PEC EPE Metallurgical cheat sheet

Pyrometallurgy relies on high-temperature reduction, roasting, and smelting; hydrometallurgy uses aqueous leaching, solvent extraction, and electrowinning; electrometallurgy applies electric current for electrolysis or arc heating.

Ellingham diagram: a metal oxide is reducible by another element whose ΔG° line lies below it at the operating temperature; CO/CO2 crossover near 710 °C explains blast-furnace indirect reduction limits.

Leaching rate follows the shrinking-core model; for diffusion control through a product layer, 1-3(1-x)^(2/3)+2(1-x) = kt, while chemical control gives 1-(1-x)^(1/3) = kt.

Cyanidation of gold: 4Au + 8NaCN + O2 + 2H2O → 4Na[Au(CN)2] + 4NaOH; typical leach pH 10-11 to suppress HCN formation.

Solvent extraction separates metals by distribution coefficient D = [M]org/[M]aq; separation factor β = D1/D2 and stage count rises sharply as β approaches 1.

Electrowinning of copper: cathode reaction Cu2+ + 2e- → Cu (E° = +0.34 V); anode oxygen evolution at ~1.23 V means cell voltage ~2 V and specific energy near 2 kWh/kg.

Froth flotation selectivity depends on collector adsorption; xanthates target sulfide minerals, while depressants such as cyanide or lime suppress pyrite and sphalerite.

Reverberatory and flash smelting separate matte from slag by density; slag viscosity falls with higher FeO and CaO but rises with SiO2 and Al2O3 content.

Gibbs phase rule F = C - P + 2 for unary and binary systems at variable T and P; at constant pressure F = C - P + 1.

Eutectic, eutectoid, and peritectic reactions: L→α+β, γ→α+β, and L+α→β respectively; the Fe-C eutectoid (pearlite) occurs at 0.76 wt% C and 727 °C.

Martensite forms by diffusionless shear; its hardness scales with carbon content up to about 0.6 wt% C, and Ms temperature falls roughly 20-30 °C per wt% C.

Tempering stages: epsilon-carbide precipitation near 100-200 °C, retained austenite decomposition near 200-300 °C, and cementite coarsening above 400 °C.

Hall-Petch relation σy = σ0 + k d^(-1/2) describes grain-boundary strengthening; finer grains raise yield strength and improve toughness.

Jominy end-quench test measures hardenability; alloying elements (Mn, Cr, Mo, Ni) shift the CCT curve right and increase the depth of full hardening.

Recrystallization temperature is roughly 0.3-0.4 Tm (absolute); cold work above the critical strain drives nucleation of new strain-free grains.

Solid solution strengthening follows mismatch in atomic radius and modulus; interstitial solutes such as C and N are far more potent than substitutional ones.

ΔG = ΔH - TΔS; a reaction is spontaneous when ΔG < 0, and equilibrium constant K = exp(-ΔG°/RT).

Van't Hoff equation d(lnK)/dT = ΔH°/(RT²): endothermic reactions are favored by higher temperature, exothermic ones by lower temperature.

Ellingham plots use ΔG° per mole of O2; the slope equals ΔS°, and phase changes produce kinks in the line.

Activity coefficient models: Raoult's law applies to solvents (a = x), Henry's law to dilute solutes (a = γx with constant γ).

Arrhenius equation k = A exp(-Q/RT); plotting ln k versus 1/T gives slope -Q/R and enables activation energy determination.

Fick's first law J = -D dC/dx; second law ∂C/∂t = D ∂²C/∂x², with D = D0 exp(-Q/RT) for thermally activated diffusion.

Mixing entropy ΔS_mix = -R Σ xi ln xi; ideal solutions have zero enthalpy of mixing and positive entropy.

Carbothermic reduction becomes favorable at high temperature because the CO product gives a large negative entropy contribution (Boudouard reaction C + CO2 → 2CO).

Blast furnace zones: lumpy zone, cohesive zone, dripping zone, raceway, and hearth; coke provides heat, reductant, and burden permeability.

Direct reduction by carbon occurs above ~1000 °C (C + CO2 → 2CO, then FeO + CO → Fe + CO2); indirect reduction dominates at lower temperature.

Basic oxygen steelmaking blows pure O2 onto hot metal; carbon, silicon, and manganese oxidize, and lime flux forms a basic slag to remove phosphorus and sulfur.

Slag basicity B = CaO/SiO2; typical BOF slag is 3-4, and higher basicity improves dephosphorization but raises slag volume.

Electric arc furnace uses scrap and direct arc heating; foamy slag practice and tap-to-tap time govern productivity and energy (about 350-450 kWh/t).

Secondary metallurgy: LF for heating and desulfurization, VD/VOD for degassing hydrogen and nitrogen, and calcium treatment to modify alumina inclusions.

Continuous casting: mold oscillation, primary shell formation, secondary spray cooling, and electromagnetic stirring control surface quality and centerline segregation.

Sulfur removal needs low oxygen potential and high slag basicity; manganese addition counters hot shortness by forming MnS instead of grain-boundary FeS.

Crystal structures: FCC has 4 atoms/cell and APF 0.74; BCC has 2 atoms/cell and APF 0.68; HCP has 6 atoms/cell and APF 0.74.

FCC metals (Cu, Al, austenite) slip on {111}<110> with 12 systems; BCC (Fe, W) has 48 systems, giving a temperature-sensitive yield stress.

Bragg's law nλ = 2d sinθ underpins XRD phase identification; peak broadening indicates small crystallite size or microstrain.

Point defects: Schottky defects dominate in ionic crystals with charge balance; Frenkel pairs preserve stoichiometry by vacancy-interstitial pairs.

Diffusion in FCC metals is generally faster than in BCC at the same homologous temperature because of the more open interstitial path.

Polymers: Tg marks the glass transition, Tm the melting point; crystallinity raises stiffness, density, and opacity.

Composites: rule of mixtures Ec = VfEf + VmEm for longitudinal loading; load transfer at the interface governs transverse and shear behavior.

Ceramics are brittle because slip systems are limited by ionic charge neutrality; toughening relies on transformation or crack bridging.

Engineering stress σ = F/A0 and strain e = ΔL/L0; true stress σt = σ(1+e) and true strain εt = ln(1+e) up to necking.

Considère criterion: necking begins when dσt/dεt = σt, equivalent to the strain-hardening exponent n equaling the true strain at UTS.

Hollomon equation σt = K εt^n; n measures formability and K the strength coefficient.

Charpy V-notch impact test measures absorbed energy; ductile-to-brittle transition is indexed at 27 J or 50% shear fracture appearance.

Fatigue: Basquin relation σa = σf'(2Nf)^b; endurance limit exists for steels but generally not for aluminum alloys.

Fracture toughness KIc relates to stress and flaw size by KI = Y σ √(πa); failure occurs when KI reaches KIc.

Hardness conversions are approximate; Vickers and Brinell correlate better with tensile strength (UTS ≈ 3.3 HB for steels) than Rockwell.

Creep stages: primary (decreasing rate), secondary (steady-state, Norton law ε̇ = A σ^n exp(-Q/RT)), and tertiary leading to rupture.

Liberation is the prerequisite for separation; comminution energy follows Bond's law W = 10 Wi (1/√P80 - 1/√F80) in kWh/t with P80 and F80 in µm.

Crushing stages: primary jaw or gyratory, secondary cone, tertiary cone or HPGR; reduction ratio per stage typically 3-5 for crushers.

Screening efficiency depends on aperture, bed depth, and moisture; moist sticky ores favor wet screening or trommels.

Gravity separation uses density difference; jigs, spirals, and shaking tables suit coal and heavy minerals, with performance described by the Tromp curve and Ep = (d75 - d25)/2.

Magnetic separation: low-intensity drums for magnetite, high-intensity or WHIMS for hematite and paramagnetic minerals.

Froth flotation reagents: collectors (xanthate), frothers (MIBC), activators (CuSO4 for sphalerite), depressants (NaCN, lime), and pH modifiers.

Dewatering sequence: thickening by settling (Stokes' law), filtration, and thermal drying; flocculants increase settling rate.

Grade-recovery trade-off: raising concentrate grade usually lowers recovery; the separation efficiency curve quantifies the compromise.

Electrochemical corrosion requires an anode, cathode, electrolyte, and metallic path; oxidation occurs at the anode (M → M^n+ + ne-).

Standard EMF series ranks metals by E°; galvanic series ranks them in a specific environment such as seawater, which is more practical.

Pourbaix diagrams plot potential versus pH and show immunity, corrosion, and passivation regions for a metal in water.

Faraday's law: mass loss m = (M I t)/(n F); corrosion rate in mm/y = 3.27 × 10^-3 (i corr in µA/cm² × equivalent weight)/density.

Forms: uniform, galvanic, crevice, pitting, intergranular (sensitization of austenitic stainless steel by Cr carbide at grain boundaries), and stress corrosion cracking.

Cathodic protection: sacrificial anodes (Zn, Al, Mg) for buried pipelines, or impressed current with inert anodes; criteria often -850 mV vs Cu/CuSO4.

Stainless steel passivity comes from a Cr-rich oxide film; chloride breaks it locally, causing pitting above a critical pitting temperature.

Corrosion rate doubles roughly per 10 °C rise in many aqueous systems; dissolved oxygen and chloride accelerate attack.

Case hardening processes: carburizing (C into austenite, 0.7-1.0 mm case), nitriding (N at 500-550 °C, no quench), carbonitriding, and induction or flame hardening.

Coating methods: electroplating, electroless Ni-P, PVD (sputtering, evaporation), CVD, thermal spray, and hot-dip galvanizing.

Adhesion and residual stress govern coating performance; PVD films often carry compressive stress, improving fatigue resistance but risking spallation.

Failure analysis sequence: collect background, inspect visually and macroscopically, examine fracture surface (SEM), metallography, chemical analysis, and simulate to test hypotheses.

Fracture surface features: beach marks from fatigue, chevron marks pointing to crack origin, dimples for ductile overload, cleavage facets for brittle failure.

Hydrogen embrittlement appears in high-strength steels after plating; baking at 190-220 °C within hours of plating mitigates it.

Galvanic corrosion in joints is mitigated by selecting metals close in the galvanic series, insulating the joint, or applying a coating on the cathodic member.

Wear mechanisms: adhesive, abrasive, erosive, and fretting; hard coatings (TiN, WC-Co, DLC) address abrasive and adhesive wear.

Chvorinov's rule: solidification time t = B (V/A)^2, so risers must have a larger modulus (V/A) than the casting section they feed.

Riser design: a riser feeds a section only if its modulus exceeds that of the section; a riser modulus about 1.2 times the casting modulus is typical.

Casting defects: shrinkage porosity (inadequate feeding), gas porosity (dissolved gas), hot tears (restraint during contraction), cold shuts, and misruns.

Sand casting: green sand molds use bentonite and water; sodium silicate (CO2) and resin-bonded molds give better dimensional accuracy.

Gating systems: sprue, runner, and ingate; a tapered sprue maintains a full flow and prevents air aspiration (continuity A1V1 = A2V2).

Fluidity is reduced by higher superheat loss, alloy freezing range, and mold thermal conductivity; long freezing-range alloys are hard to feed.

Directional solidification is achieved with chills, padding, and exothermic sleeves; the freezing front should progress toward the riser.

Casting yield = (casting weight)/(pouring weight) × 100; riser and gating weight reduce yield.

Arc welding heat input H = (η V I)/v (J/mm); higher heat input widens the heat-affected zone and can coarsen grains.

Carbon equivalent CE = C + Mn/6 + (Cr+Mo+V)/5 + (Ni+Cu)/15; CE above about 0.4 raises cold-cracking risk and requires preheat.

SMAW uses a flux-coated electrode; the flux shields the arc and forms slag, and electrodes must be baked to avoid hydrogen pickup.

GMAW (MIG/MAG) uses a continuous wire and shielding gas; short-circuit, globular, and spray transfer modes depend on current and gas.

TIG (GTAW) uses a non-consumable tungsten electrode and separate filler; it suits thin sections and reactive metals but is slow.

Resistance spot welding heat depends on I²Rt; electrode force, current, and time determine nugget size, monitored by weld-lobe diagrams.

Weld defects: lack of fusion, incomplete penetration, porosity, slag inclusion, undercut, and solidification cracking (hot cracking) in the weld metal.

Solid-state joining: friction stir welding suits aluminum and avoids fusion defects; diffusion bonding requires temperature, pressure, and time.

Professional codes require competence only within one's area of expertise, honesty in reporting, and disclosure of conflicts of interest.

Public safety takes precedence over employer or client interests when a design poses a serious risk; escalation follows internal channels before external disclosure.

Whistleblowing is justified when the harm is serious, internal remedies are exhausted, and the disclosure is proportionate and documented.

Engineers must not sign or seal work they did not prepare or supervise; the seal certifies competence and accountability.

Confidentiality obligations persist after employment ends, except for information that is legally required to be disclosed.

Safety hierarchy of controls: elimination, substitution, engineering controls, administrative controls, and PPE, in that order of effectiveness.

Hazardous atmospheres in confined spaces require testing for oxygen, flammables, and toxics; entry permits and a standby attendant are mandatory.

Sustainable practice includes life-cycle thinking, waste minimization, and compliance with environmental permits and reporting duties.

Time value of money: F = P(1+i)^n and P = F(1+i)^-n; the factor (1+i)^n is the compound amount factor.

Annuity: A = P [i(1+i)^n]/[(1+i)^n - 1] for capital recovery, and P = A [(1+i)^n - 1]/[i(1+i)^n] for present worth.

Net present value: accept projects with NPV > 0; IRR is the rate at which NPV = 0, and accept when IRR exceeds the MARR.

Depreciation methods: straight line (P - S)/n, declining balance with rate d, and MACRS accelerated schedules in the US tax code.

Payback period ignores cash flows after payback; use it as a screening tool, not a primary decision criterion.

Break-even quantity Q = FC/(p - vc), where p is unit price and vc variable cost per unit.

Inflation: real rate ≈ (1 + nominal)/(1 + inflation) - 1; use real rates with constant-dollar cash flows.

Benefit-cost ratio above 1.0 and incremental analysis are used for public-sector alternatives; mutually exclusive alternatives require incremental comparison.

Critical path method: the critical path has zero total float; total float = LS - ES = LF - EF, and any delay on the critical path delays the project.

Earned value: CV = EV - AC, SV = EV - PV, CPI = EV/AC, SPI = EV/PV; CPI below 1 means cost overrun.

Estimate at completion EAC = BAC/CPI when current performance persists; EAC = AC + (BAC - EV) when remaining work proceeds as planned.

Control charts: X-bar and R charts for variables; p and c charts for attributes; a process is in statistical control when points fall within ±3σ and show no patterns.

Process capability: Cp = (USL - LSL)/(6σ) and Cpk = min(USL - μ, μ - LSL)/(3σ); Cpk ≥ 1.33 is a common target.

Seven quality tools: Pareto chart, cause-and-effect (fishbone) diagram, check sheet, histogram, scatter diagram, control chart, and flowchart.

Risk management: identify, assess probability and impact, plan responses (avoid, mitigate, transfer, accept), and monitor throughout the project.

Scope change control requires formal change requests, impact assessment on cost and schedule, and approval before implementation.