🧮Calculators

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Pouring Velocity

Calculate the molten metal velocity at the base of the sprue by Torricelli's equation, v = √(2·g·h), from the metal column height h (m) and gravity g. The result, in m/s, is the velocity at which the metal enters the gating system by gravity, starting from the pouring basin height. It is the basis of gating system design: the velocity sets the flow rate (with the section area) and the flow regime. Velocities too high cause turbulence (air aspiration, oxidation, erosion); hence gating systems are designed to control and slow the flow. Enter the metal column height.

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Riser Modulus

Calculate the minimum modulus of a riser (feeder) by the modulus rule, M_riser = 1.2 × M_part, from the part's cooling modulus. The result, in cm, is the modulus the riser must have to solidify after the part (about 20% slower) and feed it with molten metal during solidification shrinkage, avoiding shrinkage cavities. The riser is a metal reservoir placed over the thickest region of the part; if it solidifies first, it fails its purpose. From the modulus, the riser geometry is sized. It is a fundamental rule of casting design. Enter the part's cooling modulus.

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Solidification Volumetric Shrinkage

Calculate the volumetric shrinkage on solidification of a metal, ΔV = (ρ_solid − ρ_liquid) ÷ ρ_liquid × 100%, from the metal densities in the solid and liquid states. The result, in %, is the volume reduction that occurs when the metal goes from liquid to solid — because the solid is denser (more compact) than the liquid. This shrinkage is the main cause of shrinkage cavities (internal voids) and is exactly what risers must feed with extra molten metal. Each metal has its solidification shrinkage: steel ~3%, aluminum ~6.6%, copper ~5%. Gray cast iron is an exception (graphite expands, reducing the liquid shrinkage). Enter the solid and liquid metal densities.

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Gate Area

Calculate the gating channel section area, A = Q ÷ v, dividing the desired metal flow rate Q by the metal velocity v. The result, in the consistent area unit (cm²), is the cross-section the sprue (or gate) must have to deliver the needed flow at the calculated velocity. It is the application of the continuity equation to the casting gating system. Correctly sizing the areas of the system's elements (basin, sprue, runner, gates) controls the flow rate, velocity and flow regime of the metal, avoiding turbulence and ensuring proper filling. The ratios between the areas define the system type (pressurized or unpressurized). Enter the flow rate and the velocity.

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Stoichiometric Air-Fuel Ratio

Calculate the stoichiometric air-fuel ratio (AFR) by mass of a hydrocarbon C_xH_y, AFR = (x + y/4) × 137.93 ÷ M, from the number of carbon atoms x, hydrogen atoms y and the fuel molar mass M (g/mol). The result (kg air per kg fuel) is the exact amount of air needed for complete combustion, with no leftover air or fuel. Methane (CH₄) has AFR ≈ 17.2; gasoline ≈ 14.7. The 137.93 constant comes from the air mass per mole of O₂ (32 ÷ 0.232). It is the base parameter of combustion control and mixture in engines and burners. Enter x, y and the fuel molar mass.

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Excess Air (from Flue Gas)

Calculate the excess air of a combustion from the oxygen in dry flue gas, EA = O₂ ÷ (20.9 − O₂) × 100%, from the measured O₂ percentage in the stack. The result, in %, shows how much air was supplied beyond stoichiometric — measured by the leftover oxygen in the exhaust gases. Some excess air (10-30%) is needed to ensure complete combustion (avoid CO and soot), but too much wastes energy heating useless air that leaves hot through the stack. Gas analyzers measure O₂ and compute the excess air to optimize combustion efficiency. Enter the O₂ percentage in the gases.

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Equivalence Ratio (φ)

Calculate the combustion equivalence ratio, φ = AFR_stoichiometric ÷ AFR_actual, dividing the stoichiometric air-fuel ratio by the actual one. The dimensionless result classifies the mixture: φ = 1 is stoichiometric (exact air); φ < 1 is lean (excess air, typical of industrial burners and diesel engines); φ > 1 is rich (lack of air, produces CO and soot, but more power in gasoline engines). The equivalence ratio is the preferred dimensionless parameter in combustion science, linked to excess air (φ = 1/(1+EA)). Enter the stoichiometric and actual air-fuel ratios.

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Adiabatic Flame Temperature

Estimate the adiabatic flame temperature, T_ad = T_initial + LHV ÷ (m × cp), from the lower heating value LHV (kJ/kg fuel), the mass of combustion products per kg fuel m, the average specific heat of the gases cp (kJ/kg·K) and the initial temperature. The result, in °C, is the maximum theoretical temperature the gases would reach if all the combustion energy heated the products, with no heat loss. It is an upper bound: real flames are cooler (radiation losses, dissociation, excess air). It sets the thermal severity on materials and NOx formation. Enter the LHV, the gas mass, the cp and the initial temperature.

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Stack Heat Loss (Siegert)

Calculate the heat loss through the exhaust gases by the Siegert formula, loss = K × (T_gas − T_air) ÷ CO₂, from the fuel factor K (~0.5 for natural gas, ~0.6 for oil), the gas and combustion air temperatures (°C) and the CO₂ percentage in the gases. The result, in %, is the largest energy loss of a boiler or furnace — the heat escaping hot through the stack. Lowering the gas temperature (with economizers and preheaters) and adjusting the excess air (which dilutes CO₂) minimizes this loss. The combustion efficiency is approximately 100% minus this loss. Enter the K factor, the temperatures and the CO₂.

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Theoretical Maximum CO₂

Calculate the theoretical maximum CO₂ percentage in dry flue gas from complete combustion of a hydrocarbon C_xH_y, CO₂max = x ÷ (x + (x + y/4) × 3.762) × 100%, from the carbon x and hydrogen y atoms. The result, in %, is the CO₂ obtained with perfect stoichiometric combustion (no excess air) — the reference value of gas analyzers. Methane has CO₂max ≈ 11.7%; coal, ~18-20%. Comparing the measured CO₂ with the theoretical maximum indicates the excess air: the lower the measured CO₂ relative to the maximum, the more excess air diluting the gases. Enter x and y.

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Theoretical Combustion Air Volume

Calculate the theoretical air volume needed for complete combustion of a hydrocarbon C_xH_y, V_air = (x + y/4) × 22.4 ÷ (0.21 × M), from the carbon x, hydrogen y atoms and the molar mass M. The result, in Nm³ of air per kg of fuel (at normal conditions), is the stoichiometric air — the basis for sizing fans, burners and combustion air systems. The factor 22.4 is the ideal gas molar volume (L/mol at STP) and 0.21 the volume fraction of O₂ in air. Methane needs ~13.3 Nm³/kg. Multiplied by the excess air, it gives the actual air supplied. Enter x, y and the molar mass.

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Natural Chimney Draft

Calculate the natural draft (depression) of a chimney, ΔP = 353 × h × (1/T_air − 1/T_gas), from the chimney height h (m) and the absolute temperatures of the outside air and the hot gases (K). The result, in pascals, is the pressure difference that 'pulls' the gases up and the combustion air into the burner, generated by the density difference between the hot (light) gases and the cold (dense) air — the chimney effect. Taller chimneys and hotter gases generate more draft. It is the basis of natural-draft furnace and boiler chimney design; insufficient draft requires fans (forced draft). Enter the height and the air and gas temperatures.

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Actual Air-Fuel Ratio

Calculate the actual air-fuel ratio, AFR_actual = AFR_stoichiometric × (1 + excess air ÷ 100), from the stoichiometric air-fuel ratio and the excess air (%). The result (kg air per kg fuel) is the amount of air actually supplied in practice, always greater than stoichiometric, because real combustion needs excess air to ensure complete burning (the mixture is never perfect). This value sizes the air supply (fans), the exhaust gas flow and influences the flame temperature and efficiency. Enter the stoichiometric air-fuel ratio and the excess air.

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CO₂ Volume Produced

Calculate the CO₂ volume produced in complete combustion of a hydrocarbon C_xH_y, V_CO₂ = x × 22.4 ÷ M, from the number of carbon atoms x and the fuel molar mass M (g/mol). The result, in Nm³ of CO₂ per kg of fuel, is the carbon dioxide generated by complete burning — information for emission inventories, exhaust system sizing and gas analysis. Each carbon atom in the fuel becomes one CO₂ molecule. Methane produces ~1.4 Nm³/kg. Fuels with more carbon per unit mass (coal, heavy oils) produce more CO₂. Enter x and the fuel molar mass.

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Grinding Energy (Bond Work Index)

Calculate the specific comminution (grinding/crushing) energy by Bond's law, W = 10 × Wi × (1/√P₈₀ − 1/√F₈₀), from the ore's Bond work index Wi (kWh/t), and the particle sizes passing 80% of the product (P₈₀) and feed (F₈₀), in micrometers. The result, in kWh per tonne, is the energy needed to reduce the ore from feed to product size. Comminution is mining's largest energy consumer (up to 50% of the plant). The Wi index characterizes the ore's resistance to fragmentation. It is the basis for sizing mills and energy consumption. Enter the Wi, P₈₀ and F₈₀.

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Crushing Reduction Ratio

Calculate the reduction ratio of a crusher or mill, RR = F ÷ P, dividing the feed size F by the product size P (usually F₈₀/P₈₀ or crusher openings). The dimensionless result shows how many times the material was reduced in size in one stage. Each equipment type has a typical reduction ratio range: jaw crushers 4-7, cone crushers 5-8, ball mills up to 100 or more. Since each stage has a limited ratio, reducing large blocks to fine powder requires several stages in series, whose product of ratios gives the total reduction. Enter the feed and product sizes.

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Linear Explosive Charge

Calculate the linear loading density of a blast hole, q = (π/4) × d² × ρ, from the hole diameter d (mm) and the explosive density ρ (g/cm³). The result, in kg of explosive per meter of hole, is how much explosive fits in each meter of charged column — a central parameter of rock blast design. Multiplied by the hole charge height, it gives the charge per hole; combined with the blasted rock volume, it gives the powder factor. Larger diameters and denser explosives raise the linear charge. Enter the hole diameter and the explosive density.

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Blast Burden

Calculate the burden of a blast pattern, B = k × d, multiplying a factor k (typically 25 to 40, depending on rock and explosive) by the hole diameter d. The result, in the unit of d, is the distance from the row of holes to the free rock face — one of the most critical geometric parameters of blasting. Too large a burden leaves the rock poorly fragmented (boulders) and creates toes; too small wastes explosive and causes flyrock and overpressure. Together with the hole spacing, the burden defines the drilling pattern and the resulting fragmentation. Enter the factor k and the hole diameter.

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Belt Conveyor Capacity

Calculate the mass flow capacity of a belt conveyor, Q = A × v × ρ × 3600, from the cross-section area of the load on the belt A (m²), the belt speed v (m/s) and the material's bulk density ρ (t/m³). The result, in tonnes per hour, is the conveyor's transport capacity — essential equipment in handling ore, gravel, grain and coal. The load area depends on the belt width, the material's surcharge angle and the idler configuration. Wider, faster belts and denser materials raise the capacity. It is the basis of conveying system design. Enter the load area, the speed and the density.

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Blast Hole Count

Calculate the number of holes of a blast pattern, N = area ÷ (burden × spacing), dividing the bench area to blast by the pattern area of each hole (burden B × spacing S). The result is the number of holes needed to cover the area with the specified drilling pattern. In practice, round up. It is an essential quantity calculation in blast planning: it sets the drilling time, the amount of explosive and accessories, and the operation cost. Wider patterns (larger B and S) reduce the number of holes but may worsen fragmentation. Enter the area, the burden and the spacing.

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Total Reduction Ratio

Calculate the total reduction ratio of a three-stage comminution circuit, RR_total = RR₁ × RR₂ × RR₃, multiplying the reduction ratios of each crusher/mill in series. The dimensionless result is the circuit's overall size reduction — from bench rock blocks (hundreds of mm) to fine particles (mm or µm). Since each stage has a limited reduction ratio (4 to 10 for crushers), large total reductions (100, 1000 or more) require several stages in series: primary, secondary, tertiary crushing and milling. Enter the reduction ratios of the three stages.

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Blast Subdrilling

Calculate the subdrilling of a blast hole, S_p = 0.3 × B, multiplying the burden B by a typical factor of 0.3. The result, in the unit of B, is the length the hole must drill below the desired bench floor level. This extra depth ensures the blast fragments the rock down to the floor level, avoiding toes (ledges of unfragmented rock at the bench foot) that hinder equipment operation. Insufficient subdrilling leaves toes; excessive wastes drilling and explosive and damages the rock below the floor. Enter the burden.

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Mass Recovery

Calculate the mass recovery (mass yield) of a mineral processing operation, R = (concentrate mass ÷ feed mass) × 100%, dividing the concentrate mass produced by the ore feed mass. The result, in %, is the fraction of mass reporting to the concentrate — different from metallurgical recovery (which measures the fraction of metal recovered). Low mass recovery is typical of lean ores (little concentrate from much feed); high indicates rich ore or poorly selective concentration. Combined with the grades, it closes the plant's mass balance. Enter the concentrate and feed masses.

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Detonation Velocity (VOD)

Calculate the velocity of detonation (VOD) of an explosive, VOD = L ÷ t, dividing the distance traveled by the detonation wave L (m) by the time t (s) measured between two sensors. The result, in m/s, is the speed at which the detonation reaction propagates through the explosive column — one of the most important properties of an explosive, linked to its energy and fragmentation power. High-VOD explosives (4000-7000 m/s, like emulsions and dynamites) generate high detonation pressure and are effective in hard rock; low VOD (ANFO, ~3000-4500 m/s) suits softer rock. Enter the measured distance and time.

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Elevator Motor Power

Calculate the motor power of an elevator, P = m·g·v ÷ η, from the payload m (kg), gravity g (9.81 m/s²), nominal speed v (m/s) and the system efficiency η (motor, gearbox, sheaves). The result, in watts, is the mechanical power needed to hoist the load at nominal speed. In practice, the counterweight (balancing the car plus ~45% of the load) reduces the effective power, and regenerative braking on descent can return some to the system. It is the base calculation for sizing the traction machine. Enter the load, the speed and the efficiency.

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Hoist Rope Tension

Calculate the resultant force in an elevator's hoist rope, F = (Q + M_car − M_counterweight)·g, from the payload Q, the car mass and the counterweight mass (kg). The result, in newtons, is the unbalanced effort the steel ropes must transmit, already net of the counterweight's balancing effect. It is the basis for sizing the ropes (number, diameter and safety factor, typically ≥ 12 in elevator codes) and the traction sheave. When the load is such that car + load ≈ counterweight, the force tends to zero (balanced system). Enter the load, the car mass and the counterweight mass.

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Counterweight Mass

Calculate an elevator's counterweight mass, M_cw = M_car + factor × Q_max, from the car mass, the balancing factor (typically 0.40 to 0.50) and the maximum load Q_max (kg). The result, in kg, is the mass that balances the car plus a fraction of the payload, so the motor works with the smallest average imbalance. A factor of 0.45 (45%) is common: it fully balances the car and 45% of the rated load, minimizing motor work both with a full and an empty car. Enter the car mass, the balancing factor and the maximum load.

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Round Trip Time (RTT)

Estimate an elevator's round trip time (RTT), RTT = 2·(H ÷ v) + stops × t_stop, from the travel height H (m), the speed v (m/s), the number of probable stops and the average time per stop (s, including deceleration, door opening/closing and boarding). The result, in seconds, is the time of a complete up-and-down cycle with stops — a central parameter of vertical traffic analysis. The higher the RTT, the lower the handling capacity and the longer the interval between cars. Enter the height, the speed, the number of stops and the time per stop.

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Handling Capacity (5 min)

Calculate an elevator's handling capacity over 5 minutes, HC = (300 × Q) ÷ RTT, from the car capacity Q (people) and the round trip time RTT (s). The result, in people carried per 5 minutes, is the standard vertical-traffic performance metric (building peak demand is usually measured over 5 min). The factor 300 is the seconds in 5 minutes. Multiplied by the number of elevators and compared with the building population, it tells whether the system meets demand (typically 12-15% of the population in 5 min in offices). Enter the car capacity and the RTT.

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Elevator Traffic Interval

Calculate the traffic interval (average waiting time) of an elevator group, INT = RTT ÷ N, dividing the round trip time RTT (s) by the number of elevators N in the group. The result, in seconds, is the average time between elevator arrivals at the main floor — the main service-quality indicator perceived by users (waiting time). Intervals up to 30 s are excellent; above 50-60 s, poor. More elevators in the group reduce the interval. It is the key criterion in sizing the number of elevators. Enter the RTT and the number of elevators.

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Probable Stops

Calculate the probable number of stops of an elevator, S = N × (1 − (1 − 1/N)^P), from the number of served floors N and the number of passengers P in the car. The result is how many floors, on average, the elevator actually stops at during a trip (probabilistically, two passengers may go to the same floor). It is an essential parameter of the round trip time calculation: more stops raise the RTT. The formula assumes passengers choose destination floors randomly and uniformly. With a full car, S approaches N (stops at almost all). Enter the number of floors and passengers.

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Building Population

Estimate a building's population, Pop = (area per floor × number of floors) ÷ density, from the usable area per floor (m²), the number of floors and the occupancy density (m² per person). The result, in people, is the total population to be served by the vertical transport — the starting point of elevator traffic analysis. Occupancy density varies with use: ~10 m²/person in dense offices, ~15-20 m²/person in standard offices, with specific values for hotels and residences. Compared with the elevators' handling capacity, it tells whether the system is adequate. Enter the area per floor, the number of floors and the density.

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Number of Elevators Required

Calculate the number of elevators required, N = peak demand ÷ capacity per elevator, dividing the peak transport demand (people in 5 min) by the handling capacity of a single elevator (people in 5 min). The result is the minimum number of elevators in the group to meet peak demand. In practice, round up and also check the resulting traffic interval (waiting quality). Peak demand comes from the building population times the peak percentage (12-15% in offices). Undersizing causes queues and long waits. Enter the peak demand and the capacity per elevator.

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Tunnel Convergence

Calculate a tunnel's convergence — the relative radial deformation of the excavation, ε = (u ÷ r)·100 — from the radial displacement u (the inward movement of the walls toward the center, measured by extensometers or total station) and the excavation radius r, in the same unit. Convergence is the primary monitoring indicator in NATM (New Austrian Tunnelling Method): it measures how much the rock mass deforms after excavation, reflecting stress mobilization and support effectiveness. Low, stabilized convergence indicates a stable mass; high, growing or accelerating convergence signals squeezing, instability or insufficient support, requiring immediate reinforcement. Enter the radial displacement and the tunnel radius.

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Terzaghi Rock Load Height

Estimate the rock load height over a tunnel crown by Terzaghi's classic method, Hp = Cf·(B + Ht), from the rock load factor Cf (depending on mass quality — ~0 for intact rock to >2 for heavily fractured or swelling rock), the width B and the height Ht of the excavation. Hp represents the loosened rock zone above the tunnel that effectively loads the support — Terzaghi proposed that, due to arching in the mass, only a fraction of the total overburden acts on the lining. This loosening-load model is the historic basis for rock tunnel support design. Multiplying Hp by the unit weight gives the support pressure. Enter the load factor, width and height.

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Tunnel Support Pressure

Calculate the support pressure a tunnel lining must resist, pv = γ·Hp, from the rock mass unit weight γ (kN/m³) and the rock load height Hp (m) — typically from Terzaghi's method or geomechanical classifications (RMR, Q-system). Support pressure is the vertical stress the loosened rock zone exerts on the support (shotcrete, steel sets, final lining), and it drives the structural design of the lining. In shallow tunnels the load may be the full overburden; in deep tunnels, arching reduces it to a fraction. Estimating it correctly is decisive: underestimating leads to collapse, overestimating raises cost. Enter the unit weight and the rock load height.

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Peck Settlement Trough Width

Calculate the trough-width parameter of the surface settlement induced by tunnelling, i = K·z₀, by Peck's method, from the trough-width parameter K (~0.5 for clays, ~0.25-0.35 for sands) and the tunnel axis depth z₀. The surface settlement from ground loss follows a Gaussian (inverted bell) curve, and i is its standard deviation — the horizontal distance from the tunnel axis to the inflection point, defining the trough width. Larger i means a wider, gentler trough (clays); smaller means narrower and deeper (sands). This parameter is essential to predict damage to nearby buildings in urban tunnels. Enter the K parameter and the tunnel depth.

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Maximum Surface Settlement (Tunnel)

Calculate the maximum surface settlement, over the tunnel axis, S_max = Vs ÷ (i·√(2π)), from the settlement trough volume per metre of tunnel Vs (m³/m, the lost soil volume surfacing) and the trough-width parameter i (m, Peck's method). Since the trough is Gaussian, integrating the curve gives Vs = √(2π)·i·S_max, isolating the maximum settlement, which occurs right over the axis. This is the critical value for damage assessment: compared to allowable limits (typically 10-25 mm for sensitive structures), it decides whether the excavation is safe or needs mitigation. Enter the trough volume and the width parameter.

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Tunnel Volume Loss

Calculate the volume loss of a tunnel excavation, VL = Vs ÷ (π·D²/4)·100, the percentage ratio between the settlement trough volume per metre Vs (m³/m) and the excavated cross-section area (from diameter D). Volume loss quantifies how much soil 'disappeared' relative to the theoretical tunnel volume — caused by face relaxation, overexcavation, tail-gap closure behind the TBM shield and consolidation. It is the key control parameter for urban excavation: well-run EPB/slurry TBMs achieve 0.5-1.5% in soils; values above 2-3% indicate problems and excessive settlement. Enter the trough volume and the tunnel diameter.

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Tunnel Face Pressure (EPB/Slurry)

Estimate the face support pressure needed to stabilize the excavation front of a mechanized tunnel, p = K·γ·H, from the earth pressure coefficient K (at rest K₀ ≈ 1−sinφ, or active), the soil unit weight γ (kN/m³) and the axis depth H (m). In closed-face TBMs (EPB or slurry), the pressurized chamber must balance the earth and water pressure at the front, avoiding both collapse (insufficient pressure) and blow-out (excessive pressure). Face pressure is the most critical operational parameter of a TBM, adjusted in real time per cover, water table and soil type. This gives the earth component; total pressure adds hydrostatic water pressure and a safety margin. Enter the earth pressure coefficient, unit weight and depth.

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Advance per Blast (Pull)

Calculate the effective advance per blast (pull) in drill-and-blast tunnelling, advance = L·η, from the drilled hole length L (m) and the blast efficiency η (0-1). Not all drilled depth converts to advance: part is lost because the hole bottoms do not always break fully, leaving a 'socket'. Typical efficiency is 85-95% — depending on the blast pattern, rock type and execution. Advance per blast, times the cycles per day, sets the rock face productivity. Maximizing it reduces cycles and schedule, but very long holes lose drilling accuracy and efficiency. Enter the drilled length and the blast efficiency.

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TBM Advance Rate

Calculate a tunnel boring machine's daily advance, advance = PR·U·h, from the instantaneous penetration rate PR (m/h, advance while actively boring), utilization U (0-1, the fraction of time actually boring) and operating hours per day h. The distinction between penetration and utilization is central: penetration depends on geology and cutterhead thrust/torque, but utilization — typically only 30-50% — is limited by ring building, cutter changes, maintenance, muck removal and downtime. Real advance is far below nominal penetration, and improving utilization often pays more than increasing penetration. Enter the penetration rate, utilization and hours per day.

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Spillway Discharge (Creager/Ogee)

Calculate the discharge over a Creager/ogee dam spillway, Q = C·L·H^1.5, from the discharge coefficient C (typically 2.0-2.2 in SI for well-designed ogee profiles), the crest length L (m) and the head over the crest H (m). The spillway is a dam's most critical safety structure: it releases floods safely, preventing overtopping — the leading cause of dam failure. The ogee profile follows the shape of the underside of a free nappe, maximizing discharge while keeping crest pressure near atmospheric (avoiding cavitation). The coefficient C absorbs gravity and approach effects, exceeding that of a sharp-crested weir. Spillway design starts from the design flood (often the 10,000-year flood or the PMF) and sets the required crest length. Enter the discharge coefficient, crest length and head.

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Hydraulic Jump Sequent Depth

Calculate the sequent (conjugate) depth downstream of a hydraulic jump, y₂ = (y₁/2)·(√(1 + 8·Fr₁²) − 1), from the upstream depth y₁ (supercritical) and the incoming Froude number Fr₁. The hydraulic jump is the abrupt transition from fast, shallow (supercritical) to slow, deep (subcritical) flow, with strong turbulence and energy dissipation. This Bélanger equation, from momentum conservation, is the basis for designing stilling basins downstream of spillways and gates: water descending a spillway arrives at very high (supercritical) velocity and must be decelerated before returning to the river, otherwise it erodes the bed catastrophically. The sequent depth y₂ sets the required basin depth for a stable jump. Enter the upstream depth and the Froude number.

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Hydraulic Jump Energy Loss

Calculate the specific energy dissipated in a hydraulic jump, ΔE = (y₂ − y₁)³ ÷ (4·y₁·y₂), from the upstream y₁ (supercritical) and downstream y₂ (subcritical) sequent depths. The hydraulic jump is one of the most efficient energy dissipators in hydraulics: intense turbulence in the transition converts kinetic energy to heat and sound, removing excess flow energy. This head loss ΔE is exactly what is sought downstream of spillways, gates and bottom outlets — water arrives with very high energy (able to scour the riverbed and undermine the structure), and the stilling basin induces the jump to 'burn' that energy in a controlled way. The higher the incoming Froude number, the greater the dissipated fraction — jumps with Fr > 9 dissipate up to 85%. Enter the upstream and downstream sequent depths.

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Hydraulic Jump Length

Estimate a hydraulic jump's length, L ≈ 6.9·(y₂ − y₁), by the classic empirical formula, from the upstream y₁ and downstream y₂ sequent depths. Unlike the sequent depths (from momentum), jump length is empirical, from lab tests, since the jump has no mathematically sharp end — its length is the distance from the upstream face to where the surface stabilizes. Several formulas exist (Smetana ≈ 6(y₂−y₁), USBR vs Fr, Elevatorski ≈ 6.9(y₂−y₁)); all give the order of magnitude. Jump length sets the stilling basin size downstream of a spillway: the basin must be long enough to contain the whole jump so dissipation completes within the concrete-lined structure before water returns to the natural bed. Undersizing throws the still-erosive jump tail onto the unprotected bed. Enter the upstream and downstream sequent depths.

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Critical Depth in Rectangular Channel

Calculate the critical depth of a rectangular channel, y_c = (q² ÷ g)^(1/3), from the unit discharge q (flow per unit width, m³/s/m) and gravity g. Critical depth is the depth at which specific energy is minimum, marking the boundary between the two open-flow regimes: above it the flow is subcritical (slow, deep, Fr < 1, downstream-controlled); below, supercritical (fast, shallow, Fr > 1, upstream-controlled); exactly at it, Fr = 1. Critical depth is central to channel and structure hydraulics: it defines the control section at spillways, weirs and flumes (Parshall), where flow passes through the critical regime stably and the stage-discharge relation is unique — allowing flow measurement from head. It also determines whether a hydraulic jump can form and guides water-surface profiles. Enter the channel's unit discharge.

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Hydrostatic Thrust on Dam

Calculate the horizontal hydrostatic thrust per metre of length on a dam face, E = ½·γ·H², from the unit weight of water γ (≈ 9.81 kN/m³) and the water depth H (m) at the upstream face. Since hydrostatic pressure grows linearly with depth (p = γ·h), its diagram is triangular and the resultant is its area, ½·γ·H², applied at one third of the height from the base. This thrust is the main action tending to overturn and slide the dam, and the starting point of gravity dam stability analysis: it generates the overturning moment (about the downstream toe) and the horizontal force resisted by base friction. Dam stability depends on its self-weight (providing the stabilizing moment and normal friction force) exceeding these with adequate margin, also accounting for foundation uplift. Enter the unit weight of water and the depth.

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Dam Foundation Uplift

Calculate the uplift resultant per metre of length at a gravity dam base, assuming triangular distribution, U = ½·γ_w·H·B, from the unit weight of water γ_w, the head H (upstream water height) and the base width B. Uplift is the water pressure that percolates through the foundation and concrete joints acting upward on the dam base, reducing the effective normal force and thus the sliding friction resistance — one of the most dangerous and historically underestimated factors in dam stability (the 1928 St. Francis Dam failure is a landmark). The real distribution depends on grout curtains and drains, which reduce it; the triangular hypothesis (full upstream, zero downstream) is conservative and common in preliminary design. Uplift subtracts from self-weight in the sliding check and adds overturning moment. Enter the unit weight of water, the head and the base width.

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Dam Sliding Safety Factor

Calculate the sliding safety factor of a gravity dam, FS = (μ·W) ÷ F_h, from the base friction coefficient μ (tan of the concrete-foundation friction angle, typically 0.6-0.75), the effective self-weight W (dam weight minus uplift, kN/m) and the destabilizing horizontal force F_h (hydrostatic thrust, kN/m). This factor compares the forces resisting the dam sliding on its foundation (mobilized base friction, proportional to the effective normal force) with those pushing it downstream (the reservoir thrust). It is one of the two fundamental gravity dam stability checks — the other being overturning. Codes typically require sliding FS ≥ 1.5 for normal loading. The simplified form uses friction only; fuller analyses add interface cohesion (c·B). Note how decisive uplift is: it reduces W and thus the numerator — hence the importance of foundation drainage. Enter the friction coefficient, effective weight and horizontal force.

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Reservoir Emptying Time

Calculate the time to empty a constant-surface-area reservoir through a bottom orifice, t = 2·A_s·√H ÷ (C_d·A_o·√(2g)), from the water-surface area A_s (m²), the outlet orifice area A_o (m²), the initial head H (m, water height above the orifice) and the discharge coefficient C_d (≈ 0.6 for orifices). The formula integrates Torricelli's equation over the drawdown: as orifice flow drops while the level (and head) falls, emptying decelerates, and total time results from integrating dH/dt. It is useful for designing dam bottom outlets (used to lower the reservoir in emergencies or for maintenance), emptying industrial tanks and basins. Time grows with reservoir area and the square root of head, and falls with orifice area — emptying large reservoirs takes a long time, a real limitation in dam emergency management. It assumes constant A_s; real reservoirs vary with elevation. Enter the surface area, orifice area, initial head and discharge coefficient.

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Corrected Runway Length (ARFL)

Calculate the corrected runway length from the aircraft reference field length (ARFL) and the three ICAO correction factors: L = L₀·(1 + 0.07·E/300)·(1 + 0.01·ΔT)·(1 + 0.10·S). The basic length L₀ (m) is required at sea level, ISA atmosphere and level runway; corrections increase it for: aerodrome elevation E (+7% per 300 m, as thin air reduces lift and thrust), temperature ΔT above ISA (+1% per °C, same density reason) and effective runway slope S (+10% per 1% slope, which hinders takeoff acceleration). This is fundamental in airport planning: it decides whether a runway can serve a given aircraft at a given airport. High-altitude, hot-climate airports need far longer runways — the same aircraft needs much more runway in La Paz or Brasília than at sea level. Enter the basic length, elevation, temperature difference and slope.

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Effective Runway Slope

Calculate a runway's effective slope, S = (max elevation − min elevation) ÷ length · 100, from the highest and lowest elevations along the runway centerline and its length. The effective slope is the difference between the highest and lowest points of the longitudinal profile divided by total length — a global measure of the incline the aircraft faces. It feeds directly into the runway length correction (+10% length per 1% effective slope), since an uphill runway needs more takeoff acceleration distance. ICAO limits effective slope by runway code (typically 1-2% max for higher codes) and also limits local slopes and their rate of change for safety. Geometric design minimizes effective slope and smooths transitions, balancing earthwork and drainage. Enter the maximum and minimum elevations and the runway length.

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Take-Off Distance Available (TODA)

Calculate the Take-Off Distance Available, TODA = TORA + clearway, from the Take-Off Run Available (TORA) and the clearway length. TODA is one of the four declared distances of a runway, central ICAO operational concepts. The clearway is an obstacle-free rectangular area beyond the runway over which the aircraft can complete the initial climb to a minimum height — it extends takeoff distance without extra pavement, since the aircraft is already airborne. The declared distances (TORA, TODA, ASDA, LDA) are published for each runway threshold and used by pilots and dispatchers to verify, for each takeoff, that the aircraft — with its weight, configuration and the day's conditions — fits the available runway with required margins. Enter the TORA and the clearway length.

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Accelerate-Stop Distance Available (ASDA)

Calculate the Accelerate-Stop Distance Available, ASDA = TORA + stopway, from the Take-Off Run Available (TORA) and the stopway length. ASDA is one of the four declared distances and has a critical safety role: it is the distance available to accelerate to the decision speed (V₁) and, if the pilot aborts the takeoff (engine failure or other), still stop safely. The stopway is a paved (or adequately strong) area beyond the runway, able to bear the aircraft in an emergency stop, not used in normal operation. Unlike the clearway (for the airborne aircraft), the stopway is for the aircraft on the ground, braking. ASDA is decisive in the balanced field length concept: the point where the distance to continue takeoff (one engine out) equals the distance to abort and stop defines V₁ and the required runway length. Enter the TORA and the stopway length.

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Landing Gear Wheel Load

Calculate the main landing gear wheel load, P_wheel = (W·f) ÷ n, from the aircraft weight W (N), the fraction of weight carried by the main gear f (typically ~0.90-0.95, the nose gear carries the rest) and the number of main gear wheels n. This load is the starting point of airport pavement design: it is the force each wheel transmits to the pavement, governing the required thickness and strength of runways, taxiways and aprons. Modern aircraft spread their huge weight over multi-wheel gears (4, 6 or more wheel bogies) precisely to reduce wheel load and pavement damage. The concept links to the ACN/PCN system (Aircraft/Pavement Classification Number) for compatibility checks, and to the equivalent single-wheel load (ESWL) that converts a real multi-wheel gear into one equivalent wheel for design. Enter the aircraft weight, the main gear fraction and the number of wheels.

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Runway Hourly Capacity

Estimate a runway's hourly capacity, C = 3600 ÷ T, from the average occupancy or separation time between successive operations T (seconds). A runway's capacity — the maximum operations (landings and takeoffs) per hour — is one of the most important airport planning parameters, setting the airport's traffic limit. The time T is governed by minimum wake-turbulence separation, runway occupancy time (from touchdown to clearing via a rapid-exit taxiway), air traffic control procedures and the aircraft mix. Well-run single runways reach about 40-60 operations per hour; capacity rises with parallel runways, high-speed exits (reducing occupancy time) and optimized procedures. As demand nears capacity, delays grow nonlinearly (queueing theory), driving expansions or flow management (slots). Enter the average time between operations.

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Apron Gate Positions

Estimate the number of aircraft parking positions (gates) needed at an airport apron, N = (movements per hour · average dwell time) ÷ 60, from the peak hourly aircraft arrivals and the average dwell (turnaround) time per aircraft, in minutes. The reasoning is a queueing one: if M aircraft arrive per hour and each occupies a position for t minutes, the number simultaneously occupied (and thus needed) is M·t/60. Dwell time includes deboarding, cleaning, fueling, baggage and boarding — typically 30-60 minutes for domestic and more for international. Gate count is a critical terminal sizing: too few causes aircraft waiting to park (very costly) or remote bus stands; too many wastes valuable area and capital. Real design considers aircraft size mix (a wide-body gate occupies several smaller ones), daily variation and an irregularity margin. Enter the peak hourly movements and the average dwell time.

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Taxi Time on Taxiway

Calculate an aircraft's taxi time on a taxiway, t = distance ÷ speed, from the distance to cover (m) and the taxi speed (m/s). Taxiing is the aircraft's ground movement between the runway and the apron (gate), under its own engines, at low speed. Taxi time is an important component of total operation time and cost: long taxis (at large airports with runways far from the terminal) burn fuel, generate emissions and delays, and occupy capacity-limited taxiways. Taxi speed is typically 5-15 m/s (about 20-50 km/h) on straights, slowing in curves and crossings. Taxi time feeds ground-traffic modeling, taxiway system sizing, ground fuel-burn and emissions estimates, and airport capacity studies. Efficient airports minimize taxi distances and conflicts with good geometry, well-placed runway exits and ground-traffic management. Enter the taxi distance and speed.

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Aircraft Apron Area

Estimate the total area of an aircraft apron, A = number of positions · area per position, from the number of parking positions and the average area each occupies (m²), including the aircraft, surrounding safety clearances and service/circulation lanes. The apron is the airport area where aircraft park for passenger boarding, cargo and ground services. Area per position depends strongly on aircraft size: a code-F position (such as the A380) needs a square tens of metres on a side plus safety margins, occupying several thousand square metres; regional aircraft positions are much smaller. Apron sizing is one of the biggest area consumers on an airport's airside and a high investment (reinforced pavement for parked and maneuvering aircraft loads). The simplified calculation (positions × average area) gives the planning order of magnitude; detailed design positions each gate per the aircraft mix, operation type (nose-in with pushback, or self-maneuvering) and terminal geometry. Enter the number of positions and the area per position.