An experimental and numerical study of capillary rise with evaporation

An experimental and numerical study of capillary rise with evaporation

International Journal of Thermal Sciences 91 (2015) 25e33 Contents lists available at ScienceDirect International Journal of Thermal Sciences journa...

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International Journal of Thermal Sciences 91 (2015) 25e33

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

An experimental and numerical study of capillary rise with evaporation John Polansky*, Tarik Kaya Carleton University, Department of Mechanical and Aerospace Engineering, Ottawa, ON, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 June 2014 Received in revised form 27 November 2014 Accepted 7 December 2014 Available online

An experimental and numerical study of spontaneous imbibition into capillary tubes subject to phase change is presented. A mathematical model is developed to predict the motion of a meniscus while undergoing phase change. The model addresses slip at the wall, viscous effects of the vapour in the capillary tube and transient evaporation. A set of experiments were performed for three fluids (acetone, n-pentane and iso-octane), three capillary tube diameters (0.5, 1.0, and 2.0 mm) and five heating conditions (0, 0.7, 2.7, 6.0 and 10.6 W). The experimental results demonstrated that the meniscus rise was altered by varying degrees of evaporation. A comparison of the experimental data and the mathematical model yielded a good correlation for the 1 mm capillaries, and deviated for both the 0.5 mm and 2 mm cases. It was found that an asymptotic transient mass function was unable to improve the fit to experiment. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Meniscus Capillary rise Heat pipes Imbibition LucaseWashburn equation

1. Introduction Capillary action is found to occur naturally in hydrology, anatomy and plant physiology; with industrial applications spanning textiles, biosensors, oil recovery, civil engineering and space based technologies. The static and dynamic aspects of capillary forces are of particular importance in space, as surface tension forces dominate small scale fluid flows given the reduced gravitational effects on the flow. Thus, the engineering applications of capillaries in space are common to liquid fuelled rocket motors and heat pipe based thermal control systems. In general, the utilisation of capillary structures for enhancing heat transfer presents a unique challenge. Historically the statics of a meniscus have been well studied, while the dynamics of its rise have been a more recent point of investigation. The early works of Lucas [1], Washburn [2], Bell and Cameron [3] and Bosanquet [4] founded much of the theoretical framework for capillary dynamics. Such works continue to be the basis from which more comprehensive mathematical models are being devised. Subsequent works have sought to address the problem of capillary rise dynamics using theory, numerical solutions and experimentation. Some studies have included effects such as:

* Corresponding author. E-mail address: [email protected] (J. Polansky). http://dx.doi.org/10.1016/j.ijthermalsci.2014.12.020 1290-0729/© 2015 Elsevier Masson SAS. All rights reserved.

dynamic contact angle [5e13], surfactants [14,15], gas/vapour displacement [14,16], slip [2,17,18], phase change [5,19,20], venacontracta/jet [13,21e24] and tube inclination [25]. Others have sought to capture the various regimes of capillary rise thereby leading to criteria predicting oscillatory behaviour [24,26,27]. Fries and Dreyer [28] investigated the timing of competing forces during capillary rise, followed by their systematic analysis of nondimensional governing equations describing imbibition [29]. While the majority of studies have focused on cylindrical capillaries and Newtonian fluids; Levine et al. [18] expanded the study to that of channel based imbibition, while Kornev and Neimark [30] extended to viscoelastic fluids. Complementing the theoretical and numerical studies of capillary rise dynamics, some experimentation has shed light on the true physics and dynamics. Siebold et al. [6] experimentally confirmed that the meniscus changes curvature during the dynamic portion of the rise. This changing interface shape was also confirmed by Lorenceau et al. [21], where the interface was observed to produce a liquid finger during the initial stages of capillary rise. Other factors postulated and experimentally captured include the viscous pressure drop associated with the displacement of the gas/vapour occupying the capillary. The effects of vapour displacement pressure drop were confirmed experimentally by Zhmud et at. [14], and shown to have considerable impact on the dynamics of capillary rise. Furthermore, slip at the solideliquid

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Nomenclature FM f(t) g h hfg j k L MW PVIS PVR r R Rg t T DT u

factor of merit [N/m s] arbitrary temporal function [e] gravitational acceleration [m/s2] avg. meniscus height [m] latent heat of vaporisation [J/kg] mass flux [kg/m2 s] linearised mass flux coeff. [kg/m2 s K] capillary tube length [m] molar mass [kg/mol] viscous pressure drop [Pa] vapour recoil pressure [Pa] radial coordinate [m] capillary tube radius [m] universal gas const. [J/mol K] time [s] temperature [K] superheat [K] average fluid velocity [m/s]

interface as reviewed by Neto et al. [31], can be extended to capillary flows. The potential for slip at the solideliquid interface in capillaries was put forward by Washburn [2] and experimental studies by Pit et al. [32] support the potential for slip. An experimental study of static meniscus height as a function of capillary radius, length and subject to evaporation was performed by Kuz'mich and Novikova [33]. In this experimental study, the meniscus static height was studied with only a qualitative mention as to the dynamic aspect of the meniscus motion. The oscillatory motion noted by Kuz'mich and Novikova [33] was dependent upon the capillary radius and attributed to changes in the wetting angle. The evaporation from the capillary tubes was driven by changes in ambient vapour pressure, not heating. The study of capillary rise dynamics subject to phase change has received some attention theoretically [5,19,20], and to our knowledge none experimentally. The problem of meniscus motion and phase change is a unique one lending well to flow dynamics akin to those found in capillary pumped loops and loop heat pipes. The main objective of this work is to expand upon the theoretical

Greek Symbols a accommodation coefficient [e] b slip length [m] g slip coefficient [m2] m dynamic viscosity [Pa s] n kinematic viscosity [m2/s] r density [kg/m3] s surface tension [N/m] t relaxation constant [s] Superscripts derivative with respect to time

0

Subscripts eq equilibrium F final l liquid v vapour

framework and compare our mathematical model, to experimentally measured rise dynamics subject to evaporation. In this paper, we will outline the basis of our mathematical model, the experimental setup and methods, and data extraction techniques outlined thereafter. Then, the experimental results will be given. Finally, the experimental and numerical results will be compared and analysed.

2. Mathematical model The system consists of a capillary tube of radius R and length L, oriented in the vertical such that gravity is consistent with the axis of the capillary tube as shown in Fig. 1, with a large liquid reservoir positioned below the entrance of the capillary. The atmospheric environment is assumed to be that of a saturated single species of uniform temperature. When the capillary tube makes contact with the liquid reservoir interface, the liquid begins to wet the wall thereby facilitating spontaneous imbibition. During spontaneous imbibition, surface tension draws liquid from the reservoir into the capillary tube. To simplify the problem, the liquidevapour interface is taken to be flat and at an average height h, invariant in r, for an incompressible Newtonian fluid. Therefore, the dynamics of the liquidevapour interface can be described by the equation of motion as:

r

Fig. 1. Solution domain.

d 2s ðhuÞ ¼  rgh þ PVIS þ PVR dt R

(1)

This is the LucaseWashburn equation as modified by Ramon and Oron [19]. The momentum of the liquid phase is balanced by capillarity, hydrostatics, viscous drag and vapour recoil. The capillary pressure is assumed to have a time invariant contact angle consistent with that of an ideal wetting fluid. As the liquid enters the capillary and progresses upward; the viscous drag results in a pressure drop. While the problem addresses the transients of the meniscus motion, it is assumed that the flow achieves a fully developed profile sufficiently quickly and maintains this profile throughout the solution time. The viscous pressure drop in the capillary can be affected if the fluid is able to exhibit a degree of slip at the wall interface. A slip condition for capillary flow was proposed by Washburn [2], where

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a slip condition at the wall of length b was introduced. The slip length and viscous pressure drop assumes the fully developed profile to extend the slip length b beyond the wall. The slip condition for the liquid is retained in this work, as recent experimental findings have indicated the existence of slip in micro-scale flows, most notably in cases with reduced solideliquid affinity [31,32]. As the liquid fills the tube, it displaces the vapour phase at a velocity consistent with that of the liquid phase. If however, phase change is introduced, the viscous pressure drop for the liquid and vapour fields are determined by the relative velocities of displacement and phase change. The viscous effects of each fluid phase are dictated by the relative portions of the tube occupied by each phase, thus giving:

8m 8m PVIS ¼  hu  v ðL  hÞuv g gv

(2)

The viscous pressure drop acting on each phase is afforded a slip condition unique to each phase as given by:

g ¼ R2 þ 2bR þ b2

(3a)

gv ¼ R2 þ 2bv R þ b2v

(3b)

Writing the conservation of mass across the interface, we obtain:

j ¼ rðu  h0 Þ ¼ rv ðuv  h0 Þ

(4)

The conservation of momentum across the interface results in a vapour recoil condition, which is given by:

  r  rv 2 PVR ¼ jðu  uv Þ ¼  j rrv

(5)

  d 2s 8m 8m r  rv 2 ðuhÞ ¼  rgh  hu  v ðL  hÞuv  j dt R g gv rrv

(6)

(10)

where,



arv hfg 3=2

Tv



Mw 2pRg

1=2 (11)

and where f(t) is any continuous function of time. In the case of Ramon and Oron [19], the mass flux was assumed to be a constant. Herein we permit the mass flux to vary in time, though the function which satisfies the transient mass flux is an unknown of the problem, which will be discussed later in the paper. For the purposes of this work, the time dependence of the mass flux is assumed to be of the following form:

  ðt  t0 Þ f ðtÞ ¼ 1  exp  t

(12)

To complete the model so as to numerically solve Eq. (9), a superheat is required. While superheat is not directly measurable, if we rearrange Eq. (9), the superheat is obtained as a quadratic relation. To infer the superheat from experimental data, the system can be permitted to reach a steady state condition. This further simplifies the quadratic relation for superheat to a simple measurement of meniscus height. The experiments were therefore conducted so as to permit the meniscus to reach a steady state height (heq), at which point the transient mass flux would also have achieved steady state. Applying these conditions to Eq. (9), and rearranging, we obtain a quadratic relation for the superheat as:

      mv L  heq mheq k2 r  rv 8k þ DTF þ r r rrv rg rv gv   2s ¼0 þ gheq  rR

 DTF2

Note that the vapour recoil at the interface is independent of phase change direction (evaporation or condensation), as discussed by Ramon and Oron in Ref. [19]. With the aforementioned conditions, we obtain the equation of motion for the meniscus during spontaneous imbibition with phase change.

r

j ¼ f ðtÞkDT ¼ f ðtÞkðTl  Tvsat Þ

27

(13)

In solving for the roots, we can then deduce the final liquid temperature as:

TlF ¼ Tvsat þ DTF

(14)

Finally, the relaxation constant t in Eq. (12) can be obtained from the experimental measurements. This is the time required to reach a steady state height during an experiment.

Using Eq. (4), the velocity of each phase can be written as: 3. Experimental setup and procedures

j u ¼ þ h0 r uv ¼

j þ h0 rv

(7)

(8)

Combining Eqs. (6)e(8) and rearranging, we obtain the following second order differential equation. 00

h ¼

    2s 8m j 8m ðL  hÞ j g þ h0  v þ h0 rhR rg r rhgv rv     0 j2 r  rv h0 j j þ h0    rh rrv h r r

(9)

When solving Eq. (9), the fluid properties were calculated as a function of temperature and updated at each point in time. The thermophysical relations were obtained from Lide et al. [34], with the vapour viscosities of acetone and iso-octane calculated using the Riechenberg method as outlined by Reid et al. [35]. The conditions relating to phase change are approximated using a modified form of the linearised kinetic theory model [19]:

The experimental setup as shown in Fig. 2 utilised a stainless steel bell jar vacuum chamber together with a roughing pump to maintain a working fluid saturated test environment. The vacuum chamber allowed visual observation via three viewing ports, as well as feed-throughs for thermocouples, liquid supply, electrical and mechanical connections. The setup was situated atop a level granite table and isolated by rubber padding and airbags from external vibrations. An Olympus SZH-10 optical stereomicroscope, fitted to an Olympus SZ-STU2 universal stand was used to observe the capillary tubes. The microscope used an Olympus DF Planapo 1X objective lens for the 1 mm and 2 mm capillary tubes, and an Olympus DF PL 0.54 objective lens for the 0.5 mm capillary tubes. The microscope arrangement included an Olympus coaxial vertical illuminator ILLC2 (unlit) and an Olympus photo tube SZH-PT. The MotionScope M1 high speed camera fitted with a Fujinon CF25HA1 1:1.4/25 mm TV lens and connected to the microscope via the photo tube. The capillaries were illuminated using an Edmund NT66-847 high intensity LED spotlight. The experimental setup inside the chamber utilised an adjustable stage fitted with a T-type thermocouple (Omega 5SRTC-GG-T-

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Fig. 2. Vacuum chamber experimental setup.

30-36) of ±0.5 K accuracy. The chamber was equipped with a pressure transducer (Omega PX429-015A5V) providing an accuracy of ±0.1%.The adjustable stage incorporated a kapton heater (Omega KHLV-102/10) with a resistance of 37.8 U and capillary tube array as shown in Fig. 3. The adjustable stage was fitted with a lowering arm and connected by a flexible drive assembly to the mechanical feedthrough in the base of the vacuum chamber, allowing for the lowering of capillary tubes into the liquid reservoir. The experiments were performed using three sets of open ended capillary tubes nominally sized 0.5 mm, 1.0 mm and 2.0 mm internal diameter. The capillary tubes were prepared in an ultrasonic bath of acetone for 1 h. The tubes were then dried on a hot plate at 65  C for 30 min. Five capillary tubes of each diameter were arranged into three separate arrays. The three separate arrays were

Fig. 3. Experimental setup inside the vacuum chamber (not to scale).

each epoxied (Devcon HP 250) together and thermally cured for a period of 24 h. The three epoxied capillary tube arrays were cleaned as per the procedure outline before. The internal and external diameters of the capillary tube arrays, Table 1, were measured using a Titan Centering Microscope (Model Z-1) with a Mitutoyo digital readout (Model ALC-3705W). The liquid reservoir consisted of a petri-dish and slides held together with stainless steel clips for a divider and 2 mm glass beads as arranged in Fig. 3. The glass beads and slides were installed so as to attenuate the fluid flow and surface perturbations in the reservoir due to the filling spout. All of the reservoir and its components were cleaned in a manner consistent with that used for the capillary tubes. The experimental test matrix used three different working fluids (acetone, n-pentane, and iso-octane), three capillary tube array sizes (0.5, 1.0 and 2.0 mm) and five heating levels (0, 0.7, 2.7, 6.0 and 10.6 W). The fluids were used as supplied without modification with the fluid details provided in Table 2. The vacuum chamber and mechanical setup were cleaned with acetone, dried with heat and blown down with high purity dry nitrogen gas. The experimental setup inside the chamber was then secured to the base of the vacuum chamber. A clean and dried capillary tube array was then mechanically affixed to the kapton heater and lowering arm, and aligned with gravity using a digital level with ±0.1 accuracy. The reservoir was then placed on the platform and the filling spout with splash guard were positioned over the 2 mm glass bead half of the reservoir. The mechanical arrangement was sealed in the vacuum chamber and a vacuum was drawn to a pressure less than 6.7 Pa using an Alcatel Pascal 2005 SD roughing pump. The vacuum chamber was then sealed by way of a line valve from the environment outside the chamber and the roughing pump was permitted to shut down. The liquid supply reservoir and feed line system was then filled with the working fluid. The liquid supply line was then opened, thereby charging the chamber with vapour until such time that saturation was achieved and liquid began to fill the reservoir until full. The vapour pressure, liquid reservoir and capillary tube temperatures were monitored for steady state conditions within the chamber. Any losses from the reservoir due to evaporation during the early phases of the setup were replenished. The array of capillary tubes was then inspected visually with the microscope for dryness inside the tubes. The capillary tube array was then lowered into proximity to the reservoir's liquid interface. The high speed camera was set to record at 125 frames per second, with the thermal and pressure data recorded every second

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using the Keithley 2700 DAQ 22 bit analogue-to-digital converter with an accuracy of ±0.8%. Upon triggering the data and video recordings, the adjustable stage lowered the capillaries into contact with the liquid reservoir. The menisci were observed until such time as the motion had ceased, at which point all data collecting was terminated and stored for post analysis. The capillary tube array was then raised clear of the reservoir and heated with the kapton heater until all of the liquid had evaporated. The tubes were visually inspected for dryness, and following this, the heater was turned off allowing the capillaries to cool. The capillary array temperature was monitored until steady state was achieved. Each subsequent test used the kapton heater to heat the capillary array. Thus the appropriate heater load was applied and the capillary tube temperature monitored, eventually achieving steady state. Using the procedure outlined above, the experiments were successively performed for each heater setting. Upon completion of a test set, the heater was turned off and the system permitted to cool. The vacuum chamber was then opened to atmosphere and thoroughly evacuated of all working fluid. The process was repeated for each size of capillary tube tested and fluid with decontamination being performed between each test. The data was collected using the MotionScope M1 software as individual frames in bitmap format. The image sequences were then loaded into ImageJ and cropped to size. The pixel locations of the reservoir interface and tube centres were extracted from the frames in ImageJ. The raw image sequences were loaded into a custom search algorithm constructed in MATLAB to locate the lowest position of the meniscus, and then cleaned using a custom speckle filter. The meniscus location was then translated upward by R/3 to conform to the average height employed in the mathematical model. For each particular setup, the capillary tube outer diameter was correlated with its corresponding pixel count. This ratio was unique to each set of capillary tubes. The meniscus position information was determined within a tolerance of ±1 pixel. This corresponds to a positional error of ±0.07 mm for 0.5 mm capillaries and ±0.05 mm for 1.0 mm and 2.0 mm capillaries. 4. Discussion of results As the imbibition process was captured using a high speed camera, a sequence of evenly spaced frames is shown in Fig. 4. The sequence shows a single 2.0 mm capillary tube as it makes contact with the reservoir. As soon as the tube contacts the reservoir interface, the liquid begins to climb within the tube. The first portion of the rise is fast and only apparent when the adjacent frames are differenced. As the sequence progresses the meniscus is observed to overshoot the equilibrium height leading to a damped oscillation until achieving a stable height. Furthermore, the meniscus shape appears unaltered during the flow reversal at the apex and corresponding minima of the trajectory as one might expect. The experimental results for each of the capillary tube diameters, fluids and heating conditions investigated are presented here, with only typical cases shown. The experimental results for

Table 1 Capillary dimensions with the corresponding uncertainties. Nominal ID (mm)

Manufacturer

Avg. measured ID (mm)

Avg. measured OD (mm)

Tube length (mm)

0.5 1.0 2.0

VitroCom Fisher brand VitroCom

0.511 ± 0.025 1.062 ± 0.025 2.033 ± 0.025

0.702 ± 0.025 1.459 ± 0.025 2.384 ± 0.025

100.39 ± 0.01 75.09 ± 0.01 100.24 ± 0.01

29

Table 2 Experimental fluids. Fluid

Supplier

Grade

Acetone n-pentane Iso-octane

Caledon laboratory chemicals Fisher scientific Fisher scientific

Reagent HPLC grade HPLC grade

iso-octane and three capillary diameters are presented in Figs. 5e7. Fig. 5 shows the meniscus trajectories in a 0.5 mm capillary tube for three heating conditions. It is observed that the final height of the meniscus is lower for increasing heat loads as expected. It may also be noted that the difference in equilibrium height between the 0.7 W and 2.7 W cases is larger than for that of the 2.7 W and 10.6 W cases. The menisci rise within the 0.5 mm capillary tube smoothly and without overshoot. As the heater load is increased, the rise trajectories depart progressively earlier culminating at different equilibrium heights in accordance with the heater power. The steadystate reservoir and capillary tube temperatures before dipping are shown in Table 3. When the capillary diameter is increased from 0.5 mm to that of 1.0 mm, the menisci rise at such a rate that they overshoot the equilibrium height, then to quickly damped oscillations. Once again the evaporation is observed to lower the height of the meniscus with increasing heat load. Comparing the rise with that of the 0.5 mm case shown in Fig. 5, the menisci rise at a similar rate during the initial stage and then begin to deviate as the menisci tend to their respective equilibrium heights. While the deviation was rather sharp in the case of the 0.5 mm capillary tubes, the deviation in the oscillating case of 1.0 mm is smooth. When the capillary diameter is increased to that of 2.0 mm, the oscillations continue for a minimum of five complete cycles and damping out in approximately 0.8 s. For the 1.0 mm capillary tube, two complete oscillations are observed before damping out in 0.5 s. Similar to that of the 1.0 mm tests, the initial rise dynamics are very close with the deviation to each respective height transitioning smoothly. The difference in height between the 0 W and 2.7 W is relatively small and share a very close dynamic. When comparing to the 10.6 W data, a larger disparity in height is observed in addition to a slight shift in the oscillating phase. Drawing comparison of the different fluids as shown in Fig. 8, the meniscus dynamics are observed to be unique to each fluid. The acetone dynamics are separate and distinct, while the motion of the n-pentane and iso-octane are comparable in both phase and amplitude. For the heated case as shown in Fig. 9, the dynamics of the npentane and iso-octane are still comparable with respect to that of acetone. However, the differences in the motion of n-pentane and iso-octane are more pronounced in both the amplitude and frequency of oscillation. The dynamics for all oscillating n-pentane cases were observed to have only minor changes due to evaporation, while acetone and iso-octane were more so affected. To explain the relative similarity of iso-octane and n-pentane behaviour, we introduced a factor of merit arranging the most influential fluid properties to represent the physical phenomena: FM ¼ shfg/n, as listed in Table 4. In an intuitive manner, the meniscus behaviour is expected to be directly proportional to the surface tension s and latent heat hfg (contributors for increasing height) and inversely proportional to dynamic viscosity n (contributor for decreasing height). The observed similarity of these two fluids may thus be attributed to close values of the factor of merit as it is shown in the last column of Table 4 by normalizing the FM values with that of acetone.

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Fig. 4. Image sequence of n-pentane rising within a 2.0 mm capillary tube, with the meniscus location indicated by a ‘þ’ sign.

Eq. (9) was solved using MATLAB's inbuilt RungeeKutta solver ode113 with a refinement of five. As a singularity in the governing equation exists for a height of zero, a small perturbation is required. The perturbation was chosen to be 106 m. A difference

based comparison of adjacent solutions (105, 107) was found to have an RMS of 1.61  109 m, thereby confirming that the final results are nearly independent of the choice of the perturbation value.

Fig. 5. Iso-octane in a 0.5 mm capillary tube for 0.7, 2.7, and 10.6 W heating loads.

Fig. 6. Iso-octane in a 1.0 mm capillary for 0, 0.7, 2.7 W heating loads.

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31

Fig. 7. Iso-octane in a 2.0 mm capillary 0, 2.7, 10.6 W heating loads.

Fig. 8. Three fluids in a 2.0 mm capillary with no heating.

As the slip at the wall was observed in micro/nano scale flows [2,17,18], we decided to retain an optional slip feature in our mathematical model to be able study a wide range of dimensions. The experimental conditions observed herein are similar to those of Pit et al. [32], with the flow having a Reynolds number below 50. While the experiments of Pit et al. [32] were able to capture slip at the wall of 190 mm spinning disk arrangement, the capillary tubes tested here were of a larger size and without any surface treatment. Given the differences in scale and surface treatment, the flow is assumed to have a no slip condition for the liquid and vapour phases in our calculations. The comparisons of the calculated and measured meniscus height for typical cases are shown in Fig. 10, with the conditions of a constant and transient mass flux. A close comparison was achieved between the mathematical model and experiment for the unheated case of the n-pentane in a 0.5 mm capillary tube as shown in Fig. 10(a). For the heated case, a distinct change in velocity was observed experimentally during the early portion of the capillary rise. This decline in velocity was observed in all heated cases of the 0.5 mm capillary tests, and most pronounced in the n-pentane in Fig. 10(a) and acetone in Fig. 10(d) experiments. For the 1 mm capillary tube experiment results, Fig. 10(b, e, and h), the meniscus was observed to oscillate in all cases, with the acetone seeing the least oscillation amplitude, Fig. 10(e),and npentane the largest, Fig. 10(b). The model is able to capture the dynamics of the motion relatively well for all three fluids, in both amplitude and phase. Lastly, for the 2 mm capillary tubes, Fig. 10(c, f, and i), all experiments exhibited a clear and continuous oscillation, eventually damping to a steady state height. The experimental data shows the non-linear nature of the oscillation, as the frequency of the oscillation evolves in time. In comparing the mathematical model results to those of experiments, a relatively good match was achieved in the early stages of the motion for the three cases studied. As the motion evolves, a growing discrepancy in the phases was observed. A similar trend was also observed in the unheated cases of Zhmud

Fig. 9. Three fluids in a 2.0 mm capillary tube with a heating load of 6.0 W.

et al. [14] and Lorenceau et al. [21].This phase difference occurred in all cases. In addition to this, the amplitude was slightly askew with the mathematical model. If the liquid viscosity was artificially adjusted to triple that found in literature, the amplitude difference in the oscillations was reduced while the phase difference remained. The constant mass flux condition consistently over predicts the rise height in time as compared to experiment. When a comparison is drawn between experiment and a transient mass flux model, the transient results are always greater than the constant mass flux. We believe the assumption of the transient mass flux is well founded and indirectly supported by the experimental data collected. However, it is very difficult to accurately predict the transient mass flux evolution in time. The proposed transient mass function did not improve the predictions as expected. The introduction of the transient mass flux can have a notable effect on the meniscus rise

Table 3 Initial steady state temperatures for iso-octane in 0.5 mm capillary tube.

Table 4 Fluid properties at T ¼ 293 K.

Power (W) 0.7 2.7 10.6

Tres ( C) 24.3 24.6 25.0

Ttube ( C) 37.4 61.0 123.9

Fluid Acetone n-pentane Iso-octane

s (N/m) 0.024 0.016 0.018

n (m2/s)

hfg (kJ/kg) 7

4.06  10 3.72  107 7.73  107

550.4 382.4 292.6

FM (N/m,s) 7

3.25  10 1.64  107 6.81  106

FM 1 0.51 0.21

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Fig. 10. Comparison of the mathematical model results and experimental data. The constant mass flux is a solid line and the transient mass flux is a dashed line. (a)e(c): n-pentane at 6.0 W, (d)e(f): acetone at 0.7 W, (g)e(i): iso-octane at 2.7 W for the tube diameters of 0.5 mm, 1 mm and 2 mm, respectively. A comparison of an unheated (blue) and heated (red) case is presented in Fig. 10(a). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

and amplitude of meniscus oscillation as illustrated in Fig. 10. However, the transient mass flux condition appears to have little effect on how the oscillating frequency evolves in time. This temporal evolution of the motion is present in all cases where the meniscus undergoes oscillation.

assumed in this work did not improve the calculated results as expected for either the amplitude or phase discrepancies. It was demonstrated that the proposed mathematical model can be used to predict the overall behaviour of meniscus rise dynamics in a heated capillary tube.

5. Conclusions

Acknowledgements

An experimental and numerical investigation of capillary rise with evaporation was presented. A series of environmentally controlled experiments were performed for three different fluids (acetone, n-pentane and iso-octane), three sizes of capillary tubes (0.5, 1, and 2 mm diameters) and five different thermal loads (0, 0.7, 2.7, 6.0, and 10.6 W). The experimental results were captured using a high speed camera and analysed. The results of the experiments confirmed the general trends predicted by the mathematical model. The model is capable of capturing the rise dynamics of the unheated case quite well, but begins to deviate for heated cases for the 0.5 mm capillary tubes. The model and experiment showed a great degree of agreement for the 1 mm cases with only slight differences in amplitude. Lastly, the 2 mm experiments were relatively close in amplitude and phase during the early stages of the motion, and progressively degrading in time. The evaporating mass flux was observed to be time dependent and must be accounted for in the mathematical model. The asymptotic evaporation rate

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