$1.(40)$ Consider the aluminum extrusion process, which involves heating a billet to $500$oC and then pressing it through a die $($hydraulically$,$ mechanically, or using steam$)$ to get its final shape$/$profile$.$ Watch the first $90$ seconds of the video. We use this process to produce solid circular Aluminum bars of $20$mm diameter using an extrusion speed of $0\mathrm{.}02$ m$/$s$.$ Aluminum and Air properties can be taken as given below. After leaving the Die at $500$oC$,$ the bar is exposed to airflows at $20$oC that flow perpendicular to the extruded bar at $5\mathrm{.}0$ m$/$s for cooling. Due to its high thermal conductivity, the temperature distribution $$Across$$ the bar can be assumed to be uniform but changes along the direction of the motion. Assume that radiation from the extruded rod to the environment is negligible $($i$.$e$.,$ much less than the convective loss$).$ a$.$ Watch the video for the first $5$ minutes to become familiar with the process. b$.$ Based on energy balance, develop an equation for the temperature distribution in the direction of motion. Hint: See Prob. #$2,$ Homework #$9.$ Note the simplifying differences$$ c$.$ Find the average convection coefficient $$ over its cylindrical surface exposed to airflow. d$.$ What length from the die would it take the rod to cool down to $100$oC so that it can be applied with some surface treatment? Note: Properties are given broadly, depending on whether you need them or not. Aluminum: Density $=2,700$ kg$/$m$3$ ; k $=40$ W$/$m K ; Specific heat cp $=900$ J$/$kg K $=$ cv Air: Density $=1\mathrm{.}2$ kg$/$m$3$ ; k $=0\mathrm{.}026$ W$/$m K ; Viscosity $($kinematic$)$ v $=1\mathrm{.}5$ x $10-5$ m$2/$s or $=1\mathrm{.}8$ x $10-5$ kg$/$m sec Specific heats: cp $=1,005$ J$/$kg K ; cv $=718$ J$/$kg K ; Diffusivity $=20$ x $10-6$ m$2/$s ; Pr $=0\mathrm{.}71$