Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2020 / Module NE2 / 1

Take one-dimensional steady state thermal conduction, constant thermal conductivity $k>0$, and positive volumetric heat generation rate. Write $q_0=q_0^{\prime \prime \prime }$ for brevity, and let $x$ increase away from the illuminated face. Conservation of energy and Fourier's law giveThus $T^{\prime \prime }=-(q_0/k)e^{-\mu x}<0$: the profile is strictly concave and any stationary point is its unique maximum. Two integrations and the equal-surface boundary conditions yield the exponential slab heating solutionIts derivative isDefine $\beta =\mu a$ and $C=(1-e^{-\beta })/\beta$. Setting $T^{\prime }(x_m)=0$ gives the distance from the illuminated surface:For $\beta >0$, $e^{-\beta }<C<1$, so $0<x_m<a$. More specifically $C=e^{-\beta /2}\sinh (\beta /2)/(\beta /2)>e^{-\beta /2}$, hence $x_m<a/2$: the peak is nearer the more strongly heated face.

The apparent singularity at $\mu =0$ is removable. Expansion givesThe uniform-heating limit therefore has its maximum at $a/2$, as symmetry requires.

At the maximum found in part(a), $e^{-\mu x_m}=C$ and $(1-e^{-\beta })x_m/a=C\mu x_m=-C\ln C$. Substitution into the exponential slab heating solution givesFor $0<C<1$, the function $1-C+C\ln C$ is positive: it vanishes at $C=1$ and has derivative $\ln C<0$. Thus the interior temperature exceeds the common surface temperature, consistently with positive internal heating.

A convenient dimensionless form uses $\xi =x/a$:As $\beta \to 0$, $\theta \to \xi (1-\xi )/2$, giving the check

The graph holds the normalization $q_0{a}^2/k$ fixed. Its decreasing peaks with increasing $\beta$ reflect a decreasing source away from the illuminated face under that comparison; they do not assert the same trend at fixed total absorbed power.

The two outward surface normals point in opposite directions. With positive heat flux defined as heat leaving each face, Fourier's law givesUsing the derivative from part(a), the removed fluxes per unit face area areBoth are positive because $e^{-\beta }<C<1$. Their sum satisfies the independent conservation of energy checkThe illuminated face removes more heat because heating is concentrated nearer it. Equivalently, weighting each source slice by the fraction of its heat conducted to either equal-temperature boundary givesA decreasing source weights the left face more strongly. In the uniform-heating limit, each face removes $q_{0}a/2$. If a heat removal rate rather than a flux is wanted, multiply the relevant expression by the face area.

The exponential slab heating solution separates three useful controls: the heat source, the conductive path and the boundary temperature. Reduce deposited power, improve thermal conduction, shorten paths to a cooled surface, or lower the actual surface temperature. Each proposal has to preserve the shield's radiation-protection function.

Reduce incident photon intensity, intercept part of the beam in a separately cooled upstream shield, or distribute the deposited power over a larger cooled region. These reduce the relevant volumetric heat generation rate or local heat load. Intermittent exposure can reduce average heating, but the peak then requires a transient heat equation calculation; a smaller average alone does not establish a safe pulse maximum.

Increase the effective thermal conductivity, for example through a suitable material or a well-bonded conductive path to a heat sink. Reduce contact thermal resistance in any added path. Material substitution must also satisfy shielding, mechanical and environmental constraints.

Shorten the maximum conduction distance by introducing cooled internal surfaces or dividing the shield into thinner, separately cooled layers. Reducing the thickness of a single slab lowers its temperature rise at fixed $q_0$ and $\mu$, but also reduces attenuation; it is not automatically an acceptable shield modification. A layered arrangement can preserve the required total shielding while changing the thermal boundary-value problem.

Lower the cooling-air temperature, improve forced convection by appropriate flow and surface design, enlarge the heat-exchange area, or use a more effective coolant where compatible. These can lower the actual boundary temperature when the external cooling is modeled. In the stipulated model $T_0$ is already prescribed: increasing the external heat transfer coefficient while holding $T_0$ fixed leaves the calculated interior profile unchanged. It helps physically by reducing the surface-to-coolant temperature difference under a heat load.

Do not change the attenuation coefficient independently of the heat-source normalization without explaining the comparison. At fixed incident photon energy flux, the entrance deposition $q_0$ generally changes with absorption properties. A larger $\mu$ with artificially fixed $q_0$ is a different experiment. Similarly, increasing total slab heat capacity delays heating but does not lower the steady state maximum for an unchanged source and unchanged thermal boundary conditions.