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seven general property equation of the following thermodynamics processes: a. POLYTROPIC

seven general property equation of the following thermodynamics processes: a. POLYTROPIC

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Derive the seven general property equation of the following thermodynamics processes:

a. POLYTROPIC PROCESS

5. Polytropic Process Polytropic process is an internally reversible process. The from process definition: previous four (4) processes discussed above is also a polytropic PV" = C where n = any positive real number (n # k,1, and ∞.) PV" = C P = CV-" process with various polytropic coefficient 'n'. Now, we define a special case of polytropic process wherein ´n' is any positive real number and n + k, 1, and . Thus also, we may have concluded substituting to the integral and evaluating: that Q + 0 also. And, as derived using the seven (7) general A- [V1-n PV = C V-"dV = C = PVn equations, we have: - n v Any Process relation: P2V2-P¡V1 _ mR(T2 – T1) Wn = 1-n 1-n PV = C P;V _ P¿V2 T2 Internal Energy: = Following the relation in the change in internal energy, we have The relation between these three (3) parameters can be an internal energy of: simplified AU = mC,(T2 – T1) further to determine the relation between any of the two v Heat Transferred: desired Following the relation of heat transfer, we have: parameters using: Q = AU + W, mR(T2 – T,) PV" = C where n = any positive real number (n # k,1, and co.) Q = mC,(T2 – T;) + 1-n PV™ = C From properties of ideal gas, we have: Deriving the pressure - volume relation: P,V," = P,V2" Cy k - ; R = C„(k – 1) 一 Substituting from the above equation, we have: mC, (k – 1)(T2 – T,) Q = mC,(T2 – T1) + Also to derive 'n', we may use: 1-n (k - = mc,(T, – T,) 1 + = mC,(T, – T,) In =n In 1-n Note that the relation of heat transferred under polytropic In process has a special value for specific heat. That is the specific In heat under polytropic process, Cn, with an equivalent below: Deriving the volume – temperature relation: Cn = C, ;n# 1 ; from PV relationship And heat transferred under polytropic process will be: Substituting to the previously derived pressure - volume Q = mC„(T2 – T,) relation: V Enthalpy: Following the relation in the change in enthalpy, we have an enthalpy of: 開 AH = mC,(Tz – T,) V Entropy: Deriving the pressure – temperature relation: Following the guiding equation as represented by the to obtain PV ; from relationship AS, Substituting the previously derived volume - temperature dQ mCndT AS = T relation: Evaluating the integral, we will have: n-1 原-“ n-1 AS = mC„ln 1-n Note that the other equivalent of T2/T; can be used also to determine the value of AS. That is, Simplifying, we will have: n-1 一” n-1 開- v Work Non-flow: Applying the integral to evaluate the work non-flow of the ideal gas under polytropic process, we will again employ the use of: v Work Steady flow: W. = [° rav PdV For work steady flow, we follow the integral below: W = - VdP from process definition: PV" = C where n = any positive real number (n # k,1, and ∞.) PV" = C V substituting to the integral and evaluating in below: = - 1-n Simplifying further we will arrive for the relation: P,Vz-P,V, = n mR(T, – T,) 1-n W, = n 1-n Therefore, at polytropic process, we have a work on steady flow system as: w, = nW, = n PdV
Transcribed Image Text:5. Polytropic Process Polytropic process is an internally reversible process. The from process definition: previous four (4) processes discussed above is also a polytropic PV" = C where n = any positive real number (n # k,1, and ∞.) PV" = C P = CV-" process with various polytropic coefficient 'n'. Now, we define a special case of polytropic process wherein ´n' is any positive real number and n + k, 1, and . Thus also, we may have concluded substituting to the integral and evaluating: that Q + 0 also. And, as derived using the seven (7) general A- [V1-n PV = C V-"dV = C = PVn equations, we have: - n v Any Process relation: P2V2-P¡V1 _ mR(T2 – T1) Wn = 1-n 1-n PV = C P;V _ P¿V2 T2 Internal Energy: = Following the relation in the change in internal energy, we have The relation between these three (3) parameters can be an internal energy of: simplified AU = mC,(T2 – T1) further to determine the relation between any of the two v Heat Transferred: desired Following the relation of heat transfer, we have: parameters using: Q = AU + W, mR(T2 – T,) PV" = C where n = any positive real number (n # k,1, and co.) Q = mC,(T2 – T;) + 1-n PV™ = C From properties of ideal gas, we have: Deriving the pressure - volume relation: P,V," = P,V2" Cy k - ; R = C„(k – 1) 一 Substituting from the above equation, we have: mC, (k – 1)(T2 – T,) Q = mC,(T2 – T1) + Also to derive 'n', we may use: 1-n (k - = mc,(T, – T,) 1 + = mC,(T, – T,) In =n In 1-n Note that the relation of heat transferred under polytropic In process has a special value for specific heat. That is the specific In heat under polytropic process, Cn, with an equivalent below: Deriving the volume – temperature relation: Cn = C, ;n# 1 ; from PV relationship And heat transferred under polytropic process will be: Substituting to the previously derived pressure - volume Q = mC„(T2 – T,) relation: V Enthalpy: Following the relation in the change in enthalpy, we have an enthalpy of: 開 AH = mC,(Tz – T,) V Entropy: Deriving the pressure – temperature relation: Following the guiding equation as represented by the to obtain PV ; from relationship AS, Substituting the previously derived volume - temperature dQ mCndT AS = T relation: Evaluating the integral, we will have: n-1 原-“ n-1 AS = mC„ln 1-n Note that the other equivalent of T2/T; can be used also to determine the value of AS. That is, Simplifying, we will have: n-1 一” n-1 開- v Work Non-flow: Applying the integral to evaluate the work non-flow of the ideal gas under polytropic process, we will again employ the use of: v Work Steady flow: W. = [° rav PdV For work steady flow, we follow the integral below: W = - VdP from process definition: PV" = C where n = any positive real number (n # k,1, and ∞.) PV" = C V substituting to the integral and evaluating in below: = - 1-n Simplifying further we will arrive for the relation: P,Vz-P,V, = n mR(T, – T,) 1-n W, = n 1-n Therefore, at polytropic process, we have a work on steady flow system as: w, = nW, = n PdV
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