Explanation Proof for P v k = constant : Write the given expression for isentropic expansion exponent ( k ) . k = − v P ( ∂ P ∂ v ) s (I) Here, the specific volume is v , the pressure is P , the entropy is s and the symbol ∂ indicates the partial derivative of variables. Let assume the entropy ( s ) is the functions of pressure ( P ) and specific volume ( v ) . s = s ( P , v ) (II) Partially differentiate the Equation (II) with respect to pressure and specific volume as follows. d s = ( ∂ s ∂ P ) v d P + ( ∂ s ∂ v ) P d v (III) It is given that the process is said to be isentropic i.e. the entropy is kept as constant. The change in entropy becomes zero. s = constant d s = 0 Substitute 0 for d s in Equation (III). 0 = ( ∂ s ∂ P ) v d P + ( ∂ s ∂ v ) P d v ( ∂ s ∂ P ) v d P + ( ∂ s ∂ v ) P d v = 0 (IV) Write the first Maxwell relation and second Maxell relation as follows. ( ∂ s ∂ P ) v = − ( ∂ v ∂ T ) s (V) ( ∂ s ∂ v ) P = ( ∂ P ∂ T ) s (VI) Here, the temperature is T . Substitute − ( ∂ v ∂ T ) s for ( ∂ s ∂ P ) v and ( ∂ P ∂ T ) s for ( ∂ s ∂ v ) P in Equation (IV). [ − ( ∂ v ∂ T ) s ] d P + [ ( ∂ P ∂ T ) s ] d v = 0 − ( ∂ v ∂ T ) s d P + ( ∂ P ∂ T ) s d v = 0 − ( ∂ v ∂ T ) s [ d P − ( ∂ P ∂ T ) s d v ( ∂ v ∂ T ) s ] = 0 d P − ( ∂ P ∂ T ) s d v ( ∂ v ∂ T ) s = 0 d P − ( ∂ P ∂ T ) s ( ∂ T ∂ v ) s d v = 0 d P − ( ∂ P ∂ v ) s d v = 0 (VII) Divide the Equation (VII) by the pressure P . d P − ( ∂ P ∂ v ) s d v P = 0 P d P P − 1 P ( ∂ P ∂ v ) s d v = 0 (VIII) Refer Equation (I). Substitute − k v for 1 P ( ∂ P ∂ v ) s in Equation (VIII). d P − ( ∂ P ∂ v ) s d v P = 0 P d P P − ( − k v ) d v = 0 d P P + k d v v = 0 (IX) Consider the k as constant and integrate the Equation (IX). ∫ d P P + k ∫ d v v = ∫ 0 ln P + k ln v = constant k ln ( P v ) = constant ln ( P v ) k = constant P v k = constant Thus, it has been shown that P v k = constant

8th Edition

ISBN: 9780073398174

Author: Yunus A. Cengel Dr., Michael A. Boles

Publisher: McGraw-Hill Education

See similar textbooks

Concept explainers

Thermodynamic Relations

Thermodynamics is the division of physics that deals with the transfer of any kind of energy, whether it be heat, temperature, or work, from one object to another.

Videos

Question

Chapter 12.6, Problem 88RP

To determine

Show that Pvk=constant and also the isentropic expansion exponent k reduces to the specific heat relation cp/cv for an ideal gas.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.6, Problem 87RP

Chapter 12.6, Problem 89RP

Students have asked these similar questions

الثانية Babakt Momentum equation for Boundary Layer S SS -Txfriction dray Momentum equation for Boundary Layer What laws are important for resolving issues 2 How to draw. 3 What's Point about this.

R αι g The system given on the left, consists of three pulleys and the depicted vertical ropes. Given: ri J₁, m1 R = 2r; απ r2, J2, m₂ m1; m2; M3 J1 J2 J3 J3, m3 a) Determine the radii 2 and 3.

B: Solid rotating shaft used in the boat with high speed shown in Figure. The amount of power transmitted at the greatest torque is 224 kW with 130 r.p.m. Used DE-Goodman theory to determine the shaft diameter. Take the shaft material is annealed AISI 1030, the endurance limit of 18.86 kpsi and a factor of safety 1. Which criterion is more conservative? Note: all dimensions in mm. 1 AA Motor 300 Thrust Bearing Sprocket 100 9750 เอ

Chapter 12 Solutions

Thermodynamics: An Engineering Approach

Ch. 12.6 - What is the difference between partial...Ch. 12.6 - Consider a function z(x, y) and its partial...Ch. 12.6 - Prob. 3P Ch. 12.6 - Conside the function z(x, y), its partial...Ch. 12.6 - Consider air at 350 K and 0.75 m3/kg. Using Eq....Ch. 12.6 - Consider air at 350 K and 0.75 m3/kg. Using Eq....Ch. 12.6 - 12–7 Nitrogen gas at 400 K and 300 kPa behaves as...Ch. 12.6 - Nitrogen gas at 800 R and 50 psia behaves as an...Ch. 12.6 - Prob. 9P Ch. 12.6 - Using the equation of state P(v a) = RT, verify...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Show that P v k = constant and also the isentropic expansion exponent k reduces to the specific heat relation c p / c v for an ideal gas.

Solution Summary: The author demonstrates that the isentropic expansion exponent k reduces to the specific heat ratio for an ideal gas.

Concept explainers

Videos

Show that Pvk=constant and also the isentropic expansion exponent k reduces to the specific heat relation cp/cv for an ideal gas.

Want to see the full answer?

Chapter 12 Solutions