Thermodynamic relations

(1)

1 1 Th

Therm

ermodyn

odynam

amic

ic Re

Relat

latio

ions

ns

1.1 1.1 Rel

Relati

ations

ons for

for En

Energ

ergy P

y Prope

roperti

rties

es

1.1

1.1.1 .1 InInterternal nal EneEnergy rgy ChaChangenge dU dU From ﬁrst law of thermodynamics

From ﬁrst law of thermodynamics dU

dU == dQdQ ++ dW dW (1.1)(1.1) For a reversible process

For a reversible process

dW

dW == −−PP dV dV

From second law of thermodynamics for a reversible process From second law of thermodynamics for a reversible process

dQ

dQ == TT dS dS Therefore Eqn.(1.1) becomes

Therefore Eqn.(1.1) becomes dU

dU == TT dS dS −−PP dV dV (1.2)(1.2) 1.1

1.1.2 .2 EnEnthathapy py ChaChangenge dH dH From the deﬁnition of enthalpy From the deﬁnition of enthalpy

H H == U U ++ PP V V (1.3)(1.3) Diﬀerentiating Diﬀerentiating dH dH == dU dU ++ PP dV dV ++ VV dP dP Substituting for

Substituting for dU dU from Eqn.(1.2)from Eqn.(1.2) dH

dH == VV dP dP ++ TT dS dS (1.4)(1.4) 1.1

1.1.3 .3 GibGibbs Fbs Freree Ene Enerergy Cgy Chanhangege dGdG From the deﬁnition of Gibbs free energy From the deﬁnition of Gibbs free energy

G G == H H −−TT S S Diﬀerentiating Diﬀerentiating dG dG == dH dH −−TT dS dS −− SdT SdT Substituting for

Substituting for dH dH from Eqn.(1.4)from Eqn.(1.4) dG

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1.1.

1.1.4 4 HelmHelmholtz holtz FFree ree EnerEnergy gy ChangeChange dAdA From the deﬁnition of Helmholtz free energy From the deﬁnition of Helmholtz free energy

A A == U U −−TT S S Diﬀerentiating Diﬀerentiating dA dA == dU dU −− TT dS dS −− SdT SdT Substituting for

Substituting for dU dU from Eqn.(1.2)from Eqn.(1.2) dA

dA == −−PP dV dV −−SdT SdT (1.6)(1.6)

1.2 1.2 Ma

Matha

thamat

matica

ical

l Con

Concep

cepts

ts

1.2.

1.2.1 1 ExacExact t DiﬀeDiﬀerenrential tial EquatEquationsions If

If F F == F F ((x,x, yy)) thenthen

dF dF == MMdxdx ++ NNdydy Exactness Criteria: Exactness Criteria:



∂M ∂M ∂y ∂y



x x = =



∂N ∂N ∂x ∂x



y y If

If F F == F F ((x,y,zx,y,z)) thenthen dF dF == MMdxdx ++ NNdydy ++ PP dzdz Exactness Criteria: Exactness Criteria:



∂M ∂M ∂y ∂y



x,z x,z = =



∂N ∂N ∂x ∂x



y,z y,z



∂M ∂M ∂z ∂z



x,y x,y = =



∂P ∂P ∂x ∂x



y,z y,z



∂N ∂N ∂z ∂z



x,y x,y = =



∂P ∂P ∂y ∂y



x,z x,z 1.2

1.2.2 .2 CycCyclic lic RelRelatiation on RulRulee For the function in the variables

For the function in the variables x,x, yy && zz



∂x∂x ∂y ∂y



z z



∂y∂y ∂z ∂z



x x



∂z∂z ∂x ∂x



y y = = −−11

(3)

1.2

1.2.3 .3 OthOther Rer Relaelatiotions ons of Imf Importportanceance



∂z∂z ∂x ∂x



y y = =



∂z∂z ∂w ∂w



y y



∂w∂w ∂x ∂x



y y



∂x∂x ∂y ∂y



z z = = 11



∂y∂y ∂x ∂x



zz

1.

1.3 3 Ma

Maxw

xwel

ell

l Re

Rela

lati

tion

onss

For the fundamental property relations: For the fundamental property relations:

dU dU == TT dS dS −−PP dV dV dA dA == −−PP dV dV −− SdT SdT dG dG == −−SdT SdT ++ VV dP dP dH dH == VV dP dP ++ TT dS dS Applying exactness criteria of diﬀerential equation: Applying exactness criteria of diﬀerential equation:

T T V V P P S S U U GG H H A A                d d d d d d d d d d d d d d ss PASGVHTU PASGVHTU



∂T ∂T ∂V ∂V



S S = = −−



∂P ∂P ∂S ∂S



V V



∂P ∂P ∂T ∂T



V V = =



∂S ∂S ∂V ∂V



T T



∂S ∂S ∂P ∂P



T T = = −−



∂V ∂V ∂T ∂T



P P



∂V ∂V ∂S ∂S



P P = =



∂T ∂T ∂P ∂P



S S

1.4 1.4 Rel

Relati

ations f

ons for Th

or Ther

ermody

modynam

namic Pr

ic Proper

opertie

ties in te

s in terms

rms

of

of P

P V

V T

T

and Speciﬁc heats

1.4

1.4.1 .1 DeDeﬁniﬁnitiotionsns



∂U ∂U ∂T ∂T



V V = = C C V V



∂H ∂H ∂T ∂T



P P = = C C P P

(4)

V

Volume olume expansivityexpansivity ββ

β β == 11 V V



∂V ∂V ∂T ∂T



P P Isothermal compressibility Isothermal compressibility κκ κ κ == −− 11 V V



∂V ∂V ∂P ∂P



T T 1.

1.4.4.2 2 ReRelalatitionons s foforr dU dU



∂U ∂U ∂V ∂V



T T = = T T



∂P ∂P ∂T ∂T



V V − − P P Considering

Considering U U == U U ((TT,, V V )) dU dU == C C V V dT dT ++



T T



∂P ∂P ∂T ∂T



V V − −P P



dV dV For van der Waals gas,

For van der Waals gas,

dU dU == C C V V dT dT ++ a a V V 22dV dV 1.

1.4.4.3 3 ReRelalatitionons s foforr dH dH



∂H ∂H ∂P ∂P



T T = = V V −−T T



∂V ∂V ∂T ∂T



P P Considering Considering H H == H H ((TT,, P P )) dH dH == C C P P dT dT ++



V V −−T T



∂V ∂V ∂T ∂T



P P



_dP_dP 1.

1.4.4.4 4 ReRelalatitionons s foforr dS dS



∂S ∂S ∂T ∂T



V V = = C C V V T T



∂S ∂S ∂T ∂T



P P = = C C P P T T

(5)

Considering Considering S S == S S ((TT,, V V )) dS dS == C C V V T T dT dT ++



∂P ∂P ∂T ∂T



V V dV dV For van der Waals gas,

For van der Waals gas, dS dS == C C V V T T dT dT −− R R V V −−bbdV dV Considering Considering S S == S S ((TT,, P P )) dS dS == C C P P T T dT dT −−



∂V ∂V ∂T ∂T



P P dP dP 1.4

1.4.5 .5 RelRelatiations ons for for SpecSpeciﬁc iﬁc heaheatsts C C P P == T T



∂P ∂P ∂T ∂T



S S



∂V ∂V ∂T ∂T



P P C C V V == −−T T



∂P ∂P ∂T ∂T



V V



∂V ∂V ∂T ∂T



S S Speciﬁc heat diﬀerences:

Speciﬁc heat diﬀerences: C C P P −−C C V V == −−T T



∂P ∂P ∂V ∂V



T T



∂V ∂V ∂T ∂T



22 P P For van der Waals gas,

For van der Waals gas,

C C P P −− C C V V == T T V V ββ22 κ κ Speciﬁc heat ratio:

Speciﬁc heat ratio:

C C P P C C V V = = ((∂P/∂V ∂P/∂V ))S S ((∂P/∂V ∂P/∂V ))T T Speciﬁc heat variations:

Speciﬁc heat variations:



∂C ∂C P P ∂P ∂P



T T = = −−T T



∂ ∂ 2 2_V_V ∂T ∂T 22



P P



∂C ∂C V V ∂V ∂V



T T = = T T



∂ ∂ 2 2_P_P ∂T ∂T 22



V V

(6)

1.

1.5 5 Tw

Two-

o-ph

phas

ase

e Sy

Syst

stem

emss

Equilibrium in a closed system of constant composition: Equilibrium in a closed system of constant composition:

d

d((nGnG) ) = = ((nV nV ))dP dP −−((nS nS ))dT dT During phase change

During phase change T T andand P P remairemains constantns constant. . TherThereforeefore,, dd((nGnG) ) = = 00.. Since

Since dndn = = 00,, dGdG = = 00.. For two phases

For two phases αα andand ββ of a pure species coexisting at equilibrium:of a pure species coexisting at equilibrium: G

Gαα == GGββ where

where GGαα andand GGββ are the molar Gibbs free energies of the individual phases.are the molar Gibbs free energies of the individual phases. dG dGαα == dGdGββ dP dP satsat dT dT == ∆ ∆S S αβαβ ∆ ∆V V αβαβ 1.5

1.5.1 .1 ClaClapeypeyon on equequatiationon dP dP satsat dT dT == ∆ ∆H H αβαβ T T ∆∆V V αβαβ 1.5.

1.5.2 2 ClausClausius-Clius-Clapeyron apeyron equatequationion

ln ln P P sat sat 2 2 P P satsat 1 1 = = ∆∆H H R R



11 T T 11 − − 11 T T 22



1.5

1.5.3 .3 VVapor Papor Presressursure vse vs. . TTempeemperatratureure From Clausius-Clapeyron equation

From Clausius-Clapeyron equation ln

ln P P satsat == AA −− BB

T T A satisfactory relation given by

A satisfactory relation given by AntoineAntoine _{is of the form}_{is of the form} ln

T

T ++ C C The values of the constants

The values of the constants A,A, BB andand C C are are readireadily ly avavailablailable e for for manymany species.

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1.6 1.6 Gib

Gibbs F

bs Free En

ree Ener

ergy as a

gy as a Gen

Genera

eratin

ting F

g Fun

uncti

ction

on

d d



GG RT RT



₌₌ 11 RT RT dGdG−− G G RT RT 22dT dT Substituting for

Substituting for dGdG from fundamental property relation, and from the deﬁ-from fundamental property relation, and from the deﬁ-nition of nition of GG:: d d



GG RT RT



₌₌ V V RT RT dP dP −− H H RT RT 22dT dT This is a dimensionless equation.

This is a dimensionless equation. V V RT RT ==



∂ ∂ ((G/RT G/RT )) ∂P ∂P



T T H H RT RT == −−T T



∂ ∂ ((G/RT G/RT )) ∂T ∂T



P P When

When G/RT G/RT is known as a function of is known as a function of T T andand P P ,, V/RT V/RT andand H/RT H/RT fol- fol-low by simple differentiation. The remaining properties are given by defining low by simple differentiation. The remaining properties are given by defining equations. equations. S S R R == H H RT RT −− G G RT RT U U RT RT == H H RT RT −− P P V V RT RT

1.

1.7 7 Re

Resi

sidu

dual

al Pr

Prope

opert

rtie

iess

Any extensive property

Any extensive property M M is given by:is given by: M

M == M M igig ++ M M R R where

where M M igig _{is ideal gas value of the property, and}_{is ideal gas value of the property, and M}_MR R _{is the residual value}_{is the residual value} of the property.

of the property.

For example for the extensive property For example for the extensive property V V ::

V V == V V igig ++ V V R R == RT RT P P ++ V V R R Since Since V V == ZRT/P ZRT/P V V R R == RT RT P P ((Z Z −−1)1)

(8)

For Gibbs free enrgy For Gibbs free enrgy d d



GG R R RT RT



₌₌ V V R R RT RT dP dP −− H H R R RT RT 22dT dT (1.7)(1.7) V V R R RT RT ==



∂ ∂ ((GGR R _/RT_/RT )) ∂P ∂P



T T (1.8) (1.8) H H R R RT RT == −−T T



∂ ∂ ((GGR R /RT /RT )) ∂T ∂T



P P (1.9) (1.9) From the deﬁnition of

From the deﬁnition of GG:: G GR R == H H R R −−TT S S R R S S R R R R == H H R R RT RT −− G GR R RT RT (1.10)(1.10) At constant

At constant T T Eqn.(1.7) becomesEqn.(1.7) becomes d d



GG R R RT RT



₌₌ V V R R RT RT dP dP Int

Integrategration ion from zero from zero presspressure ure to to the the arbitarbitrary pressurerary pressure P P givesgives G GR R RT RT ==



P P 0 0 V V R R RT RT dP dP (const.(const. T T ) ) ((11..1111)) where at the lower limit, we have set

where at the lower limit, we have set GGR R _/RT_{/RT equal to zero on the basis that}_{equal to zero on the basis that} the zero-pressure state is an ideal-gas state. (

the zero-pressure state is an ideal-gas state. (V V R R = = 00)) Since

Since V V R R ₌_{= ((RT/P}_RT/P₎₍_)(Z_Z₋₋₁₎_{1) Eqn.(1.11) becomes}_{Eqn.(1.11) becomes} G GR R RT RT ==



P P 0 0 ((Z Z −−1)1)dP dP P P (const.(const. T T ) ) ((11..1122)) Diﬀer

Diﬀerententiatiniating g Eqn.(Eqn.(1.12) at 1.12) at with respect with respect toto T T at constantat constant P P



∂ ∂ ((GGR R /RT /RT )) ∂T ∂T



P P = =



P P 0 0



∂Z ∂Z ∂T ∂T



P P dP dP P P H H R R RT RT == −−T T



P P 0 0



∂Z ∂Z ∂T ∂T



P P dP dP P P (const. T (const. T ) ) ((11..1133)) Combining Eqn.(1.12) and (1.13) and from Eqn.(1.10)

Combining Eqn.(1.12) and (1.13) and from Eqn.(1.10) S S R R R R == −−T T



P P 0 0



∂Z ∂Z ∂T ∂T



P P dP dP P P −−



P P 0 0 ((Z Z −−1)1)dP dP P P (const.(const. T T ) ) (1(1.1.14)4)

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1.8 1.8 Gen

Generali

eralized

zed Corr

Correlati

elations

ons of

of Ther

Thermodyna

modynamic

mic Prop

Prop--erties for Gases

erties for Gases

Of the two hinds of data needed for the evaluation of thermodynamic Of the two hinds of data needed for the evaluation of thermodynamic proper-ties, heat capacities and

ties, heat capacities and PP VV T T data, the latter are most frequently missing.data, the latter are most frequently missing. Fortunately, the generalized methods developed for compressibility factor Fortunately, the generalized methods developed for compressibility factor Z Z are also applicable to residual properties.

are also applicable to residual properties. Substituting for

Substituting for P P == P P ccP P rr and T and T == T T ccT T rr,, dP dP == P P ccdP dP rr and dT and dT == T T ccdT dT rr H H R R RT RT cc = = −−T T _r_r22



P P rr 0 0



∂Z ∂Z ∂T ∂T rr



P P rr dP dP rr P P rr (const. (const. T T rr) ) ((11..1155)) S S R R R R == −−T T rr



P P rr 0 0



∂Z ∂Z ∂T ∂T rr



P P rr dP dP rr P P rr − −



P P rr 0 0 ((Z Z −− 1)1)dP dP rr P P rr (const. (const. T T rr)) (1.16) (1.16) 1.8

1.8.1 .1 ThrThree ee PaParamrameteeter Mr Modelodelss

From three-parameter corresponding states principle developed by

From three-parameter corresponding states principle developed by Pitzer Pitzer Z

Z == Z Z 00 ++ ωZ ωZ 11 Simila

Similar r equatequations ions forfor H H R R andand S S R R are:are: H H R R RT RT cc = = ((H H R R ₎₎00 RT RT cc + + ωω((H H R R ₎₎11 RT RT cc S S R R R R == ((S S R R ))00 R R ++ ωω ((S S R R ))11 R R Calculated values of the quantities

Calculated values of the quantities ((H H R R ₎₎00 RT RT cc ,, ((H H R R ₎₎11 RT RT cc ,, ((S S R R ₎₎00 R R andand ((S S R R ))11 R R are shown by plots of these quantities vs.

are shown by plots of these quantities vs. P P rr for various values of for various values of T T rr.. ((H H R R ₎₎00 RT RT cc and and ((S S R R ₎₎00 R

R used alone provide two-parameter corresponding statesused alone provide two-parameter corresponding states corr

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1.8.

1.8.2 2 CorrCorrelatelations froions from Redlim Redlich/ch/KwoKwong Equang Equation otion of Statef State Z Z == 11 1 1 −−hh −− 4 4..934934 T T 11..55 r r



hh 1 + 1 + hh



where where h h == 00..0866408664P P rr ZT ZT rr and and T T rr == T T T T cc P P rr == P P P P cc

1.9 1.9 De

Dev

velo

elopin

ping

g T

Tabl

ables of

es of The

Thermod

rmodyna

ynamic Prope

mic Propert

rties

ies

from Experimental Data

Experimental Data: Experimental Data:

(a)

(a) VVapor pressurapor pressure data.e data. (b)

(b) PressPressure, speciﬁc volumeure, speciﬁc volume, , tempertemperature (ature (PP VV T T ) data in the vapor re-) data in the vapor re-gion.

gion. (c)

(c) DensiDensity of ty of saturasaturated liquid and ted liquid and the criticthe critical pressure and temperatural pressure and temperature.e. (d)

(d) Zero pressZero pressure speciﬁc heat data for the vure speciﬁc heat data for the vapor.apor.

From these data, a complete set of thermodynamic tables for the saturated From these data, a complete set of thermodynamic tables for the saturated liqui

liquid, d, satursaturated vaporated vapor, , and super-heatand super-heated vapor can ed vapor can be be calcucalculated as lated as per per thethe steps below:

steps below: 1.

1. RelatRelation foion forr lnln P P satsat _vs._{vs. T}_{T such as}_{such as}

ln

T

T ++ C C 2.

2. EquatEquation of ion of state for the state for the vvapor that apor that accuaccuratelrately represeny represents thets the PP VV T T data.

data. 3.

3. StaState: te: 11 Fix values for

(11)

4.

4. StaState: te: 22

Enthalpy and entropy changes during vaporization are calculated from Enthalpy and entropy changes during vaporization are calculated from Clapeyron equation using the

Clapeyron equation using the lnln P P satsat _vs._{vs. T}_{T data as:}_{data as:}

dP dP satsat dT dT == ∆ ∆H H lvlv T T ((V V vv −−V V ll)) and and ∆ ∆S S lvlv == ∆ ∆H H lvlv T T Here

Here V V ll shall be measured, andshall be measured, and V V vv is calculated from the relation ob-is calculated from the relation ob-tained in step-2.

tained in step-2. From these values of

From these values of ∆∆H H vlvl andand ∆∆S S vlvl obtaine the values of H obtaine the values of H andand S S at state: 2

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5.

5. StaState: te: 33 Follow the constant pressure line.Follow the constant pressure line. dH dH == C C P P dT dT ++



V V −−T T



∂V ∂V ∂T ∂T



P P



_dP_dP dS dS == C C P P dT dT T T −−



∂V ∂V ∂T ∂T



P P dP dP

Here for the speciﬁc heat of vapor corresponding to the pressure at Here for the speciﬁc heat of vapor corresponding to the pressure at state: 2 is obtained from the relation:

state: 2 is obtained from the relation:



∂C ∂C P P ∂P ∂P



T T = = −−T T



∂ ∂ 2 2_V_V ∂T ∂T 22



and from the zero pressure speciﬁc heat data. and from the zero pressure speciﬁc heat data. With the value of

With the value of C C P P for this state as calculated above,for this state as calculated above, S S andand H H values at state: 3 are calculated.

values at state: 3 are calculated. 6.

6. SSttaattee: : 44, , 5 & 5 & 66 ThThe e aboabove ve cacalclcululatatioion n cacan n be be dodone ne alalonong g cocon- n-sta

stant nt temtemperaperaturture e linlineaneand d the vthe valualues at es at stastates 4, 5 and 6 can be tes 4, 5 and 6 can be ob- ob-tained.

tained. 7.

7. StaState: te: 77 The calculations made for state: 2 can de done for the tem-The calculations made for state: 2 can de done for the tem-perature at state: 6.

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1.1

1.10 0 The

Thermod

rmodyna

ynamic

mic Dia

Diagra

grams o

ms of Im

f Import

portanc

ancee

1.10.1

(14)

1.10.2

(15)

1.10.3

(16)

2 2 Th

Therm

ermodyn

odynam

amic

ics

s of

of Fl

Flo

ow

w Pr

Proces

ocesses

ses

2.

2.1 1 Co

Cons

nser

erv

vat

atio

ion o

n of M

f Mas

asss

dm dm dt

dt + ∆(+ ∆(ρuAρuA) ) = = 00 (2.1)(2.1) where the symbol

where the symbol ∆∆ denotes the difference between exit and entrance streams.denotes the difference between exit and entrance streams. For steady flow process

For steady ﬂow process

∆(

∆(ρuAρuA))fsfs = = 00 (2.2)(2.2)

Since speciﬁc volume is the reciprocal of density, Since speciﬁc volume is the reciprocal of density,

˙˙ m

m == uAuA V

V == ccoonnssttaanntt. . ((22..33)) This is the equation of continuity.

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2.2 2.2 Con

Conser

serv

vati

ation

on of

of En

Energ

ergy

y

d d((mU mU )) dt dt + ∆[(+ ∆[(U U ++ 1 1 2 2uu 2 2 ₊_{+ zg}_zg₎_{) ˙˙}_m_{m] ] =}_{= ˙˙}_Q_Q ₋₋_W_W_˙˙ _(2.4)_(2.4) W W == W W ss+ ∆[(+ ∆[(PP V V ) ) ˙˙mm]] (2.5)(2.5) d d((mU mU )) dt dt + ∆[(+ ∆[(H H ++ 1 1 2 2uu 2 2 ₊_{+ zg}_zg₎_{) ˙˙}_m_{m] ] =}_{= ˙˙}_Q_Q₋₋ _W_W_˙˙ s s (2.6)(2.6) For most applications, kinetic- and potential-energy changes are For most applications, kinetic- and potential-energy changes are negligi-ble. Therefore ble. Therefore d d((mU mU )) dt dt + ∆(+ ∆(H H ˙˙mm) ) == ˙˙QQ −−W W ˙˙ss (2.7)(2.7) Energy balances for steady state ﬂow processes:

Energy balances for steady state ﬂow processes: ∆[( ∆[(H H ++ 11₂₂uu22 ++ zgzg) ) ˙˙mm] ] == ˙˙QQ−−W W ˙˙ss (2.8)(2.8) Bernoulli’s equation: Bernoulli’s equation: P P ρ ρ ++ u u22 2 2 ++ gzgz = = 00 (2.9)(2.9)

2.3 2.3 Flo

Flow in P

w in Pipes o

ipes of Con

f Consta

stan

nt Cr

t Cross

oss-se

-secti

ction

on

∆ ∆H H ++ ∆∆uu 2 2 2 2 = = 00 (2.10)(2.10) In diﬀerential form In diﬀerential form dH dH == −−uduudu (2.11)(2.11)

Equation of continuity in diﬀerential form: Equation of continuity in diﬀerential form:

d

d((uA/V uA/V ) ) = = 00 (2.12)(2.12) Since

Since AA is a constant,is a constant, dd((u/V u/V ) ) = = 00. Therefore. Therefore du du V V −− udV udV V V 22 = = 00

(18)

or or du du == udV udV V V (const.(const. AA) ) ((22..1133)) Substituting this in Eqn.(2.11)

Substituting this in Eqn.(2.11) dH dH == −−uu 2 2_dV_dV V V (2.14)(2.14)

From the fundamental property relations From the fundamental property relations

T T dS dS == dH dH −− VV dP dP Therefore Therefore T T dS dS == −−uu 2 2_dV_dV V V −− VV dP dP (2.15)(2.15) As gas flows along a pipe in the direction of decreasing pressure, its specific As gas flows along a pipe in the direction of decreasing pressure, its specific vo

volumlume e incincreareasesses, , and and alsalso o the the vevelocilocity ty (as(as ˙˙mm == uA/V uA/V ). ). ThThus ius in thn thee direction of increasing velocity,

direction of increasing velocity, dP dP is negative,is negative, dV dV is positive, and the twois positive, and the two terms of Eqn.(2.15) contribute in opposite directions to the entropy change. terms of Eqn.(2.15) contribute in opposite directions to the entropy change. According to second law

According to second law dS dS ≥≥ 00.. u u22_max_maxdV dV V V ++ VV dP dP = = 00 (const.(const. S S )) Rearranging Rearranging u u22_max_max == −−V V 22



∂P ∂P ∂V ∂V



S S (2.16) (2.16) This is the speed of sound in ﬂuid.

This is the speed of sound in ﬂuid.

2.4 2.4 Gen

General

eral Relat

Relations

ionship be

hip betw

tween

een V

Velocit

elocity and

y and Cross

Cross--sectional Area

sectional Area

d d((uA/V uA/V ) ) = = 00 1 1 V

V ((udAudA ++ AduAdu)) −−uAuA dV dV V

(19)

or or

udA

udA ++ AduAdu uA uA == V V dV dV V V 22 From the fundamental property relation for

From the fundamental property relation for dH dH and from steady ﬂow energyand from steady ﬂow energy equation

equation

−

−VV dP dP == uduudu (const.(const. S S )) i.e.,

i.e., V V == −−udu/dP udu/dP at constantat constant S S . Therefore. Therefore dA dA A A ++ du du d d == udu udu − −V V 22_((∂P/∂V_{∂P/∂V ))} S S

From the relation for velocity of sound, the above equation becomes From the relation for velocity of sound, the above equation becomes

dA dA A A ++ du du u u == udu udu u u22 sonic sonic Therefore Therefore dA dA A A == udu udu u u22 sonic sonic − − dudu u u ==



uu22 u u22 sonic sonic − −11



dudu u u The ratio of actual velocity to the velocity of sound is called the

The ratio of actual velocity to the velocity of sound is called the Mach Number Mach Number M M.. dA dA A A = = (M(M 2 2 ₋₋ ₁₎₁₎dudu u u (2.17)(2.17) Depending on whether

Depending on whether MM is greater than unity (supersonic) or less thanis greater than unity (supersonic) or less than unity (subsonic), the cross sectional area increases or decreases with velocity unity (subsonic), the cross sectional area increases or decreases with velocity increase. increase. includegraphicssupersonic.eps includegraphicssupersonic.eps includegraphicssubsonic.eps includegraphicssubsonic.eps includegraphicsconvergdiverg.eps includegraphicsconvergdiverg.eps

2

2.5 .5 N

No

ozz

zzle

less

u u22₂₂ −−uu₁2₁2 == −−22



P P 22 P P 11 V V dP dP == 22γγP P 11V V 11 γ γ −−11



1 1 −−



P P 22 P P 11



((γ γ −−1)1)/γ /γ



(2.18) (2.18)

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From the deﬁnition of sound velocity From the deﬁnition of sound velocity

u u22_max_max == −−V V 22



∂P ∂P ∂V ∂V



S S and from the evaluation of the derivative

and from the evaluation of the derivative ((∂P/∂V ∂P/∂V ))S S for the isentropic ex-for the isentropic ex-pansion of ideal gas with constant heat capacities from the relation

pansion of ideal gas with constant heat capacities from the relation PP V V γ γ == const,

const,

u

u22_throat_throat == γP γP 22V V 22 (2.19)(2.19) Substituting this value of the throat velocity for

Substituting this value of the throat velocity for uu22 in Eqn.(2.18) and solvingin Eqn.(2.18) and solving for the pressure ratio with

for the pressure ratio with uu11 = = 00 givesgives P P 22 P P 11 = =



22 γ γ + 1+ 1



γ/γ/((γ γ −−1)1) (2.20) (2.20) The speed of sound is attained at the throat of a conerging/diverging nozzle 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 only when the pressure at the throat is low enough that the critical value of

of P P 22/P /P 11 is reacheis reached. d. If insufficienIf insufficient pressure drop is t pressure drop is avavailablailable in e in the nozzlethe nozzle for the velocity to become sonic, the diverging section of the nozzle acts as for the velocity to become sonic, the diverging section of the nozzle acts as a diffuser.

a diﬀuser.

2.

2.6 6 T

Tur

urbi

bine

ness

˙˙ W

(21)

and and W W ss == −−∆∆H H (2.22)(2.22) W W ss(isentropic)(isentropic) == −−(∆(∆H H ))S S (2.23)(2.23) η η == W W ss W W ss(isentropic)(isentropic) = = ∆∆H H (∆ (∆H H ))S S (2.24) (2.24)

2.7 2.7 Th

Throt

rottli

tling

ng Pr

Proces

ocesses

ses

∆ ∆H H = = 00 Joule-Thomson Coeﬃcient: Joule-Thomson Coeﬃcient: µ µJ J ==



∂T ∂T ∂P ∂P



H H = = T T ((∂V/∂T ∂V/∂T ))P P −−V V C C P P

(22)

For ideal gases

For ideal gases µµJ J = = 00. For a real gas. For a real gas uuJ J can be positive, zero or negative.can be positive, zero or negative. Any gas for which volume is linear with temperature along an isobar will Any gas for which volume is linear with temperature along an isobar will hav

have a e a zero Joule-zero Joule-ThomsThomson coeﬃcienon coeﬃcient. t. i.e., if i.e., if VV /T /T == constantconstant == φφ((P P )),, µ

µJ J = = 00..

Inversion curve:

Inversion curve: T T −−P P diagram. The points in the curve corresponddiagram. The points in the curve correspond to

to µµJ J = = 00. In the region inside the curve. In the region inside the curve µµJ J is positive.is positive.

2.

2.8 8 Co

Comp

mpre

ress

ssio

ion

n

W W == −−



P P 22 P P 11 V V dP dP For reversible-adiabatic compression

For reversible-adiabatic compression W W == γP γP 11V V 11 γ γ −−11



1 1−−



P P 22 P P 11



γ γ −−γ γ 11



Eﬀect of clearance on work of compression: Eﬀect of clearance on work of compression:

W W == γγP P 11V V I I γ γ −−11



1 1−−



P P 22 P P 11



γ γ −−γ γ 11



(23)

Multistage compression Multistage compression::

Optimum compression ratio per stage

Optimum compression ratio per stage ==



P P 22 P P 11



11/n/n W W == nγnγP P 11V V I I γ γ −−11



1 1 −−



P P 22 P P 11



γ γ −γnγn−11



Relation between

Relation between V V DD andand V V I I :: V V I I == V V DD



1 + 1 + C C −− C C



P P 22 P P 11



11/γ /γ



where

where C C == V V C C /V /V DD For compression in multistages,For compression in multistages, V V I I == V V DD



1 + 1 + C C 11 −−C C 11



P P 22 P P 11



nγ nγ 11



where

(24)

2.