A solid part (see figure) of polymer material is to be fabricated by additive manufacturing (AM) in square-shaped layers starting from the bottom of the part working upwards. The nozzle diameter of the AM machine is a/10 mm and the nozzle follows a linear serpentine path parallel to the sides of the square layers with a feed rate of a/5 mm/min.

Ignore any tool path motions other than those involved in adding material, and any other delays between layers or the serpentine scan lines. The time taken to fabricate this part is ___________ minutes. (Answer in integer)

Correct Answer: The time taken to fabricate this part is 9000 minutes.

Step-by-Step Explanation:

1. Calculate the Total Volume of the Solid Part:
From the provided image, we can identify that the solid part is composed of three stacked square blocks (bottom, middle, and top):

• Bottom Block:
  Dimensions: Length = $3a$ mm, Width = $3a$ mm, Height = $1.5a$ mm.
  Volume of the bottom block, $V_1$:
  $V_1=3a\times 3a\times 1.5a=13.5a^3{\text{mm}}^3$
  
• Middle Block:
  Dimensions: Length = $2a$ mm, Width = $2a$ mm, Height = $a$ mm.
  Volume of the middle block, $V_2$:
  $V_2=2a\times 2a\times a=4a^3{\text{mm}}^3$
  
• Top Block:
  Dimensions: Length = $a$ mm, Width = $a$ mm, Height = $0.5a$ mm.
  Volume of the top block, $V_3$:
  $V_3=a\times a\times 0.5a=0.5a^3{\text{mm}}^3$
  

Adding the volumes of all three sections gives the total volume $V$ of the solid part:
$V=V_1+V_2+V_3$
$V=13.5a^3+4a^3+0.5a^3=18a^3{\text{mm}}^3$

2. Determine the Volumetric Deposition Rate:
In additive manufacturing, the volume of material deposited per unit time depends on the cross-sectional area of the track (road) and the scan speed (feed rate).
The nozzle diameter is:
$d=\frac{a}{10}\text{mm}$
Assuming the layer thickness and road width are both equal to the nozzle diameter $d$, the cross-sectional area of a single deposited road is:
$A=d^2={\left(\frac{a}{10}\right)}^2=\frac{a^2}{100}{\text{mm}}^2$
The feed rate (linear velocity of deposition) is:
$v=\frac{a}{5}\text{mm/min}$
The volumetric deposition rate $Q$ is:
$Q=A\times v=\frac{a^2}{100}\times \frac{a}{5}=\frac{a^3}{500}{\text{mm}}^3\text{/min}$

3. Calculate the Total Fabrication Time:
The total time $T$ required to fabricate the solid part is given by the ratio of the total volume to the volumetric deposition rate:
$T=\frac{V}{Q}$
Substituting the expressions for $V$ and $Q$:
$T=\frac{18a^3}{\left(\frac{a^3}{500}\right)}$
Simplifying the expression by cancelling $a^3$:
$T=18\times 500=9000\text{minutes}$