In a 1000 cc four-stroke IC engine with a crank running at 1000 rpm, if the mean effective pressure is 400 kPa and the efficiency of the engine is 0.5, then the brake power of the engine is
5/3 kW
Engine Power Calculation: Four-Stroke IC Engine
This problem requires us to calculate the brake power of a four-stroke internal combustion (IC) engine given its displacement volume, crank speed, mean effective pressure, and efficiency. We will first determine the indicated power and then use the efficiency to find the brake power.
Key Concepts for Engine Power
- Indicated Power (IP): This is the theoretical power produced inside the engine cylinders due to the combustion of fuel. It's the power developed by the gases on the piston.
- Brake Power (BP): This is the actual usable power available at the engine's crankshaft. It's the power delivered by the engine to the transmission or output shaft, after accounting for mechanical losses like friction within the engine.
- Mean Effective Pressure (MEP - \(P_m\)): This is a hypothetical constant pressure that, if acting on the piston during the power stroke, would produce the same net work per cycle as the actual varying pressures. It's a measure of the engine's ability to do work per unit of displacement.
- Engine Efficiency (\(\eta\)): In this context, it refers to the mechanical efficiency, which is the ratio of brake power to indicated power. \[ \eta = \frac{\text{Brake Power (BP)}}{\text{Indicated Power (IP)}} \]
- Displacement Volume (\(V_d\)): The total volume displaced by all the pistons in the engine as they move from Bottom Dead Center (BDC) to Top Dead Center (TDC). For a multi-cylinder engine, this is the sum of the swept volumes of all cylinders. The question states 1000 cc, which is the total displacement volume.
Given Data for the IC Engine
Let's list the given parameters for the 1000 cc four-stroke IC engine:
- Displacement Volume (\(V_d\)): 1000 cc
- Crank Speed (N): 1000 rpm
- Mean Effective Pressure (\(P_m\)): 400 kPa
- Engine Efficiency (\(\eta\)): 0.5
Unit Conversion for Accurate Calculation
Before proceeding with the calculations, it's crucial to convert all units to a consistent system (SI units). We'll convert cubic centimeters to cubic meters and kilopascals to pascals.
- Displacement Volume: 1 cc = $10^{-6}$ m³ So, 1000 cc = $1000 \times 10^{-6}$ m³ = $10^{-3}$ m³
- Mean Effective Pressure: 1 kPa = $10^3$ Pa So, 400 kPa = $400 \times 10^3$ Pa
- Crank Speed (N): 1000 rpm (revolutions per minute)
Calculating Indicated Power (IP)
For a four-stroke engine, there is one power stroke for every two revolutions of the crankshaft. The formula for indicated power (IP) using total displacement volume (\(V_d\)) is:
\[ IP = \frac{P_m \times V_d \times N}{2 \times 60} \]Where:
- \(P_m\) is the mean effective pressure in Pascals (Pa)
- \(V_d\) is the displacement volume in cubic meters (m³)
- \(N\) is the crank speed in revolutions per minute (rpm)
- The factor of 2 accounts for two revolutions per power stroke in a four-stroke engine.
- The factor of 60 converts minutes to seconds, giving power in Watts (W).
Now, let's substitute the converted values into the formula:
\[ IP = \frac{(400 \times 10^3 \, \text{Pa}) \times (10^{-3} \, \text{m}^3) \times (1000 \, \text{rpm})}{2 \times 60} \] \[ IP = \frac{400 \times 10^3 \times 10^{-3} \times 1000}{120} \] \[ IP = \frac{400 \times 1 \times 1000}{120} \] \[ IP = \frac{400000}{120} \] \[ IP = \frac{40000}{12} \] \[ IP = \frac{10000}{3} \, \text{W} \]To express IP in kilowatts (kW), we divide by 1000:
\[ IP = \frac{10000}{3 \times 1000} \, \text{kW} \] \[ IP = \frac{10}{3} \, \text{kW} \]Calculating Brake Power (BP)
The relationship between brake power, indicated power, and engine efficiency is given by:
\[ BP = \eta \times IP \]We are given the engine efficiency (\(\eta\)) as 0.5 and we have calculated the indicated power (IP) as \(\frac{10}{3}\) kW.
Substitute these values into the formula:
\[ BP = 0.5 \times \frac{10}{3} \, \text{kW} \] \[ BP = \frac{1}{2} \times \frac{10}{3} \, \text{kW} \] \[ BP = \frac{10}{6} \, \text{kW} \]Simplify the fraction:
\[ BP = \frac{5}{3} \, \text{kW} \]Conclusion
Based on our calculations, the brake power of the 1000 cc four-stroke IC engine is \(\frac{5}{3}\) kW.
| Parameter | Value | Unit |
|---|---|---|
| Displacement Volume (\(V_d\)) | $10^{-3}$ | m³ |
| Mean Effective Pressure (\(P_m\)) | $400 \times 10^3$ | Pa |
| Crank Speed (N) | 1000 | rpm |
| Engine Efficiency (\(\eta\)) | 0.5 | (dimensionless) |
| Indicated Power (IP) | $\frac{10}{3}$ | kW |
| Brake Power (BP) | $\frac{5}{3}$ | kW |
This result matches option 1.
Important Questions from Power and Efficiency
The total power developed by combustion of fuel in the combustion chamber is called:
A transmission dynamometer measures
Choose the correct answer from the following four options.
S1: Higher volumetric efficiency due to more time for mixture intake in a four-stroke engine.
S2: Lower volumetric efficiency due to the lesser time for mixture intake in a two-stroke engine.
An I.C. engine develops an indicated power of 150 kW. If the mechanical efficiency of the engine is 80%, then the brake power delivered is
The actual power generated in an engine cylinder is known as:
