Differential equation for the rate of capillary action of water on a single brand tissue paper

Ruivivar, Sylvester S.*BS Chemical Engineering 3, January13, 2015

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1. Introduction

Capillary action, or capillarity, is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity. [1]  It is defined as the movement of water within the spaces of a porous material due to the forces of cohesion, adhesion, and surface tension. [2]

The mutual attractive force that exists between like molecules of a particular liquid is called cohesion. This force is responsible for holding a raindrop together as a single unit. Cohesion allows for the development of surface tension, the capacity of a substance to withstand

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being ruptured when placed under tension or stress. [3]

When an attractive force exists between two unlike materials, such as liquid and a solid container, the attractive force is known as adhesion. Adhesion is the force that causes water to stick to the inside of a glass. If the adhesive force between the liquid and solid is greater than the cohesive force within the liquid, the liquid is said to wet the surface and the surface of the liquid near the edge of the container will curve upward. In cases where the cohesive force is greater than adhesion, the liquid is said to be non-wetting and the surface will curve downward near the edge of the container.

Fig-1. Non-wet and wet surfaces

Bicol UniversityCollege of EngineeringDEPARTMENT OF CHEMICAL ENGINEERINGLegazpi City

ABSTRACT

A single brand tissue paper was subjected to an experiment to determine the differential equation

of the rate of capillarity action dhdt

of water on the said material. A single ply of Toily 2-ply bathroom

tissue was used as a medium. With three trials, the time and height of water rise were acquired. Each trial lasted only for an hour. The data were tabulated and graphed separately. The graph for the average of the three trials and the equation of the curve was then obtained. From the equation of the curve, the differential equation for the rate of capillary action of water on the said medium (having a width of 100mm) was

formulated and was found out to be dhdt

=1.2518t

, wherein when integrated will have a value for c equal

to 2.1641 . Values obtained from this DE were close enough to the actual value gathered during the

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The combination of the adhesive forces and the surface tension that arises from cohesion produces the characteristic upward curve in a wetting fluid. Capillarity is the result of cohesion of water molecules and adhesion of those molecules to the solid material forming the void. As the edges of the container are brought closer together, such as in a very narrow tube, the interaction of these phenomena causes the liquid to be drawn upward in the tube. The more narrow the tube, the greater the rise of the liquid. Greater surface tension and increased ratio of adhesion to cohesion also result in greater rise.[4]

A familiar example of capillary action is the tendency of a dry paper towel to absorb a liquid by drawing it into the narrow openings between the fibers. Paper towels are permeable and porous, meaning that they contain small spaces that both liquid and air may pass through. [5]

In the experiment performed, a single brand bathroom tissue was used as a medium for capillary action. This experiment aims to obtain a differential equation for the said mechanism, the differential of height h (in centimeters) of the water over the differential of time t (in minutes).

2. Materials and Methods

2.1 Materials

The set up for the experiment used a meter stick, stopwatch, wooden stick and a glass container which contains the water. The bathroom tissue which was used as the medium for capillary action is of single brand (Toily, 2-ply bathroom tissue), measuring 100mm x 100mm. To avoid the effects of the gap between the 2-ply tissue paper, only a single ply of the said medium was used.

2.2 Preparation of the Fluid

Water was placed on a shallow glass container whose mouth is wide enough to accommodate the tip of the tissue paper. The fluid on the container was about 5cm in depth.

2.3 Preparation of the Tissue Holder

A wooden stick was attached perpendicular to the meter stick at 30cm mark. This will serve as the holder of the tissue paper.

2.4 Procedure

The tissue paper with a length of 30cm and with of 100mm was hanged on the tissue holder. It hangs from the 30 cm mark down to 4cm mark on the meter stick. The glass container which contains the water of about 5cm in depth was placed under the tip of the tissue paper. The tip of the bathroom tissue was submerged on the water (at about 1cm).

2.5 Data Collection

The height (in centimeters) of the water rise was recorded on a table in a span of an hour (60 minutes) only. The experiment was done three times using the same materials and set up.

2.6 The Set-up

Fig-2. The Set up

3. Results and Discussion

3.1. Obtaining the Height of Water Rise in a span of an hour (60 min) to determine the Rate of Capillarity

The experiment was done three times. The water on the container has a depth of 5 cm. The tip of the tissue paper at about 1 cm was submerged on the water. The rise of the water on the tissue paper was determined starting from the 5cm mark.

Table 1 shows the data collected on the three trials.

Table-1.. Height of Fluid (cm) VS. Time (min). 5cm as the baseline

Time(minutes)

Height in Centimeter1st Trial 2nd Trial 3rd

Trial0 5.0 5.0 5.01 7.5 7.5 7.02 8.0 8.0 8.0

3 8.3 8.5 8.34 8.6 9.0 8.55 9.0 9.2 8.76 9.3 9.5 9.07 9.5 9.9 9.48 9.5 10 9.59 9.7 10.2 9.5

10 9.9 10.5 9.711 10.1 10.8 9.912 10.3 11.0 10.013 10.5 11.0 10.214 10.6 11.2 10.515 10.7 11.3 10.520 11.0 11.5 10.925 11.3 11.9 11.030 11.5 12.3 11.240 11.5 12.5 11.350 11.5 12.5 11.560 11.5 12.5 11.5

For convenience, the actual rise of the water was tabulated on a zero baseline. This was shown in Table 2.

Table-2. Height of Fluid (cm) VS. Time (min). 0cm as the baseline

TimeIn minutes

Height in Centimeter1st Trail 2nd trial 3rd trial

0 0 0 01 2.5 2.5 2.02 3.0 3.0 3.03 3.3 3.5 3.34 3.6 4.0 3.55 4.0 4.2 3.76 4.3 4.5 4.07 4.5 4.9 4.48 4.5 5 4.59 4.7 5.2 4.5

10 4.9 5.5 4.711 5.1 5.8 4.912 5.3 6.0 5.013 5.5 6.0 5.2

14 5.6 6.2 5.515 5.7 6.3 5.520 6.0 6.5 5.925 6.3 6.9 6.030 6.5 7.3 6.240 6.5 7.5 6.350 6.5 7.5 6.560 6.5 7.5 6.5

On Table 3, the mean or average of the three trials was determined and were used to determine differential equation of the capillarity.

Table-3. Average of the Three Trials

TimeIn

minutes

Height in Centimeter1st

Trial2nd Trial

3rd Trial

Ave

0 0 0 0 01 2.5 2.5 2.0 2.332 3.0 3.0 3.0 3.003 3.3 3.5 3.3 3.374 3.6 4.0 3.5 3.705 4.0 4.2 3.7 3.976 4.3 4.5 4.0 4.277 4.5 4.9 4.4 4.608 4.5 5 4.5 4.679 4.7 5.2 4.5 4.80

10 4.9 5.5 4.7 5.0311 5.1 5.8 4.9 5.2712 5.3 6.0 5.0 5.4313 5.5 6.0 5.2 5.5714 5.6 6.2 5.5 5.7715 5.7 6.3 5.5 5.8320 6.0 6.5 5.9 6.1325 6.3 6.9 6.0 6.4030 6.5 7.3 6.2 6.6740 6.5 7.5 6.3 6.7750 6.5 7.5 6.5 6.8360 6.5 7.5 6.5 6.83

From the three tables above, it is shown that the water rise was observed for 15 minutes in a 1 minute interval. This is because the rise of

water is rapid on the said interval. As time passes it becomes slow.

3.2 Determination of Rate of Capillarity

The data on the table were graphed separately to show the behavior of the curve. The x- axis corresponds for the time t (in seconds) while the y-axis corresponds to the height h (in centimeters).

The data do not form consistent curves so trend lines were used to show logarithmic relationship between variables.

The rate of capillarity on the first, second and third trials as well as the average of the three trials were illustrated on the graphs below.

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7f(x) = 1.154999155735 ln(x) + 2.28391035517f(x) = 1.154999155735 ln(x) + 2.28391035517

Trial 1

Time (min)

Heig

ht (c

m)

Fig-3. Rate of Capillarity in Trial 1

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8f(x) = 1.41888113034 ln(x) + 2.18935238195

Trial 2

Time (min)

Heig

ht (c

m)

Fig-4. Rate of Capillarity in Trial 2

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7f(x) = 1.18232808398 ln(x) + 2.04367866897

Trial 3

Time (min)

Heig

ht (c

m)

Fig-5. Rate of Capillarity in Trial 3

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

f(x) = 1.251775776603 ln(x) + 2.173319457196

Average

Time (min)He

ight

(cm

)

Fig-6. Average Rate of Capillarity

Using the equation obtained from the graph which is

y=1.2518 ln (x)+2.1733 (Eq 1)

where y is the height in centimeters and x is the time in minutes, we compute for height ( given x=1,2,3…) and error to verify whether the equation is valid.

%error

¿ actual value−computed valueactual value

(100 % )

Fig-7. Formula for Percent Error

The table below shows the new values of height (H) using equation 1 as well as the percent error for the obtained values.

Table-4. Percent Error for the Values (obtained using Eq 1) of H

Time (min)

Height (cm)Actual

Height (cm) computed

% Error

1 2.33 2.17 6.872 3.00 3.04 1.333 3.37 3.55 5.344 3.70 3.91 5.685 3.97 4.19 5.546 4.27 4.42 3.517 4.60 4.61 0.218 4.67 4.77 2.349 4.80 4.92 2.50

10 5.03 5.06 0.6011 5.27 5.17 1.9012 5.43 5.28 2.7613 5.57 5.38 3.4114 5.77 5.48 5.0315 5.83 5.56 4.6320 6.13 5.92 3.4325 6.40 6.20 3.1330 6.67 6.43 3.6040 6.77 6.79 0.3050 6.83 7.07 3.5160 6.83 7.30 6.88

From the table above, it can be seen that the highest % error is 6.88 and the lowest is 0.21. Since the errors are low, we will accept the equation.

3.3 Formulation of Differential Equation

By getting the derivative of the Equation 1 with respect to time, the differential equation for the capillary action was obained.

y=1.2518 ln (x)+2.1733

h=1.2518 ln(t )+2.1733

dhdt

=1.2518t

Fig-8. Differential Equation

To check, we integrate:

∫ dh=∫ 1.2518t

dt

h=1.2518 ln ( t )+c

To get the value of c, we assign h= 4.60 at t=7. As shown from Table 4, the new value of h at t=7 gave the lowest error.

4.60=1.2518 ln (7 )+c

c=¿2.1641

Now we have a new equation for the graph which is :

h=1.2518 ln(t )+2.1641(Eq. 2)

Using Equation 2, we compute for the new value of height and the percent error to determine whether the new equation as well as the differential equation formulated is valid:

Table-5. Percent Error for the Values (obtained using Eq 2) of H

Time (min)

Height (cm)actual

Height (cm) computed

% Error

1 2.33 2.16 7.292 3.00 3.03 13 3.37 3.54 5.044 3.70 3.90 5.415 3.97 4.18 5.296 4.27 4.40 3.047 4.60 4.60 08 4.67 4.77 2.149 4.80 4.91 2.29

10 5.03 5.05 0.4011 5.27 5.17 1.9012 5.43 5.27 2.9513 5.57 5.37 3.6014 5.77 5.47 5.2015 5.83 5.55 4.8020 6.13 5.91 3.59

25 6.40 6.19 3.2830 6.67 6.42 3.7540 6.77 6.78 0.1550 6.83 7.06 3.3760 6.83 7.28 6.59

The new values of the height obtained using equation 2 yielded low percent errors. Therefore the new equation is still valid.

The three values (actual, from Eq1, Eq2), was graphed to visualize their relationships.

Fig-9. Comparison of Values Obtained

4. Conclusion

Based from the data above, the differential equation for the rate of capillary action of water on a single brand tissue paper Toily (single ply) with a width of 100 millimeters is:

dhdt

=1.2518t

This equation, when integrated, will have a value for c which is equal to2.1641. When graphed, it will form a curve.

5. Recommendations

The experiment can also be done to other brands of tissue papers with variation in thickness and width. Other liquids can also be used.

6. Documentation

The following pictures were taken during the conduct of the experiment.

Acknowledgements

The author is truly grateful to his family for their moral and financial support, to Engr. Junel Borbo for sharing his expertise in differential equations, and to his classmates and to all those people who helped and shared their insights for the completion of this journal.

References

[1]https://www.boundless.com/physics/textbook s/boundless-physics-textbook/fluids-10/cohesion-and-adhesion-94/surface-tension-and-capillary-action-347-6310/[2]http://water.usgs.gov/edu/capillaryaction.html[3] Boundless.” Water’s Cohesive and Adhesive Properties.” Boundless Biology. Boundless, 03 Jul.2014. retrieved 04 Jan.2015. From https://www.boundless.com/biology/textbook/boundless-biology-textbook/the-chemical-foundation-of-life-2/water-51/water-s-cohesive-and-adhesive-properties-286-11419/[4]http://science.jrank.org/pages/1182/Capillary-Action.html[5]http://www.education.com/science-fair/article/find-absorbent-brand-paper-towels/

Figures

Fig-1. Non-wet and wet surfacesFig-2. The Set-upFig-3. Rate of Capillarity in Trial 1Fig-4. Rate of Capillarity in Trial 2Fig-5. Rate of Capillarity in Trial 3Fig-6. Average Rate of Capillarity Fig-7. Formula for Percent ErrorFig-8. Differential EquationFig-9. Comparison of Values Obtained

Tables

Table-1. Height of Fluid (cm) VS. Time (min). 5cm as the baseline

Table-2. Height of Fluid (cm) VS. Time (min). 0cm as the baseline

Table-3. Average of the Three TrialsTable-4. Percent Error for the Values (obtained using Eq 1) of H Table-5. Percent Error for the Values (obtained using Eq 2) of H