Procesy cieplne i aparaty 8370858643, 9788370858643 [PDF]

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Q = α1 A ( T1 − Ts 1 ) Q =

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Rλ

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(4.29)

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α2

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(4.30)

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T1 − T2 Rk

(4.31)

5yZQDQLH WR PR*QD WH* ]DSLVDü Z SRGVWDZRZHM SRVWDFL Q = kA ( T1 − T2 )

(4.32)

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kA =

1 1 = i =n Rk 1 1 s +∑ i + α1 A i =1 λi A α 2 A

(4.33)

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Q = Ts ( n +1 ) − T2 α 2 Œd z L

(4.39)

3R GRGDQLX VWURQDPL L Z\FLJQL FLX SU]HG QDZLDV Z\UD]X Q /(πL) otrzymujemy Q 1 1 + Rλ r + ( ) = T1 − T2 ŒL α1 d w α2 dz

(4.40)

: QDZLDVLH SR]D RSRUHP SU]HZRG]HQLD FLHSáD Rλr mamy opory wnikania opisane wzorami Rα r 1 =

1 α1 d w

Rα r 2 =

1 α2 dz

(4.41)

(4.42)

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Rkr = Rα r1 + Rλ r + Rα r 2 .

(4.43)

RWU]\PDP\ Z]yU QD VWUXPLH FLHSáD SU]HQLNDMF\ SU]H] FLDQ UXURZ zwany równaniem Pecleta dla tej przegrody Q = ŒL

T1 − T2 Rkr

(4.44)

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∑

di +1 di 1 + 2λi α2dz

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(4.45)

ln

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:\SURZDG]RQH UyZQDQLH VWDQRZL Z]yU QD VWUXPLH FLHSáD FLDQ UXURZ zwany równaniem Pecleta dla tej przegrody.

(4.46)

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##  

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a2E1 RGELMD ]D HQHUJL  – a2)E1. Z tej energii z kolei powierzchnia 1 absorbuje a1(1 – a2)E1, odbija natomiast (1 – a1)(1 – a2)E1 3RZLHU]FKQLD  SRFKáDQLD HQHr2 JL a2(1 – a1)(1 – a2)E1, a odbija (1 – a1)(1 – a2) E1 6XPXMF RWU]ymane w ten sposób NROHMQH LORFL ]DDEVRUERZDQHM HQHUJLL SU]H] Sá\W  RWU]\PXMHP\ V]ereg: qa 2 = a2 E1 + a2 E1 (1 − a1 )(1 − a2 ) + a2 E1 (1 − a1 ) (1 − a2 ) 2

(

+ ⋅⋅⋅ = a2 E1 1 + p + p 2 + ⋅⋅⋅

2

)

(10.24)

gdzie p = (1 – a1)(1 – a2). *

6]HUHJ Z QDZLDVLH GDMH Z\UD HQLH 

qa 2 =

W

– p  ] F]HJR Z\QLND ]ZL]HN a2 E1 1− p

(10.25)

-H*HOL SU]HSURZDG]LP\ WDN VDP DQDOL] SURPLHQLRZDQLD Z\V\áDQHJR SU]H] Sáy 2, otrzymamy podobne UyZQDQLH RNUHODMFH HQHUJL ]DDEVRUERZDQ SU]H] SáyW 

qa 1 = .R

a1 E2 1− p

FRZ\ ELODQV HQHUJLL VWDQRZL FLHSáR X]\VNDQH SU]H] Sá\W 

(10.26)

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Podobnie dla CO2 QDOH*\ XZ]JO GQLü SRSUDZN βCO2 U\V  Z ]DOH*QRFL RG FLQLHQLD FDáNRZLWHJR L LORF]\QX FLQLHQLD F]VWNRZHJR &22 L JUXERFL ZDUVWZ\ 8Z]JO GQLHQLH REHFQRFL FLDQ\ ELRUFHM XG]LDá Z Z\PLDQLH FLHSáD ]QDF]QLH XWUXGQLD ]DGDQLH 'OD REOLF]H WHFKQLF]Q\FK ]DOHFD VL ]DOH*QRü qg − s

4   Tg 4  Ts   ′  = ε s C0 ε g   − ag   ,  100     100 

W/m 2

(10.46)

gdzie: qg–s – FLHSáR Z\PLHQLDQH PL G]\ JD]HP D FLDQ QD  P2 powierzchni, Tg – temperatura gazu, Ts – WHPSHUDWXUD FLDQ\ ELRUFHM XG]LDá Z Z\PLDQLH FLeSáD ε s′ = (εs + 1)/2 jest efektywn ]GROQRFL HPLV\MQ FLDQ\ εs – WHRUHW\F]Q ]GROQRFL HPiV\MQ FLDQ\ εg – ]DVW SF] ]GROQRFL HPLV\MQ JD]X ] Z\NUHVyZ Z WHPSHUDWXU]H gazu, ag – ]DVW pF] ]GROQRFL DEVRUSF\MQ JD]X Z WHPSHUDWXU]H FLDQ\ : REOLF]HQLDFK WHFKQLF]Q\FK PR*QD SU]\Mü *H ag = εg Z WHPSHUDWXU]H FLDQ\ L RGF]\WDü MH ] Z\NUHVyZ

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ε g = β CO2 ε g (CO2 ) + β H2 O ε g (H2 O) − ∆ε g

(10.47)

• GOD WHPSHUDWXU\ FLDQ\  Tg  ag = βCO2 ε g (CO2 )    Ts 

0 ,65

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0 ,45

− ∆ε g

(10.48)

:DUWRFL SRSUDZHN ∆εg QDOH*\ RGF]\WDü ] RGSRZLHGQLFK Z\NUesów (rys. 10.16) w zaOH*QRFL RG WHPSHUDWXU\ gazu.

11. Obliczanie wymienników cieSáD .ODV\ILNDFMDZ\PLHQQLNyZFLHSáD


        
  
   

         
  
odzaje.        
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  lub zakrzywionej. Wymiana energii zachod   
  
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yn  ywa   

" 
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ddaje     
   
 

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* regeneratory pieców Siemensa–Martina oraz niektóre generatory gazu.
   
   

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ktorem. &   
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*   • 
     
      
  
ików jest zgodny, • 
    
   
 
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193

•
 
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  urze, przedstawiony schematycznie na rys. 11.1a. Schemat wymienników z  
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*    
   
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*    
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 * ûta(przeciwp.) ûta(wspólp.)

194

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∆9

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,     
 
   

  
        
 
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 rnych kilkaset   *    mperatury topnienia. 2   *    
*
      

 
        
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    iwego

 p W = mc

(11.1)

$*  *


 
      la

 
 
      Q a −b = Wa (Ta 1 − Ta 2 ) = ±Wb (Tb 1 − Tb 2 )

(11.2)

3  

      –   du. ,      "# Q a −b = m a c pa (Ta 1 − Ta 2 ) = ± m b c pb (Tb 1 − Tb 2 )

(11.3)

     

Wb T − Ta 2 = ± a1 Wa Tb 1 − Tb 2

(11.4)

4      

   
     

 
   

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      ,      Q = kAûT

 
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(11.5) 



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' *  
    
 
 
  * owierzchni i  

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   yparka, w                 ycona      *  
   6    *  
ynów ∆T = Ta – Tb   *    
      
 / i 7         
   *
  e   *  

   $

  *
           *
 
∆Tm 
 
  anie Pecleta

Q = kAûTm

(11.6)

 
       *  
   

   6  *     
  dA  

 aje w       *      dQa: dQ a = − m a c pa dTa

(11.7)

1 
     
     A, tzn. na drodze od przekroju 1 do przekroju 2 (rys. 11.5), wynosi

Ta 2

Q a = − m a

∫c

pa dTa

(11.8)

Ta 1

W równaniach tych mamy znak –   

   8   du. !    *     
   

ami:

• w odniesieniu do elementu powierzchni dA •  

dQ a −b = kdAûT

(11.9)

Q a −b = A ( kûT )m

(11.10)

     A

'  "9#   
    dA =

dQ a −b kûT

+ * 
 * 
 

  d Q a

(11.11) = dQ a −b oraz Q a = Q a −b .

Po podstawieniu wzoru (11.7) do (11.11), otrzymujemy dA = −

m a c pa dTa kûT

(11.12)

197

    Ta 2

c pa dTa

∫

A = − m a

(11.13)

kûT

Ta 1

,       
*   ":# Ta 2

Q a −b

=

− m a

c pa dTa

∫

kûT

Ta 1

( kûT )m

(11.14)

i porównaniu równania (11.14) z równaniem (11.8) otrzymujemy Ta 2

− m a

∫c

Ta 2 pa dTa

∫

= − m a

Ta 1

c pa dTa kûT

Ta 1

( kûT )m

(11.15)

     Ta 2

( kûT )m

=T

∫c

pa dTa

Ta 1 a2

∫

Ta 1

(11.16) c pa dTa kûT

  * * cp = const mamy: ( kûT )m

=

Ta 2 − Ta 1 Ta 2

∫

Ta 1

(11.17)

dTa kûT

dla k ;    ûTm =

Ta 2 − Ta 1 Ta 2

∫

Ta 1

(11.18)

dTa ûT

$*  *

 * k = const, to z równania (11.10) otrzymujemy równanie Pec-

leta w ogólnej postaci:

Q = kAûTm

(11.19)

4  
*    *  
    o  *  ∆T = f (Ta# 3*        
             
   
 
  

 $*

198

 *

    *    rzymujemy: ûTm =

∆T = aTa + b (tj. równanie y = ax + b), to po prze-

Ta 2 − Ta 1 Ta 2

∫

Ta 1

T − Ta 1 Ta 2 − Ta 1 aT − aTa 1 = T a2 = = a2 a2 1 aTa 2 + b ûT dTa dTa ln ln 2 a aT b ûT1 + a1 aTa + b ûT T

(11.20)

∫

a1

       
 b i –b mamy ûTm =

ûT2 − ûT1

ln

ûT2

(11.21)

ûT

$    
   *  

  
*  <   * 
    
   " Tb), jak np. podczas   
         

  <    e
 
  =
 
       
 
     wnaniem (11.6) lub (11.7). W s   
 
 

* "k∆T) zmienia   *   Ta lub Tb    
     a
  ( kûT )m

=

( kûT )2 − ( kûT )1 ( kûT )2 ln ( kûT )1

1   
 k   *  
∆T   
 

cjami temperatury czynnika a 5      1  ( kûT )m

=

k1 ûT2 − k2 ûT1 k1 ûT2 k2 ûT1

(11.22)


 k(11.23)

1   *

       
 
* 
 etodami. 3UG\VNU]\*RZDQHLPLHV]DQH

6  

 
  
     
 y
 3  
  
 
*
     
 >   *  
 
   
      o 
   ε. Poprawka ε  ( 
 
  ymiany  
   
      
 
*owego * 

pokazano na rys. 11.7.

199

5\V  :\PLHQQLN FLHSáD ] SU]HSá\ZHP NU]\*RZ\P %

∆9PD[

∆9$

∆9

%

∆9

∆9%

%







5\V  5R]NáDG Uy*QLF WHPSHUDWXU\ GOD ]DVW SF]HJR XNáDGX ] SU]HSá\ZHP NU]\*RZ\P

$*   *  
  
 
*  e   ûTm′ = ε ûTm = ε

ûT2 − ûT1

ln

ûT2

(11.24)

ûT1

   



  A=

Q kε ûTm

(11.25)

   *   *
 
 
 
*
    *
 
  
  w  ∆Tm. ,    *
   X i Z.

to poprawka ε

200

5\V  =DOH*QRFL ε = f (X, Z GOD Uy*Q\FK przySDGNyZ SU]HSá\ZX NU]\*RZHJR ZHZQWU] rurek i prostopDGOH GR QLFK ZHGáXJ VFKHPDWX

5\V  =DOH*QRFL ε = f (X, Z) GOD Uy*Q\FK SU]\SDGNyZ SU]eSá\ZX PLHV]DQHJR

Parametr X 
    *
 
       owej czynnika b (ogrzanie czynnika b, ∆Tb#  
  *
eratury w   ∆Tmax, parametr Z      *
 
     i    
 a "    
 a, ∆Tb#  
  *
eratury ∆Tmax:

201

X=

Tbk − Tbp Tap − Tbp

Z=

Tap − Tak Tbk − Tbp

=

ûTb ûTmax

(11.26

ûTa ûTb

(11.27)

=



 *   ε = f (X, Z   
    *  o    
  
 

 
      e  !" # $
 



  
        % lda (jedna rurka we 

  &    
   

 Hoblera [4].

2EOLF]HQLDZLHONRFLZ\PLHQQLND oparte na jego sprawQRFL


  

  '"    
  '"    

 $ 

 
 

    $    
 
  Number of Transfer Units – ()* +*  ,-  (       i
  .
  
 

  η /.
 0  0 
 n

       
   
j  &  
$
& 
$



  


  *

 
 
 

 kowicie 
  *


∆T = T – T w tym wymienniku max

η=

a1

b1

Q Q max

(11.28)

5\V  5R]NáDG WHPSHUDWXU\ GOD SU]HFLZSUGRZ\FK Z\PLHnniNyZ FLHSáD a) Wa > Wb, Wb = Wmin; b) Wb > Wa, Wa = Wmin

)    



 ∆T  *   
 

  o
 



    

 
 

 
   *  


  & 
   *

 
 
 

 e

 *
&       $ 
  +   
 


  o*

 
 W        

 +* 
 emax

min

202


& * W = W   
 -    &   


 enie *   


 
 
 b  
  $     urze wlo 
 a 1

  .

 
b

  



min

Q max = Wmin (Tap − Tbp ) = Wmin ûTmax

(11.29)

Q = Wmax (Tap − Tak ) = Wmax ûTmin

(11.30)

oraz

η=

Wa (Tap − Tak )

Wb (Tbk − Tbp )max

=

Wmax (Tap − Tak )

(11.31)

Wmin (Tap − Tbp )

+ 

 
 

&   
 - $&  

 
*enie 
  0 η=

Wb (Tbk − Tbp )

Wa (Tap − Tak )max

=

Wmax (Tbk − Tbp )

(11.32)

Wmin (Tap − Tbp )

2

*  

  
  
   Q ,  *
a 0& * 

Q = ηWmin (Tap − Tbp )

(11.33)

+* 
 
 
 

&      
 



mamy:

Q = Wb (Tb 2 − Tb 1 ) = kA

(Ta 1 − Tb 1 ) − (Ta 2 − Tb 2 )  T −T  ln  a 1 b 1   Ta 2 − Tb 2 

(11.34)

 

     0 
 

3

Z równania (11.33) otrzymujemy wy *

η=

 * 
 
& * W = W b

Wb (Tb 2 − Tb 1 )

Wmin (Ta 2 − Tb 1 )

(11.35)

min

Ta 2 = Tb 1 +

  

  
  T

b2,

to

1 (Tb 2 − Tb1 ) η

(11.36)

203

Ta 2 − Tb 2 = Tb 1 − Tb 2 +

2

*

1  1 (Tb 2 − Tb 1 ) =  − 1 (Tb 2 − Tb 1 ) η η 

Wb Ta 2 − Ta 1 = Wa Tb 2 − Tb 1

(11.37)

(11.38)

   

 



Ta 1 = Ta 2 −

Wmin (Tb 2 − Tb 1 ) Wmax

(11.39)

oraz Ta 1 − Tb 1 = Ta 2 − Tb 1 −

1  

 ,4 

 * 0 Ta 2 − Tb 1 =

  
  

 !-3 Ta 1 − Tb 1 =

Wmin (Tb 2 − Tb 1 ) Wmax

1 (Tb 2 − Tb 1 ) η

(11.40)

(11.41)

1 W  1 W (Tb 2 − Tb 1 ) − min (Tb 2 − Tb 1 ) =  − min  (Tb 2 − Tb 1 ) (11.42) η Wmax  η Wmax 

  

  *  ,5) i (11.42) podstawiamy do równania (11.34) 1 Wmin − η Wmax kA = ln 1 Wmin −1 η

 Wmin  1 −   Wmax 

(11.43)

2 $
  
  ceniu otrzymamy ostatecznie  kA  Wmin   1 − exp  − 1 −   Wmin  Wmax   η=  kA  Wmin W 1 − min exp  − 1 − Wmax  Wmin  Wmax

  

(11.44)


*

 kA/W 

$   
 
()* 6  W /W oznaczymy jako R i dla przeciwpr 
 

  

 

3 min

min

max

204

ηp =

1 − e−

NTU (1− R )

1 − Re−

(11.45)

NTU (1− R )

7   
 

   


 $  


 '& '" 
  
*

 ηw =

( 1− e 1+ R

− NTU 1+ R )

(11.46)

 *    (0  

 


  *    a 
    
 

  5R]NáDGWHPSHUDWXU\Sá\QyZLFLDQ\ 7HPSHUDWXUDSá\QyZ

# 

  

 *  
  
ienniku     

  $ 

  

 *



&  
 
 o  


 
 +       ,&  

 T = f (A) lub T = f (L), gdzie A    

& L    y 

  *
od stosu       *

     
  m      
   !  
 *

  

  
   
   $
  
 

  (z 

    otoczenia, dQ = dQ = dQ) w przeciwprdowym i 
 
  
 +* 
 
 

   
 8&  dla 
   *
 0 *


$    

  h

*

 dA): a

b

dQ = − m a c pa dTa

(11.47)

dQ = − m b c pb dTb

(11.48)

dQ = −Wa dTa

(11.49)

dQ = −Wb dTb

(11.50)

2 

 .

  *

 
  

   
 t-

szej postaci:



*

    *

 
3

i otrzymam

205

dTa = −

dQ Wa

(11.51)

dTb = −

dQ Wb

(11.52)

2 *

z

 *




3 ∆T = T – T 
 

a

b

d ( ûT ) = dTa − dTb

(11.53)

1   1 d (ûT ) = − dQ  −   Wa Wb 

(11.54)

dQ = kdA∆T

(11.55)

1   1 d ( ûT ) = − kdAûT  −   Wa Wb 

(11.56)

  
   8 8'

2 

 *

   

 2  otrzymujemy

  

 
d (ûT )

ûT

1  1 = −k  −  Wa Wb

  dA 

9 

  
   

   A 
& * k = const)

1

ln

ûT2 ûT1

= 0 do A2 oraz od ∆T1 do ∆T2

    = − kA2  −   Wa Wb 

(11.57)

  

 

1   1 − kA2  −   Wa Wb 

 (11.58) +    
*



  
  
 

  .   *



   &    

     e  *

    :  '" ûT2 = ûT1

206

ûT2 = ûT1 H− kA /W 2

(11.59)

z

+*       $   *

 
ynów w dowolnym prze 
 

    x   &  *
 $ 
0    ∆T   

 
   


 

 1

ûTx = ûT1H− kA /W x

(11.60)

z

# 0     *

   
 y*   *

    
  *

 
 

 $
0  t

 7   & 
W < W , mamy zatem: 1

*  W > W , to 1

2

1 1 1 = − Wz W1 W2

(11.61)

1 1 1 = − Wz W2 W1

(11.62)

2

  
 
 

 

   

&    
w 

 

   


Dla  * przeciwne znaki (równania (11.47)–(11.52)); dTb * y * 1 1 1 = + Wz W1 W2

(11.63)

Wz = W1

(11.64)

Wz = W2

(11.65)

+*     




  

  &   podczas skrapla

 
    $ 


  
 &  

& *  o 0   

 

 * W = ∞   
 *

 
prze
  
     

 w skraplac

 

We wzorach Hudlera (11.59)–(11.60) ∆T  *

0  

   
   
 

& 
    &  

  kszej *


mperatury. 1   
 *
$ 0 

*   
 
 1  

    
& * *  
     
 
    &  
 
T     #$ 
1

najpierw ∆T2:

b1

207

ûT2 = Ta 2 − Tb 2

(11.66)

Tb 2 = Ta 2 − ûT2

(11.67)

   T

b2

Po wyeliminowaniu temperatury na podstawie równania bilansu cieplnego (11.2) mamy: Ta 1 − Ta 2 =

Wb (Tb 1 − Tb 2 ) Wa

(11.68)

Ta 2 = Ta 1 −

Wb (Tb 1 − Tb 2 ) Wa

(11.69)

Wb W Tb 1 + b Tb 2 − ûT2 Wa Wa

(11.70)

  



 

   

 45

Po podstawieniu tego wy *

Tb 2 = Ta 1 −



 

Wb Tb 1 Wa  Wb 1 − W a 

Ta 1 − Tb 2 =

− ûT2

(11.71)

  

2

* *


∆T *
 0  *    ∆T &  
 * 0 . 
  2

1

otrzy

Tb 2 = f

(ûT1  Ta 1  Tb 1 )

(11.72)

2$

 *

 0  T lub T w dowolnym przekroju wy 
    *

  
0  

 ∆T 
 &   

  0      o  b

a

miennika: Tb = f (∆T1, Ta1, Tb1 w

Wa (Ta 1 − Ta ) = ±Wb (Tb 1 − Tb )

(11.73)

1    
   &         du. (    

 5 *
 0 
    * i 
 
   
 
 

 

208

11.6.2. TemperatuUDFLDQNL

# 

 
       

   
 $o    
  2

& * 


 a     czynnik b

Q = α a A (Ta − Tsa )

(11.74)

Q = α b A (Tsb − Tb )

(11.75)

Q = kA∆T

(11.76)

 $    

 
 

 2 

  

  
  
    Tsa = Ta −

k ∆T αa

(11.77)

Tsb = Tb +

k ∆T αb

(11.78)

oraz

+* 
 
& * T

sa

≅ Tsb ≅Ts

&    

 

Ta − Ts α b = Ts − Tb α a

  



Ts =

α a Ta + α b Tb αa + αb

(11.79)

(11.80)

)    T *
  

 $

*
0&    


 



   

 
   

ego. s

11.:VND]yZNLGRSURMHNWRZDQLDZ\PLHQQLNyZFLHSáD

2

 
 

  

 $0

 
 $ 

 
 any    &   

* 



  
  $    o   #$ 

      
 3  2 $   

 $       
  



   *
 0     ariantów.

209

;     
 

 $ 

 
 $  
          
 8&  
3 T > T , T > T )

1.1. skupienia (w przeci  opisuje znane nam równanie

a1

a2

Q = m a c pa (Ta 1 − Ta 2 ) = m b c pb (Tb 1 − Tb 2 )

b1

b2

(11.81)


 
&  0     
a   

  
    emperatury. 1.2. Gdy po jednej stronie wymiennika zachodzi zmiana stanu skupienia czynni&      

  T > T ) lub w wyparce ogrzewanej spa  & $    
  0 w którym indeksy a i b

b2

b1

Q = m a ra = m b c pb (Tb 2 − Tb 1 )

(11.82)

Q = m a ra = m b rb

(11.83)

, +* i zmiana stanu skupienia zachodzi po obu stronach, np. w wyparce   &  
!  
 
& 

 

     

  o 


     

 
&     

  a


&  

 $   
& *3 T > T , T > T , i > i ) przyjmuje po0 a1

a2

b2

b1

a1

a2

Q = m i c pi (Ti 1 − Ti 2 ) + m i ( xa 1 ia 1 − xa 2 ia 2 ) − Kik = m b c pb (Tb 2 − Tb 1 ) (11.84)

    0 
 


 
*            & i& +@ % $
 !    "  :17 :DUV]DZD  >@ . + # 6DXHUODHQGHU $* $DUDX 6FKZHL]  >@ % >@ :


 6 76 : 
 

>@ .




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