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21-Mat-A2 Materials Transport Phenomena · December 2019

Question 4 of 5: Mould-Resistance-Controlled Solidification

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2019 — 12-MTL-A2 Transport Phenomena in Materials Engineering. Three-hour, open-book exam (one textbook of the candidate's choice permitted, with margin notations, no loose notes); any non-communicating calculator permitted. Each of the five questions is worth 25 points, and any four constitute a complete paper — only the first four questions as they appear in the answer book are marked. All five are solved below for completeness. Candidates were told to state all assumptions clearly.

Reference texts: Bird, R. B., Stewart, W. E. & Lightfoot, E. N., Transport Phenomena — power-law non-Newtonian flow between parallel plates and annular fully-developed duct flow (Questions 1, 3), matching this paper's own Appendix A conservation-equation tables; Incropera, F. P. et al., Fundamentals of Heat and Mass Transfer — natural-convection correlations for a horizontal and a vertical flat plate (Question 2); Geiger, G. H. & Poirier, D. R., Transport Phenomena in Materials Processing — mould-resistance-controlled solidification of castings (Question 4); Shewmon, P. G., Diffusion in Solids — the Boltzmann–Matano graphical method for a concentration-dependent interdiffusion coefficient (Question 5).

Question 4: Mould-Resistance-Controlled Solidification (25 marks: (a) 8, (b) 3, (c) 9, (d) 5)

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.

Given.

QuantitySymbolValue
Mould (ceramic) conductivity$k_m$$0.7\ \text{W/m}\cdot\text{K}$
Outside convection coefficient$h$$150\ \text{W/m}^2\text{K}$
Mould wall thickness$L$$10\ \text{mm}=0.010\ \text{m}$
Latent heat of fusion, nickel$L_f$$291\ \text{kJ/kg}$
Density, nickel$\rho_{Ni}$$7.85\ \text{g/cm}^3=7850\ \text{kg/m}^3$
Casting thickness / diameter—38 mm plate; 38 mm ø cylinder

Find. (a) Solidified-layer thickness $s(t)$, flat mould; (b) solidification time for the 38 mm plate; (c) $r(t)$ for a cylindrical mould; (d) solidification time for the 38 mm diameter cylinder.

liquidmould, Lmould, Lflat mould (2-sided cooling)h, T∞ (both faces)liquid core, r(t)cylindrical mouldrRc+L
Fig. 3 — flat mould with symmetric two-face cooling (left) and cylindrical mould solidifying inward from a fixed cavity radius (right); in both cases the mould's own geometry — and hence its thermal resistance — stays fixed while the metal solidifies.
Check — assumed temperature driving force
The source gives no pouring/freezing temperature for nickel, only the outside ambient implied elsewhere on this paper (295 K, Q2). Parts (b) and (d) use the standard nickel melting point $T_{melt}=1455^\circ\text{C}=1728\ \text{K}$ (no superheat, i.e. the liquid/solid metal is taken isothermal at the freezing point) together with $T_\infty=295\ \text{K}$, giving $\Delta T=1433\ \text{K}$. This is a stated, flagged assumption, not a given datum.

Approach. Because only the mould's own conductivity, thickness and outside $h$ are given (not the metal's own conductivity or diffusivity), the intended model is mould-resistance-controlled solidification: the liquid and already-solid metal are both taken isothermal at the freezing point (zero internal resistance), so all the resistance to heat flow sits in the mould wall + outside convection, in series, and — because the mould's own geometry never changes as the casting solidifies — this series resistance is constant in time.

  1. Part (a) — flat mould, energy balance at the moving front. Latent heat released per unit area as the front advances by $ds$ must conduct across the fixed-geometry mould resistance $R''=L/k_m+1/h$: $$\rho_{Ni}L_f\,\frac{ds}{dt}=\frac{\Delta T}{R''}=\frac{\Delta T}{L/k_m+1/h}\quad(\text{constant, since }R''\text{ does not depend on }s)$$ Integrating from $s(0)=0$: $$\boxed{s(t)=\frac{\Delta T}{\rho_{Ni}L_f\left(L/k_m+1/h\right)}\,t}\qquad(\text{linear in }t,\text{ not the }\sqrt t\text{ Chvorinov law of a metal-conduction-controlled layer})$$
  2. Part (b) — flat-plate solidification time. A plate mould cools symmetrically from both faces, so the two fronts meet at mid-thickness after each has advanced $s=19\ \text{mm}=0.019\ \text{m}$. With $$R''=\frac{L}{k_m}+\frac{1}{h}=\frac{0.010}{0.7}+\frac{1}{150}=0.02095\ \text{m}^2\text{K/W}$$ $$t=\frac{s\,\rho_{Ni}L_f\,R''}{\Delta T}=\frac{(0.019)(7850)(291\,000)(0.02095)}{1433}$$ $$\boxed{t\approx 635\ \text{s}\ (\approx10.6\ \text{min})}$$
  3. Part (c) — cylindrical mould, energy balance. For a cylinder of fixed casting (cavity) radius $R_c$, mould outer radius $R_c+L$, the mould's own resistance per unit length is fixed geometry: $R'_m=\ln[(R_c+L)/R_c]/(2\pi k_m)$, $R'_{conv}=1/[h\,2\pi(R_c+L)]$, giving a constant heat rate per unit length $q'=\Delta T/(R'_m+R'_{conv})$. The solid–liquid interface at radius $r(t)$ moves inward from $R_c$; the latent-heat balance per unit length is $$\rho_{Ni}L_f\left(-2\pi r\,\frac{dr}{dt}\right)=q'\quad\Rightarrow\quad r\,dr=-\frac{q'}{2\pi\rho_{Ni}L_f}\,dt$$ Integrating from $r(0)=R_c$: $$\boxed{r(t)^2=R_c^2-\frac{q'}{\pi\rho_{Ni}L_f}\,t}\ ,\qquad\text{complete at }r=0:\ \ t_{solid}=\frac{\pi\rho_{Ni}L_f R_c^2}{q'}$$
  4. Part (d) — cylinder solidification time. With $R_c=0.019\ \text{m}$, $R_c+L=0.029\ \text{m}$: $$R'_m=\frac{\ln(0.029/0.019)}{2\pi(0.7)}=0.0961\ \text{K}\cdot\text{m/W},\qquad R'_{conv}=\frac{1}{150\cdot2\pi(0.029)}=0.0366\ \text{K}\cdot\text{m/W}$$ $$q'=\frac{1433}{0.0961+0.0366}=1.080\times10^{4}\ \text{W/m}$$ $$t_{solid}=\frac{\pi(7850)(291\,000)(0.019)^2}{1.080\times10^{4}}$$ $$\boxed{t_{solid}\approx 240\ \text{s}\ (=4.00\ \text{min})}$$
QuantityResult
Flat mould: $s(t)$$\dfrac{\Delta T}{\rho_{Ni}L_f(L/k_m+1/h)}\,t$ (linear in $t$)
(b) 38 mm plate solidification time635 s (10.6 min)
Cylindrical mould: $r(t)^2$$R_c^2-\dfrac{q'}{\pi\rho_{Ni}L_f}\,t$
(d) 38 mm ø cylinder solidification time240 s (4.00 min)
Check
Assumes: liquid and solid nickel are both isothermal at the freezing point (zero internal thermal resistance in the casting itself); the mould's cavity geometry is fixed (its own resistance does not change as the casting solidifies); the flat casting is cooled symmetrically from both faces; no superheat above the melting point; $\Delta T=1433\ \text{K}$ per the flagged assumption above.