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EXPERIMENTAL DESIGN FOR ADSORPTION

(BOX WILSON DESIGN)

AN ASSIGNMENT SUBMITTED

BY

NWOKOMA DARLINGTON, B.

(REG. NO.: 20074609578)

IN PARTIAL FULFILLMENT FOR THE REQUIREMENT FOR THE

COURSE CHE 705: APPLIED STATISTICS IN CHEMICAL ENGINEERING

RESEARCH

FEDERAL UNIVERSITY OF TECHNOLOGY, OWERRI (FUTO)

IMO STATE, NIGERIA.


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1.0 INTRODUCTION1.1 Research Background

The Design of Experiment (DoE) is an efficient procedure of combining two or more

variables and point out the correlation amongst them. It involves planning experiments so

that the data obtained can be analysed to yield valid and objective conclusion. There are

different types of DoE and their choices depend on the purpose of the research (comparative

aim, screening or building a model). The most popular of DoE is the Box-Wilson design aka

the Central Composite Design (CCD). The Box-Wilson experimental design is a response

surface method used for evaluation of a dependent variable as a function of independent

variable. In order word, it is an empirical modeling technique dedicated to evaluating the

relationship of a set of controlled experimental factors and their observed responses.

1.2 Research Objective and Scope

The aim is to obtain a well-validated quadratic model which would allow prediction of

adsorptivity values in an adsorption experiment, as a function of time and sorbent mass,

within the working limit of those variables. The objective of this work is to apply Box-

Wilson experimental design to adsorption experiment data and investigate the response of

yield to varied adsorption parameters. In this work, the variation of adsorption capacity as a

function of time and mass of adsorbent of an adsorbent that is washed with two different

kinds of solvents (5%H2SO4 and Water) is investigated. Thus, it implies studying a real

function with two independent variables, preferably using the Box-Wilson design, which is a

surface response method.

1.3 General Overview

Chapter 1 is an introduction of the research work. Chapter 2 discusses the theoretical

background of the work. The application of Box-Wilson design to analyzing data obtained

from adsorption experiment is divulged in Chapter 3, while Chapter 4 highlights and

analyses the results obtained from the application of Box-Wilson design. Chapter 5

concludes and makes recommendations based on the findings.


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2.0 THEORITECAL BACKGROUND2.1 Design of Experiments: an synopsisDesign of Experiment (DoE) is a structured, systematic and rigorous approach to problem

solving that applies principles and techniques at the data collection stage, so as to ensure the

generation of valid, defensible and supportable conclusions (Offurum and Chukwu, 2011;

Matthews, P. G., 2005). It is a planned approach for determining cause and effect

relationships (Liza, Z. R., 2004).It is the laying out of detailed experimental plan in

advance of doing the experiment (NIST/SEMATECH, 2010). In chemical engineering

industry, it is a method used to determine the empirical relationship between the different

factors affecting a process and the output of that process. Establishing such relations enable

specification of variables mixture that would achieve some practical benefits. In an

experiment, one or more process factors (or variable) are deliberately changed in order to

observe the effect the changes have on one or more response variables. The statistical design

of experiments is an efficient procedure for planning experiments so that the data obtained

can be analyzed to yield valid and objective conclusion.

2.2 Methodology of Design of ExperimentDesign of experiments (DoE) deals with experimental methods, thus it has its own

terminology, methodology and subject of research. The methodology of DoE was

introduced by Fisher in 1935 with six basic principles (Mason et al., 2003;Lazic, R. Z.,2004; Vahid et al., 2011; Fisher, R., 1935) namely; comparison, replication, blocking,

randomization, orthogonality and factorial experiments. Comparison is the use of a control

which acts as a baseline. Replication is the use of multiple measurements for each

experiment which is good mean for the estimation of casual variation around the results.

Blocking is the arrangement of experiments into groups (also called blocks). Blocking

reduces sources of variation, thus allowing a more precise estimate of the output of the

process. Randomization is the process of assigning the various levels of the investigated

factors to the experimental units in random order. When a set of experiments is divided into

different groups or blocks, randomization gives each experiment the same choice of being

assigned to any of the groups. Orthogonality which means that there are several set of

experiments or some levels of the factors that are independent of one another and can be

combined to derive all the combination of the design. Factorial experiments, i.e. the use of


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significant combinations amongst the different factors, instead of evaluating one factor at a

time.

Box et al., (2005) gave a systematic description of the step-by-step procedure for effective

DoE approach. Figure 1 illustrates a robust DoE approach for chemical engineering

processes.

Determine dependent variable

Determine independent variable

Select the problem

Data collection

Modeling

Determine possible combination

Validation of the Model

Figure 1: Step-by-step procedure for DoE approach in Chemical Processes

Depending on the purpose and the number of factors to be tested, the basic types of

experimental design are full factorial design, fractional factorial design, central composition

design, which is also known as Box-Wilson, Plackett-Burman design and Box-Behneken

design. The purpose of the DoE may be to screen/study the effects of two or more factors on

a process and its output or the building of model/optimizing the output of a process.

2.3 Model for Design of ExperimentThe purpose of statistically designing an experiment is to collect the maximum amount of

relevant information with a minimum expenditure of time and resources. Experimental data

are used to derive an empirical (approximation) model linking the outputs and inputs. Such

models usually contain first and second order terms. The most common empirical models fit


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to the experimental data take either a linear form or quadratic form. A linear model with two

factors,X1 andX2, can be expressed as

where Y = response for given levels of the main effects X1 and X2, X1X2 =possible

interaction effect betweenX1 andX2, 0 (constant) = response of Y when both main effects

are 0,1and2=linear coefficients, = cross product coefficient, = experimental error.A quadratic model is a second order model. Such model with two factors, X1 andX2, can be

expressed as

where =

,

= quadratic coefficients.

A second order model (polynomial) is formed during second order designs for describing a

response surface. The second order designs are practicable in situations when the linear

model is insufficient for a mathematical description of a research subject with adequate

precision. We also have higher order polynomials like the cubic or third order polynomials,

which is defined by

2.4 RESPONSE SURFACE METHODSThe response surface method basically involves three major steps. The first step is to

properly design the experiment in order to evaluate model parameters efficiently after

performing experiments. The second step is to develop a polynomial equation to which the

experimental data through regression is fitted (i.e. response surface modeling through

regression). Then test the correlation fitness by applying statistical criteria and finally

evaluating the response by the fitted model (i.e. predicting the response and checking the

adequacy of the model). The response surface method enables the representation of

independent process parameters in quantitative form as:

where = predicted response (yield), = response function, =independent variables, and = experimental error.


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The surface response is obtained by plotting the expected response of . The form of isunknown and may be very complicated. Thus, a response surface method aims at

approximating by a suitable lower-ordered polynomial in some region of the independentprocess variables. If the response can be well modeled by a linear function of the

independent variables, the function in Equation (3) can be written as (Lazic, 2004; Mason et

al., 2003; Montgomery, 2005;Vahid et al., 2011):

where k = number of factors of the design, = ith linear coefficient, = ith quadraticcoefficient, = ith interaction coefficient, = independent variable.The predicted response (

) is therefore correlated to the set of regression coefficients (

):

the intercept (), linear (), interaction () and quadratic coefficients ().2.5 BOX-WILSON EXPERIMENTAL DESIGNThe Bow-Wilson design is a response surface method which is an empirical modeling

technique devoted to the evaluation of the relationship of a set of controlled experimental

factors and observed results (Myers and Montgomery, 2002). This popular second order

rotatable experimental design, which is also known as Central Composite Design

(CCD),enables experiment to be designed in such a way that factors are varied on three or

more levels, rather than on only two level. The Box-Wilson design does not require large

number of design points, therefore the cost and time needed for performing experiment is

reduced.

The Box-Wilson design involves five different levels () for each factorand contains a factorial design, added with some combinations containing thecentral point (i.e. the mean value of each factor) and some star or axial points()(i.e. the minimum and the maximum of the range) to allow a good estimation of thecurvature of the output. A central composite design always contains twice as many star (or

axial) points as there are factors in the design. The axial points are the new extreme values.

There are three types of Box-Wilson Design (Circumscribed, Inscribed and Face-Centered),

depending on the distance of the levels from the average value of each factor; the most used

in chemical engineering is the Circumscribed Design, typically referred to as Central


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Composite Design (CCD). In a CCD the distance from the centre of the design space to a

factorial point is 1 unit for each factor, the distance from the centre of the design space to a

star (or axial) point is with || The precise value of depends on certain propertiesdesired for the design and on the number of factors involved. If the design is rotatable (i.e.

the levels can be put on a circumference and the central point is the centre of this

circumference), the value of, can be calculated as; []

2.5.1 Step-by-Step Procedure for developing Box-Wilson design

1. Define the variables (i.e. the factors)

2. Define the levels of the variables (i.e. choose a range for each factor), which is

usually set/coded as -and + for the minimum and maximum respectively. Then, themean value (i.e. the central point of the range) coded 0, is considered. Finally, the meanvalue of the ranges - /0 and 0/+ are estimated (i.e. -1 and +1, referred to as thelevels). For example, if the researcher is studying the combined effect of pH and

temperature on microbial degradation of BOD in wastewater treatment plant. Knowing that

microbial activity strive in the temperature range of 30oC - 35

oC, therefore, 32 and 35 will

be referred to as the coded levels - and +, respectively. Then, the mean value of thisrange (32.5

oC) is the central point of the experimental design (level 0). Finally, the

researcher calculates the mean values of the ranges 30oC - 32.5

oC and 32.5

oC - 35

oC, that

are 31.25oC and 33.5

oC; these values are referred to as -1 and +1.

3. Define the number the number of combinations of the design: The number of

experiments in Box-Wilson design contains three sets including; factorial runs () orfractional factorial, studying factors at -1 and +1 level; centre point runs (examining factors at the centre point of the design space, helping in understanding of

curvature and replicating them to evaluate pure errors and axial or star point runs ()setting all factors to 0 (i.e. the centre point) except one, which has the value

and +

(Lizac, 2005).The number of experimental combinations Ncan be calculated as(Lizac, 2005,

Montgomery, 2005):

(6)where = number of central point runs.


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A 2-factor CCD requires at least 11 combinations (normally 13), a 3-factor CCD requires at

least 15 (normally 17), whereas a 4-factor design are at least 25 (normally 30).

4. Write the combinations, adding to the design the so called cube or factorial points,

the star/axial point and null/central points. The cube/factorial points are the combinations of

the design in which all factors are set to the levels -1 or +1, following a binary

procedure; a central point is a kind of control of the CCD, because all the factors are set to

the mean value of the range, i.e. the level 0. The star point contain the factor at the

minimum or maximum (i.e. the level - and +) and the other set to the central value.3.0 PROBLEMSTATEMENT AND SOLUTION3.1 Box-Wilson Experimental Design for AdsorptionThe experimental data collection points (Box-Wilson) are shown in Tables 15 below.

Table 1: EXPERIMENTAL DATA COLLECTION POINTS (BOX WILSON)

LEVEL

PROCESS

VARIABLES DESCRIPTION CODE

ACTUAL VALUE

YIELD

TIME CONC TIME Mass g/l

1 5 100 Extreme -K 5.0000 0.500

2 13.0546 158.5786 Intermediate -1 13.0546 0.793

3 32.5000 300.0000 Centre Point 0 32.5000 1.500

4 51.9454 441.4214 Intermediate 1 51.9454 2.210

5 60.0000 500.0000 Extreme K 60.0000 2.500

Table 2: AXIAL POINTS

CODED FORM ACTUAL VALUES

TIME CONC S/NO TIME Mass g/l

-K 0 1 5.0000 1.5

K 0 2 60.0000 1.5

0 -K 3 32.5000 0.5

0 K 4 32.5000 2.5

Table 3: FACTORIAL POINTS


TIME MASS g/l TIME Mass g/l

-1 0 5.0000 1.5

-1 0 60.0000 1.5

1 -K 32.5000 0.5

1 K 32.5000 2.5


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Table 4 3 CENTRE POINTS


TIME CONC TIME Mass g/l

-K 0 5.0000 1.5

K 0 60.0000 1.5

0 -K 32.5000 0.50 K 32.5000 2.5

Table 5 COLLECTIONS

S/NO

CODED ACTUAL

TIME MASS g/l TIME MASS g/l

1 -K 0 5 1.5

2 K 0 60 1.5

3 0 -K 32.5 0.5

4 0 K 32.5 2.55 -1 -1 13.1 0.793

6 -1 1 13.1 2.21

7 1 -1 51.95 0.793

8 1 1 51.95 2.21

9 0 0 32.5 1.5

10 0 0 32.5 1.5

11 0 0 32.5 1.5

12 0 0 32.5 1.5

X1 = 5, 13, 33, 52, 60

X2 = 0.5, 0.793, 1.5, 2.21, 2.5

3.1.1 Adsorption with Adsorbent (ZW1)

Tables 6shows the experimental data collection point (Box-Wilson) for adsorption by

adsorbent (ZW1) washed with 5%H2SO4, respectively, at temperature, T = 32oC, stirring

time X1, mass of adsorbent (g) and initial concentration of 7167.5mg/l.

Tables 7 shows the experimental data collection point (Box-Wilson) for adsorption by

adsorbent (ZW1) washed with water at temperature, T = 32oC, stirring time X1, mass of

adsorbent (g) and initial concentration of 7167.5mg/l.


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Table 6: Experimental Data Collection Points (Box-Wilson) for Adsorbent washed

with 5%H2SO4at T = 32oC (ZW1)

CODED VARIABLES ACTUAL

VALUE Y

S/N Time

(Min)X1

Mass of

AdsorbentX2

Time

(Min)X1

Mass of

AdsorbentX2

Conc.

(mg/l)

1 -K 0 5 1.5 1687.95

2 K 0 60 1.5 1218.48

3 0 -K 32.5 0.5 1435.29

4 0 K 32.5 2.5 828.74

5 -1 -1 13.1 0.793 1664.29

6 -1 1 13.1 2.21 1257.83

7 1 -1 51.95 0.793 1338.17

8 1 1 51.95 2.21 633.06

9 0 0 32.5 1.5 1364.51

10 0 0 32.5 1.5 1364.5111 0 0 32.5 1.5 1364.51

12 0 0 32.5 1.5 1364.51


with water at T = 32oC (ZW1)

CODED VARIABLES ACTUAL

VALUE Y

S/N Time

(Min)X1

Mass of

AdsorbentX2

Time

(Min)X1

Mass of

AdsorbentX2

Conc.

(mg/l)

1 -K 0 5 1.5 1759.62

2 K 0 60 1.5 1318.82

3 0 -K 32.5 0.5 1416.30

4 0 K 32.5 2.5 1236.40

5 -1 -1 13.1 0.793 1650.03

6 -1 1 13.1 2.21 1598.35

7 1 -1 51.95 0.793 1382.75

8 1 1 51.95 2.21 1226.36

9 0 0 32.5 1.5 1332.22

10 0 0 32.5 1.5 1332.2211 0 0 32.5 1.5 1332.22

12 0 0 32.5 1.5 1332.22


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3.1.2 Adsorption with Adsorbent (ZW2)


adsorbent (ZW1) washed with 5%H2SO4, respectively, at temperature, T = 32oC, stirring

time X1, mass of adsorbent (g) and initial concentration of 7167.5mg/l.


adsorbent (ZW1) washed with water at temperature, T = 32oC, stirring time X1, mass of

adsorbent (g) and initial concentration of 7167.5mg/l.

Table 8: Experimental Data Collection Points (Box-Wilson) for Adsorbent washedwith 5%H2SO4 at T = 32

oC (ZW2)

CODED VARIABLES ACTUAL VALUE

Y

S/N Time

(Min)

X1

Mass of

Adsorbent

X2

Time

(Min)

X1

Mass of

Adsorbent

X2

Conc.

(mg/l)

1 -K 0 5 1.5 1698.702 K 0 60 1.5 566.23

3 0 -K 32.5 0.5 1644.22

4 0 K 32.5 2.5 151.24

5 -1 -1 13.1 0.793 1812.00

6 -1 1 13.1 2.21 1164.72

7 1 -1 51.95 0.793 1218.48

8 1 1 51.95 2.21 129.02

9 0 0 32.5 1.5 759.04

10 0 0 32.5 1.5 759.04

11 0 0 32.5 1.5 759.04

12 0 0 32.5 1.5 759.04


with water at T = 32oC (ZW2)

CODED VARIABLES ACTUAL VALUEY

S/N Time

(Min)

X1

Mass of

Adsorbent

X2

Time

(Min)

X1

Mass of

Adsorbent

X2

Conc.

(mg/l)

1 -K 0 5 1.5 1770.37

2 K 0 60 1.5 1297.32

3 0 -K 32.5 0.5 1462.89

4 0 K 32.5 2.5 1230.66

5 -1 -1 13.1 0.793 1696.55

6 -1 1 13.1 2.21 1591.90

7 1 -1 51.95 0.793 1417.02

8 1 1 51.95 2.21 1247.15

9 0 0 32.5 1.5 1340.32

10 0 0 32.5 1.5 1340.32

11 0 0 32.5 1.5 1340.32

12 0 0 32.5 1.5 1340.32


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3.2 Analysis of Box-Wilson designBatch experiments were carried out randomly at five different levels: extreme (k),

intermediate (-1), null/centre point (0), intermediate (+1), and extreme (+K) with reference

to two independent factors: stirring time (X1) and mass of adsorbent (X2). The dependent

variable is the adsorbed concentration (Y). The value of K, from Equation (5) is:

[] [] = = 1.414 (7)Coded values of the process variables and their variation intervals are as shown in Table 10.

Table 10: Coded values of Process variables and their variation intervals

Variable Symbol Levels Variation

interval

-1.414 -1 0 +1 +1.414

Stirring time X1 5.0 13.0546 32.500 51.9454 60.000 19.4454

Mass of

Adsorbent, g

X2 0.500 0.793 1.500 2.210 2.500 0.707

Concentration,

mg/l

Y 100 158.5786 300.00 441.4214 500.00 141.4214

With the adsorption results, surface response studies were conducted. The total number of

design points N for the Box-Wilson DoE was determined using Equation (6):

Thus, as depicted in Tables 2 4, we have a 22

factorial experimental design (k = 2), with

four axial points (-1.414, 0, +1.414, 0) and four replicates of the central/null points (),totaling 12 experiments. Table 5 shows the experimental plan (coded and actual values) for

the 12 experiments, with replications (as illustrated in Tables 6 - 9) to establish the

homogeneity of the experimental method.

3.2.1 Mathematical Modeling of Box-Wilson Design Using 3rd Order PolynomialThe statistical model for Box-Wilson experimental design with a third order polynomial,

having two factors (k = 2) could be expressed as:

(8)Using the actual values, the regression coefficients are calculated as shown in Table 11.


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Table 11: Estimated Regression Coefficients for the various Experiments

Experiment

Regression Coefficient

ZW1(5%H2SO4) 812.8 36.9 1470.5 -13.3 -1.5 -756.9 -0.15 5.98 0.02 65.9

ZW1(H2O) 743 45 1489.6 -40 -1.3 -656.9 0.14 9.8 0.014 77.8

ZW2(5%H2SO4) 741.8 97.7 1626.5 -89 -2.5 -759.4 0.46 17.1 0.02 84.1

ZW2(H2O) 776 47 1521.6 -44 -1.3 -684.9 0.15 11.1 0.014 79.3

Substituting the estimated regression coefficients into Equation (8), we obtain the

mathematical models for each of the experimental data as follows:

I. Model 1: For the adsorbent washed with 5%H2SO4 (ZW1):

(9)II. Model 2: For the adsorbent washed with H2O (ZW1):

(10)III. Model 3: For the adsorbent washed with 5%H2SO4 (ZW2):

(11)IV. Model 4: For the adsorbent washed with H2O (ZW2): (12)3.3.2 Analysis of Variance (ANOVA)

The goodness of fit is conducted by correlating the observed and predicted values. The

correlation coefficient of multiple determination and its square (coefficient of multipledetermination) are calculated by the expressions (Montgomery, 2005):


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where = the error sum of squares, = sum of square of predicted values ( ).Equations (9) - (12) are used to predict the surface responses (

) to the factors X1 and

X2.Then, the error sum of squares (or residual sum of square) is calculatedusing the expression:

The experimental error variance (i.e. the mean error sum of squares) is obtained by:

where = degree of freedom for error variance = , (N= number of experimentalruns/trial = 12, = number of coefficients in the model = 10).The estimated variance of coefficient is then calculated by the following equation:

The F-value is calculated by the expression:

The significant of effects, which is also the significant coefficient, is estimated by

comparing the values of the ratio with the critical value of the F-distribution at 95%confidence level( ). If the ratio , the effect is significant.4.0 RESULTS AND DISCUSSIONThe predicted yield () obtained from Equations (9) (12) for the different experimentstheir deviations from the experimental values (,the error sum of squares and variance areshown in Tables 12 - 15.The results of the correlation coefficient of multiple determination

and its square (coefficient of multiple determination) calculated from Equation (13)and (14) are given in Table 16. The results showed that the model fitted with the


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experimental values in the range of the variables investigated. Multiple regression analysis

results indicated a high degree of correlation between the experimental values and those

predicted by the model. The variance analysis of variable effects is shown on Tables 17 - 20.

Table 12: Predicted Yield from Model 1for Sorbent washed with5%H2SO4 at T = 32o

C (ZW1)

CODED VARIABLES ACTUAL VALUE Conc.

(mg/l)

Conc.

(mg/l) S/N Time

(Min)

X1

Mass of

Adsorbent

X2

Time

(Min)

X1

Mass of

Adsorbent

X2

1 -1.414 0 5 1.5 1687.95 1650.59 1395.77

2 1.414 0 60 1.5 1218.48 1652.24 188147.7

3 0 -1.414 32.5 0.5 1435.29 1474.56 1542.1

4 0 1.414 32.5 2.5 828.74 880.33 2661.53

5 -1 -1 13.1 0.793 1664.29 1706 1739.7

6 -1 1 13.1 2.21 1257.83 1297.34 1561

7 1 -1 51.95 0.793 1338.17 1669.97 110091.2

8 1 1 51.95 2.21 633.06 980.57 120763.2

9 0 0 32.5 1.5 1364.51 1443.44 6229.95

10 0 0 32.5 1.5 1364.51 1443.44 6229.95

11 0 0 32.5 1.5 1364.51 1443.44 6229.95

12 0 0 32.5 1.5 1364.51 1443.44 6229.95

Table 13: Predicted Yield from Model 2 for Sorbent washed with Water at T = 32

oC (ZW1)

CODED VARIABLES ACTUAL VALUE Conc.(mg/l)

Conc.

(mg/l) S/N Time(Min)X1

Mass of

Adsorbent

X2

Time

(Min)

X1

Mass of

Adsorbent

X2

1 -1.414 0 5 1.5 1759.62 1771.7 145.93

2 1.414 0 60 1.5 1318.82 1284.95 1147.18

3 0 -1.414 32.5 0.5 1416.30 1406.83 89.68

4 0 1.414 32.5 2.5 1236.40 1257.28 435.97

5 -1 -1 13.1 0.793 1650.03 1632.09 321.84

6 -1 1 13.1 2.21 1598.35 1586.37 143.52

7 1 -1 51.95 0.793 1382.75 1314.03 4722.43

8 1 1 51.95 2.21 1226.36 1187.73 1492.28

9 0 0 32.5 1.5 1332.22 1320.36 140.6610 0 0 32.5 1.5 1332.22 1320.36 140.66

11 0 0 32.5 1.5 1332.22 1320.36 140.66

12 0 0 32.5 1.5 1332.22 1320.36 140.66


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Table 14: Predicted Yield from Model 3 for Sorbent washed with 5%H2SO4at T = 32oC (ZW2)

CODED VARIABLES ACTUAL VALUE Conc.

(mg/l)

Conc.

(mg/l) S/N Time

(Min)

X1

Mass of

Adsorbent

X2

Time

(Min)

X1

Mass of

Adsorbent

X2

1 -1.414 0 5 1.5 1698.70 1727.36 821.40

2 1.414 0 60 1.5 566.23 -278.76 714008.10

3 0 -1.414 32.5 0.5 1644.22 1532.53 12474.66

4 0 1.414 32.5 2.5 151.24 53.93 9469.24

5 -1 -1 13.1 0.793 1812.00 1770.72 1704.04

6 -1 1 13.1 2.21 1164.72 1122.86 1752.26

7 1 -1 51.95 0.793 1218.48 605.20 376112.36

8 1 1 51.95 2.21 129.02 -467.97 356397.06

9 0 0 32.5 1.5 759.04 618.43 19771.17

10 0 0 32.5 1.5 759.04 618.43 19771.17

11 0 0 32.5 1.5 759.04 618.43 19771.17

12 0 0 32.5 1.5 759.04 618.43 19771.17

Table 15: Predicted Yield from Model 4 for Sorbent washed with Water at T = 32oC (ZW2)CODED VARIABLES ACTUAL VALUE

Conc.

(mg/l)

Conc.

(mg/l) S/N Time

(Min)

X1

Mass of

Adsorbent

X2

Time

(Min)

X1

Mass of

Adsorbent

X2

1 -1.414 0 5 1.5 1770.37 1788.86 341.88

2 1.414 0 60 1.5 1297.32 1296.61 0.50

3 0 -1.414 32.5 0.5 1462.89 1464.56 2.794 0 1.414 32.5 2.5 1230.66 1247.69 290.02

5 -1 -1 13.1 0.793 1696.55 1669.85 712.89

6 -1 1 13.1 2.21 1591.90 1565.57 693.27

7 1 -1 51.95 0.793 1417.02 1358.05 3477.46

8 1 1 51.95 2.21 1247.15 1203.72 1886.16

9 0 0 32.5 1.5 1340.32 1323.43 285.27

10 0 0 32.5 1.5 1340.32 1323.43 285.27

11 0 0 32.5 1.5 1340.32 1323.43 285.27

12 0 0 32.5 1.5 1340.32 1323.43 285.27


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Table 16: Calculated values of Correlation Coefficients

Experiment Adsorbent washed with

5%H2SO4 (ZW1)

45288.03 226411.08 25081573.08 0.99 0.99

Adsorbent washed with

water (ZW1)

9061.47 4530.74 23641008.09 0.999 0.999

Adsorbent washed with5%H2SO4 (ZW2)

1551823.79 775911.90 11924385.87 0.97 0.94

Adsorbent washed with

water (ZW2)

4273.03 24121401.87 0.999 0.999Table 17: Variance Analysis of Variable Effects for Model 1

Effect

F-Value

( )

812.8

15703.325 36.9 14.418 94.438 S 31.026 1470.5 7297.463 296.318 S 40352.563 -13.3 5.611 31.527 S 34280579.227 -1.5 0.0066 340.669 S 118 -756.9 1918.738 298.580 S 86770312.154 -0.15 0.0026 8.623 S 150673.576 5.98 1.503 23.798 S 93050289024.849 0.02 0.000002 164.392 S 546.012 65.9 414.663 10.473 STable 18: Variance Analysis of Variable Effects for Model 2

Effect F-Value( ) 743 15703.325 45 0.289 7018.552 S 31.026 1489.6 146.030 15194.835 S 40352.563 -40 0.112 14250.233 S 34280579.227 -1.3 0.00013 12786.913 S 118 -656.9 38.396 11238.579 S

86770312.154

0.14 0.00005 375.369 S

150673.576 9.8 0.030 3193.891 S 93050289024.849 0.014 0.000000049 4025.359 S 546.012 77.8 8.298 729.444 S


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Table 19: Variance Analysis of Variable Effects for Model 3

Effect

F-Value

( )

741.8 15703.325 97.7 49.411 193.183 S 31.026 1626.5 25008.44 105.784 S 40352.563 -89 19.228 411.944 S 34280579.227 -2.5 0.023 276.131 S 118 -759.4 6575.525 87.702 S 86770312.154 0.46 0.0089 23.663 S 150673.576 17.1 5.150 56.783 S

93050289024.849

0.02 0.000008 47.969 S

546.012 84.1 1421.052 4.977 STable 20: Variance Analysis of Variable Effects for Model 4

Effect F-Value( )

776 15703.325 47 0.272 8118.044 S

31.026

1521.6 137.724 16810.895 S

40352.563 -44 0.106 18282.708 S 34280579.227 -1.3 0.000125 13558.102 S 118 -684.9 36.212 12953.896 S 86770312.154 0.15 0.000049 456.896 S 150673.576 11.1 0.028 4344.573 S 93050289024.849 0.014 0.000000046 4268.132 S 546.012 79.3 7.826 803.549 SFrom the analysis of variance for adsorbent washed with 5%H2SO4(ZW1), comparison of

the F-value with the critical value of F-distribution at 95% confidence level as shown in

Table 17 indicates that all the effects are significant.


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Table 18 indicates that all the effects on the adsorbent washed with water (ZW1) are

significant. The analysis of variance as shown in Table 19 for the adsorbent washed with

5%H2SO4 (ZW2) indicates that all the effects are significant. The analysis of variance as

shown in Table 20 for the adsorbent washed with water (ZW2) indicates that all the effects

are significant. The optimum operating conditions are determined using the following

Matlab programs:

Model 1: For the adsorbent washed with 5%H2SO4 (ZW1):

function y = myfun2(X)y = -(812.8 + 36.9*X(1)+1470.5*X(2)-(13.3*X(1)* X(2))-

(1.5*(X(1)^2))-(756.9*(X(2)^2))-(0.15*(X(1)^2)* X(2))+

(5.98*(X(1)*(X(2)^2))+ (0.02*(X(1)^3))+ (65.9*X(2)^3)));

X0 = [1.0, 1.0];

[X,fval] = fminunc(@myfun2,X0)

Model 2: For the adsorbent washed with H2O (ZW1):

function y = myfun4(X)y = -(743+(45*X(1))+(1489.6*X(2))-(40*X(1)*X(2))-(1.3*(X(1)^2))-

(656.9*(X(2)^2))+(0.14*(X(1)^2)*X(2))+(9.8*X(1)*(X(2)^2))+(0.014*(X(

1)^3))+(77.8*(X(2)^3)));

X0 = [1,1];


Model 3: For the adsorbent washed with 5%H2SO4 (ZW2):

function y = myfun5(X)y = -((741.8)+(97.7*X(1))+(1626.5*X(2))-(89*X(1)*X(2))-

(2.5*(X(1)^2))-

(759.4*(X(2)^2))+(0.46*(X(1)^2)*X(2))+(17.1*X(1)*(X(2)^2))+(0.02*(X(

1)^3))+(84.1*(X(2)^3)));

X0 = [2,0.5]; % 0r X0 = [4,0.5];


Model 4: For the adsorbent washed with H2O (ZW2):

function y = myfun6(X)y = -(776+(47*X(1))+(1521.6*X(2))-(44*X(1)*X(2))-(1.3*(X(1)^2))-

(684.9*(X(2)^2))+(0.15*(X(1)^2)*X(2))+(11.1*X(1)*(X(2)^2))+(0.014*(X

(1)^3))+(79.3*(X(2)^3)));

X0 = [5,0.5];



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