GATE Environmental Science & Engineering (ES) Calculators
Calculators for the numerical questions in GATE Environmental Science and Engineering, from water chemistry and hydraulics to treatment, air pollution and noise, each showing the working.
42 calculators · 5 converters · common conversions
Mathematics Foundation
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Eigenvalue Calculator
det(A − λI) = 0
Σλ = trace A, Πλ = det A
Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.
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Newton-Raphson Method Calculator
xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
quadratic convergence near a simple root
Example: f(x) = x³ − 2x − 5 starting at x₀ = 2: Root x ≈ 2.09455148154.
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- Lagrange Interpolation Calculator – Data points and x → interpolated value, Lagrange basis values and the polynomial.
- Numerical Integration Calculator – f(x), limits and intervals → trapezoidal, Simpson 1/3 and 3/8 results with a reference value.
- Euler Method Calculator – An ODE, initial value, step and end point → y by Euler, Heun, RK2 or RK4 with every step.
- Standard Deviation Calculator – Paste a list of numbers → standard deviation, variance, mean, median, MAD and CV with steps.
- Correlation & Regression Calculator – Paste x and y values → correlation r, R², slope, intercept and regression equation.
- Bayes’ Theorem Calculator – Priors and likelihoods → total probability and posterior probabilities, with a Bayes table.
- Normal Distribution Calculator – Mean, standard deviation and bounds → normal probabilities, z-scores, density and percentiles.
Environmental Chemistry
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pH Calculator
pH = −log₁₀[H⁺], pH + pOH = 14
[H⁺] hydrogen-ion concentration (mol/L), [OH⁻] hydroxide concentration; pKw = 14 at 25 °C
Example: A solution with [H⁺] = 0.001 mol/L (0.001 M HCl): pH = 3.
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Water Hardness Calculator
TH = 2.497 Ca + 4.118 Mg (mg/L as CaCO₃)
carbonate hardness = min(TH, alkalinity); non-carbonate = TH − carbonate
Example: Water with 60 mg/L calcium, 24 mg/L magnesium and 150 mg/L alkalinity: Total hardness = 248.669 mg/L as CaCO₃.
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- Sodium Adsorption Ratio Calculator – Sodium, calcium and magnesium → SAR and sodium hazard class.
- Molarity Calculator – Mass, molar mass and volume → molarity; or the mass to weigh out for a target molarity.
- Molar Mass Calculator – A chemical formula → molar mass and the mass percent of each element.
- Combined Gas Law Calculator – Five of initial and final pressure, volume and temperature → the sixth.
Environmental Microbiology
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Microbial Growth Calculator
N = N₀ e^(μt), t_d = ln 2 / μ
generations n = log₂(N/N₀)
Example: A culture growing from 1000 to 64,000 cells in 6 hours: Specific growth rate μ = 0.693147 per h.
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Monod Equation Calculator
μ = μmax S / (Ks + S)
Ks the substrate concentration at μ = μmax/2; doubling time ln 2/μ
Example: A culture with μmax = 0.5 per hour and Ks = 0.2 g/L at S = 1 g/L: Specific growth rate μ = 0.416667 per h.
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- Michaelis-Menten Calculator – Km, Vmax and [S] (and inhibitor) → reaction rate; or rate data → Km and Vmax.
- Disinfection CT Calculator – Disinfectant concentration, contact time and rate constant → log removal, CT and survivors.
Water Resources and Hydraulics
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Darcy's Law Calculator
Q = K i A, i = Δh/L
Darcy velocity q = Ki; seepage (linear) velocity = q/nₑ
Example: An aquifer with K = 10 m/day, 2 m head drop over 100 m and 50 m² cross-section: Discharge Q = 10 m³/day.
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Well Yield Calculator
Confined: Q = 2πKb(h₂ − h₁)/ln(r₂/r₁)
unconfined: Q = πK(h₂² − h₁²)/ln(r₂/r₁)
Example: A confined aquifer, K = 20 m/day, 30 m thick, with heads 40 m at the 0.15 m well and 45 m at 300 m: Well discharge Q = 2479.91 m³/day.
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- Manning's Equation Calculator – Channel shape, depth, slope and roughness → discharge, velocity and flow regime.
- Critical Depth Calculator – Discharge per unit width (and a depth) → critical depth, minimum specific energy and flow regime.
- Hydraulic Jump Calculator – Upstream depth and velocity → sequent depth, energy loss, power and jump type.
- Bernoulli Equation Calculator – Pressure, velocity and elevation at 1, velocity and elevation at 2 → pressure at 2 and heads.
- Hydrostatic Force Calculator – Rectangle or circle, depth and angle → total hydrostatic force and centre of pressure.
- Reynolds Number Calculator – Velocity, diameter, density and viscosity in any units → Re and flow regime.
- Pipe Pressure Drop Calculator – Flow + pipe size + material + fluid → pressure drop, velocity, Re and friction factor.
Water and Wastewater Treatment
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BOD Calculator
BOD_t = L₀(1 − e^(−kt)), BOD₅ = (D₁ − D₂)/P
k_T = k₂₀ × 1.047^(T − 20); P = sample volume ÷ bottle volume
Example: Wastewater with ultimate BOD 250 mg/L and k = 0.23 per day at 20 °C, after 5 days: BOD at day 5 = 170.841 mg/L.
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Oxygen Sag Curve Calculator
D_t = (k_d L₀/(k_r − k_d))(e^(−k_d t) − e^(−k_r t)) + D₀e^(−k_r t)
t_c = ln[(k_r/k_d)(1 − D₀(k_r − k_d)/(k_d L₀))]/(k_r − k_d); D_c = (k_d/k_r)L₀e^(−k_d t_c)
Example: A river with mixed L₀ = 20 mg/L, D₀ = 1 mg/L, k_d = 0.23 and k_r = 0.46 per day: Critical (maximum) deficit D_c = 5.26316 mg/L.
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- Mixing Mass Balance Calculator – Two flows and their concentrations (and temperatures) → mixed concentration, flow and temperature.
- Population Forecast Calculator – A series of census populations → forecasts by arithmetic, geometric and incremental increase.
- Stokes' Law Settling Calculator – Particle size and density, fluid density and viscosity → settling velocity and Reynolds number.
- Sedimentation Tank Calculator – Flow and tank dimensions → overflow rate, detention time, horizontal velocity and removal.
- CSTR and PFR Reactor Calculator – Rate constant and detention time (or target removal) → outlet concentration and reactor sizes.
Air and Noise Pollution
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Gaussian Plume Calculator
C = Q/(2πuσ_yσ_z) e^(−y²/2σ_y²)[e^(−(z−H)²/2σ_z²) + e^(−(z+H)²/2σ_z²)]
σ_y, σ_z from Briggs open-country formulas; ground reflection included
Example: 100 g/s from a 50 m effective stack, 5 m/s wind, neutral (D) stability, 1 km downwind at ground level: Concentration C = 923.238 µg/m³.
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Cyclone Separator Calculator
d₅₀ = √(9μW / (2πNₑVᵢ(ρₚ − ρ_g)))
η = 1/(1 + (d₅₀/d)²) for particle diameter d
Example: A cyclone with 0.25 m inlet, 5 turns, 15 m/s inlet velocity, for 10 µm particles of 1500 kg/m³: Collection efficiency η = 62.919 %.
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- Electrostatic Precipitator Calculator – Drift velocity, gas flow and plate area → collection efficiency; or the area for a target.
- Noise Level Calculator – Sound levels and times or distances → combined level, Leq, Ldn or level at a new distance.
- Decibel Calculator – Power or voltage ratio ↔ dB, and watts ↔ dBm.
Global Environmental Issues
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CO₂-Equivalent Calculator
CO₂e = mass × GWP₁₀₀
AR6 GWP-100: CH₄ 27.0 (non-fossil) or 29.8 (fossil), N₂O 273, SF₆ 25,200
Example: 10 tonnes of biogenic methane: CO₂-equivalent = 270 t CO₂e.
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Half-Life Calculator
N = N₀ (½)^(t / T½)
N amount left, N₀ initial amount, t elapsed time, T½ half-life; λ = ln 2 / T½
Example: 100 g of carbon-14 (half-life 5730 years) after 11,460 years: Amount remaining N = 25.
Unit Converters for GATE Environmental (ES)
- Flow Rate Converter – CFM, m³/h, L/min, L/s, GPM, sccm, SLPM
- Volume Converter – L, mL, m³, ft³, in³, US gal, imp gal
- Pressure Converter – Pa, kPa, psi, bar, inH₂O, mmHg, torr, atm
- Temperature Converter – °C, °F, K, °R
- Weight and Mass Converter – kg, g, mg, lb, oz, st, t, ton, oz t, ct
Common GATE Environmental (ES) Conversions
How to use these GATE Environmental Science calculators
GATE Environmental Science and Engineering mixes chemistry, microbiology, hydraulics and process design, and its numerical questions repeat a core set of equations: first-order BOD and the Streeter–Phelps oxygen sag, flow-weighted mixing, hardness and the sodium adsorption ratio, Darcy’s law and the Thiem well equations, Manning’s equation and open-channel flow, Stokes settling and overflow rates, plug-flow and mixed reactors, Chick’s law, the Gaussian plume, cyclones, electrostatic precipitators and decibel arithmetic.
Each calculator states its convention: rate constants in base e with the base-10 value alongside, temperature corrections with the usual θ values, Briggs rural coefficients for dispersion and IPCC AR6 global warming potentials. Questions sometimes use other conventions, so compare the constants before comparing the answers.
Solve the question by hand, then confirm here. Unit slips are the most common error: mg/L versus meq/L, days versus hours in rate constants, and m³/day versus m³/s in overflow rates.