To determine the average power involved in the adiabatic expansion of the gases in a motorcycle engine, we can start by calculating the work done during the expansion and then find the average power based on the cycle duration. Let’s break this down step by step.
Understanding the Engine Cycle
Motorcycle engines operate in cycles, and for a typical four-stroke engine, the cycle consists of intake, compression, power, and exhaust strokes. Since the question specifies that we are interested in the adiabatic expansion phase right after combustion, we will focus on that aspect of the cycle.
Key Variables
- Gauge Pressure (P): 15.0 atm
- Initial Volume (V1): 50.0 cm³
- Final Volume (V2): 250 cm³
- Engine Speed: 4000 rpm
- Time of Expansion: Half of the cycle time
Calculating Work Done during Expansion
The work done during an adiabatic process can be calculated using the formula:
W = (P1 * V1) * [(V2/V1)^(γ) - 1] / (γ - 1)
Where γ (gamma) is the heat capacity ratio. For diatomic gases, γ is approximately 1.4. First, we need to convert the gauge pressure from atm to pascals (Pa) for our calculations:
P1 = 15.0 atm = 15.0 * 101325 Pa = 1519875 Pa
Volume in cubic meters
Next, convert the volumes from cm³ to m³:
V1 = 50.0 cm³ = 50.0 * 10^-6 m³
V2 = 250 cm³ = 250 * 10^-6 m³
Substituting into the Work Formula
Now we substitute the values into the work formula:
W = (1519875 * 50.0 * 10^-6) * [(250 * 10^-6 / 50.0 * 10^-6)^(1.4) - 1] / (1.4 - 1)
Simplifying the volume ratio:
V2/V1 = 250/50 = 5
Calculating the power:
W = (1519875 * 50.0 * 10^-6) * [5^(1.4) - 1] / (0.4)
5^(1.4) ≈ 7.529
W = (1519875 * 50.0 * 10^-6) * [7.529 - 1] / (0.4)
W = (1519875 * 50.0 * 10^-6) * 6.529 / 0.4
W ≈ 1519875 * 50.0 * 10^-6 * 16.3225
W ≈ 1237.5 J
Finding Average Power
Now, let's determine the average power. The engine is running at 4000 rpm, which means there are 4000 revolutions per minute. Since this is a four-stroke engine, each complete cycle occurs every two revolutions:
Time for one cycle = 1 minute / (4000 / 2) = 0.0005 minutes = 0.03 seconds
The time for the adiabatic expansion (half of the cycle time) is:
t = 0.03 / 2 = 0.015 seconds
Next, we find the average power using the formula:
Power (P) = Work (W) / Time (t)
P = 1237.5 J / 0.015 s ≈ 82500 W
Final Result
Therefore, the average power involved in the adiabatic expansion of the gases during this phase of the engine cycle is approximately 82,500 watts or 82.5 kW.
This calculation illustrates the significant energy conversion happening within your motorcycle engine during the combustion and expansion phases, showcasing the impressive mechanics behind how such machines operate efficiently.
