Table 3 Summary of parameters employed in analytical models for predicting grain size and PCG formation in extruded profiles.

(1) \(Pd = \frac{{Gb^{2} }}{{10}}\left[ {\rho {\text{i(1}} - {\text{ln(10B}}\rho {\text{i}}^{{0,5}} {\text{)) + }}\frac{{2\theta }}{{b\delta }}*\left( {1 + \ln \left( {\frac{{\theta c}}{\theta }} \right)} \right)} \right]\)

\(\:Pd:\) stored energy (J/mol*K)

\(\:\text{G}\): material shear modulus (2.05 × 10¹⁰ Pa)

\(\:\text{B}\): Burgers vector (2.86 × 10^− 10 m)

\(\rho i\): internal dislocation density

\(\:\delta\:\): subgrain size (µm)

\(\:\varTheta\:\): subgrain misorientation angle

\(\:\varTheta\:c\): misorientation angle limit (15°)

C: material constant (3.36 × 10 –9 m^− 1)

\(\:n\): material constant (5.577)

\(\:Z\): Zener-Hollomon parameter

Q: activation energy (182000 J/mol*K)

R: universal gas constant (8.341 J/mol)

T: temperature (K)

\(\:\dot{\epsilon\:}\): maximum strain rate of point along extrusion flow path (s^− 1)

\(\:{X}_{DRX}\): percentage dynamic recrystallization (%)

\(\:\beta\::\) constant (1.823)

\(\:\epsilon\:\): strain

\(\:{\epsilon\:}_{c}\): critical pinch-off strain (3)

\(\:{\epsilon\:}_{s}\): saturation strain (16)

\(\:m:\:\)constant (1.109)

\(\:{d}_{t}\): cross-sectional diameter of DRX grain (µm)

\(\:{\delta\:}_{ss}\): subgrain size under steady-state conditions (6.4 μm)

\(\:{k}_{1}:\:\)material constant (\(\:0.4\))

\(\:{d}_{0}\): billet grain diameter (µm)

(2) \(\:\frac{1}{\delta\:}=C\:{\left(ln\left(Z\right)\right)}^{n}\)

(3) \(\:Z=\dot{\epsilon\:}\:exp\left(\frac{Q}{RT}\right)\)

(4) \(\:{X}_{DRX}=1-\text{e}\text{x}\text{p}\left[-\beta\:{\left(\frac{\epsilon\:-{\epsilon\:}_{c}}{{\epsilon\:}_{\text{s}}}\right)}^{m}\right]\)

(5) \(\:{d}_{t}=\left({d}_{0}-2.5\:{\delta\:}_{ss}\right){\left({k}_{1}\right)}^{\epsilon\:}+2.5\:{\delta\:}_{ss}\)