Stirling engines utilize regeneration, a process during which heat is transferred to a thermal

energy storage device $($called a regenerator$)$ during one part of the cycle and is transferred

back to the working fluid during another part of the cycle. The regenerator commonly

comprises a packed bed of solid spheres, through which a hot gas, in this case air, flows if the

regenerator is being charged $($i$.$e$.$ heat is transferred to the spheres$),$ or a cold gas if it is being

discharged $($i$.$e$.$ heat is transferred back to the working fluid$).$ Consider a packed bed of $7\mathrm{.}5-$

cm$-$diameter spheres and a charging process for which a hot gas enters the regenerator at a

temperature of $300\backslash$deg C and with a velocity of $1\mathrm{.}6$ m$/$s$.$ The initial temperature of the spheres is

$25\backslash$deg C$.$ You are required to analyse and evaluate two potential materials for the spheres in the

regenerator: $$pure aluminium$$ and $$Pyrex glass$.$ Begin by drawing a labelled schematic

diagram of the regenerator and illustrate the expected temperature profile in the sphere.

Determine the length of time it would take a sphere near the inlet of the regenerator to

accumulate $90\%$ of the maximum possible thermal energy and the corresponding temperature

at the centre of that sphere. Based on your analysis make a recommendation as to which

material should be used.

Note: The Nusselt number correlation for heat transfer to$/$from a single sphere in a packed bed is Nu $=5\mathrm{.}15$e$0\mathrm{.}425$Pr$13$ where the symbols have their usual meaning.