Transport Phenomena and Rate Processes: Momentum, Heat and Mass Transfer, Kinetics and Polarization

Fourier’s law, q = −k dT/dx, makes the heat flux proportional to the temperature gradient. In one-dimensional steady conduction with no generation the temperature is linear through a plane wall and q = kΔT/L; it is convenient to write the wall as a thermal resistance R = L/(kA), so that layers in series add. A furnace wall of firebrick backed by insulating brick is the standard case: the heat flux is the overall temperature difference divided by the sum of the resistances, and the interface temperature follows from the drop across the first layer — which is how one checks that the insulating brick is not being run above its service limit. For a cylindrical shell the resistance is ln(r₂/r₁)/(2πkL). A convective film at a surface adds a resistance 1/(hA).

Convection is described by Newton’s law of cooling, q = h(T_s − T_∞), where the heat transfer coefficient h lumps together the conduction through the thermal boundary layer and the flow that renews it. h is found from correlations in dimensionless form, Nu = hL/k = f(Re, Pr) for forced convection and f(Gr, Pr) for natural convection. For fully developed turbulent flow in a pipe the Dittus-Boelter correlation Nu = 0.023 Re^0.8 Pr^n (n = 0.4 for heating the fluid, 0.3 for cooling) is the one to know; fully developed laminar pipe flow has constant Nu, 3.66 at constant wall temperature and 4.36 at constant heat flux. When a body’s internal resistance is small compared with its surface resistance — Biot number Bi = h(V/A)/k < 0.1 — its temperature is uniform and it cools by the lumped-capacitance law, ρcV dT/dt = −hA(T − T_∞), so that (T − T_∞)/(T₀ − T_∞) = exp(−hAt/(ρcV)) and the instantaneous cooling rate is hA(T − T_∞)/(ρcV).

Every surface above absolute zero radiates. A black body emits E_b = σT⁴, with the Stefan-Boltzmann constant σ = 5.67 × 10⁻⁸ W/(m² K⁴); a real grey surface emits εσT⁴ with emissivity ε < 1. A small grey body in large surroundings at T_sur exchanges a net q = εσ(T_s⁴ − T_sur⁴) per unit area. Kirchhoff’s law states that at thermal equilibrium the absorptivity of a surface equals its emissivity, α = ε, at each wavelength; for a grey surface this holds over all wavelengths, which is what allows ε to be used for both emission and absorption. Because of the fourth power, radiation dominates heat loss from anything at steelmaking temperatures: a surface at 1600 °C emits (1873/1273)⁴ ≈ 4.7 times as much per unit area as one at 1000 °C.

Fick’s first law, J = −D dC/dx, is the mass-transfer counterpart of Fourier’s and Newton’s laws: the diffusive flux (mol m⁻² s⁻¹) is proportional to the concentration gradient, with the diffusivity D in m²/s. Combining it with conservation of mass gives Fick’s second law, ∂C/∂t = D ∂²C/∂x² for constant D, the equation behind carburising, decarburisation and homogenisation, and solved with the error function in the physical-metallurgy chapter. Diffusivities in liquid metals are of order 10⁻⁹ m²/s and in gases of order 10⁻⁵ m²/s, which is why gas-phase transport is rarely the slow step in a gas-metal reaction.

Between a surface and a flowing fluid the flux is written with a mass transfer coefficient, J = k_m (C_s − C_b), exactly as convection is written with h. Three models relate k_m to D. The film model imagines a stagnant film of thickness δ, giving k_m = D/δ, so k_m ∝ D. Higbie’s penetration model assumes fluid elements that touch the surface for a time t_e before being swept away, giving k_m = 2√(D/(πt_e)), so k_m ∝ D^½ — the better model for a gas bubble rising through liquid steel or a stirred bath surface. The surface-renewal model gives the same square-root dependence. Correlations are written as Sh = k_mL/D = f(Re, Sc); for a sphere, Ranz-Marshall gives Sh = 2 + 0.6 Re^½ Sc^⅓, whose limit Sh = 2 is pure diffusion into a stagnant medium. The Chilton-Colburn analogy, j_D = j_H = f/2, carries a known friction factor or heat transfer coefficient over to mass transfer.

Buckingham’s π theorem says that a relation among n variables involving m independent dimensions can be rewritten as a relation among n − m dimensionless groups. Its practical value is twofold: a correlation measured on a water model applies to liquid steel if the governing groups are matched (the basis of physical modelling of ladles and tundishes, usually by matching the Froude number), and every transport result can be read as one dimensionless number as a function of others.

A first-order reaction has rate −dC/dt = kC, so ln(C₀/C) = kt and C = C₀e^(−kt); the rate constant k has units of s⁻¹, the half-life t½ = ln 2/k is independent of the starting concentration, and a plot of ln C against t is a straight line of slope −k. Decarburisation of liquid steel at low carbon, where the rate is controlled by carbon transport to the bubble or slag interface, behaves as first order in [%C]. The temperature dependence of k is the Arrhenius relation k = A exp(−E_a/RT): ln k against 1/T is a line of slope −E_a/R, and two measurements give E_a = R ln(k₂/k₁)/(1/T₁ − 1/T₂). The same form governs diffusivities, creep rates and the rate of any thermally activated process.

Most metallurgical reactions are heterogeneous — gas-solid (reduction of iron ore pellets, roasting), gas-liquid (degassing, decarburisation), liquid-liquid (slag-metal refining) — and proceed through steps in series: transport of the reactant through the fluid boundary layer, diffusion through a porous product layer if one forms, chemical reaction at the interface, and transport of products away. At steady state the flux through every step is the same, so resistances add: for a first-order interfacial reaction with rate constant k_r and a mass transfer coefficient k_m, the overall coefficient is 1/k = 1/k_m + 1/k_r. The slowest step is rate-controlling. The shrinking-core model of a dense particle reacting from the outside in gives distinct time laws: under chemical-reaction control, t/τ = 1 − (1 − X)^⅓; under product-layer diffusion control, t/τ = 1 − 3(1 − X)^⅔ + 2(1 − X); under gas-film control, t/τ = X — so plotting conversion data against each form identifies the controlling step. A high activation energy (above roughly 100 kJ/mol) points to chemical control; a low one, and sensitivity to gas velocity or particle size, points to transport control.

High-temperature oxidation of metals follows one of three rate laws depending on the scale. If the oxide is porous or cracks, oxygen reaches the metal freely and the thickness grows linearly, x = k_l t. If the oxide is dense and adherent, growth is controlled by diffusion of ions through it and the flux falls as the scale thickens: the parabolic law x² = k_p t (Wagner’s theory), which describes iron, nickel, chromia-forming and alumina-forming alloys. Very thin films at low temperatures follow a logarithmic law. The Pilling-Bedworth ratio, PBR = (volume of oxide formed)/(volume of metal consumed) = M_oxide ρ_metal/(n M_metal ρ_oxide) for an oxide MₙOₘ, predicts which: PBR < 1 (magnesium, alkali metals) gives a porous, non-protective oxide; PBR between about 1 and 2 (aluminium, nickel, chromium) a protective one; PBR well above 2 a scale that spalls under compressive stress.

When a current flows through an electrode its potential moves away from the equilibrium (Nernst) value; the displacement η = E − E_eq is the overpotential, and the phenomenon is polarization. Three kinds add. Activation polarization is the energy barrier of the charge-transfer step itself; the Butler-Volmer equation describes it, and at high overpotential it reduces to the Tafel equation η = β log(i/i₀), where i₀ is the exchange current density (the equal and opposite forward and back currents at equilibrium) and β the Tafel slope, typically about 0.1 V per decade. A large i₀ means a reaction that polarizes little — hydrogen evolution on platinum — and a small i₀ one that polarizes strongly — hydrogen on zinc or lead. Concentration polarization appears when the reacting species cannot be supplied fast enough by diffusion; the current cannot exceed the limiting current density i_L = nFDC_b/δ, reached when the surface concentration falls to zero. Resistance (ohmic) polarization is the IR drop through the electrolyte and any surface film.

Polarization is what turns the Pourbaix diagram’s “can it corrode?” into “how fast?”. In mixed-potential theory a corroding metal carries an anodic reaction (M → Mⁿ⁺ + ne⁻) and a cathodic one (hydrogen evolution or oxygen reduction) on the same surface; with no external current they must balance, and the potential at which the anodic and cathodic polarization curves intersect is the corrosion potential E_corr, the current there the corrosion current i_corr. Anything that raises i₀ of the cathodic reaction, or its limiting current (more dissolved oxygen, more stirring), raises the corrosion rate. Passivation shows as an anodic curve that, above a critical current density, suddenly drops to a small passive current — the basis of anodic protection and of stainless steel’s behaviour.