a study of factors that affect capillary · crown shape. shaoshan deng discussed that the changes...

2011/6/1

——DOE Project Report | 杨路怡陆泓宇陈驰

GROUP 1 A STUDY OF FACTORS THAT AFFECT

CAPILLARY

A Study of Factors That Affect Capillary Group 1: 杨路怡陆泓宇陈驰

Content Abstract ..................................................................................................................... 4

Experiment Background & Objective ........................................................................ 4

Background ........................................................................................................ 4

Objective ............................................................................................................ 5

Experiment Principles ............................................................................................... 5

Phenomenon Description .................................................................................... 5

Cause Analysis ................................................................................................... 5

Theoretical derivation ......................................................................................... 6

Literature Review ...................................................................................................... 7

Experiment Design .................................................................................................... 8

Initial Variable Analysis & Selection ................................................................... 8

Variable Table ..................................................................................................... 9

Fishbone Diagram ............................................................................................. 11

Full Factorial Design ......................................................................................... 11

Conducting the Experiment ..................................................................................... 13

Analysis and Results ............................................................................................... 14

Effects of solution type ..................................................................................... 21

Effects of Tube diameter ................................................................................... 24

Effects of experimental method ........................................................................ 25

Effects of moisture............................................................................................ 27

Effects of inclination angle ............................................................................... 27

Three-factor interaction .................................................................................... 28

Regression Model ............................................................................................. 28

Conclusions and Discussions ................................................................................... 30

Reference ................................................................................................................ 31

Figure Content

Figure 1: Schematic illustration of the theoretical derivation .................................... 7

Figure 2: Fishbone diagram that illustrates cause and effect in capillarity ................ 11


Figure 3: Snapshots of the experiment .................................................................... 14

Figure 4: main effects plot of five factors ............................................................... 16

Figure 5: Interaction plot of five factors ................................................................. 16

Figure 6: Normal probability plot of the effects for the experiment ........................ 17

Figure 7: Normal probability plot of the significant effects ..................................... 19

Figure 8: Half normal probability plot of the significant effects .............................. 20

Figure 9: Pareto plot of the significant effects ........................................................ 20

Figure 10: Residual plot of the factorial design ...................................................... 21

Figure 11: Main effect solution type ....................................................................... 22

Figure 12: Interaction of solution type and experimental method............................ 22

Figure 13: Contour plot of solution type and experimental method ......................... 23

Figure 14: Response surface plot of solution type and experimental method .......... 24

Figure 15: Main effect of tube diameter ................................................................. 25

Figure 16: Main effect of experimental method ...................................................... 26

Figure 17: Interaction of experimental method and solution type............................ 26

Figure 18: Three-factor interaction of solution type, tube diameter and

experimentalmethod............................................................................................... 28

Table Content

Table 1: Response variables for initial selection ........................................................ 9

Table 2: Control variables for initial selection ........................................................... 9

Table 3: Constant variables of the experiment ......................................................... 10

Table 4: Noise variables of the experiment ............................................................. 10

Table 5: Test Matrix ................................................................................................ 12

Table 6: Factor levels.............................................................................................. 13

Table 7: Data of the height response under experimental conditions ....................... 14

Table 8: ANOVA table of the close full model ........................................................ 17

Table 9: ANOVA table for the reduced model ......................................................... 20

Table 10: Regression model coefficients ................................................................. 28


Abstract

Various factors may affect the capillary phenomenon at different levels. Although

there have been formulas designed to address the issue, for instance,

, none is free from criticism. They either fail to go in perfect conformity with the

reality, or contain variables that are hard to get. Formulas, therefore, stay mostly at the

theoretical genre and fall in short in terms of practical application. The full-factorial

designed experiment identifies the fluid type, the tube diameter and the experimental

method as the main effects. Two-factor interactions and three-factor interactions are

presented. The practical implications of the findings are discussed.

Key words: Capillary, Coefficient of the Surface Tensile Force, Fluid Height

Experiment Background & Objective

Background

Capillary phenomenon occurs everywhere in our daily life, often bringing great

convenience while sometimes also causing severe embarrassment. When you are

forced to take a blood test, it is the capillary phenomenon that enables the scaring

suction tube to get your blood away; when you put a napkin into water, it is the

capillary phenomenon again that enables you to create your own wet tissue; in society,

also, there exists “social capillary phenomenon” in which some petty people attach

themselves to those in power and secretly make personal profits. Due to the great

opportunity offered by the course Experiment Design from Department of Industrial

Engineering, this project plans to do some further research into this extremely

widespread phenomenon and unveil its mysteries based on the application of

scientific experiment design methods—by exploring its effect factors as well as

conducting all-sided, systematic experiments and data analysis.


Objective

This project, by examining all the factors that affect the capillary phenomenon of

paper fiber materials, determines the correlations and various levels of influence

between the different materials, different factors, and degrees of capillary

phenomenon (shown by height and speed). Experimental design method is used in

this project, helping to study several controllable factors’ influence on these response

variables. During this process, by collecting and analyzing a huge amount of data, we

find out the most important effect factor of the capillary phenomenon.

Experiment Principles

Phenomenon Description

a) Putting several thin glass tubes with different inner diameters into water, we can

see that the fluid level in the tubes is higher than that in the water container, and

that the smaller the tube’s inner diameter, the bigger the gap between the two

surfaces of water.

b) Putting several thin glass tubes with different inner diameters into Hg, we can see

that the fluid level in the tubes is lower than that in the water container.

Cause Analysis

The surface of liquid resembles a tightened rubber membrane. If the surface is bent, it

has the tendency to become flat. Therefore, a concave surface pulls the liquid under it

upward, while a convex surface pushes the liquid under it downward. The infiltrating

liquid has a concave surface in the capillary tube and is pulling the liquid upward,

forcing it to rise along the tube wall. The rising process stops when the pulling force

becomes equal to the gravity of the liquid in the tube—a balance is reached. It may

also explain why non-infiltrating liquid’s surface falls in the capillary tube.


Theoretical derivation

The capillary phenomenon is directly caused by the tensile force of the liquid surface.

Define the effect of liquid ball by surface tension is additional pressure.

To liquid drop (solid liquid)：

To soap bubble (hollow liquid):

.

R stands for the radius of the sphere，σ for the coefficient of the surface tensile force,

which is defined as

To further explain the formula above，G stands for Gibbes free energy; A, the area; ,

the free energy increment of liquid surface; , for the increment of liquid surface.

According to the formula, it is not difficult to understand the coefficient of the surface

tensile force σ, under the condition with the same temperature T and pressure force p.

Based on the coefficient of the surface tensile force, we can deduce the height of

liquid of the capillary phenomenon.

As showed in Figure 1, we have:

And

Therefore

According to ，it follows

For concave surface，，the liquid surface rises；for convex surface，，

the liquid surface drops.

app:ds:theoretical

app:ds:derivation


Figure 1: Schematic illustration of the theoretical derivation

Literature Review

Although we deduced the formula of capillary liquid height from the theories above,

the literature we consult show that this formula is based on ideal conditions, its

application scope is still controversial. Yang Cheng (2010) pointed out that Laplace

equation is used in the derivation of the formula, so a basic assumption is sphere

crown shape. Shaoshan Deng discussed that the changes along with the effects

of temperature, circumstance and the length of the glass tube in fact, rather than the

settings in the theory that it is only affected by the attribute of the liquid. Mingzheng

Hu, Rongliang He (2008) considered the capillary phenomenon of short capillary (i.e.

the length of the capillary is a little smaller than the theoretical calculated height),

they stated that Gibbs function should be used to analyze the capillary phenomenon.

Zhongren Huang (1999) doubted of the rise method in capillary phenomenon

experiment. He pointed out that the height of liquid cannot reach the calculated value

in theory and they deduced that the maximum estimated error of can reach .

Shaoshan Zheng (2001) and Renzhong Huang (1999) found that the moisture level of

inner capillary also affects the liquid height.

Besides, capillary experiment plays an important role in physical experiments,

because it’s the early method of measuring surface tension. Wenhui Ren, Zhiqun Lin,


Daolin Peng (2004) discussed the effects of temperature, liquid density on the surface

tension, qualitatively analyzed the type and degree of correlation between the

variables. So when considering the capillary phenomenon under different liquid

density, the liquid heights are different, this difference cannot be reflected in the

theoretical formula. And with limited experiment conditions and data, σ value is hard

to determine and value cannot e measured. That’s the original intention of our

group: when avoiding the ideal model, find out the factors affect the liquid height,

build a more practical and rougher model.

Experiment Design

Initial Variable Analysis & Selection

By literature searching we find that many theories try to explain Capillary

Phenomenon, but they still have deviation with the reality. One relatively acceptance

formula is:

According to a synthesis of the formula above and the literature description, we

summarize the main factors affect the height of rising water in the capillary

phenomenon and the rise rate, they are:

Liquid density

Internal diameter of the capillary

Local gravity acceleration

The angle of the capillary and the liquid level in it


Liquid surface tension

The moisture inside the capillary

The experiment methods (the rising method and the descending method)

The temperature and the air pressure around

Variable Table

a) Response variable

Table 1: Response variables for initial selection

Response Variable

(units)

Normal

Operating Level

& Range

Measure

precision,

Accuracy

Relationship with the

Experiment Objective

Height h(mm) 0~200.0mm 0.1mm Determine the factors

affect height

Average Rate v(mm/s) 10.0~20.0mm 0.1mm Determine the factors

affect average rate

b) Control variable

Table 2: Control variables for initial selection

Control

variable

Normal

Operating Level

& Range

Measure

precision,

Accuracy

Recommended

experiment

settings

Predicted effects

Liquid

density 0%~100% 0.1% Glue head dropper

larger density,

higher height and

smaller rate

Angle 10~90º 1º Protractor

The height and

rate decrease

along with the

decrease of the

angle


Capillary

diameter 0.3~1.0mm 0.01mm

Ex-factory

parameter

Larger diameter,

smaller height and

rate

Moisture Yes/No — —

with moisture, the

height and the rate

increase

c) Constant variable

Table 3: Constant variables of the experiment

Constant

variable

Normal

Operating

Level & Range

Measure

precision,

Accuracy

Recommended

experiment

settings

Predicted effects

Experiment

place

The same

experiment

place

— Fixed measuring

conductors

Avoid the effect

due to the change

of gravity

acceleration

Measuring

tools

d) Noise variable

Table 4: Noise variables of the experiment

Noise

variable

Normal

Operating

Level &

Range

Measure

precision,

Accuracy

Relevant strategy Predicted

effects

Temperature fixed No

measurement

Shorten the experiment

time and measure in the

same place, avoid the

change of temperature.

No effect


Air pressure fixed No

measurement Ditto No effect

Smoothness of

the inner

capillary

fixed No

measurement

select the capillary with

higher quality and clean

them before the

experiment

No effect

Fishbone Diagram

The factors that could potentially affect the capillarity are summarized in Figure 2 in

accordance to their priorities. Based on this, the factors that are considered to be vital

and can be controlled given the experimental conditions are selected in the following

full factorial design.

Figure 2: Fishbone diagram that illustrates cause and effect in capillarity

Full Factorial Design

Five factors are identified as ones that may substantially affect capillarity: the type of

solution (organic versus inorganic), the diameter of the tube (thick versus thin),

experimental method (ascending method versus descending method), moisture (wet

versus dry) and angle of inclination (45。versus 90

。(perpendicular)). A 2

5 (five factors,


two levels, 16 tests) full factorial design is used for the experiments. The randomized

experiment matrix is shown in Table 5 and the factor levels are listed in table 2. The

response variable is the height of capillarity.

Table 5: Test Matrix

Standard

Sequence

Run

Sequence

Solution

type

Tube

diameter

Experimental

method Moisture

Inclination

angle

29 1 - - - - +

5 2 - - - + -

17 3 - - + + +

31 4 - + - - +

18 5 + - + + +

15 6 - + - - -

23 7 - + - + +

3 8 - + + + -

28 9 + + + - +

30 10 + - - - +

21 11 - - - + +

1 12 - - + + -

12 13 + + + - -

14 14 + - - - -

8 15 + + - + -

13 16 - - - - -

2 17 + - + + -

9 18 - - + - -

10 19 + - + - -

19 20 - + + + +

24 21 + + - + +

32 22 + + - - +

6 23 + - - + -


7 24 - + - + -

25 25 - - + - +

26 26 + - + - +

4 27 + + + + -

27 28 - + + - +

11 29 - + + - -

22 30 + - - + +

16 31 + + - - -

20 32 + + + + +

Table 6: Factor levels

Factor Low Level (-) High Level (+) Note

Solution type water ethanol drink water, pure

ethanol

Tube diameter thin thick thin:0.5mm ; thick:

1mm

Experimental

method descending ascending

Moisture dry wet

Inclination angle 45。

90。

Conducting the Experiment

Date 2011/5/14 Time 19:00-22:00

Experimenters 邵一桓、陈驰、陆泓宇、杨路怡 Place Dormitory

Equipment pipette, capillary tubes, beaker, Dixie cup, ruler, protractor

Notes

1. The ascending method is to make the liquid ascend in the tube as high as possible;

the descending method is first let the liquid ascend to a level higher than the actual

height of capillarity, then let it fall to the actual stationary level. In the experiment,


the tube is slanted to let more liquid flow into the tube, then reposition the tube to

the desirable angle.

2. We define moisture as the initial state of the tube before the experiment is started.

3. The angle of inclination is the included angle between the tube and the liquid

level.

Figure 3: Snapshots of the experiment

Analysis and Results

The test data are shown in Table 7. The software Minitab is used to process the data.

Table 7: Data of the height response under experimental conditions

Solution

type

Tube

diameter

Experimental

method Moisture

Inclination

angle

Height

(mm)

- - - - + 68.5

- - - + - 62.4

- - + + + 35.5

- + - - + 31.5

+ - + + + 39.5

- + - - - 29.5

- + - + + 32

- + + + - 29.7

+ + + - + 10

+ - - - + 24.2


- - - + + 69

- - + + - 57.6

+ + + - - 10.7

+ - - - - 23.9

+ + - + - 9.9

- - - - - 58.7

+ - + + - 22.6

- - + - - 25.1

+ - + - - 21.9

- + + + + 28.5

+ + - + + 8.5

+ + - - + 9

+ - - + - 23

- + - + - 22.6

- - + - + 33.5

+ - + - + 23.6

+ + + + - 9.5

- + + - + 21.5

- + + - - 26.5

+ - - + + 22.5

+ + - - - 9.2

+ + + + + 9.5

From Figure 4, we can see the main effects of solution type, tube diameter and

experimental method are rather significant. Capillarity is higher with water than with

ethanol, with the thin tube than the thick and with the descending method than the

ascending method.

Figure 5 shows that the interactions between most factors are negligible except for

that between solution type and experimental method.


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溶液

平均

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管直径上升下降

是否浸润倾斜角

Height（mm）主效应图数据平均值

Figure 4: main effects plot of five factors

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1

下降

上升

-1

1

浸润

是否

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倾斜角

Height（mm）交互作用图数据平均值

Figure 5: Interaction plot of five factors

Now we would like to examine what main effects and interactions are significant. Bu

producing the normal probability plot, as shown in Figure 6, we conclude the


significant terms are solution type (A), tube diameter (B), experimental method (C),

two-factor interaction AC and three-factor interaction ABC.

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ABC

AC

B

A

效应的正态图（响应为 Height（mm），Alpha = .05)

Lenth 的 PSE = 2.83125

Figure 6: Normal probability plot of the effects for the experiment

To confirm the preliminary conclusion drawn from Figure 6, an ANOVA table is made

is check the significant terms. Note that since this is a single replicate experiment, the

error term may fall short of the degree of freedom if all terms are added. Therefore, an

assumption is proposed that high-order interaction terms (4 and 5) are insignificant;

thus the ANOVA can be made.

Table 8: ANOVA table of the close full model

拟合因子: Height（mm）与溶液, 管直径, 上升下降, 是否浸润, 倾斜角

Height（mm）的效应和系数的估计（已编码sss单位）

系数标


项效应系数准误 T P

常量 28.43 1.173 24.23 0.000

溶液 -22.16 -11.08 1.173 -9.44 0.000

管直径 -19.59 -9.79 1.173 -8.35 0.000

上升下降 -6.20 -3.10 1.173 -2.64 0.038

是否浸润 3.44 1.72 1.173 1.46 0.193

倾斜角 1.50 0.75 1.173 0.64 0.546

溶液*管直径 3.98 1.99 1.173 1.69 0.141

溶液*上升下降 8.34 4.17 1.173 3.55 0.012

溶液*是否浸润 -1.88 -0.94 1.173 -0.80 0.455

溶液*倾斜角 0.51 0.26 1.173 0.22 0.834

管直径*上升下降 5.41 2.71 1.173 2.31 0.061

管直径*是否浸润 -3.15 -1.57 1.173 -1.34 0.228

管直径*倾斜角 -1.14 -0.57 1.173 -0.48 0.645

上升下降*是否浸润 4.01 2.01 1.173 1.71 0.138

上升下降*倾斜角 -1.75 -0.88 1.173 -0.75 0.484

是否浸润*倾斜角 -0.54 -0.27 1.173 -0.23 0.826

溶液*管直径*上升下降 -6.78 -3.39 1.173 -2.89 0.028

溶液*管直径*是否浸润 1.21 0.61 1.173 0.52 0.624

溶液*管直径*倾斜角 -1.45 -0.73 1.173 -0.62 0.559

溶液*上升下降*是否浸润 -1.85 -0.92 1.173 -0.79 0.460

溶液*上升下降*倾斜角 4.21 2.11 1.173 1.80 0.123

溶液*是否浸润*倾斜角 2.27 1.14 1.173 0.97 0.370

管直径*上升下降*是否浸润 -2.18 -1.09 1.173 -0.93 0.390

管直径*上升下降*倾斜角 -0.34 -0.17 1.173 -0.14 0.890

管直径*是否浸润*倾斜角 1.87 0.94 1.173 0.80 0.455

上升下降*是否浸润*倾斜角 -0.81 -0.41 1.173 -0.35 0.741

The underlined terms are significant, which are exactly what has been obtained from


Figure 6: A, B, C, AC and ABC.

Now, a reduced model with only the identified significant terms can be analyzed.

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B 管直径

C 上升下降

因子名称

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显著

效应类型

ABC

AC

C

B

A

标准化效应的正态图（响应为 Height（mm），Alpha = .05)

Figure 7: Normal probability plot of the significant effects

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效应类型

ABC

AC

C

B

A

标准化效应的半正态图（响应为 Height（mm），Alpha = .05)


Figure 8: Half normal probability plot of the significant effects

Figure 7, Figure 8 and Figure 9 essentially point to the same conclusion that these

identified significant terms are indeed significant.

C

ABC

AC

B

A

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2.056

A 溶液

B 管直径

C 上升下降

因子名称

标准化效应的 Pareto 图（响应为 Height（mm），Alpha = .05)

Figure 9: Pareto plot of the significant effects

Note that in Table 9, all P-values of the identified main effects and interactions are

significant, indicating that these are truly significant effects.

Table 9: ANOVA table for the reduced model

Source DF Seq SS SS Adj MS F P

Main effects 3 7306.3 7306.3 2435.43 47.80 0.000

2-factor interaction 1 556.1 556.1 556.11 10.92 0.003

3-factor interaction 1 367.2 367.2 367.21 7.21 0.012

Residual 26 1324.7 1324.7 50.95

Lack of fit 2 360.8 360.8 180.38 4.49 0.022

Pure error 24 963.9 963.9 40.16

Total 31 9554.3

Figure 10 reveals that the normal probability plot of residuals is generally satisfactory


since strong linearality is observed, indicating the residuals do follow a normal

distribution.

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正态概率图与拟合值

直方图与顺序

Height（mm）残差图

Figure 10: Residual plot of the factorial design

Effects of solution type

Figure 11 illustrates the main effect of capillarity height with the solution type. The

height is more substantial with the inorganic solution (water) than the organic solution

(ethanol). This is consistent with the theoretical result, since under the same

temperature, the surface tension coefficient of water is much larger than that of

ethanol and the theoretical model dictates that the height of capillarity is proportional

to surface tension coefficient.

As shown in Figure 12, the two-factor interactions of solution type and experimental

method are significant. At the low level of experimental method (the descending

method), the change in solution type causes a large change in the height of capillarity

than at the high level of experimental method (the ascending method). There are no

significant interaction effects between solution type and other factors.


1-1

40

35

30

25

20

溶液

平均

值


Figure 11: Main effect solution type

1-1

50

45

40

35

30

25

20

15

溶液

平均

值

-1

1

下降

上升


Figure 12: Interaction of solution type and experimental method

Figure 13 presents the contour plot of height with solution type and experimental

method whereas Figure 14 presents the response surface plot. Notice that because


significant interactions exist, the contour lines are curves and the response surface is a

twisted plane. From examining the contour plot, it can be seen that height increases as

the solution type is chosen as water and the descending method is applied.

溶液

上升

下降

1.00.50.0-0.5-1.0

1.0

0.5

0.0

-0.5

-1.0

管直径 -1

是否浸润 -1

倾斜角 -1

保持值

>

–

–

–

< 20

20 30

30 40

40 50

50

Height（mm）

Height（mm）与上升下降, 溶液的等值线图

Figure 13: Contour plot of solution type and experimental method


120

40

60

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0

-11

H e i g h t（m m）

上升下降溶液

管直径 -1

是否浸润 -1

倾斜角 -1

保持值

Height（mm）与上升下降, 溶液的曲面图

Figure 14: Response surface plot of solution type and experimental method

Effects of Tube diameter

The main effect of tube diameter on the height of capillarity is shown in Figure 15. It

can be seen that when the tube diameter increases, the height is reduced. This is

consistent with the theoretical model previously discussed, wherein

. In

particular, note that the tube diameter at the low level is 0.5mm and the tube diameter

at the high level is 1mm. Not surprisingly, the main effect of the tube diameter at the

high level (18.63) is half of that at the low level (38.21), which corroborates the

theory.


1-1

40

35

30

25

20

管直径

平均

值


Figure 15: Main effect of tube diameter

Effects of experimental method

The main effect of the experimental method on the capillarity height is shown in

Figure 16. It can be seen that height is larger when the experimental method is at the

low is level. In other words, it is easier to achieve higher capillarity when the

descending method is employed, as opposed to the ascending method.


1-1

32

31

30

29

28

27

26

25

上升下降

平均

值


Figure 16: Main effect of experimental method

1-1

50

45

40

35

30

25

20

15

上升下降

平均

值

-1

1

溶液


Figure 17: Interaction of experimental method and solution type

As shown in Figure 17, the two-factor interactions of experimental method and

solution type are significant. At the low level of solution type (water), the change in


experimental method causes a large change in the height of capillarity than at the high

level of solution type (ethanol). There are no significant interaction effects between

the experimental method and other factors.

Effects of moisture

The main effect and interaction effects of moisture are shown in Figure 4 and Figure 5.

Although the main effect plot suggests that capillarity is higher with the wet condition

than the dry one, no significant effect of moisture on height is detected within the

range studied. This is counterintuitive since it is expected that moisture should have

main effects according to the literature. It is surmised that moisture (wet/dry) are

sometimes difficult to quantify and in the experiment, confined by the number of

tubes available, an electric hair dryer was used to dry the tubes, vicariously blurring

the distinction between dry and wet. Another conjecture is that moisture’s effect will

be confounded when the descending method is used. We define moisture as the initial

state of the tube before the experiment is started, but with the descending method, the

tube is supposedly soaked in the process, therefore nullifying the effects of moisture

in the first place. However, an analysis of the data qualified to the ascending method

does not real any essential difference. This is, moisture is still insignificant. This

unsettling result leads us to the third possible explanation: measurement error is too

large to render the moisture effect significant.

Effects of inclination angle

The main effect and interaction effects of angle of inclination are shown in Figure 4

and Figure 5. Although the main effect plot suggests that capillarity is slightly higher

with the perpendicular position than the inclined position, no significant effect of

inclination angle on height is detected within the range studied. However, the

discovery of no change does amount to crucial implications. Though the height

remains the same, there is actually more liquid in the tube. This evident observation

has profound practical significance. If the purpose of using capillarity is to fill the


tube as much as possible, in the case of blood test, for example, an inclined tube is

preferred to extract sufficient blood sample.

Three-factor interaction

The three-factor interaction of solution type, tube diameter and experimental method

is significant. As shown in Figure 18, the best scenario for increasing the height of

capillarity is the combination of water (inorganic solution), thinner tube and

descending method. The worst case occurs with ethanol (organic solution), thicker

tube and ascending method.

Figure 18: Three-factor interaction of solution type, tube diameter and experimental method

Regression Model

Table 10: Regression model coefficients

拟合因子: Height（mm）与溶液, 管直径, 上升下降

Height（mm）的效应和系数的估计（已编码单位）


系数标

项效应系数准误 T P

常量 28.43 1.262 22.53 0.000

溶液 -22.16 -11.08 1.262 -8.78 0.000

管直径 -19.59 -9.79 1.262 -7.76 0.000

上升下降 -6.20 -3.10 1.262 -2.46 0.021

溶液*上升下降 8.34 4.17 1.262 3.30 0.003

溶液*管直径*上升下降 -6.78 -3.39 1.262 -2.68 0.012

S = 7.13780 PRESS = 2006.57

R-Sq = 86.14% R-Sq（预测） = 79.00% R-Sq（调整） = 83.47%

Height（mm）的异常观测值

拟合值标准化

观测值标准序 Height（mm）拟合值标准误残差残差

12 12 57.6000 38.6437 3.0908 18.9563 2.95R

18 18 25.1000 38.6437 3.0908 -13.5437 -2.11R

R 表示此观测值含有大的标准化残差

Judging from the considerably small P-values for all the items incorporated in the

model in Table 10, all the included factors, solution type (A), tube diameter(B),

experimental method(C), AC and ABC are significant, reaffirming the aforesaid

findings. Hence, the regression model relating height to the significant coded

variables is

This model is considered meaningful as it accounts for about 80% of the data


variability (based on the three R-Sq values).

Conclusions and Discussions

Five-factor two-level full factorial design is used to conduct an empirical study of

capillarity. The main effects and the two-factor interactions of these five factors

(solution type, tube diameter, experimental method, moisture and angle of inclination)

on the height of capillarity are obtained.

The following conclusions can be drawn from this study:

1. The main effects of solution type, tube diameter and experimental method are

significant. It becomes more difficult to observe capillarity as the solution type

becomes ethanol (as opposed to water), or as the tube diameter increases, or as

the traditional ascending method is used (as opposed to the descending method).

2. The two-factor interactions between the solution type and experimental method

are also significant. The effect of capillarity height is greatly enhanced at the low

level of solution type, i.e. water, and at the low level of experimental method, i.e.

the descending method. Also, the effect of solution change is larger when the

descending method is applied than that when the ascending method is used.

3. Moisture and angle of inclination do not show any significant effects at either

main effect or two-factor interactions.

4. The three-factor interaction of solution type, tube diameter and experimental

method is significant. As shown in Figure 18, the best scenario for increasing the

height of capillarity is the combination of water (inorganic solution), thinner tube

and descending method. The worst case occurs with ethanol (organic solution),

thicker tube and ascending method.

5. While the main effects of solution type and tube diameter is widely

acknowledged (included directly or vicariously in the theoretical formula), the

significant main effect of different experimental method (ascending versus

descending) discovered is not so well known, albeit studied. More critically, the


experiment reveals the two-factor interaction between the solution type and

experimental method as well as three-factor interaction of solution type, tube

diameter and experimental method, which can lay a strategic foundation for

future studies of capillarity.

Reference

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a study of factors that affect capillary · crown shape. shaoshan deng discussed that the changes...

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