equations.doc

8/11/2019 equations.doc

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College of Engineering and Computer ScienceMechanical Engineering Department

Mechanical Engineering 370Thermodynamics

Spring 2003 Ticket: 57010 Instructor: Larry Caretto

Summary of Important Equations in Thermodynamics

Basic notation and definition of terms

Fundamental dimensions o mass! lengt"! time! temperature! and amount o su#stance $mol% are

denoted! respecti&ely as '! L! T( ! and in t"e a##re&iations #elo)( For e*ample! t"e

dimensions o pressure are 'L+1T+2( T"ermodynamic properties are said to #e e*tensi&e i t"eydepend on t"e si,e o t"e system( -olume and mass are e*amples o intensi&e properties.intensi&e properties! suc" as temperature and pressure! do not depend on t"e si,e o t"e system(T"e ratio o t)o e*tensi&e properties is an intensi&e property( T"e ratio o an e*tensi&e property!suc" as &olume! to t"e mass o t"e system! is called t"e speciic property. e(g(! t"e ratio o&olume to mass is called t"e speciic &olume(

' molecular )eig"t $'/%

m mass $'%

M

mn = num#er o moles $%

nergy or general e*tensi&e property

m

Ee= Speciic molar energy $energy per unit mass% or general e*tensi&e property per

unit mass

eMn

Ee

== Speciic energy $energy per unit mole% or general e*tensi&e property per unit mole

pressure $'L+1T+2%

- &olume $L3%. )e also "a&e t"e speciic &olume or &olume per unit mass! & $L 3'+1%

and t"e &olume per unit mole v $L3+1%

T temperature $%

density $'L+3%. 1/&(

* uality

su#scripts and & denote saturated liuid and stauarated &apor! respecti&ely

4 t"ermodynamic internal energy $'L2T+2%. )e also "a&e t"e internal energy per unit

mass! u $L2T+2%! and t"e internal energy per unit mole! u $'L2T+2+1%

4 6 - t"ermodynamic ent"alpy $'L2T+2%. )e also "a&e t"e ent"alpy per unit mass! " u

6 & $dimensions: L2T+2% and t"e internal energy per unit mole h $'L2T+2+1%

S entropy $'L2T+2+1%. )e also "a&e t"e entropy per unit mass! s$L2T+2+1% and t"e

internal energy per unit mole s $'L2T+2+1+1%

)ork $'L2T+2%

8 "eat transer $'L2T+2%

ngineering 9uilding oom 1333 'ail Code "one: ;1;(


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uW : t"e useul )ork rate or mec"anical po)er $'L2T+3%

m : t"e mass lo) rate $'T+1%

2

2V

: t"e kinetic energy per unit mass $L2T+2%

g,: t"e potential energy per unit mass $L2T+2%

tot: t"e total energy m$u 62

2V

6 g,% $'L2T+2%

Q : t"e "eat transer rate $'L2T+3%

: t"e rate o c"ange o energy or t"e control &olume( $'L2T+3%

Mass and energy balance equations

T"e euations #elo) use t"e standard t"ermodynamic sign con&ention or "eat and )ork( eatadded to a system is positi&e and "eat re?ected rom a system is negati&e( T"e oppositecon&ention "olds or )ork( ork done #y a system is positi&e. )ork done on $added to% a systemis negati&e( In terms o t"e in! out! 8in! and 8outterms used in t"e te*t )e can make t"eollo)ing statements(

8 8in+ 8out out+ in

T"ese euations may #e su#stituted or any o t"e "eat! 8! or )ork! terms #elo)(

elation o mass lo) to &elocity! speciic &olume and area:v

AVAVm

==

@eneral irst la):

+++

++=

inlet

i

2

i

ii

outlet

i

2

i

iiu

cv gz2

Vhmgz

2

VhmWQ

dt

dE

'ass #alance euation: =outlet

i

inlet

icv mm

dt

dm

Steady+lo) assumptions: 0 and 0

Steady+lo) irst la):

+++

++=

inlet

i

2

i

ii

outlet

i

2

i

iiu gz2

Vhmgz

2

VhmQW

'ass #alance or steady lo)s: =outlet

i

inlet

i mm

Steady+lo) irst la) )it" negligi#le A B : +=inlet

ii

outlet

iiu hmhmQW

Second law of thermodynamics and entropy

9asic deinition o entropy:T

PdVdUdS

+=



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@eneral second la):T

WL

T

Qsmsm

dt

dS cvcv

inlet

ii

outlet

iicv

+=+

Second la) ineuality:T

Qsmsm

dt

dS cv

inlet

ii

outlet

iicv

+

dia#atic steady+lo)! second la) ineuality: 0smsminlet

iioutlet

ii

Ideal-gas equations

1( T"e gas constant

R is t"e uni&ersal gas constant )it" dimensions o energy di&ided #y $moles times

temperature%! andM

RR= is t"e engineering gas constant )it" dimensions o energy di&ided

#y $mass times temperature%( T"e energy dimensions are sometimes e*pressed as energy

units $e(g(! D%( o)e&er! or +&+T calculations! t"e energy dimensions are e*pressed inpressure units times &olume units $e(g(! kaEm3( Some &alues o t"e uni&ersal gas constant ares"o)n #elo)(

Rlbmol

ftpsi!"#$%#0

Rlbmol

lbft"&%#&'&

()mol

m)P"#'%*

()mol

)+"#'%*R

"f

"

=

=

=

=

2( +-+T calculations )it" mass or moles( T"e e*ample in t"e irst euation is air $ 0(2;7kD/kg+A%( 9ot" e*amples are or an ideal gas t"at occupies a &olume o 1 m3at a pressure o100 ka and a temperature o 25 C 2;(15 A(

m 1(1


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c( elations"ips or internal energy and ent"alpy

du c&dT d" cpdT u2G u1 2

#

T

T

vdTc "2G "1

2

#

T

T

PdTc

For constant "eat capacity: u2G u1 c&$T2G T1% and "2G "1 c$T2G T1%

From ideal gas ta#les: u2G u1 uo$T2% G u

o$T1% and "2G "1 "o$T2% G "

o$T1%

=( ntropy c"anges

a( constant "eat capacity

s2+ s1 cpln + ln c&ln + ln

#( &aria#le "eat capacity #y integration

s2+ s1 + ln 6 ln

c( &aria#le "eat capacity #y ideal gas ta#les

s2+ s1 so$T2% + s

o$T1% + ln

5( nd states o an isentropic process or an ideal gas )it"

a( constant "eat capacity $k cp/c&%

T2 T1

/k T2 T1 2 1

k

#( &aria#le "eat capacity using t"e air ta#les

c( &aria#le "eat capacity using general ideal gas ta#les

so$T2% so$T1% + ln

d( &aria#le "eat capacity #y integration o cp$T% or c&$T% euation

ln or ln

Cycles and efficiencies

In an engine cycle a certain amount o "eat! H8H! is added to t"e cyclic de&ice at a "ig"

temperature and a certain amount o )ork HH is perormed byt"e de&ice( T"e dierence

#et)een H8H and HH is H8LH! t"e "eat re?ected rom t"e de&ice at a lo) temperature(

In a rerigeration cycle a certain amount o "eat! H8LH! is added to t"e cyclic de&ice at a lo)

temperature and a certain amount o )ork HH is perormed ont"e de&ice( T"e sum o H8LH and H

H is H8H! t"e "eat re?ected rom t"e de&ice at a "ig" temperature(

For #ot" de&ices: H8H HH 6 H8LH



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ngine cycle eiciency:1Q1

1W1

=

erigeration cycle coeicient o perormance:W

Q/P

L=

T"e isentropic eiciency! s! compares t"e actual )ork! H)aH! to t"e ideal )ork t"at )ould #e done

in an isentropic $re&ersi#le adia#atic% process! H)sH( 9ot" t"e actual and t"e isentropic process

"a&e t"e same initial states and t"e same inal pressure(

For a )ork output de&ice:131

131

s

s=

For a )ork input de&ice:131

131

s

s=


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