thermo toolbox

8/6/2019 Thermo Toolbox

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Universal Balance Equation for Any Extensive Property

Accumulation = transport + generation

Inte rated form for some eriod of time:

Review of Engineering Thermodynamics

Rate form:

final initial amount amount amount amount

amount amount entering leaving generated consumed

! " ! " ! " ! " ! " ! "# $ # % #& ' & ' & ' & ' & ' & '

( ) ( ) ( ) ( ) ( ) ( )

rate of rate of rate of rate of rate of

change transport in transport out generation consumption

! " ! " ! " ! " ! "$ # % #& ' & ' & ' & ' & '

( ) ( ) ( ) ( ) ( )

Controlvolume

or system

Mass

entering (1)

Mass

leaving (2)

Q W

Geof Silcox, Chemical

Engineering, University of Utah

Mass Balance• Unsteady balance for CV

** #$exits

e

inlets

iCV mm

dt

dm!! ** #$#$+

exits

e

inlets

iCV mmmmm 12

• Steady balance for CV

• Balance for closed system

0$dt

dm sys012 $#$+ mmm sys

** #$exits

e

inlets

i mm !!0 ** #$exits

e

inlets

i mm0

• Averaged flow

avav

avavav

v

V

v

AVel AVel !m

!

! $$$





Energy Balance

• Unsteady balance for CV

Pvuh gz Vel ue%$%%$

,2

1 2


** ,, -

.//0

1 %%#,,

-

.//0

1 %%%%$

exits e

e

inlets i

iin,net in,net CV gz

Vel hm gz

Vel hmW Q

dt

dE

22

22

!!!!

** ,, .

//1

%%#,, .

//1

%%%%$ eiin,net in,net gz Vel

hm gz Vel

hmW Q22

022

!!!!


net innet in

sysW Q

dt

dE ,,

!! %$ net innet in sys W Q E ,, %$+



ei

Entropy Balance

• Unsteady balance for CV

There is only one form of entropy.

rocessleirreversib02S !


gen

exits

ee

inlets

ii

n

j j

jinCV S sm smT

Q

dt

dS !!!

!

%#%$ ***$1

,

processimpossible

processreversible

0

0

3

$

gen

gen

S

S

!

!

n jin

S sm smQ

!!!

!

%#%$ ,0


gen

n

j j

jin sysS

T

Q

dt

dS !

!

%$ *$1

, gen

in sys S

T

Q s smS %$#$+ 4

2

1

12 )(5



gen

exits

ee

inlets j jT $1



Combined Entropy and Energy Balance

wqdu 5 5 %$ gendsT

qds %$ 5

T

qds rev5

$revrev wqdu 5 5 %$

Tds du Pdv $ %du Pdv

dsT T

$ %or

Pdvwrev #$5

n a erna e orm o ows rom e re a on

vdP Pdv Pvd %$)(

vdP dhTds #$ or T

vdP

T

dhds #$



Combined Entropy and Energy Balance

for Ideal Gases

2 2 v dT

$2

2 P dT $ #v

11T v

p1

1T P

2 2 v ,av

T v s c ln R ln+ $ % 2 2

p,av

T P s c ln R ln+ $ #

For constant or averaged heat capacities,

1 1 1 1

where 1 2 av av av

T T T and c c(T )

2

%$ $





Equations for Work

Reversible boundary work, closed system:

Steady-flow, reversible work, open system:

4#$2

1,

v

vinrev Pdvw

!

6 71212

,2

2

1

z z g ee

vdP m

w P

P inrev #%

#%$$ 4

!



Models of Working Substances

Solid/

Liquid

C p = C v = C

GasesPhase-change

fluids (water &

refrigerants)

Mixtures

(advanced)

vvm #+u = + h = C +T

Perfect gas

Pv = RT

P and T are absolute.

C p = constant

C = C – R

Ideal gas

Pv = RT


C p = f(T).

C = C – R

Real gas

Pv

v

v P T f Z

actual

ideal

actual R R

$

$$ ),(

f g tot

vap

vvm x

#$$

v

+u = C v +T +h = C p+T .

RT

c

R

c

R P

P P

T

T T $$


44 $+$+T

T v

T

T p

R R

dT C udT C h

This chart was inspired by Subrata Bhattacharjee,The Expert

System for Thermodynamics, Prentice Hall, Upper Saddle

River, NJ, 2003. ISBN 0-13-009235-5. www.thermofluids.net.



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