Spherical polymer beads $1\mathrm{.}0cm$ in diameter have been hot$-$molded and then

dropped into a large, tall tank of water at $20C$ to cool. The beads quickly

reach a terminal velocity of $3\mathrm{.}0c\frac{m}{s}($that is$,$ a constant settling velocity$),$

cooling as they fall.

We want to calculate the heat flux from a bead when the surface

temperature of the bead is $100C.$ Possibly relevant correlations are given

below; each is valid over a broad range of Reynolds, Rayleigh and Prandtl

numbers. Physical properties for these correlations should be evaluated at

the film temperature. The standard definitions of the dimensionless groups

involved are given below the correlations.

${}_D=2+0\mathrm{.}43R{a}_D^{\frac{1}{4}}$

${}_D=2+0\mathrm{.}6R{e}_D^{\frac{l}{2}}P{r}^{\frac{l}{3}}$

Where:

$Re=Dv\frac{}{}=D{v}^{\text{'}}v$

$Pr=\frac{v}{}=\frac{C_p}{k}$

$Gr=g{}^2{L}^{3}T{}_{}{r}^2=g{L}^{3}T{v}^2$

$Ra=GrPr$

a$)$ Which correlation should be used to find the heat transfer coefficient for this system? Briefly explain the reasoning

for your correlation choice.

If you can't tell for sure which of the two correlations would be the best to use without doing calculations or getting

more information, briefly state you would need to make a confident decision.

You do NOT need to do any calculations for part a$.$

b$)$ Although this is not necessarily the correct answer to part a$),$ for this question, use the following correlation to find $.$

${}_D=2+0\mathrm{.}43R{a}_D^{\frac{l}{4}}$

c$)$ Calculate the heat flux $(\frac{W}{m^2})$ from the sphere.

d$)$

What is the physical significance of the $\text{'}2\text{'}$ in each $$ correlation?