A Designed Experiment - Tsinghua · PDF fileThe full-factorial designed experiment ... effervescent tablet as effects. ... Analysis and result Experiment data

Factor Analysis for Dissolution

Time of Effervescent Tablets

A Designed Experiment

Niu Jinshu (2010010837)

Wu Xianyu (2010010838)

Cao Liuxing (2010010836)

Wu Yining (2010010835)

Dai Yibo (2010010833)

DOE Project – Factor analysis for dissolution time of effervescent tablets

1

Content

Abstract ....................................................................................................................................................... 3

Experiment Background & Objectives ....................................................................................................... 3

Background ......................................................................................................................................... 3

Objectives ............................................................................................................................................ 3

Experiment Design ...................................................................................................................................... 4

Initial Variable Analysis & Selection .................................................................................................. 4

Variable Table ...................................................................................................................................... 4

Response Variable ....................................................................................................................... 4

Control variable ........................................................................................................................... 5

Constant variable ......................................................................................................................... 5

Noise variable.............................................................................................................................. 6

Fishbone Diagram ............................................................................................................................... 6

Full factorial design .................................................................................................................................... 7

Test matrix ........................................................................................................................................... 7

Factor levels ........................................................................................................................................ 8

Conducting the experiment ......................................................................................................................... 9

Experiment operations ........................................................................................................................ 9

Snapshots of experiments .................................................................................................................. 10

Analysis and result .................................................................................................................................... 10

Experiment data ................................................................................................................................ 10

Effects of factors ............................................................................................................................... 13

Significant factors ..................................................................................................................... 13

Effect of each factor .................................................................................................................. 18

2-factor-interacion ............................................................................................................................. 20

3-factor-interacion ............................................................................................................................. 23

Regression model .............................................................................................................................. 26

Conclusion and discussion ........................................................................................................................ 27

Model validation ....................................................................................................................................... 27

Figure Content

Figure 1 Fishbone diagram of dissolution rate .................................................................................... 7

Figure 2 Snapshot of experiments ..................................................................................................... 10

Figure 3 Result cubes ........................................................................................................................ 12

Figure 4 Main effect plot for the 4 factors ........................................................................................ 13

Figure 5 Interaction plot for the 4 factors.......................................................................................... 14


2

Figure 6 The normality plot for effects ............................................................................................. 14

Figure 7 ANOVA table ...................................................................................................................... 15

Figure 8 Reduced Model Normality plot .......................................................................................... 15

Figure 9 Half normal probability plot ............................................................................................... 16

Figure 10 Pareto plot of the reduced model ...................................................................................... 16

Figure 11 ANOVA for the reduced model ......................................................................................... 17

Figure 12 Residual plot ..................................................................................................................... 17

Figure 13 Main effect plot of flavor .................................................................................................. 18

Figure 14 Main effect plot of volume ............................................................................................... 19

Figure 15 Main Effect Plot of temperature ....................................................................................... 20

Figure 16 Interaction plot for temperature and flavor ....................................................................... 21

Figure 17 The contour plot for flavor and temperature ..................................................................... 22

Figure 18 The response surface ......................................................................................................... 22

Figure 19 Cube plot of three factor interaction ................................................................................. 23

Figure 20 The interactions between BC and D ................................................................................. 24

Figure 21 The interactions between CD and B ................................................................................. 24

Figure 22 The interactions between BD and C ................................................................................. 25

Figure 23 The regression model ........................................................................................................ 26

Figure 24 paired-t test ....................................................................................................................... 28

Table Content

Table 1 Response variable ................................................................................................................... 4

Table 2 Control variables .................................................................................................................... 5

Table 3 Constant variables .................................................................................................................. 5

Table 4 Noise variables ....................................................................................................................... 6

Table 5 Test matrix of experiment ....................................................................................................... 7

Table 6 Factor levels ........................................................................................................................... 8

Table 7 Experiment operations ............................................................................................................ 9

Table 8 Experiment result ................................................................................................................. 10

Table 10 Interactions analysis of three factor .................................................................................... 23


3

Abstract

It is generally accepted that the dissolution rate of effervescent tablets varies with own

properties as well as other environment factors. However, among various possible influential

factors, there are factors, such as volume of water, whose impact on dissolution rate is unclear.

In order to explore the impact of some factors, we design this experiment to gain observations.

Moreover, some factors, such as liquid temperature, are commonly known as a key factor. We

also include it in the experiment to examine its impact. The full-factorial designed experiment

identifies the temperature of water, volume of water, surface area of liquid and flavor of

effervescent tablet as effects. Two-factor interactions and three-factor interactions are involved.

Based on the collected data and result, we have some discussion, and summarize the

implication.

Key words: Dissolution Rate, Effervescent Tablet, Design of Experiment

Experiment Background & Objectives

Background

Effervescent tablets are popular among young people. It contains vitamins and has a pleasant

taste. If dropped into water, the tablet starts effervesce, goes up and down, and gradually

dissolves, and finally disappears. We observe that sometimes it takes only about one minute to

dissolute completely, while it may sometimes take several minutes. Therefore, we are curious

about what factors are having significant effects on the dissolution rate. Thanks to the

opportunity of DOE project, we are able to design a systematic experiment to probe into this

problem.

Objectives

The project aims at determining what the influential factors of effervescent tablets’ dissolution

rate are and their level of influence. Also, we are looking into the correlation between these

factors. Experimental design method is employed to study how response variable reacts to

controllable factors. We collect fairly reasonable amount of data, and then analyze them to find

out the influential factors of dissolution rate of effervescent tablets. Finally, we give our own

explanation on the result and implication presented.


4

Experiment Design

Initial Variable Analysis & Selection

The dissolution rate of the effervescent tablet under the same circumstance varies a lot because

of effervesce, which will release a huge amount of carbon dioxide and lead to the acceleration

of dissolution. Based on the observation of the dissolution of the effervescent tablet, we assume

that the influence of effervesce can be minimized carefully and will not affect the initial

intention of our experiment.

We believe that there may be several factors which will affect the rate of dissolution.

1. Composition of the effervescent tablet

2. Type of liquid

3. Volume of water

4. Temperature of water

5. Surface area of liquid (related to the release of carbon dioxide)

6. Surface area of the effervescent tablet

7. Temperature and air pressure around

8. Experiment method (whether stir is included)

9. Place of experiment

10. Time of ups and downs of the effervescent tablet

Variable Table

Response Variable

Table 1 Response variable

Response

Variable (units)

Normal

Operating

Level & Range

Measure

Precision

Recommended

Experiment

Settings

Relationship with the

Experiment Objective

Dissolution Time

(s) 0 ~ 300 s 0.1 s Stopwatch

Determine the

dissolution rate of the

effervescent tablet


5

Control variable

Table 2 Control variables

Control

Variable

(units)

Normal

Operating

Level & Range

Measure

Precision

Recommended

Experiment Settings Predicted Effects

Volume of

water (ml) 0 ~ 500 ml 1 ml Measuring Pot

Larger the volume

of water is, larger

the rate of

dissolution will be.

Temperature of

water (℃) 0 ~ 100 ℃ 0.1 ℃

Electronic

Thermometer

Higher the

temperature of

water is, larger the

rate of dissolution

will be.

Surface area of

liquid (cm2) 0 ~ 200 cm2 1 cm2

Calculating by

measuring the volume

of water with

measuring pot and the

height of the container

with ruler

Larger the surface

area of liquid is,

larger the rate of


Flavor of the

effervescent

tablet

Orange/Peach --- --- ---

Constant variable

Table 3 Constant variables

Control

Variable

(units)

Normal

Operating

Level & Range

Measure

Precision

Recommended

Experiment

Settings

Predicted Effects

Type of liquid --- --- Water ---

Composition of

the effervescent

tablet

--- --- Looking into

the list of

ingredients

effervescent tablet

from the same brand

will have the familiar

composition

Surface area of --- --- Surface area of Larger the surface


6

the effervescent

tablet

the original

effervescent

tablet

area of the

effervescent tablet is,

larger the rate of


Experiment

method

--- --- No stir included Stir will accelerate

the dissolution

Place of

experiment

--- --- Fixed place ---

Noise variable

Table 4 Noise variables

Control

Variable

(units)

Normal

Operating

Level & Range

Measure

Precision

Recommended Experiment

Settings

Predicted

Effects

Temperature

around

Fixed Unnecessary Shorten the experiment time

and do the experiment at the

same place to avoid the

change of temperature around

No effect

Air pressure

around

Fixed Unnecessary Ditto No effect

Time of ups

and downs of

the effervescent

tablet

0 ~ 3 times,

length of time

varies

Unnecessary --- No effect

Fishbone Diagram

Factors that could potentially affect the dissolution rate are summarized in the fishbone diagram

below. Based on this, the factors that are considered able to control given the experimental

conditions are selected in the following full factorial design.


7

Figure 1 Fishbone diagram of dissolution rate

Full factorial design

A four-factor two-level design is employed to reveal the main effects and the interaction effects

of four factors on dissolution time:

Tablet flavor (A)

Surface area of water (B)

Volume of water (C)

Temperature of water (D)

Test matrix

Table 5 Test matrix of experiment

Standard

Sequence

Running

Sequence

Flavor Surface

area

Volume

of water

Temperature

of water

11 1 -1 1 -1 1

1 2 -1 -1 -1 -1

7 3 -1 1 1 -1

32 4 1 1 1 1

27 5 -1 1 -1 1

17 6 -1 -1 -1 -1

19 7 -1 1 -1 -1

31 8 -1 1 1 1

12 9 1 1 -1 1

10 10 1 -1 -1 1


8

13 11 -1 -1 1 1

22 12 1 -1 1 -1

20 13 1 1 -1 -1

18 14 1 -1 -1 -1

23 15 -1 1 1 -1

16 16 1 1 1 1

2 17 1 -1 -1 -1

15 18 -1 1 1 1

24 19 1 1 1 -1

30 20 1 -1 1 1

25 21 -1 -1 -1 1

8 22 1 1 1 -1

14 23 1 -1 1 1

9 24 -1 -1 -1 1

28 25 1 1 -1 1

21 26 -1 -1 1 -1

26 27 1 -1 -1 1

3 28 -1 1 -1 -1

6 29 1 -1 1 -1

29 30 -1 -1 1 1

4 31 1 1 -1 -1

5 32 -1 -1 1 -1

Factor levels

Table 6 Factor levels

-1 1

Flavor Orange peach

Surface area small large

Volume of water 150 ml 450 ml

Temperature of water 25 degrees 70 degrees


9

Conducting the experiment

Experiment operations

Table 7 Experiment operations

Date 2013/5/27 Time 18:00-21:00

Experimenters Girls in ZJ8# 612B and their friend Place Dormitory

Equipment

and materials

500 ml measuring cup; digital thermometer (-50~300 degree); two

transparent cups (with small and large surface areas); effervescent

tablets (orange and peach flavors); hot and cold water

Operation

steps

1. Mixing the hot and cold water in the measuring cup and measure

the current temperature in the cup using the digital thermometer

2. Adding more hot water or cold water according to the degrees

shown on the thermometer

3. When the temperature of the water is appropriate, putting defined

amount of water into the transparent cup needed

4. Putting defined tablets into that cup and then start timing

immediately

5. Stop timing when the tablet completely dissolves


10

Snapshots of experiments

Figure 2 Snapshot of experiments

Analysis and result

Experiment data

Using Minitab, we design the order of each replication. And the experiment was conducted by

the operation order shown in Table 1. The time is the dissolution time of the tablets. More

explanation of the meaning of “-1/1” is shown in table 2. We did 2 replications for each

experiment conditions, and the average time is shown in Figure 1.

Table 8 Experiment result

标准序运行序中心点口味表面积水量水温时间(s)

11 1 1 -1 1 -1 1 107

1 2 1 -1 -1 -1 -1 254.8


11

7 3 1 -1 1 1 -1 249.1

32 4 1 1 1 1 1 93.2

27 5 1 -1 1 -1 1 115.7

17 6 1 -1 -1 -1 -1 240.5

19 7 1 -1 1 -1 -1 251.9

31 8 1 -1 1 1 1 100.7

12 9 1 1 1 -1 1 107.2

10 10 1 1 -1 -1 1 99.2

13 11 1 -1 -1 1 1 106.9

22 12 1 1 -1 1 -1 189.2

20 13 1 1 1 -1 -1 200.3

18 14 1 1 -1 -1 -1 223.8

23 15 1 -1 1 1 -1 243.9

16 16 1 1 1 1 1 108.4

2 17 1 1 -1 -1 -1 208.7

15 18 1 -1 1 1 1 103

24 19 1 1 1 1 -1 195.4

30 20 1 1 -1 1 1 108.8

25 21 1 -1 -1 -1 1 112.1

8 22 1 1 1 1 -1 201.7

14 23 1 1 -1 1 1 94.8

9 24 1 -1 -1 -1 1 113.1

28 25 1 1 1 -1 1 109.6

21 26 1 -1 -1 1 -1 207.2

26 27 1 1 -1 -1 1 95.9

3 28 1 -1 1 -1 -1 244

6 29 1 1 -1 1 -1 187.9

29 30 1 -1 -1 1 1 99.7

4 31 1 1 1 -1 -1 211.6

5 32 1 -1 -1 1 -1 213.1


12

Figure 3 Result cubes

1-1

1

-1

1

-1

1-1

水温

水量

表面积

口味

100.80

101.80103.30

101.85

108.40

97.55112.60

111.35

198.55

188.55210.15

246.50

205.95

216.25247.65

247.95

时间的立方图（数据平均值）


13

Effects of factors

Significant factors

Figure 4 Main effect plot for the 4 factors

From the figure above, we can see that the surface area have little main effect on the dissolution

time, while the flavor, volume of water, temperature have significant main effects on the

dissolution time. And the temperature has the largest main effect.

For the flavor factor, the peach flavor tablets have shorter resolving time than the orange ones.

And if the volume of the water is large, the resolving time is shorter than that of the small

volume. As temperature rises, the dissolution time gets shorter.

If no interaction is detected between the surface area and other factors, this factor can be

negligible.

1-1

200

150

100

1-1

1-1

200

150

100

1-1

口味

平均

值

表面积

水量水温

时间主效应图数据平均值


14

Figure 5 Interaction plot for the 4 factors

From the figure above, we can find there is interaction between the flavor and temperature

factors. To test whether the main effects from each factor and the interaction are significant, we

check the normality plot.

Figure 6 The normality plot for effects

From Figure 6, we get the conclusion that the significant terms are flavor (A), volume(C),

temperature (D), two-factor interaction AD and three-factor interaction B(surface area)CD.

1-1 1-1

200

150

100

200

150

100

200

150

100

1-1

200

150

100

1-1

口味

表面积

水量

水温

-1

1

口味

-1

1

表面积

-1

1

水量

-1

1

水温

时间交互作用图数据平均值

100-10-20-30-40-50

99

95

90

80

70

60

50

40

30

20

10

5

1

标准化效应

百分

比

A 口味

B 表面积

C 水量

D 水温

因子名称

不显著

显著

效应类型

BCD

AD

D

C

A

标准化效应的正态图（响应为时间，Alpha = .01)


15

To confirm the finding, we use ANOVA analysis to check the effects of the factors and the

interaction of the factors. And on a significance level of 99, we get the same result.

Figure 7 ANOVA table

We form a reduced model with only the significant terms, and get the following figures,

including the normality plot, half normal probability plot and Pareto Plot.

Figure 8 Reduced Model Normality plot

0-10-20-30-40

99

95

90

80

70

60

50

40

30

20

10

5

1

标准化效应

百分

比

A 口味

B 表面积

C 水量

D 水温

因子名称

不显著

显著

效应类型

BCD

AD

D

C

A

标准化效应的正态图（响应为时间，Alpha = .01)


16

Figure 9 Half normal probability plot

Figure 10 Pareto plot of the reduced model

And the ANOVA result from the reduced model is shown in Figure 9. The left factors and

interactions all have a p<=0.01. The result is quite satisfied.

403020100

98

95

90

85

80

70

60

50

40

30

20

10

0

绝对标准化效应

百分

比

A 口味

B 表面积

C 水量

D 水温

因子名称

不显著

显著

效应类型

BCD

AD

D

C

A

标准化效应的半正态图（响应为时间，Alpha = .01)

B

BCD

C

AD

A

D

403020100

项

标准化效应

2.79

A 口味

B 表面积

C 水量

D 水温

因子名称

标准化效应的 Pareto 图（响应为时间，Alpha = .01)


17

Figure 11 ANOVA for the reduced model

Figure 12 Residual plot

1050-5-10

99

90

50

10

1

残差

百分

比

250200150100

8

4

0

-4

-8

拟合值

残差

840-4-8

4.8

3.6

2.4

1.2

0.0

残差

频率

3230282624222018161412108642

8

4

0

-4

-8

观测值顺序

残差

正态概率图与拟合值

直方图与顺序

时间残差图


18

Effect of each factor

Flavor

Figure 13 Main effect plot of flavor

The orange tablets take longer time to dissolve than the peach ones. We checked the content of

each type of the tablets, and found the types of pigment added in the tablets are not the same.

Maybe the dissolving ability of the pigment are not the same, thus causes the difference in the

dissolution time of the two flavors of tablets.

1-1

175

170

165

160

155

150

口味

平均

值



19

Volume

Figure 14 Main effect plot of volume

From the figure above, we draw conclusion that as the volume of water increases, the

dissolution time becomes shorter. From the research of others, we found little information of

the common relationship between the resolve speed and the volume of water, but from the study

of Shuhong Wang, “Analysis on relationship between Cyclophosphamide dissolving speed and

amount of solvent”, she found that if the amount of solvent is larger, the speed of dissolve is

higher.

And in our experiment, we found a phenomenon that when the tablets were dropped in larger

amount of water, the tablets would go up and down several times in the water, which gave the

tablets more opportunities to contact with the solvent, thus increase the dissolve speed.

1-1

170.0

167.5

165.0

162.5

160.0

157.5

155.0

水量

平均

值



20

Temperature

Figure 15 Main Effect Plot of temperature

It is clear that as the temperature rises, the dissolution time will be shorter. This result matches

our common sense well. And among the three factors, the temperature plays the most important

role in determination of the dissolution time.

2-factor-interacion

From the interaction plot of the flavor and temperature (the following figure), we detect the

lines are not parallel; there may be interaction between the two factors.

1-1

220

200

180

160

140

120

100

水温

平均

值



21

Figure 16 Interaction plot for temperature and flavor

The contour plot (Figure 15) also confirms the interaction between these two factors. The

contour is curved, so the response is determined by the two factors together. And the response

surface (Figure 16) is a little bit twisted, leads to the same conclusion. The reason why these

two factors have interaction may be the feature of the pigment. We guess the pigment added in

the peach flavor have different dissolving ability against the pigment added in the orange flavor

at different level of temperature. But since the pigments are not the main components, so the

interaction between flavor and temperature is not so obvious.

1-1

240

220

200

180

160

140

120

100

水温

平均

值-1

1

口味

时间交互作用图数据平均值


22

Figure 17 The contour plot for flavor and temperature

Figure 18 The response surface

口味

水温

1.00.50.0-0.5-1.0

1.0

0.5

0.0

-0.5

-1.0

>

–

–

–

–

–

–

< 100

100 125

125 150

150 175

175 200

200 225

225 250

250

时间

时间与水温, 口味的等值线图

10

100

150

200

250

-1

0

-11

口味

时间

水温

时间与口味, 水温的曲面图


23

3-factor-interacion

Figure 19 Cube plot of three factor interaction

From the cube plot (Figure 17) of the three-factor interaction, we can have a basic conception

of the data, but it is not so obvious that the three-factor terms BCD is significant in the effect of

the dissolution time. So we continue to do the interaction plot of “B and CD”, “C and BD”, “D

and BC” as figure 18, 19, 20.

Table 9 Interactions analysis of three factor

BC D Time

1 1 103.2

1 -1 227.2375

-1 1 106.2125

-1 -1 213.15

CD B Time

1 1 164.1375

-1 1 167.25

1 -1 166.2

-1 -1 152.2125

BD C Time

1 1 150.3375

1 -1 170.9125

1

-1

1

-1

1-1

水温

水量

表面积

101.325

109.875105.075

102.550

222.525

226.950231.950

199.350

时间的立方图（数据均值）


24

-1 1 162.5375

-1 -1 166.0125

Figure 20 The interactions between BC and D

Figure 21 The interactions between CD and B

1-1

200

150

100

1-1

200

150

100

BC

D

-1

1

BC

-1

1

D

time 交互作用图数据均值

1-1

165

160

155

150

1-1

165

160

155

150

CD

B

-1

1

CD

-1

1

B



25

Figure 22 The interactions between BD and C

From the above figure, we can perceptional confirm that interactions between B and CD, C and

BD are significant at 5% confidence level. As a result, the three factor interaction is significant.

However，it may require a reasonable explanation of the three factor interaction as the main

effect of the surface area is not significant and the high order interaction is not so much familiar

in the normal experiment. We went back to check the data and find that the response factor is

largely influenced by the factor D (water temperature). And the cross-sectional area of the

experimental cup, namely the factor B surface area, and the water volume has relationship

physically. As we all know,

V = S × h = π𝑟2 × h

S𝑇 = C × h + 2S = 2πrh + 2π𝑟2

Then we guess this may contribute some hidden effects of three factor interaction.

The room temperature is about25𝑜𝐶. The material of the cup is plastic. According to the

previous study of the scientists, water temperature decreases faster with higher differences with

the outer temperature, and the total surface area of the water body experimented may have

larger effect on the cooling process. Checking the interaction plot of BC and D, the difference

of the two lines slopes are not so big, while the interaction between BD and C, CD and B are

much significant.

It is sure that the dissolution time is shorter with high water temperature, the interaction plot

also shows that the response factor also has difference performances under the condition of

different total surface area (BC), and this kind of difference is more obvious when temperature

is higher. So when the water temperature is high, the hidden effect of BC becomes significant.

1-1

170

165

160

155

150

1-1

170

165

160

155

150

BD

C

-1

1

BD

-1

1

C



26

And this kind of interactions cannot be rejected in our data, we try to do normality of the

response time, the BCD is always in the accept family with near the boundary value (p=0.05).

This may be kind of Marginal utility.

Regression model

Figure 23 The regression model

The final model is:

时间 = 162 - 10.2 口味 + 2.72 表面积 - 6.01 水量 - 57.7 水温 + 7.65 口味*水温

- 4.28 表面积*水量*水温

Although there are two outliers, we find R-Sq is quite large, and there are only 32 observations,

so we do not eliminate the outliers.


27

Conclusion and discussion

The following conclusions can be drawn from this study:

a. The main effects of flavor, volume of water and temperature of water are significant.

b. The two-factor interactions between flavor and temperature is also significant.

c. Surface area does not show any significant effects at either main effect or two-factor

interactions.

d. The three-factor interaction of flavor, volume of water and temperature of water is

significant.

e. 时间 = 162 - 10.2 口味 + 2.72 表面积 - 6.01 水量 - 57.7 水温 + 7.65 口味*水温-

4.28 表面积*水量*水温

Model validation

From the above test, we form a model of the factors and interactions:

Time = 162 − 10.2 口味 + 2.72 表面积 − 6.01 水量 − 57.7 水温 + 7.65 口味 ∗水

− 4.28 表面积 ∗水量 ∗水温.

To evaluate the model, we conduct 8 more runs to compare the anticipated time and the real

time. The settings of the level of the factors are shown below with the anticipated value and

experiment value:

Table 10 Evaluation of the model

标准序运行序水量口味水温表面积（大）预测实际

6 1 1 -1 1 1 99.28 103.0

4 2 1 1 -1 1 202.84 206.0

5 3 -1 -1 1 1 119.86 112.4

7 4 -1 1 1 1 114.76 104.8

1 5 -1 -1 -1 1 242 246.3

2 6 1 -1 -1 1 238.54 229.9

8 7 1 1 1 1 94.18 90.1

3 8 -1 1 -1 1 206.3 208.2

Using a 95 significant level paired-t test, we check whether the anticipated value matches the

actual time. From the result, we find the difference of the two sets of data will fall into the

interval (-2.92, 7.19). And we cannot reject the hypothesis that the anticipate value and actual

value are equal.


28

Figure 24 paired-t test

Documents

A Designed Experiment - Tsinghua · PDF fileThe full-factorial designed experiment ... effervescent tablet as effects. ... Analysis and result Experiment data