DrumBinTransfer

Drum/bin solid transfer model for batch operations (steady-state and dynamic).

Overview

The DrumBinTransfer class models batch solid material transfer operations using drums or bins with conveyor-based discharge systems. This model is commonly used in pharmaceutical, food, and chemical processing industries for transferring powders, granules, and other solid materials between process units.

Use Case

The drum/bin transfer system is employed when:

• Batch processing requires controlled material handling

• Materials need to be transferred between different elevation levels

• Dust containment is important

• Process requires intermediate storage capacity

• Materials have varying flowability characteristics

Mathematical Model

Steady-State Model

The steady-state calculation determines the actual transfer rate and batch completion time based on:

1. Available Material Mass: Calculated from container fill level and material density

2. Effective Discharge Rate: Considers flowability factor and discharge efficiency

3. Transfer Rate Limiting: Accounts for material availability and discharge capacity

4. Batch Time Calculation: Includes discharge, transport, and handling times

Key equations:

\[\text{flowability\_factor} = 0.5 + 0.5 \times \text{flowability}\]

\[\text{max\_effective\_rate} = \text{transfer\_rate\_max} \times \text{flowability\_factor} \times \text{discharge\_efficiency}\]

\[\text{actual\_rate} = \min(\text{rate\_setpoint}, \text{max\_effective\_rate})\]

\[\text{total\_time} = \frac{\text{available\_mass}}{\text{actual\_rate}/60} + \frac{\text{transfer\_distance}}{\text{conveyor\_speed}} + \text{handling\_time}\]

Dynamic Model

The dynamic model tracks:

• Transfer rate response with first-order dynamics

• Container level changes based on mass balance

• System stops when container is empty

State equations:

\[\frac{d(\text{transfer\_rate})}{dt} = \frac{\text{rate\_ss} - \text{transfer\_rate}}{\tau_{\text{rate}}}\]

\[\frac{d(\text{fill\_level})}{dt} = -\frac{\text{volume\_flow\_rate}}{\text{container\_capacity}}\]

where \(\tau_{\text{rate}} = 10.0\) s is the discharge rate response time constant.

Parameters

Model Parameters

Parameter	Range	Unit	Description
container_capacity	0.1 - 2.0	m³	Container volume capacity
transfer_rate_max	10.0 - 500.0	kg/min	Maximum discharge rate
material_density	200.0 - 2000.0	kg/m³	Bulk density of material
discharge_efficiency	0.5 - 1.0		Discharge mechanism efficiency
handling_time	60.0 - 300.0	s	Setup and handling time per batch
conveyor_speed	0.1 - 2.0	m/s	Conveyor belt speed
transfer_distance	1.0 - 50.0	m	Transfer distance

Examples

Basic Usage

from transport.batch.solid.DrumBinTransfer import DrumBinTransfer
import numpy as np

# Create pharmaceutical transfer system
pharma_transfer = DrumBinTransfer(
    container_capacity=0.3,      # 300 L container
    transfer_rate_max=80.0,      # 80 kg/min max rate
    material_density=600.0,      # Pharmaceutical powder density
    discharge_efficiency=0.9,    # Good discharge efficiency
    handling_time=90.0,          # 1.5 min handling time
    conveyor_speed=0.4,          # 0.4 m/s conveyor speed
    transfer_distance=8.0        # 8 m transfer distance
)

# Steady-state calculation
u = np.array([0.8, 70.0, 0.7])  # [fill_level, setpoint, flowability]
result = pharma_transfer.steady_state(u)
transfer_rate, batch_time = result
print(f"Transfer rate: {transfer_rate:.1f} kg/min")
print(f"Batch time: {batch_time:.1f} s")

Dynamic Simulation

# Dynamic simulation
import matplotlib.pyplot as plt

time_span = np.linspace(0, 600, 301)  # 10 minutes
dt = time_span[1] - time_span[0]

# Initial conditions
x = np.array([0.0, 1.0])  # [transfer_rate=0, fill_level=1.0]
u = np.array([1.0, 70.0, 0.8])  # [target_fill, setpoint, flowability]

# Storage for results
transfer_rates = []
fill_levels = []

# Euler integration
for t in time_span:
    transfer_rates.append(x[0])
    fill_levels.append(x[1])

    if x[1] > 0:  # Continue only if material remains
        dx_dt = pharma_transfer.dynamics(t, x, u)
        x = x + dx_dt * dt
        x[1] = max(0.0, x[1])  # Prevent negative fill level
    else:
        break

# Plot results
plt.figure(figsize=(10, 6))
plt.subplot(2, 1, 1)
plt.plot(time_span[:len(transfer_rates)]/60, transfer_rates)
plt.ylabel('Transfer Rate (kg/min)')
plt.title('DrumBinTransfer Dynamic Response')

plt.subplot(2, 1, 2)
plt.plot(time_span[:len(fill_levels)]/60, np.array(fill_levels)*100)
plt.xlabel('Time (min)')
plt.ylabel('Fill Level (%)')
plt.show()

Visualization Results

Dynamic Response

Dynamic response showing transfer rate and container fill level during batch operation.

Detailed Analysis

Detailed analysis showing flowability effects, fill level impacts, setpoint tracking, and batch time calculations.

Example Output

Complete example output

=== DrumBinTransfer Example ===

1. Creating DrumBinTransfer instances:
----------------------------------------
Pharmaceutical transfer: PharmaPowderTransfer
  Container capacity: 0.3 m³
  Max transfer rate: 80.0 kg/min

Food ingredient transfer: FoodIngredientTransfer
  Container capacity: 1.5 m³
  Max transfer rate: 300.0 kg/min

2. Steady-state analysis:
-------------------------
Pharmaceutical transfer results:
  Full container, good flow:
    Transfer rate: 60.0 kg/min
    Batch time: 290.0 s

  Half full, moderate flow:
    Transfer rate: 57.6 kg/min
    Batch time: 203.8 s

  Low level, poor flow:
    Transfer rate: 46.8 kg/min
    Batch time: 133.1 s

  High setpoint, excellent flow:
    Transfer rate: 72.0 kg/min
    Batch time: 230.0 s

3. Dynamic simulation:
--------------------
Simulating batch transfer with pharma_transfer:
Initial conditions: rate=0 kg/min, fill=100%
Setpoint: 70.0 kg/min, flowability: 0.8

Dynamic simulation results:
  Final time: 182.0 s
  Final fill level: 0.000
  Average transfer rate: 59.8 kg/min

4. Flowability sensitivity analysis:
-----------------------------------
Food transfer flowability sensitivity (80% fill, 80 kg/min setpoint):
  Flowability 0.1: 80.0 kg/min
  Flowability 0.2: 80.0 kg/min
  Flowability 0.3: 80.0 kg/min
  Flowability 0.4: 80.0 kg/min
  Flowability 0.5: 80.0 kg/min
  Flowability 0.6: 80.0 kg/min
  Flowability 0.7: 80.0 kg/min
  Flowability 0.8: 80.0 kg/min
  Flowability 0.9: 80.0 kg/min
  Flowability 1.0: 80.0 kg/min

5. Model introspection:
--------------------
Model type: Drum/Bin Solid Transfer
Description: Batch solid material transfer using drums or bins with conveyor discharge

Key parameters:
  container_capacity: 0.3 m³ - Container volume capacity
  transfer_rate_max: 80.0 kg/min - Maximum discharge rate
  material_density: 600.0 kg/m³ - Bulk density of material
  discharge_efficiency: 0.9 - - Discharge mechanism efficiency
  handling_time: 90.0 s - Setup and handling time per batch

Operating ranges:
  container_capacity: 0.1 - 2.0
  transfer_rate_max: 10.0 - 500.0
  material_density: 200.0 - 2000.0
  discharge_efficiency: 0.5 - 1.0
  fill_level: 0.0 - 1.0

6. Creating visualization plots...
Plots saved as DrumBinTransfer_example_plots.png and DrumBinTransfer_detailed_analysis.png

=== Example completed successfully ===

Literature References

1. Perry, R.H., Green, D.W. (2019). “Perry’s Chemical Engineers’ Handbook”, 9th Edition, McGraw-Hill, Chapter 21: Solid-Solid Operations.

2. Schulze, D. (2008). “Powders and Bulk Solids: Behavior, Characterization, Storage and Flow”, Springer, ISBN: 978-3-540-73768-1.

3. Marinelli, J., Carson, J.W. (1992). “Solve solids flow problems in bins, hoppers, and feeders”, Chemical Engineering Progress, 88(5), 22-28.

4. Jenike, A.W. (1964). “Storage and Flow of Solids”, Bulletin No. 123, University of Utah Engineering Experiment Station.

5. BMHB (2003). “The Design of Hoppers, Silos and Bunkers”, Institution of Chemical Engineers, Rugby, UK.

6. Roberts, A.W. (2001). “Particle Technology - Storage and Flow of Particulate Solids”, TUNRA Bulk Solids Research Associates, University of Newcastle.