21-Mat-A2 Materials Transport Phenomena · December 2013
Question 9 of 9: Ladle-Shroud Draining Velocity
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2013 — Met-A2, Metallurgical Rate Phenomena. Three-hour, closed-book exam using an approved (Casio or Sharp) calculator, one double-sided aid sheet permitted; candidates were told to state any interpretive assumptions. Candidates answer Question 1 plus any four of Questions 2–9 — all nine are solved below for completeness. All questions are of equal value (20 marks each, five questions = 100%).
Reference texts: Geankoplis, C. J., Transport Processes and Separation Process Principles — mass/heat/momentum transfer fundamentals (Fick's/Newton's/Fourier's laws, boundary-layer correlations); Szekely, J. & Themelis, N. J., Rate Phenomena in Process Metallurgy — gas-halo diffusion, ladle/tundish fluid flow, wire injection; Turkdogan, E. T., Fundamentals of Steelmaking — BOF/EAF heat and mass balances; Callister, W. D., Materials Science and Engineering — TTT/CCT diagrams and phase transformations; Incropera, F. P. & DeWitt, D. P., Fundamentals of Heat and Mass Transfer — liquid-metal (low-Pr) convection correlations.
Check: the source gives no friction factor or liquid-steel viscosity for this question. Density $\rho=7000\ \text{kg/m}^3$ and viscosity $\mu=7\ \text{mPa s}$ are carried over from the identical “liquid steel” property set stated for Questions 7–8 of the same paper, and the Fanning friction factor is estimated from a smooth-pipe turbulent correlation (Blasius) rather than assumed as a fixed value, since neither is separately given here.
Given. $H_o=3$ m, shroud length $L=1.5$ m, shroud diameter $D=0.08$ m, ladle ID $=4$ m ($\gg D$, so ladle free-surface velocity $\approx0$), $\rho=7000\ \text{kg/m}^3$, $\mu=7\times10^{-3}$ Pa s (Q7–8 values, carried over — see the check note), sharp entrance, abrupt exit into air.
Find. A general expression for exit velocity $V$ including friction, and its numerical value.
Ladle draining through a shroud into the tundish: free surface (point 1, $V\approx0$) to shroud exit (point 2, velocity $V$), total driving head $H_o+L$.
Approach. Apply the steady-flow (modified Bernoulli) energy equation between the quiescent ladle free surface and the shroud exit, with gravity driving the flow against the stated friction loss term; because the friction factor depends on the (unknown) velocity through the Reynolds number, solve iteratively (or note the simpler frictionless limit as a bounding check).
General expression (SFEE / modified Bernoulli), free surface to shroud exit. With $V_1\approx0$ (large ladle cross-section), exit to atmosphere ($P_1=P_2=P_{atm}$), datum at the shroud exit, total elevation drop $H=H_o+L$: $$gH=\dfrac{V^2}{2}+E_f=\dfrac{V^2}{2}+2f\left(\dfrac{L}{D}\right)V^2$$ $$\boxed{V=\sqrt{\dfrac{2gH}{1+4f(L/D)}}}\qquad H=H_o+L$$
Estimate the friction factor. With $D=80$ mm and $\mu=7\times10^{-3}$ Pa s, flow is strongly turbulent even at moderate $V$; using the Blasius smooth-pipe correlation $f=0.079\,Re^{-0.25}$ (Fanning) and iterating with the velocity equation above (since $Re=\rho VD/\mu$ depends on $V$):
Converged solution. $$Re\approx6.85\times10^5,\qquad f\approx0.00275,\qquad E_f=2f(L/D)V^2\approx7.54\ \text{J/kg}$$ $$\boxed{V\approx8.56\ \text{m/s}}$$ — about 9% below the frictionless Torricelli bound, consistent with a short (L/D≈19), moderately smooth shroud passage.
