🧮Calculators

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DVA — Debit Valuation Adjustment

Computes the DVA (Debit Valuation Adjustment), the mirror image of CVA seen from the other side. While CVA discounts value for the risk of the counterparty failing, DVA recognizes the counterintuitive benefit of the institution's own default risk: if it might not pay its obligations, that effectively reduces the liability. It's the piece that makes derivative pricing symmetric between the two parties. The calculation follows the same structure as CVA, but uses the expected negative exposure and the own default probability. Enter the exposure and credit data.

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Up-and-In Barrier Option (Call)

Computes the price of an up-and-in barrier call option. It's a knock-in barrier option: it only comes to life if the asset price touches a barrier above the current level before expiry; if it never touches, it expires worthless, however deep in the money it might have been. Because of that extra condition, it costs less than a plain call. The pricing uses the Reiner-Rubinstein closed-form formulas. Enter price, strike, barrier, rate, dividend, volatility and term.

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Down-and-Out Barrier Option (Put)

Computes the price of a down-and-out barrier put option. It's a knock-out barrier option: it works like a normal put, but is cancelled the moment the asset price touches a barrier below the current level. Because it disappears precisely when the put would be gaining the most value, it's usually worth much less than a plain put, making it cheaper for those betting on moderate declines. The pricing uses the Reiner-Rubinstein formulas. Enter price, strike, barrier, rate, dividend, volatility and term.

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Option Price (Heston Model)

Prices a European call under the Heston (1993) model, where the asset volatility is itself stochastic and mean-reverting — reproducing the volatility smile and skew that constant-vol Black-Scholes misses. The price comes out in semi-closed form by numerically integrating the characteristic function (Little Heston Trap formulation, Gauss-Legendre quadrature). Enter spot, strike, maturity, rate, dividend and the five model parameters: initial variance, kappa, theta, vol of vol and correlation.

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SABR Implied Volatility (Hagan)

Computes the Black/lognormal implied volatility from Hagan's (2002) closed-form SABR approximation, used on rates and FX desks to interpolate the volatility smile and quote out-of-the-money options consistently. The resulting vol feeds straight into Black-76. Enter the forward, strike, maturity and the four calibrated parameters: alpha (initial vol), beta (CEV elasticity), rho (correlation) and nu (vol of vol).

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Jump-Diffusion Option (Merton)

Prices a European call under Merton's (1976) jump-diffusion model: on top of continuous Brownian motion, the price can take lognormal jumps arriving as a Poisson process, capturing fat tails and price gaps. The premium is a Poisson-weighted sum of Black-Scholes prices. Enter spot, strike, maturity, rate, diffusion vol, the jump intensity and the mean and standard deviation of the log jump.

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CDS Par Spread (Reduced-Form)

Computes the par (breakeven) spread of a Credit Default Swap in the reduced-form model with a constant hazard rate, setting the present value of the protection leg — which pays 1−R on default — equal to that of the premium leg. It shows the credit-triangle relationship s ≈ (1−R)·λ in practice. Enter the hazard rate, recovery rate, maturity, payment frequency and risk-free rate; the result is in basis points.

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Z-spread (Zero-Volatility Spread)

Computes a bond's Z-spread: the constant spread added to the entire zero (spot) rate curve so the present value of its cashflows equals the market price. Unlike the nominal spread, which uses a single point, it accounts for the whole shape of the curve; for an option-free bond the Z-spread equals the OAS. Enter the cashflow times and amounts, the zero rate at each node and the price; the result is in basis points.

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Bootstrap the Zero Curve

Builds the zero (spot) rate curve from par rates using sequential bootstrapping: at each maturity it uses the par-bond identity to strip out the discount factor and converts it to the annual zero rate. This is the step that turns observed market rates into the discount curve used to price any cashflow. Enter the list of annual par rates; the output is the zero rate at each maturity.

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Standard Resistor Value (E6 to E192 Series)

Find the resistor value that actually exists in the E6, E12, E24, E48, E96 or E192 series, the error against your calculated value, and the matching colour code.

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Group Expense Settle-Up

Everyone paid for part of the trip or the dinner. Enter the expenses and the page works out who owes what and the transfers that settle the whole thing.

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Brazil Import Tax Calculator (Remessa Conforme)

Work out the import duty and ICMS a parcel pays on arrival in Brazil under the Remessa Conforme rules, before you check out on an overseas store.

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Whittaker Beta Diversity

Calculates Whittaker's beta diversity (βw), the total species richness of the sample set divided by the mean number of species per sample, minus one. It measures species turnover between samples: zero means identical composition, and high values indicate communities that differ sharply. Enter the total richness and the per-sample mean.

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Hedström Number

Calculates the Hedström number (He), a dimensionless group combining density, yield stress, pipe diameter and plastic viscosity of a Bingham fluid. Used together with the Bingham Reynolds number, it locates the laminar-to-turbulent transition for drilling fluids, mineral slurries and cement pastes. Enter the four quantities.

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Leverett J-Function

Calculates the Leverett J-function, which normalises capillary pressure by interfacial tension, contact angle and the square root of the permeability-to-porosity ratio. It collapses capillary pressure curves from samples with different properties onto a single rock-type curve. Enter the capillary pressure, interfacial tension, contact angle, permeability in millidarcy and porosity.

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White Liquor Sulfidity (Kraft)

Calculates the sulfidity of white liquor in a kraft pulp mill: sodium sulfide divided by active alkali, which is sulfide plus sodium hydroxide, as a percentage. Both terms have to be expressed on the same basis — mixing Na₂O with NaOH is the classic mistake here. Enter the two concentrations.

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Mean Fragment Size (Kuz-Ram)

Estimates the mean fragment size X₅₀ of a blast with the Kuz-Ram model, from the rock factor, the volume broken per hole, the explosive mass per hole and the relative weight strength of the explosive. It is the screen size half the muckpile passes, the number that decides whether crushing will struggle. Enter the four blast design parameters.

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Metacentric Height (GM)

Calculates the metacentric height GM of a vessel by adding the centre of buoyancy height to the metacentric radius (waterplane moment of inertia divided by displaced volume) and subtracting the centre of gravity height. A positive GM means stable equilibrium; negative means a heeled hull will not right itself. Enter the moment of inertia, displaced volume, KB and KG.

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Gross Tonnage (GT — IMO 1969)

Calculates a vessel's gross tonnage GT with the formula from the 1969 International Convention on Tonnage Measurement of Ships: the moulded volume of all enclosed spaces multiplied by a coefficient that grows with the logarithm of that volume. GT is dimensionless, not a mass, and it drives port dues, crewing requirements and regulatory bracket. Enter the moulded volume.

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Breguet Range (Jet Aircraft)

Calculates a jet aircraft's cruise range with the Breguet equation: speed divided by thrust specific fuel consumption, times the aerodynamic efficiency, times the natural logarithm of the ratio between weight at the start and at the end of cruise. Valid for cruise with V, specific fuel consumption and L/D held constant — in practice the cruise-climb, at fixed Mach and lift coefficient, or step-climb flight. Enter the speed, TSFC, L/D and both weights.

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Temperature-Humidity Index (THI) for Cattle

Calculates the temperature-humidity index (THI) used to gauge heat stress in cattle, in the metric form of the 1971 NRC formula, from air temperature and relative humidity. Below 72 the animal is comfortable; 72 to 78 is alert, 79 to 88 is danger and above 88 is emergency. Enter temperature and relative humidity.

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Aggregate Fineness Modulus

Calculates the fineness modulus of an aggregate by adding the cumulative percentages retained on the eight standard sieves and dividing by one hundred. Note that the input is the CUMULATIVE retained percentage, not the passing nor the individual retained — that is where the calculation usually goes wrong. Concrete sand normally lands between 2.2 and 3.1. Enter the eight percentages.

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CBR — California Bearing Ratio

Calculates a soil's CBR by comparing the pressure measured in the penetration test against the standard crushed stone: 6.9 MPa at 2.54 mm and 10.3 MPa at 5.08 mm. By the standard the HIGHER of the two governs, not just the 2.54 mm one — the trap that shows up most often in subgrade reports. Enter both measured pressures.

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Transformer K-Factor (Harmonics)

Calculates a transformer's K-factor per ANSI/IEEE C57.110: the sum of squared harmonic currents weighted by the squared harmonic order, divided by the sum of squared currents. A purely sinusoidal load gives K equal to one; you pick the commercial class immediately above the computed value. Enter the fundamental and the odd harmonics up to the 13th.

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Karlovitz Number

Computes the Karlovitz number of a turbulent premixed flame, Ka = (u'/S_L)^1.5 · (ℓ_t/δ_L)^−0.5, comparing the chemical time of the flame front with the turnover time of the smallest eddies (Kolmogorov scale). Ka below 1 means the laminar flame structure survives the turbulence (wrinkled and corrugated regimes); between 1 and 100 eddies penetrate the preheat zone and thicken the flame; above 100 the reaction zone itself is broken. Together with the Damköhler number it forms the axes of the Borghi diagram, used to pick combustion models in CFD. Peters' form is adopted; integral scale and flame thickness must share the same unit. Enter the velocity fluctuation, the laminar flame speed, the integral length scale and the laminar flame thickness.

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Zeldovich Number

Computes the Zeldovich number of a flame, β = E_a·(T_b − T_u) ÷ (R·T_b²), the activation energy made dimensionless by the temperature rise across the flame front. It measures how sensitive the reaction rate is to a small temperature change: a high β (typically 8 to 12 for hydrocarbons) means the reaction is concentrated in a very thin layer near the flame temperature, which justifies the large-activation-energy assumption of asymptotic flame theory and the extinction and cellular-instability criteria. The universal gas constant R = 8.314 J/(mol·K) is adopted, with activation energy in J/mol and temperatures in kelvin. Enter the activation energy, the burned gas temperature and the unburned gas temperature.

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Elenbaas Number

Computes the Elenbaas number of a channel formed by two heated vertical parallel plates, El = g·β·ΔT·b⁴ ÷ (ν·α·H), which is the Rayleigh number based on the plate spacing b multiplied by the ratio b/H. It governs natural convection in finned heat sinks, electronics enclosures and solar collectors: a very small El means a narrow, tall channel where the boundary layers merge and choke the flow, while a large El means the plates behave as isolated. The spacing that maximises a heat sink's total dissipation falls near El ≈ 46 for isothermal plates, a classic natural-convection fin design criterion. Gravity is taken as 9.80665 m/s², β is the fluid thermal expansion coefficient and H the plate height. Enter the expansion coefficient, the temperature difference, the spacing, the kinematic viscosity, the thermal diffusivity and the plate height.

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Clay Activity (Skempton)

Computes clay activity as defined by Skempton, A = PI ÷ (% of particles finer than 2 μm), the ratio of the soil's plasticity index to the truly clay-sized fraction. It separates the clay mineral's effect from the mere amount of fines: two soils with the same PI behave very differently if one owes its plasticity to a little highly active clay and the other to a lot of inert clay. The usual classification is A < 0.75 inactive (kaolinite), 0.75 to 1.25 normal (illite) and A > 1.25 active (montmorillonite), the range where the expansive soils that warp pavements and shallow foundations are found. Enter the plasticity index and the clay fraction.

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Seed Cultural Value

Computes the cultural value of a seed lot, CV = purity × germination ÷ 100, with both percentages taken from the laboratory analysis report. The result is the percentage of the lot's weight that is pure, live seed — what will actually become a plant: a lot with 98.5% purity and 92% germination delivers 90.6% useful seed, and the remaining 9.4% is inert matter and dead seed you are paying for. It is the basis for correcting the seeding rate (target kg/ha ÷ CV × 100) and for comparing prices between lots of different quality; the English equivalent is pure live seed (PLS). Enter the physical purity and the germination percentage.

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Crop Growth Rate (CGR)

Computes the crop growth rate, CGR = (W₂ − W₁) ÷ (Δt × A), the canopy's dry-matter gain per unit of ground area per day between two destructive samplings. Unlike relative growth rate, which measures efficiency per gram of existing plant, CGR measures the productivity of the LAND — it is what you compare across row spacings, seeding densities and fertiliser levels, because it answers how much biomass each square metre of field produces per day. Peak values in well-managed C4 crops fall around 20 to 30 g/(m²·day), and the integral of the CGR curve over the season is total biological yield. Enter the initial and final dry masses, the interval between samplings and the ground area sampled.

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Pulp Dilution Water

Computes how much water must be added to take a pulp from one mass percent solids to a lower one, Water = M × (C₁/C₂ − 1), where M is the incoming pulp mass (or mass flow). It follows from the mass balance: the solids mass does not change on dilution, so the final pulp mass is M·C₁/C₂ and the difference is water. This is the most routine operation in a mineral processing plant — grinding, desliming, flotation and thickening each demand their own percent-solids range, and getting the dilution water wrong throws off residence time, viscosity and reagent consumption. Both percentages are by mass (weight of solids per weight of pulp), and the result comes out in the same unit entered for M. Enter the pulp mass or flow, the current percent solids and the target percent solids.

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Arc Flash Incident Energy (Lee Method)

Computes arc flash incident energy by the Ralph Lee method, E = 5.12×10⁵ × V × I_bf × t ÷ d², with voltage in kV, bolted fault current in kA, fault clearing time in seconds and working distance in millimetres. Lee's method models an open-air arc as an ideal radiant heat source, ignoring the energy an enclosure reflects back; IEEE 1584 therefore keeps it only as the legacy model, recommended for open-air arcs and for voltages above 15 kV where the empirical equations do not apply. The result in cal/cm² sets the PPE category: 1.2 cal/cm² is the second-degree burn threshold and the value that bounds the arc flash boundary. The cal/cm² form is adopted (the J/cm² variant uses 2.142×10⁶ and is 4.184 times larger). Enter the voltage, the fault current, the clearing time and the working distance.

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TDD — Total Demand Distortion (IEEE 519)

Converts total harmonic current distortion into total demand distortion, TDD = THD_I × I₁ ÷ I_L, where I₁ is the fundamental current at the moment of measurement and I_L the installation's maximum demand current. The difference matters a lot: THD has the instantaneous fundamental in its denominator, so a drive running at 20% load can read 60% THD while injecting negligible harmonic current into the grid. IEEE 519 sets its limits in TDD for exactly this reason, referencing everything to maximum demand — typically 5% TDD for installations with an I_sc/I_L ratio below 20, which is the criterion the utility enforces. Enter the measured current THD, the fundamental current at measurement and the maximum demand current.

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Number of Stages by the Gilliland Correlation

Estimates the number of theoretical stages of a distillation column with the Gilliland correlation, which links excess reflux to excess stages: with X = (R − R_min)/(R + 1) and Y = (N − N_min)/(N + 1), you get N = (Y + N_min)/(1 − Y). It is the third step of the FUG shortcut method, after N_min from the Fenske equation and R_min from Underwood, and it settles in one line the preliminary sizing that would otherwise need a McCabe-Thiele diagram or a simulator. Eduljee's analytical fit is adopted, Y = 0.75·(1 − X^0.5668), the usual form for hand calculation; the Molokanov correlation is more accurate at the extremes and gives a result a few percent different. Enter the operating reflux ratio, the minimum reflux ratio and the minimum number of stages.

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Shale Volume from Gamma Ray (Larionov)

Estimates the shale volume of a formation from the gamma ray log, the first step in any petrophysical well evaluation. The calculation normalizes the zone reading between the cleanest sand and the most radioactive shale in the interval, giving the gamma ray index IGR = (GR − GR min) ÷ (GR max − GR min), and then applies the non-linear Larionov curve, Vsh = 0.083 × (2^(3.7 × IGR) − 1). The result is the fraction of rock volume occupied by clay, as a percentage: intervals above 30 to 40% are usually discarded as reservoir, and the value later feeds the porosity and water saturation corrections for shaly sands. The Larionov curve for Tertiary, poorly consolidated rocks was adopted, which is the usual one in young sedimentary basins; for Mesozoic or older rocks the literature uses Vsh = 0.33 × (2^(2 × IGR) − 1), which returns far larger volumes for the same IGR. Enter the zone gamma ray reading, the minimum reading and the maximum reading of the interval.

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Absolute Permeability by the Timur Correlation

Estimates the absolute permeability of a sandstone from porosity and irreducible water saturation, when no core sample is available for laboratory testing. The Timur correlation (1968), fitted on 155 Alaskan sandstone samples, is k = 0.136 × porosity^4.4 ÷ irreducible saturation², with permeability in millidarcy. The result tells how well the reservoir can flow: below 1 mD the formation is considered tight, between 10 and 100 mD it is moderate and above 500 mD it is excellent. The convention that causes most errors here is the unit: the constants 0.136 and 4.4 were fitted with porosity and saturation in percent, not as fractions — entering 0.22 instead of 22 drops the result by several orders of magnitude, which is why both fields ask for percent. Enter the effective porosity and the irreducible water saturation.

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Ship Roll Period (IMO IS Code)

Computes the natural roll period of a ship using the empirical formula of the IMO International Code on Intact Stability (IS Code 2008), used in the weather criterion and in the inclining experiment. The period is T = 2 × C × beam ÷ square root of GM, with the coefficient C = 0.373 + 0.023 × (beam ÷ draught) − 0.043 × (length ÷ 100), so that rolling gets faster as the metacentric height grows. The number indicates comfort and safety: short periods, below some 8 seconds, reveal a stiff ship that rolls with high acceleration and punishes cargo and crew; long periods indicate a tender ship, with little stability reserve. The IMO empirical form was adopted rather than the pendulum expression T = 2π × radius of gyration ÷ square root of (g × GM), because the roll radius of gyration is rarely known on board — which is exactly what the coefficient C estimates from hull geometry. Enter the beam, the draught, the waterline length and the metacentric height GM.

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Three-Phase Voltage Unbalance (NEMA)

Measures the unbalance of the three line voltages of a three-phase system by the NEMA MG-1 criterion, the same one used by motor derating curves. The calculation takes the average of the three line voltages, finds the largest absolute deviation between any voltage and that average, and divides this deviation by the average, as a percentage. The number has a direct consequence: NEMA forbids operating induction motors above 5%, recommends derating from 1% on (at 2% the derating factor is already about 0.95) and warns that 1% of voltage unbalance can become 6 to 10% of current unbalance, with extra rotor heating. The NEMA definition was adopted (largest deviation divided by the average, also called LVUR) rather than the IEC and IEEE unbalance factor, which is the ratio between negative and positive sequence components and requires full phasors, not just magnitudes. Enter the three measured line voltages.

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Emergence Speed Index (Maguire)

Computes the seedling emergence speed index by the Maguire (1962) formula, the most widely used vigour test in seed analysis laboratories. The index sums, over each counting date, the number of seedlings that emerged on that day divided by the number of days since sowing: ESI = n1÷t1 + n2÷t2 + n3÷t3. Because early emergences enter the sum divided by a smaller number, the index rewards the lot that emerges fast and uniformly — two lots may end with the same final emergence percentage and have very different indices, and it is that difference which predicts field performance. The convention adopted is Maguire's original one: each count takes the NEW seedlings of that day, not the cumulative total; using the cumulative total inflates the index because it counts the same seedling several times. This version works with three counting dates. Enter the number of new seedlings and the day of each of the three counts.

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Net Assimilation Rate (Gregory Formula)

Computes the net assimilation rate of a plant by the classic Gregory formula, the core of plant growth analysis. The rate is the dry matter gain per day multiplied by the ratio between the difference of the natural logarithms of the two leaf areas and the difference of the areas themselves: NAR = [(W2 − W1) ÷ interval] × [ln(A2) − ln(A1)] ÷ (A2 − A1). The result measures net photosynthetic efficiency per unit of leaf area, with respiration already discounted — typical values for annual crops in full growth lie between 5 and 15 grams per square metre of leaf per day, and a decline along the cycle indicates canopy self-shading. Gregory's (1926) logarithmic form was adopted rather than the approximation NAR = mass gain ÷ (mean leaf area × interval), because the former is exact when leaf area grows linearly with dry mass over the interval, which is the standard assumption of classic growth analysis. Enter the initial and final dry masses, the initial and final leaf areas and the interval between the two samplings.

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Overburden-Corrected SPT (N1)60 — Liao and Whitman

Corrects the SPT blow count already normalized to 60% energy for the effect of vertical effective stress, producing the (N1)60 required by liquefaction and relative density correlations. The overburden factor adopted is that of Liao and Whitman (1986), CN = square root of (100 ÷ vertical effective stress) with stress in kilopascal, capped at 1.7; the result is (N1)60 = N60 × CN. The correction exists because the same soil, at the same density, resists penetration more when it is deeper: without it, a loose sand at 20 metres would look denser than the same loose sand at 3 metres, and the liquefaction potential would be underestimated. The reference pressure of 100 kPa (one atmosphere) and the cap of 1.7 recommended by the 1997 NCEER report were adopted, because without a cap the correction blows up at shallow depths; part of the literature uses 95.76 kPa, which is 1 tsf, and a cap of 2.0, changing the result by a few percent. Enter the measured N60 and the vertical effective stress at the test depth.

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Voids in the Mineral Aggregate (VMA) of an Asphalt Mix

Computes the voids in the mineral aggregate of a compacted asphalt mix, the volumetric parameter that decides whether a Marshall or Superpave mix design is approved. VMA is the space between aggregate grains inside the compacted specimen, that is, the air plus the effective binder: VMA = 100 − (bulk specific gravity of the mix × percentage of aggregate) ÷ bulk specific gravity of the aggregate. The number drives pavement durability: a VMA that is too low leaves no room for a thick binder film and the mix ages and cracks, while a VMA that is too high leaves the mix unstable and prone to rutting — normative ranges sit around 13 to 15% depending on the nominal maximum aggregate size. The critical convention is which aggregate gravity goes in the denominator: here it is the bulk dry gravity (Gsb), as required by Asphalt Institute MS-2; using the effective gravity (Gse), which is larger and sits in the denominator, overestimates VMA by more than a percentage point and may wrongly approve a failing mix. Enter the bulk specific gravity of the compacted mix, the aggregate percentage by mass and the bulk dry specific gravity of the aggregate.

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Kraft Cooking H-Factor

Computes the H-factor of a kraft cook, the index that combines time and temperature into a single number to control the digester. The H-factor is the integral of the relative delignification rate over the cook, and in the isothermal form used on the shop floor it equals H = time × exp(43.20 − 16113 ÷ absolute temperature), with time in hours and temperature in kelvin; the two constants come from Vroom's (1957) fit and were chosen so that the relative rate equals 1 at 100 °C. The number lets you trade time for temperature without changing the outcome: two cooks with the same H-factor and the same alkali charge reach the same kappa number, so raising the temperature allows shortening the plateau, and this is how production is recovered from a late digester. The isothermal form was adopted, considering only the time at the temperature plateau; the full H-factor also integrates the heating ramp and comes out 10 to 20% larger, depending on the ramp rate. Enter the time at the plateau and the cooking temperature.

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Overall Plate Efficiency (O'Connell Correlation)

Estimates the overall plate efficiency of a distillation column by the O'Connell correlation, the step that converts theoretical stages into actual trays in preliminary design. The correlation uses a single combined variable, the product of liquid viscosity and mean relative volatility, and reads efficiency = 0.492 × (viscosity × relative volatility) raised to −0.245, with viscosity in centipoise. What governs it is the product μ·α: a low-viscosity liquid with relative volatility near 1 lands in the 70 to 80% band, and efficiency falls to 40% or less once the liquid is viscous or the relative volatility is high. Note that a difficult separation, with α close to 1, gives the highest efficiency per tray — that column is tall because of the theoretical stage count, not the efficiency, which there actually dampens the height. The analytical fit 0.492 × (μ·α)^−0.245 was adopted, the usual form for hand calculation; the Kessler and Wankat polynomial on the logarithm of the product differs by a few percentage points at the ends of the range. Enter the liquid viscosity at the mean column temperature and the mean relative volatility.

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Buller-Woodrow Loss Factor

Estimates the loss factor of a distribution feeder from its load factor using the empirical Buller-Woodrow relation: loss factor = k × load factor + (1 − k) × load factor squared. The loss factor is the ratio of average loss to peak loss over the period, and it is what turns the instantaneous loss measured at peak hour into energy lost over the month without needing a recorded load curve. Because Joule loss varies with the square of the current, the loss factor always sits below the load factor, and the lower the load factor the lower the ratio between them: at a load factor of 0.20 the loss factor is under half of it, while at 0.80 it sits around 86% of its value. The coefficient k is an input rather than fixed at 0.30, the classic Buller-Woodrow value for distribution networks, because utilities recalibrate k between 0.15 and 0.50 according to the feeder load profile. Enter the load factor for the period and the coefficient k.

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Axle Load Equivalency Factor

Computes how many passes of the standard axle are equivalent to one pass of the real axle, using the power law of pavement design: factor = (axle load ÷ standard axle load) raised to the damage exponent. This factor is what converts a traffic count into the number N of standard axle repetitions, which in Brazil is the 8.2 tf, or 80 kN, single axle with dual wheels. The exponent amplifies overload brutally: an axle 20% heavier than the standard does not consume 20% more pavement but 2.07 times as much, which is why a single overloaded truck weighs more on the life of the road than thousands of cars, whose factor is practically zero. The exponent is an input rather than fixed at 4, the AASHTO value known as the fourth power law, because rigid pavement and fatigue cracking models work with exponents between 3 and 5 and the result shifts by a whole level depending on the choice. Enter the axle load, the standard axle load and the damage exponent.

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Acid Dew Point of Flue Gas

Computes the temperature at which sulphuric acid starts to condense on the cold surfaces of a boiler, using the Verhoff and Banchero correlation: the reciprocal of the absolute dew point temperature is a combination of the logarithms of the partial pressures of water vapour and sulphur trioxide in the flue gas, plus the product of those two logarithms. The result is the thermal floor of the design — keeping the stack, the economiser and the air preheater above it is what prevents the acid corrosion that eats steel in a few weeks, and it is why heavy fuel oil boilers throw away up the stack heat they could otherwise recover. SO₃ is what rules here, not humidity, and the reason is the range each one spans: water vapour barely leaves the 5% to 15% band in a flue gas, which accounts for 11 °C end to end, while SO₃ varies by orders of magnitude with the sulphur in the fuel — going from 1 to 10 ppm alone raises the dew point by almost 22 °C. The Verhoff and Banchero correlation was adopted, with partial pressures in millimetres of mercury at atmospheric pressure, as it is the one most used in boiler design, in its original form with the interaction term between the two logarithms; the later Okkes correlation returns 1 to 8 °C lower for the same composition, so treat the value as a reference and not as an exact limit. Enter the water vapour content and the SO₃ content of the flue gas.

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Weld Cooling Time t8/5

Computes how long the heat affected zone takes to cool from 800 °C to 500 °C, the t8/5 parameter of EN 1011-2, from the heat input, the preheat temperature and the joint shape factor. The formula multiplies the term (6700 − 5 × preheat temperature) by the heat input, by the difference between the reciprocals of (500 − T₀) and (800 − T₀), and by the shape factor tabulated in the standard, which is 1.0 for a bead deposited on a plate and drops to about 0.9 for a butt weld and 0.67 for a fillet weld on a T-joint. Austenite decomposes in that range, so t8/5 decides the microstructure of the joint: cooling too fast forms martensite and opens the door to cold cracking, cooling too slowly coarsens the grain and destroys impact toughness, and most structural steels call for something between 5 and 25 seconds. The three-dimensional heat flow equation was adopted, valid when the plate is thick relative to the weld bead; in thin plate the flow is two-dimensional and t8/5 grows with the square of the heat input rather than in proportion to it. Enter the heat input, the preheat temperature and the joint shape factor.

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Clinker Lime Saturation Factor (LSF)

Computes the lime saturation factor of raw meal or clinker, the index that tells how close the lime present sits to the maximum that silica, alumina and iron oxide could combine with: LSF = 100 × CaO ÷ (2.8 × SiO₂ + 1.18 × Al₂O₃ + 0.65 × Fe₂O₃), with contents as mass percentages. It is the number one parameter in cement kiln control because it governs the split between alite and belite: a value near 100 means nearly all the lime combines and the clinker comes out rich in C₃S, with good early strength, but it demands a hotter burn and a narrow operating margin. Above 100 free lime is left over, hydrating late and expanding in concrete; below 90 the clinker is poor in alite and 3-day strength drops. The form without free lime correction was adopted, as used in raw meal control; on burnt clinker some laboratories subtract free CaO from the numerator, which lowers the index by one or two points. Enter the CaO, SiO₂, Al₂O₃ and Fe₂O₃ contents of the sample.

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Clinker C3S Content (Bogue)

Estimates the tricalcium silicate content, the alite or C₃S, of a clinker using the Bogue equation, the mass balance that splits the four main oxides among the mineral phases: C₃S = 4.071 × CaO − 7.600 × SiO₂ − 6.718 × Al₂O₃ − 1.430 × Fe₂O₃, with contents as mass percentages. Alite is the phase that gives cement its early strength, and ordinary Portland clinker sits between 50% and 65% — below that the 3-day and 7-day strengths collapse, above that kiln fuel consumption rises and the heat of hydration becomes a problem in mass concrete. The negative coefficients are large, so the result is very sensitive to the chemical analysis: half a point more silica knocks 3.8 points off C₃S, and a composition outside the clinker range can even return a negative value, which simply means that mixture has not enough lime to form alite. The classic four-term Bogue equation was adopted, the one for clinker without gypsum; for finished cement the ASTM C150 version also subtracts 2.852 × SO₃ and discounts free lime from CaO. Enter the CaO, SiO₂, Al₂O₃ and Fe₂O₃ contents.

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Soil Group Index (HRB/AASHTO)

Computes the group index of the HRB/AASHTO M 145 classification, the number in parentheses that follows the soil symbol in a soil report: GI = 0.2a + 0.005ac + 0.01bd, where a and b come from the percentage passing the No. 200 sieve and c and d come from the liquid limit and the plasticity index. Each term is truncated — a and b from 0 to 40, c and d from 0 to 20 — and the result is rounded to an integer and never negative, which makes the index range from 0 to 20. The higher the group index, the worse the soil as a subgrade: 0 points to clean, well behaved granular material, values above 12 point to plastic clay that only works once replaced or stabilised, and this is the number that feeds the pavement thickness charts. The complete formula with both terms was adopted; for subgroups A-2-6 and A-2-7 the standard calls for the 0.01bd term alone, and the result is the same: every A-2 soil has at most 35% passing the No. 200 sieve, which is exactly where the a term goes to zero, so the first two terms drop out on their own. Enter the percentage passing the No. 200 sieve, the liquid limit and the plasticity index.

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Cheese Yield (Van Slyke Formula)

Estimates how many kilograms of cheese come out of 100 kg of milk using the Van Slyke formula: 93% of the milk fat is added to the casein content, 0.1 point is discounted as loss to the whey, the sum is multiplied by 1.09 to include the salt and ash retained in the curd, and all of it is divided by (1 minus the cheese moisture). The result is the theoretical yield, the reference against which the real plant loss is measured: a gap above half a point between the theoretical figure and the vat balance is almost always fat escaping into the whey or curd cut at the wrong moment. Note that moisture sits in the denominator and dominates the result — the same milk yields 11.2 kg in a soft cheese at 45% moisture and only 9.5 kg in a hard cheese at 35%, and that difference is water, not solids, so a high yield on its own does not mean a better process. The classic Van Slyke form was adopted, with the 0.93 fat retention coefficient and the 1.09 factor; dairies usually recalibrate both numbers for their own process, and for milk standardised by ultrafiltration the formula underestimates the yield. Enter the milk fat, the milk casein and the target cheese moisture.

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Larson-Skold Index (Water Corrosivity)

Computes the Larson-Skold index, the ratio between the aggressive and the protective anions in a water: chloride plus sulphate divided by alkalinity, all converted to milliequivalents per litre with the equivalent weights 35.45 for chloride, 48.03 for sulphate and 50.04 for alkalinity expressed as CaCO₃. The reading is direct: below 0.8 alkalinity dominates and the carbonate film protects carbon steel; between 0.8 and 1.2 corrosion stops being negligible; above 1.2 chloride and sulphate break the film and the localised corrosion rate takes off, the typical scenario of cooling tower makeup water running at many cycles of concentration. Unlike the Langelier index, this one does not say whether the water will scale — it measures only the corrosive power of the anions, which is why the two readings complement each other rather than compete. Total alkalinity was adopted as the input, instead of separate bicarbonate and carbonate, because that is what a routine laboratory reports, and converting it through the CaCO₃ equivalent returns exactly the sum of the two in milliequivalents per litre. Enter the chloride, the sulphate and the total alkalinity.

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Silt Density Index (SDI)

Computes the silt density index of ASTM D4189, the test that measures the fouling potential of the feed water of a reverse osmosis membrane: the sample is filtered through a 0.45 µm membrane at 207 kPa, the time to collect 500 mL is measured at the start and again at the end of the test, and the index is the percentage of flow loss divided by the duration. The expression is SDI = (1 − initial time ÷ final time) × 100 ÷ duration, and the number tells how much the filter plugged per minute of test: membrane makers usually require under 5 for a spiral wound element and under 3 for extended warranty, and water above that forces stronger coagulation or filtration upstream. The test is only valid if the flow loss stays below 75%; if the filter plugs before that, run it again with a shorter duration. The duration was adopted as an input instead of fixing 15 minutes, the standard value, precisely because poor water demands repeating at 10 or 5 minutes — results from different durations are not comparable, which is why the value is recorded as SDI₁₅, SDI₁₀ or SDI₅. Enter the initial time, the final time and the test duration.

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Ball Pass Frequency, Outer Race (BPFO)

Computes the ball pass frequency of the outer race (BPFO), the signature a localized defect on a bearing outer ring leaves in the vibration spectrum: BPFO = (number of elements ÷ 2) × shaft rotation frequency × (1 − element diameter ÷ pitch diameter × cosine of the contact angle). The result, in hertz, is the frequency at which a peak appears every time a ball or roller rides over the flaw; because it is not an integer multiple of shaft speed, it is distinguishable from unbalance and misalignment, which show up at 1× and 2× rotation. Since the races slip slightly, the measured frequency usually falls 1% to 2% below the theoretical one, so look for the band rather than the exact line. A 6205 deep-groove ball bearing has a BPFO near 3.6 times shaft speed, and it is always worth checking that BPFO plus the inner race frequency equals exactly the number of elements times the rotation frequency. Enter the number of rolling elements, the shaft rotation frequency, the rolling element diameter, the pitch diameter and the contact angle.

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Compressed Air Receiver Volume

Sizes a compressed air receiver from the maximum number of motor starts per hour the compressor allows: V = 15 × free air delivery in m³/min × atmospheric pressure in bar ÷ (starts per hour × load-unload pressure band in bar). The constant 15 comes from the worst case, in which demand is exactly half the compressor capacity: the receiver then fills and drains at half flow, each of the two legs lasts twice what it would at full flow, and the whole cycle is four times that unit time — the condition that maximizes the number of starts, and the 15 is the 60 minutes in an hour divided by that 4. The result, in cubic metres, is the minimum volume: a smaller receiver makes the motor start more often than the manufacturer allows and overheats the winding, while a larger one costs nothing but money and floor space. Because volume is inversely proportional to the pressure band, widening the unload differential from 1 to 2 bar halves the tank, which is usually cheaper than buying a bigger vessel. Enter the compressor free air delivery, the maximum number of starts per hour, the load-unload pressure band and the local atmospheric pressure.

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Acoustic Barrier Attenuation (Maekawa)

Computes how much an acoustic barrier cuts the noise reaching a receiver, using Maekawa's empirical formula, Attenuation = 10 × log₁₀(3 + 20N), where N is the Fresnel number, equal to twice the path difference divided by the wavelength — that is, N = 2 × path difference × frequency ÷ speed of sound. The path difference is the extra distance sound must travel to go over the top of the barrier instead of straight from source to receiver, and it is the only geometric input the formula needs. The result, in decibels, is what the barrier subtracts from the level that would arrive without it: with N equal to zero, meaning the receiver sits exactly on the line of sight to the top edge, attenuation is already 4.8 dB, and in practice the gain saturates between 20 and 24 dB because sound eventually flanks around the sides and passes through the panel. Since N grows with frequency, the same barrier is far more effective at high frequencies than at low ones: once N is large, every octave adds about 3 dB, which is why enclosing a compressor kills the hiss and barely touches the rumble. Enter the path difference, the frequency and the speed of sound.

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Box Compression Strength (McKee)

Estimates the vertical compression strength of a corrugated box with the simplified McKee formula, BCT = 5.87 × ECT × √(board caliper × box perimeter), where ECT — the edge crush resistance measured per TAPPI T 811 (ISO 3037) — is in newtons per millimetre and both dimensions are in millimetres. The result, in newtons, is the load an empty box withstands in the laboratory compression test, with the board conditioned at 23 °C and 50% relative humidity. Because strength grows with the square root of the perimeter and linearly with ECT, doubling the perimeter buys only 41% more, while switching to a flute with 30% higher ECT buys the full 30% — upgrading the board usually beats reshaping the box. For real pallet stacking, divide the BCT by a safety factor of 3 to 5, which covers the stiffness lost to ambient humidity, the creep of board under load over weeks and the misalignment between boxes in the column. Enter the ECT, the board caliper and the box perimeter.

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Leaching Requirement (Irrigation)

Computes the leaching requirement of an irrigated field — the fraction of the applied depth that must pass through the root zone and drain away to flush out the salts the irrigation water leaves behind: LR = water EC ÷ (5 × tolerable saturation extract EC − water EC), with both electrical conductivities in decisiemens per metre. The tolerable EC comes from the crop salt tolerance table — beans sit near 1 dS/m, maize near 1.7 and barley above 8. The result, as a percentage, feeds the gross depth calculation, which is the net depth divided by (1 − LR): a requirement of 13.6%, for instance, forces you to apply about 16% more water than the crop consumes. The saltier the water relative to what the crop tolerates, the larger the fraction; and when the water EC approaches five times the tolerable EC the value blows up, a sign that this water is unusable for that crop without artificial drainage or a change of species. Enter the electrical conductivity of the irrigation water and the tolerable electrical conductivity of the soil saturation extract.

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Stand Density Index (Reineke)

Computes Reineke's stand density index, SDI = trees per hectare × (quadratic mean diameter ÷ 25)^1.605, which expresses the stocking of a forest stand as the equivalent number of trees per hectare it would hold if every one measured 25 cm DBH — the 10 inches of the original 1933 work, rounded in the metric version. The exponent 1.605 is the slope of the self-thinning line Reineke fitted empirically, and it is precisely what makes the index nearly independent of age and site quality, unlike a plain trees-per-hectare count. The number guides thinning decisions when compared with the species maximum SDI, which for most species falls between 1,000 and 1,200: competition mortality typically starts around 55% to 60% of the maximum, and the recommended management zone runs from 35% to 55%. In the example, 559 against a maximum of 1,100 gives about 51%, meaning the stand is still below the self-thinning threshold but already at the top of the management zone. Enter the number of trees per hectare and the quadratic mean diameter.