GATE Metallurgical Engineering (MT) Calculators
Calculators for the numerical questions in GATE Metallurgical Engineering, from thermodynamics and diffusion to crystallography, mechanical behaviour and processing, each showing the working.
41 calculators · 6 converters · common conversions
Engineering Mathematics
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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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- Bisection Method Calculator – f(x) and an interval where it changes sign → root by bisection or false position, with every iteration.
- Secant Method Calculator – f(x) and two starting points → root by the secant method with each iteration.
- Numerical Integration Calculator – f(x), limits and intervals → trapezoidal, Simpson 1/3 and 3/8 results with a reference value.
- Normal Distribution Calculator – Mean, standard deviation and bounds → normal probabilities, z-scores, density and percentiles.
- Correlation & Regression Calculator – Paste x and y values → correlation r, R², slope, intercept and regression equation.
Metallurgical Thermodynamics
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Gibbs Free Energy Calculator
ΔG = ΔH − TΔS, K = exp(−ΔG/RT)
R = 8.314 J/(mol·K); ΔG = 0 at T = ΔH/ΔS
Example: A reaction with ΔH = −200 kJ/mol and ΔS = −100 J/(mol·K) at 1000 K: Gibbs free energy ΔG = -100 kJ/mol.
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Lever Rule Calculator
Wα = (Cβ − C₀) / (Cβ − Cα)
Fe–C: eutectoid 0.76 wt% C, ferrite 0.022, cementite 6.7 wt% C
Example: An alloy of 35 wt% B between α at 20 wt% and β at 60 wt%: Fraction of α, Wα = 62.5 %.
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- Gibbs Phase Rule Calculator – Number of components and phases → degrees of freedom.
- Free Energy of Mixing Calculator – Composition, temperature and interaction parameter → ΔG_mix, ΔH_mix, ΔS_mix and activity.
- Nernst Equation Calculator – Ion, concentrations and temperature → equilibrium (Nernst) potential.
- Faraday's Law Electrolysis Calculator – Metal, current, time and current efficiency → mass deposited and electrochemical equivalent.
Transport Phenomena and Rate Processes
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Arrhenius Equation Calculator
k = A e^(−Eₐ/RT)
ln(k₂/k₁) = (Eₐ/R)(1/T₁ − 1/T₂)
Example: A process with A = 10¹³ and Eₐ = 150 kJ/mol at 800 K: Rate constant k = 1607.67.
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Diffusion Calculator
(Cₓ − C₀)/(Cₛ − C₀) = 1 − erf(x / (2√(Dt)))
D = D₀ exp(−Q/RT); diffusion distance ~ √(Dt)
Example: Carburising 0.2 % C steel at 950 °C with a 1.2 % C surface for 10 hours: Concentration at depth Cₓ = 0.774281 wt%.
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- Heat Conduction Calculator – Geometry, conductivities and temperatures → heat flow, thermal resistance and interface temperatures.
- Radiation Heat Transfer Calculator – Temperatures, emissivities and area → net radiation, emissive power and Wien peak wavelength.
- Reynolds Number Calculator – Velocity, diameter, density and viscosity in any units → Re and flow regime.
- Bernoulli Equation Calculator – Pressure, velocity and elevation at 1, velocity and elevation at 2 → pressure at 2 and heads.
Physical Metallurgy
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Bragg's Law Calculator
nλ = 2d sin θ, d = a/√(h² + k² + l²)
Cu Kα₁ λ = 1.5406 Å; cubic crystals
Example: The (110) peak of α-iron at 2θ = 44.67° with Cu Kα radiation: Interplanar spacing d = 2.02699 Å.
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Crystal Structure and Metal Properties Calculator
ρ = nA / (V_c Nₐ)
n atoms per cell (BCC 2, FCC 4, HCP 6), V_c cell volume; BCC r = √3a/4, FCC r = √2a/4
Example: α-iron, body-centred cubic with a = 2.8665 Å: Theoretical density ρ = 7.87422 g/cm³.
Open the crystal structure and metal properties calculator →
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- Vacancy Concentration Calculator – Formation energy, temperature, density and atomic mass → vacancy fraction and number per m³.
- Nucleation Calculator – Interfacial energy, latent heat, melting point and undercooling → critical radius and nucleation barrier.
- Avrami Equation Calculator – Rate constant and Avrami exponent → fraction transformed at a time, or time for a fraction.
- ASTM Grain Size Calculator – Grain count, G or mean intercept → ASTM grain size number, grains per mm² and intercept length.
- Corrosion Rate Calculator – Coupon mass loss + area + time + density → corrosion rate in mm/y, mpy, µm/y (ASTM G1).
Mechanical Behaviour of Materials
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True Stress-Strain Calculator
σ_T = σ_e(1 + ε_e), ε_T = ln(1 + ε_e)
Hollomon σ_T = Kε_Tⁿ; necking begins at ε_T = n
Example: A tensile test reading 400 MPa at 20 % engineering strain: True stress σ_T = 480 MPa.
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Hall-Petch Calculator
σ_y = σ₀ + k_y / √d
σ₀ lattice friction stress, k_y Hall–Petch coefficient, d grain diameter
Example: Mild steel (σ₀ = 70 MPa, k_y = 0.74 MPa·m^½) with 25 µm grains: Yield strength σ_y = 218 MPa.
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- Von Mises Stress Calculator – Plane or principal stresses and yield strength → von Mises and Tresca stresses and factors of safety.
- Mohr’s Circle Calculator – σx, σy and τxy → principal stresses, maximum shear, principal angle and stresses on a plane.
- Fracture Toughness Calculator – Geometry factor, stress and crack length (or K_Ic) → stress intensity, critical stress or crack length.
- Paris Law Calculator – Paris constants, stress range and crack lengths → fatigue cycles to grow the crack.
- Larson-Miller Parameter Calculator – Temperature and rupture time → Larson–Miller parameter, or rupture life at another temperature.
- Tensile Stress Calculator – Load + round, rectangular or tube section → tensile or yield stress in MPa, N/mm², psi, ksi.
- Young's Modulus Calculator – Stress and strain → Young's modulus, or E with one of them → the other (Hooke's law).
Manufacturing Processes
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Chvorinov's Rule Calculator
t = B (V/A)ⁿ
B mould constant, n ≈ 2; riser modulus ≥ 1.2 × casting modulus
Example: A 200 × 100 × 50 mm block with B = 2 s/mm² and n = 2: Solidification time t = 6.80272 min.
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Rolling Calculator
Δh_max = μ²R, L = √(RΔh)
F ≈ σ̄ w L (no friction hill); true strain ln(h₀/h₁)
Example: Rolling 25 mm plate to 20 mm with 300 mm radius rolls and μ = 0.1: Draft Δh = 5 mm.
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Mesh to Micron Converter
opening ≈ 25.4 mm ÷ mesh − wire diameter
mesh wires per inch; standard sieve openings are fixed by ASTM E11
Example: No. 200 sieve = 75 µm, No. 100 = 150 µm.
Materials Science and Non-Metallic Materials
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Rule of Mixtures Calculator
Isostrain: E_c = V_fE_f + V_mE_m
isostress: 1/E_c = V_f/E_f + V_m/E_m
Example: Carbon fibre (230 GPa) in epoxy (3.5 GPa) at 60 % fibre volume: Longitudinal modulus (isostrain) E_c = 139.4 GPa.
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Polymer Molecular Weight Calculator
Mₙ = Σ xᵢMᵢ, M_w = Σ xᵢMᵢ² / Σ xᵢMᵢ
xᵢ number fraction; PDI = M_w/Mₙ ≥ 1; DP = Mₙ/m₀
Example: Fractions of 10, 20, 30 and 40 kg/mol with number fractions 0.1, 0.3, 0.4 and 0.2: Number-average molecular weight Mₙ = 27000 g/mol.
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- Radius Ratio Calculator – Cation and anion radii → radius ratio, coordination number and typical structure.
- Thermal Expansion Calculator – Material or α + length + temperature change → expansion ΔL and new length.
Unit Converters for GATE Metallurgy (MT)
- Density Converter – kg/m³, g/cm³, g/mL, lb/ft³, lb/gal, mg/L
- Temperature Converter – °C, °F, K, °R
- Pressure Converter – Pa, kPa, psi, bar, inH₂O, mmHg, torr, atm
- Corrosion Rate Converter – mpy, mm/y, µm/y, ipy
- Thermal Expansion Coefficient Converter – 1/K, 1/°C, 1/°F, ppm/K, ppm/°F
- Mesh to Micron Converter – US sieve number (mesh) ↔ opening in µm, mm and inches, with the full ASTM E11 table.
Common GATE Metallurgy (MT) Conversions
- gram per cubic centimeter to kilogram per cubic meter
- kilogram per cubic meter to gram per cubic centimeter
- Celsius to Fahrenheit
- Fahrenheit to Celsius
- pascal to psi
- psi to pascal
- mil per year to millimetre per year
- millimetre per year to mil per year
- ppm per kelvin to ppm per degree Fahrenheit
- ppm per degree Fahrenheit to ppm per kelvin
How to use these GATE Metallurgical Engineering calculators
GATE Metallurgy questions draw on thermodynamics, kinetics, crystallography and mechanics in roughly equal measure. This hub collects calculators for Gibbs energy and equilibrium constants, the lever rule and the Fe–C diagram, the phase rule and regular solutions, Faraday’s law, Arrhenius kinetics, Fick’s second law and carburising, Avrami kinetics, Bragg diffraction, ASTM grain size, nucleation, Hall–Petch strengthening, vacancies, crystal densities, fracture toughness, Paris-law fatigue, Larson–Miller creep, true stress, composites, rolling, Chvorinov’s rule, polymer molecular weights and the radius-ratio rule.
The crystal-structure tool doubles as a lookup: choose a common metal to see its lattice, lattice parameter, theoretical and measured density, melting point, Young’s modulus and thermal conductivity from the CRC Handbook. The diffusion tool uses an accurate error function, so carburising problems can be solved in either direction.
Work each question by hand, then check it here. The usual traps are degrees Celsius in Arrhenius and Larson–Miller expressions (both need kelvin), kJ versus J in activation energies, and crack length versus half-length in fracture mechanics.