numerical Optimisation of one and two variables

Unit - 6
OPTIMUM DESIGN AND
DESIGN STRATEGY

• An optimum design is based on the best or most
favorable conditions. In almost every case, these
optimum conditions can ultimately be reduced to a
consideration of costs or profits. Thus, an
optimum economic design could be based on
conditions giving the least cost per unit of time
or the maximum profit per unit of production.
When one design variable is changed, it is often
found that some costs increase and others
decrease. Under these conditions, the total cost
may go through a minimum at one value of the
particular design variable, and this value would be
considered as an optimum

• An example illustrating the principles of an
optimum economic design is presented in Fig.
11-1. In this simple case, the problem is to
determine the optimum thickness of insulation
for a given steam-pipe installation. As the
insulation thickness is increased, the annual
fixed costs increase, the cost of heat
loss decreases, and all other costs remain
constant. Therefore, as shown in Fig.
11-1, the sum of the costs must go through a
minimum at the optimum insulation thickness.

• INCREMENTAL COSTS
• Consideration of incremental costs shows that a final
recommended design does not need to correspond to
the optimum economic design, because the
incremental return on the added investment may
become unacceptable before the optimum point is
reached.
• However, the optimum values can be used as a basis
for starting the incremental-cost analyses. This chapter
deals with methods for determining optimum
conditions, and it is assumed that the reader
understands the role of incremental costs in
establishing a final recommended design.

INTANGIBLE AND PRACTICAL CONSIDERATIONS
• The various mathematical methods for determining
optimum conditions, as presented in this chapter, represent
on a theoretical basis the conditions that best meet the
requirements. However, factors that cannot easily be
quantitized or practical considerations may change the final
recommendation to other than the theoretically correct
optimum condition. Thus, a determination of an “optimum
condition,” as described in this chapter, serves as a base
point for a cost or design analysis, and it can often be
quantitized in specific mathematical form. From this point,
the engineer must apply judgment to take into account
other important practical factors, such as return on
investment or the fact that commercial equipment is often
available in discrete intervals of size.

• As an example, consider the case where an engineer has made
an estimation of the optimum pipe diameter necessary to
handle a given flow stream based on minimizing the costs due
to fixed charges and frictional pumping costs. The
mathematical result shows that the optimum inside pipe
diameter is 2.54 in. based on costs for standard (schedule 40
steel pipe. Nominal pipe diameters available commercially in
this range are 2; in. (ID of 2.469 in.) and 3 in. (ID of 3.069
in.). The practical engineer would probably immediately
recommend a nominal pipe diameter of 2 in. without going to
the extra effort of calculating return on investment for the
various sizes available. This approach would normally be
acceptable because of the estimations necessarily involved in
the optimum calculation and because of the fact that an
investment for pipe represents only a small portion of the total
investment.

• Intangible factors may have an effect on the degree of
faith that can be placed on calculated results for optimum
conditions. Perhaps the optimum is based on an assumed
selling price for the product from the process, or it might
be that a preliminary evaluation is involved in which the
location of the plant is not final obviously, for cases of this
type, an analysis for optimum conditions can give only a
general idea of the actual results that will be obtained in
the final plant, and it is not reasonable to go to extreme
limits of precision and accuracy in making
recommendations. Even for the case of a detailed and firm
design, intangibles, such as the final bid from various
contractors for the construction, may make it impractical
to waste a large amount of effort in bringing too many
refinements into the estimation of optimum conditions.

• GENERAL PROCEDURE FOR DETERMINING
OPTIMUM CONDITIONS
• The first step in the development of an optimum
design is to determine what factor is to be optimized.
• Typical factors would be total cost per unit of
production or per unit of time, profit, amount of
final product per unit of time, and percent
conversion.
• Once the basis is determined, it is necessary to
develop relationships showing how the different
variables involved affect the chosen factor. Finally,
these relationships are combined graphically or
analytically to give the desired optimum conditions.

• PROCEDURE WITH ONE VARIARLE
• There are many cases in which the factor being optimized is a
function of a single variable. The procedure then becomes very
simple. Consider the example presented in Fig. 11-1, where it is
necessary to obtain the insulation thickness which gives the least
total cost. The primary variable involved is the thickness of the
insulation, and relationships can be developed showing how this
variable affects all costs. Cost data for the purchase and installation
of the insulation are available, and the length of service life can be
estimated.
• Therefore, a relationship giving the effect of insulation thickness on
fixed charges can be developed.
• Similarly, a relationship showing the cost of heat lost as a function
of insulation thickness can be obtained from data on the value of
steam, properties of the insulation, and heat-transfer considerations.
• All other costs, such as maintenance and plant expenses, can be
assumed to be independent of the insulation thickness. The two cost
relationships obtained might be expressed in a simplified form
similar to the following

• The graphical method for determining the
optimum insulation thickness is shown in Fig.
11-1. The optimum thickness of insulation is
found at the minimum point on the curve
obtained by plotting total variable cost versus
insulation thickness. The slope of the total-
variable-cost curve is zero at the point of
optimum insulation thickness. Therefore, if
Eq. (3) applies, the optimum value can be
found an analytically by merely setting the
derivative of C, with respect to x equal to zero
and solving for X.

• PROCEDURE WITH TWO OR MORE
VARIABLES
When two or more independent variables
affect the factor being optimized, the
procedure for determining the optimum
conditions may become rather tedious;
however, the general approach is the same as
when only one variables involved.

GRAPHICAL PROCEDURE.
• The relationship among CT, x, and y could be shown as a curved
surface in a three-dimensional plot, with a minimum value of C,
occurring at the optimum values of x and y.
• However, the use of a three-dimensional plot is not practical for
most engineering determinations. The optimum values of x and y
in Eq. (8) can be found graphically on a two-dimensional plot by
using the method indicated in Fig. 11-2. In this figure, the factor
being optimized is plotted against one of the independent
variables (x), with the second variable (y) held at a constant
value. A series of such plots is made with each dashed curve
representing a different constant value of the second variable. As
shown in Fig. 11-2, each of the curves (A, B, C, D, and E) gives
one value of the first variable x at the point where the total cost is
a minimum. The curve NM represents the locus of all these
minimum points, and the optimum value of x and y occurs at the
minimum point on curve NM.

• Similar graphical procedures can be used when
there are more than two independent variables.
For example, if a third variable z were included
in Eq. (8), the first step would be to make a plot
similar to Fig. 11-2 at one constant value of z.
• Similar plots would then be made at other
constant values of z. Each plot would give an
optimum value of x, y, and C, for a particular z.
Finally, as shown in the insert in Fig. 11-2, the
overall optimum value of x, y, z, and C, could
be obtained by plotting z versus the individual
optimum values of CT.

• At the optimum conditions, both of these partial
derivatives must be equal to
zero; thus, Eqs. (9) and (10) can be set equal to
zero and the optimum values
of x = (cb/a2
)1/3 and y = (ab/c2
)1/3 can be
obtained by solving the two
simultaneous equations. If more than two
independent variables were involved,
the same procedure would be followed, with the
number of simultaneous
equations being equal to the number of
independent variables.

• THE BREAK-EVEN CHART FOR
PRODUCTION SCHEDULE AND ITS
SIGNIFICANCE FOR OPTIMUM ANALYSIS
In considering the overall costs or profits in a plant
operation, one of the factors that has an important
effect on the economic results is the fraction of
total available time during which the plant is in
operation. If the plant stands idle or
operates at low capacity, certain costs, such as
those for raw materials and labor, are reduced, but
costs for depreciation and maintenance continue at
essentially the same rate even though the plant is
not in full use.

• There is a close relationship among operating
time, rate of production, and selling price. It is
desirable to operate at a schedule which will
permit maximum utilization of fixed costs while
simultaneously meeting market sales
demand and using the capacity of the plant
production to give the best economic results.
Figure 11-3 shows graphically how production
rate affects costs and profits. The fixed costs
remain constant while the total product cost, as

• OPTIMUM PRODUCTION RATES IN PLANT
OPERATION
The same principles used for developing an optimum design
can be applied when determining the most favorable
conditions in the operation of a manufacturing plant. One of
the most important variables in any plant operation is the
amount of product produced per unit of time. The
production rate depends on many factors, such as the
number of hours in operation per day, per week, or per
month; the load placed on the equipment; and the sales
market available. From an analysis of the costs involved
under different situations and consideration of other factors
affecting the particular plant, it is possible to determine an
optimum rate of production or a so-called economic lot
size.

• When a design engineer submits a complete plant design, the study
ordinarily is based on a given production capacity for the plant. After the
plant is put into operation, however, some of the original design factors
will have changed, and the optimum rate of production may vary
considerably from the “designed capacity.”
• For example, suppose a plant had been designed originally for the batch
wise production of an organic chemical on the basis of one batch every 8
hours. After the plant has been put into operation, cost data on the actual
process are obtained, and tests with various operating procedures are
conducted. It is found that more total production per month can be
obtained if the time per batch is reduced. However, when the shorter
batch time is used, more labor is required, the percent conversion of
raw materials is reduced, and steam and power costs increase. Here is
an obvious case in which an economic balance can be used to find the
optimum production rate.
• Although the design engineer may have based the original
recommendations on a similar type of economic balance, price and market
conditions do not remain constant, and the operations engineer now has
actual results on which to base an economic balance. The following
analysis indicates the general method for determining economic production
rates or lot sizes.

• The total product cost per unit of time may be
divided into the two classifications of
• 1. Operating costs
• 2. Organization costs.
Operating costs depend on the rate of production
and include expenses for direct labor, raw
materials, power, heat, supplies and similar items
which are a function of the amount of material
produced. Organization costs are due to expenses
for directive personnel, physical equipment, and
other services or facilities which must be
maintained irrespective of the amount of material
produced. Organization costs are independent of
the rate of production.

• It is convenient to consider operating costs on the basis of
one unit of production. When this is done, the operating
costs can be divided into two types of expenses as follows:
• (1) Minimum expenses for raw materials, labor, power,
etc., that remain constant and must be paid for each unit of
production as long as any amount of material is produced;
• (2) Extra expenses due to increasing the rate of production.
These extra expenses are known as super production costs.
They become particularly important at high rates of
production. Examples of super production costs are extra
expenses caused by overload on power facilities,
additional labor requirements, or decreased efficiency
of conversion. Super production costs can often be
represented as follows:

• OPTIMUM PRODUCTION RATE FOR MINIMUM COST PER
UNIT OF PRODUCTION
• It is often necessary to know the rate of production which will
give the least cost on the basis of one unit of material produced.
• This information shows the selling price at which the company
would be forced to cease operation or else operate at a loss.
• At this particular optimum rate, a plot of the total product cost
per unit of production versus the production rate shows a
minimum product cost; therefore, the optimum production rate
must occur where dCT/dP = 0. An analytical solution for this
case may be obtained from Eq. (121, and the optimum rate Po
giving the minimum cost per unit of production is found as
follows:

• OPTIMUM PRODUCTION RATE FOR MAXIMUM
TOTAL PROFIT PER UNIT OF TIME
• In most business concerns, the amount of money earned
over a given time period is much more important than the
amount of money earned for each unit of product sold.
Therefore, it is necessary to recognize that the production
rate for maximum profit per unit of time may differ
considerably from the production rate for minimum cost
per unit of production. Equation (15) presents the basic
relationship between costs and profits. A plot of profit per
unit of time versus production rate goes through a
maximum. Equation (19), therefore, can be used to find an
analytical value of the optimum production rate. When the
selling price remains constant, the optimum rate giving the
maximum profit per unit of time is

• OPTIMUM CONDITIONS IN CYCLIC
OPERATIONS
Many processes are carried out by the use of cyclic
operations which involve periodic shutdowns for
discharging, cleanout, or reactivation. This type of
operation occurs when the product is produced by a
batch process or when the rate of production
decreases with time, as in the operation of a plate-
and-frame filtration unit. In a true batch operation,
no product is obtained until the unit is shut down for
discharging. In semi continuous cyclic operations,
product is delivered continuously while the unit is in
operation, but the rate of delivery decreases with
time.

• Thus, in batch or semi continuous cyclic
operations, the variable of total time
required per cycle must be considered when
determining optimum conditions. Analyses
of cyclic operations can be carried out
conveniently by using the time for one cycle
as a basis. When this is done, relationships
similar to the following can be developed to
express overall factors, such as total annual
cost or annual rate of production:

• SEMICONTINUOUS CYCLIC OPERATIONS
Semicontinuous cyclic operations are often encountered in
the chemical industry, and the design engineer should
understand the methods for determining optimum cycle
times in this type of operation.
• Although product is delivered continuously, the rate of
delivery decreases with time owing to scaling, collection
of side product, reduction in conversion efficiency, or
similar causes.
• It becomes necessary, therefore, to shut down the
operation periodically in order to restore the original
conditions for high production rates. The optimum
cycle time can be determined for conditions such as
maximum amount of production per unit of time or
minimum cost per unit of production.

Scale Formation in Evaporation
• During the time an evaporator is in operation,
solids often deposit on the heat-transfer surfaces,
forming a scale. The continuous formation of the
scale causes a gradual increase in the resistance
to the flow of heat and, consequently, a reduction
in the rate of heat transfer and rate of
evaporation if the same temperature-difference
driving forces are maintained. Under these
conditions, the evaporation unit must be shut
down and cleaned after an optimum operation
time, and the cycle is then repeated.

• Scale formation occurs to some extent in all types of evaporators,
but it is of particular importance when the feed mixture contains a
dissolved material that has an inverted solubility.
• The expression inverted solubility means the solubility
decreases as the temperature of the solution is increased. For a
material of this type, the solubility is least near the heat-transfer
surface where the temperature is the greatest. Thus, any solid
crystallizing out of the solution
does so near the heat-transfer surface and is quite likely to form a
scale on this surface.
• The most common scale-forming substances are calcium
sulfate, calcium hydroxide, sodium carbonate, sodium sulfate,
and calcium salts of certain organic acids. When true scale
formation occurs, the overall heat-transfer coefficient may be
related to the time the evaporator has been in operation by the
straight-line equation

• CYCLE TIME FOR MINIMUM COST PER
UNIT OF HEAT TRANSFER
• There are many different circumstances which may
affect the minimum cost per unit of heat transferred in
an evaporation operation. One simple and commonly
occurring case will be considered. It may be assumed
that an evaporation unit of fixed capacity is available,
and a definite amount of feed and evaporation must be
handled each day. The total cost for one cleaning and
inventory charge is assumed to be constant no matter
how much boiling time is used. The problem is to
determine the cycle time which will permit operation
at the least total cost

• The total cost includes (1) fixed charges on the
equipment and fixed overhead expenses, (2) steam,
materials, and storage costs which are proportional to
the amount of feed and evaporation, (3) expenses for
direct labor during the actual evaporation operation, and
(4) cost of cleaning.
• Since the size of the equipment and the amounts of feed
and evaporation are fixed, the costs included in (1) and
(2) are independent of the cycle time. The optimum
cycle time, therefore, can be found by minimizing the
sum of the costs for cleaning and for direct labor during
the evaporation. If C, represents the cost for one
cleaning and S, is the direct labor cost per hour during
operation, the total variable costs during H of operating
and cleaning time must be

• The optimum cycle time determined by the preceding
methods may not fit into convenient operating schedules.
Fortunately, as shown in Figs. 11-4 and 11-5, the
optimum points usually occur where a considerable
variation in the cycle time has little effect on the factor
that is being optimized. It is possible, therefore, to adjust
the cycle time to fit a convenient operating schedule
without causing much change in the final results. The
approach described in the preceding sections can be
applied to many different types of semicontinuous cyclic
operations. An illustration showing how the same
reasoning is used for determining optimum cycle times in
filter-press
operations is presented in Example 4.

• FLUID DYNAMICS (OPTIMUM ECONOMIC
PIPE DIAMETER)
• The investment for piping and pipe fittings can amount to
an important part of the total investment for a chemical
plant. It is necessary, therefore, to choose pipe sizes which
give close to a minimum total cost for pumping and fixed
charges. For any given set of flow conditions, the use of an
increased pipe diameter will cause an increase in the fixed
charges for the piping system and a decrease in the pumping or
blowing charges. Therefore, an optimum economic pipe
diameter must exist. The value of this optimum diameter can
be determined by combining the principles of fluid dynamics
with cost considerations. The optimum economic pipe
diameter is found at the point at which the sum of pumping or
blowing costs and fixed charges based on the cost of the
piping system is a minimum.

• Pumping or Blowing Costs
• For any given operating conditions involving
the flow of a noncompressible fluid through a
pipe of constant diameter, the total
mechanical-energy balance can be reduced to
the following form:

• THE STRATEGY OF LINEARIZATION FOR
OPTIMIZATION ANALYSIS
• In the preceding analyses for optimum conditions, the
general strategy has been to establish a partial derivative
of the dependent variable from which the absolute
optimum conditions are determined. This procedure
assumes that an absolute maximum or minimum occurs
within attainable operating limits and is restricted to
relatively simple conditions in which limiting constraints
are not exceeded. However, practical industrial problems
often involve establishing the best possible program to
satisfy existing conditions under circumstances where
the optimum may be at a boundary or limiting condition
rather than at a true maximum or minimum point.

• A typical example is that of a manufacturer who
must determine how to blend various raw
materials into a final mix that will
meet basic specifications while simultaneously
giving maximum profit or least
cost. In this case, the basic limitations or
constraints are available raw materials,
product specifications, and production schedule,
while the overall objective (or
objective function) is to maximize profit.

numerical Optimisation of one and two variables

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