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Formula solvers — T

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t-Squared Fire Growth

Q˙=α t2\dot{Q} = \alpha \, t^{2}

Fire Protection & Fire DynamicsThe design fire curve every performance-based analysis starts from: heat release rate grows as the square of the time since ignition. One coefficient sets how fast, and the four named rates — slow, medium, fast, ultra-fast — are just four values of it.

Tafel Equation for Overpotential

η=a+blog⁡10i\eta = a + b \log_{10} i

Corrosion & ProtectionJulius Tafel's 1905 observation, still the backbone of electrode kinetics: the overpotential driving an electrode reaction rises with the logarithm of the current density, in a straight line whose slope is the Tafel slope.

Takt Time

Tt=TaDT_t = \frac{T_a}{D}

Industrial EngineeringThe pace the customer sets: available production time in a period divided by the units the customer wants in that period. One unit must leave the line every takt.

Talbot IDF Rainfall Intensity

i=at+bi = \frac{a}{t + b}

Water & WastewaterCivil & SurveyingFluid MechanicsTalbot's two-parameter IDF form, the three-parameter curve with the exponent fixed at one. Two regional constants instead of three, fitted to a local record for a single return period; a is stated here for intensity in mm/h against duration in minutes, and the duration you type is converted before the fit is applied.

Tank Loads to Cover a Field

N=A VTN = \frac{A \, V}{T}

Crop ProductionHow many tankfuls a field will take: the area times the application rate, divided by the tank volume. A fractional answer is the part-tank to mix last.

Target Heart Rate (Karvonen Method)

THR=HRrest+I (HRmax−HRrest)\mathrm{THR} = \mathrm{HR}_{rest} + I\,(\mathrm{HR}_{max} - \mathrm{HR}_{rest})

Everyday & HealthTraining heart rate at a chosen fraction of heart rate reserve, the range between resting and maximum rate.

Taylor Number (rotating-flow instability)

Ta=Ω2 r d3ν2\mathrm{Ta} = \frac{\Omega^{2} \, r \, d^{3}}{\nu^{2}}

Fluid MechanicsCentrifugal driving against viscous damping in the gap between rotating cylinders. Past a critical value the smooth circular flow gives way to stacked Taylor vortices — and it is why a journal bearing has a speed above which its oil film stops being simple laminar shear.

Taylor Tool Life Equation

V T n=CV \, T^{\,n} = C

Machining & TurningThe relation Frederick Taylor drew out of twenty-six years of cutting trials: cutting speed times tool life raised to a small exponent is a constant. It is why a modest increase in speed collapses tool life, and it is the arithmetic behind every decision about whether to run hard and change inserts often or run gently and leave them in.

TDS Estimated from Conductivity (TDS = k × EC)

TDS=k×EC\mathrm{TDS} = k \times \mathrm{EC}

Water TreatmentChemistryEstimates total dissolved solids in mg/L from a conductivity reading in µS/cm using an empirical factor k, typically 0.55 to 0.70 for natural waters.

Terminal Velocity

vt=2mgρACdv_t = \sqrt{\frac{2 m g}{\rho A C_d}}

MechanicsPhysicsSteady falling speed at which drag balances weight, from mass, air density, frontal area, and the drag coefficient.

Terzaghi Rock Load Height

Hp=k (B+Ht)H_p = k\,(B + H_t)

Rock MechanicsTerzaghi's 1946 estimate of how deep a block of loosened rock hangs above a tunnel: a load factor read from his rock condition table, multiplied by the sum of the tunnel's width and height. It was calibrated on drill-and-blast tunnels held up by steel sets on timber blocking, and it is deliberately conservative.

Terzaghi Support Pressure

pv=γ Hpp_v = \gamma\,H_p

Rock MechanicsThe weight of Terzaghi's loosened rock block, expressed as a pressure on the support: the rock load height multiplied by the unit weight of the rock. This is where a rock load height becomes a number a steel set or a lining can be designed against, and it is the second half of the 1946 method.

Terzaghi Ultimate Bearing Capacity (Strip Footing)

qu=c Nc+q Nq+12 γ B Nγq_u = c\,N_c + q\,N_q + \tfrac{1}{2}\,\gamma\,B\,N_\gamma

Soil MechanicsMechanicsTerzaghi's three-term ultimate bearing capacity of a shallow strip footing, summing the cohesion, surcharge and footing-width contributions.

The Airway Square Law

p=R Q2p = R \, Q^{2}

Rock MechanicsThe single most consequential fact in mine ventilation: pressure drop rises with the SQUARE of the airflow. Doubling the air through an opening quadruples the pressure the fan must raise, and because power is pressure times flow, it octuples the power bill. Plot it and you have the mine characteristic curve that a fan curve is read against.

Theoretical Minimum Number of Stations

Nmin⁡=∑tiCN_{\min} = \frac{\sum t_i}{C}

Industrial EngineeringFewest workstations a line could possibly need: total work content divided by the cycle time. Round the answer UP to the next whole station — the fraction cannot be staffed.

Theoretical Surface Roughness — Turning

Rt=f28 rεR_t = \frac{f^{2}}{8 \, r_\varepsilon}

Machining & TurningThe peak-to-valley height a round tool nose leaves between one feed mark and the next: feed squared over eight times the nose radius. Pure geometry — the best a perfect tool on a perfect machine could do, and a floor the real surface never quite reaches.

Thermal Conductivity of Snow (Sturm 1997)

keff=0.138−1.01 ρ∗+3.233 ρ∗2k_{\mathrm{eff}} = 0.138 - 1.01\,\rho^{*} + 3.233\,\rho^{*2}

Snow & IceHeat TransferThermodynamicsEffective thermal conductivity of seasonal snow from its density alone, using the quadratic fit Sturm and colleagues published in 1997 from 488 measurements. This is the equation behind the fact that a snow shelter works: at 300 kg/m³ snow conducts about the same heat as softwood, and the air trapped between the grains is doing almost all of it.

Thermal Damping Depth

d=αPπd = \sqrt{\frac{\alpha P}{\pi}}

Heat TransferThe depth at which a repeating temperature swing has faded to 1/e — about 37% — of its size at the surface. It is the natural ruler for every periodic heat problem: a wall thin compared with it barely delays anything, and a wall several times thicker than it has effectively swallowed the cycle.

Thermal Diffusivity

α=kρ c\alpha = \frac{k}{\rho \, c}

Heat TransferHow fast a temperature change travels through a material, as distinct from how much heat the material lets through. Conductivity divided by volumetric heat capacity: the numerator carries heat forward, the denominator soaks it up along the way. Every periodic and transient result on this page is built on it.

Thermal Efficiency

η=WQh\eta = \frac{W}{Q_h}

ThermodynamicsPhysicsFraction of heat input that a heat engine converts into useful work.

Thermal Linear Expansion

ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

ThermodynamicsPhysicsLength change of a solid caused by a temperature change, via the linear expansion coefficient.

Thermal Resistance of a Plane Wall

R=LkAR = \frac{L}{k A}

Heat TransferThermodynamicsConduction resistance of a flat slab in kelvin per watt, from its thickness, thermal conductivity and the area heat crosses.

Thermal Resistances in Series

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferThermodynamicsTotal resistance of a composite wall, where the same heat crosses each layer in turn so the layer resistances simply add.

Thermal Shock Resistance Parameter R

R=σf (1−ν)E αR = \frac{\sigma_f\,(1-\nu)}{E\,\alpha}

Ceramics & GlassKingery's 1955 ranking parameter for how much of a temperature drop a brittle material can take. It has the units of a temperature difference and it is tempting to read it as one — which is the mistake this page exists to prevent.

Thermal Shock Resistance Parameter R′

R′=k σf (1−ν)E αR' = \frac{k\,\sigma_f\,(1-\nu)}{E\,\alpha}

Ceramics & GlassKingery's R with the thermal conductivity multiplied in, for the far more common case of a merely brisk temperature change rather than an instantaneous quench. It has the units of power per unit length, and it is the parameter that finally explains why glass cracks in the sink and alumina does not.

Thermal Stress in a Restrained Member

σ=EαΔT\sigma = E \alpha \Delta T

Strength of MaterialsMechanicsPhysicsStress raised in a fully restrained member that is heated or cooled — the cause of rail sun kinks and cracked pipe anchors.

Thermal Time Lag

φ=x2Pπα\varphi = \frac{x}{2}\sqrt{\frac{P}{\pi \alpha}}

Heat TransferHow many hours late the peak of a repeating temperature swing arrives at depth x. This is the number behind every thick-wall building on earth: the afternoon heat that lands on the outside face does not reach the room until the evening, and by then the outside has cooled and takes it back.

Thermal Time Lag from Damping Depth

φ=xP2πd\varphi = \frac{x P}{2 \pi d}

Heat TransferThe same time lag, expressed through the damping depth instead of the diffusivity. In this form the structure of the answer is plain: the delay is the number of damping depths travelled, divided by 2π, times the period. One damping depth costs one radian of phase, and one radian of a 24-hour cycle is 3.82 hours.

Thevenin Resistance from an Open-Circuit and Loaded Measurement

RTh=RL(VOCVL−1)R_{Th} = R_{L} \left( \frac{V_{OC}}{V_{L}} - 1 \right)

Electricity & MagnetismElectrical TradeInternal (Thevenin) resistance of any source, found by measuring its open-circuit voltage and then the voltage it holds under a known load.

Thiem Steady-State Well Drawdown

s=Q2πTln⁡Rrs = \frac{Q}{2 \pi T} \ln\frac{R}{r}

Water & WastewaterCivil & SurveyingSteady-state drawdown around a well fully penetrating a confined aquifer, by the Dupuit-Thiem equilibrium solution. Transmissivity is taken in square metres per day and the pumping rate is converted to a daily volume inside the brain.

Thin Lens Equation

1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

OpticsPhysicsRelates a thin lens's focal length to its object and image distances.

Thin-Film Constructive Interference (Bright Reflection)

2nt=(m+12)λ2 n t = \left(m + \tfrac{1}{2}\right)\lambda

OpticsWaves & OscillationsPhysicsBright-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a half-odd number of wavelengths.

Thin-Film Destructive Interference (Dark Reflection)

2nt=mλ2 n t = m \lambda

OpticsWaves & OscillationsPhysicsDark-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a whole number of wavelengths.

Thomas Flashover Correlation

Q˙fo=7.8 AT+378 A0H0\dot{Q}_{fo} = 7.8 \, A_T + 378 \, A_0 \sqrt{H_0}

Fire Protection & Fire DynamicsPhilip Thomas's 1981 estimate of the heat release rate at which flashover becomes likely in a compartment: one term for the heat the boundaries absorb, one for the heat the opening carries away. It is a screening figure, and flashover is a transition rather than a threshold.

Thread Tensile Stress Area

At=π4(d−ktp)2A_{t} = \frac{\pi}{4} \left( d - k_{t} p \right)^{2}

Machine DesignThe effective cross-section a threaded fastener pulls on: a circle of diameter partway between the pitch and minor diameters, not the shank and not the root. The area every bolt strength figure is quoted against.

Three-Phase Apparent Power

S=3 VLILS = \sqrt{3} \, V_{L} I_{L}

Electrical TradeElectricity & MagnetismApparent power in volt-amperes for a balanced three-phase load — the quantity that sizes transformers, cables and breakers.

Three-Phase Motor Full-Load Current

I=Pout3 V PF ηI = \frac{P_{out}}{\sqrt{3} \, V \, \text{PF} \, \eta}

Electrical TradeElectricity & MagnetismLine current of a three-phase motor from its shaft output power, voltage, power factor and nameplate efficiency in percent.

Three-Phase Real Power

P=3 VLIL PFP = \sqrt{3} \, V_{L} I_{L} \, \text{PF}

Electrical TradeElectricity & MagnetismReal power drawn by a balanced three-phase load from its line-to-line voltage, line current, and power factor.

Thrust-to-Weight Ratio

TW=Fm g\frac{T}{W} = \frac{F}{m \, g}

Aerospace & FlightThrust divided by weight. Below 1 the vehicle cannot leave the pad at all; above 1, the excess is what accelerates it. This is the one place in rocketry where LOCAL gravity is the right number to use.

Time Dilation

Δt=Δt01−β2\Delta t = \frac{\Delta t_0}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsA moving clock's proper time Δt₀ stretches to Δt for a stationary observer; β = v/c.

Time Factor for Consolidation

Tv=cv tHdr2T_v = \frac{c_v\,t}{H_{dr}^{2}}

Soil MechanicsFluid MechanicsDimensionless time factor of Terzaghi consolidation theory, with the coefficient of consolidation entered in m²/s and the longest drainage path.

Time Factor from Degree of Consolidation (U ≤ 60%)

Tv=π4(U100)2T_v = \frac{\pi}{4}\left(\frac{U}{100}\right)^{2}

Soil MechanicsFluid MechanicsTerzaghi's parabolic approximation relating the time factor to the average degree of consolidation, valid for U of 60 percent or less.

Time to Cover a Field

t=ACt = \frac{A}{C}

Farm OperationsThe hours a field takes at a given effective field capacity. The calculation behind every decision about whether the crop gets off before the weather turns.

Tip Offset to Spin Ratio

Rω0v0=5b2R\frac{R\omega_0}{v_0} = \frac{5b}{2R}

Billiards & Cue SportsMechanicsHow much spin a given tip offset puts on the cue ball. Strike the ball a distance b off centre and the same impulse that gives it speed also gives it angular momentum about its centre — and because a solid sphere has I = ⅖mR², the ratio of surface spin speed to travel speed comes out as 5b/2R, independent of how hard you hit it.

Tip-Speed Ratio

λ=ωRv\lambda = \frac{\omega R}{v}

Solar & Wind PowerHow fast the blade tips travel compared with the wind coming at them. The single number that decides whether a rotor is running at its best power coefficient, and the reason large turbines turn slowly while small ones spin fast.

Titration: Concentration of an Unknown

Ca=n CbVbVaC_a = \frac{n\,C_b V_b}{V_a}

ChemistryFinds an analyte's concentration from the titre volume, titrant concentration, and the reaction's mole ratio at the equivalence point.

Tons of Refrigeration from BTU/hr

T=Q˙12,000 BTU/hrT = \frac{\dot{Q}}{12{,}000\ \text{BTU/hr}}

HVAC & HydronicsThermodynamicsConverts a cooling load in BTU/hr (or kW) to tons of refrigeration, where one ton is 12,000 BTU/hr or 3.5169 kW.

Torque

τ=rFsin⁡θ\tau = r F \sin\theta

MechanicsPhysicsTurning effect of a force applied at distance r from a pivot, at angle θ to the lever arm.

Torque with a Lever Arm (τ = rF sin θ)

τ=rFsin⁡θ\tau = r F \sin\theta

MechanicsPhysicsTorque produced by a force applied at distance r from the pivot, at angle θ to the lever.

Torricelli's Law (v = √(2gh))

v=2ghv = \sqrt{2 g h}

Fluid MechanicsWater TreatmentPhysicsSpeed of fluid jetting from an opening a depth h below the free surface, with g = 9.80665 m/s².

Torsional Shear Stress (τ = Tr/J)

τ=TrJ\tau = \frac{T r}{J}

Strength of MaterialsMechanicsPhysicsTorsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J entered in m⁴ as a plain number.

Torus Surface Area

A=4π2RrA = 4\pi^2 R r

GeometrySurface area of a ring torus: the tube's circumference swept once around the ring, by Pappus's centroid theorem.

Torus Volume

V=2π2Rr2V = 2\pi^2 R r^2

GeometryVolume of a ring torus from the centre-to-tube radius R and the tube radius r, by Pappus's centroid theorem.

Total Absorption (Sabins)

A=S1α1+S2α2+S3α3A = S_1\alpha_1 + S_2\alpha_2 + S_3\alpha_3

Acoustics & NoiseAdd up the room: every surface contributes its area multiplied by how much of the sound striking it never comes back. The total, in metric sabins, is the A that Sabine's reverberation equation divides into the volume.

Total Alkalinity as CaCO₃ from Species

TA=0.8202 HCO3+1.6679 CO3+2.9425 OH\mathrm{TA} = 0.8202\,\mathrm{HCO_3} + 1.6679\,\mathrm{CO_3} + 2.9425\,\mathrm{OH}

Water TreatmentChemistrySums bicarbonate, carbonate and hydroxide reported as their own ions into one total alkalinity expressed as milligrams per litre of CaCO₃.

Total Chip Power: Dynamic plus Leakage

Ptot=αCV2f+VIleakP_{tot} = \alpha C V^2 f + V I_{leak}

Semiconductors & ChipsWhat a chip actually burns: the switching term plus the current that flows through transistors that are supposed to be off. Below about 65 nm the second term stopped being a rounding error and became a design constraint.

Total Delta-v Across Stages

Δvtot=Δv1+Δv2+Δv3\Delta v_{tot} = \Delta v_1 + \Delta v_2 + \Delta v_3

Aerospace & FlightDelta-v simply adds across stages, while the mass ratios that produce it multiply. That mismatch between a sum and a product is the entire argument for staging, and it is worth seeing written down.

Total Digestible Nutrients Supplied

TDN=M fTDN = M \, f

Livestock & FeedThe digestible energy a feed delivers, as the mass of total digestible nutrients in it. TDN is quoted as a percentage of the DRY MATTER, so the feed weight has to be on a dry matter basis before this multiplication means anything.

Total Dynamic Head

TDH=hs+hf+hvTDH = h_s + h_f + h_v

HVAC & HydronicsFluid MechanicsWater TreatmentThe head a pump must develop: static lift plus friction losses plus velocity head, all expressed in feet or metres of the pumped liquid.

Total Hardness as CaCO₃

TH=2.497 Ca+4.118 Mg\mathrm{TH} = 2.497\,\mathrm{Ca} + 4.118\,\mathrm{Mg}

Water TreatmentChemistryConverts a lab report's calcium and magnesium ion results into a single total hardness expressed as milligrams per litre of CaCO₃.

Total Harmonic Distortion

THD=IHI1\text{THD} = \frac{I_{H}}{I_{1}}

Electrical TradeRatio of the harmonic content of a distorted wave to its fundamental — the single number that says how far a current or voltage is from a sine.

Total Interest Paid Over a Loan

I=M n−PI = M\,n - P

Money & BusinessEverything a loan costs beyond the amount borrowed: total of all payments minus the principal.

Total R-Value of an Assembly

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferHVAC & HydronicsThermodynamicsTotal R-value of a wall, roof or floor built up from three layers in series, where the same heat crosses each layer so the R-values simply add.

Total Vertical Stress (σ = γz)

σv=γz\sigma_v = \gamma z

Soil MechanicsMechanicsTotal vertical stress at depth in a uniform soil layer, the weight of the overburden column standing on one unit of area.

Total Water Treatment Program Cost

C=Cw+Cp+CeC = C_w + C_p + C_e

Water TreatmentChemistryFluid MechanicsTotal operating cost of a treated cooling system: the water and sewer bill, the chemical invoice and the energy bill added together.

Totient of a Prime Power

φ(pk)=pk−1(p−1)\varphi(p^k) = p^{k-1}(p - 1)

Computer ScienceEuler's totient when the number is a single prime raised to a power. Of the p^k numbers below it, only the multiples of p share a factor, and there are p^(k−1) of those.

Trace of a 2×2 Matrix

tr⁡(M)=a11+a22\operatorname{tr}(M) = a_{11} + a_{22}

Vectors & MatricesAlgebraAdds the two entries on the main diagonal of a square matrix, a quantity unchanged by any change of basis.

Tractor Wheel Slip

s=d0−dLd0s = \frac{d_0 - d_L}{d_0}

Farm OperationsWheel slip measured the way it is measured in the field: mark a wheel, count the same number of revolutions loaded and unloaded, and compare the distances travelled. The check that decides ballast.

Transfer Units for Dilute Absorption

NOG=ln⁡[y1y2(1−1A)+1A]1−1AN_{OG} = \frac{\ln \left[ \dfrac{y_1}{y_2} \left( 1 - \dfrac{1}{A} \right) + \dfrac{1}{A} \right]}{1 - \dfrac{1}{A}}

Chemical EngineeringThe number of transfer units a dilute absorber demands — the packed-column counterpart of a stage count. Colburn's 1939 integration of the design equation for a straight equilibrium line, and the number that multiplies HTU to give a height.

Transformer Full-Load Current

IFL=Sk VI_{FL} = \frac{S}{k \, V}

Electrical TradeRated current on either side of a transformer from its kVA and that side's voltage, with one factor selecting single-phase or three-phase.

Transformer Percent-Impedance Voltage Drop

Vd=%Z100⋅SLSR⋅VRV_{d} = \frac{\%Z}{100} \cdot \frac{S_{L}}{S_{R}} \cdot V_{R}

Electrical TradeElectricity & MagnetismVolts lost inside a transformer's own windings at a given loading, taken straight from its nameplate percent impedance.

Transformer Voltage Ratio

VsVp=NsNp\frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}}

Electricity & MagnetismPhysicsRelates the primary and secondary voltages of an ideal transformer to its turns ratio.

Transmissivity from Conductivity and Thickness

T=K bT = K \, b

Water & WastewaterCivil & SurveyingTransmissivity of a confined aquifer as the hydraulic conductivity multiplied by the saturated thickness, returned in square metres per day for a conductivity entered in any speed unit.

Transverse Shear Stress (τ = VQ/Ib)

τ=VQIb\tau = \frac{V Q}{I b}

Strength of MaterialsCivil & SurveyingMechanicsShear stress at any height in a beam cross-section, peaking at the neutral axis. For a rectangle it works out to exactly 1.5 times the average V/A.

Trapezoid Area

A=a+b2⋅hA = \frac{a + b}{2} \cdot h

GeometryArea of a trapezoid as the average of its two parallel sides times the perpendicular height.

Traverse Closure Error

Ec=(ΣLat)2+(ΣDep)2E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}

Civil & SurveyingGeometryLinear misclosure of a closed traverse, combining the residual sums of the latitudes and departures as a right triangle.

Traverse Precision Ratio

N=PEcN = \frac{P}{E_c}

Civil & SurveyingExpresses traverse accuracy as one part in N, dividing the total perimeter walked by the linear closure error.

Tree Basal Area from DBH

BA=πD24BA = \frac{\pi D^{2}}{4}

Forestry & Forest EcologyThe cross-sectional area of a stem at breast height, from its diameter. Nothing more than the area of a circle, given a job: it is the currency forestry counts stocking, competition and growth in, because it is the one dimension of a standing tree you can measure accurately with a tape.

Tree Biomass from an Allometric Equation

B=a D bB = a \, D^{\,b}

Forestry & Forest EcologyThe power law every forest carbon inventory rests on: oven-dry biomass as a fitted coefficient times diameter at breast height raised to a fitted exponent. There is no physics in it. It is a regression through trees somebody cut down and weighed, and everything difficult about it follows from that.

Triangle Area (Two Sides and Included Angle)

A=12 absin⁡CA = \tfrac{1}{2}\,ab\sin C

TrigonometryGeometryComputes a triangle's area from two sides and the angle between them, with no height needed.

Triangle Area from Two Vectors

A=12∣axby−aybx∣A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the triangle formed by two plane vectors from a shared vertex, using only their components — the shoelace formula.

Triangle Perimeter

P=a+b+cP = a + b + c

GeometryPerimeter of a triangle as the sum of its three sides.

Triangular Prism Volume

V=12bhtLV = \tfrac{1}{2} b h_t L

GeometryVolume of a triangular prism — the triangular end area times the prism length, from the triangle's base and height.

Tributary Area Pillar Stress

σp=σv(Wp+B)2Wp2\sigma_p = \sigma_v \frac{(W_p + B)^{2}}{W_p^{2}}

Rock MechanicsThe load a room-and-pillar layout puts on each pillar: every pillar carries the full weight of the rock over its own area plus the rock over the openings around it. It is a statics argument with no rock mechanics in it at all, which is both why it is trustworthy and why it is crude.

Trickling Filter Hydraulic Loading

Lh=Q+QrAL_h = \frac{Q + Q_r}{A}

Water & WastewaterWater TreatmentFluid MechanicsHydraulic loading on a trickling filter including recirculation — total flow per unit of media surface area, in metres per day.

Trickling Filter Recirculation Factor

F=1+R(1+0.1R)2F = \frac{1 + R}{(1 + 0.1R)^2}

Water & WastewaterWater TreatmentNRC recirculation factor for trickling filter design, converting a recirculation ratio into the effective number of passes through the media.

Truck Loads from Bank Volume and Swell

N=VB(1+S100)CN = \frac{V_B\left(1 + \frac{S}{100}\right)}{C}

Trades & ConstructionTurns an in-place excavation quantity into truck loads: bank volume swelled to its loose volume, divided by the capacity of one truck box.

True Stress from Engineering Stress

σt=σe (1+e),εt=ln⁡(1+e)\sigma_t = \sigma_e \, (1 + e), \qquad \varepsilon_t = \ln(1 + e)

Metallurgy & Heat TreatmentA tensile machine divides load by the ORIGINAL area, because that is the only area it knows. True stress divides by the area the bar actually has at that instant, which is smaller — so true stress is always the larger number, and the gap grows with every percent of strain. Every plasticity relation on this site wants the true values.

Truncated Hash Strength

scoll=b2spre=bs_{\mathrm{coll}} = \frac{b}{2} \qquad s_{\mathrm{pre}} = b

Cryptography & Key SizingSecurity a hash retains when its output is truncated to b bits, following NIST SP 800-107 Rev. 1: collision resistance falls to b/2 bits while preimage resistance stays at the full b. Truncation is permitted and normal — take the leftmost bits — but the two resistances shrink at different rates, and confusing them is how a digest ends up too short.

Tsai–Hill Failure Index for a Lamina

FI=(σ1X)2−σ1σ2X2+(σ2Y)2+(τ12S)2FI = \left( \frac{\sigma_1}{X} \right)^{2} - \frac{\sigma_1 \sigma_2}{X^{2}} + \left( \frac{\sigma_2}{Y} \right)^{2} + \left( \frac{\tau_{12}}{S} \right)^{2}

Strength of MaterialsHill's anisotropic yield condition reduced to plane stress, which is the standard way to ask whether a single ply survives a combined stress state. It is interactive: the longitudinal, transverse and shear stresses are not checked one at a time but together, so a ply can fail with every individual stress comfortably below its own allowable. What it gives back is an index, not a margin, and it will not tell you how the ply failed.

Tsiolkovsky Rocket Equation

Δv=veln⁡ ⁣(m0mf)\Delta v = v_e \ln\!\left(\frac{m_0}{m_f}\right)

Aerospace & FlightThe velocity a rocket gains is its exhaust velocity times the natural logarithm of the ratio of its starting mass to its burnout mass. Derived by Konstantin Tsiolkovsky in 1903, and still the equation that decides whether a mission is possible.

Turbine Isentropic Efficiency

ηisen=h1−h2h1−h2s\eta_{isen} = \frac{h_1 - h_2}{h_1 - h_{2s}}

ThermodynamicsIsentropic efficiency of a steam turbine: the actual enthalpy drop divided by the ideal drop a frictionless expansion to the same exhaust pressure would give.

Turnover Number k_cat and the Specificity Constant

kcat=Vmax[E]tk_{cat} = \frac{V_{max}}{[E]_t}

Chemical EngineeringV_max tells you how fast a particular tube of enzyme goes; k_cat tells you how good the enzyme is. Divide the maximum velocity by the enzyme you actually put in and you get molecules of substrate turned over per molecule of enzyme per second — the catalytic constant, and the only measure of an enzyme's speed that survives being scaled, diluted or purified.

Twist Factor — Metric and Tex Systems

αm=TNm\alpha_m = \frac{T}{\sqrt{N_m}}

Textiles & FabricThe same square-root law as the English twist multiplier, expressed in turns per metre over the square root of the metric count. Typical values run near 60 for a soft knitting yarn, 100 for weft, 120 to 140 for warp — larger numbers than TM only because the length unit is a metre rather than an inch.

Twist Multiplier — English System

TM=TNeTM = \frac{T}{\sqrt{N_e}}

Textiles & FabricThe number a spinner actually specifies. Turns per inch alone says nothing useful, because a fine yarn needs far more turns than a coarse one to reach the same helix angle at its surface. Dividing by the square root of the English cotton count removes the fineness and leaves a figure that describes the CHARACTER of the yarn: about 3 for a soft hosiery yarn, 4 for weft, 4.5 to 5 for warp, 6 and up for a hard voile.

Two Airways in Parallel

1Req=1R1+1R2\frac{1}{\sqrt{R_{eq}}} = \frac{1}{\sqrt{R_1}} + \frac{1}{\sqrt{R_2}}

Rock MechanicsWhen the air can choose between two openings it takes both, dividing itself so that the pressure drop across each is the same. The equivalent resistance that results is far below either branch — and unlike electrical resistors, the combination goes as the reciprocal SQUARE ROOT, because the airway law is quadratic rather than linear.

Two Airways in Series

Req=R1+R2R_{eq} = R_1 + R_2

Rock MechanicsAir that must pass through one opening and then the next carries the same quantity through both, so the pressure drops add and the resistances add with them. It is the simplest relation in ventilation network analysis and the one that makes the pointlessness of enlarging the wrong airway obvious.

Two Capacitors in Parallel

Ct=C1+C2C_{t} = C_{1} + C_{2}

Electricity & MagnetismPhysicsCombines two parallel capacitors by simply adding their capacitances.

Two Capacitors in Series

Ct=C1C2C1+C2C_{t} = \frac{C_{1} C_{2}}{C_{1} + C_{2}}

Electricity & MagnetismPhysicsCombines two series capacitors into a total that is smaller than either one.

Two Resistors in Parallel

Rt=R1R2R1+R2R_{t} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}

Electricity & MagnetismPhysicsTotal resistance of two resistors connected side by side: product over sum, always less than either branch.

Two Resistors in Series

Rt=R1+R2R_{t} = R_{1} + R_{2}

Electricity & MagnetismPhysicsTotal resistance of two resistors connected end to end is simply their sum.

Two-Film Overall Mass Transfer Coefficient

1KOG=1kG+mkL\frac{1}{K_{OG}} = \frac{1}{k_G} + \frac{m}{k_L}

Chemical EngineeringTwo films in series, and the resistances add. Lewis and Whitman's 1924 picture of a gas film and a liquid film meeting at an interface in equilibrium is still the working model for every absorber and stripper built.

Two-Sample Z-Test Statistic

z=xˉ1−xˉ2σ12n1+σ22n2z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{\sigma_1^{2}}{n_1} + \dfrac{\sigma_2^{2}}{n_2}}}

StatisticsAlgebraCompares two independent sample means when both population standard deviations are known, scaled by the combined standard error.

Tyre Contact Area

A=PpA = \frac{P}{p}

Axle Loads & PavementThe area of the patch a loaded tyre presses onto the pavement, taken as the wheel load divided by the contact pressure. Contact pressure is NOT inflation pressure: van Vuuren measured it as much as 30% lower, and the textbook rule of adding ten per cent was found completely unacceptable.