EL

Otto Cycle and Thermal Efficiency

Energy Conservation Principle and Otto Cycle Processes

  • Fundamental Equation for Closed System (based on Energy Conservation Principle):

    • The core equation is derived from the energy conservation principle for a closed system.

    • It states that the change in total internal energy of the system is equal to the net energy transferred into the system.

    • m \cdot (u{\text{final}} - u{\text{initial}}) = E{\text{in}} - E{\text{out}}

      • Here, m is the mass of the working fluid.

      • u (lowercase) represents the specific internal energy (internal energy per unit mass).

      • U (uppercase) represents the total internal energy (U = m \cdot u).

      • E_{\text{in}} is the energy entering the system.

      • E_{\text{out}} is the energy leaving the system.

  • Application to Otto Cycle Processes:

    • Process 1-2: Isentropic Compression

      • Description: Piston moves from bottom dead center (BDC) to top dead center (TDC), compressing the air inside. This involves work input.

      • Energy Transfer: Work input (W_{12}).

      • Initial State: State 1. Final State: State 2.

      • Equation Application:

        • m \cdot (u2 - u1) = W{12} (since E{\text{in}} = W{12} and E{\text{out}} = 0)

        • Rearranging to find work per unit mass: \frac{W{12}}{m} = u2 - u_1

    • Process 2-3: Constant Volume Heat Addition

      • Description: Heat is added to the system at constant volume, increasing temperature and pressure.

      • Energy Transfer: Heat addition (Q_{23}).

      • Initial State: State 2. Final State: State 3.

      • Equation Application:

        • m \cdot (u3 - u2) = Q{23} (since E{\text{in}} = Q{23} and E{\text{out}} = 0)

        • Rearranging to find heat per unit mass: \frac{Q{23}}{m} = u3 - u_2

    • Process 3-4: Isentropic Expansion

      • Description: High-pressure, high-temperature air pushes the piston from TDC to BDC, performing work output.

      • Energy Transfer: Work output (W_{34}).

      • Initial State: State 3. Final State: State 4.

      • Equation Application:

        • m \cdot (u4 - u3) = -W{34} (since E{\text{in}} = 0 and E{\text{out}} = W{34} for work output, a negative sign is used if W is defined as work done by the system)

        • Rearranging to find work per unit mass: \frac{W{34}}{m} = u3 - u_4

    • Process 4-1: Constant Volume Heat Rejection

      • Description: Heat is rejected from the system at constant volume, decreasing temperature and pressure.

      • Energy Transfer: Heat rejection (Q_{41}).

      • Initial State: State 4. Final State: State 1.

      • Equation Application:

        • m \cdot (u1 - u4) = -Q{41} (since E{\text{in}} = 0 and E{\text{out}} = Q{41})

        • Rearranging to find heat per unit mass: \frac{Q{41}}{m} = u4 - u_1

Net Work Output and Thermal Efficiency

  • Network Output per unit mass (W_{\text{net,out}}/m):

    • Based on Work difference:

      • W{\text{net,out}}/m = W{34}/m - W_{12}/m (Work output minus work input)

      • Substituting the specific internal energy expressions: \frac{W{\text{net,out}}}{m} = (u3 - u4) - (u2 - u_1)

    • Based on Heat difference:

      • W{\text{net,out}}/m = Q{23}/m - Q_{41}/m (Heat addition minus heat rejection)

      • Substituting the specific internal energy expressions: \frac{W{\text{net,out}}}{m} = (u3 - u2) - (u4 - u_1)

  • Thermal Efficiency (\eta_{\text{th}}):

    • Definition: The thermal efficiency of the Otto cycle is the ratio of the net work output to the heat addition.

    • \eta{\text{th}} = \frac{W{\text{net,out}}}{Q{\text{in}}} = \frac{Q{\text{in}} - Q{\text{out}}}{Q{\text{in}}} = 1 - \frac{Q{\text{out}}}{Q{\text{in}}}

    • For the Otto cycle:

      • Q{\text{in}} = Q{23} (Heat added during process 2-3)

      • Q{\text{out}} = Q{41} (Heat rejected during process 4-1)

      • Therefore, \eta{\text{th}} = 1 - \frac{Q{41}}{Q_{23}}

      • In terms of specific internal energies: \eta{\text{th}} = 1 - \frac{m(u4 - u1)}{m(u3 - u2)} = 1 - \frac{u4 - u1}{u3 - u_2}

Compression Ratio (r)

  • Definition: The compression ratio is a key parameter for the Otto cycle.

  • It is defined as the ratio of the maximum volume to the minimum volume in the cycle.

  • r = \frac{V{\text{max}}}{V{\text{min}}}

  • Using Otto Cycle States:

    • State 1: Piston at BDC (maximum volume, V_1).

    • State 2: Piston at TDC (minimum volume, V_2).

    • Therefore, r = \frac{V1}{V2}

  • Considering Constant Volume Processes:

    • Process 2-3 is constant volume heat addition: V3 = V2

    • Process 4-1 is constant volume heat rejection: V4 = V1

    • Thus, the compression ratio can also be expressed as: r = \frac{V4}{V3}

Effect of Compression Ratio on Thermal Efficiency

  • Analysis using Temperature-Entropy (T-S) Diagram:

    • T-S Diagram Axes: Temperature (T) on the y-axis, Entropy (s) on the x-axis.

    • Original Otto Cycle: Represented by 1-2-3-4-1.

      • 1-2: Isentropic Compression (vertical line upwards)

      • 2-3: Constant Volume Heat Addition (curved line upwards and right)

      • 3-4: Isentropic Expansion (vertical line downwards)

      • 4-1: Constant Volume Heat Rejection (curved line downwards and left)

    • Comparing with an Increased Compression Ratio Cycle (Cycle Prime):

      • Let's consider a new cycle, 1-2'-3'-4-1, where the compression ratio is increased from the original cycle.

      • Both cycles share the same initial state (State 1) and the same heat rejection process (4-1).

    • Impact on Temperature at State 2:

      • On the T-S diagram, state 2' is above state 2, meaning T{2'} > T2

      • This indicates that with greater compression (more work input), the temperature at the end of the compression process is higher.

    • Impact on Compression Ratio:

      • Since T{2'} > T2 for the same initial state and an isentropic compression process, more compression has been applied in the prime cycle.

      • This implies a smaller minimum volume (V{2'} < V2) when the same initial volume (V_1) is considered.

      • Therefore, the compression ratio of the prime cycle is greater than the original cycle: r' > r

    • Comparison of Heat Rejection (Q_{\text{out}}):

      • Both cycles share the same heat rejection process (4-1).

      • On a T-S diagram, the area underneath a process line represents the heat transfer.

      • Since the process 4-1 is identical for both cycles, the area underneath this line is the same.

      • Conclusion: Q{\text{out}} = Q{\text{out}}' (Heat rejected is the same for both cycles).

    • Comparison of Heat Addition (Q_{\text{in}}):

      • For the original cycle, Q_{\text{in}} is represented by the area underneath the process 2-3 curve.

      • For the prime cycle, Q_{\text{in}}' is represented by the area underneath the process 2'-3' curve.

      • Visually, the area underneath 2'-3' is greater than the area underneath 2-3.

      • Conclusion: Q{\text{in}}' > Q{\text{in}} (Heat added is greater for the cycle with higher compression ratio).

    • Impact on Thermal Efficiency:

      • Recall: \eta{\text{th}} = 1 - \frac{Q{\text{out}}}{Q_{\text{in}}}

      • As Q{\text{out}} remains constant, and Q{\text{in}} increases with a higher compression ratio.

      • The ratio \frac{Q{\text{out}}}{Q{\text{in}}} decreases.

      • Therefore, 1 - \frac{Q{\text{out}}}{Q{\text{in}}}$$ increases.

      • Conclusion: Increasing the compression ratio increases the thermal efficiency of the Otto cycle.

  • Take-Home Message: To improve the thermal efficiency of an Otto cycle, one effective method is to increase its compression ratio.