Sea urchins are typically spiny, globular echinoderms. About 950 species live on the seabed, inhabiting all oceans and depth zones from the intertidal to 5,000 meters.
For studying the effect of temperature acclimation on the metabolic rate of sea urchins, a Ph.D. student fabricated a laboratory “artificial urchin”: she filled a thin-walled table-tennis ball (diameter: 38 mm), with hand wanner reactants, so to produce a slightly-exothermic unifonn reaction (volumetric heat generation: 370 kW/m3, thermal conductivity of products/reactants: 2.28 W/m•K), simulating the urchin metabolism. She then enclosed the ball into a finned shell (material: magnesite), composed by 114 parabolic pin fins (fin length: 52 mm; base diameter: 2.3 mm; fin base considered as flat) attached on a 1.8- mm-thick spherical sleeve. Finally, she placed the “artificial urchin” in a small tank, with seawater calmly flowing around the device (heat transfer coefficient: 2340 W/m2•K). Knowing that the maximum temperature inside the device should never exceed 32.8 °C for not jeopardizing the experiment, but having the possibility to place a thennocouple just in the seawater tank, at which alarm-temperature the setup must be set? State all the assumptions that you consider for the calculations so that all the formulas are properly justified.