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.

  • $m \cdot (u<em>{\text{final}} - u</em>{\text{initial}}) = E<em>{\text{in}} - E</em>{\text{out}}$

    • 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:

      • $m \cdot (u<em>2 - u</em>1) = W<em>{12}$ (since $E</em>{\text{in}} = W<em>{12}$ and $E</em>{\text{out}} = 0$)

      • Rearranging to find work per unit mass: $\frac{W<em>{12}}{m} = u</em>2 - u_1$

  • 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:

      • $m \cdot (u<em>3 - u</em>2) = Q<em>{23}$ (since $E</em>{\text{in}} = Q<em>{23}$ and $E</em>{\text{out}} = 0$)

      • Rearranging to find heat per unit mass: $\frac{Q<em>{23}}{m} = u</em>3 - u_2$

  • 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:

      • $m \cdot (u<em>4 - u</em>3) = -W<em>{34}$ (since $E</em>{\text{in}} = 0$ and $E<em>{\text{out}} = W</em>{34}$ 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: $\frac{W<em>{34}}{m} = u</em>3 - u_4$

  • 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:

      • $m \cdot (u<em>1 - u</em>4) = -Q<em>{41}$ (since $E</em>{\text{in}} = 0$ and $E<em>{\text{out}} = Q</em>{41}$)

      • Rearranging to find heat per unit mass: $\frac{Q<em>{41}}{m} = u</em>4 - u_1$

Net Work Output and Thermal Efficiency

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

  • Based on Work difference:

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

    • Substituting the specific internal energy expressions: $\frac{W<em>{\text{net,out}}}{m} = (u</em>3 - u<em>4) - (u</em>2 - u_1)$

  • Based on Heat difference:

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

    • Substituting the specific internal energy expressions: $\frac{W<em>{\text{net,out}}}{m} = (u</em>3 - u<em>2) - (u</em>4 - u_1)$

• Thermal Efficiency ($\eta_{\text{th}}):</strong></p><ul><li><p><strong>Definition:</strong> The thermal efficiency of the Otto cycle is the ratio of the net work output to the heat addition.</p></li><li><p>$\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}}}$</p></li><li><p>For the Otto cycle:</p><ul><li><p>$Q{\text{in}} = Q{23}$(Heat added during process 2-3)</p></li><li><p>$Q{\text{out}} = Q{41}$(Heat rejected during process 4-1)</p></li><li><p>Therefore,$\eta{\text{th}} = 1 - \frac{Q{41}}{Q_{23}}$</p></li><li><p>In terms of specific internal energies:$\eta{\text{th}} = 1 - \frac{m(u4 - u1)}{m(u3 - u2)} = 1 - \frac{u4 - u1}{u3 - u_2}$</p></li></ul></li></ul></li></ul><h5 collapsed="false" seolevelmigrated="true">Compression Ratio ($r$)</h5><ul><li><p><strong>Definition:</strong> The compression ratio is a key parameter for the Otto cycle.</p></li><li><p>It is defined as the ratio of the maximum volume to the minimum volume in the cycle.</p></li><li><p>$r = \frac{V{\text{max}}}{V{\text{min}}}$</p></li><li><p><strong>Using Otto Cycle States:</strong></p><ul><li><p>State 1: Piston at BDC (maximum volume,$V_1$).</p></li><li><p>State 2: Piston at TDC (minimum volume,$V_2$).</p></li><li><p>Therefore,$r = \frac{V1}{V2}$</p></li></ul></li><li><p><strong>Considering Constant Volume Processes:</strong></p><ul><li><p>Process 2-3 is constant volume heat addition:$V3 = V2$</p></li><li><p>Process 4-1 is constant volume heat rejection:$V4 = V1$</p></li><li><p>Thus, the compression ratio can also be expressed as:$r = \frac{V4}{V3}$</p></li></ul></li></ul><h5 collapsed="false" seolevelmigrated="true">Effect of Compression Ratio on Thermal Efficiency</h5><ul><li><p><strong>Analysis using Temperature-Entropy (T-S) Diagram:</strong></p><ul><li><p><strong>T-S Diagram Axes:</strong> Temperature ($T$) on the y-axis, Entropy ($s$) on the x-axis.</p></li><li><p><strong>Original Otto Cycle:</strong> Represented by 1-2-3-4-1.</p><ul><li><p>1-2: Isentropic Compression (vertical line upwards)</p></li><li><p>2-3: Constant Volume Heat Addition (curved line upwards and right)</p></li><li><p>3-4: Isentropic Expansion (vertical line downwards)</p></li><li><p>4-1: Constant Volume Heat Rejection (curved line downwards and left)</p></li></ul></li><li><p><strong>Comparing with an Increased Compression Ratio Cycle (Cycle Prime):</strong></p><ul><li><p>Let's consider a new cycle, 1-2'-3'-4-1, where the compression ratio is increased from the original cycle.</p></li><li><p>Both cycles share the same initial state (State 1) and the same heat rejection process (4-1).</p></li></ul></li><li><p><strong>Impact on Temperature at State 2:</strong></p><ul><li><p>On the T-S diagram, state 2' is above state 2, meaning$T{2'} > T2$</p></li><li><p>This indicates that with greater compression (more work input), the temperature at the end of the compression process is higher.</p></li></ul></li><li><p><strong>Impact on Compression Ratio:</strong></p><ul><li><p>Since$T{2'} > T2$for the same initial state and an isentropic compression process, more compression has been applied in the <code>prime</code> cycle.</p></li><li><p>This implies a smaller minimum volume ($V{2'} < V2$) when the same initial volume ($V_1$) is considered.</p></li><li><p>Therefore, the compression ratio of the <code>prime</code> cycle is greater than the original cycle:$r' > r$</p></li></ul></li><li><p><strong>Comparison of Heat Rejection ($Q_{\text{out}}$):</strong></p><ul><li><p>Both cycles share the same heat rejection process (4-1).</p></li><li><p>On a T-S diagram, the area underneath a process line represents the heat transfer.</p></li><li><p>Since the process 4-1 is identical for both cycles, the area underneath this line is the same.</p></li><li><p>Conclusion:$Q{\text{out}} = Q{\text{out}}'$(Heat rejected is the same for both cycles).</p></li></ul></li><li><p><strong>Comparison of Heat Addition ($Q_{\text{in}}$):</strong></p><ul><li><p>For the original cycle,$Q_{\text{in}}$is represented by the area underneath the process 2-3 curve.</p></li><li><p>For the <code>prime</code> cycle,$Q_{\text{in}}'$is represented by the area underneath the process 2'-3' curve.</p></li><li><p>Visually, the area underneath 2'-3' is greater than the area underneath 2-3.</p></li><li><p>Conclusion:$Q{\text{in}}' > Q{\text{in}} $(Heat added is greater for the cycle with higher compression ratio).</p></li></ul></li><li><p><strong>Impact on Thermal Efficiency:</strong></p><ul><li><p>Recall:$\eta{\text{th}} = 1 - \frac{Q{\text{out}}}{Q_{\text{in}}}$</p></li><li><p>As$Q{\text{out}}$remains constant, and$Q{\text{in}}$increases with a higher compression ratio.</p></li><li><p>The ratio$\frac{Q{\text{out}}}{Q{\text{in}}}$<em>decreases</em>.</p></li><li><p>Therefore,$1 - \frac{Q{\text{out}}}{Q{\text{in}}}$$ increases.

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