Subtopic: Sources of power on the farm – human, animal, mechanical, electrical, wind, solar and biomass
Question Statement:
A dairy farm has $25$ crossbred cows producing an average of $10\text{ kg}$ of fresh dung per cow per day. The fresh dung yields $0.040\text{ m}^3$ of biogas per kilogram of wet dung. The biogas has a calorific value of $20\text{ MJ/m}^3$. If this biogas is utilized to run a dual-fuel diesel generator having an overall brake thermal efficiency of $\eta = 25\%$, the total electrical energy generated per day in $\text{kWh}$ is ________ (round off to 2 decimal places).
Answer Key:
13.89
Step-by-Step Solution & Explanation
```
Biogas Energy Conversion:
Number of cows: 25
Dung per cow: 10 kg/day
Total dung: 250 kg/day
Gas yield: 0.040 m^3/kg
CV: 20 MJ/m^3
Thermal efficiency: 25%
```
**Step-by-step Calculation:**
1. Total daily fresh dung collected:
$M = 25 \times 10\text{ kg} = 250\text{ kg/day}$
2. Daily biogas volume generated:
$V_{\text{gas}} = 250\text{ kg} \times 0.040\text{ m}^3/\text{kg} = 10.0\text{ m}^3/\text{day}$
3. Total daily heat energy available in the biogas ($Q$):
$Q = V_{\text{gas}} \times CV = 10.0\text{ m}^3 \times 20\text{ MJ/m}^3 = 200\text{ MJ/day}$
4. Electrical energy output ($E_{\text{elec}}$) with $\eta = 25\% = 0.25$:
$E_{\text{thermal, MJ}} = 200 \times 0.25 = 50.0\text{ MJ}$
5. Converting $\text{MJ}$ to $\text{kWh}$ ($1\text{ kWh} = 3.6\text{ MJ}$):
$E_{\text{elec}} = \frac{50.0\text{ MJ}}{3.6\text{ MJ/kWh}} = 13.8889\text{ kWh} \approx 13.89\text{ kWh}$