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Dynamics of liquid rise in a vertical capillary tube
Journal of Colloid and Interface Science 389 (2013) 268–272
Contents lists available at SciVerse ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Dynamics of liquid rise in a vertical capillary tube Reza Masoodi a,⇑, Ehsan Languri b, Alireza Ostadhossein c a
School of Design and Engineering, Philadelphia University, 4201 Henry Ave., Philadelphia, PA 19144, USA Applied Research Associates, 421 Oak Ave., Panama City, FL 32401, USA c Department of Engineering Science and Mechanics, Penn State University, State College, PA 16801, USA b
a r t i c l e
i n f o
Article history: Received 5 June 2012 Accepted 3 September 2012 Available online 21 September 2012 Keywords: Oscillation Capillary model Capillary pressure Lucas–Washburn equation Capillarity Liquid rise Wicking
a b s t r a c t The governing equation for capillary rise in a vertical tube is derived using energy balance. The derived governing equation includes kinetic, gravity, and viscous effects. Through normalizing different terms in the governing equation, a form of nonlinear ordinary differential equation (ODE) with a positive dimensionless parameter was obtained. The ODE equation was solved numerically and the numerical results were compared with some published experimental data. The derived governing equation was found to be quite accurate for predicting the liquid rise and oscillation in a capillary tube. The effect of a dimensionless parameter on the behavior of the liquid rise was explored numerically. A simple critical condition, which leads to the oscillation of the liquid column in the capillary tube, was found in the form of a dimensionless parameter in the governing equation. Published by Elsevier Inc.
1. Introduction The mathematical theory of the capillary rise phenomenon in a tube was first proposed by Green and Ampt [1], and later independently developed by Lucas [2] and Washburn [3]. They used the momentum balance equation after neglecting the gravity and kinetic effects to derive a governing equation, known as Lucas– Washburn equation (LWE), for meniscus height versus time in a tube. The Lucas–Washburn equation (LWE) balances the viscous pressure drop in a capillary tube with the capillary pressure at the meniscus that pulls the liquid up the tube [4]. The Lucas– Washburn equation (LWE) may also called capillary model in the porous media literature, since a porous medium is modeled as a bundle of aligned capillary tubes [4]. The driving force that leads to the capillarity is capillary pressure generated due to the difference in the energies of dry and wet surfaces in a tube [5]. The capillary pressure is obtained by the Young–Laplace equation, which relates the capillary pressure at the interface between two static immiscible fluids in a capillary tube to the radius of the tube, the surface energy of the fluid, and the dynamic contact angle [5]. Several researchers tried to include the missing or neglected terms to modify the LWE (e.g. [4,6–10]). The missing terms are the gravity effects, the kinetic effects, the viscous loss in liquid below the tube, and the viscous loss associated with the entrance ef⇑ Corresponding author. Fax: +1 215 951 2651. E-mail address: [email protected] (R. Masoodi). 0021-9797/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jcis.2012.09.004
fects, etc. [10]. The main approach that was used to derive/modify LWE was applying momentum balance to a control volume inside the capillary tube [6–11]. Previous research showed that including kinetic effect to the traditional LWE is sufficient to match experimental results (e.g. [5,9–11]). Such modified LWE was used to predict the wicking delay at the beginning as well as oscillatory behavior around the equilibrium height [10–12]. The transition in flow pattern between different zones where the kinetic, gravity, and viscous forces are dominant is also studied (e.g. [10–14]). Such studies show that kinetic effects are dominant just at the very beginning of capillarity. The gravity effects are significant near the equilibrium height or Jurin height, where meniscus stops rising [12,15]. The viscous effect is insignificant at the beginning where height of liquid column is small [10,14]. The oscillatory behavior of liquid column was studied by some researchers (e.g. [10,12,16–18]). It was shown that kinetic force is the main reason that may cause the oscillation of liquid column. The viscous force is the main reason for damping the oscillatory energy out. Hamraoui and Nylander suggested a critical value for capillary radius, bellow which the oscillation disappears [16]. It means for a defined liquid and capillary tube, there is a critical radius bellow which the viscous dissipation is strong enough to absorb the kinetic force and prevent oscillation of liquid column. In this paper, we have used energy balance to derive the governing equation for liquid rise in a capillary tube. The governing equation was converted to a dimensionless equation with a parameter. The suggested dimensionless governing equation has just one parameter and is similar to an equation reported by some authors
R. Masoodi et al. / Journal of Colloid and Interface Science 389 (2013) 268–272
269
Nomenclature x
Roman letters A area (m2) b a coefficient defined by Eq. (28) Bo Bond number (Eq. (25)) C perimeter (Eq. (3)) E energy (Eq. (1)) F force (N) g acceleration due to gravity (m/s2) h height of liquid in a capillary tube (Fig. 1) h_ =dh/dt € h =d2h/dt2 H dimensionless height (Eq. (17)) H_ =dH/dT € H =d2H/dT2 L length (m) m mass (kg) Oh Ohnesorge number (Eq. (24)) p pressure (Pa) Q volume flow rate (m3/s) R radius (m) t time (s) T dimensionless time (Eq (18)) u velocity toward z (Fig. 1) V velocity (m/s)
Greek letters h contact angle (°) l viscosity of liquid (kg/m s) specific time (Eq. (20)) s q density (kg/m3) c surface tension (N/m) x a dimensionless coefficient (Eq. (22)) Subscripts 1 an index referring to a coordinate axis 2 an index referring to a coordinate axis 3 an index referring to a coordinate axis c capillary cr critical e equilibrium g gravity k kinetic l liquid s solid sl solid–liquid v viscosity
(i.e. [10,12,24]). The dimensionless equation was solved numerically and results compared well with the experimental data. The critical condition that leads to oscillation was explored by numerical solution of governing equation. 2. Mathematical modeling In order to find the governing equation for liquid rise we apply the energy balance principle to the liquid column in a vertical capillary tube. The amount of energy needed to raise the liquid in the tube is equal to the sum of the viscous energy dissipated by the fluid, the kinetic energy spent on accelerating the fluid from zero to the capillarity speed and to maintain its speed, and the energy needed to overcome the gravity. As the liquid moves through a capillary tube, it reduces the dry solid–air interface area and thus increases the wetted solid–liquid interface area. If cs stands for the surface energy of the dry surface, and csl stands for surface energy of the wetted surface areas, then energy balance leads to [19].
cs dAsl csl dAsl ¼ dEv þ dEg þ dEk
a coordinate
ð1Þ
in which dAsl is the area of solid–liquid interface, Ev is viscous work done by the flow of the liquid, Eg is the work done by gravity, and Ek
is kinetic energy. According to Young’s equation [20], the relation between the contact angle and surface energy is given by:1
cos h ¼
cs csl cl
ð2Þ
Fig. 1 shows liquid-rise in a vertical capillary tube as a result of capillarity. The interfacial area dAsl is the wetted surface of the capillary tube that is related to the height of liquid rise, h, and the inside perimeter in a cross-section of the capillary tube, C, through following expression:
dAsl ¼ Cdh
ð3Þ
Using Eqs. (2) and (3) in Eq. (1) leads to the following expression:
C cl cosðhÞ ¼
dEv dEg dEk þ þ dh dh dh
ð4Þ
Right hand side of Eq. (4) has three terms that are viscous, gravitational, and kinetic forces respectively.
C cl cosðhÞ ¼ F v þ F g þ F k
ð5Þ
The differential equation of mass balance for a fluid is [22].
dV 1 dV 2 dV 3 þ þ ¼0 dx1 dx2 dx3
ð6Þ
where V1, V2, and V3 are velocity components toward x1, x2, and x3 axes. The mass balance in the capillary tube shown in Fig. 1 has just one term:
du ¼0 dz
Fig. 1. A schematic of liquid rise in a capillary tube.
ð7Þ
1 Note that Eq. (2) is equilibrium contact angle, which is valid for static contact angle. However, dynamic contact angle is a function of both capillary number and static contact angle [21]. Here, the contact angle variation is neglected and dynamic contact angle is assumed to be identical to static contact angle. However, it is possible to use dynamic contact angle in the capillary pressure that will appear in the governing equation [5].
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where u is the average velocity of liquid inside the capillary tube and toward z. According to Eq. (7), the velocity is constant throughout a capillary tube, which means:
Eq. (15) and dividing all terms by Pc lead to the following expression.
u ¼ h_
1¼
ð8Þ
Eq. (8) indicates that the rate of liquid rise in the tube is identical to the average velocity of liquid inside the tube. The viscous pressure drop can be found using the Hagen–Poiseuille law as [22].
DP ¼
8lLQ
pR
4
ð9Þ
where l is the dynamic viscosity, L is the length of tube, Q is the volume flow rate, R is the radius, and Dp is the pressure drop inside the tube. For the tube shown in Fig. 1, the viscous pressure drop can be expressed as:
DP ¼
8l _ hh R2
ð10Þ
Therefore, the viscous force associated with the above viscous pressure is:
8l _ F v ¼ 2 hhA R
ð11Þ
where A is the cross-sectional area of the tube. The gravitational force is the weight of the liquid column inside the capillary tube and can be expressed as:
F g ¼ qgAh
ð12Þ
The kinetic force acts on the column of liquid, which depends on mass and velocity of the liquid. Since both mass and velocity of liquid column change ith time, so:
Fk ¼
d d _ ðmuÞ ¼ qA ðhhÞ dt dt
ð13Þ
Substituting expressions for viscous, gravitational, and kinetic forces into Eq. (5) leads to the following expression for liquid-front movement in a capillary tube:
2c cosðhÞ 8l _ d _ ¼ 2 hh þ qgh þ q ðhhÞ 2A=C dt R
ð14Þ
The left hand side of Eq. (14) is the definition of capillary pressure [5], so:
Pc ¼
8l _ d _ hh þ qgh þ q ðhhÞ dt R2
ð15Þ
Eq. (15) indicates that the capillary pressure is identical to the summation of the viscous drop pressure, hydrostatic pressure, and pressure due to kinetic force. Eq. (15) is the governing equation for flow in a capillary tube with following initial conditions.
hð0Þ ¼ 0
ð16aÞ
_ hð0Þ ¼0
ð16bÞ
3. Dimensionless parameters
T¼
h he t
s
Pc R
2
s
HH_ þ
qghe Pc
Hþ
qh2e d _ ðHHÞ Pc s2 dT
ð17Þ
ð18Þ
where he is a reference height and s is a reference time used to normalize height and time. Using dimensionless variables H and T in
ð19Þ
The first two coefficients on right side of Eq. (19) are assumed to be unity to find he and s as follows:
Pc
he ¼
s¼
ð20aÞ
qg
8l P c q 2 g 2 R2
ð20bÞ
Note that the he, as found by Eq (20a), is Jurin height. Assuming the third coefficient in Eq (20a) to be x, Eq (19) transforms to the following equation
HH_ þ H þ x
_ dðHHÞ ¼1 dT
ð21Þ
where x is a positive dimensionless number, obtained by
q2 gR4 64l2 he
x¼
ð22Þ
Note that H and s are identical to h and t reported by Fries [10] and Fries and Dreyer [24]. The parameter defined by Eq. (22) is identical to X12 reported by Fries [10], Quéré et al. [12], and Fries and Dreyer [24]. Note that a small value of x indicates a small influence of the kinetic forces in Eq. (21). The coefficient x is related to the other recognized dimensionless numbers through following expression:2
2 1 Bo 128 cosðhÞ Oh
x¼
ð23Þ
where Bo is the Bond number and Oh is the Ohnesorge number that are defined as:
l
Oh ¼ pffiffiffiffiffiffiffiffiffi qRc Bo ¼
qgR2 c
ð24Þ
ð25Þ
The boundary conditions for Eq. (21) are:
Hð0Þ ¼ 0:
ð26aÞ
_ Hð0Þ ¼ 0:
ð26bÞ
Eq. (15) was derived for upward moving liquid in a capillary tube. As discussed by Quéré et al. [12], Fries [10], and Lorenceau et al. [23], to use this equation for downward moving liquid in a capillary tube, term h_ 2 should be omitted. Consequently, term H_ 2 should be omitted from Eq. (21) to use it for downward moving flow. Therefore, the general form of dimensionless governing equation for both upward and downward moving liquid is:
HH_ þ H þ xðHH_ þ bH_ 2 Þ ¼ 1
In order to simplify the governing equation, following dimensionless variables are introduced:
H¼
2
8lhe
ð27Þ
where b is a coefficient that is different for upward and downward velocity as following:
b¼
1 if dH P 0 0
if dH < 0
ð28Þ
2 It is possible to find other relations between x and dimensionless numbers. According to Fries [10] and Fries and Dreyer [24], x ¼ 129BoGa cosðhÞ, where Ga is Galileo number. This coefficient can also be related to Ca, Bo, and Fr, where Fr is Froude number [4].
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R. Masoodi et al. / Journal of Colloid and Interface Science 389 (2013) 268–272
4. Results and discussion The standard forth-order Runge–Kutta method was used to solve the Eq. (27) with initial conditions Eqs. (26a) and (26b). Figs. 2 and 3 compare the numerical predictions of Eq. (27) with the experimental data. The liquid height and time were normalized using Eqs. (17) and (18). Note that x = 1.4 for the data reported by Zhmud et al. [18] and x = 21.4 for the data reported by Quéré et al. [12]. The accuracy of numerical results is very good at the
beginning but the error increases slightly as time passes. It should be related to the neglected terms/effects, such as the entrance effect, the difference between descending and ascending contact angles, and Vena contracta effect [6,10,19]. Fig. 4 shows the numerical prediction for small values of x. Since the differences between different curves are significant for short times, so the results are restricted to T < 2. When x = 0, the predictions of Eq. (27) matches predictions obtained by LWE. Note that the kinetic effect is not included in LWE, which is identical to Eq. (21) when x = 0 [9]. Fig. 5 shows the numerical results for cases of significant kinetic effect (higher values of x). Figs. 4 and 5 show that at the beginning of the capillary motion, the liquid velocity is slows down as x increases. As x increases, more energy is needed
Fig. 2. A comparison of numerical predictions using Eq. (27) with experimental data reported by Zhmud et al. [18]. Here, density, capillary radius, contact angle, surface tension, and viscosity are 710 kg/m3, 0.5 mm, 0, 16.7 mN/m, and 0.6 mPa s, respectively [18].
Fig. 5. Oscillatory behavior of liquid column.
Fig. 3. A comparison of numerical predictions using Eq. (27) with experimental data reported by Quéré et al. [12]. Here, density, Jurin height, contact angle, surface tension, and viscosity are 710 kg/m3, 7.1 mm, 0, 16.6 mN/m, and 0.3 mPa s respectively [12].
(a) A comparison of curves for
and
(b) A closer look at curves crossing H=1 Fig. 4. The dimensionless height versus dimensionless time for low values of x and short times.
Fig. 6. Critical value of x which leads to oscillation behavior in Eq. (27).
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R. Masoodi et al. / Journal of Colloid and Interface Science 389 (2013) 268–272
to overcome the kinetic effects, so liquid velocity is slower at the beginning of the motion. However, as liquid reaches the equilibrium height (Jurin height), this kinetic force prevents liquid column from stopping and so liquid rises higher than the equilibrium height. It takes time for this kinetic force to damp out by viscous forces, which leads to an oscillatory behavior at the liquid front around the Jurin height. Fig. 6 shows that there is a critical value of x between 0.2 and 0.3 above which oscillation occurs. The detail study of numerical data shows that the oscillation appears when x > 0.25, which means xcr = 0.25 is the critical value. This critical condition is identical to X 6 2 that is reported by Fries [10], Quéré et al. [12], and Fries and Dreyer [24]. However, they used equal sign for oscillation to occur while our numerical data does not support equal sign. Using Eq. (22), it can be shown that the critical radius, Rcr, which is related to xcr is:
Rcr ¼ 2
l
2
1 c cosðhÞ 5 q2 g 2
ð29Þ
The Eq. (29) is identical to an expression reported by Hamraoui and Nylander [16]. The above equation indicates that the oscillations appears if R > Rcr. In the other words, when x > 0.25 or R > Rcr, the kinetic force is strong enough to rise liquid up the equilibrium height, which leads to oscillation around Jurin height. 5. Summary and conclusion We used energy balance to derive the governing equation for liquid rise in a capillary tube. A simple method was used to derive dimensionless governing equation, which has just one parameter. The standard fourth-order Runge–Kutta method was employed to solve the nonlinear dimensionless governing equation. The governing equation and numerical method were validated by comparing the numerical results with some published experimental data. The dimensionless parameter (x) in the dimensionless governing equation determines the oscillatory behavior of the liquid column. We showed that x > xcr leads to the oscillation of the liquid
column in the capillary tube where xcr = 0.25. There is also a relevant critical capillary radius for xcr, which is a function of liquid properties and contact angle. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jcis.2012.09.004. References [1] [2] [3] [4]
[5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
W.H. Green, G.A. Ampt, J. Agric. Sci. 4 (1911) 1–24. R. Lucas, Kollid Z. 23 (1918) 15–22. E.V. Washburn, Phys. Rev. 17 (1921) 273–283. R. Masoodi, K.M. Pillai, Traditional theories of wicking: capillary models, in: Masoodi, Pillai (Ed.), Wicking in Porous Materials: Traditional and Modern Modeling Approaches, Taylor & Francis, 2012 (Chapter 3). R. Masoodi, K.M. Pillai, J. Porous Media 15 (8) (2012) 775–783. J. Szekely, A.W. Neumann, Y.K. Chuang, J. Colloid Interface Sci. 35 (2) (1971) 273–278. B. Lavi, A. Marmur, J. Bachmann, Langmuir 24 (5) (2008) 1918–1923. R. Chebbi, J. Colloid Interface Sci. 315 (2007) 255–260. R. Masoodi, K.M. Pillai, P.P. Varanasi, Role of hydraulic and capillary radii in improving the effectiveness of capillary model in wicking, in: ASME Summer Conference, Jacksonville, FL, USA, August 10–14, 2008. N. Fries, Capillary Transport Processes in Porous Materials – Experiment and Model, Cuvillier Verlag Göttingen, 2010. M. Stange, M.E. Dreyer, H.J. Rath, Phys. Fluids 15 (9) (2003) 2587–2601. D. Quéré, E. Raphaël, J.Y. Ollitrault, Langmuir 15 (1999) 3679–3682. R. Masoodi, K.M. Pillai, P.P. Varanasi, AIChE J. 53 (11) (2007) 2769–2782. N. Fries, M. Dreyer, J. Colloid Interface Sci. 320 (1) (2008) 259–263. J. Cai, B. Yu, M. Mei, L. Luo, Chin. Phys. Lett. 27 (5) (2010) 054701. A. Hamraoui, T. Nylander, J. Colloid Interface Sci. 250 (2002) 415–421. D. Quéré, Europhys. Lett. 39 (5) (1997) 533–538. B.V. Zhmud, F. Tiberg, K. Hallstensson, J. Colloid Interface Sci. 228 (2000) 263– 269. S. Chwastiak, J. Colloid Interface Sci. 42 (2) (1973) 298–309. J.C. Berg, Wettability, Marcel Dekker, New York, 1993. M. Bracke, F. De Voeght, P. Joos, Prog. Colloid Polym. Sci. 79 (1989) 142–149. F.M. White, Fluid Mechanics with Student DVD, seventh ed., McGraw Hill, New York, 2010. E. Lorenceau, D. Quéré, J.Y. Ollitrault, C. Clanet, Phys. Fluids 14 (6) (2002) 1985–1992. N. Fries, M. Dreyer, J. Colloid Interface Sci. 338 (2) (2009) 514–518.
Contents lists available at SciVerse ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Dynamics of liquid rise in a vertical capillary tube Reza Masoodi a,⇑, Ehsan Languri b, Alireza Ostadhossein c a
School of Design and Engineering, Philadelphia University, 4201 Henry Ave., Philadelphia, PA 19144, USA Applied Research Associates, 421 Oak Ave., Panama City, FL 32401, USA c Department of Engineering Science and Mechanics, Penn State University, State College, PA 16801, USA b
a r t i c l e
i n f o
Article history: Received 5 June 2012 Accepted 3 September 2012 Available online 21 September 2012 Keywords: Oscillation Capillary model Capillary pressure Lucas–Washburn equation Capillarity Liquid rise Wicking
a b s t r a c t The governing equation for capillary rise in a vertical tube is derived using energy balance. The derived governing equation includes kinetic, gravity, and viscous effects. Through normalizing different terms in the governing equation, a form of nonlinear ordinary differential equation (ODE) with a positive dimensionless parameter was obtained. The ODE equation was solved numerically and the numerical results were compared with some published experimental data. The derived governing equation was found to be quite accurate for predicting the liquid rise and oscillation in a capillary tube. The effect of a dimensionless parameter on the behavior of the liquid rise was explored numerically. A simple critical condition, which leads to the oscillation of the liquid column in the capillary tube, was found in the form of a dimensionless parameter in the governing equation. Published by Elsevier Inc.
1. Introduction The mathematical theory of the capillary rise phenomenon in a tube was first proposed by Green and Ampt [1], and later independently developed by Lucas [2] and Washburn [3]. They used the momentum balance equation after neglecting the gravity and kinetic effects to derive a governing equation, known as Lucas– Washburn equation (LWE), for meniscus height versus time in a tube. The Lucas–Washburn equation (LWE) balances the viscous pressure drop in a capillary tube with the capillary pressure at the meniscus that pulls the liquid up the tube [4]. The Lucas– Washburn equation (LWE) may also called capillary model in the porous media literature, since a porous medium is modeled as a bundle of aligned capillary tubes [4]. The driving force that leads to the capillarity is capillary pressure generated due to the difference in the energies of dry and wet surfaces in a tube [5]. The capillary pressure is obtained by the Young–Laplace equation, which relates the capillary pressure at the interface between two static immiscible fluids in a capillary tube to the radius of the tube, the surface energy of the fluid, and the dynamic contact angle [5]. Several researchers tried to include the missing or neglected terms to modify the LWE (e.g. [4,6–10]). The missing terms are the gravity effects, the kinetic effects, the viscous loss in liquid below the tube, and the viscous loss associated with the entrance ef⇑ Corresponding author. Fax: +1 215 951 2651. E-mail address: [email protected] (R. Masoodi). 0021-9797/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jcis.2012.09.004
fects, etc. [10]. The main approach that was used to derive/modify LWE was applying momentum balance to a control volume inside the capillary tube [6–11]. Previous research showed that including kinetic effect to the traditional LWE is sufficient to match experimental results (e.g. [5,9–11]). Such modified LWE was used to predict the wicking delay at the beginning as well as oscillatory behavior around the equilibrium height [10–12]. The transition in flow pattern between different zones where the kinetic, gravity, and viscous forces are dominant is also studied (e.g. [10–14]). Such studies show that kinetic effects are dominant just at the very beginning of capillarity. The gravity effects are significant near the equilibrium height or Jurin height, where meniscus stops rising [12,15]. The viscous effect is insignificant at the beginning where height of liquid column is small [10,14]. The oscillatory behavior of liquid column was studied by some researchers (e.g. [10,12,16–18]). It was shown that kinetic force is the main reason that may cause the oscillation of liquid column. The viscous force is the main reason for damping the oscillatory energy out. Hamraoui and Nylander suggested a critical value for capillary radius, bellow which the oscillation disappears [16]. It means for a defined liquid and capillary tube, there is a critical radius bellow which the viscous dissipation is strong enough to absorb the kinetic force and prevent oscillation of liquid column. In this paper, we have used energy balance to derive the governing equation for liquid rise in a capillary tube. The governing equation was converted to a dimensionless equation with a parameter. The suggested dimensionless governing equation has just one parameter and is similar to an equation reported by some authors
R. Masoodi et al. / Journal of Colloid and Interface Science 389 (2013) 268–272
269
Nomenclature x
Roman letters A area (m2) b a coefficient defined by Eq. (28) Bo Bond number (Eq. (25)) C perimeter (Eq. (3)) E energy (Eq. (1)) F force (N) g acceleration due to gravity (m/s2) h height of liquid in a capillary tube (Fig. 1) h_ =dh/dt € h =d2h/dt2 H dimensionless height (Eq. (17)) H_ =dH/dT € H =d2H/dT2 L length (m) m mass (kg) Oh Ohnesorge number (Eq. (24)) p pressure (Pa) Q volume flow rate (m3/s) R radius (m) t time (s) T dimensionless time (Eq (18)) u velocity toward z (Fig. 1) V velocity (m/s)
Greek letters h contact angle (°) l viscosity of liquid (kg/m s) specific time (Eq. (20)) s q density (kg/m3) c surface tension (N/m) x a dimensionless coefficient (Eq. (22)) Subscripts 1 an index referring to a coordinate axis 2 an index referring to a coordinate axis 3 an index referring to a coordinate axis c capillary cr critical e equilibrium g gravity k kinetic l liquid s solid sl solid–liquid v viscosity
(i.e. [10,12,24]). The dimensionless equation was solved numerically and results compared well with the experimental data. The critical condition that leads to oscillation was explored by numerical solution of governing equation. 2. Mathematical modeling In order to find the governing equation for liquid rise we apply the energy balance principle to the liquid column in a vertical capillary tube. The amount of energy needed to raise the liquid in the tube is equal to the sum of the viscous energy dissipated by the fluid, the kinetic energy spent on accelerating the fluid from zero to the capillarity speed and to maintain its speed, and the energy needed to overcome the gravity. As the liquid moves through a capillary tube, it reduces the dry solid–air interface area and thus increases the wetted solid–liquid interface area. If cs stands for the surface energy of the dry surface, and csl stands for surface energy of the wetted surface areas, then energy balance leads to [19].
cs dAsl csl dAsl ¼ dEv þ dEg þ dEk
a coordinate
ð1Þ
in which dAsl is the area of solid–liquid interface, Ev is viscous work done by the flow of the liquid, Eg is the work done by gravity, and Ek
is kinetic energy. According to Young’s equation [20], the relation between the contact angle and surface energy is given by:1
cos h ¼
cs csl cl
ð2Þ
Fig. 1 shows liquid-rise in a vertical capillary tube as a result of capillarity. The interfacial area dAsl is the wetted surface of the capillary tube that is related to the height of liquid rise, h, and the inside perimeter in a cross-section of the capillary tube, C, through following expression:
dAsl ¼ Cdh
ð3Þ
Using Eqs. (2) and (3) in Eq. (1) leads to the following expression:
C cl cosðhÞ ¼
dEv dEg dEk þ þ dh dh dh
ð4Þ
Right hand side of Eq. (4) has three terms that are viscous, gravitational, and kinetic forces respectively.
C cl cosðhÞ ¼ F v þ F g þ F k
ð5Þ
The differential equation of mass balance for a fluid is [22].
dV 1 dV 2 dV 3 þ þ ¼0 dx1 dx2 dx3
ð6Þ
where V1, V2, and V3 are velocity components toward x1, x2, and x3 axes. The mass balance in the capillary tube shown in Fig. 1 has just one term:
du ¼0 dz
Fig. 1. A schematic of liquid rise in a capillary tube.
ð7Þ
1 Note that Eq. (2) is equilibrium contact angle, which is valid for static contact angle. However, dynamic contact angle is a function of both capillary number and static contact angle [21]. Here, the contact angle variation is neglected and dynamic contact angle is assumed to be identical to static contact angle. However, it is possible to use dynamic contact angle in the capillary pressure that will appear in the governing equation [5].
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where u is the average velocity of liquid inside the capillary tube and toward z. According to Eq. (7), the velocity is constant throughout a capillary tube, which means:
Eq. (15) and dividing all terms by Pc lead to the following expression.
u ¼ h_
1¼
ð8Þ
Eq. (8) indicates that the rate of liquid rise in the tube is identical to the average velocity of liquid inside the tube. The viscous pressure drop can be found using the Hagen–Poiseuille law as [22].
DP ¼
8lLQ
pR
4
ð9Þ
where l is the dynamic viscosity, L is the length of tube, Q is the volume flow rate, R is the radius, and Dp is the pressure drop inside the tube. For the tube shown in Fig. 1, the viscous pressure drop can be expressed as:
DP ¼
8l _ hh R2
ð10Þ
Therefore, the viscous force associated with the above viscous pressure is:
8l _ F v ¼ 2 hhA R
ð11Þ
where A is the cross-sectional area of the tube. The gravitational force is the weight of the liquid column inside the capillary tube and can be expressed as:
F g ¼ qgAh
ð12Þ
The kinetic force acts on the column of liquid, which depends on mass and velocity of the liquid. Since both mass and velocity of liquid column change ith time, so:
Fk ¼
d d _ ðmuÞ ¼ qA ðhhÞ dt dt
ð13Þ
Substituting expressions for viscous, gravitational, and kinetic forces into Eq. (5) leads to the following expression for liquid-front movement in a capillary tube:
2c cosðhÞ 8l _ d _ ¼ 2 hh þ qgh þ q ðhhÞ 2A=C dt R
ð14Þ
The left hand side of Eq. (14) is the definition of capillary pressure [5], so:
Pc ¼
8l _ d _ hh þ qgh þ q ðhhÞ dt R2
ð15Þ
Eq. (15) indicates that the capillary pressure is identical to the summation of the viscous drop pressure, hydrostatic pressure, and pressure due to kinetic force. Eq. (15) is the governing equation for flow in a capillary tube with following initial conditions.
hð0Þ ¼ 0
ð16aÞ
_ hð0Þ ¼0
ð16bÞ
3. Dimensionless parameters
T¼
h he t
s
Pc R
2
s
HH_ þ
qghe Pc
Hþ
qh2e d _ ðHHÞ Pc s2 dT
ð17Þ
ð18Þ
where he is a reference height and s is a reference time used to normalize height and time. Using dimensionless variables H and T in
ð19Þ
The first two coefficients on right side of Eq. (19) are assumed to be unity to find he and s as follows:
Pc
he ¼
s¼
ð20aÞ
qg
8l P c q 2 g 2 R2
ð20bÞ
Note that the he, as found by Eq (20a), is Jurin height. Assuming the third coefficient in Eq (20a) to be x, Eq (19) transforms to the following equation
HH_ þ H þ x
_ dðHHÞ ¼1 dT
ð21Þ
where x is a positive dimensionless number, obtained by
q2 gR4 64l2 he
x¼
ð22Þ
Note that H and s are identical to h and t reported by Fries [10] and Fries and Dreyer [24]. The parameter defined by Eq. (22) is identical to X12 reported by Fries [10], Quéré et al. [12], and Fries and Dreyer [24]. Note that a small value of x indicates a small influence of the kinetic forces in Eq. (21). The coefficient x is related to the other recognized dimensionless numbers through following expression:2
2 1 Bo 128 cosðhÞ Oh
x¼
ð23Þ
where Bo is the Bond number and Oh is the Ohnesorge number that are defined as:
l
Oh ¼ pffiffiffiffiffiffiffiffiffi qRc Bo ¼
qgR2 c
ð24Þ
ð25Þ
The boundary conditions for Eq. (21) are:
Hð0Þ ¼ 0:
ð26aÞ
_ Hð0Þ ¼ 0:
ð26bÞ
Eq. (15) was derived for upward moving liquid in a capillary tube. As discussed by Quéré et al. [12], Fries [10], and Lorenceau et al. [23], to use this equation for downward moving liquid in a capillary tube, term h_ 2 should be omitted. Consequently, term H_ 2 should be omitted from Eq. (21) to use it for downward moving flow. Therefore, the general form of dimensionless governing equation for both upward and downward moving liquid is:
HH_ þ H þ xðHH_ þ bH_ 2 Þ ¼ 1
In order to simplify the governing equation, following dimensionless variables are introduced:
H¼
2
8lhe
ð27Þ
where b is a coefficient that is different for upward and downward velocity as following:
b¼
1 if dH P 0 0
if dH < 0
ð28Þ
2 It is possible to find other relations between x and dimensionless numbers. According to Fries [10] and Fries and Dreyer [24], x ¼ 129BoGa cosðhÞ, where Ga is Galileo number. This coefficient can also be related to Ca, Bo, and Fr, where Fr is Froude number [4].
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4. Results and discussion The standard forth-order Runge–Kutta method was used to solve the Eq. (27) with initial conditions Eqs. (26a) and (26b). Figs. 2 and 3 compare the numerical predictions of Eq. (27) with the experimental data. The liquid height and time were normalized using Eqs. (17) and (18). Note that x = 1.4 for the data reported by Zhmud et al. [18] and x = 21.4 for the data reported by Quéré et al. [12]. The accuracy of numerical results is very good at the
beginning but the error increases slightly as time passes. It should be related to the neglected terms/effects, such as the entrance effect, the difference between descending and ascending contact angles, and Vena contracta effect [6,10,19]. Fig. 4 shows the numerical prediction for small values of x. Since the differences between different curves are significant for short times, so the results are restricted to T < 2. When x = 0, the predictions of Eq. (27) matches predictions obtained by LWE. Note that the kinetic effect is not included in LWE, which is identical to Eq. (21) when x = 0 [9]. Fig. 5 shows the numerical results for cases of significant kinetic effect (higher values of x). Figs. 4 and 5 show that at the beginning of the capillary motion, the liquid velocity is slows down as x increases. As x increases, more energy is needed
Fig. 2. A comparison of numerical predictions using Eq. (27) with experimental data reported by Zhmud et al. [18]. Here, density, capillary radius, contact angle, surface tension, and viscosity are 710 kg/m3, 0.5 mm, 0, 16.7 mN/m, and 0.6 mPa s, respectively [18].
Fig. 5. Oscillatory behavior of liquid column.
Fig. 3. A comparison of numerical predictions using Eq. (27) with experimental data reported by Quéré et al. [12]. Here, density, Jurin height, contact angle, surface tension, and viscosity are 710 kg/m3, 7.1 mm, 0, 16.6 mN/m, and 0.3 mPa s respectively [12].
(a) A comparison of curves for
and
(b) A closer look at curves crossing H=1 Fig. 4. The dimensionless height versus dimensionless time for low values of x and short times.
Fig. 6. Critical value of x which leads to oscillation behavior in Eq. (27).
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to overcome the kinetic effects, so liquid velocity is slower at the beginning of the motion. However, as liquid reaches the equilibrium height (Jurin height), this kinetic force prevents liquid column from stopping and so liquid rises higher than the equilibrium height. It takes time for this kinetic force to damp out by viscous forces, which leads to an oscillatory behavior at the liquid front around the Jurin height. Fig. 6 shows that there is a critical value of x between 0.2 and 0.3 above which oscillation occurs. The detail study of numerical data shows that the oscillation appears when x > 0.25, which means xcr = 0.25 is the critical value. This critical condition is identical to X 6 2 that is reported by Fries [10], Quéré et al. [12], and Fries and Dreyer [24]. However, they used equal sign for oscillation to occur while our numerical data does not support equal sign. Using Eq. (22), it can be shown that the critical radius, Rcr, which is related to xcr is:
Rcr ¼ 2
l
2
1 c cosðhÞ 5 q2 g 2
ð29Þ
The Eq. (29) is identical to an expression reported by Hamraoui and Nylander [16]. The above equation indicates that the oscillations appears if R > Rcr. In the other words, when x > 0.25 or R > Rcr, the kinetic force is strong enough to rise liquid up the equilibrium height, which leads to oscillation around Jurin height. 5. Summary and conclusion We used energy balance to derive the governing equation for liquid rise in a capillary tube. A simple method was used to derive dimensionless governing equation, which has just one parameter. The standard fourth-order Runge–Kutta method was employed to solve the nonlinear dimensionless governing equation. The governing equation and numerical method were validated by comparing the numerical results with some published experimental data. The dimensionless parameter (x) in the dimensionless governing equation determines the oscillatory behavior of the liquid column. We showed that x > xcr leads to the oscillation of the liquid
column in the capillary tube where xcr = 0.25. There is also a relevant critical capillary radius for xcr, which is a function of liquid properties and contact angle. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jcis.2012.09.004. References [1] [2] [3] [4]
[5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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