Heat-Generating Cylindrical Conductor — Maximum Generation Rate and Surface Temperature: 17-Phys-B6 Applied Thermodynamics and Heat Transfer, December 2018, Q6

Question 6: Heat-Generating Cylindrical Conductor — Maximum Generation Rate and Surface Temperature

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. A solid cylindrical conductor with uniform internal heat generation $\dot q_{gen}$ (W/m³), radius $r_o=37.5\text{ mm}$, $k=70\text{ W/m}\cdot{}^{\circ}\text{C}$, submerged in fluid at $T_\infty=27\,{}^{\circ}\text{C}$ with $h=568\text{ W/m}^2\cdot{}^{\circ}\text{C}$; the hottest point (the centreline) must not exceed $T_c=540\,{}^{\circ}\text{C}$.

Given data
Quantity	Symbol	Value
Diameter	$D$	75 mm
Thermal conductivity	$k$	70 W/m·°C
Fluid temperature	$T_\infty$	27°C
Surface coefficient	$h$	568 W/m²·°C
Max. allowable (centreline) temperature	$T_c$	540°C

Find. (a) The maximum heat generation rate per unit length $q'_{gen}$; (b) the corresponding surface temperature $T_s$.

Radial temperature profile inside a solid cylinder with uniform heat generation: parabolic from the centreline ($T_c$) down to the surface ($T_s$), then a further drop across the convective film to the fluid.

Approach. Two standard results for uniform volumetric generation in a solid cylinder with surface convection: the centre-to-surface temperature drop $T_c-T_s=\dot q_{gen}r_o^2/(4k)$ (from solving the conduction equation), and the surface-to-fluid drop $T_s-T_\infty=\dot q_{gen}r_o/(2h)$ (from an overall energy balance: heat generated per unit length equals heat convected from the surface). Combine both with the ONE given constraint $T_c=540\,{}^{\circ}\text{C}$ to solve simultaneously for $\dot q_{gen}$ and $T_s$.

Combine the two temperature-drop relations. $$T_c-T_\infty=(T_c-T_s)+(T_s-T_\infty)=\dot q_{gen}\left(\frac{r_o^2}{4k}+\frac{r_o}{2h}\right)$$ $$\frac{r_o^2}{4k}=\frac{0.0375^2}{4\times70}=5.022\times10^{-6},\qquad \frac{r_o}{2h}=\frac{0.0375}{2\times568}=3.301\times10^{-5}$$ $$540-27=\dot q_{gen}\times(5.022\times10^{-6}+3.301\times10^{-5})=\dot q_{gen}\times3.803\times10^{-5}$$ $$\boxed{\dot q_{gen}=1.349\times10^{7}\text{ W/m}^3}$$
Maximum heat generation rate per unit length, part (a). $$q'_{gen}=\dot q_{gen}\times\pi r_o^2=1.349\times10^{7}\times\pi\times0.0375^2$$ $$\boxed{q'_{gen}=59.6\text{ kW/m}}$$
Surface temperature, part (b). $$T_s=T_\infty+\dot q_{gen}\frac{r_o}{2h}=27+1.349\times10^{7}\times3.301\times10^{-5}$$ $$\boxed{T_s=472.3\,{}^{\circ}\text{C}}$$ Check: $T_c-T_s=540-472.3=67.7\,{}^{\circ}\text{C}$, and independently $\dot q_{gen}r_o^2/(4k)=1.349\times10^7\times5.022\times10^{-6}=67.8\,{}^{\circ}\text{C}$ — consistent.

Question 6 — results
Quantity	Value
Volumetric generation rate $\dot q_{gen}$	$1.349\times10^7$ W/m³
(a) Max. heat generation rate per unit length	59.6 kW/m
(b) Surface temperature $T_s$	472.3°C

Topic. One-dimensional radial conduction with uniform internal heat generation in a solid cylinder, convectively cooled at the surface.

Key relations. Centre-to-surface drop $T_c-T_s=\dot q_{gen}r_o^2/(4k)$; surface energy balance $\dot q_{gen}\pi r_o^2=h(2\pi r_o)(T_s-T_\infty)\Rightarrow T_s-T_\infty=\dot q_{gen}r_o/(2h)$.

Why this works. The problem gives only ONE temperature (the centreline maximum) but has TWO unknowns ($\dot q_{gen}$ and $T_s$) — the trick is recognising the two standard temperature-drop formulas share the SAME $\dot q_{gen}$, so adding them together eliminates $T_s$ and leaves one equation in one unknown; $T_s$ then falls out from either relation once $\dot q_{gen}$ is known.

Common pitfall. Using the diameter instead of the radius in either formula (both are written in terms of $r_o$, not $D$); forgetting the factor of 4 in the centreline term (it comes from integrating the 1-D generation equation in cylindrical coordinates, not the factor of 2 used for plane walls).

Source. F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed., Ch. 3 (heat generation in a solid cylinder with surface convection).

Question 6 of 8: Heat-Generating Cylindrical Conductor — Maximum Generation Rate and Surface Temperature

Question 6: Heat-Generating Cylindrical Conductor — Maximum Generation Rate and Surface Temperature