AIS — Where Geometry Breathes

Geometry as a Control Element

Each cell at the container bottom is a Venturi-style truncated pyramid. The bottom is divided into three concentric zones, each with a distinct chamfer angle. The model now shows both the flow equations and the pruning-preview values that drive the graph.

Interactive bottom schematic

Central zone

\(u_{\text{eff}} \approx \dfrac{u_0}{\cos(15^\circ)}\)

Transition zone

Moderate speed, moderate drying effect.

\(u_{\text{eff}} \approx \dfrac{u_0}{\cos(25^\circ)}\)

Peripheral zone

Aggressive drying, high local speed.

\(u_{\text{eff}} \approx \dfrac{u_0}{\cos(40^\circ)}\)

Model parameters

Units: mm, m/s

Top hole D1 (mm)

Cone base Dbase (mm)

Internal hole Dint (mm)

Number of internal holes Nint

Bottom diameter Dbottom (mm)

Number of outer holes Nout

Vavg — under-bottom air-flow speed

0.68 m/s

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Y format Y step Show Vavg line Line color

Hysteresis ms Show delay ms Hide delay ms

k_a f_focus t* α Arrow amplitude 0.80 Arrow speed 1.00

Flow equations used by the graph

\(A=\pi (d/2)^2\)

\(Q=N_{out}\cdot A_{D1}\cdot V_0\)

\(V_{int}=Q/A_{int,tot}\)

\(V_{avg}=Q/A_{bottom}\)

Pruning preview updates with the model.

📐 1. Geometry as a Control Element

Base: 20 × 20 mm
Top: 10 × 10 mm
Height: 8 mm
Chamfer angle: \(\theta = \arctan\!\left(\frac{20-10}{2 \cdot 8}\right) \approx 32^\circ\)

\[ u_{\text{eff}} \approx \frac{u_0}{\cos(32^\circ)} \approx 1.18 \cdot u_0 \;\Rightarrow\; \text{air accelerates by } \sim 18\% \] \[ u_{\text{eff}} \approx \frac{u_0}{\cos(\theta)} \]

💨 2. Bottom Microclimate: Air Bag and Controlled Drying

The gap \(g = 2\text{–}5\,\text{cm}\) between the bottom and the stand forms an adjustable air chamber. At \(g \geq 20\,\text{mm}\), airflow saturates:

\[ m_{u,b} \approx 0.2 + 0.6 \cdot \tanh\!\left(\frac{g}{10}\right) \quad \Rightarrow \quad u_b = m_{u,b} \cdot u_{\text{out}} \] \[ \varphi_b \approx m_{\varphi,b} \cdot \varphi_{\text{out}} + (1 - m_{\varphi,b}) \cdot \varphi_{\text{sub}} \] \[ t_{\text{dry},b} \approx \frac{\rho_w \cdot \delta}{k_m \cdot \Delta \rho_v}, \quad k_m = \frac{Sh \cdot D_v}{d}, \quad Re = \frac{u_{\text{eff}} \cdot d}{\nu} \]

→ All governed by geometry, gap height, and climate.

🌱 3. Apex Emergence and Pruning Efficiency

\[ P_{\text{exit},b} = 1 - \exp(-k_a \cdot \varphi_{\text{cap}}) \]

\[ \varphi_{\text{cap}} = \left(\frac{\pi \cdot r_{\text{eff},b}^2}{s_b^2}\right) \cdot f_{\text{focus}} \]

\[ q_b = \frac{1}{1 + \left(\frac{t_{\text{dry},b}}{t^*}\right)^\alpha} \]

→ With optimal geometry and drying, apices emerge and are pruned in the target zone.

📊 4. Daily Laterals and Vertical Profile

\[ N_{\text{events},b} = J_{\downarrow} \cdot P_{\text{exit},b}, \quad J_{\downarrow} = \frac{N_{\downarrow} \cdot v_z}{H_{\text{col}}} \]

\[ N_{\text{lat},b} = N_{\text{events},b} \cdot b \cdot q_b \]

\[ \rho_{\text{lat},b}(r) \propto P_{\text{exit},b} \cdot q_b \cdot w(r) \]

→ With chamfer and localized drying, the branching peak shifts downward, creating programmable root architecture.