Question

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) 

 

Hint:

Estimating AM build time can be done in several ways. The volume-based approach (\(T = V/Q\)) is often the simplest. Be careful about how the volumetric rate \(Q\) is defined. A common approximation is \(Q = f \cdot w \cdot t\), where \(f\) is feed rate, \(w\) is bead width (often assumed to be nozzle diameter), and \(t\) is layer thickness.

Answer 1

Step 1: Understanding the Concept:
This problem asks for the total fabrication time in an additive manufacturing process. The time is determined by the total volume of the part and the volumetric deposition rate of the material. The volumetric deposition rate can be calculated from the feed rate, nozzle diameter, and layer thickness. Due to ambiguity in the question's formulation regarding layer thickness, the official answer was "Marks to All", but a logical solution can be derived with a reasonable assumption.
Step 2: Key Formula or Approach:
1. Calculate the total volume (\(V_{total}\)) of the part by summing the volumes of the three square blocks.
2. Assume a layer thickness (\(t\)). A common and logical assumption is that the layer thickness is equal to the nozzle diameter (\(d_n\)).
3. Calculate the volumetric deposition rate (\(Q\)). This is the volume of material deposited per unit time. It can be approximated as \(Q = (\text{Area covered per unit time}) \times (\text{layer thickness})\). The area covered per unit time is \((\text{feed rate } f) \times (\text{nozzle diameter } d_n)\). \[ Q = f \cdot d_n \cdot t \] 4. Calculate the total time: \[ \text{Time} = \frac{V_{total}}{Q} \] Step 3: Detailed Calculation:
1. Given Data:
- Nozzle diameter, \(d_n = a/10\) mm.
- Feed rate, \(f = a/5\) mm/min.
- Dimensions of the blocks (side x side x height):
- Bottom block: \(3a \times 3a \times 1.5a\)
- Middle block: \(2a \times 2a \times a\)
- Top block: \(a \times a \times 0.5a\)
2. Calculate Total Volume: \[ V_{total} = (3a \times 3a \times 1.5a) + (2a \times 2a \times a) + (a \times a \times 0.5a) \] \[ V_{total} = 13.5a^3 + 4a^3 + 0.5a^3 = 18a^3 \text{ mm}^3 \] 3. Calculate Volumetric Deposition Rate:
- Assume layer thickness \(t = d_n = a/10\) mm.
- The cross-sectional area of the deposited bead is approximately \(d_n \times t = (a/10) \times (a/10) = a^2/100\) mm².
- Volumetric rate \(Q = (\text{bead cross-section area}) \times (\text{feed rate})\) \[ Q = \left(\frac{a^2}{100}\right) \times \left(\frac{a}{5}\right) = \frac{a^3}{500} \text{ mm}^3/\text{min} \] 4. Calculate Total Fabrication Time: \[ \text{Time} = \frac{V_{total}}{Q} = \frac{18a^3 \text{ mm}^3}{a^3/500 \text{ mm}^3/\text{min}} \] \[ \text{Time} = 18 \times 500 = 9000 \text{ minutes} \] Step 4: Final Answer:
The time taken to fabricate this part is 9000 minutes.
Step 5: Why This is Correct:
This solution follows a standard method for estimating build time in additive manufacturing by dividing the total part volume by the material deposition rate. The result is consistent and derived logically from the problem statement, assuming the layer height equals the nozzle diameter. The official "Marks to All" indicates the question was likely deemed ambiguous, but this approach represents the most reasonable interpretation.