21-Mat-A2 Materials Transport Phenomena · December 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2018 — Met-A2 Metallurgical Rate Phenomena. Three-hour, open-book exam; any Casio or Sharp approved calculator permitted. Candidates answer Question 1 (compulsory) plus any four of Questions 2–8; the five answered count equally (20 marks each). All eight are solved below for completeness. Candidates were told to state any interpretive assumptions where doubt exists.
Reference texts: Geiger, G. H. & Poirier, D. R., Transport Phenomena in Materials Processing (TMS) — the primary reference for the momentum-, heat- and mass-transfer analyses in Questions 3, 5, 6, 7 and 8; Szekely, J., Evans, J. W. & Sohn, H. Y., Rate Phenomena in Process Metallurgy — reactor-network and interfacial mass-transfer modelling (Questions 2, 3); Welty, J. R. et al., Fundamentals of Momentum, Heat and Mass Transfer — dimensional analysis and pipe-friction data (Questions 4, 7); Gaskell, D. R., Introduction to the Thermodynamics of Materials — Fe–C phase relations (Question 1h).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
(a) True. $\dot N''_z=-D_{A/B}\,\dfrac{\partial C_A}{\partial x}$ is exactly Fick's first law for one-dimensional diffusive molar flux: the flux is proportional to the concentration gradient, directed down-gradient (hence the minus sign), with $D_{A/B}$ the binary diffusivity of A in B.
(b) Ambiguous. Fick's law in its basic form, $N''=-D\,\partial C/\partial x$, describes only the diffusive contribution to flux; it does not by itself contain a convective (bulk-flow) term. For a binary mixture with a net molar velocity, the total flux is $N_A=-D_{AB}\,\partial C_A/\partial x + C_A v^*$ — convection can be added to the diffusive term, but Fick's law proper is silent on it, so the statement is true only if "Fick's law" is read loosely as the whole flux equation.
(c) True (with a caveat). Newton's equation of viscosity is $\tau_{yx}=-\mu\,\dfrac{du_x}{dy}$: shear stress is proportional to the velocity gradient, with $\mu$ the (constant) dynamic viscosity. Liquid metals and alloys are, in normal processing ranges, Newtonian fluids (low-molecular, non-polymeric liquids), so the equation applies directly — the caveat is that near-eutectic slurries or partially solidified (semi-solid) alloys carrying a solid fraction can turn strongly non-Newtonian.
(d) False. Kinetic theory gives gas viscosity increasing with temperature (roughly $\mu\propto T^{0.5\text{–}0.7}$, since faster molecules transport momentum more effectively across a shear layer) but essentially independent of pressure over the normal engineering range (mean free path shortens as $1/P$ but molecular flux rises as $P$, and the two cancel) — only at very low pressure (mean free path comparable to the vessel) or very high pressure does $\mu$ become pressure-dependent.
(e) False. Thermal conductivity of a metal drops on melting for most metals: the electronic contribution dominates conduction in both phases, but the loss of the ordered lattice on melting shortens the electron mean free path and increases scattering, cutting $k$ by roughly 30–50% even though the liquid remains a far better conductor than any non-metallic liquid.
(f) Ambiguous / requires a sketch. A contact angle of 180° (measured through the liquid) means the liquid is completely non-wetting toward the solid — equivalently, the gas phase preferentially wets the container wall. A bubble entering through a small orifice therefore does not pinch off as a compact sphere; it spreads laterally into a thin, wide cap hugging the floor of the container (large basal contact area, small height), because the gas "prefers" contact with the solid over the liquid does. The sketch below shows this spread cap versus the compact sphere expected at a normal (small) contact angle.
(g) True. Oxygen has a small but genuinely finite equilibrium solubility in liquid and solid iron (parts-per-million range, governed by $\underline{O}\rightleftharpoons\tfrac12 O_2(g)$ equilibria and Sievert's law in the melt) — it is not zero, which is precisely why deoxidation practice (Al, Si, Mn additions) is needed in steelmaking.
(h) True, essentially. The Fe–Fe$_3$C eutectic occurs at 1147°C (close to the 1140°C quoted) with the maximum solubility of carbon in austenite ($\gamma$-Fe, FCC) at that temperature equal to 2.11 wt% C — the “~2% at ~1140°C” statement is the standard textbook approximation for this maximum-solubility point, provided “solid steel” is read as austenite (ferrite, BCC, holds far less — only ~0.02 wt% C at 727°C).
(i) False. For a positive-order reaction (including first order), a plug-flow reactor (PFR) achieves higher conversion than a continuous well-mixed (CSTR) reactor of the same volume, because the CSTR operates everywhere at the low, already-diluted outlet concentration (minimum driving force throughout), while the PFR sees the full concentration gradient from inlet to outlet. The PFR is therefore the more volume-efficient (and rate-efficient) of the two for a given duty.
(j) True. $Fr=U^2/(gL)$ compares inertial to gravitational (buoyancy/free-surface) forces; $Re=\rho UL/\mu$ compares inertial to viscous forces — both are stated correctly.
(k) True. Fourier's second law, $\partial\theta/\partial t=\alpha\,\partial^2\theta/\partial x^2$, is written in terms of the thermal diffusivity $\alpha=k/(\rho C_p)$, which explicitly contains the thermal conductivity $k$ (together with density and heat capacity).
(l) True. $Bi=hL/k_{solid}$ uses the conductivity of the solid being heated or cooled (ratio of internal conduction resistance to surface convection resistance), whereas $Nu=hL/k_{fluid}$ uses the conductivity of the fluid film (ratio of convective to conductive transport in the boundary layer). The two numbers look alike algebraically but compare $h$ against the conductivity of two different phases.
(m) True. Liquid metals have very low Prandtl number ($Pr=\nu/\alpha\sim0.005$–$0.03$) because their thermal diffusivity $\alpha$ (driven by high $k$) vastly exceeds their momentum diffusivity $\nu$. Since $\delta_t/\delta\sim Pr^{-1/2}$ for laminar boundary layers, the thermal boundary layer in a liquid metal is many times thicker than the momentum (velocity) boundary layer — and thicker than the thermal layer of an ordinary liquid ($Pr\sim1$–$10$) growing under the same flow.
(n) False. Incident radiation on a real surface is generally split three ways — transmitted, reflected, and absorbed — with $\alpha+\rho+\tau=1$. Omitting absorption is incorrect for any real (non-transparent, non-perfectly-reflecting) surface; absorption is in fact the mechanism by which radiative heat transfer heats the receiving body.
