Thermodynamic relations
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1 1.1 1.1.1 1.1.1
Therm Thermodyn odynam amic ic Relat Relatio ions ns Relati Relations ons for Energ Energy y Prope Properti rties es Inter Internal nal Energy Energy Change Change dU
From first law of thermodynamics dU = dQ + dW
(1.1)
For a reversible process dW = −P dV From second law of thermodynamics for a reversible process dQ = T dS Therefore Eqn.(1.1) becomes dU = T dS − P dV
1.1.2 1.1.2
(1.2)
Entha Enthapy py Change Change dH
From the definition of enthalpy H = U + P V
(1.3)
Differentiating dH = dU + P dV + V dP Substituting for dU from Eqn.(1.2) dH = V dP + T dS
1.1.3 1.1.3
(1.4)
Gibbs Gibbs Fre Free e Ener Energy gy Chan Change ge dG
From the definition of Gibbs free energy G = H − T S Differentiating dG = dH − T dS − SdT Substituting for dH from Eqn.(1.4) dG = V dP − SdT 1
(1.5)
1.1.4 1.1.4
Helmholtz Helmholtz Free Energy Energy Change dA
From the definition of Helmholtz free energy A = U − T S Differentiating dA = dU − T dS − SdT Substituting for dU from Eqn.(1.2) dA = −P dV − SdT
1.2 1.2.1 1.2.1
Matha Mathamat matica icall Concep Concepts ts Exact Exact Differen Differential tial Equations Equations
If F = F (x, y ) then dF = M dx + N dy Exactness Criteria: ∂M ∂y
∂N ∂x
=
x
y
If F = F (x , y , z ) then dF = M dx + N dy + P dz Exactness Criteria: ∂M ∂y ∂M ∂z ∂N ∂z
1.2.2 1.2.2
= x,z
=
x,y
=
x,y
∂N ∂x ∂P ∂x ∂P ∂y
y,z
y,z
x,z
Cyclic Cyclic Relati Relation on Rule Rule
For the function in the variables x, y & z ∂x ∂y
∂y ∂z
∂z ∂x
z
x
2
= −1 y
(1.6)
1.2.3 1.2.3
Other Other Rela Relatio tions ns of of Import Importance ance ∂z ∂x
∂z ∂w
∂w ∂x
=
y
∂x ∂y
1.3 1.3
y
y
1
=
z
∂y ∂x z
Maxw Maxwel elll Rela Relati tion onss
For the fundamental property relations:
= = = =
dU dA dG dH
P
T dS − P dV −P dV − SdT −SdT + V dP V dP + T dS
Applying exactness criteria of differential equation: ∂T ∂V ∂P ∂T ∂S ∂P ∂V ∂S
1.4 1.4.1 1.4.1
= S
=
V
=
T
=
P
∂P − ∂S ∂S ∂V T ∂V − ∂T ∂T ∂P S
U
A d s d d d d d d
T
H PASGVHTU
V
P
Relati Relations ons for for Ther Thermody modynam namic ic Proper Propertie tiess in terms terms of P V T and Specific heats Defini Definitio tions ns ∂U ∂T ∂H ∂T
= C V V
V
= C P P
P
3
S
G
V
Volume expansivity β
1 ∂V β= V ∂T
P
Isothermal compressibility κ
1 ∂V κ=− V ∂P
1.4. 1.4.2 2
T
Rela Relati tion onss for for dU ∂U ∂V
= T T
∂P ∂T
V
∂P ∂T
V
− P
Considering U = U (T , V )
dU = C V V dT + T
− P dV
For van der Waals gas, dU = C V V dT +
1.4. 1.4.3 3
a dV V 2
Rela Relati tion onss for for dH ∂H ∂P
= V − T T
∂V ∂T
P
Considering H = H (T , P )
dH = C P P dT + V − T
1.4. 1.4.4 4
∂V ∂T
Rela Relati tion onss for for dS ∂S ∂T ∂S ∂T
C V V T C P P = T
=
V
P
4
P
dP
Considering S = S (T , V ) C V V dS = dT + T
∂P ∂T
dV V
For van der Waals gas, dS =
C V R V dT − dV T V − b
Considering S = S (T , P ) C P P dS = dT − T
1.4.5 1.4.5
∂V ∂T
dP
P
Relati Relations ons for Specific Specific heats heats ∂P ∂V ∂T S ∂T P ∂P ∂V = −T ∂T V ∂T
C P P = T C V V
S
Specific heat differences: ∂P C P P − C V V = −T ∂V
∂V ∂T
2
T
For van der Waals gas, V β2 C P P − C V V = T κ Specific heat ratio:
(∂P/∂V )S C P P = (∂P/∂V )T C V V Specific heat variations: ∂C P P ∂P ∂C V V ∂V
T
T
∂ 2 V = −T ∂T 2 P ∂ 2 P = T ∂T 2 V
5
P
1.5 1.5
TwoTwo-ph phas ase e Syst System emss
Equilibrium in a closed system of constant composition: d(nG) = (nV )dP − (nS )dT During phase change T and P remains remains constant. constant. Therefore Therefore,, d(nG) = 0. Since dn = 0, dG = 0. For two phases α and β of a pure species coexisting at equilibrium: Gα = Gβ where Gα and Gβ are the molar Gibbs free energies of the individual phases. dGα = dGβ
∆S αβ dP sat = ∆V αβ dT 1.5.1 1.5.1
Clapey Clapeyon on equati equation on ∆H αβ dP sat = dT T ∆V αβ
1.5.2 1.5.2
Clausius-Cl Clausius-Clapeyron apeyron equation equation ∆H 1 1 P 2sat − ln sat = P 1 R T 1 T 2
1.5.3 1.5.3
Vapor Pres Pressur sure e vs. vs. Temperat emperature ure
From Clausius-Clapeyron equation
ln P sat = A − A satisfactory relation given by
Antoine
B T
is of the form
ln P sat = A −
B T + C
The values of the constants A, B and C are readily readily availabl availablee for many species. 6
1.6
Gibbs Gibbs Free Free Ener Energy gy as a Genera Generatin ting g Fun Functi ction on d
G RT
=
1 G dG − dT RT RT 2
Substituting for dG from fundamental property relation, and from the definition of G: d
G RT
=
V H dP − dT RT RT 2
This is a dimensionless equation. V ∂ (G/RT ) = RT ∂P T H ∂ (G/RT ) = −T RT ∂T
P
When G/RT is known as a function of T and P , P , V/RT and H/RT follow by simple differentiation. The remaining properties are given by defining equations. S H G = − R RT RT U H P V = − RT RT RT
1.7 1.7
Resi Residu dual al Prope Propert rtie iess
Any extensive property M is given by: M = M ig + M R where M ig is ideal gas value of the property, and M R is the residual value of the property. For example for the extensive property V : V = V ig + V R =
RT + V R P
Since V = ZRT/P V R =
RT (Z − 1) P 7
For Gibbs free enrgy d
GR RT
V R H R = dP − dT RT RT 2
V R ∂ (GR /RT ) = RT ∂P T R R H ∂ (G /RT ) = −T RT ∂T
(1.7)
(1.8)
(1.9)
P
From the definition of G: GR = H R − T S R S R H R GR = − R RT RT At constant T Eqn.(1.7) becomes d
GR RT
(1.10)
V R = dP RT
Integrat Integration ion from zero pressure pressure to the arbitrary arbitrary pressure P gives GR = RT
P
0
V R dP RT
(const. T ) T )
(1.11)
where at the lower limit, we have set GR /RT equal to zero on the basis that the zero-pressure state is an ideal-gas state. (V ( V R = 0) Since V R = (RT/P )(Z − 1) Eqn.(1.11) becomes GR = RT
P
(Z − 1)
0
dP P
(const. T ) T )
(1.12)
Different Differentiatin iatingg Eqn.(1.12) Eqn.(1.12) at with respect to T at constant P
∂ (GR /RT ) ∂T
H R = −T RT
P
∂Z ∂T
P
0
=
0
P
∂Z ∂T
dP P P
dP P P (const. T ) T )
(1.13)
Combining Eqn.(1.12) and (1.13) and from Eqn.(1.10) S R = −T R
P
∂Z ∂T
0
dP − P P
P
(Z − 1)
0
8
dP P
(const. T ) (1.14) 4) T ) (1.1
1.8
Generali Generalized zed Correlati Correlations ons of Thermodyna Thermodynamic mic PropProperties for Gases
Of the two hinds of data needed for the evaluation of thermodynamic properties, heat capacities and P V T data, the latter are most frequently missing. Fortunately, the generalized methods developed for compressibility factor Z are also applicable to residual properties. Substituting for P = P cP r and T = T cT r , dP = P cdP r and dT = T cdT r
H R = −T r2 RT c S R = −T r R
1.8.1 1.8.1
P r
P r
0
∂Z ∂T r
0
∂Z ∂T r
P r
P r
dP r − P r
dP r P r
(const. T r )
P r
(Z − 1)
0
dP r P r
(1.15)
(const. T r ) (1.16)
Three Three Param Paramete eterr Model Modelss
From three-parameter corresponding states principle developed by
Pitzer
Z = Z 0 + ωZ 1 Similar Similar equations equations for H R and S R are:
(H R )0 (H R )1 H R = +ω RT c RT c RT c R R 0 (S ) (S R )1 S = +ω R R R (H R )0 (H R )1 (S R )0 (S R )1 Calculated values of the quantities and , , RT c RT c R R are shown by plots of these quantities vs. P r for various values of T r . (H R )0 (S R )0 and used alone provide two-parameter corresponding states RT c R correlati correlations ons that quickly yield coarse coarse estimates estimates of the residual residual properties. properties.
9
1.8.2 1.8.2
Correlat Correlations ions from from Redlich/ Redlich/Kwo Kwong ng Equation Equation of of State 1
4.934 − Z = 1−h T r1.5
h 1+ h
where h=
0.08664P r ZT r
and T T c P = P c
T r = P r
1.9
Dev Developin eloping g Tables ables of Thermod Thermodyna ynamic mic Propert Properties ies from Experimental Data
Experimental Data: (a) Vapor pressure pressure data. (b) Pressure, Pressure, specific volume, volume, temperature temperature (P ( P V T ) T ) data in the vapor region. (c) Density Density of saturated saturated liquid and the critical critical pressure and temperature. temperature. (d) Zero pressure pressure specific heat data for the vapor. vapor. From these data, a complete set of thermodynamic tables for the saturated liquid, liquid, saturated saturated vapor, vapor, and super-heated super-heated vapor can be calculated calculated as per the steps below: 1. Relation Relation for for ln P sat vs. T such as
ln P sat = A −
B T + C
2. Equation Equation of state for the vapor that accuratel accurately y represents represents the P V T data.
State: 1 3. State: Fix values for H and S of saturated-liquid at a reference state. 10
4. State: State: 2 Enthalpy and entropy changes during vaporization are calculated from Clapeyron equation using the ln P sat vs. T data as:
∆H lv dP sat lv = dT T (V v − V l) and
∆H lv lv T Here V l shall be measured, and V v is calculated from the relation obtained in step-2. ∆S lv lv =
From these values of ∆H vl vl and ∆S vl vl obtaine the values of H and S at state: 2
11
5. State: State: 3 Follow the constant pressure line. dH = C P P dT + V − T dS = C P P
dT − T
∂V ∂T
∂V ∂T
dP
P
dP
P
Here for the specific heat of vapor corresponding to the pressure at state: 2 is obtained from the relation: ∂C P P ∂P
T
∂ 2 V = −T ∂T 2
and from the zero pressure specific heat data. With the value of C P P for this state as calculated above, S and H values at state: 3 are calculated. 6. State: 4, 5 & 6 Th Thee above above calc calcul ulat atio ion n can can be done done alon alongg conconstant stant tempera temperatur turee linean lineand d the valu values es at states states 4, 5 and 6 can be obtained.
State: 7 The calculations made for state: 2 can de done for the tem7. State: perature at state: 6.
12
1.10 1.10
Thermod Thermodyna ynamic mic Diagra Diagrams ms of of Import Importanc ance e
1.10.1
T − S diagram
13
1.10.2
P − H diagram
14
1.10.3
H − S diagram
15
2
Therm Thermodyn odynam amic icss of Flo Flow Proces Processes ses
2.1 2.1
Cons Conser erv vatio ation n of of Mas Masss
dm + ∆( ρuA) = 0 dt
(2.1)
where the symbol ∆ denotes the difference between exit and entrance streams. For steady flow process
∆(ρuA)fs = 0
(2.2)
Since specific volume is the reciprocal of density,
˙ = m
uA = constant. V
This is the equation of continuity. 16
(2.3)
2.2
Conser Conserv vation ation of Energ Energy y d(mU ) ˙ − W ˙ + ∆[(U + 12 u2 + zg )m ˙]=Q dt
(2.4)
˙] W = W s + ∆[(P V )m
(2.5)
d(mU ) ˙s + ∆[(H + 12 u2 + zg )m ˙ ] = Q˙ − W dt
(2.6)
For most applications, kinetic- and potential-energy changes are negligible. Therefore d(mU ) ˙s + ∆( H m ˙ ) = Q˙ − W dt
(2.7)
Energy balances for steady state flow processes:
˙s ∆[(H + 12 u2 + zg )m ˙ ] = Q˙ − W
(2.8)
Bernoulli’s equation: P u2 + + gz = 0 2 ρ
2.3
(2.9)
Flow Flow in Pipes Pipes of of Consta Constan nt Cross Cross-se -secti ction on ∆u2 ∆H + =0 2
(2.10)
dH = −udu
(2.11)
In differential form
Equation of continuity in differential form: d(uA/V ) = 0 Since A is a constant, d(u/V ) = 0. Therefore du udV =0 − V V 2 17
(2.12)
or du =
udV (const. A) V
(2.13)
Substituting this in Eqn.(2.11) u2 dV dH = − V
(2.14)
From the fundamental property relations T dS = dH − V dP Therefore u2 dV T dS = − − V dP V
(2.15)
As gas flows along a pipe in the direction of decreasing pressure, its specific ˙ = uA/V ). volum volumee increa increases ses,, and also also the veloci velocity ty (as m ) . Th Thus us in in the the direction of increasing velocity, dP is negative, dV is positive, and the two terms of Eqn.(2.15) contribute in opposite directions to the entropy change. According to second law dS ≥ 0. 2 umax dV + V dP = 0 (const. S ) V
Rearranging 2
2
umax = −V
∂P ∂V
(2.16) S
This is the speed of sound in fluid.
2.4
General General Relations Relationship hip betw between een Velocity elocity and CrossCrosssectional Area d(uA/V ) = 0
1 dV (udA + Adu) − uA 2 = 0 V V 18
or udA + Adu V dV = uA V 2 From the fundamental property relation for dH and from steady flow energy equation −V dP = udu (const. S )
i.e., V = −udu/dP at constant S . Therefore dA du udu + = −V 2 (∂P/∂V )S A d From the relation for velocity of sound, the above equation becomes dA du udu + = 2 A u usonic Therefore dA udu du = 2 − = A usonic u
u2 −1 2 usonic
du u
The ratio of actual velocity to the velocity of sound is called the M.
Mach Number
dA du = (M2 − 1) A u
(2.17)
Depending on whether M is greater than unity (supersonic) or less than unity (subsonic), the cross sectional area increases or decreases with velocity increase. includegraphicssupersonic.eps includegraphicssubsonic.eps includegraphicsconvergdiverg.eps
2.5
Nozz zzle less u22 − u21 = −2
P 2
P 1
V dP =
2γ P 1 V 1 1− γ − 1 19
P 2 P 1
(γ −1)/γ 1)/γ
(2.18)
From the definition of sound velocity 2 = −V 2 umax
∂P ∂V
S
and from the evaluation of the derivative (∂P/∂V )S for the isentropic expansion of ideal gas with constant heat capacities from the relation P V γ = const, 2 = γP 2 V 2 uthroat
(2.19)
Substituting this value of the throat velocity for u2 in Eqn.(2.18) and solving for the pressure ratio with u1 = 0 gives P 2 = P 1
2 γ + 1
γ/( γ/(γ −1)
(2.20)
The speed of sound is attained at the throat of a conerging/diverging nozzle only when the pressure at the throat is low enough that the critical value of P 2 /P 1 is reached. reached. If insufficient insufficient pressure drop is availabl availablee in the nozzle for the velocity to become sonic, the diverging section of the nozzle acts as a diffuser.
2.6 2.6
Tur urbi bine ness
˙ s = −m ˙ ∆H W 20
(2.21)
and W s = −∆H
(2.22)
W s(isentropic) = −(∆H )S
(2.23)
η=
2.7
∆H W s = (∆H )S W s(isentropic)
Throt Throttli tling ng Proces Processes ses ∆H = 0
Joule-Thomson Coefficient: µJ =
∂T ∂P
= H
T (∂V/∂T )P − V C P P
21
(2.24)
For ideal gases µJ = 0. For a real gas uJ can be positive, zero or negative. Any gas for which volume is linear with temperature along an isobar will have have a zero Joule-Thoms Joule-Thomson on coefficient. coefficient. i.e., if V /T = constant = φ(P ), µJ = 0. Inversion curve: T − P diagram. The points in the curve correspond to µJ = 0. In the region inside the curve µJ is positive.
2.8 2.8
Comp Compre ress ssio ion n
P 2
W = −
V dP
P 1
For reversible-adiabatic compression
γ −1 γ
γP 1 V 1 1− W = γ − 1
P 2 P 1
Effect of clearance on work of compression: γ −1 γ
γ P 1 V I 1− W = γ − 1 22
P 2 P 1
Multistage compression: P 2 P 1
Optimum compression ratio per stage = nγP nγ P 1 V I 1− W = γ − 1
P 2 P 1
γ −1 γn
Relation between V D and V I : V I = V D
P 2 1 + C − C P 1
1/γ
where C = V C /V D For compression in multistages,
V I = V D 1 + C 1 − C 1 where C 1 is the clearance in the first stage.
23
1
P 2 P 1
nγ
1/n
2.9 2.9
Ejec Ejecto tors rs
24
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