Formula Summary¶

Quick formula summary¶

Profile area¶

\[ A_1 = \frac{w}{\rho} \]

\[ A_{total} = A_1 n \]

Container area¶

\[ A_c = \frac{\pi D_c^2}{4} \]

Extrusion ratio¶

\[ RE = \frac{A_c}{A_{total}} \]

Volume constancy¶

\[ V_r A_c = V_e A_{total} \]

Ram speed from exit speed¶

\[ V_r = V_e \frac{A_{total}}{A_c} \]

Billet area¶

\[ A_b = \frac{\pi D_b^2}{4} \]

Billet mass per millimetre¶

\[ m_{b,mm} = \frac{A_b \rho}{1000} \]

Physical butt and billet length¶

With areas in m², lengths in mm and C_B,effective in kg/mm:

\[ M_{butt}=\rho A_c L_{butt}/1000,\qquad L_{butt,eq}=M_{butt}/C_{B,effective} \]

\[ L_{billet}=M_{extruded}/C_{B,effective}+L_{butt,eq},\qquad M_{billet\ saw}=C_{B,effective} k_{billet} \]

C_B,effective is geometric by default or the user/plant incoming-billet override. One butt is counted per billet, not per pull. See Geometry for the unchanged upsetting relation and override limitation.

Canonical trim and physical event geometry¶

All lengths below are in metres. Let d_total = trim_total_per_billet_m, p = profiles_per_billet, and d_j = d_total / p for each actual contribution. The total is not a front-only or metallurgical charge-weld prediction.

Before calculating mass/time, positive-trim geometry uses:

\[ L_{pull}=\sum_j[n_{c,j}L_c+d_j+(n_{c,j}+1)k_f]+k_{puller} \]

For common commercial positions n_c, N_final = p N_B(n_c+1). For p=1, N_internal = N_B - N_pulls and N_final = N_B n_c + N_pulls + N_internal. Actual partial groups matter. See trim topology for the separate zero-trim convention. Saw events are not multiplied by simultaneous exits; mass uses combined linear weight.

For zero trim, retain N_final = p * N_B * n_c + N_pulls; no extra rejection frontier is added at an internal transition. A zero kerf keeps event counts but gives zero associated kerf length and mass. Commercial cuts is not N_final.

Nominal gross productivity¶

\[ Q_{gross,nominal} = w_{total}V_e60 \]

Technical extrusion time¶

\[ T_{extrusion} = \frac{L_{extruded}}{V_e} \]

Net productivity¶

\[ Q_{net,real} = \frac{M_{good,manufactured}}{T_{technical,h}} \]

AA6063 direct-extrusion analytical estimates¶

For a single AA6063 direct-extrusion operation point, PyExtrusion composes the following validated analytical pieces without adding porthole, bridge-die or Integral Profile corrections.

Equivalent extrudate diameter¶

\[ D_E = \frac{D_C}{\sqrt{R}} \]

Modified Feltham mean strain rate¶

\[ \dot{\bar\varepsilon} = \frac{ 6 V_R D_C^2 (0.171 + 1.86 \ln R)\tan\alpha }{ D_C^3 - D_E^3 } \]

with:

\[ \alpha = 38.7 + 6.9 \ln R \]

Zener-Hollomon and Sheppard-Wright flow stress¶

\[ \ln Z = \ln \dot{\bar\varepsilon} + \frac{Q}{RT} \]

\[ Z = A[\sinh(\alpha_s \sigma)]^n \]

For the built-in AA6063 model:

\[ \alpha_s=0.040\;MPa^{-1},\quad n=5.385,\quad Q=141550\;J/mol,\quad \ln A=22.5 \]

Axisymmetric pressure and force¶

\[ p_{steady} = \bar\sigma (0.171 + 1.86 \ln R) + \bar\sigma\frac{4mL_B}{\sqrt{3}D_C} \]

\[ \Delta p_{breakthrough} = 6.62 + \frac{0.921\ln(Z/A)}{\alpha_s n} \]

\[ F_{baseline} = p_{peak} A_C \]

Stüwe surface exit-temperature estimate¶

\[ T_{exit,surface} = T_B + \Delta T_1 + \Delta T_2 + \Delta T_3 \]

\[ \Delta T_1= \frac{\bar\sigma\ln R} {\sqrt3\,\rho C_p} \]

\[ \Delta T_2= \frac{\bar\sigma} {4\sqrt3\,\rho C_p} \sqrt{\frac{V_RL_B}{a}} \]

\[ \Delta T_3= \frac{\bar\sigma} {4\sqrt3\,\rho C_p} \sqrt{\frac{V_EL_D}{a}}, \qquad a=\frac{k}{\rho C_p} \]

This is a limited analytical surface estimate, not a complete transient thermal field and not a production-grade exit-temperature predictor.

Saha local thermal source terms¶

PyExtrusion exposes selected local source terms from Saha without converting them into an exit temperature:

\[ U'''=\bar\sigma\dot{\bar\varepsilon} \]

\[ q_{container}=\frac{\bar\sigma V_R}{\sqrt3} \]

\[ q_{DMZ}=\frac{\bar\sigma V_{m,j}}{\sqrt3} \]

\[ q_{bearing}=\tau_f V_E \]

For complete sticking at the die bearing:

\[ \tau_f=\frac{\bar\sigma}{\sqrt3} \]

These are local heat-generation/heat-flux terms. A transient heat balance, heat partition and boundary conditions are still required before an exit temperature can be predicted.

Sheppard billet/tooling interface temperature¶

With thermal effusivity

\[ e=\sqrt{k\rho C_p} \]

Sheppard Eq. 2.25 gives the interface relation

\[ \frac{T_B-T_i}{T_i-T_C}=\frac{e_C}{e_B} \]

or

\[ T_i=\frac{T_B+(e_C/e_B)T_C}{1+e_C/e_B} \]

This relation predicts an interface temperature under its stated conduction assumptions. It does not define the partition of newly generated friction heat.

Force interpretation¶

The pressure/force result above is deliberately named \(F_{baseline}\). It is an equivalent-axisymmetric mechanical screening result. It does not include shaped-profile or bridge/porthole-die pressure corrections and must not be presented as the final force requirement of a real porthole die.

When the baseline is compared with a configured scalar press limit, the arithmetic difference is:

\[ F_{reserve}=F_{press}-F_{baseline} \]

\(F_{reserve}\) is a screening indicator only. It is not guaranteed remaining capacity, available hydraulic capacity, a safety margin, force available for a porthole die or a prediction of the real die load.