3TE/formulas_reference.md

# 4NK Waste & Water - Complete Calculation Formulas Reference

## 1. Infrastructure Capacity Calculations

### 1.1 Daily Processing Capacity
```
Daily_processing_capacity (T/day) = Module_capacity (T) × Number_of_modules
Daily_processing_capacity = 67T × 21 = 1,407T/day
```

### 1.2 Water Content Calculation
```
Water_content (T) = Total_waste (T) × Water_percentage (%)
Water_content = 67T × 75% = 50.25T water per module
```

### 1.3 Total Containers
```
Total_containers = Number_of_modules × Containers_per_module
Total_containers = 21 × 4 = 84 containers
```

## 2. Processing Time Calculations

### 2.1 Mesophilic Digestion Duration
```
Mesophilic_duration (days) = Hygienization_duration + Additional_days
Mesophilic_duration = 18 days + 3 days = 21 days
```

### 2.2 Drying and Bioremediation Duration
```
Drying_bioremediation_duration (days) = Drying_duration + (Bioremediation_phases × Phase_duration)
Drying_bioremediation_duration = 21 days + (3 phases × 21 days) = 84 days (variable)
```

### 2.3 Thermophilic Digestion and Composting Duration
```
Thermophilic_composting_duration (days) = Thermophilic_duration + Composting_duration
Thermophilic_composting_duration = 18 days + 3 days = 21 days
```

### 2.4 Spirulina Cycle Duration
```
Spirulina_cycle_duration (hours) = 72 hours
Spirulina_cycle_duration (days) = 72 / 24 = 3 days
```

## 3. Methane Production Calculations

### 3.1 Dry Matter Calculation (VS - Volatile Solids)
```
Dry_matter (kg VS) = Waste_quantity (kg) × (1 - Water_percentage (%))
Dry_matter_percentage (%) = 100% - Water_percentage (%)
```

**Example**:
- Waste quantity: 67,000 kg (67T)
- Water percentage: 75%
- Dry matter: 67,000 × (1 - 0.75) = 67,000 × 0.25 = 16,750 kg VS
- Dry matter percentage: 100% - 75% = 25%

### 3.2 Methane Production from BMP
```
Methane_production (m³/day) = BMP (Nm³ CH₄/kg VS) × Dry_matter (kg VS/day) × Efficiency_factor
```

**Parameters**:
- BMP: Biochemical Methane Potential (Nm³ CH₄/kg VS) - varies by waste type
- Dry_matter: Calculated from waste quantity and water percentage
- Efficiency_factor: Processing efficiency (typically 0.7-0.9)

### 3.3 Origin Units to Methane Conversion
```
Methane_per_1000m³ = Number_of_origin_units × Conversion_factor
```
**Parameter**: Number of origin units to produce 1000m³ of methane (varies by waste origin)

## 4. Gas Production Calculations

### 4.1 Biogas Composition
```
Biogas_total (m³/day) = Methane_production (m³/day) / Methane_percentage_in_biogas
```

**Biogas Composition**:
- Methane (CH₄): 40% of biogas
- CO₂: 60% of biogas

### 4.2 Methane from Biogas
```
Methane_production (m³/day) = Biogas_total (m³/day) × 0.40
```

### 4.3 CO₂ Production from Biogas
```
CO2_production (m³/day) = Biogas_total (m³/day) × 0.60
```

**Alternative calculation**:
```
CO2_production (m³/day) = Methane_production (m³/day) × (0.60 / 0.40)
CO2_production (m³/day) = Methane_production (m³/day) × 1.5
```

### 4.4 Total Gas Production
```
Total_gas_production (m³/day) = Methane_production (m³/day) + CO2_production (m³/day)
Total_gas_production (m³/day) = Biogas_total (m³/day)
```

## 5. Energy Calculations

### 5.1 Heat Energy from Biogas
```
Heat_energy (kJ/day) = Methane_production (m³/day) × Methane_energy_content (kJ/m³) × Combustion_efficiency
```
**Parameters**:
- Methane_energy_content: 35,800 kJ/m³ (lower heating value)
- Combustion_efficiency: 0.90 (90%)

### 5.2 Heat Energy Conversion to kWh
```
Heat_energy (kW.h/day) = Heat_energy (kJ/day) / 3600
```

### 5.3 Electrical Power from Biogas Generator
```
Electrical_power_biogas (kW) = Methane_production (m³/day) × Methane_energy_content (kJ/m³) × Electrical_efficiency / (3600 × 24)
```
**Parameter**: Electrical_efficiency: 0.40 (40% for combined heat and power systems)

### 5.4 Electrical Power from Solar Panels
```
Electrical_power_solar (kW) = Solar_panel_surface (m²) × Solar_irradiance (kW/m²) × Panel_efficiency
```
**Parameters**:
- Solar_irradiance: Varies by location and season (typically 0.1-1.0 kW/m²)
- Panel_efficiency: Typically 0.15-0.22

### 5.5 Total Electrical Power
```
Total_electrical_power (kW) = Electrical_power_biogas (kW) + Electrical_power_solar (kW)
```

### 5.6 Module Electrical Consumption
```
Module_electrical_consumption (kW) = Sum of all equipment consumption:
  - 1 pump (méthanisation): 0.5 kW
  - 1 séchoir du gaz: 2.0 kW
  - 1 compresseur: 3.0 kW
  - 1 lampe UV-C × 12m: 0.3 kW
  - 5 racloires électriques × 3: 1.5 kW
  - 1 pompe (spiruline): 0.5 kW
  - 5 × 12m LED de culture: 0.6 kW
  - 3 pompes eau: 1.5 kW
  - Capteurs: 0.1 kW
  - 1 serveur: 0.2 kW
  - 1 borne Starlink: 0.1 kW
  - 1 tableau élec: 0.1 kW
  - 1 convertisseur panneaux solaires: 0.1 kW
  Total per module: ~10.5 kW
```

```
Total_modules_consumption (kW) = Module_electrical_consumption (kW) × Number_of_modules
Total_modules_consumption (kW) = 10.5 kW × Number_of_modules
```

### 5.7 Net Electrical Power
```
Net_electrical_power (kW) = Total_electrical_power (kW) - Total_modules_consumption (kW)
```

## 6. Bitcoin Mining Calculations

### 6.1 Number of Flex Miners
```
Number_of_flex_miners = Available_electrical_power (kW) / Power_per_miner (kW)
Number_of_flex_miners = Available_electrical_power (kW) / 2 kW
```

### 6.2 Bitcoin Production
```
Bitcoins_BTC_per_year = 79.2 × 0.0001525 / flex_miner
```
**Formula**: `BTC/year = 79.2 × 0.0001525 / number_of_flex_miners`

**Parameters**:
- 79.2: Constant factor
- 0.0001525: BTC per flex miner factor
- flex_miner: Number of 4NK flex miners (2kW each)

### 6.3 Bitcoin Value
```
Bitcoin_value (€) = Bitcoin_quantity (BTC) × Bitcoin_price (€/BTC)
```
**Parameter**: Bitcoin_price = 100,000 €/BTC

## 7. Material Output Calculations

### 7.1 Water Output
```
Water_output (t/day) = Water_input (t/day) - Water_consumed_in_processes (t/day) + Water_from_spirulina (t/day)
```

**Water Input**:
```
Water_input (t/day) = Waste_quantity (t/day) × Water_percentage (%)
Water_input = 67T × 75% = 50.25T water per module per day
```

### 7.2 Fertilizer Production
```
Fertilizer_output (t/day) = Composting_output (t/day) × Fertilizer_yield_factor
```
**Parameter**: Fertilizer_yield_factor: 1.0 (100% - all compost becomes standardized fertilizer)

## 8. Valorization Calculations

### 8.1 Waste Treatment Valorization
```
Waste_treatment_valorization (€/year) = Waste_quantity (t/year) × 100 €/t
```
**Parameter**: 100 €/t

### 8.2 Fertilizer Valorization
```
Fertilizer_valorization (€/year) = Fertilizer_quantity (t/year) × 215 €/t
```
**Parameter**: 215 €/t

### 8.3 Heat Valorization
```
Heat_valorization (€/year) = Heat_quantity (t/year) × 0.12 €/t
```
**Parameter**: 0.12 €/t

### 8.4 Carbon Equivalent - Burned Methane (CH₄)
```
CH4_carbon_valorization (€/year) = CH4_quantity (tCO₂e/year) × 172 €/tCO₂e
```
**Parameters**:
- 630 €/tC ≈ 172 €/tCO₂e
- Conversion: 1 tC = 3.67 tCO₂e
- Formula: `172 = 630 / 3.67`

### 8.5 Carbon Equivalent - Sequestered CO₂
```
CO2_carbon_valorization (€/year) = CO2_sequestered (tCO₂e/year) × 27 €/tCO₂e
```
**Parameters**:
- 100 €/tC ≈ 27 €/tCO₂e
- Conversion: 1 tC = 3.67 tCO₂e
- Formula: `27 = 100 / 3.67`

### 8.6 Carbon Equivalent - Avoided Electricity Consumption
```
Energy_carbon_valorization (€/year) = Electricity_avoided (kW/year) × 0.12 €/kW
```
**Parameter**: 0.12 €/kW

### 8.7 Land Valorization (Brownfield)
```
Land_valorization (€) = Brownfield_area (m²) × Valorization_rate (€/m²)
```
**Parameter**: 4000 m² brownfield (valorization rate configurable)

## 9. Financial Calculations

### 9.1 Total Revenues
```
Total_Revenues (€/year) = Sum of all revenue items:
  - Raw_rental
  - Biological_waste_treatment_service
  - Bitcoin_management_service
  - Provision_of_standardized_fertilizers_service
  - Provision_of_waste_heat_service
  - Provision_of_carbon_credit_indices_service
  - Brownfield_redevelopment_service
  - Transport_service
  - Commercial_partnerships
  - Other_revenues
```

### 9.2 Total Variable Costs
```
Total_Variable_Costs (€/year) = Sum of all variable cost items:
  - Rental_and_services
  - Commissions_intermediaries_import
  - Other_variable_costs
  - Transport
```

### 9.3 Gross Margin
```
Gross_Margin (€/year) = Total_Revenues (€/year) - Total_Variable_Costs (€/year)
```

### 9.4 Total Fixed Costs (OPEX)
```
Total_Fixed_Costs (€/year) = Sum of all fixed cost items:
  - Salaries_and_social_charges
  - Marketing_communication_expenses
  - R&D_product_development
  - Administrative_legal_fees
  - Other_general_expenses
```

### 9.5 Operating Result (EBITDA)
```
EBITDA (€/year) = Gross_Margin (€/year) - Total_Fixed_Costs (€/year)
```

### 9.6 Cash Flow
```
Cash_Flow (€/year) = EBITDA (€/year) - Non_cash_adjustments (€/year) - Working_capital_changes (€/year)
```

### 9.7 Total Investments (CAPEX)
```
Total_Investments (€) = Sum of all investment items:
  - Equipment_machinery
  - Technology_development
  - Patents_IP
```

### 9.8 Funding Need
```
Funding_Need (€) = Total_Investments (€) - Available_Cash (€) - Cash_Flow (€)
```

### 9.9 Use of Raised Funds
```
Total_Use_of_Funds (€) = Sum of all fund utilization items:
  - Product_development_POC_MVP
  - Marketing_customer_acquisition
  - Team_strengthening_recruitment
  - Structure_administrative_fees
```

## 10. Key Performance Indicators (KPIs)

### 10.1 Customer Acquisition Cost (CAC)
```
CAC (€) = Marketing_Costs (€) / Number_of_New_Customers
```

### 10.2 Lifetime Value (LTV)
```
LTV (€) = Average_Revenue_per_Customer (€) × Average_Customer_Lifespan (years)
```

### 10.3 Break-even Point
```
Break_even_days = Fixed_Costs (€) / (Daily_Revenue (€/day) - Daily_Variable_Costs (€/day))
```

### 10.4 Break-even Point (Alternative)
```
Break_even_days = Fixed_Costs (€) / Daily_Gross_Margin (€/day)
```

## 11. Service Pricing Calculations

### 11.1 Service Pricing per Module per Year
```
Service_price_per_module_per_year (€) = Base_price (€) × Module_multiplier
```

### 11.2 Service Pricing over 10 Years
```
Service_price_10_years (€) = Service_price_per_module_per_year (€) × Number_of_modules × 10
```

### 11.3 First Year Prototype Pricing
```
First_year_price (€) = Service_price_per_module_per_year (€) × Prototype_discount_factor
```
**Parameter**: Prototype_discount_factor: Typically 0.5-0.8 (50-80% of standard price)

## 12. Conversion Factors

### 12.1 Energy Conversions
```
1 kW.h = 3600 kJ
1 kJ = 1/3600 kW.h
```

### 12.2 Carbon Conversions
```
1 tC (tonne of carbon) = 3.67 tCO₂e (tonnes of CO₂ equivalent)
1 tCO₂e = 1/3.67 tC ≈ 0.272 tC
```

### 12.3 Time Conversions
```
1 day = 24 hours
1 year = 365 days (or 366 for leap years)
```

### 12.4 Mass Conversions
```
1 tonne (t) = 1000 kg
1 kg = 0.001 t
```

## 13. Processing Efficiency Factors

### 13.1 Anaerobic Digestion Efficiency
```
Methane_efficiency = Actual_methane_production / Theoretical_methane_production
```
**Typical range**: 0.70 - 0.90 (70-90%)

### 13.2 Electrical Conversion Efficiency
```
Electrical_efficiency = Electrical_power_output / Energy_input
```
**Typical range**: 0.35 - 0.45 (35-45% for CHP systems)

### 13.3 Solar Panel Efficiency
```
Solar_panel_efficiency = Electrical_power_output / (Solar_irradiance × Panel_area)
```
**Typical range**: 0.15 - 0.22 (15-22%)

## 14. Water Balance Calculations

### 14.1 Water Input from Waste
```
Water_input (t/day) = Waste_quantity (t/day) × Water_percentage (%)
```

### 14.2 Water Consumption in Processes
```
Water_consumption (t/day) = Sum of water used in:
  - Mesophilic_digestion
  - Drying_process (evaporation)
  - Bioremediation
  - Thermophilic_digestion
  - Composting
  - Water wall cooling (evaporation)
```

### 14.3 Water from Spirulina Cycle
```
Spirulina_cycle_duration = 21 days
Spirulina_cycles_per_day = 1 / 21 = 0.0476 cycles/day

Water_from_spirulina (t/day) = Spirulina_cycle_water_output (t/cycle) × Spirulina_cycles_per_day
```

**Spirulina Cycle Management**:
- Spirulina culture: 21 days cycle
- After 21 days: Water returns to thermophilic anaerobic digestion
- Water output from spirulina: Depends on culture volume and evaporation

### 14.4 Water Evaporation
```
Water_evaporation (t/day) = Water_wall_surface (m²) × Evaporation_rate (m/day) × Water_density (t/m³)
```
**Parameters**:
- Evaporation_rate: Depends on temperature, humidity, wind (typically 0.001-0.01 m/day)
- Water_density: 1 t/m³

### 14.5 Net Water Output
```
Net_water_output (t/day) = Water_input (t/day) - Water_consumption (t/day) - Water_evaporation (t/day) + Water_from_spirulina (t/day)
```

## 15. Module and Container Calculations

### 15.1 Modules per Year
```
Modules_per_year = Total_modules / Project_duration (years)
```

### 15.2 Containers per Module
```
Containers_per_module = 4 (fixed: mesophilic, drying/bioremediation, thermophilic/composting, water/spirulina)
```

### 15.3 Total Container Capacity
```
Total_container_capacity = Number_of_modules × Containers_per_module × Container_capacity
```

## Notes on Formula Display

All formulas must be displayed with:
- **Monospace font** (JetBrains Mono, Fira Code, or Courier New)
- **Clear variable names** with units
- **Parameter values** shown explicitly
- **Calculation steps** when applicable
- **Input values** used in the calculation
- **Result** with appropriate units

## Formula Validation Rules

- All input values must be positive (where applicable)
- Division by zero must be prevented
- Unit conversions must be consistent
- Efficiency factors must be between 0 and 1
- Percentages must be between 0 and 100
- Time values must be positive
- Mass/volume values must be positive