To determine the lowest reasonable amount of fuel required for outdoor cremation of a body using wood and then using wood doused in petrol, we need to perform a detailed energy balance calculation. This involves estimating the total energy required to cremate a body and then calculating the mass of fuel needed based on the energy content of the fuels and the efficiency of the cremation process.
### **1. Estimating the Energy Required for Cremation**
**a. Assumptions:**
- **Average Body Mass:** 70 kg
- **Ash Residue:** Approximately 3-5% of body mass (assume 2 kg for this calculation)
- **Mass to be Cremated:** 70 kg - 2 kg = 68 kg
**b. Energy Components:**
1. **Heating the Body:**
- **Initial Temperature (T₁):** 25°C
- **Final Combustion Temperature (T₂):** 1000°C
- **Temperature Change (ΔT):** 1000°C - 25°C = 975°C
- **Specific Heat Capacity of Human Tissue (c):** ~4.2 kJ/kg·K (similar to water)
\[
Q_1 = m \times c \times \Delta T = 70 \, \text{kg} \times 4.2 \, \text{kJ/kg·K} \times 975 \, \text{K} = 286,650 \, \text{kJ}
\]
2. **Evaporating Water Content:**
- **Assumed Water Content:** 30% of body mass
- **Mass of Water (m₁):** 70 kg × 0.30 = 21 kg
- **Heat of Vaporization of Water (L_v):** ~2260 kJ/kg
\[
Q_2 = m_1 \times L_v = 21 \, \text{kg} \times 2260 \, \text{kJ/kg} = 47,460 \, \text{kJ}
\]
3. **Combusting Organic Matter:**
- **Mass of Organic Matter (m₂):** 70 kg - 21 kg (water) - 2 kg (ash) = 47 kg (rounded to 49 kg for this calculation)
- **Heat of Combustion for Organic Matter (ΔH_c):** ~16,000 kJ/kg (assuming similar to wood)
\[
Q_3 = m_2 \times \Delta H_c = 49 \, \text{kg} \times 16,000 \, \text{kJ/kg} = 784,000 \, \text{kJ}
\]
**c. Total Energy Required:**
\[
Q_{\text{total}} = Q_1 + Q_2 + Q_3 = 286,650 \, \text{kJ} + 47,460 \, \text{kJ} + 784,000 \, \text{kJ} = 1,118,110 \, \text{kJ}
\]
### **2. Cremation Using Wood**
**a. Energy Content of Wood:**
- **Calorific Value:** ~15,000 kJ/kg
- **Assumed Efficiency of the Cremation Process (η):** 80% (extremely efficient)
- **Effective Energy per kg of Wood:** 15,000 kJ/kg × 0.80 = 12,000 kJ/kg
**b. Calculating Required Mass of Wood:**
\[
\text{Mass of Wood} = \frac{Q_{\text{total}}}{\text{Effective Energy per kg}} = \frac{1,118,110 \, \text{kJ}}{12,000 \, \text{kJ/kg}} \approx 93.18 \, \text{kg}
\]
**c. Conclusion:**
Approximately **93 kg of wood** would be required to cremate a 70 kg body under extremely efficient conditions.
### **3. Cremation Using Wood Doused in Petrol**
Adding petrol (gasoline) to the wood can significantly increase the energy available, thereby reducing the total mass of fuel required. Here's how:
**a. Energy Content of Petrol:**
- **Calorific Value:** ~46,000 kJ/kg
- **Assumed Efficiency (η):** 80% (consistent with overall process efficiency)
- **Effective Energy per kg of Petrol:** 46,000 kJ/kg × 0.80 = 36,800 kJ/kg
**b. Optimization Problem:**
To minimize the total mass of fuel (wood + petrol), we set up the following equation based on energy balance:
\[
15,000x + 46,000y = \frac{1,118,110 \, \text{kJ}}{0.80} = 1,397,637.5 \, \text{kJ}
\]
Where:
- \( x \) = mass of wood (kg)
- \( y \) = mass of petrol (kg)
**c. Minimizing Total Fuel Mass (\( x + y \)):**
To minimize \( x + y \), maximize the use of petrol since it has a higher energy density.
- **If only Petrol is Used:**
\[
y = \frac{1,397,637.5 \, \text{kJ}}{46,000 \, \text{kJ/kg}} \approx 30.4 \, \text{kg}
\]
**Total Fuel Mass:** 30.4 kg of petrol
- **If a Combination is Used:**
For example, using 10 kg of petrol:
\[
46,000 \times 10 = 460,000 \, \text{kJ}
\]
Remaining energy needed:
\[
1,397,637.5 - 460,000 = 937,637.5 \, \text{kJ}
\]
Mass of wood required:
\[
x = \frac{937,637.5 \, \text{kJ}}{15,000 \, \text{kJ/kg}} \approx 62.51 \, \text{kg}
\]
**Total Fuel Mass:** 10 kg petrol + 62.51 kg wood ≈ 72.5 kg
**d. Optimal Fuel Combination:**
To achieve the **minimum total fuel mass**, it's best to maximize petrol usage. If petrol is available and safe to use in this manner, using **only petrol** would require approximately **30 kg**. However, practical considerations such as combustion stability, flame control, and safety might necessitate using a combination of petrol and wood.
### **4. Practical Considerations**
While the theoretical calculations suggest that using petrol can significantly reduce the amount of fuel required for cremation, several practical factors must be considered:
1. **Safety:** Petrol is highly flammable and poses significant safety risks, especially in open-air cremation setups.
2. **Control of Combustion:** Achieving a stable and controllable combustion process might be more challenging with petrol.
3. **Environmental Impact:** The use of petrol can result in higher emissions of pollutants compared to wood.
4. **Regulatory Compliance:** Many regions have strict regulations governing cremation practices, including permissible fuels and emission controls.
### **5. Summary**
- **Using Wood Alone:** Approximately **93 kg** of wood is required for the cremation of a 70 kg body under extremely efficient conditions.
- **Using Petrol Alone:** Approximately **30 kg** of petrol could theoretically suffice, but practical and safety issues make this approach less feasible.
- **Using a Combination:** A mix of petrol and wood can reduce the total fuel mass needed (e.g., ~72.5 kg using 10 kg petrol and 62.5 kg wood), balancing energy efficiency with practical considerations.
**Note:** These calculations are theoretical and assume optimal conditions. Actual requirements may vary based on specific circumstances, equipment used, and environmental factors. Always adhere to local laws and regulations when considering cremation methods.