Appendix C. Minimization of the Nonlinear Cost Equation
(2.C-2) where T0Src is the exergy dissipated in region r and ir is the exergy
stored in that region.
Finally, for convenience, we define an exergy transport ibl at boundary region b:
£bl = ib* + s\w + <?lM + tiT + <?ic + <?bd. (2.B-14) Substitution of (2.B-14) into (2.B-13) yields the following convenient form of the general exergy balance:
D^V = X(^ + ^ ) . (2.B-15)
b r
For a steady-state system, £r = 0 , and the general exergy balance reduces to
2 ) ^ = 2 (2.B-16)
b r
Thermoeconomic Considerations of Sea Water Demineralization 71 Hence from (2.C-1) and (2.C-2) the minimum cost for the linear case is
J m i n = 2 V A B + C . (2.C-3)
Equations (2.22) and (2.23) are examples of (2.C-2) and (2.C-3).
Equation (2.C-2) may be extended to account for the nonlinear case merely by inserting a factor / (which is solved for below) as follows:
* oPt = / ^ / £ , (2.C-4)
from which the minimum cost for (2.C-1) in the nonlinear case becomes
ym Xn = 2F V A B + C , (2.C-5)
where
1 /l
( 7 + / ) . (2.C-5a) 2\f
Equations (2.C-4) and (2.C-5) reduce to the linear case, (C-2) and (2.C-3), w h e n F = / = 1.
W e will first solve for the factor / in the case where A = A(x) and Β = B ( # ) , while C is constant with respect to x. For this case, differen
tiation of (2.C-1) [the differentiability of A = Α(Λ?) and Β = B(x) being discussed below] yields
dy = d(A/x) d(Bx) „
dx dx dx
— ? ( - ^ ) + · ( τ ^ ) · <2 C-6' >
Setting the derivative dyjdx equal to zero in (2.C-6') and solving for the value xo pt of x, which thereby minimizes y , one obtains (2.C-4) with / given by the following expression:
' - " k - ( 2'c-7 )
where
Ah where
or, alternatively,
where
| = - ^ + B / „ (2.C-8-)
Setting the derivative dy/dx equal to zero in (2.C-8") and solving for
^opt yields (2.C-4) with the following expression for / [ i t being observed that (2.C-8'") is redundant when dy/dx = 0 since fc' = l//c in this case]:
( 2-c-9 )
where
fc=4r (Jc 2C"9a)
W e will adhere to the convention of positive root extraction throughout all the equations of this appendix, so that / is thereby a positive number (only in abnormal cases involving negative costs or conductances can the radicals ever be negative, so that / will never be imaginary under normal circumstances). It follows that the minimum value ofF= J(l// + / ) m (2.C-5a) is unity. Hence the linear expression, ( 2 . C - 3 ) , provides a lower bound cost in the nonlinear case.
In the case where C also varies differentiably with x, (2.C-6') is extended to read
which from (2.C-7a) and (2.C-7b) is
(2.C-8)
(2.C-8')
(2.C-8")
(2.C-8"a)
Thermoeconomic Considerations of Sea Water Demineralization 73 The factors fA and fB are seen from (2.C-7a) and (2.C-7b) to be merely the geometric slopes of plots on log-log paper (i.e., fA is the negative slope of a plot of A/x vs. x, while fB is the slope of a plot of Bx vs. x). The factor fc in (2.C-9a) is obtained by measuring the slope of C versus either χ or I fx on ordinary graph paper and solving either (2.C-8"a) or (2.C-8"'a). Having prepared these graphs, (2.C-4) is then solved by the iterative procedure of first setting / = 1 and solving for
* oPt t 0 y i ^ d a new value of / in (2.C-9). This new value of / is then used in (2.C-4) and the process is repeated, this procedure being continued until (2.C-9) yields the same value of / w h i c h was previously used in (2.C-4). It is to be observed that C is to be read from the graph of C vs. χ or l/x [i.e., C = C ( *O P F C) ] in the final minimum cost expres
sion, (2.C-5).
The only mathematical restriction on the use of (2.C-4), (2.C-5), and (2.C-9) is that the lowest minimum point for y must be a point for which the function y(x) is at least once differentiate in x. If there should be more than one point where dy/dx = 0, then an additional criterion for the lowest minimum would of course be needed [such a criterion being provided by simple Taylor's expansions of (2.C-1) in the regions dy/dx = 0 ] . Since equipment tends to come in discrete sizes, it may sometimes occur that the lowest minimum is in a region where at least one of the functions A = A ( # ) , Β = B(x), or C = C(x) is not differentiate. This can never happen if these functions are determined statistically, since A(x)y B ( # ) , and C(x) would then be expectation values which by definition are always differentiate. Thus in any complete statistical treatment of thermoeconomics, (2.C-4), (2.C-5), and (2.C-9) will always be valid. In a nonstatistical treatment, the occurrence of a nondifferentiable lowest minimum point would require some smoothing operation in order to make use of (2.C-4), (2.C-5), and (2.C-9).
The subminimization of the parameters A , B , and C can usually be carried out independently whenever the factor F in (2.C-5) is not very sensitive to the variations involved. For example, the optimization of the heat exchanger on p. 41 has been found to meet this criterion. Such suboptimization is usually rendered feasible because of the convenient additive property of exergy dissipation.
ACKNOWLEDGMENT
This work was supported, in part, by a grant from the National Science Foundation for studies in the engineering applications of information theory.
Appendix A was written by Robert B. Evans.
L I S T OF S Y M B O L S
In this chapter, the writers have presented their main results in the form of relationships among dimensionless ratios. However, if desired, the units given in the following list may be used to render all water costs directly in cents per kilogallon.
Conventions Σ Summation of d Differential of
ξ Equal to by definition
> Greater than or equal to
—* Approaches or reduces to
(Placed over a symbol) means time derivative
~ (Placed over a symbol) means per unit amount of matter
— (Placed over a symbol) means "partial
— p e r unit amount of matter"
C * Cost quantity
˜ Delta—a change or difference in a function
Cp Heat capacity—constant pressure, kw-hr/kilogal-°R
˜˙ Heat of vaporization, kw-hr/kilogal / Entropy change (or information),
˝ Molal amount of matter, moles Pressure, lb/in2
U Unit thermal conductance, kw-hr/
d a y - f t2- ° R
* Denotes an economic quantity which depends upon the marketplace
Thermoeconotnic Considerations of Sea Water Demtneralization 75
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