A long cylindrical fiber of a plastic material with thermal conductivity k = 0.13 W/m K, density ρ = 1,400 kg/m³, and heat capacity Cp = 1,050 J/kg K is continuously extruded and cooled by pulling it through a chamber where air at 10°C flows perpendicular to the fiber at a speed of 10 m/s as sketched in Fig. P.5.8. The fiber, having a diameter of 3.0 mm, enters the chamber at a uniform temperature of 100°C, and is pulled through the chamber at the speed of 0.5 m/s. The convective heat transfer coefficient at the fiber surface is estimated to be h = 100 W/m2 K.

(a) Write down the differential energy balance and the proper initial and boundary conditions needed to describe the fiber cooling process. Include only the relevant terms assuming that the fiber is very long compared to its radius.

There is no need to solve the equation.

(b) Using available solutions, calculate the length of the chamber needed to attain a centerline fiber temperature of 30°C. Note that time = length/velocity.

(c) Calculate the surface temperature of the fiber when it leaves the chamber.

FIGURE P.5.8:

Transcribed Image Text:

Air flow