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Exercise The cost of constructing a distillation column can be written

where C = Total cost, $ , Cp = cost per square foot of plate area, $/ft2 , A = column crosssectional area, ft2 , N = number of plates , Nmin, = minimum number of plates ,C = cost of shell, $/ft3 , H = distance between plates, ft , Cf = cost of feed pump, $ , Cd = cost of distillate pump, $ , Cb = cost of bottoms pump, $ , CL = cost of reflux pump, $ , Cx = other fixed costs, $. The problem is to minimize the total cost, once produce specifications and the throughput are fixed and the product and feed pumping costs are fixed; that is, Cf, Cd, CL, and Cb, are fixed. After selection of the material of construction, the costs are determined; that is, Cp, Cs, Cx are also fixed The process variables can be related through two empirical equations:

For a certain separation and distillation column the following parameters are known to apply

The pump cost for the reflux stream can be expressed as

(a) Determine the process decision or independent variables. Which variables are dependent? (b) Find the minimum total cost and corresponding values of the variables.

Solution The problem is

C  C p AN  Cs HAN  C f  Cd  Cb  CL  Cx

Minimize: Subject to:

 L  L  1    D 1  N min / N   D min A = K (L + D) N > Nmin With numerical values, this becomes (Nmin = 5)

C  50 AN  0.7 L  22,000

Minimize:

(a)

Subject to:

L

1000 N N 5

(b)

A

L  1000 100

(c)

N>5 a.

(d)

The variables are A, L, and N. Although N is an integer we will assume it to be a continuous variable. A and L may be eliminated using equations (b) and (c) to get a cost function in terms of N only (the independent variable):

 10 N   1, 000 N  C  50 N   10   0.7    22, 000  N 5   N 5 

C

=

1, 000 N 2  20, 200 N  110, 000 N 5

Thus, N is the independent variable, and A and L are dependent variables. To obtain the minimum, Use The use Chain rule and Quotient rule

(

)(

) ( (

) )

dC 1, 000 N 2  10, 000 N  9, 000  0 dN ( N  5)2 N = 1 or N = 9 Because of constraint (d), select N = 9 (

)(

)

(

) (

)

Take common (

)⌈ (

)(

) (

(

) ⌉

)

Re arrange (

) (

(

d 2C dN 2 b.

)

)

= 500

 500  0. This is a minimum N 9

C*  $38200 , N *  9 plates, L* = 2250 lb/hr, A* = 32.5 ft2

(

)