Factorial Design of Experiments in Ceramics, I by DON SMITH and P. R. JONES

Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia

A factorial design of experiment is one in which a variable is evaluated at all levels of all other variables. This type of experiment was applied to the unit operations in body preparation and forming by extrusion. Drying and firing unit operations were not considered, and the vari- ance due to these operations was minimized by duplication. The variables or factors were (1 ) particle-size distribution at three levels, (2) water of plasticity at two levels, (3) entrapped air at two levels, and (4) two replications. Analysis of variance was used to determine the significant effects. It was found that 95%. of the total variance of the drying shrinkage was accounted for by the particle-size distribution, water of plasticity, and two interactions; 95% of the total variance of the dry modulus of rupture was accounted for by particle-size distribution, water of plasticity, entrapped air, and two interactions ; 52% of the total variance of the fking shrinkage was accounted for by the particle-size distribu- tion; 87% of the total variance of the fired modulus of rupture was accounted for by the par- ticle-size distribution, water of plasticity, en- trapped air, and three interactions. Since a tunnel kiln makes it possible to control the firing operation closely and with no changes in com- position, close control of the particle-size distri- bution, water of plasticity, and entrapped air should make it possible to produce more uniform ware. The slope constants for the factors were also determined which show the most efficient path leading to the development of optimum conditions necessary for maximizing a desired

property.

1. Introduction XPERIMENTS based on statistical methods have been ap- plied in many different fields and offer a number of ad- E vantages. A most important advantage is the ability

to identify the variables responsible for changes in the physical properties. Another important advantage is that it is possible to determine the direction and amount of change in the base conditions in order to maximize some physical property. Still another important advantage is that it is possible to identify interactions or an effect of a simultaneous change of two or more variables. Statistical methods can reduce the costs of bench-scale and pilot-plant studies, and close control over the variables can result in lower manufac- turing costs.

Presented at the Fifty-Ninth Annual Meeting, The American Ceramic Society, Dallas, Texas, May 6, 1957 (White Wares Division, No. 3). Received March 28, 1957; revised copy re- ceived November 15,1957.

At the time this work was done, the authors were, respectively, graduate student and professor of ceramic engineering, Depart- ment of Chemical Engineering, West Virginia University. Don Smith is now serving in the armed services.

S O H I Po

H

S Fig. 1. Design of experiment. S = particle size, H = water of plar.

ticity, and P == entrapped air.

II. Factorial Design In the classical design of experiments, variables are studied

one by one. In the factorial design, each variable is evalu- ated at several levels of all other variables. In other words, the factorial design views the process as a whole. When no interactions are present, factorial design gives the maxi- mum efficiency. When interactions do exist, and no infor- mation is known concerning their relations, the factorial design must be used to avoid misleading conclusions. The factorial design is particularly useful when data are subject to fluctuations or errors of the same order of magnitude as the effect.

111. Design of the Experiment The initial application of factorial design was applied to

body preparation and to extrusion unit operations. In this part of the manufacturing process, three factors or variables were considered. These were particle-size distribution, water of plasticity, and entrapped air.

The particle-size distribution m-as attained by Ajax clays supplied by the Georgia Kaolin Company. This factor was at three levels, the term “level” referring to the value of particle-size distribution. The particle-size distribution of these clays was classified into two parts: plates (particles below 2 p ) and stacks (particles larger than 2 p ) ; this offered a convenient method of incorporating the particle-size dis- tribution factor into the factorial design. It was found that the high level or largest particle-size distribution would not extrude very well and therefore a constant amount of ball clay was added to overcome this difficulty.

The low level of water of plasticity was at 22% and the high level was at 28%. This range was the extreme that would permit good forming. The low level of entrapped air was at 21 in. Hg and the high level was a t atmospheric conditions.

All possible combinations of the factors are illustrated in Fig. 1 ; each corner represents a set of test conditions, twelve in all. Duplicate tests of each test condition were made, making a total of 24 tests. The test conditions are shown in Table I.

The remaining two factors were at two levels.

110

March 1958 Factorial Design of Exfieriments in Ceramics, I 111

cooling to room temperature, the firing shrinkage, fired modulus of rupture, and apparent porosity were determined.

The duplicate test bars were fired separately, the second firing operation duplicating the first as closely as possible.

V. Analysis of the Data A factorial design of an experiment is a means of regres-

sion analysis and essentially amounts to fitting the data to linear and quadratic curves. The data can be fitted to linear curves when two levels are used and to both linear and quad- ratic curves when three levels are used. Should the base conditions be some distance from the optimum values, linear effects will predominate. If the optimum values are not known, it is usually better to design the experiment with two levels of each of the variables in order to determine the location of the optimum values with respect to the initial base conditions.

The treatment combinations in the first column have been coded as follows: S = particle-size distribution, H = water of plasticity, P = entrapped air, and R = replication. The levels of the variables have been coded as follows: 0 = low level, 1 = intermediate level, and 2 = high level. Thus a treatment combination of 1001 refers to a run having an intermediate level of particle-size distribution, a low level of water of plasticity, a low level of entrapped air, and the second replication.

The response column is the per cent drying shrinkage ob- tained for each treatment combination, and each value is an average of ten observations.

Columns A, B, C, and D are the work columns and are obtained successively. Column A is obtained by first adding pairs of numbers in the response column. For example, the first two numbers in the response column are 3.50 and 4.11. The sum of these equals 7.61, the first number in column A.

Table I1 shows the analysis for the drying shrinkage.

Table 1. Factors, levels, and Body Composition Factors Level Value Symbol

Particle-size distribution 0 96%plates S

Water of plasticity 0 22 % H

Entrapped air 0 21 in. Hg P

1 60% plates 2 30% plates

1 28%

1 Atmospheric B O D Y COMPOSITION (%)

Test clay 47

Feldspar 9

Ball clay 15 Nepheline syenite 29

IV. Experimental Procedure The batches were dry mixed in a Simpson mixer for 15

minutes, and then the correct amount of distilled water was added for the particular test run, and mixing continued for an additional 15 minutes. The body was extruded and cut into test bars either under vacuum or atmospheric pressure as required. The bars were l/z by 1 by 7 in. Ten bars were made of each run and the bodies were made up in a random order.

After the bars were marked for identification, shrinkage marks were inscribed and the bars were weighed. The bars were air dried for 24 hours and finally dried to constant weight at 105°C. After drying, the drying shrinkage and water of plasticity were determined. The dry modulus of rupture was determined on a 3-in. span.

The larger part of the test bars from the dry modulus test was inscribed with shrinkage marks and fired to cone 5 in a direct gas-fired laboratory kiln in about 15 hours. After

Table II. Drying Shrinkage Treatment Operation

__A--_ Response Mean S H P R (%) A B C D Factor Effect square

0 0 0 0 3.50 7.61 16.36 33.77 75.69 24 T 238.7073 n n n i 4.11 8.75 17.41 29.08 -2.33 24 R 0.2262

4.64 8.88 11.27 12.84 -1.09 24 P L 0.0495 0 0 1 1 4.11 8.53 17.81 -0.27 -1.19 24 PLR 0.0590 0 1 0 0 4.51 5.78 4.17 -0.44 12.09 24 HL 6.0903 ** * 0 1 0 1 4.37 5.49 8.67 -1.62 -2.29 24 HLR 0.2185

0 0 1 0

0 1 1 0 4.37 9.02 0.08 0.79 -1.69 24 PLHL 0.1190 0 1 1 1 4.16 8.79 -0.35 -0.52 0.73 24 PLHLR 0.0222

1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1

2.89 2.89 2.80 2.69 4.48 4.54 4.59 4.20

1.25 1.11 0.%3

2.36 -0.11 -0.33

-1.36 -1.21 -0.56

0.58 1.05 6 .54 4.50

-0.43

-0.22 -1.64 -1.49

0.06

-20.93 -1.35

16 16 16

SL S1.R

27.3791 * 0.1139 0.2889 0.2003 0.7439* 0.0915 0.0946 0.0716

** I .8i 4.74 3.93 0.61

0.Oi -1.63

1.14 -0.35

-2.15 1.79 3.45

S 9 L SLPLR SLHL SLHLR SLPLHL SLPLHLR

_.

16 16 16 16

i i o 0 1 1 0 1 1 1 1 0

-0.53 -0.14 -0.21

0.00 -0.11

0.06 -0.39 -0.14

-0.96 -0.67

0.15

-1.21 1.23

-1.07

-11.55 -1.01

0.47 0.49

-7.53 -1.63 -1.87

1.75

-0.29 -0.23

-0.55 -0.81 -1.14 -0.07 -0.11 -0.45

0.29 0.29

16 1 1 1 1

2 0 0 0 2 0 0 1 2 0 1 0 2 0 1 1 2 1 0 0 2 1 0 1 2 1 1 0 2 1 1 1

48 SQ 48 SaR 48 SQPL 48 SQPLR 48 SoHr.

2.7792*** 0.0213 0.0046 0.0050 I . 1813**

0.98 2.85 1.89 2.30

-0.26 1.07

-0.34 0.00

48 GHLR 0.0554 48 SQPLHL 0.0729 48 SQPLHLR 0.0638 1.63

~

33 33 13 79

- .28 .60 .04 .92

Sum of mean squares 278.6591. Sum of R terms 1.0985 Sum of R terms/l2 0.0915

Subtotal (0) 39.01 Subtotal (1) 36.68 Subtotal (2) Column total 75.69

Check total 73.36

37.82 35.54

73.36

71.08

31.12 39.96

71.08

79.92

38.80 Significance factors

5% = 0.4348; 1% = 0.9473; 0.1% = 1.706 *) (**I (***)

38.80 . .

Sum of responsesa = 278.6591

112 Journal of The American Ceramic Society-Smith and Jones Vol. 41, No. 3 greater than 1.706 are significant a t the 0.1% level. The significant factors are marked with asterisks, and the effects associated with these factors are shown in the effect column.

y2 - yo 2h

SLOPE -

Ye

Yl

Yo

XO X I x 2

A. LINEAR SLOPE

D Y p +YO - 2 y ,

2h2

xo X I x 2

8 . QUADRATIC SLOPE

Fig. 2. Method of determining slope constants.

Table 111. Summary of Drying Shrinkage Significant Mean

effects square

HL SL SLHL

SQHL SQ

6.0903 .. .-

27.3791 0.7439 2.7792 1.1813

38.1738

The second half of column A is obtained by subtracting the first number from the second. Column B is obtained by the addition and subtraction of pairs of numbers in column A. Column C is obtained by the addition and subtraction of pairs of numbers in column B. Column D is obtained from column C by (1) the addition of triplets, ( 2 ) the subtraction of the first number of the triplet from the third number of the triplet, and (3) the addition of the first and third numbers minus twice the middle number. This process is illustrated in Fig. 2, where xo is the low level of the variable being studied and h is the uniform increment of x. Because uniform levels were used, the slope constants can be obtained in the manner outlined. The values in column D represent the slope con- stants and each must be divided by an appropriate factor in order to arrive at the average value.

Although the determination of the slope constant was one of the main objectives of the analysis, the identification of the significant factors was made first and was accomplished by variance analysis. The variance is shown in the last column and was obtained by squaring the values in column D and dividing by the factors shown. The mean squares thus ob- tained were tested for significance by the F test; values greater than 0.4348 are significant at the 5% level, values greater than 0.9473 are significant at the 1% level, and values

VI. Results

(I) Drying Shrinkage The results of the analysis of Table I1 are summarized in

Table 111. The sum of the mean squares of the significant effects is

that part of the total variance of the system that can be accounted for. The total variance of the system is deter- mined as follows :

Sum of mean squares = 278 6591 Correction for the mean = 238.7073

Total variance of the system = 39.9518

Variance accounted for = 38.1738

Variance unaccounted for = 1.7780

or __ 1'7780 X 100 = 4.5% unaccounted-for variance

In other words, 95.5% of the variance is accounted for by the significant effects and drying shrinkage depends on particle- size distribution and water of plasticity only.

Having found the significant effects, the slope constants for each of these effects are determined next. This is done by dividing the appropriate value in column D by a factor, taking into consideration the sign before each value. The empirical relation for the drying shrinkage is as follows:

39.9518

y = 3.15 + 0.5~1 + 0.22~1~3 - 0.72(xa2 - 2/3) - 0.47(xa2 - 2/3)~s - 1.3~3 (1)

y = drying shrinkage (70). xl = water of plasticity. xa = particle-size distribution.

This relation shows how the average value is affected by the various factors. The slope constants are most useful, however, for determining the magnitude and direction of change of the base conditions to maximize or minimize a property. Because of this, factorial design is a very efficient tool, and the optimum conditions are found most efficiently by proceeding along the most advantageous path.

The dry and fired modulus of rupture, firing shrinkage, and apparent porosity have been analyzed in the same way.

(2) Dry Modulus of Rupture The results of the analysis of Table I V show that 95y0 of

the total variance was accounted for by the particle-size dis- tribution, water of plasticity, and entrapped air only. As in the case of drying shrinkage, several interactions were found to be significant. The empirical relation for the dry modulus of rupture is as follows:

lo0 = 3.72 - 0.24~1 - 0.43~2 - 1 . 8 5 ~ ~ - 0.2421X2Xa - 1.08(xaZ - 2/3) -k O.42(xa2 - 2/3)~2 (2)

x2 = entrapped air.

(3) Firing Shrinkage The results of the analysis of Table V show that the firing

shrinkage was a linear function of particle size only and this significant effect accounted for 51.5% of the total vari- ance.

y = 9.83 - 0 . 7 2 ~ ~ (3)

(4) Fired Modulus of Rupture The results of the analysis of Table V I show that 86.7%

of the total variance was accounted for by the particle-size distribution, water of plasticity, entrapped air, and several interactions. Since the firing was hand controlled, the firing treatment was not duplicated exactly, introducing another

The empirical relation is as follows:

March 1958 Factorial Design of Experiments in Ceramics, I 113

Table IV. Dry Modulus of Rupture Operation - Mean

S H P R in./100) A B C D Factor Effect square

Treatment Response _ _ ~ .A -___ (lb./sq.

0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0

6.97 5.93 4.17 5.04 4.59 5.30 4.49 5.20

5.63 5.46 3.87 3.93 5.24 4.32 3.41 3.73

1.56 1.83 1.89 1.30 1.55 1.48 1.46 1.06

12.92 9.21 9.89 9.69

11.06 7.80 9.56 7.14

3.39 3.19 3.03 2.52

0.87 0.71 0.71

0.06

0.32 0.27

-1.04

-0.14

-0.92

-0.59 -0.07 -0.40

23.11 19.58 18.86 16.70 6.58 5.55

-0.17 1.42

-0.08 -0.60 -0.32 -0.47 -3.69 -0.20 -3.26 -2.42

-0.20 -0.51

1.91 0.00 0.20 1.24

-0.86 -0.33

41.69 35.56 12.13 1.25

-0.63 -0.79 -3.89 -5.68

-0.71 1.91 1.44

-1.19 -2.53 -2.16 -1.03

1.59

-0.52 -0.15

3.49 0.84

-0.31 -1.91

1.04 0.53

89.38 -0.22

-10.28 2.16

-5.72 0.92 4.02

-0.34

-29.56 -2.04

3.18 -3.10

1.50 -1.74 -3.80

2.44

1.82 6.76

-2.16 0.76 2.48 1.50

-3.46

-17.20

24 24 24 24 24 24 24 24

16 16 16 16 16 16 16 16

48 48 48 48 48 48 48 48

T R P L PLR

332.8660 0.0020 4.4033*** 0.1944

HL 1.3633* HLR 0.0353 PLHL 0.6734 PLHLR 0.0048

0 1 0 1 0 1 1 0 0 1 1 1

1 0 0 0 1 0 0 1 1 0 1 0

S L SLR Sr.Pr.

54.6121*** 0.2601 0.6320

i o i i 1 1 0 0 1 1 0 1 1 1 1 0

_ _ SLPLR 0.6006 SLHL 0.1406 SLHLR 0.1892 SLPLHL 0.9025*

1 1 1 1

2 0 0 0

SLPLHLR 0.3721

SO 6.2352*** 2 O O i 2 0 1 0 2 0 1 1 2 1 0 0

SiR SQPL SQPLR SQHL SOHLR

0.0690 0.9520* 0.0972 0.0120 0.1281 2 1 0 1

2 1 1 0 2 1 1 1

SQPLHL 0.0469 SQPLHLR 0.2494

~~

Subtotal (0, 44.80 48.64 41.08 41.60 Sum of mean squares 405.04 15

Column total 89.38 89.16 81.04 79.92 37.20

Check total 89.16 81.04 79.92 37.23 5% = 0.8716; 1 % = 1.712; 0.1% = 3.42

Subtotal (1) 44.58 38.96 39.96 29.84 Sum of R terms 2.0220 Subtotal (2) 8.48 Sum of R terms/l2 0.1853

Sum of responsese = 405.0416 ( *) (**I (***I Significance factors

Table V. Firing Shrinkage Treatment Operation

-_h__ Response 7 7 Mean S H P R (%I A B C D Factor Effect square

0 0 0 0 0 0 0 1

10.94 11.11 10.23 10.40

22.05 20.63 21.25 21.47

42.68 42.72 37.11 39.54 37.52 36.39 0.34 2.72

-0.01 1.14 2.52 1.39

- 1 .42

85.40 76.65 73.81 3.06 1.13 3.91

-1.20 -0.91

-0.13 -0.54

2.85 -0.63

0.04

235.86 24 24 24 24 24 24 24 24

16 16 16 16 16

T 2317.9142 R

~~

8.10

1.68 1.24

-2.24 P L PLR HL Hr.R

0 0 1 0 0 0 1 1 0 1 0 0 9.81

11.44 10.19 11.28

9.67 8.86 8.89 9-69

18.53 18.58 20.25 19.29

18.71 18.81 18.26 18.03

0 1 0 1 0 1 ' 1 0 0 1 1 1

2.40 0.30

-0.30

-11.59 0.85 1.07

-0.09 -1.27 -3.51 -1.97

1.15

5.91 4.71 0.49

-6.87 -6.05 -1.05

3.33 0.81

0.0037 0.0037

1 0 0 0 1 0 0 1 1 0 1 0

S L SLR SLPL

8.3955* ** 0.0452 0.0716

i o i i 1 1 0 0 1 1 0 1 1 1 1 0

SLPLR 0.0005 ST.Hr. 0,1008 iO.ii

10.10 9.05

10.24

0.17 0.17 1.63 1.09

- ~~

0.22 0.05

-0.96

2.43 -1.23

2.38

16 16 16

S E H ~ R SLPLHL SLPLHLR

0.7700 0.2426 0.0826 1 1 1 1

2 0 0 0 -0.81 0.80 0.05 1.19 1.57 0.95 0.70 0.69

0.10 -0.23

0.00 -0 .54

1.61 1.24

-0.62 -0.01

1.15 -1.13

1.64 -1.01 -0.33 -0.54 -0.37

0.61

48 48 48 48 48 48 48 48

SQ SaR

0.7276 0.4622 0.0050 0.9833 0.7625 0.0230

8.57 10.14 8.93 9.88 8.78 9.48 8.67 9.36

Z o o i 2 0 1 0 2 0 1 1 2 1 0 0 2 1 0 1 2 1 1 0 2 1 1 1

0.2310 SQPLHLR 0.0137

Subtotal (0) 113.88 122.26 119.88 90.24 Sum of mean squares 2334.2034 Subtotal (1 ) 121.98 121.70 123.52 81.92 Sum of R terms 5.4758 Subtotal (2) 74.88 Sum of R terms/l2 0.4563 Column total 235.86 243.96 243.40 247.04 232.96

Sinnificance factors Check total 243.96 243.40 247.04 232.96

Sum of responses2 = 2334.2032 5% = 2.1675; 1% = 4.2574; 0.1% = 8.5056

( *) (**I (***I

114 Journal of The American Ceramic Society-Smith and Jones Vol. 41, No. 3

Table VI. Fired Modulus of Rupture Treatment Operation

Mean A B C D Factor Effect square

Response (lb./sq. in./1000)

1.76 1.81 2.20 1.52

S H P R

0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0

3.57 3.72 3.32

7.29 6.65 7.61

13.94 16.78 11.71

42.43 -0.59 1.97

-0.61 2.01 0.55 0.71 1.29

-2.23 1.31 0.09

24 T 24 R 24 P L

75.0127 0.0145 0.1617* 0.0155 3.33

3.53 4.08 4.08 5.09 2.69 2.62 3.04 3.36 0.05

-0.68 -0.28 -0.05 0.13

-0.02 -0.06 -0.03 '

0.01

9.17 5.31 6.40

-0.63 -0.33 0.11

-0.09 -0.05 0.40 0.15 0.01 0.55 1.01

-n n7

-0.96 0.02

24 PLR i.80 1.52 1.69

- 24 HT. 0.1683*

0.35 0.16 1.56

24 H;R 0.0126 0.0210 0.0693 1.64

1.70 1.83

0.25 -0.50 -n. 12

16 S L 0.3108* * 16 SLR 0.1073 16 Sr.Pr. 0.0005

i o o i 1 0 1 0 1 0 1 1 1 1 0 0

~ -~ 2.05 2.03 2.07

0.01 0.51 -0.64 1.73 1.56 0.15

16 SLPLR 0.0163 16 SLHL 0.1871 * 16 SLHLR 0.0014 16 SLPLHL 0.0176

1 1 0 1 2.01 2.56 2.53

1 1 1 0 1 1 1 1

1.09 0.30

-0.20 0.45

-0.14 0.46 0.39

0.53 -0.81 -7.91 -0.65 -2.71 -0.25 -2.67

-0.67 1.15

0.75

16 SLPLHLR 0.0410

2 0 0 0 2 0 0 1 2 0 1 0 2 0 1 1

1.34 1.35 1.34

48 SQ 48 SQR 48 SQPL 48 SaPLR

1.3035*** 0.0088 0.1530* 0.0013 0.1485*

0.32 -0.73 0.23 1.28

1.44 2 1 0 0 2 1 0 1 2 1 1 0 2 1 1 1

48 SOHT. -0.15

-0.07 0.03

0.08 19.32 23.88

43.20

47.76

i.60 1.56 1.80

-0.60 0.16 0.24

0.96 0.18 0.15

48 S~HLR 0.0276 48 SQPLHL 0.0094 48 SQPGHLR 0.0117

~~

Sum of mean squares Sum of R terms Sum of R terms/l2

77.8214 0.3273 0.0272

Subtotal (0) 21.51 20.24 21.60

41.84

43.20

13.12. 20.24 14.40 47.76

36.08

Subtotal (ij 20.92 Subtotal (2) Column total 42.43 36.08

Significance factors

5% = 0.1295; 1% = 0.2544; 0.ly - 0.5083 ( *) (**) O(*<*)

Check total 41.84

Sum of responses2 = 77.8213

Table VII. Apparent Porosity Treatment Operation

_--L__- Response ----- 7 Mean S H P R (%) A B C D Factor Effect square

~ ~

0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 I 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1

2 0 0 0 2 0 0 1 2 0 1 0 2 0 1 1 2 1 0 0

0.97 . 0.92 1.16 1.28

1.89 2.44 2.49 3.38 7.88

4.33 5.87 16.40 15.32 23.47 41.06 0.07 1.53

10.20 31.72 64.53

106.45 19.31

24 24

T R PL PLR Hi.

472.1501 15.5365 0.1962 5.2547

2.17 11.23 18.05 3.71

-3.07 -1.61 54.33 10.09 0.77 2.57 16.05 -1.63 -0.99 -6.95

24 24 24 24 24 24 16 16 16

1.60 6.02 11.69 1.44

-1.48 2.21 2.04 4.58

i.33 1.16 0.84 2.54

13.5751 * n .5735 . -~

8.52 8.72 6.60

0.3927 0.1080

11.02 12.45 20.14 20.93 -0.05 0.12

-0.17 1.70

1.80 4.22 5.93

S L SLR Sr.Pr.

184.4843*** 6.3630 0.0371

3.82 4.06 3.48 5.04 4.12 4.60 4.43 5.17

. ..

5.76 0.55 0.89

~ ~~

4.61 1.54

17.59 -1.08

16 16 16

_ _ SLPLR 0.4128 SLHL 16.1002* SLHLR 0.1661 SLPLHL 0.0613 0.64

-2.12 16 16 48 48 48 48 48 48 48 48

1.46 2.43

0.34 -0.17

-2.76 -0.65 1.70 1.94

-5.25

SLPLHLR 3.0189

SQ 2.6555 SQR 0.0326 SQPL 0.9103 SQPLR 0.1313 SQHL 9.4430 SQHLR 0.2625 SQPLHL 0.5655 SQPLHLR 1.1050

5.26 5.76 3.51

0.24 1.56

1.43 0.78 0.17

11.29 1.25 6.61 0.48

3.74 0.50 5.43

8.94 1.87 1.32 3.26 4.93

-0.32

-2.51 21.29 -3.55 5.21

-7.43

8.55 11.59 9.10 11.82

2 i O i 2 1 1 0 2 1 1 1

- . ~~

3.04 2.72

Subtotal (0) 48.57 56.18 61.04 2q.32 Sum of mean squares 733.581 1

Subtotal (2) 94.56 Sum of R terms/l2 2.7509 Column total 106.45 125.76 139.16 156.24 262.64

Check total 125.76 139.16 156.24 262.64

Subtotal (1) 62 88 69.58 78.12 41.36 Sum of R terms 33.0108

Significance factors

5% = 13.067; 1% = 26.599; 0.1% = 51.277 Sum of responsesZ = 733.5811 ( *) (**) (***)

March 1958 Factorial Design of Experiments in Ceramics, I 115

source of error. of rupture is as follows:

1000

The empirical relation for the fired modulus

-- = 1.77 + 0.08~1 + 0.08x2 - 0.14~3 + 0.11~1~1 - 0 . 4 9 ( ~ 1 ~ - 2 / 3 ) - 0.17(~32 - 2/3)X2 -

0.17 ( 3 ~ 3 ~ - 2/3)X1 (4)

(5) Appcrrenfporosity The results of the analysis of Table VI I show that 82%

of the total variance was accounted for by particle-size dis- tribution, water of plasticity, and one interaction. The empirical relation for the apparent porosity is as follom-s :

y = 4.44 f O.75Xi + 3.04~3 + 1 . 0 ~ 1 ~ 1 ( 5 )

VII. Conclusions Within the limits of this experiment, the following con-

clusions were reached : ( I ) The only factors affecting the drying and firing

shrinkages, the dry and fired modulus of rupture, and the

apparent porosity are the particle-size distribution, water of plasticity, and entrapped air.

(2 ) Particle-size distribution was an important and sen- sitive factor in all properties measured. Most of the vari- ance was accounted for by particle-size distribution.

Close control of these three factors, especially par- ticle-size distribution, should result in more uniform produc- tion of ware.

(3)

VIII. Summary Factorial design of experiments has been applied to the

unit operations of body preparation and forming by extru- sion. The drying and firing unit operations were not studied. Variations due to the drying and firing unit operations were minimized by duplication. It was shown that the factors affecting the physical properties up to the drying operation were particle-size distribution, water of plasticity, and en- trapped air only. The most efficient path leading to the development of optimum conditions was also determined.

Appendix

Simplified Example of a Factorial Design

In the following example, the design is known as a 2* factorial, i.e., two factors a t two levels each (see Table VIII). A and B are the two factors and the minus and plus signs are the low and high levels respectively. The four trials are all possible combi- nations of the factors and levels.

For example, trial No. 1 consists of the low levels of both factors; trial No. 2 consists of the high level of A and the low level of B. All properties of each trial can be determined. For simplicity, assume that just one (compressive strength) is to be analyzed, so that the compressive strength of each trial has been determined. Let yl, yz, y3, and y, represent the compressive strength. The following calculations indicate the principle used in the main body of this paper.

Colculationa ya - y3 = change in strength as a result of changing the level of

A a t the high level of B. y2 - yl = change in strength as a result of changing the level of

A a t the low level of B.

(1) Therefore (” - + (” - 2

= average effect of the

change in A over the range of B.

(’’‘ - ”) + (” - ’1) = average effect of the change in B 2 ( 2 )

over the range of A. (” - ”) - (” - = effect of the simultaneous change

2 (3)

in both A and B and is the interaction AB. Notice that the signs now become the directions for the treat-

ment of the data. For example, the effect of A can be determined by the algebraic sum of the data as indicated in column A. The Yates method does this but in a slightly different way. If it is specified that the increment between levels is uniform, then these calculations.become the slope constants of the effects.

Table VIII. 22 Factorial Design Trial No. A B AB

1 2 3 4

Levels and Slope Constanis The determination of the slope constant of the change in a fac-

tor is made easy by the factorial design. Simple arithmetic can be used and is based on the “level” concept. A level refers to a change in the amounts of the factors in a special way; i.e., the in- crements between levels must be uniform. Figure 2 ( A ) illustrates the method for the linear slope using three points. Although two points and three points were used in the paper, three points show the general method more clearly and both linear and qvadratic slopes can be determined. With two points, only a linear slope can be determined. The reason that simple arithmetic can be used to determine the slope constant is derived as follows:

when x = xo, yo = a + bxo

The general linear equation is y = a + bx

XI = xo + h, yl = a + b(x0 + h) xz = xo + 2h, y~ = a + b(xo + 2h)

Therefore

y2 = a + b ( x ~ + 2h) yo = a + b x ~

y2 - yo = b(2h) or b = = linear slope 2h

Figure 2 ( 9 ) illustrates the determination of the quadratic slope.

116 Journal of The American Ceramic Society-Discussions and Notes Vol. 41, No. 3

The general quadratic equation is y = a + bx + CX’, and the derivation is as follows:

39 = a + b(x0 + 2h) + C(%2 + &oh + 4h2)

2Yl = 2a + b(2xo + 2h) + c(2x02 + 4xoh + 2h2)

yo u + bxo + C X O ~

, ~ 2 + yo = 2a + b(2xo + 2h) + ~ ( 2 x 0 ~ + 4Xoh + 4h’)

y2 + yo - 2y1 = c(2W ’* + - 2y1 = quadratic slope Or = 2h2

Bibliography

(1) J. R. Bainbridge, “Factorial Experiments i 1 Pilot-Plant Studies.” Ind. Eng. Chem.. 43 [6] 1300-1306 (1951).

(2) C. A. Bennett and N. L. Franklin, Statistical Analysis in Chemistry and the Chemical Industry. John Wiley & Sons, Inc., New York, 1954. (3) 0. L. Davies (editor), Design and Analysis of Industrial

Experiments. Hafner Publishing Company, New York, 1954. 636 pp.

(4) F. Yates, Design and Analysis of Factorial Experiments. Imperial Bureau of Soil Science, London, 1937.

724 pp.

Acknowledgments This research project was conducted under a fellowship granted

by The Edward Orton, Jr., Ceramic Foundation of Columbus, Ohio, to whom the authors wish to express their sincere appre- ciation.

Discussions and Notes

Note on Thermal Expansion of Neutron-Irradiated Silica

by IVAN SIMON

T IS now well known’ that the final product of extensive irradi- ation of quartz by fast neutrons is an amorphous solid very similar to the thermally fused silica both in its bulk properties

(isotropy, density, refractive index) and in its microstruc- turea (radial distribution of electron density). The present writer has now determined the thermal-expansion coefficient of

I

Received November 1, 1957. The author is physicist at Arthur D. Little, Incorporated,

Cambridge, Massachusetts. (a) W. Primak, L. H. Fuchs, and P. Day, “Effects of Nuclear

Reactor Exposure on Some Properties of Vitreous Silica and Quartz,” J . Am. Ceram. Soc., 38 [4] 135-39 (1955).

(b) M. Wittels and F. A. Sherrill, “Radiation Damage in Silica Structures,’:,Phys. Rev., 93 [5] 1117-18 (1954).

Ivan Simon, Structure of hleutron-Irradiated Quartz and Vitreous Silica,” J . Am. Ceram. Soc., 40 [5] 150-53 (1957).

this amorphous form of silica produced by irradiation of quartz to a total integrated flux of 1.4 X lom fast neutrons per sq. cm. and found it to have a value of 5.4 X deg.-l between 25” and 200OC. This is practically identical with the value measured on thermally fused vitreous silica. Smyth3 pointed out that the anomalously low (or negative) expansion of vitreous silica can be explained by the contribution of vibrational modes, the frequency of which decreases with decreasing volume (increasing density). This explanation is consistent with the observation* that the neutron-disordered silica has a greater density (2.30 gm. per crn.l) than ordinary vitreous silica (2.21 gm. per ~ m . ~ ) , whereas the vibrational frequency of the 9.1-p infrared band is lower (1110 cm.-l) tban that of vitreous silica (1125 cm.-l).

3 H. T. Smyth, “Thermal Expansion of Vitreous Silica,” J . Am. Ceram. SOC., 38 [4] 140-41 (1955).