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23-Chem-B1 Transport Phenomena · Undated paper

Question 4 of 6: B2 — Slug flow with uniform wall flux: show that $Nu=8$

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

Notes on this paper

Paper. National Examination (Engineers Canada) — 16-Chem-B1 Transport Phenomena, May 2019. Open book, 3 hours. Six 25-point problems in three sections — A Fluid Mechanics (A1, A2), B Heat Transfer (B1, B2), C Mass Transfer (C1, C2); one problem from each section must be attempted, plus a fourth from any section. All six problems are solved and fully worked below.

Reference texts: R. B. Bird, W. E. Stewart & E. N. Lightfoot, Transport Phenomena (2nd ed., Wiley) — the equations of change, the power-law slit-flow momentum balance and the film/annular diffusion balances (A2, B2, C2); J. R. Welty, C. E. Wicks, R. E. Wilson & G. L. Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (Wiley) — pipe friction, the Moody chart, and the external-flow convection/mass-transfer correlations (A1, B1, C1); F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (8th ed., Wiley) — Churchill–Chu free convection, horizontal-plate correlations, slug-flow internal convection and air properties (B1, B2); C. J. Geankoplis, Transport Processes and Separation Process Principles (Prentice Hall) — boundary-layer mass transfer and the film model (C1); R. H. Perry & D. W. Green, Perry’s Chemical Engineers’ Handbook — transport properties.

Question 4: B2 — Slug flow with uniform wall flux: show that $Nu=8$ (25 marks)

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. Circular tube of radius $R$ (diameter $D=2R$), fluid thermal conductivity $k$; uniform axial velocity $v_z$ (slug/plug flow); constant wall heat flux, printed $q_o$ and written $q_s$ below; wall temperature, printed $T_R$ and written $T_s$ below; thermally fully-developed. Find. The Nusselt number $Nu=hD/k$, showing it equals 8.

r = R (wall, qₛ) r = 0 slug: uniform v T(r) − T₀ = (qₛ/2kR) r² Tₛ T₀ (centre)
Fig. B2: With a flat (slug) velocity profile and a uniform wall flux, the fully-developed temperature profile is exactly parabolic, $T(r)-T_0=(q_s/2kR)\,r^2$. Area-averaging (uniform velocity) gives $T_s-T_m=q_sR/4k$, hence $h=4k/R=8k/D$ and $Nu=8$.

Approach. Impose a constant-flux energy balance to fix the axial temperature rise, reduce the fully-developed energy equation to an ODE in $r$, solve for $T(r)$, form the flux-weighted mean $T_m$ (which for slug flow is the plain area average), and evaluate $h$.

  1. Axial gradient from an energy balance. Fully developed with constant $q_s$ means $\partial T/\partial z=dT_m/dz=$ const. A slice balance $q_s(2\pi R)=\rho c_p v_z(\pi R^2)\,dT_m/dz$ gives $$\frac{dT_m}{dz}=\frac{2q_s}{\rho c_p v_z R}.$$
  2. Reduce the energy equation. For slug flow with negligible axial conduction, $$\rho c_p v_z\frac{\partial T}{\partial z}=\frac{k}{r}\frac{d}{dr}\!\left(r\frac{dT}{dr}\right).$$ Substituting $\partial T/\partial z=dT_m/dz$ from Step 1 makes the left side constant: $$\frac{1}{r}\frac{d}{dr}\!\left(r\frac{dT}{dr}\right)=\frac{2q_s}{kR}.$$
  3. Integrate for the profile. Integrating twice with $dT/dr$ finite at $r=0$ and taking $T(0)=T_0$: $$T(r)-T_0=\frac{q_s}{2kR}\,r^2,\qquad\text{so } T_s=T(R)=T_0+\frac{q_sR}{2k}.$$
  4. Mean temperature (slug ⇒ area average). Because $v_z$ is uniform, the flux-weighted bulk mean reduces to the plain area average: $$T_m=\frac{\displaystyle\int_0^R T\,v_z\,2\pi r\,dr}{\displaystyle\int_0^R v_z\,2\pi r\,dr}=T_0+\frac{2}{R^2}\!\int_0^R\!\frac{q_s}{2kR}r^2\,r\,dr=T_0+\frac{q_sR}{4k}.$$
  5. Coefficient and Nusselt number. Then $T_s-T_m=\dfrac{q_sR}{2k}-\dfrac{q_sR}{4k}=\dfrac{q_sR}{4k}$, so $$h=\frac{q_s}{T_s-T_m}=\frac{4k}{R}=\frac{8k}{D}\quad\Longrightarrow\quad\boxed{\,Nu=\frac{hD}{k}=8.\,}$$
QuantityResult
Temperature profile$T(r)-T_0=(q_s/2kR)\,r^2$
Wall − mean temperature$T_s-T_m=q_sR/4k$
Heat-transfer coefficient$h=4k/R=8k/D$
Nusselt number$Nu=hD/k=8$
Check — why 8, and how it compares

The result is exact for idealised slug flow and follows purely from geometry — no fluid properties survive. It is the plug-flow companion of the two classic laminar (parabolic-velocity) constant-property results: $Nu=48/11\approx4.36$ for uniform wall flux and $Nu=3.66$ for uniform wall temperature. Slug flow gives a higher $Nu$ than parabolic flow because the flat velocity profile carries cool fluid closer to the wall, steepening the near-wall temperature gradient.