Fully Developed Flow and Heat Transfer in a Heated Annulus: 21-Mat-A2 Materials Transport Phenomena, December 2019, Q3

Question 3: Fully Developed Flow and Heat Transfer in a Heated Annulus (25 marks: (a) 15, (b) 7, (c) 3)

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.

Quantity	Symbol	Value
Inner cylinder radius (heated, flux $q_1$)	$R_1$	symbolic
Outer cylinder radius (held at $T_0$)	$R_2$	symbolic
Fluid	$\rho,\mu$	constant (Newtonian)
Flow & thermal state	—	steady, fully developed (both velocity and temperature profiles)

Find. (a) $V_z(r)$; (b) the governing energy equation and its assumptions; (c) the boundary conditions needed to solve it.

Fig. 2 — annular duct: fluid enters at $T_0$ and flows upward ($z$-direction) between the heated inner wall ($R_1$, flux $q_1$) and the outer wall held at $T_0$ ($R_2$).

Approach. Part (a): differential momentum balance in cylindrical coordinates → second-order ODE in $V_z(r)$, integrated twice and closed with no-slip at both walls. Parts (b)–(c): specialize the general cylindrical energy equation (Appendix Table A.3 of this exam) to this flow's assumptions, then state the physical boundary conditions at each wall.

Part (a) — differential momentum balance. For steady, fully developed, axisymmetric flow with only $u_z(r)$ nonzero, the $z$-component of the Navier–Stokes/motion equation (Appendix Table A.2, cylindrical) reduces to $$\begin{aligned}0&=-\frac{dP}{dz}+\mu\frac{1}{r}\frac{d}{dr}\!\left(r\frac{dV_z}{dr}\right)\\[4pt] \frac{1}{r}\frac{d}{dr}\!\left(r\frac{dV_z}{dr}\right)&=-\frac{G}{\mu},\qquad G\equiv-\frac{dP}{dz}\ (\text{modified-pressure driving gradient})\end{aligned}$$
Integrate twice. Two integrations give $V_z(r)=-\dfrac{G}{4\mu}r^2+C_1\ln r+C_2$. Applying no-slip at both stationary walls, $V_z(R_1)=V_z(R_2)=0$, and solving the resulting $2\times2$ system for $C_1,C_2$: $$\boxed{V_z(r)=\frac{G R_2^2}{4\mu}\left[1-\left(\frac{r}{R_2}\right)^2-\frac{1-(R_1/R_2)^2}{\ln(R_2/R_1)}\ln\frac{R_2}{r}\right]}\qquad R_1\le r\le R_2$$ which vanishes at $r=R_1$ and $r=R_2$ and is positive and single-peaked in between.
Part (b) — energy equation. Starting from the general cylindrical energy equation for incompressible media (this exam's own Appendix Table A.3b) and applying: steady state ($\partial T/\partial t=0$), axisymmetric ($\partial/\partial\theta=0$), only $u_z\ne0$ (no radial or tangential convection), fully developed temperature so $\partial T/\partial z$ is a constant independent of $r$, negligible axial conduction compared with radial conduction and convection ($\partial^2T/\partial z^2\approx0$), no internal heat generation and negligible viscous dissipation: $$\boxed{\rho c_P\,V_z(r)\,\frac{\partial T}{\partial z}=\frac{k}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial T}{\partial r}\right)}$$ with $\partial T/\partial z=$ constant (a standard result for thermally fully developed flow under a constant wall heat flux).
Part (c) — boundary conditions. Two radial boundary conditions close the second-order ODE in $r$ obtained from the reduced energy equation: $$\boxed{-k\left.\frac{\partial T}{\partial r}\right|_{r=R_1}=q_1\ \ (\text{prescribed inward-facing flux leaving the heated inner wall into the liquid})}$$ $$\boxed{T(R_2)=T_0\ \ (\text{outer wall held at constant temperature})}$$

Quantity	Result
Velocity profile, $V_z(r)$	$\dfrac{GR_2^2}{4\mu}\Big[1-(r/R_2)^2-\dfrac{1-(R_1/R_2)^2}{\ln(R_2/R_1)}\ln\dfrac{R_2}{r}\Big]$
Reduced energy equation	$\rho c_P V_z\,\partial T/\partial z=\dfrac{k}{r}\dfrac{\partial}{\partial r}\big(r\,\partial T/\partial r\big)$
BC at inner wall ($r=R_1$)	$-k\,\partial T/\partial r=q_1$
BC at outer wall ($r=R_2$)	$T=T_0$

Check

Assumes both cylinder walls are stationary (no-slip, no wall motion), the flow is laminar and fully developed with no swirl ($u_\theta=0$), and that "fully developed temperature" under a constant heat flux means $\partial T/\partial z$ is independent of $r$ — a standard, exam-appropriate simplification for a thermally fully developed constant-flux duct flow.

Concept & Theory

Topic. Fully developed laminar flow and heat transfer in a concentric annular duct with a constant-heat-flux inner wall and constant-temperature outer wall.

Key relations. Momentum: $\mu\frac{1}{r}\frac{d}{dr}(r\,dV_z/dr)=dP/dz$; energy: $\rho c_P V_z\,\partial T/\partial z=\frac{k}{r}\frac{\partial}{\partial r}(r\,\partial T/\partial r)$; flux BC $-k\,\partial T/\partial r|_{R_1}=q_1$.

Why this works. Because the geometry and boundary conditions have no $\theta$ or $t$ dependence, both the momentum and energy equations collapse from a 3-D PDE to a 1-D ODE in $r$ alone (with $z$ entering the energy equation only through the constant axial gradient $\partial T/\partial z$) — the hallmark of a "fully developed" duct-flow problem.

Common pitfall. Forgetting that a uniform wall heat flux (as opposed to a uniform wall temperature) forces $\partial T/\partial z$ to be constant but nonzero, so the energy equation cannot be reduced to Laplace's equation the way a uniform-wall-temperature problem could.

Source. Bird, Stewart & Lightfoot, Transport Phenomena, Ch. 2 & 10 (annular duct flow; forced convection with constant wall flux).

Question 3 of 5: Fully Developed Flow and Heat Transfer in a Heated Annulus

Question 3: Fully Developed Flow and Heat Transfer in a Heated Annulus (25 marks: (a) 15, (b) 7, (c) 3)