A STUDY OF FACTORS THAT AFFECT CAPILLARY

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1 20/6/ GROUP A STUDY OF FACTORS THAT AFFECT CAPILLARY DOE Project Report 杨路怡陆泓宇陈驰

2 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... Full Factorial Design... Conducting the Experiment... 3 Analysis and Results... 4 Effects of solution type... 2 Effects of Tube diameter Effects of experimental method Effects of moisture Effects of inclination angle Three-factor interaction Regression Model Conclusions and Discussions... Reference... 3 Figure Content Figure : Schematic illustration of the theoretical derivation... 7 Figure 2: Fishbone diagram that illustrates cause and effect in capillarity...

3 Figure 3: Snapshots of the experiment... 4 Figure 4: main effects plot of five factors... 6 Figure 5: Interaction plot of five factors... 6 Figure 6: Normal probability plot of the effects for the experiment... 7 Figure 7: Normal probability plot of the significant effects... 9 Figure 8: Half normal probability plot of the significant effects Figure 9: Pareto plot of the significant effects Figure 0: Residual plot of the factorial design... 2 Figure : Main effect solution type Figure 2: Interaction of solution type and experimental method Figure 3: Contour plot of solution type and experimental method Figure 4: Response surface plot of solution type and experimental method Figure 5: Main effect of tube diameter Figure 6: Main effect of experimental method Figure 7: Interaction of experimental method and solution type Figure 8: Three-factor interaction of solution type, tube diameter and experimentalmethod Table Content Table : Response variables for initial selection... 9 Table 2: Control variables for initial selection... 9 Table 3: Constant variables of the experiment... 0 Table 4: Noise variables of the experiment... 0 Table 5: Test Matrix... 2 Table 6: Factor levels... 3 Table 7: Data of the height response under experimental conditions... 4 Table 8: ANOVA table of the close full model... 7 Table 9: ANOVA table for the reduced model Table 0: Regression model coefficients... 28

4 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.

5 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.

6 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, we have: And Therefore According to,it follows For concave surface,,the liquid surface rises;for convex surface,, the liquid surface drops.

7 Figure : 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 (200) 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 (999) 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 (200) and Renzhong Huang (999) 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,

8 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

9 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 : 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.mm Average Rate v(mm/s) 0.0~20.0mm 0.mm Determine the factors affect height 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%~00% 0.% Glue head dropper larger density, higher height and smaller rate The height and rate decrease Angle 0~90º º Protractor along with the decrease of the angle

10 Capillary diameter 0.3~.0mm 0.0mm Ex-factory parameter Larger diameter, smaller height and rate with moisture, the Moisture Yes/No 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 Shorten the experiment Temperature fixed No measurement time and measure in the same place, avoid the No effect change of temperature.

11 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,

12 two levels, 6 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

13 Table 6: Factor levels Factor Low Level (-) High Level (+) Note Solution type water ethanol Tube diameter thin thick drink water, pure ethanol thin:0.5mm ; thick: mm Experimental method descending ascending Moisture dry wet Inclination angle Conducting the Experiment Date 20/5/4 Time 9:00-22:00 Experimenters 邵一桓 陈驰 陆泓宇 杨路怡 Place Dormitory Equipment pipette, capillary tubes, beaker, Dixie cup, ruler, protractor Notes. 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,

14 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)

15 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.

16 平均值 Height(mm) 主效应图数据平均值 40 溶液 管直径 上升下降 是否浸润 倾斜角 Figure 4: main effects plot of five factors Height(mm) 交互作用图数据平均值 50 溶液 溶液 0 管直径 50 管直径 上升下降 是否浸润 倾斜角 上升下降 是否浸润 倾斜角 0 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

17 百分比 significant terms are solution type (A), tube diameter (B), experimental method (C), two-factor interaction AC and three-factor interaction ABC. 效应的正态图 ( 响应为 Height(mm),Alpha =.05) AC 因子 A B C D E 效应类型不显著显著 名称溶液管直径上升下降是否浸润倾斜角 0 5 B ABC A 效应 Lenth 的 PSE = 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

18 项 效应 系数 准误 T P 常量 溶液 管直径 上升下降 是否浸润 倾斜角 溶液 * 管直径 溶液 * 上升下降 溶液 * 是否浸润 溶液 * 倾斜角 管直径 * 上升下降 管直径 * 是否浸润 管直径 * 倾斜角 上升下降 * 是否浸润 上升下降 * 倾斜角 是否浸润 * 倾斜角 溶液 * 管直径 * 上升下降 溶液 * 管直径 * 是否浸润 溶液 * 管直径 * 倾斜角 溶液 * 上升下降 * 是否浸润 溶液 * 上升下降 * 倾斜角 溶液 * 是否浸润 * 倾斜角 管直径 * 上升下降 * 是否浸润 管直径 * 上升下降 * 倾斜角 管直径 * 是否浸润 * 倾斜角 上升下降 * 是否浸润 * 倾斜角 The underlined terms are significant, which are exactly what has been obtained from

19 百分比 百分比 Figure 6: A, B, C, AC and ABC. Now, a reduced model with only the identified significant terms can be analyzed. 标准化效应的正态图 ( 响应为 Height(mm),Alpha =.05) 效应类型不显著显著 B C ABC AC 因子 A B C 名称溶液管直径上升下降 20 0 A 标准化效应 Figure 7: Normal probability plot of the significant effects 标准化效应的半正态图 ( 响应为 Height(mm),Alpha =.05) 98 效应类型不显著显著 A 因子 A B C 名称溶液管直径上升下降 B AC ABC C 绝对标准化效应

20 项 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. 标准化效应的 Pareto 图 ( 响应为 Height(mm),Alpha =.05) A 因子 A B C 名称溶液管直径上升下降 B AC ABC C 标准化效应 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 factor interaction factor interaction Residual Lack of fit Pure error Total Figure 0 reveals that the normal probability plot of residuals is generally satisfactory

21 频率 残差 百分比 残差 since strong linearality is observed, indicating the residuals do follow a normal distribution. Height(mm) 残差图 99 正态概率图 20 与拟合值 残差 拟合值 直方图 20 与顺序 残差 观测值顺序 Figure 0: Residual plot of the factorial design Effects of solution type Figure 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 2, 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.

22 平均值 平均值 Height(mm) 主效应图数据平均值 溶液 Figure : Main effect solution type Height(mm) 交互作用图数据平均值 上升下降 溶液 Figure 2: Interaction of solution type and experimental method Figure 3 presents the contour plot of height with solution type and experimental method whereas Figure 4 presents the response surface plot. Notice that because

23 上升下降 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. Height(mm) 与上升下降, 溶液的等值线图 Height(mm) < > 保持值 管直径 是否浸润 倾斜角 溶液 Figure 3: Contour plot of solution type and experimental method

24 Height(mm) 与上升下降, 溶液的曲面图 保持值 管直径 是否浸润 倾斜角 60 H e i g h t (m m ) 溶液 0 上升下降 Figure 4: 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 5. 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 mm. Not surprisingly, the main effect of the tube diameter at the high level (8.63) is half of that at the low level (38.2), which corroborates the theory.

25 平均值 40 Height(mm) 主效应图数据平均值 管直径 Figure 5: Main effect of tube diameter Effects of experimental method The main effect of the experimental method on the capillarity height is shown in Figure 6. 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.

26 平均值 平均值 32 Height(mm) 主效应图数据平均值 上升下降 Figure 6: Main effect of experimental method Height(mm) 交互作用图数据平均值 溶液 上升下降 Figure 7: Interaction of experimental method and solution type As shown in Figure 7, the two-factor interactions of experimental method and solution type are significant. At the low level of solution type (water), the change in

27 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

28 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 8, 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 8: Three-factor interaction of solution type, tube diameter and experimental method Regression Model Table 0: 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 )

29 系数标 项 效应 系数 准误 T P 常量 溶液 管直径 上升下降 溶液 * 上升下降 溶液 * 管直径 * 上升下降 S = PRESS = R-Sq = 86.4% 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 R R 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 0, 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

30 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:. 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 8, 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

31 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 [] 程阳 (200), 由于毛细现象液面上升高度计算的思考 [J] 数理医药学杂志, 200 年第 23 卷第 4 期, [2] 郑少山 (200), 毛细现象和接触角 [J] 甘肃高师学报, 第 6 卷第 2 期 (200), 28-3 [3] 胡明政, 何荣良 (2008), 用吉布斯函数讨论短毛细管的毛细现象 [J] 中国新技术新产品,2008 NO.09 ( 下 ),75 [4] 黄仁忠 (999), 毛细管中液柱为什么升不到 预期 的高度 与李传文等同志商榷 [J] 大学物理, 第 8 卷第 8 期,23 [5] 任文辉, 林智群, 彭道林, 液体表面张力系数与温度和浓度的关系 [J] 湖南农业大学学报 ( 自然科学版 ) 2004,(0)

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