Apart from the energy consumption mentioned in Sections 6.2.1–6.2.4, some additional energy is also consumed by the computer, workstation and some amount of energy are also lost during the process. This extra amount of miscellaneous energy (Emisc) is considered to be 5% of the total energy:
0.05 .
misc laser heater roller piston
E E E E E (6.17)
extruder. The pressure drop is dependent on the viscosity of the molten material and several geometrical parameters like length and diameter of the liquefier, diameter of the filament, length and diameter of the nozzle, and nozzle angle. Figure 6.2 illustrates different zones of the extruder where the filament is melted and extruded from the nozzle. The material behaviour of thermoplastic filament obeys shear thinning where the viscosity of the material decreases with an increase in the shear rate (Bellini et al. 2004). A power law is adopted to model the dependence of viscosity on shear rate (Turner et al. 2014):
1, K n
(6.20) where η is the viscosity of the material,is the shear rate, K and n are the power law fit parameters. Also, the flow of the molten material in the extruder is non-isothermal. Hence, the dependence of viscosity on temperature should also be considered along with shear rate (Bellini et al. 2004, Turner et al. 2014):
T0
, H T (6.21) where H(T) is the temperature-dependent term for viscosity. The term T0 is the reference temperature at which the power law fit parameters K and n are determined. The Arrhenius relation to model the temperature-dependent term is given by (Turner et al. 2014)
0
1 1
exp ,
H T T T
(6.22) where α is the activation energy and T is the temperature at the end of the extruder.
Figure 6.2 Different zones of the extruder. With permission from Turner et al. (2014).
Copyright 2014, Emerald Publishing Company.
Applying Eqs. (6.21) and (6.22) to the momentum balance on the extruder, Bellini et al. (2004) estimated the pressure drop in the extruder by dividing it into three different zones as illustrated in Figure 6.2. For simplicity, the model was developed considering several assumptions: (a) the melt, i.e. the molten material, is incompressible, (b) no-slip boundary condition is applicable at the wall of the extruder, and (c) the flow of the melt is uniform, steady and laminar. The pressure drops at each sections of the extruder are given by (Bellin et al. 2004, Turner el al. 2014)
1 1
1 1 1
1 0
2 1 1 3
2 3 3
0
2 1
1
1
3 3
3 1 1
2 exp ,
2
2 1 1 1 1
3 2 exp ,
3 tan 2 2
3 2
2
f f
f
f f
f f
f
m m
f f
m
m f m
f
m m
n
m f
f
v m
P l
T T d
m d
P m
T T
d d
m d
P l v
2 1
3 1 0
1 1
exp ,
2
f
f
m
m T T
d
(6.23)
where ∆P1, ∆P2 and ∆P3 are the pressure drops in zone I, II and III, respectively, l1 and l3 are the lengths of the extruder corresponding to zone I and III, respectively, vf is the velocity of the filament at the entry, βn is the nozzle angle, ϕ and mf are the material constants, and d1 and d2 are diameters of the filament and nozzle, respectively. The constants ϕ and mf represent the fluidity and flow exponent, respectively. ϕ and mf are related to the power law fit parameters as (Bellin 2002)
1
, 1.
n
f
K
m n
(6.24)
The total pressure drop (∆P) is given by the algebraic summation of the pressure drops at all the three zones:
1 2 3.
P P P P
(6.25) The force (Fe) required to push the molten thermoplastic material through the extruder is given by
e f,
F PA (6.26) where Af is the cross-sectional area of the filament. The energy required in extrusion (Eextrusion) is given by
extrusion e m,
E F s (6.27)
where sm is the distance travelled by the molten material in the extruder. The distance travelled (sm) is given by
m e e,
s v t (6.28) where ve is the extrusion velocity and te is the extrusion time. The extrusion time (te) is assumed to be equal to the printing time. The energy required in extrusion (Eextrusion) is provided by the motor:
e,
extrusion m
E E
(6.29) where ηm is the efficiency of the motor.
6.3.3 Energy consumed in heating the baseplate
The extruded material is deposited in a baseplate that is heated by the resistive heater. The baseplate where the printing takes place is made of a thermosetting plastic. The heat input (Qb) to raise the temperature of the base plate during printing is given by
,b b pb f i
Q m c T T (6.30) where mb is the mass of the base plate, cpb is specific heat of the baseplate, Tf is the final attainable temperature by the baseplate and Ti is the initial temperature of the baseplate.
The heat input provided by the resistive heater and the energy consumption (Ebaseplate) is given by
b.
baseplate r
E Q
(6.31) 6.3.4 Energy consumed by the movable platform
After a layer is formed, the baseplate moves in the downward direction according to the predetermined layer thickness. The baseplate along with the platform moves and stops repeatedly according to the number of layers present in the part. Assuming that no energy is required in stopping, the mechanical energy of the movable platform (Epl) is given by the summation of potential and total kinetic energies:
1 2
2 ,
pl pl pl pl pl l
E m gh m v N (6.32) where mpl is the mass of platform, g is the acceleration due to gravity, hpl is the total height travelled, vpl is the velocity of the platform and Nl is the number of layers present in the part. The energy required by the platform (Eplatform) is provided by the motor:
pl,
platform m
E E
(6.33) where ηm is the efficiency of the motor.
6.3.5 Miscellaneous energy consumptions
Apart from the energy consumption sources mentioned in Sections 6.3.1–6.3.4, energy is also consumed by the computer, workstation and some energy are also lost in the form of heat and machine error (Song and Telenko 2017, Yosofi et al. 2018). The energy consumed by miscellaneous sources (Emisc) is approximated as 5% of the total energy consumed by all other sources, i.e.,
0.05 .
misc melting baseplate platform extrusion
E E E E E (6.34)