Q. 4.18 Derive equation 4.10聽for the ef... [FREE SOLUTION]

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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 4: Q. 4.18 (page 133)

Derive equation $4.10$ for the efficiency of the Otto cycle.

Short Answer

Expert verified

The efficiency of the Otto cycle is $畏 = 1 鈭� {(\frac{V_{2}}{V_{1}})}^{纬鈭� 1}$

Step by step solution

Introduction

Efficiency $(畏) = \frac{Net work done (W_{net})}{Heat supplied (Q_{1})}$

Heat supplied:

$Q_{1} = m C_{v} (T_{3} - T_{2}) - - - (1)$

Hear rejected:

$\begin{aligned} Q_{2} = m C_{v} (T_{4} 鈭� T_{1}) 鈭� 鈭� 鈭� (2) \\ 畏 = \frac{Q_{1} 鈭� Q_{2}}{Q_{1}} = 1 鈭� \frac{Q_{2}}{Q_{1}} 鈭� 鈭� (3) \\ From (1) and (2) \\ 畏 = 1 鈭� \frac{m C_{v} (T_{4} 鈭� T_{1})}{m C_{4} (T_{3} 鈭� T_{2})} \\ 畏 = 1 鈭� \frac{(T_{4} 鈭� T_{1})}{(T_{3} 鈭� T_{2})} 鈭� 鈭� (4) \end{aligned}$

Calculation

As it is an adiabatic process:

$\begin{aligned} T_{2} V_{2}^{纬鈭� 1} = T_{1} V_{1}^{纬鈭� 1} \\ \frac{T_{2}}{T_{1}} = {(\frac{V_{1}}{V_{2}})}^{纬鈭� 1} \end{aligned}$

Similarly,

$\begin{aligned} T_{2} V_{2}^{纬鈭� 1} = T_{1} V_{1}^{纬鈭� 1} \\ \frac{T_{3}}{T_{4}} = {(\frac{V_{4}}{V_{3}})}^{纬鈭� 1} = {(\frac{V_{1}}{V_{2}})}^{纬鈭� 1} \\ 鈭� \frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{4}} \end{aligned}$

$\frac{T_{3}}{T_{2}} = \frac{T_{4}}{T_{1}} 鈭� 鈭� 鈭� 鈭� (5)$

Subtract the above equation by 1 both sides:

$\begin{aligned} \frac{T_{3}}{T_{2}} 鈭� 1 = \frac{T_{4}}{T_{1}} 鈭� 1 \\ \frac{T_{3} 鈭� T_{2}}{T_{2}} = \frac{T_{4} 鈭� T_{1}}{T_{1}} \\ \frac{T_{1}}{T_{2}} = \frac{T_{4} 鈭� T_{1}}{T_{3} 鈭� T_{2}} = {(\frac{V_{2}}{V_{1}})}^{纬鈭� 1} \end{aligned}$

Substitute the above equation in (4):

$畏 = 1 - {(\frac{V_{2}}{V_{1}})}^{纬 - 1}$

Conclusion

Efficiency of the Otto cycle is $畏 = 1 - {(\frac{V_{2}}{V_{1}})}^{纬 - 1}$ .

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91影视

Derive equation $4.10$ for the efficiency of the Otto cycle.

Short Answer

Step by step solution

Introduction

Calculation

Conclusion

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91影视

Derive equation 4.10for the efficiency of the Otto cycle.

Short Answer

Step by step solution

Introduction

Calculation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Wave Optics

Electrostatics

Mechanics and Materials

Dynamics

Oscillations

Study anywhere. Anytime. Across all devices.

Derive equation $4.10$ for the efficiency of the Otto cycle.