If the plastic bead covering the tip of the thermocouple described in Problem 1.4 is quite...

 

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If the plastic bead covering the tip of the thermocouple described in Problem 1.4 is quite large, and since plastic usually sustains a very low value of thermal conductivity, then the simple lumped model solution becomes quite inaccurate. To improve the model, we need to account for thermal conductivity in the (assumed) spherical shape of the plastic bead. (a) Assuming the bead is a perfect sphere, contacted everywhere by external fluid of temperature T, perform a shell balance on an element of volume 4mr?Ar and show that 1 a aT pC, at ar (b) Perform an elemental heat balance at the surface of the sphere and deduce -사-A(T-T) h(T- T) at r = R ar where R is the radius of the plastic sphere. If the plastic bead covering the tip of the thermocouple described in Problem 1.4 is quite large, and since plastic usually sustains a very low value of thermal conductivity, then the simple lumped model solution becomes quite inaccurate. To improve the model, we need to account for thermal conductivity in the (assumed) spherical shape of the plastic bead. (a) Assuming the bead is a perfect sphere, contacted everywhere by external fluid of temperature T, perform a shell balance on an element of volume 4mr?Ar and show that 1 a aT pC, at ar (b) Perform an elemental heat balance at the surface of the sphere and deduce -사-A(T-T) h(T- T) at r = R ar where R is the radius of the plastic sphere.