$(40)$ Consider the aluminum extrusion process,

which involves heating a billet to $500C$ 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 $0\mathrm{.}02\frac{m}{s}.$ Aluminum and

Air properties can be taken as given below.

a$.$ After leaving the Die at $500C,$ the bar is exposed to airflows at $20C$ that flow perpendicular to the

extruded bar at $5\mathrm{.}0\frac{m}{s}$ 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$).$ 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 $(3)$

c$.$ Find the average convection coefficient $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{\text{r}\text{o}\text{d}}$ over its cylindrical surface exposed to airflow.

d$.$ What length from the die would it take for the rod to cool down to $100C$ so that it can be applied with

some surface paint?

Note: Properties are given broadly, depending on whether you need them or not.

Aluminum: Density $=2,700k\frac{g}{m^3}$;$k=0\mathrm{.}025\frac{W}{m}K$; Specific heat $c_p=900\frac{J}{k}gK=c_v$

Air: Density $=1\mathrm{.}2k\frac{g}{m^3}$;$k=0\mathrm{.}026\frac{W}{m}K$; Viscosity $($kinematic$)v=1\mathrm{.}5{10}^{-5}\frac{m^2}{s}$ or $=1\mathrm{.}8{10}^{-5}k\frac{g}{m}sec$

Specific heats: $,c_p=1,005\frac{J}{k}gK$;$c_v=718\frac{J}{k}gK$; Diffusivity $=20{10}^{-6}\frac{m^2}{s}$;$Pr=0\mathrm{.}71(40)$ Consider the aluminum extrusion process,

which involves heating a billet to $500C$ 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 $0\mathrm{.}02\frac{m}{s}.$ Aluminum and

Air properties can be taken as given below.

a$.$ After leaving the Die at $500C,$ the bar is exposed to airflows at $20C$ that flow perpendicular to the

extruded bar at $5\mathrm{.}0\frac{m}{s}$ 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$).$ 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 $(3)$

c$.$ Find the average convection coefficient $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{\text{r}\text{o}\text{d}}$ over its cylindrical surface exposed to airflow.

d$.$ What length from the die would it take for the rod to cool down to $100C$ so that it can be applied with

some surface paint?

Note: Properties are given broadly, depending on whether you need them or not.

Aluminum: Density $=2,700k\frac{g}{m^3}$;$k=0\mathrm{.}025\frac{W}{m}K$; Specific heat $c_p=900\frac{J}{k}gK=c_v$

Air: Density $=1\mathrm{.}2k\frac{g}{m^3}$;$k=0\mathrm{.}026\frac{W}{m}K$; Viscosity $($kinematic$)v=1\mathrm{.}5{10}^{-5}\frac{m^2}{s}$ or $=1\mathrm{.}8{10}^{-5}k\frac{g}{m}sec$

Specific heats: $,c_p=1,005\frac{J}{k}gK$;$c_v=718\frac{J}{k}gK$; Diffusivity $=20{10}^{-6}\frac{m^2}{s}$;$Pr=0\mathrm{.}71$

$(40)$ Consider the aluminum extrusion process,

which involves heating a billet to $500C$ 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 $0\mathrm{.}02\frac{m}{s}.$ Aluminum and

Air properties can be taken as given below.

a$.$ After leaving the Die at $500C,$ the bar is exposed to airflows at $$