Chapter 5 Modeling Thermal Equipment

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Optimal Design of Energy Systems

Chapter 5 Modeling Thermal Equipment

Min Soo KIM

Department of Mechanical and Aerospace Engineering Seoul National University

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5.1 Using Physical Insight

Heat exchanger (HX) Distillation separator Turbomachinery

Design condition Off-design condition

Chapter 5. Modeling Thermal Equipment

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5.2 Selecting and simulating a Heat-Exchanger

(a) Shell and tube liquid : liquid

(b) Finned coil liquid : vapor(gas)

(c) Compact > 700m²/m³

(4)

5.3 Counterflow Heat-Exchanger – most favorable ΔT

1 _p1( 1,_i 1,_o) q  w c t  t

2 _p2( 2 ,_o 2 ,_i) q  w c t  t

q  U A t lm ^1, ^{2 ,} ^1, ^{2 ,}

1, 2 ,

( ) ( )

ln

i o o i

lm

i o

o i

t t t t

t t t

t t

  

  



(5)

5.3 Counterflow Heat-Exchanger – most favorable ΔT Known :

Unknown :

1,_i 2,_i 1( 1 _p1) 2( 2 _p2)

t t W  w c W  w c UA

1,o 2,o

t t q

1(1,_i 1,_o) 2( 2,_o 2,_i) (1)

W t t W t t

1, 2, 1, 2,

1 1, 1,

1, 2, 1, 2,

( ) ( )

( ) (2)

ln[( ) / ( )

i o o i

i o

i o o i

t t t t

W t t UA

t t t t

  

 

 

1

2, 2, 1, 1,

2

(1) _o _i W ( _i _o) (3)

t t t t

  W 

1

1, 2, 1, 1, 1, 2,

2

1 1, 1,

1

1, 2, 1, 1,

2

1, 2,

( ) ( )

(3) (2) ( )

( )

ln

i i i o o i

i o

i i i o

o i

t t W t t t t

W t t UA W

t t W t t

W t t

  

     

 

 

 

  

 

   

 



(6)

5.3 Counterflow Heat-Exchanger – most favorable ΔT

1

1, 2, 1, 1,

2

1, 2, 1 2

( )

1 1

ln

i i i o

o i

t t W t t

W UA D

t t W W

 

         

1

1, 2,

2 1,

1 2

1 (1 )

(4)

D

i i

o

D

t W t e

t W

W e

W

 

  

 

 





(4)(3) t2,_o (5) (4), (5) q

(7)

5.4 Special Case of Counterflow Heat-Exchanger

1 2

W W W

1, 1, 1, 2,

1 2

(4) ( ) 1

/

D

o i i i D

t t t t e

W W e

    



2

1 1

2

1 1 1 2

1 2 1 2

2 2

1 2

1 1 1 1 1 1 2

2 1 2 1 2

1 1 1 1 1 1

( / ) ( / ) (1 / )

1 2 2

/ 1 1 ( / ) 1( / ) (1 / )

1 1 1 2

2

D D

W W

UA UA

UA W UA W W W

W W W W

e

W W e W UA W UA W UA W UA W W W

W W W W W

      

 

              

    

                

    

 

 

 

1 2

1 /

/ 1 /

D D

e UA W

W W e UA W

 

 

1, 1, 1, 2,

( ) /

1 /

o i i i

t t t t UA W

    UA W



(8)

5.4 Special Case of Counterflow Heat-Exchanger

1 , 2 ,

1 ,

/

1 /

i i

o

t t U A W

t U A W

 



1 ,

6 5 3 0 4

1 2 2 .1 4

2 .1 4 2 .1

4 2 .9 0

1 2 .1

o

t o C

   



2 ,_o 3 0 ( 6 5 4 2 .1) 5 2 .9 ^o

t     C

1 2 .1^o

t C

 

4 .1 9 / ( )

cp  k J k g K

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5.5 Evaporators and Condensers

Refrigerant - working fluid

during evaporation and condensation

→ constant temperature (→ constant pressure)

Vapor Liquid↑

Vapor Liquid↓

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5.5 Evaporators and Condensers

Known : Unknown :

c i

t t UA W

to

ln ^c ⁱ

c o

t t U A

W t t

 



( )

ex p ^c ô ^c ⁱ ô ⁱ 1 ô ⁱ

c i c i c i

t t t t t t t t

U A

W t t t t t t

    

      

    

 

( ) 1 ex p

o i c i

t t t t U A

W

  

       

( )

lm p o i

q  U A t  w c t  t

( ) ( )

ln

c i c o

lm

c i

c o

t t t t

t t t

t t

  

  



(11)

5.6 Heat-Exchanger effectiveness

Maximum possible heat transfer rate (infinite heat transfer area)

m ax a ctu a l

q

  q



1 p1 1,i

w c t

2 p2 2,i

w c t

1,max 1 _p1( 1,_i 2,_i)

q  w c t t q_2,max  w c₂ _p₂(t_1,_i t_2,_i)

max min( 1,max, 2,max) ( _p)min( _{hot i}, _{cold i}, )

q  q q  wc t t

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<Example> What is q^{m ax}?

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Case 1)

Water leaves at Case 2)

Oil leaves at

Case 2 is impossible !

, m ax ( 2 .6 / )[ 2 .2 ( )](6 0 3 0 ^o ) 1 7 1 .6

qo il  kg s kJ kg K  C  kW

max min( _oil,max, _water,max) _oil,max

q q q q

  

1 7 1 .6

3 0 5 7 .3

(1 .5 / )[ 4 .1 9 ( )]

o kW o

C C

kg s kJ kg K

 



, m ax (1 .5 / )[ 4 .1 9 ( )](6 0 3 0 ^o ) 1 8 8 .6

w a ter

q  kg s kJ kg K  C  kW

1 8 8 .6

6 0 2 7

( 2 .6 / )[ 2 .2 ( )]

o kW o

C C

kg s kJ kg K

 



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5.7 Temperature profiles

(w c_p)1  m in (w c_p )₂  m in

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5.8 Effectiveness of a Counterflow heat exchanger If (w c_p )1  (w c_p )m in

(1) (1)

1 , ,

( 2 ) (1)

m ax m in , ,

( ) ( )

p co ld o u t co ld in

a ctu a l

p h o t in co ld in

w c t t

q

q w c t t

   ^



1

1, 2 ,

2 1,

1 2

1 (1 ^D )

i i

o

D

t W t e

t W

W e

W

 

  

 

 





1 1

1, 1, 2 ,

2 2

1, 1, 2 , 1,

1 1

2 2

( 1) (1 )

( ) 1

D D

i i i D

o i i i

D D

W W

t e t t e

W W e

t t t t

W W

e e

W W

 

        

   

 

(16)

5.8 Effectiveness of a Counterflow heat exchanger

1 2

1 ^D

D

e

W e

W

 ^

 



1 2

1 1

D U A

W W

 

   

 

Wm in

(17)

5.9 Number of Transfer Unit (NTU) q  U A t lm

m in

N T U U A

 W

q   qm ax

t1,i

t2,i

t2,o

t1,o

1 p1 1,i

w c t

2 p2 2,i

w c t

(18)

5.10 Binary Solutions, P-T-x diagram

vap liq

A+B A+B

y x

m a ss o f A m a ss fra ctio n o f A

m a ss o f A m a ss o f B

 

a a

a b

m m o le fra ctio n o f A M

m m

M M



 molecular mass weight ex> petroleum

cryogenic separation food

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5.10 Binary Solutions, P-T-x diagram

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5.12 Developing T-x diagram Sat. P – Sat. T relation

ln D

P C

  T

T d s  d h  vd P sd T d g vd P

  

f f f

s d T v d P d g

  

g g g

s d T v d P d g

  

2

( )

f g fg

f g f g

fg fg

f

s s h

d P

d T v v T v v

h h P

T v R T

  

 

 

,

a a sa t a

P  x P

Raoult’s law Dalton’s law

a b

a a

b b

P P P

P y P

 



vapor pressure

in mixture sat. P of

pure A

mole fraction of A in the liq. phase

mole fraction of A in the vap. phase

total P

equal

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5.13 Condensation of a binary mixture

x T ^vap.

liq.

(22)

5.14 Single stage distillation

x T

P

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5.14 Turbomachinery

speed Pressure

ratio

2 1

P P

Volume flow rate ¹

2 1

w c Tp

D P