$3\mathrm{.}90$ One method that is used to grow nanowires $($nanotubes with solid cores$)$ is to initially deposit a small droplet of a liquid catalyst onto a flat surface. The surface and catalyst are heated and simultaneously exposed to a higher$-$temperature, low$-$pressure gas that contains a

mixture of chemical species from which the nanowire is to be formed. The catalytic liquid slowly absorbs the species from the gas through its top surface and converts these to a solid material that is deposited onto the underlying liquid$-$solid interface, resulting in construction of the nanowire. The liquid catalyst remains suspended at the tip of the nanowire.

Consider the growth of a $15-$nm$-$diameter silicon carbide nanowire onto a silicon carbide surface. The surface is maintained at a temperature of $T_s=2400K,$ and the particular liquid catalyst that is used must be maintained in the range $2400K{T}_{c}3000K$ to perform its function. Determine the maximum length of a nanowire that may be grown for conditions characterized by $h={10}^5\frac{W}{m^2}*K$ and $T_{}=8000K.$ Assume properties of the nanowire are the same as for bulk silicon carbide.