In car engines, write out what $$ S / $$ stands for. A four-stroke 2 -cylinder internal combustion engine running at the sp of $$ 240 \mathrm{r} / \mathrm{min} $$, has a bore and stroke length of $$ 0,380 \mathrm{~m} $$ and $$ 0,585 \mathrm{~m} $$ respectiv The engine develops a brake torque of $$ 11,860 \mathrm{kNm} $$, while the volume efficiency is 0,85 . The air-fuel ratio by volume is $$ 7: 1 $$. The fuel used in this ens has a calorific value of $$ 38600 \mathrm{~kJ} / \mathrm{m}^{3} $$. Calculate: 5.2.1 The engine's brake power in kW 5.2.2 The mean speed of the engine's pistons in $$ \mathrm{m} / \mathrm{s} $$ 5.2.3 The engine's swept volume in $$ \mathrm{m}^{3} / \mathrm{s} $$ i.2.4 The engine's induced volume in $$ \mathrm{m}^{3} / \mathrm{s} $$ 2.5 The brake thermal efficiency

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To solve the problems related to the four-stroke 2-cylinder internal combustion engine, we will use the following formulas and steps:

### Given Data:
- Engine speed, $$ N = 240 \, \text{r/min} $$
- Bore, $$ D = 0.380 \, \text{m} $$
- Stroke length, $$ L = 0.585 \, \text{m} $$
- Brake torque, $$ T = 11.860 \, \text{kNm} = 11,860 \, \text{Nm} $$
- Volume efficiency, $$ \eta_v = 0.85 $$
- Air-fuel ratio, $$ \text{AFR} = 7:1 $$
- Calorific value of fuel, $$ CV = 38,600 \, \text{kJ/m}^3 $$

### 5.2.1 The Engine's Brake Power in kW
The brake power (BP) can be calculated using the formula:
$$
BP = \frac{2 \pi N T}{60}
$$
Where:
- $$ N $$ is the engine speed in r/min
- $$ T $$ is the brake torque in Nm

Substituting the values:
$$
BP = \frac{2 \pi (240) (11,860)}{60}
$$
$$
BP = \frac{2 \pi (240) (11,860)}{60} \approx  15,000 \, \text{W} \approx 15 \, \text{kW}
$$

### 5.2.2 The Mean Speed of the Engine's Pistons in $$ \mathrm{m/s} $$
The mean speed of the pistons can be calculated using the formula:
$$
v = \frac{N \cdot L}{2 \cdot 60}
$$
Where:
- $$ L $$ is the stroke length in meters

Substituting the values:
$$
v = \frac{240 \cdot 0.585}{2 \cdot 60} = \frac{240 \cdot 0.585}{120} \approx 2.87 \, \text{m/s}
$$

### 5.2.3 The Engine's Swept Volume in $$ \mathrm{m}^3 $$
The swept volume (V_s) for a 2-cylinder engine can be calculated using the formula:
$$
V_s = \frac{\pi D^2 L}{4} \cdot n
$$
Where:
- $$ n $$ is the number of cylinders

Substituting the values:
$$
V_s = \frac{\pi (0.380)^2 (0.585)}{4} \cdot 2
$$
$$
V_s \approx \frac{\pi (0.1444) (0.585)}{4} \cdot 2 \approx 0.067 \, \text{m}^3
$$

### 5.2.4 The Engine's Induced Volume in $$ \mathrm{m}^3/s $$
The induced volume (V_i) can be calculated using the formula:
$$
V_i = V_s \cdot \frac{N}{2 \cdot 60} \cdot \eta_v
$$
Substituting the values:
$$
V_i = 0.067 \cdot \frac{240}{2 \cdot 60} \cdot 0.85
$$
$$
V_i \approx 0.067 \cdot 2 \cdot 0.85 \approx 0.114 \, \text{m}^3/s
$$

### 5.2.5 The Brake Thermal Efficiency
The brake thermal efficiency ($$ \eta_b $$) can be calculated using the formula:
$$
\eta_b = \frac{BP}{\dot{Q}}
$$
Where $$ \dot{Q} $$ is the heat input per second, calculated as:
$$
\dot{Q} = \frac{V_i \cdot \text{Density of air} \cdot \text{Calorific value}}{\text{AFR}}
$$
Assuming the density of air is approximately $$ 1.225 \, \text{kg/m}^3 $$:
$$
\dot{Q} = \frac{0.114 \cdot 1.225 \cdot 38,600}{7}
$$
Calculating $$ \dot{Q} $$:
$$
\dot{Q} \approx \frac{0.114 \cdot 1.225 \cdot 38,600}{7} \approx 0.114 \cdot 1.225 \cdot 5,514.29 \approx 0.114 \cdot 6760.71 \approx 770.5 \, \text{kW}
$$
Now substituting into the efficiency formula:
$$
\eta_b = \frac{15}{770.5} \approx 0.0194 \text{ or } 1.94\%
$$

### Summary of Results:
- **5.2.1 Brake Power:** $$ \approx 15 \, \text{kW} $$
- **5.2.2 Mean Speed of Pistons:** $$ \approx 2.87 \, \text{m/s} $$
- **5.2.3 Swept Volume:** $$ \approx 0.067 \, \text{m}^3 $$
- **5.2.4 Induced Volume:** $$ \approx 0.114 \, \text{m}^3/s $$
- **5.2.5 Brake Thermal Efficiency:** $$ \approx 1.94\% $$

Please verify the calculations and assumptions, especially regarding the density of air and the calorific value, as they can vary based on conditions.
- 5.2.1 The engine's brake power is approximately 15 kW. - 5.2.2 The mean speed of the engine's pistons is approximately 2.87 m/s. - 5.2.3 The engine's swept volume is approximately 0.067 m³. - 5.2.4 The engine's induced volume is approximately 0.114 m³/s. - 5.2.5 The brake thermal efficiency is approximately 1.94%.

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