On the Application of a Classical Fixed Point Method in the Optimization of a Multieffect Evaporator [2010 MSC: 46B25]

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RESEARCH ARTICLE
On the Application of a Classical Fixed Point Method in the Optimization of a
Multieffect Evaporator [2010 MSC: 46B25]
Eziokwu C. Emmanuel
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria
Received: 10-09-2018; Revised: 10-10-2018; Accepted: 10-01-2019
ABSTRACT
This work on classical optimization reveals the Newton’s fixed point iterative method as involved in
the computation of extrema of convex functions. Such functions must be differentiable in the Banach
space such that their solution exists in the space on application of the Newton’s optimization algorithm
and convergence to the unique point is realized. These results analytically were carried as application
into the optimization of a multieffect evaporator which reveals the feasibility of theoretical and practical
optimization of the multieffect evaporator.
Key words: Non-linear Programming, convex subsets and the continuous function, Euler–Lagrange
equation, newton’s optimization algorithm, convergence, multieffect evaporator
INTRODUCTION
The non-linear programming problem for (P) is
defined for (P) if K={v ∈ X⁄(φi
(v)) ≤ 0, 1 ≤ I ≤ m’,
φi
(v) = 0, m’+ 1 ≤ 1 ≤ m}. If i
 and J are convex
functionals, then P
( ) is called a convex
programming while P
( ) is a quadratic
programming if for X Rn
= ,
K v R v d i m
n
i
= ∈ ( ) ≤ ≤ ≤
{ }
/ ;
ϕ 1
J v Av v b v
( ) = −
1
2
, , ,
Where, A aij
= { }, an n n
× positive definite
matrix and ( )
1
n
i ij ji
j
v a v

=
= ∑ .
Definition [Peterson, J and Bayazitoglu, Y.
(1991)]: Let A be a subset of a normed space X
and f a real valued function on A . f is said to
have a local or relative minimum (maximum) at
x A
0 ∈ if there is an open sphere S x
r 0
( ) of X
such that f x f x f x f x
0 0
( )≤ ( ) ( ) ≤ ( )
( ) holds for
Address for correspondence:
Eziokwu C. Emmanuel,
E-mail: okereemm@yahoo.com
all x S x
r
∈ ( )
0 ∩ . If f has either a relative
minimum or a relative maximum at x0 , then f is
said to have a relative extremum.
Theorem 1.1 [Kaliventzeif, B (1991)]: Let
f X R
: → be a Gateaux differentiable functional
at x X
0 ∈ and f have a local extremum at x0.
Then, Df x t
0 0
( ) = for all t X
∈ .
Proof
For every t X
∈ , the function ( )
0
f x t

+ (of a
real variable function) has a local extremum at
0.
 = Since it is differentiable at 0 , it follows
from ordinary calculus that
d
d
f x t
α
α
α
0
0
0
+
( )





 =
=
This means that Df x t
0 0
( ) = for all t X
∈ which
proves the theorem.
Remark 1.1: Given a real-valued function on a
solution of P
( ) on a convex set K and if f is a
Gateaux differentiable at x0 , then
( )( )
0 0 0;
Df x x x x K
− ≥ ∀ ∈

Emmanuel: On the application of a classical fixed point method
AJMS/Jan-Mar-2019/Vol 3/Issue 1 9
Theorem 1.2 (Existence of Solution in R)
[Kaliventzeif, B. (1991)]: Let K be a non-empty
closed convex subset of Rn
and J R R
n
: → a
continuous function which is coercive if K is
unbounded. Then, there exists at least one
solution of P
( ).
Proof
Let Uk
{ } be a minimizing sequence of J; that is
a sequence satisfying conditions u K
k ∈ for every
integer k and lim
k u K
Inf J u
→∞ ∈
( ).
This sequence is necessarily bounded since the
functional J is coercive so that it is possible to
find a subsequence Uk’
{ } which converges to an
element v K
∈ ( K being closed). Since J is
continuous, J u J U Inf J v
k k
v K
( ) = ( )= ( )
→∞ ∈
lim
’
’ , which
proves the existence of a solution of P
( ).
Theorem 1.3 (Existence of Solution in Infinite
Dimensional Hilbert Space):[43]
Let K be a non-
empty convex closed subset of a separable Hilbert
space H and J H R
: → a convex, continuous
functional which is coercive if K is unbounded.
Then, (P) has at least one solution. Proof[8]
[see
A.H. Siddiqi (1993)].
MINIMIZATION OF ENERGY
FUNCTIONAL
In this section, we employ the use of classical
calculus of variation which is a special case of
P
( ) where we look for the extremum of functional
of the type
( ) ( ) ( )
' '
, , ,
b
a
du
J u F x u u dx u x
dx
 
=
 
 
∫ (2.1)
Which is twice differentiable on a b
,
[ ] and F has
a continuous partial derivative with respect to u
and u’
. Also considered is the functional
( ) ( ) ( )
1
,
2
J v v v F v

= − (2.2)
Where a .,.
( ) is a bilinear and continuous form on
a Hilbert space X and F is an element of the dual
space X *
of X which is an energy functional on
a quadratic functional.
Theorem 2.1:[29]
A necessary condition for the
functional J u
( ) to have an extremum at u is that
u must satisfy the Euler–Lagrange equation
∂
∂
−
∂
∂





 =
F
u
d
dx
F
u’
0
In a b
,
[ ] with the boundary condition in
( ) ( )
and
u a u b
 
= = .
Proof
Let u a
( ) = 0 and u b
( ) = 0, then
( ) ( )
( )
( )
'
'
, ,
, ,
b
a
F x u v u v
J u v J u dx
F u u
 


 
+ +
 
+ − =
 
−
 
∫
Using the Taylor series expansion
F x u v u v
F x u u v
F
u
v
F
u
v
F
u
v
( , , )
( , , )
!
+ ′ + ′ =
′ +
∂
∂
+
∂
∂ ′






+
∂
∂
+
 

2
2
∂
∂
∂ ′





 +
F
u
2
 (2.3)
It follows from (2.3) that
( ) ( ) ( )( )
( )( )
2
2
2!
J u v J u dJ u v
d J u v
 

+ = +
+ +… (2.4)
Where, the first and second Frechet differentials
are given by
( ) '
'
b
a
F F
dJ u v u u dx
u u
∂ ∂
 
= +
 
 
∂ ∂
∫
( )
2
2 '
'
b
a
F F
d J u v v v dx
u u
∂ ∂
 
= +
 
 
∂ ∂
∫
The necessary condition for the functional J to
have an extremum at u is that JJ u v
( ) = 0 for all
v C a b
∈ [ ]
2
, such that v a v b
( ) = ( ) = 0 that is
0 = ( ) =
∂
∂
+
∂
∂






∫
dJ u v v
F
u
v
F
u
dx
a
b
’
’
(2.5)

Integrating the second term in the integrand (2.5)
by parts, we get
a
b
a
b
F
u
d
dx
F
u
vdx v
F
u
∫
∂
∂
−
∂
∂











 +
∂
∂





 =
’ ’
0
Since v a v b
( ) = ( ) = 0 , the boundary terms vanish
and the necessary conditions become
[ ]
2
'
0 ,
b
a
F d F
vd for all v C a b
u dx u
 
∂ ∂
 
− = ∈
 
 
 
∂ ∂
 
∫
For all functions v C a b
∈ [ ]
2
, vanishing at a and b.
This is possible only if
∂
∂
−
∂
∂





 =
F
u
d
dx
F
u’
0
Thus, the desired result is achieved.
Theorem 2.2: Let ( )
.,.
 be coercive and
symmetric, and K a non-empty closed convex
subset of X . Then, P
( )for J in (2.2) has a unique
solution in K.
Proof
The bilinear form induces an inner product over
the Hilbert space equivalent to the norm induced
by the inner product of X . In fact, the equations
imply that
( )
( )
1
2
|| || , || || || ||
v v v v
  
≤ ≤
Since F is a linear continuous form with this new
norm, the Riez representation theorem exists and
has a unique element u X
∈ such that
( ) ( )
,
F v u v

= for every u X
∈ .
Hence,
( ) ( ) ( )
1
, ,
2
J v v v u v
 
= −
( ) ( )
1 1
, ,
2 2
v u v u u u
 
= − − −
= − − −
1
2
1
2
v u v u u u
, , for all v K
∈ and a
unique u.
Therefore, Infv K J v
∈ ( ) is equivalent to
Inf v u
v K
∈ −
|| ||. Thus, in the present situation, P
( )
amounts to looking for the projection x of the
element u onto subset K . Therefore, P
( ) has a
unique solution.
Optimization algorithm and the convergence
theorems[1]
The iterative method for this research is the
Newton’s method stated below. For the function
: ,
F U R R U
⊂ → the open subset of R , the
Newton’s method is
( )
( )
1 '
, 0,
k
k k
k
F u
u u k
F u
+ =
− ≥
u0 an arbitrary starting point in the open set U .
Hence, the sequence is defined by
u u F u F u
k k k k
+
−
= − ( )
{ } ( )
1
1
’
under the assumption that all the points lie in U
and if ( )
, , 0
n n
X R Y R F u
= = = is equivalent to
( ) ( )
1 1 2
0, , , , n
n
F u u u u u R
= = … ∈
F u
2 0
( ) =
F u
3 0
( ) =
F u
n ( ) = 0
Where, : , 1,2, ,
n
F R R i n
→ = …
Theorem 2.2.1 (convergence): [2-7]
Let X be a
Banach Space, U an open subset of X Y
,
normed linear space, and F U X Y
: ⊂ →
differentiable over U. Suppose that there exists
three constants , ,
   such that 0
 and
( ) { }
0 0
/
S u u X u u U
 
= ∈ − ≤ ⊆
i. Sup Sup A
k u S u k B X Y
≥ ∈ ( )
−
[ ]≤
0
1
0
α
β
|| ,
|| ,
A u A B X Y
k k
( ) = ∈ [ ]
, is bijective
ii. ( ) ( ) ( ) [ ]
'
0
'
0 ,
|| ||
1
k k B X Y
u S u
Sup Sup F x A x
and




≥ ∈
−
≤

iii. || ( ) ||
F u0 1
≤ −
( )
α
β
γ
Then, the sequence defined by
( ) ( )
'
1 '
1 , 0
k k k
k
u u A u F u k k
−
+ = − ≥ ≥
is entirely contained within the ball and converges
to a zero of F in S u
x 0
( ) which is unique.
Furthermore,
|| ||
|| ||
u u
u u
k
k
− ≤
−
−
1 0
1 γ
γ
Theorem 2.2.2 (convergence): [9-15]
Let X be a
Banach space, U an open subset of
X F U X Y
. : ⊂ → and Y a normed space.
Furthermore, let F be continuously differentiable
over U. Suppose that u is a point of U such that
( ) ( )
'
0, :
F u A F u X Y
= = → bounded linear and
bijective
[ ]
[ ]
0 , 1
,
|| ||
|| ||
k k B X Y
B X Y
Sup A A
A

≥ −
− ≤ and 1

Then, there exists a closed ball S u
r 0
( ) with center
u and radius r such that for every point
u S u
r
0 ∈ ( ), the sequence Uk
{ } defined by
( )
1
1 , 0
k k k k
u u A F u k
−
+ =
− ≥
is constrained in S u
r ( ) and converges to point u,
which is the only zero of F in the ball S u
r ( ).
Furthermore, there exists a number  such that
0
1 and || || ||,
| 0
|
k
k
u u u u k
 
− ≤ − ≥
APPLICATION TO THE OPTIMIZATION
OF A MULTIEFFECT EVAPORATOR
When a process requires an evaporation step,
the problem of evaporator design needs serious
examination. Although the subject of evaporation
and the equipment to carry out evaporation have
been studied and analyzed for many years, each
application has to receive individual attention.
No evaporation configuration and its equipment
can be picked from a stock list and be expected to
produce trouble-free operation.[16-19]
An engineer working on the selection of optimal
evaporation equipment must list what is “known,”
“unknown,” and “to be determined.” Such analysis
should at least include the following:
Known
• Production rate and analysis of product
• Feed flow rate, feed analysis, and feed
temperature
• Available utilities (steam, water, gas, etc.)
• Disposition of condensate (location) and its
purity
• Probable materials of construction.
Unknown
• Pressures, temperatures, solids, compositions,
capacities, and concentrations
• Number of evaporator effects
• Amount of vapor leaving the last effect
• Heat transfer surface.
Features to be determined
• Best type of evaporator body and heater
arrangement
• Filtering characteristics of any solid or crystals
• Equipment dimensions arrangement
• Separator elements for purity overhead vapors
• Materials, fabrication details, and
instrumentation.
Utility consumption
• Steam
• Electric power
• Water
• Air.
In multiple effect evaporation, as shown in
Figure 1a, the total capacity of the system of
evaporation is no greater than that of a single effect
evaporator having a heating surface equal to one
effect and operating under the same terminal
conditions. The amount of water vaporized per
unit surface area in n effects is roughly
1
n
that of
a single effect. Furthermore, the boiling point
elevation causes a loss of available temperature
drop in every effect, thus reducing capacity. Why
then are multiple effects often economic? It is

because the cost of an evaporator per square foot
of surface area decreases with total area (and
asymptotically becomes a constant value) so that
to achieve a given production, the cost of heat
exchangecanbebalancedwiththesteamcosts.[20-24]
Steady-state mathematical models of single and
multiple effect evaporators involving material
energy balances can be found in McCabe et al.
(1993), Yanniotis and Pilavachi (1996), and
Esplugas and Mata (1983).The classical simplified
optimization problem for evaporators (Schweyer,
1995) is to determine the most suitable number of
effects given[25-32]
(1) An analytical expression for the fixed costs in
terms of the number of effects n
(2) The steam (variable) cost also in terms of n.
Analytic differentiation yields an analytical
solution for the optimal n*
, as shown here.
Assume we are concentrating an organic salt in
the range of 0.1 to 1.0 wt% using a capacity of
0.1–10 million gallons/day. Initially, we treat the
number of stages n as a continuous variable.
Figure1b shows a single effect in the process.
Before discussions of the capital and operating
costs, we need to define the temperature driving
force for heat transfer in Figure 1c.[33,34]
By
definition the log mean temperature difference
lm
T
∆ is
∆T
T T
T T
lm
i d
i d
=
−
( )
In / (a)
Let Ti be equal to constant K for a constant
performance ratio P . Because T T
T
n
d i
f
= −
∆
∆
∆
T
T
n
K
K
T
n
lm
f
f
=
−












In
(b)
Let A = Condenser heat transfer areas ft2
cp = Liquid heat capacity,
( )( )
1.05
m
Btu
lb ℉
Cc = Cost per unit area of condenser,
6 25
2
.
ft
CE = Cost per evaporator (including partitions),
7000
stage
Cs = cost of steam, $lb at the brine heater (first
stages)
Fout = liquid flow out of evaporator,
lb
h
K = Ti , a constant (T T T
i d
= −
∆ at inlet)
n = number of stages
Figure 1a: Multiple effect evaporator with forward feed [Singiresu, S.R. (1996)]

P = Performance ratio, lb of H O
2 evaporated/Btu
supplied to brine heater
Q = heat duty, 9 5 108
. ×
Btu
h
(a constant)
qe = total 2O
H
lb evaporated/h
qr = total lb steam used/h
r = Capital recovery factor
S = lb steam supplied/h
Tb = boiling point rise, 4.3 F
°
f
T
∆ = flash down range, 250 F
°
U = overall heat transfer coefficient (assumed to
be constant),
( )( )( )
2
625Btu
ft h ℉
vap
H
∆ = heat of vaporization of water, about
1000Btu
lb
The optimum number of stages is n*
. For a
constant performance ratio, the total cost of the
evaporator is
f C n C A
E c
1 = + (c)
For A, we introduce
A
Q
U Tlm
=
( )
∆
Then, we differentiate f1 in equation (c) with
respect to n and set the resulting expression equal
to zero (Q and U are constant)
Figure 1b: Individual effect evaporator with forward feed
Figure 1c: Boiling Point effect

C C
Q
U
T
n
E c
lm
P
+
∂
∂





 =
( / )
1
0
∆
(d)
With the use of equation (b)
∂( )
∂





 =
−






−
−
( )
1 1
1
1
/ ∆
∆
∆
∆
T
n
nK
T
nK
In T
T
lm
P f
f
f
(e)
Substituting equation (e) into (d) plus introducing
the values of ,
, , ,
f E
Q U T C
∆ and Cc , we get
7000
6 25 9 5 10
625
1
1
1
8
−
( ) ×
( )








−
( )
+
−
( )
. .
/
/
nK T nk
In T nK
f
f
∆
∆
∆T
Tf








= 0
Rearranging
625 7000 250
6 25 9 5 10
0 184
250
250
1
250
8
( )( )( )
( ) ×
( )
= =
−
+ +

. .
.
nK nK
In






(f)
In practice, as the evaporation plant size changes
(for constantQ ), the ratio of the stage condenser
area cost to the unit evaporator cost remains
essentially constant so that the number 0 184
. is
treated as a constant for all practical purposes.
Equation (f) can be solved for nK for
constant P.[35-40]
nK = 590 (g)
Next, we eliminate K from equation (g) by
replacing K with a function of P so that n
becomes a function of P . The performance ratio
(with constant liquid heat capacity at 347 F
° ) is
defined as
( )( )
( ) ( )
first stage
1000
1.05 4.3
vap e e
out
out pF heater
H q q
P
K F
F C T
∆
=
+
∆

(h)
The ratio
q
F
e
can be calculated from
q
F
e
out
= −
−
−





 =
1
1194 322
1194 70
0 31
1 49
.
.
Where,
∆H Btu lb
vap 355 143 1194
°
( ) =
F psi
, /
( )
2 0 350 322 /
liq
H H Btu lb
∆ =
℉
( )
2
70
0 100
liq
Btu
H H
lb
∆ =
℉
Equations (g) and (h) can be solved together to
eliminate K and obtain the desired relation
300
4 3
590
P n
− =
. *
(i)
Equation (i) shows how the boiling point rise
(Tb = °
4 3
. F) and the number of stages affects the
performance ratio.
Optimal performance ratio
The optimal plant operation can be determined by
minimizing the total cost function, including
steam cost, with respect to P (liquid pumping
costs are negligible)
f C A C n r C S
c E s
2 = +
[ ] + (j)
rC
A
P
rC
n
P
C
S
P
c E c
=
∂
∂
+
∂
∂
+
∂
∂
= 0 (k)
The quantity for
∂
∂
A
P
can be calculated using the
equations already developed and can be expressed
in terms of a ratio of polynomials in P such as
a P
bP
1 1
1
2
+
( )
−
( )
/
Where, a and b are determined by fitting
experimental data. The relation for
∂
∂
n
P
can be
determined from equation (i). The relation for
∂
∂
S
P
can be obtained from equation (l)
P
q
Q
q
H
q
S
e e
vap
e
= = =
∆ 1000
or
S
lb
h
q
P
e





 =
1000

or
( )
( )
8760
1000
e
q
S lb
P

= (l)
Where,  is the fraction of hours per year (8760)
during which the system operates. Equation (k)
given the cost cannot be explicitly solved for P*
,
but P*
can be obtained by any effective root
finding technique.
If a more complex mathematical model is
employed to represent the evaporation process,
you must shift from analytic to numerical methods.
The material and enthalpy balances become
complicated functions of temperature (and
pressure). Usually, all of the system parameters
are specified except for the heat transfer areas in
each effect ( n unknown variables) and the vapor
temperatures in each effect excluding the last one
( n −1 unknown variables). The model introduces
n independent equations that serve as constraints,
many of which are non-linear, plus non-linear
relations among the temperatures, concentrations,
and physical properties such as enthalpy and the
heat transfer coefficient.[41-44]
The number of evaporators represents an integer-
valued variable because many engineers use tables
and graphs as well as equations for evaporator
calculations.
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