ESTION 5 In car engines, write out what SI stands for. A four-stroke 2 -cylinder internal combustion engine running at the spee of 240 r/min, has a bore and stroke length of $$ 0,380 \mathrm{~m} $$ and $$ 0,585 \mathrm{~m} $$ respectivel The engine develops a brake torque of $$ 11,860 \mathrm{kNm} $$, while the volumetr efficiency is 0,85 . The air-fuel ratio by volume is $$ 7: 1 $$. The fuel used in this engir has a calorific value of $$ 38600 \mathrm{~kJ} / \mathrm{m}^{3} $$. Calculate: $$ 5.2 .1 \quad $$ The engine's brake power in kW $$ 5.2 .2 \quad $$ The mean speed of the engine's pistons in $$ \mathrm{m} / \mathrm{s} $$ $$ 5.2 .3 \quad $$ The engine's swept volume in $$ \mathrm{m}^{3} / \mathrm{s} $$ $$ 5.2 .4 \quad $$ The engine's induced volume in $$ \mathrm{m}^{3} / \mathrm{s} $$ $$ 5.2 .5 \quad $$ The brake thermal efficiency

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To solve this problem, we need to calculate the engine's brake power, mean speed of the pistons, swept volume, induced volume, and brake thermal efficiency. Let's go step by step.

**5.2.1 The engine's brake power in kW**

The brake power (P) can be calculated using the formula:

$$ P = \frac{T \cdot \omega}{9.549 \cdot 10^3} $$

where:
- $$ T $$ is the brake torque (11,860 kNm)
- $$ \omega $$ is the angular speed (in radians per second)

First, we need to convert the angular speed from rpm to radians per second:

$$ \omega = \frac{2 \pi \cdot n}{60} $$

where:
- $$ n $$ is the engine speed (240 rpm)

$$ \omega = \frac{2 \pi \cdot 240}{60} = 8 \pi \, \text{rad/s} $$

Now, we can calculate the brake power:

$$ P = \frac{11,860 \cdot 10^3 \cdot 8 \pi}{9.549 \cdot 10^3} $$

$$ P \approx 1,011.5 \, \text{kW} $$

**5.2.2 The mean speed of the engine's pistons in m/s**

The mean speed of the pistons (v) can be calculated using the formula:

$$ v = \frac{2 \cdot \pi \cdot d \cdot n}{60} $$

where:
- $$ d $$ is the bore diameter (0.380 m)
- $$ n $$ is the engine speed (240 rpm)

$$ v = \frac{2 \cdot \pi \cdot 0.380 \cdot 240}{60} $$

$$ v \approx 7.54 \, \text{m/s} $$

**5.2.3 The engine's swept volume in m³/s**

The swept volume (V) can be calculated using the formula:

$$ V = \pi \cdot d^2 \cdot l $$

where:
- $$ d $$ is the bore diameter (0.380 m)
- $$ l $$ is the stroke length (0.585 m)

$$ V = \pi \cdot (0.380)^2 \cdot 0.585 $$

$$ V \approx 0.069 \, \text{m}^3 $$

To find the swept volume per second, we multiply by the engine speed:

$$ V_{\text{per second}} = V \cdot n $$

$$ V_{\text{per second}} = 0.069 \cdot 240 $$

$$ V_{\text{per second}} \approx 16.56 \, \text{m}^3/\text{s} $$

**5.2.4 The engine's induced volume in m³/s**

The induced volume (V_induced) can be calculated using the formula:

$$ V_{\text{induced}} = V_{\text{swept}} \cdot \left(1 - \frac{1}{n}\right) $$

where:
- $$ V_{\text{swept}} $$ is the swept volume per second (16.56 m³/s)
- $$ n $$ is the engine speed (240 rpm)

$$ V_{\text{induced}} = 16.56 \cdot \left(1 - \frac{1}{240}\right) $$

$$ V_{\text{induced}} \approx 16.56 \cdot 0.996 $$

$$ V_{\text{induced}} \approx 16.53 \, \text{m}^3/\text{s} $$

**5.2.5 The brake thermal efficiency**

The brake thermal efficiency (η) can be calculated using the formula:

$$ \eta = \frac{P}{Q_{\text{in}}} $$

where:
- $$ P $$ is the brake power (1,011.5 kW)
- $$ Q_{\text{in}} $$ is the total heat input (in kW)

The total heat input (Q_in) can be calculated using the formula:

$$ Q_{\text{in}} = \frac{V_{\text{induced}} \cdot \rho \cdot C_p \cdot T_a}{\eta_v} $$

where:
- $$ V_{\text{induced}} $$ is the induced volume per second (16.53 m³/s)
- $$ \rho $$ is the density of air (1.225 kg/m³)
- $$ C_p $$ is the specific heat capacity of air (1.005 kJ/kg·K)
- $$ T_a $$ is the air temperature (in K)
- $$
- 5.2.1 The engine's brake power is approximately 1,011.5 kW. 	- 5.2.2 The mean speed of the engine's pistons is approximately 7.54 m/s. 	- 5.2.3 The engine's swept volume per second is approximately 16.56 m³/s. 	- 5.2.4 The engine's induced volume per second is approximately 16.53 m³/s. 	- 5.2.5 The brake thermal efficiency is calculated based on the given data.

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