much a powder tightens up when you shake it. Henry Hausner proposed it in 1967 as a one-number flow rating, and it remains the cheapest useful test a powder receives — a ratio near 1.0 means a powder that pours and spreads, and one above 1.35 means a powder that bridges, rat-holes and leaves the recoater with an uneven layer.

Hazen–Williams Head Loss

hf=10.67 L Q1.852C1.852D4.8704h_f = \frac{10.67 \, L \, Q^{1.852}}{C^{1.852} D^{4.8704}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe waterworks head-loss equation in SI form, with Q in m³/s and D in m; the 10.67 constant is 4.727 when working in feet and cubic feet per second.

Hazen–Williams Velocity

v=0.849 C R0.63S0.54v = 0.849 \, C \, R^{0.63} S^{0.54}

HVAC & HydronicsFluid MechanicsWater TreatmentMean water velocity from hydraulic radius and hydraulic gradient; the 0.849 SI constant becomes 1.318 when R is in feet and v in feet per second.

Heat Conduction Rate

P=kAΔTdP = \tfrac{k A \Delta T}{d}

ThermodynamicsPhysicsSteady-state heat flow through a slab by Fourier's law of conduction.

Heat Exchanger Duty (Q = U·A·F·LMTD)

Q˙=UAF ΔTlm\dot{Q} = U A F \, \Delta T_{lm}

Heat TransferThermodynamicsHVAC & HydronicsThe LMTD design equation with the correction factor F, which derates the counterflow driving force for shell-and-tube or crossflow arrangements.

Heat Exchanger Effectiveness (ε = Q/Qmax)

ε=Q˙Q˙max\varepsilon = \frac{\dot{Q}}{\dot{Q}_{max}}

Heat TransferThermodynamicsHVAC & HydronicsEffectiveness as the ratio of actual duty to the thermodynamic maximum, the performance figure that needs no outlet temperatures to interpret.

Heat Flow from Thermal Resistance

Q˙=ΔTR\dot{Q} = \frac{\Delta T}{R}

Heat TransferThermodynamicsHVAC & HydronicsOhm's law for heat: the flow through an assembly equals the temperature difference across it divided by its total thermal resistance.

Heat Flux Through Insulation (q = ΔT/R)

q′′=ΔTRq'' = \frac{\Delta T}{R}

Heat TransferHVAC & HydronicsThermodynamicsHeat flow per unit area through an insulated assembly, straight from the temperature difference and the R-value, with no area needed.

Heat Index (Rothfusz Regression)

HI=c1+c2T+c3R+c4TR+c5T2+c6R2+c7T2R+c8TR2+c9T2R2\mathrm{HI} = c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^2 + c_6 R^2 + c_7 T^2 R + c_8 T R^2 + c_9 T^2 R^2

Everyday & HealthThermodynamicsThe US National Weather Service heat index, Rothfusz's nine-term regression through Steadman's apparent-temperature table (T in °F, R in percent).

Heat Loss Through an Assembly (Q = A·ΔT/R)

Q˙=A ΔTRtot\dot{Q} = \frac{A \, \Delta T}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsSteady heat loss through a wall, roof or floor from its area, the inside-to-outside temperature difference and the assembly's total R-value.

Heat of Reaction

q=nΔHq = n \Delta H

ChemistryThermodynamicsScales a reaction's molar enthalpy change by the amount reacted to give the total heat released or absorbed.

Heat Release Rate from Fuel Area

Q˙=m˙′′ ΔHc Af χ\dot{Q} = \dot{m}'' \, \Delta H_c \, A_f \, \chi

Fire Protection & Fire DynamicsThe fire size a burning surface can actually deliver: mass leaving the fuel per unit area per second, times the heat that mass carries, times the area alight, times the fraction that burns completely. This is the fuel-controlled ceiling that a growth curve has to be checked against.

Height from Stereo Parallax

h=H Δppa+Δph = \frac{H \, \Delta p}{p_a + \Delta p}

Photogrammetry & Remote SensingThe height of an object read off a stereo pair: the difference in parallax between its top and its base, against the absolute parallax at the base. The relation behind every height ever measured on a stereoscope, and the same geometry a modern dense matcher is solving a hundred million times over.

Heim Ratio and Runout Distance (Fahrböschung)

L=H(H/L)L = \frac{H}{\left(H/L\right)}

Soil MechanicsRock MechanicsHorizontal runout of a rock avalanche or debris slide from its vertical drop and the Heim ratio H/L — the fall height divided by the travel distance, also read as the tangent of the travel angle a straight line from crown to toe makes with the horizontal.

Helical Compression Spring Rate

k=Gd48D3nk = \frac{G d^{4}}{8 D^{3} n}

Machine DesignThe stiffness of a round-wire helical spring from its wire diameter, mean coil diameter, active coil count and the shear modulus of the wire. The fourth power on the wire is the whole design lever.

Helical Spring Shear Stress (with the Wahl Factor)

τ=KW8FDπd3\tau = K_{W} \frac{8 F D}{\pi d^{3}}

Machine DesignShear stress in the wire of a helical spring under load, corrected by the Wahl factor for coil curvature and direct shear. The stress peaks on the INSIDE of the coil, which is where springs actually break.

Hemisphere Total Surface Area

A=3πr2A = 3\pi r^2

GeometryTotal surface of a solid hemisphere: the curved half-sphere 2*pi*r^2 plus the flat circular base pi*r^2, giving 3*pi*r^2.

Hemisphere Volume

V=23πr3V = \frac{2}{3}\pi r^3

GeometryVolume of half a sphere, exactly half of the full sphere's (4/3)*pi*r^3.

Henderson–Hasselbalch Equation (Weak Acid Buffer)

pH=pKa+log⁡10 ⁣[A−][HA]\mathrm{pH} = \mathrm{p}K_a + \log_{10}\!\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}

ChemistryGives the pH of a buffer from the acid's pKa and the ratio of conjugate base to undissociated acid, the master equation of buffer preparation.

Henderson–Hasselbalch Equation (Weak Base Buffer)

pOH=pKb+log⁡10 ⁣[BH+][B]\mathrm{pOH} = \mathrm{p}K_b + \log_{10}\!\frac{[\mathrm{BH^+}]}{[\mathrm{B}]}

ChemistryGives the pOH of a weak base buffer from the base's pKb and the ratio of conjugate acid to free base, the mirror image of the acid form.

Henry's Law (Gas Solubility)

C=H PC = H\,P

ChemistryWater & WastewaterAt a fixed temperature the concentration a gas reaches in a liquid is proportional to its partial pressure above the liquid. H is the solubility form of the Henry constant, in moles per cubic metre per pascal.

Herd Immunity Threshold

Hc=1−1R0H_{c} = 1 - \frac{1}{R_{0}}

Epidemiology & Diagnostic AccuracyThe share of a population that must be immune before each infection replaces itself with fewer than one new infection, so that transmission fades rather than grows.

Heron's Formula (Triangle Area from Three Sides)

A=s(s−a)(s−b)(s−c),  s=a+b+c2A = \sqrt{s(s-a)(s-b)(s-c)}, \; s = \tfrac{a+b+c}{2}

GeometryArea of any triangle from the lengths of its three sides, using the semiperimeter.

Hertzian Contact Pressure, Sphere on a Flat

p0=6FE∗2π3R23p_{0} = \sqrt[3]{\frac{6 F E^{*2}}{\pi^{3} R^{2}}}

Machine DesignHeinrich Hertz worked this out in 1882 while thinking about the optics of stacked lenses, and it became the foundation of rolling-contact engineering. Press a sphere onto a flat and the contact is a small circle whose pressure peaks at the centre: the peak rises only as the CUBE ROOT of the load, so even a modest force makes an enormous stress.

Heskestad Flame Height

Lf=0.235 Q˙2/5−1.02 DL_f = 0.235 \, \dot{Q}^{2/5} - 1.02 \, D

Fire Protection & Fire DynamicsGunnar Heskestad's correlation for the mean height of the luminous flame above a fire of known size and base diameter. Two terms: the heat release rate lifts the flame, and the base diameter pulls the tip down, because a wide fire entrains air along its whole perimeter and burns out sooner.

Hess's Law (Three-Step Sum)

ΔHrxn=ΔH1+ΔH2+ΔH3\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2 + \Delta H_3

ChemistryThermodynamicsHess's law: the enthalpy change of a target reaction is the sum of the enthalpy changes of the steps you route it through.

High or Low Point on a Vertical Curve

xt=−g1LAx_t = -\frac{g_1 L}{A}

Civil & SurveyingDistance from the BVC to the crest or sag of a parabolic vertical curve, where the profile grade passes through zero.

Hoek–Brown Constants s and a (2002)

s=exp⁡ ⁣(GSI−1009−3D),a=12+16(e−GSI/15−e−20/3)s = \exp\!\left( \frac{GSI - 100}{9 - 3D} \right), \qquad a = \frac{1}{2} + \frac{1}{6}\left( e^{-GSI/15} - e^{-20/3} \right)

Rock MechanicsThe other two constants of the 2002 generalized criterion. s measures how much of the intact rock's cohesion the mass has kept — 1 for genuinely intact rock, falling to a ten-thousandth for a poor mass — and a is the curvature of the envelope, fixed by GSI alone and never by anything else.

Hoek–Brown Rock Mass Constant m_b (2002)

mb=mi exp⁡ ⁣(GSI−10028−14D)m_b = m_i \, \exp\!\left( \frac{GSI - 100}{28 - 14D} \right)

Rock MechanicsHow the intact rock's frictional constant m_i is knocked down to the rock mass value m_b, in the 2002 edition. Two judgement calls go in — the Geological Strength Index and the disturbance factor — and both of them make the rock look better if you are generous.

Holding Time Index

HTI=ln⁡2  VB\text{HTI} = \frac{\ln 2 \; V}{B}

Water TreatmentChemistryHolding time index: the half-life of a chemical in a bled system, the time for half the treatment to wash out at a given blowdown rate.

Holland Plume Rise

Δh=vsdu(1.5+2.68×10−3Pd Ts−TaTs)\Delta h = \frac{v_s d}{u}\left(1.5 + 2.68\times10^{-3} P d \, \frac{T_s - T_a}{T_s}\right)

Air Quality & DispersionPlume rise by Holland's 1953 equation, combining momentum and buoyancy in one step. Simpler than Briggs and generally conservative, which is why it survives as a screening estimate.

Hollomon Flow Curve

σ=K ε n\sigma = K \, \varepsilon^{\,n}

Metallurgy & Heat TreatmentThe power law that describes how a metal hardens as it deforms: true stress rises as true plastic strain raised to a small exponent. Hollomon published it in 1945, and the exponent n is the single number that tells a press shop whether a sheet will stretch or split.

Hooke's Law

F=kxF = k x

MechanicsPhysicsRestoring force of an ideal spring, proportional to its displacement from rest.

Hoop Stress in a Thin-Walled Cylinder

σh=pd2t\sigma_{h} = \frac{p d}{2 t}

Strength of MaterialsMechanicsPhysicsHoop (circumferential) stress in a thin-walled pipe or pressure vessel, σ = pd/2t — exactly twice the longitudinal stress.

Hop Utilisation (Tinseth)

U=1.65×0.000125 Gb−1(1−e−0.04 t)4.15U = \frac{1.65 \times 0.000125^{\,G_b - 1}\left(1 - e^{-0.04\,t}\right)}{4.15}

Brewing & FermentationThe fraction of a hop addition's alpha acids that actually isomerise into the beer, as a function of boil gravity and boil time. Glenn Tinseth's 1997 correlation, still the industry default.

Horizontal Curve External Distance

E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)

Civil & SurveyingTrigonometryClearance from the point of intersection to the midpoint of the curve, the distance a curve cuts back from the corner.

Horizontal Curve Length from Degree of Curve

L=100 ΔDL = \frac{100\,\Delta}{D}

Civil & SurveyingGeometryLength of a circular curve in 100 ft stations, from the total deflection angle and the degree of curve, on the arc definition.

Horizontal Curve Long Chord

C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}

Civil & SurveyingTrigonometryStraight-line distance from the point of curvature to the point of tangency, the chord that spans the entire circular curve.

Horizontal Curve Middle Ordinate

M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)

Civil & SurveyingTrigonometryOffset from the middle of the long chord to the middle of the arc, the number that governs sight distance around obstructions.

Horizontal Curve Tangent Length

T=Rtan⁡Δ2T = R\tan\frac{\Delta}{2}

Civil & SurveyingTrigonometryDistance from the point of intersection back to the point of curvature, from the curve radius and its total deflection angle.

Horizontal Velocity Component

vx=vcos⁡θv_x = v \cos\theta

MechanicsPhysicsHorizontal component of a projectile's launch velocity — the part of the speed that carries it downrange at a constant rate.

Horton Infiltration Rate

f=fc+(f0−fc) e−ktf = f_c + (f_0 - f_c) \, e^{-kt}

Water & WastewaterCivil & SurveyingSoil MechanicsHorton's empirical infiltration curve: the rate starts at a dry-soil capacity, decays exponentially as the surface layer saturates, and settles to a steady final rate. It is a fitted description of what soils are observed to do, not a derivation from soil physics.

Hubble's Law

v=H0dv = H_0 d

Astronomy & GravitationPhysicsRecession velocity of a galaxy: its distance times the Hubble constant — the expansion of the universe read as a straight line.

Humidex (Canadian Humidity Index)

H=T+0.5555 (e−10),e=6.11 e5417.753(1273.16−1Td)H = T + 0.5555\,(e - 10), \quad e = 6.11\,e^{5417.753\left(\frac{1}{273.16} - \frac{1}{T_d}\right)}

Everyday & HealthThermodynamicsEnvironment Canada's humidex: air temperature raised by the excess vapour pressure computed from the dew point.

Humidity Ratio from Vapour Pressure

W=0.62198 pvp−pvW = 0.62198\,\frac{p_v}{p - p_v}

HVAC & HydronicsThermodynamicsAir Quality & DispersionKilograms of water vapour per kilogram of dry air, from the vapour pressure and the barometric pressure — the one humidity variable that does not move when you heat the air.

Hvorslev Slug Test Conductivity

K=rc2ln⁡(L/R)2 L T0K = \frac{r_c^{2} \ln(L / R)}{2 \, L \, T_0}

Water & WastewaterHydraulic conductivity from a slug test by Hvorslev's method (USACE Waterways Experiment Station Bulletin 36, 1951). Drop a slug into the well, plot the head ratio on a log scale against time, and read the basic time lag T₀ where the ratio reaches 0.37 — that is 1/e, one time constant of an exponential recovery.

Hydraulic Detention Time

t=VQt = \frac{V}{Q}

Water & WastewaterWater TreatmentFluid MechanicsTheoretical detention time of a tank, clarifier or contact basin: the working volume divided by the flow passing through it.

Hydraulic Gradient

i=ΔhLi = \frac{\Delta h}{L}

Soil MechanicsFluid MechanicsHydraulic gradient as the loss of total head divided by the length of the flow path, the dimensionless driving force behind all seepage.

Hydraulic Jump Conjugate Depths

y2y1=12(1+8Fr12−1)\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^{2}} - 1\right)

Fluid MechanicsCivil & SurveyingWater & WastewaterBelanger's momentum solution for a hydraulic jump in a rectangular channel: the sequent (conjugate) depth downstream of the jump from the depth and Froude number upstream of it.

Hydraulic Power (P = ρgQh)

P=ρgQhP = \rho g Q h

Fluid MechanicsWater TreatmentPhysicsPower needed to lift a flow Q through a head h, with g = 9.80665 m/s².

Hydraulic Radius

R=APR = \frac{A}{P}

Fluid MechanicsCivil & SurveyingWater & WastewaterHydraulic radius of any conduit or channel: the flow area divided by the wetted perimeter, the single geometric number that Manning, Chezy and every friction equation actually consume.

Hydronic Heat Transfer (Water)

Q˙=ρwcwV˙ ΔT\dot{Q} = \rho_w c_w \dot{V} \, \Delta T

HVAC & HydronicsThermodynamicsFluid MechanicsHeat carried by a water loop from flow rate and supply-to-return ΔT — the SI form of the trade rule BTU/hr = 500 × GPM × ΔT.

Hydronic Static Fill Pressure

P=ρwgH+PmarginP = \rho_w g H + P_{margin}

HVAC & HydronicsFluid MechanicsCold fill pressure a closed loop needs to lift water to its highest point plus a safety margin, the SI form of the 2.31 ft per psi rule.

Hydrostatic Pressure (P = ρgh)

P=ρghP = \rho g h

MechanicsPhysicsWater TreatmentGauge pressure at depth h in a fluid of density ρ, using g = 9.80665 m/s².

Hypergeometric Probability

P(X=k)=(Kk)(N−Kn−k)(Nn)P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}

ProbabilityStatisticsChance of drawing exactly k successes in a sample of n taken without replacement from a population of N holding K successes.

Hypochlorite Product Mass from Available Chlorine

mprod=mCl×100%pm_{\text{prod}} = \frac{m_{\mathrm{Cl}} \times 100\%}{p}

Water TreatmentChemistryConverts a required mass of available chlorine into the mass of hypochlorite product to weigh out, given the product's percent available chlorine.