Paste extruder Flyingbear tornado

Hi everyone,

This is the paste extruder I designed for flyingbear tornado.
Syringue:
https://www.aliexpress.com/item/150ML-Transparent-Reusable-Large-Big-Plastic-Hydroponics-Nutrient-Measuring-Syringe/32845092700.html?spm=a2g0s.9042311.0.0.dbLT2w
Ball screw:
https://www.aliexpress.com/item/Free-shipping-T8-Lead-screw-300-mm-8mm-brass-copper-nut-KP08-or-KFL08-bearing-Bracket/32596099063.html?spm=a2g0s.9042311.0.0.tIquex
Bearing: extracted from ball screw kit
Linear rail:
https://www.aliexpress.com/item/Top-Quality-4pcs-OD-8mm-x-500mm-Cylinder-Liner-Rail-Linear-Shaft-Optical-Axis-chrome-For/32765021606.html?spm=a2g0s.9042311.0.0.tIquex
Linear bearing (4 parts needed):
https://www.aliexpress.com/item/Free-Shipping-12pcs-lot-LM8UU-8mm-Linear-Bushing-CNC-Linear-Bearings/32434013744.html?spm=a2g0s.9042311.0.0.wu3MNC
Stepper motor:
https://www.aliexpress.com/item/42-Stepper-Motor-Nema-17-4-Lead-42BYGH-1-7A-17HS4401S-with-4-Pin-Dupont-Cable/32814608446.html?spm=a2g0s.9042311.0.0.wu3MNC

Note: in order to keep same firmware for both filament and past extruder I trick filament diameter. Indeed, a parameter from firmware set the length extruded per step motor. This parameter should change depending on the changing of extruder.
This tricking is simple: putting a corrected diameter of filament in cura (different than syringue inside diameter) in orderto correct extruder "wrong" parameter.
I hope you get it.
This calculation is in calculation file included. Please confirm my calculation.

I will put more details on summary soon...