manipulation of cooling water flow rates in a cooling ... · 169 hwahak konghak vol. 41, no. 2,...

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167 HWAHAK KONGHAK Vol. 41, No. 2, April, 2003, pp. 167-173 * 449-701 1 * 608-739 100 (2002 9 5 , 2003 2 15 ) Manipulation of Cooling Water Flow Rates in a Cooling Water Heat Exchangers Network Jin-Kyeong Kim, Gyeongbeom Yi* and Bomsock Lee Department of Chemical Engineering, Kyunghee University, Seocheonli 1, Giheung-eup, Yongin 449-701, Korea *Department of Chemical Engineering, Pukyung University, San 100, Yongdang-dong, Nam-gu, Busan 608-739, Korea (Received 5 September 2002; accepted 15 February 2003) . , , . . , GAMS/ MINOS . Abstract - In a large, complex cooling water heat exchangers network, cooling water flow rates are manipulated by adjust- ing the valves located at the rear of each heat exchanger. Since the heat exchangers are connected to a network, the manipu- lation of the cooling water flow rate in a heat exchanger by adjusting the valve results in the changes of the cooling water flow rates in the other heat exchangers. Therefore, the manipulation techniques of cooling water flow rates in a cooling water heat exchangers network are required in order to operate the cooling water networks efficiently. This paper presents the analysis and the manipulation method of the cooling water flow rates in all heat exchangers in a network simultaneously. A nonlinear pro- gramming(NLP) formulation is proposed to determine the optimal adjustment of the valves for the required flow rates of the cooling water in the heat exchangers. Solution of this formulation is obtained with a commercial NLP solver (GAMS/MINOS). Key words: GAMS/MINOS, Optimization, Heat Exchanger Network Cooling Water Flow Rates 1. (cooling water network) . ( ) . . . . , [11], . . To whom correspondence should be addressed. E-mail: [email protected]

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HWAHAK KONGHAK Vol. 41, No. 2, April, 2003, pp. 167-173

��� � ��� � �� ��� �� ��

�������*����†

����� �����449-701 � � � ��� 1��

*����� �����608-739 ��� �� �� � 100

(2002� 9� 5� ��, 2003� 2� 15� � )

Manipulation of Cooling Water Flow Rates in a Cooling Water Heat Exchangers Network

Jin-Kyeong Kim, Gyeongbeom Yi* and Bomsock Lee†

Department of Chemical Engineering, Kyunghee University, Seocheonli 1, Giheung-eup, Yongin 449-701, Korea*Department of Chemical Engineering, Pukyung University, San 100, Yongdang-dong, Nam-gu, Busan 608-739, Korea

(Received 5 September 2002; accepted 15 February 2003)

� �

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C 01�� RS� TU< VW�X3. � � ��� YI�� ��� YI '< Z[L\� RS� *+ ,- �.6

!��� =�> ]^_ RS� 4`< Z�a$%, 4`� RS� /6 I�� =�> RS� b cCdef GAMS/

MINOS6 � �X3.

Abstract − In a large, complex cooling water heat exchangers network, cooling water flow rates are manipulated by adjust-

ing the valves located at the rear of each heat exchanger. Since the heat exchangers are connected to a network, the manipu-

lation of the cooling water flow rate in a heat exchanger by adjusting the valve results in the changes of the cooling water flow

rates in the other heat exchangers. Therefore, the manipulation techniques of cooling water flow rates in a cooling water heat

exchangers network are required in order to operate the cooling water networks efficiently. This paper presents the analysis and

the manipulation method of the cooling water flow rates in all heat exchangers in a network simultaneously. A nonlinear pro-

gramming(NLP) formulation is proposed to determine the optimal adjustment of the valves for the required flow rates of the

cooling water in the heat exchangers. Solution of this formulation is obtained with a commercial NLP solver (GAMS/MINOS).

Key words: GAMS/MINOS, Optimization, Heat Exchanger Network� Cooling Water Flow Rates

1. � �

����� ���� � � ��� ���� ��� �� �

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) %* +) #$, � ����� -./ ��� � ��� 0

(cooling water network)1 2) ,3. ���� ��� �� � ��

��� ��� �4 5(6� ��� ��/ ��� 5)� 78 �

��9 � %&': /3. ;<� => ���� �� ��%-�

?�@ �� A9%� BC� ��, � ��� 01 D�E � %

� ���� ��� � ����� �4 5F GH I� 5�� J

K) ,3. </ BC ��� � ���� JK� ��� 51 L

M�� N� OP � ���9 ��� 0�� #$%& ���� �

(� QR� 3S OP � ���� JK� ��� 5� T�� 9'

U) ,3. 78� � � ���� .�E ��/ 5VW ���

51 LM�� N� XC &Y@ �3. B>�, 6� �Z�[ �

OP � ���� 5\� ]^�E � � ������ ��� 5

1 _��� N� `9a�� QR�[11], ��� � ��� 01 JK

� �bbc��� ��� 5F de1 fg�� N ��� 5

LM� h ij Lk ,3. � lm� ��� � ��� 01 JK

� ���� �n o=1 �p���� \q�E: /3. r� fq s†To whom correspondence should be addressed.E-mail: [email protected]

167

168 �����������

tuv��� 5 fg� 9 w xy%� =z{��� Hazen-

Williams {1 � � |�} {1 xy�E de ~� �m� J

K� !"��� ��� J�1 �) ��� � |3[3].

� #4��� ��� � ��� 0�� n � b� !"9 #$

%� H�1 node� ����3. n node �1 JK� ���� 5�

n nodes �� de ~� !"� �BF �� �m� 4� � |

�}, Q xy%� =z{ Hazen-Williams {3. �/ ��� �

�"��� ���� � (-� ����� ���9 �'�9�

node��� ��� ��� .m�� ��� r� �H{1 �Y�E

: /3. ��� � ��� 0 �� � � ���� JK� ���

51 LM�� lm�� � � ��� ��� �%&' |� ���

L��E: /3. Q OP � ���9 ��� 0�� #$%& ��

�� � (� QR� ��� � ���� ./ ��� 51 LM�

� l�E ; � ��� ��� |� ��� L�� BC 3S OP �

���� JK� ���� 5� ��1  ) ,3. ;<¡� OP �

���� JK� ���� 51 � � ���� .�E ��/ ��

� 5� G�¢ LM�� lm�� OP � ���� ./ ����

J�1 £¤� �Y�E: /3.

� ��� ��� |� ��� ¥¦ ��� �m� � ���� JK

� ���� 51 LM� � |3. ��� ¥¦ ��� ;� m§%

� !"� �9 �� �¨© � |3[8]. �/ Hazen-Williams {

�� !"� �9 �� ��� T�� xy%&H*, ;Q xy%

� � ��� ��� !" �� => !"� �� �� �9

�� ª�� �¨�& xy�) ,3.

��� � ��� 01 JK� ��� 51 LM�� l�E � #

4�� �«/ ¬�� O­� LM T�� � � ��� �� !"�

�9 %* � ��� �%&® ��� ¥¦ ��� �¨�� T

�� ��/3. ¯�°�� � � ������ ��� �4 5F \

q, 5F� ~�1 ¬±��� N3. >²Lk� � node���

r� �H{�F � node x� JK� ��� 51 ���� Hazen-

Williams {*, � >² Lk{�� ³´µ¡�, ¬�� O­�

m� 4�� l�E ¬�� by "�;¶[ GAMS/MINOS[1]� xy

��3.

��� � ��� 0 O­� ¬�� m��� � � ���� ��/

��� 51 On V·�� � ��� ��� !" �9 �� 4

� � |3. E�� $�%&® !"� �9 �� ¸ � � ���

��� �, ��� ¥¦ ��� ���¡�, $¹ � � ���� �

�/ ��� 51 LM� � |� � ��� ��� ��� ¥¦ �

�� $�� � |3.

2. ��� � � � ��� ��

2-1. ����

!"� JK� º� 5F deF� »\� �¨�� w� =z

{ ¼�� ½¾ 51 deF !"� �� �B�� �¤/ Hazen-

Williams {� �¿ÀF �Á°�� [m XC y�) xy,3[4,

7, 9]. � ��� 01 4À�� |� !"� �BF ��� �) Â

� |�¡�, � #4��� ���� J�1 ���� l�E ÃÄ {

(1)� ��%&H� Hazen-Williams {1 y��3.

(1)

E�� ∆H� º9 JK� n H��� �n ~(6� de~)[m]

� �¨�* D� L� �� !"� �BF �[m]� �¨Å3. \

q%&® 5 Q� [m3/s]� Æl� Ç�3. Hazen-Williams {��

-È� \�[ CH� »� É�, xy #/� 78 �� 3S Ê1 Ç

�3. Table 1��� Hazen-Williams {�� xy%� -� \�[

CH� Ë 9H Ê�1 �¨�Ì3.

� ��� 0� OP node� .�E ��� � 5F � 5�

͵1 2&: /3. � r� �H{� �m ��Π� |�}, n

�� node� 2&® 0� .�E node j�� node i� JK� ���

� J�� .�E node i�� r� �H{� ÃÄ� { (2)� Ï ��

,3.

(2)

E�� Qji� node j�� node i� JK� ���� 51 �¨�*

(+) �Ð� node j�� node i� �Ñ1 Ò��, (-) �Ð� node i�

� node j� �Ñ1 Ò/3. Qext,i� node i�� Ó���� � �

� �%� 5(Ô: ��� � �"���� �, ������

� �� � ���� ���[ blow down)1 ��/3. �/ Qext,i

� �� BC (+) Ê1, �[ BC (-)Ê�� ��,3. ;p� node

i� #$%H I� node j� .m� Qji� 0 ,3.

Õ�� ]Ö/ Hazen-Williams {� � node �1 JK� ���

51 Ô_�� }� xy,3. n�� node� 2&® � ��� 0�

node i��� r� �H{� { (1)1 { (2)� .��× 3Ø { (3)��

�¨�& ®3.

(3)

E�� Hj� Hi� node j� i�� �n(6� de)[m]� �¨�* ;

� node� � ڰ/3. Dji� Lji� �� node j� i� #$��

!"� �BF �[m]� �¨Å3.

n�� node� ./ � ³´µ #Û Ü�{� ¬�� O­� >²

Lk{ ,3.

2-2. ����

� ��� 0�� => � � ���� JK� ���� 5� �

� ����� �4�� ��� 5� ¬./ Ýx��¢ LM%&:

/3. ;<¡� ¬�� O­� ¯�°�� � � ����� ��/ �

�� 5F ��� O­� �m \q, 5F� U~� ¬±���

N ,3. { (4)� � #4�� �«, ¬�� O­� ¯�°�� �

¨Å3.

Minimize (4)

Q CHD2.63 H∆6.819L-----------------

0.54

=

Qjij 1=

n

∑ Qext i,+ 0 i 1 … n,,=( )=

CHDji2.63 H j Hi–

6.819Lji

--------------------- 0.54

j 1=

n

∑ Qext i,+ 0 i 1… n,,=( )=

Qjid Qji

c–

Qjid

--------------------j i,( ) S∈∑

Table 1. Values of roughness coefficient for Hazen-Williams equation[10]

Pipe material CH

Asbestos cement 140Ductile iron

Cement lined 130 to 150New, unlined 1305-year-old, unlined 12020-year-old, unlined 100

Concrete 130Copper 130 to 140Plastic 140 to 150New welded steel 120New riveted steel 110

���� �41� �2� 2003� 4�

� � � � �� ��� � ��� �� 169

E�� ު S� node j� node i9 �� � ���� �4� �4�

�¨�� BC� ��/3. Q node j� node i� ß� !"� �

���� ���* ; �9 �� � ��� ]\ ¤� $�, ��

� �4 5 Qijd, (j, i)àSF deá� ]\Ê ;p� � ��� ��

�� ���9 DF�� � �B1 Hazen-Williams {� �

.��E 4/ !"� � Ê1 xy/3. � ./ \q Ô� Ô

>1�� ]Ö��3. �/, Qijc, (j, i)àS� ��� O­� mg1 D�

E \q, � � ���� JK� ���� 51 ��/3.

2-3. ��

� #4� ¬�� O­�� LMT�� xy%� N� � � ���

� JK� ���� 51 LM� � |� � ��� ��� �,

��� ¥¦ ��3. ��� ¥¦ ��� ��� äå� 78 3æ/

!" �9 �� �¨© � |3[5, 8]. � #4� Ô>� xy, �

��� ��� �, ��� )t ��8� � Q ��9 �, !

"� �BF �� ¥¦ ��� 7S �9 �� Table 2� �¨�Ì

3. >² Lk1 On V·�×� ¯�°�� Ê1 ¬±��� ¬��

m�� LMT�� Ê�� ��� ¥¦ ��� �¨�� �9 ��

4mH) ,3. ¬�� m�� 4/ � ��� ��� !" ��

Table 2� y/ ��� ¥¦ ��� 4°��ç � � ���� JK

� ��� �4 5� G� � ��� ��� �, ��� ¥¦ �

�� $�� � |) ,3.

2-4. NLP ��� �

Õ� ]Ö/ >² Lk{F ¯�°�� Lª/ ¬�� O­� 3ØF

Ï ��� � |3.

Minimize (5)

Subject to (6)

(7)

E�� Þª (j, i)àS� node j� node i9 � ���� �4� �4�

�¨�� BC� ��/3. l ¬�� O­�� b��  &H� Ê�

� ¯�°��� è[ � � ���� ��� �4 5 Qijd, (j, i)àSF

� ��� 01 4À�� !"� �B Dji� ��9 �, � ��

� �� !"� >Ó/ OP !"� � Lji*, Q \q, �

�� 51 ���9 JK� ���� �� |� !"� Æ×��

� �é× ���� ´ê�9 4mH) ,3. => Lë��� ��

�� �� !"� JK� º� ´ê�� ��� ìí1 l�E �

��[ @îLk(Ô: 1-3 m/s)1 Hï: �¡� � ./ >²Lk1

¬�� O­� A9/3× 3ØF Ï� Lk{1 A9� � |3. ;

Ó� b��  &H� Ê�� ���1 �¨�� node� �n(Fig. 1

�� H40) ;p� ��� � �"��� � %� ð ��� 5

(Fig. 1�� Qext,1)3.

(8)

l ³´µ ¬�� O­� ¬�� m� bëy ¬�� "�;¶[

GAMS/MINOS 5.3[1]� y�E 4� � |3. ¬�� m� 4°��

ç Ê $�%&H� T��� � � ���� DF�� \q, ��

� 5 Qijd, (j, i)àSF ���1 >Ó/ � node�� �n Hi(i=1,...,n-

1)� ����� �%� ��� 5(Fig. 1�� Qext,40) ;p� � �

�� �� !"� �3. ¬�� m�� 4/ � ��� �� !

"� �� !"� => �� � ��� ��� �, ���

�9 �� ª/ �¡� ¬�� m�� 4/ ��� � ���

�� !"� => �� ñ Ê ��� �9 �3. +) 4

Qjid Qji

c–

Qjid

--------------------j i,( ) S∈∑

CHDji2.63 H j Hi–

6.819Lji

--------------------- 0.54

j 1=

n

∑ Qext i,+ 0 i 1… n,,=( )=

Qjic CHDji

2.63 Hj H i–6.819Lji

--------------------- 0.54

i j,( ) S⊂=

vmin

Qjic

π4---D ji

2---------- vmin i j,( ) S⊂≤ ≤

Table 2. Equivalent lengths of gate valve for degrees of valve closing[8]

Equivalent length of gate valve, Lv [m]

Degree of valveclosing000

Diameter D [m]

75%closed

50%closed

25%closed

0%closed

0.0254 21.34 5.18 1.07 0.180.0318 27.43 6.71 1.37 0.240.0889 76.20 18.29 3.66 0.610.1524 128.02 30.48 6.10 0.98

Fig. 1. The heat exchangers network for example 1.

HWAHAK KONGHAK Vol. 41, No. 2, April, 2003

170 �����������

ized

/ ��� �9 �� Table 2� y�E ��� ¥¦ ��� 4°�

�ç � � ���� �4%� ��� 51 G� � |� � ���

¥¦ ��� $�� � |) ,3. � ��� 0��� ��� 5 L

M� ./ => xy Ô>� 3Ø �� ]Ö��3.

4. �� ��� ��

4-1. �� 1

Fig. 1F Ï 10�� � ���9 ��� 0�� #$%& ����

� (� ��� � ��� 0 |3. 01 4À�� »� 5ò £

ó xy,  ô»3. Table 1� �m Hazen-Williams {� -È� \

�� 1201 xy��� /3. ��� � �"(node 1)� D�E ��

� � ��� 0� �%� ���� 5� Qext,1=0.0891 m3/s� �

��*, ���9 �%� ���1 �¨�� node 40��� �n�

õb H40=103 m9 H,3. � � ���� ���� )t ��9

��%& |3. ö� bc�� �� ¥¦ ��� 25% closed 8� 9�

��3. ¬�� O­�� LM T�� xy%� �� ¥¦ ÷l� =>

��� xy%� 25-75% closed� /���3. Õ� ]Ö/ ø� Ï

� � ���� �4%� ��� �4 5(Qijd, (j, i)àS)F � ��

�� DF�� ���� deá� ]\Ê(∆Hji , (j, i)àS), ;p� ��

�9 DF�� � ��� â�� �B(Dji , (j, i)àS)1 Hazen-Williams

{� ã�� .��E � � ���� m§%� !" �9 � (Lji , (j, i)

àS)� 4�E Table 3� �¨�Ì3.

Table 4��� � � ��� ��� ��� On 25% ¥� ö� b

c� BC� .�E Õ� ]Ö/ ¬�� O­� >/ Lk{ (6)F (7)

1 xy�E � � ���� JK� ���� 51 \q�E �¨�

Ì3. ö� bc�� \q, ��� 5� � ������ �4��

��� �4 5F b§/ U~� ùú1 Table 4�� û � |3. �

� ���� JK� ��� 5� ./ U~� ¬±��� l�E �

#4�� �«/ ¬�� O­1 ¬�� bëy "�;¶[ GAMS/

MINOS�� ü& ¬�� m� 4��3.

¬�� m�� 4/ � ��� �� !"� �� !"� =>

�� � ��� ��� �, ��� �9 �� ª/ �¡�

¬�� m�� 4/ ��� � ��� �� !"� => �� ñ

Ê ��� �9 �3. +) 4/ ��� �9 �� Table 2

� y�E 4/ ��� ¥¦ ��9 25-75% x� �H I� BC

�� m§%� � ��� �� !"� �B1 �º�E ¬�� m�

3¤ 4�� Üý1 xy��3. � ��� ��� !"� �º�E

��� ¥¦ ��9 => @î ÷l[ 25-75% x� On � QþH

�ÿ��� ¬�� m� 4°��ç � � ���� �4%� ���

51 LM� � |� � ��� ¥¦ ��� $�� � |) %Ì3.

; $F� Table 5� �¨�Ì3.

Table 6��� �º, � ��� �� !"� �BF $�, ��

� ¥¦ ��� 7S � � ���� DF�� \q, ��� 5F

� � ���9 ��� �� ��� �4 5F� ~� �¨�Ì3.

¬�� O­� 4/ \q, ���� 5� � � ����� ���

�� ��� �4 5F� ~9 �� �) �±/ N1  � |3.

� Table 5� �¨� |� � ��� �� !" �B� �º� ��

� ¥¦ ��� @î� BC � ��� 0� #$, 10�� � ���

� .�E � � ���� JK� ���� 51 ��� �4 5�

XC Ý�¤� � |Ø1  � |3.

Table 3. Equivalent length and desired flow rate for each heat exchangerof example 1

Heat exchanger

Desired flow rate Qji

d [m3/s]Pressure drop

Hji [m]Tube size Dji [m]

Equivalent length Lji [m]

A(node 11-12) 0.0019 3.20 0.0508 081.0B(node 13-14) 0.0230 13.860 0.1524 734.0C(node 15-16) 0.0205 2.67 0.1524 175.0D(node 17-18) 0.0205 6.39 0.1524 419.0E(node 19-20) 0.0014 6.06 0.0508 270.0F(node 21-22) 0.0010 3.25 0.0508 270.0G(node 23-24) 10.00080 6.66 0.0381 206.0H(node 25-26) .0.0078. 7.10 0.1016 386.0I(node 27-28) 0.0100 9.53 0.1016 327.0J(node 29-30) 10.00220 1.89 0.0381 009.0

Table 4. Initial state(each valve 25% closed) of example 1

Heat exchanger

Valve stateDesired flow rate Qd [m3/s]

Calculated flow rate Qc

[m3/s]

Error [%]

A 25% closed 0.0019 10.00137 27.9B 25% closed 0.0230 0.0198 13.9C 25% closed 0.0205 00.03036 48.1D 25% closed 0.0205 00.01896 17.5E 25% closed 0.0014 10.00115 17.9F 25% closed 0.0010 10.00075 25.0G 25% closed 10.00080 10.00071 11.3H 25% closed .0.0078. 10.00583 25.3I 25% closed 0.0100 10.00699 30.1J 25% closed 10.00220 0.0.00319 45.0

Table 5. The degrees of valve closing after adjustment for example 1

Node Pipe diameter*

D [m]Le [m] L v [m] Valve adjustment

from to

12 39 0.0254 22.70 111.07 29% closed14 35 0.0889 164.20 139.27 64% closed16 35 *0.0889* 113.72 145.79 66% closed18 37 0.0889 116.73 158.80 70% closed20 34 0.0318 34.70 125.07 73% closed22 33 0.3048 267.69 256.06 74% closed24 32 *0.0254* 21.41 114.70 49% closed26 31 0.0762 33.69 118.39 56% closed28 31 0.3810 359.28 337.98 75% closed30 36 *0.0381* 126.08 130.83 73% closed

*Changed diameter: new pipe diameter when the model was not optimwith the old pipe diameter

Table 6. The optimized results for example 1

Heat exchanger

Desired flow rate Qd [m3/s]

Calculated flow rate after adjustment Qc [m3/s]

Error [%]

A 0.0019 0.0019 0B 0.0230 0.0230 0C 0.0205 0.0205 0D 0.0205 0.0205 0E 0.0014 0.0014 0F 0.0010 00.00105 5.0G 10.00080 0.0008 0H .0.0078. 0.0078 0I 0.0100 0.00995 0.5J 10.00220 0.0022 0

���� �41� �2� 2003� 4�

� � � � �� ��� � ��� �� 171

4-2. �� 2

Ô> 2��� Fig. 2� ùE® ø� Ï 30�� � ���� 120�

� node9 ÿ��) #$, � ��� 0� .�E �ù�� /3.

� ��� 01 #$�� !"� É�� ?  ô»��, Table 1��

� Hazen-Williams -� \�� 1301 xy��� /3. � ���

���� ��� 51 LM��} xy%� ��� )t ���,

ö� bc�� �� ¥¦ ��� 25% closed* L� ÷l� 25-60%

closed� /3. ¬�� m� �m $�%� �� ¥¦ ��9 V² L�

÷l[ 25-60% closed x� &�� BC�� � ��� �� !

"� �º¤ï L� ÷l �� Ê�� $�%�¢ ¬�� R>� m�

%ü�E 4/3. îº � ��� 5� Qext,1=11,852.0 m3/s� �

��*, n �� ���1 D�E ���9 �,3. Q node 118�

��� I� �n9 H118=103 m� ���* node 120� ��� II� �

n� H120=94 m� ���3. � � ���9 ��� �� ��� �4

5, de á� ]\Ê, ���9 DF�� ���� â�� �B, ;

p� � Ê1 { (1)� ã�� .��E 4/ � ����� �9 �

� Table 7� �¨�Ì3.

�� ö� bc�� \q, ���� 5F �4 5F� ~9

& ��[H !�m ù�¢ /3. Table 8� Ô> 2� � ��� 0

� |� OP ���1 25% closed bc�� >² Lk{ (6)F (7)

1 xy�E 4/ � � ���� JK� ���� 51 �¨�Ì

3. Table 8� �¨� ø� Ï \q, ��� 5F �4 5F

� U~9 10%� �� � ���9 3� �É/3. � #4�� �

«/ ¬�� O­� ¬�� m� 4°��ç � ��� �� !"�

�9 �� 4��� Ê�� !"� => �� �× ��� ¥

¦ ��� $��� �9 �� 4� � |3. �ç 4/ ��� �

9 �� Table 2� y�× ��� ¥¦ ��� 4� � |�} �

Table 9� �¨�Ì3. �/ Table 9�� ùEH� Ê�� � ��� �

� !"� �B1 �º�� �� ¥¦ ��� @î��1 Q, � � �

��� JK� ���� 51 Table 10� �¨�Ì3. Table 10� $

F� �ù× � � ���� JK� \q, ��� 5F �4 5

Fig. 2. The heat exchangers network for example 2.

Table 7. Equivalent length and desired flow rate for each heat exchangerof example 2

Heat exchanger

Equivalent lengthLji [m]

Tube sizeDji [m]

Desired flow rateQd

ji [m3/s]

H1 206 0.0387 20.0H2 206 0.0387 13.0H3 206 0.0387 16.0H4 206 0.0387 18.0H5 206 0.0387 10.0H6 206 0.0387 16.0H7 320 0.1524 85.0H8 235 0.4318 1975.0H9 73 0.1524 166.0H10 206 0.0387 10.0H11 206 0.0387 8.0H12 206 0.0387 15.0H13 206 0.0387 15.0H14 235 0.4318 1975.0H15 52 0.0762 24.0H16 54 0.0387 20.0H17 54 0.0387 13.0H18 54 0.0387 16.0H19 54 0.0387 18.0H20 54 0.0387 10.0H21 54 0.0387 16.0H22 270 0.0504 85.0H23 75 0.2032 1975.0H24 749 1.2954 166.0H25 749 1.6510 10.0H26 182 0.1524 8.0H27 184 0.1524 15.0H28 289 0.3048 15.0H29 189 0.3048 1975.0H30 838 1.0161 24.0

HWAHAK KONGHAK Vol. 41, No. 2, April, 2003

172 �����������

F� U~� On 10% �� �&�ÌØ1  � |3.

4-3. ��

H�þH E< �� � ���� 2&® � ��� 01 JK� �

�� 51 LM�� l/ ¬�� Üý1 n �� Ô>� �& ]Ö�

�3. n Ô>�� Ô� xy/ � ��� 0� => ����� x

y%� � ��� 0� ��f1 «�/ N��� !"� �B, �

;p� � � ����� ��� �� ��� �4 5 � #$ µc

�� £�/ Ê1 xy��3. ;<� => �� ��� xy%� �

�� � ��� 0� 100�� 9þ@ � ���� 2&® -./ 0

��� � �R�� Ô� P � ��� 0� ³m� ���) v3. Ô

> 1��� 10�� � ���� 2&® 3± �Æ/ � ��� 0�

�� ¬�� m� 4°��ç �&® � � ��� ��� 5� �

������ ��� �� ��� �4 5F -� �^�� Ê(Table

6)1 �1 � |Ì3. 3V Table 5� |� �� ¥¦Ê1 ù× 70%

b ��� ¥� bc� @î�� � ���9 10� ¼ 5�� ~H��

} => ��� ��� � Ï F��) ¥� bc�� @î��

N� y[%�� �HV ø���3� û �� �1 N3. � ¤�

�� lm�� F��) w ¥� ��9 �, � ��� �� !

"� �B1 �º�� Üó >¤Î � |3. +) �Y× �º, !

" �B Ê1 ¬�� O­� .��E ¬�� O­1 3¤ ü&: �

� i-��1 &:/3.

20�� ����� 2&® 3± ÿ�/ 4L� ���� 01 �¨

�� Ô> 2��� ��� ¥¦ ��� ���� @î ÷l� ù3

9!) 60%� /�H� ÷l ��� ¬�� m� 4��� � ���

��� ��� �� ��� �4 5F LM, ��� 5F� U~

� On 10% �� ��¢ 3ØF Ï� Lk{1 A9��3.

(9)

Table 10� $F� ù× � � ���� JK� LM, ����

5 �4 5ÊF On 10% U~ ÷l �� "� V·/ Ê�� L

M, N1  � |3. U~ ÷l� 10%ù3 # �� ¬�� m� 4

�� lm�� { (9)� �¨�&H� � ���� ��� 5� U~

$y÷l� #% �) �× 9am ®3. 3V ¬�� R>� m� 4

�� lm�� \q ¤� w &9�) ,3. => ��� xy%

� -./ � ��� 0� BC(���� ��=100�) ��� 5�

LMÊ1 �4 5F `F 10% U~ ÷l �� GA� lm�� ¬�

� m� 4�� l/ '(� \q ¤�� ���) ±�,3.

� �R�� xy, ���� 5F de á�� �¨�� =z{[

Hazen-Williams { )º9 Ç� |� U~� ¬�� O­� m� 4�

E® ��� 5� U~� [m � ����� ��� �� ����

5� à �¿�) � %H� I1 N3. �*/ ��� 5�

LM ��/ BC�� � �R�� >¤�� ¬�� Üý�� $y

U~ ÷l��� ��� 51 LM�� => Lë ��� +,,

Qjid Qji

c–

Qjid

-------------------- 0.1 j i,( ) S⊂<

Table 8. Initial state(each valve 25% closed) of example 2

Heat exchanger

Valve state

Desired flow rateQd [m3/s]

Calculated flow rate Qc

[m3/s]

Error [%]

H1 25% closed 20.0 20.3 1.5H2 25% closed 13.0 14.9 14.6H3 25% closed 16.0 14.9 6.9H4 25% closed 18.0 18.2 1.1H5 25% closed 10.0 10.0 0H6 25% closed 16.0 14.9 6.9H7 25% closed 85.0 50.9 40.1H8 25% closed 1975.0 1963.3 0.6H9 25% closed 166.0 173.5 4.5H10 25% closed 10.0 10.3 3.0H11 25% closed 8.0 7.7 3.8H12 25% closed 15.0 12.8 14.7H13 25% closed 15.0 12.5 16.7H14 25% closed 1975.0 2281.8 15.5H15 25% closed 24.0 16.2 32.5H16 25% closed 13.0 11.2 13.8H17 25% closed 13.0 11.1 14.6H18 25% closed 15.0 20.0 33.3H19 25% closed 13.0 11.1 14.6H20 25% closed 12.0 11.0 8.3H21 25% closed 13.0 14.5 11.5H22 25% closed 15.0 16.5 10.0H23 25% closed 310.0 290.2 6.4H24 25% closed 2100.0 2005.8 4.5H25 25% closed 2450.0 2300.2 6.1H26 25% closed 170.0 198.2 16.6H27 25% closed 240.0 237.4 1.1H28 25% closed 651.0 797.9 22.6H29 25% closed 915.0 816.1 10.8H30 25% closed 556.0 539.6 2.9

Table 9. The degrees of valve closing after adjustment for example 2

Node Changed diameterD [m]

Le [m] Lv [m] Valve adjustmentfrom to

5 9 0.0229 24.68 6.68 58% closed8 9 0.0178 18.82 5.82 60% closed12 13 0.0178 13.63 0.63 25% closed16 17 0.0203 16.63 3.63 52% closed20 21 0.0152 15.37 2.37 50% closed23 24 0.0178 13.63 0.63 25% closed30 31 0.1270 108.59 65.59 60% closed28 33 0.1651 60.10 25.10 49% closed37 38 0.0762 118.56 24.56 60% closed43 44 0.0203 11.84 4.84 56% closed48 44 0.0178 12.66 2.66 49% closed51 52 0.0635 16.24 2.24 25% closed54 52 0.1524 83.11 49.11 60% closed56 46 0.1651* 106.78 5.78 25% closed59 60 0.0635 35.68 20.68 60% closed64 65 0.0152 10.53 0.53 25% closed69 70 0.0152 10.53 0.53 25% closed73 74 0.0178* 15.67 5.67 60% closed77 78 0.0152 10.53 0.53 25% closed81 82 0.0152 10.53 0.53 25% closed85 86 0.0152 8.27 3.27 54% closed88 86 0.0203 26.46 6.46 60% closed92 93 0.0762 76.66 2.66 25% closed96 97 1.1430 99.05 46.05 28% closed99 97 0.8890 91.12 31.12 25% closed107 105 0.0635 50.68 20.68 60% closed104 105 0.0635 29.41 9.41 49% closed109 110 0.1016 75.96 32.96 60% closed113 111 0.1397 144.90 4.90 25% closed115 116 0.5461* 558.06 148.06 56% closed

*Diameter changed for the optimization

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n,

ng

-

-

Lë)9 � ����� ��, 5\(6� -�\)� ù� �*�)

��� 51 LM�E: � N3.

5. � �

� �R��� � ��� 01 JK� ��� 51 LM�� l/

¬�� Üý1 �«��3. � lm ��� � ��� 01 4À��

� !"� #$ H�1 DF�� ��� 5� ./ r� �H{1

*C� !"� JK� ��� 5F !"� �B, � ;p� d

e á�� »/ »\{��� Hazen-Williams =z{1 xy��3. �

/ � ��� ��� �%&® ��� �� ¥Ø��ç � � ���

� JK� ���� 51 LM�). ¬�� O­1 V�Ì3. ��

� � ��� 01 ���� ¬�� O­� ¬�� m� 4°��ç $

�%&® �9 �� D�E � � ��� ��� �� ¥¦ ���

$�� � |Ì�, +) $�%&® �� ¥¦ ��� ��E n �

� Ô>�� ùE® ø� Ï V·/<@ U~ ÷l ��� ���

5 LM%Ì3. � #4� $F� => ���� � ���

xy%� -./ ��� � ��� 0� .m�� �y 9a�* �

��� 51 fg�� � � ���� JK� ���� 51 LM

�� }� y�) �y� |1 N�� �.,3.

� �

�R� /¹F�ÉÆ(F>iÐ: 1999-1-307-002-3)� �f�[ É

� H0� ��E �1%Ì�¡� � �x"Û23.

����

CH : roughness coefficient in Hazen-Williams equation

D : pipe diameter [m]

H : heads [m]

L : pipe length [m]

Qd : desired cooling water flow rates for heat exchanger [m3/s]

Qc : calculated cooling water flow rates for heat exchanger [m3/s]

S : set of node connections which denotes heat exchangers

����

e : equivalent

v : valve

n : number of nodes in network

i : any node

j : any node connected to node i

ext : external flow rate into(or from) the node

����

1. Brooke, A., Kendrick, D. and Meeraus, A., GAMS: A User's Guide,

Release 2.25, The Scientific Press, Massachusetts(1992).

2. Carnahan, B., Luther, H. A. and Wilkes, J. O., Applied Numerical

Methods, John Wiley & Sons, New York, NY(1969).

3. Foust, A. S., Wenzel, L. A., Clump, C. W., Maus, L. and Anderse

L. B., Principles of Unit Operations, John Wiley & Sons, New York,

NY(1980).

4. Hammer and Mark, J., Water and Wastewater Technology, John

Wiley & Sons, New York, NY(1986).

5. Lee, H., Lee, B. and Lee, I. B., “Anaysis and Operation of Cooli

Water Flows in a Heat Exchanger Network,” Korean J. Chem. Eng.,

14(4), 257-262(1997).

6. Lee, I. B., Lee, E. S., Chung, C., Yi, G., Lee, B. and Shin, D., Chem-

ical Process Optimization, A-Jin, Seoul(1999).

7. King, H. W., Wisler, O. W. and Woodburn, G. W., Hydraulics, Rob-

ert E. Krieger, Florida(1980).

8. Ludwig, E. E., Applied Process Design for Chemical and Petro

chemical Plants, Gulf Publishing Co., Texas(1977).

9. Male, J. W. and Walski, T. M., Water Distribution Systems: A Trou-

bleshooting Manual, Lewis Publishers, Chelsea, Mich.(1990).

10. Mays, L. W. and Tung, Y. K, Hydrosystems Engineering and Man

agement, McGraw-Hill, New York, NY(1992).

11. Simon, A. L., Hydraulics, John Wiley & Sons, New York, NY(1986).

Table 10. The optimized results for example 2

Heat exchanger

Desired flow rate Qd [m3/s]

Real flow rate afteradjustment Qc [m

3/s]Error[%]

H1 20.0 20.0 0H2 13.0 13.8 6.2H3 16.0 15.8 1.3H4 18.0 18.0 0H5 10.0 10.0 0H6 16.0 15.9 0.6H7 85.0 89.8 5.6H8 1975.0 1987.5 0.6H9 166.0 175.7 5.8H10 10.0 10.0 0H11 8.0 8.0 0H12 15.0 14.0 6.7H13 15.0 13.7 8.4H14 1975.0 2014.0 2H15 24.0 24.0 0H16 13.0 12.0 7.7H17 13.0 12.0 7.7H18 15.0 15.0 0H19 13.0 12.0 7.7H20 12.0 12.0 0H21 13.0 13.1 0.8H22 15.0 15.7 4.7H23 310.0 309.8 0.1H24 2100.0 2128.4 1.4H25 2450.0 2450.9 0H26 170.0 183.2 7.8H27 240.0 240.5 0.2H28 651.0 664.0 2H29 915.0 876.1 4.3H30 556.0 567.2 2

HWAHAK KONGHAK Vol. 41, No. 2, April, 2003