Experimental study and modelling of the
evaporation in capillary tubes simulating a porous
media
Frédéric Debaste
Frank Dubois
Véronique Halloin
Chemical Engineering
Microgravity
Chemical Engineering
Department
Research Centre
Department
Free University of Brussels, Av. Franklin Roosevelt, 50 CP 165/67
B-1050 Brussels, Belgium
Tel : +32(2)6504096, Fax : +32(2)6502910
[email protected]
1
Introduction
Drying modeling is classically done threw continuous volume averaged models [1]. In
some applications, continuous averaged hypothesis aren’t satisfied [2]. Then, discrete
approach, like porous network modeling, can be used. They are directly based on
the phenomena at the pore scale [3]. Thus, a good knowledge of the evaporation at
that scale is fundamental for enhancing those kind of models [4].
The present work focus on the evaporation of a pure liquid from a capillary
tube of a circular cross-section, with the perspective of using the results in a porous
network model.
Three distincts models are presented. Each model studies different physical
phenomena.
The first approach deals with convection of vapor in the gas phase inside the
capillary tubes. Mass and momentum transport equations are solved numerically for
the gas phase to evaluate the importance of convective mass transport for different
fluids.
The second approach is about thermal gradients inside the capillary tube. Mass
and energy transport equation are solved in liquid, gas and solid phase to evaluate
the importance of the cooling of the meniscus on the global evaporation rate.
The last approach investigates the effect of thin liquid films trapped on the walls
on the drying rates. This model is based on lubrification equation with disjunction
pressure modification [5].
Results of those models are compared to experimental results of evaporation in
circular section capillaries.
2
Experimental setup
Evaporation is studied in a vertical capillary tube of a circular cross-section whose
external wall is kept at constant temperature. The capillary entrance is opened to
stagnant air; the other end of the capillary is sealed. The temperature around the
capillary is maintained by placing the capillary in a thermostatic cell presented on
figure 1. The two largest walls of the cell are made of optical glass, three others
are made of steal and contain a cooling water circuit. The last one is left open to
allow evaporation from the capillary. The inner part of the cell is filled with liquid
to enhance heat transfer.
Figure 1: Experimental cell
Different set of capillaries and parameters are used :
• Capillary have a diameter of 0.1, 0.2, 0.3 or 0.5 mm
• The cell temperature is maintained at 30, 40 or 50 ◦ C
• The fluids tested are acetone, cyclohexane, ethanol, methanol and water.
A first set of measurement is realized simply by following with video recorder
the evolution of global position of the meniscus. The level of the meniscus can be
directly connected to the evaporation rate. A second set of measurement uses a
Mach-Zendher interferometer to investigate more precisely the thin film left by the
receding meniscus. Using appropriate lens, this device offers the ability to follow
the film thickness to 100 nm [6].
3
3.1
Convective transport modeling
Hypothesis
The first model investigates the apparition of a convective mass transport in a capillary due to the evaporation. Some general hypothesis are made :
2
• the system is axisymetric. The rotation axis z is the axis of the capillary (as
shown on figure 2),
Figure 2: Axis references
• evaporating liquid is a pure substance,
• gas phase is composed of an inert gas and of the vapor coming from the evaporation of the liquid . They act like a mix of perfect gases,
• liquid phase is still,
• the gaseous solution have the properties of the inert gas. This can only be used
when vapor concentration are low or when the vapor has properties similar to
those of the inert gas,
• gas phase is incompressible,
• gravity can be neglected.
Some more specific hypothesis are also taken to highlight the apparition of convection :
• the system is supposed to be isotherm. In other words, cooling due to evaporation is instantly compensated by heat conduction from the environnement
• adsorbed thin liquid film have no influence,
• gaz-liquid interface is planar,
• at the meniscus, the gas is saturated in vapor,
• the interface position is at a fixed distance from the aperture of the capillary,
• the system is steady state.
3
3.2
Equations
Under those hypothesis, equations of transport for the gas phase can be written as
[7]:
Continuity
∇v = 0
(1)
ρg v∇v = µg ∆v − ∇pg
(2)
Navier-Stokes
Convection-diffusion of vapor
v∇ρv = D∆ρv
(3)
The boundary conditions are the following : no mass flux and non slip wall on
the capillary wall, fixed partial pressure of vapor at the aperture of the capillary. At
the meniscus the partial pressure is set to saturation. Velocity are calculated threw
a mass balance at the interface [8]:
v0 = −
∂ρv D
∂z 1 − ρρvg
(4)
We simplify this latest expression by approaching the gradient with it’s average value
:
ρsat − ρ0 D
(5)
v0 = −
L
1 − ρρsat
g
This simplification is rigourously applicable only when diffusion is dominating but
is helpful to exhibit the beginning of convection.
Those equations are rewritten in a dimensionless form using :
x
y
u
v
α
p
=
=
=
=
=
=
r/L
z/L
vr /v0
vz /v0
ρv /ρg
pg /ρg v02
(6)
(7)
(8)
(9)
(10)
(11)
This lead to the following set of dimensionless equations
Continuity
∇v = 0
Navier-Stokes
v∇v =
1
∆v − ∇p
ReL
4
(12)
(13)
x = R/L
y=0
y=1
u
u=0
u=0
v
v=0
v=1
α
=0
αsat = ρsat /ρg
α0 = ρ0 /ρg
∂α
∂x
Table 1: Dimensionless boundary conditions
Convection-diffusion
v∇α = Bs∆α
(14)
The dimensionless boundary conditions are summarized on table 1.
Equation (14) contains a Bodenstein number Bs that in our study can be reformulated in term of saturation fraction of vapor αsat :
Bs =
3.3
v0 L
αsat − α0
=
D
1 − αsat
(15)
Simulations
Simulations of equations (12),(13) and (14) with the boundary conditions presented
in table 1 are done with the FEMLAB 3.1 software.
Figure 3 shows the vapor fraction along the capillary symmetry axis for different
Bodenstein numbers with a α0 = 0. The curves are linear for Bs < 1. When getting
to bigger values, the profile changes because of the apparition of convection.
Figure 3: Vapor fraction along the capillary symmetry axis for different Bodenstein
numbers with α0 = 0
The Bs < 1 condition can be directly linked to the saturating fraction of vapor
with relation (15). If the limit case α0 = 0 is considered, the condition resume
to αsat = 0.5. Considering a fluid at a given temperature and pressure, we may
directly deduce if convection may occur. If αsat < 0.5, whatever the vapor pressure
5
is outside the capillary, convection won’t appear. If αsat > 0.5, convection might
begin to influence evaporation rate. We can verify that at standard pressure and
temperature water doesn’t exhibit any convection.
4
4.1
Heat transfer modeling
Hypothesis
The second model investigates the thermal effect in the capillary. Energy transport
equation is solved for gas, liquid and solid phase. Mass transfer in the gas phase is
supposed to occur only by diffusion. Most of the hypothesis are the same as those
presented on point 3.1. In this case non planar interface and unsteady-state evaporation are considered. This latest condition would imply to follow the mouvement
of the interface with time. To avoid that, an other simplification is done : diffusion
equation is supposed to attain steady-state instantly compared to the recession of
the meniscus.
4.2
Equations
Transport equation of mass and energy are set in a dimensionless form like in point
3.2, for the temperature the following dimensionless form is used :
Tad = T
cpl
∆Hvap
(16)
The general dimensionless equation are then (without the
ad
underscript)
Mass Diffusion
∆α = 0
(17)
∆T = 0
(18)
Thermal Conduction
The boundary conditions for the mass diffusion equation are the same as in the
first model. For the energy equation, temperature is supposed to be uniform at the
external limit of the capillary. At the inner boundary continuity of the temperature
is assumed. At the meniscus we set the temperature to integrate the cooling due to
evaporation :
kg (∇T )g ~n = kl (∇T )l~n + D∇α~nCpl ρg
(19)
The saturation pressure of the meniscus is given by the Clausius-Clapeyron relation
[9]:
p = p0sat exp
∆Hvap Mv 1
1
( − )
<
T
T0
6
(20)
4.3
Validation
Simulation are done with the FEMLAB 3.1 software coupled with MATLAB. For
a given meniscus position, FEMLAB solve the equations and compute the global
evaporation rate. The MATLAB software deduce a new position for the meniscus
and generate a new geometry for FEMLAB. The simulation begins with a capillary
full of liquid and ends when it’s dry. Those simulations are compared to experimental
datas obtained with the setup presented in 2. Typical results for different fluids
are presented on figure 4. We see a good agreement for most fluids but model
underestimate the evaporation rate. The difference between experiment and model
increase when volability of the fluid augment.
Figure 4: Comparaison of experimental datas (cross) with model (continious line)
for different fluids. The graphic show apex to meniscus distance as a function of
time.
The result of the model can also be compared to a simplified one dimension
diffusion model considering no thermal effects. Such a model lead to the following
relation [10] :
q
L = (Dt(ρsat − ρ0 )/ρg )
(21)
The figure 5 show the result of the two models compared to experiments in
the case of water. The simplified model gives better results than the full model.
The simplified model doesn’t take into account the cooling of the interface. This
effect tend to diminish the global evaporation rate. The experimental result seems
to indicate that thermal effect are counterbalanced by an other phenomenon. One
possibility could be influence of the thin liquid film. Therefor, a third modeling
approach is derived.
7
Figure 5: Comparaison of experimental datas (cross) with model (continious line)
for different fluids. The graphic show apex to meniscus distance as a function of
time.
5
Thin films modeling
This third model focus on the influence of the thin liquid film left by the receding
meniscus. The main goal of this model is to evaluate wether or not , thin liquid film
could explain the differences between our second model and experiments.
5.1
Hypothesis
Because of numerical problem, this model cannot be simply be added to the second
model. Therefor we make the following hypothesis :
• The gaz phase is perfectly mixed in the radial direction except next to the
capillary wall where the thin film appear. In other words,diffusion of vapor
and heat transfer is supposed to be only axial in the gas phase.
• The thin liquid film evaporate threw a stagnant limit layer of thickness Lf .
• The meniscus temperature is fixed. This temperature is not necessarily the
same as the temperature outside the capillary. So the meniscus cooling can be
taken into account without having to solve all the energy equations again.
8
The film considered here is schematized on figure 6. This film is supposed to be
stationary. It is feeded by the meniscus on the left. The feeding is induced by the
evaporation and the disjoining pressure. This last term is used to take into account
the influence of the solid surface on the fluid. It’s study is still an actual subject of
research [11]. In this work we simply take a 2 term disjoining pressure including a
polar and a non-polar term [12] :
d0 − δ
A
pd = 3 + Kexp
δ
l0
!
(22)
The effect of this pressure is taken into account in the momentum balance of the
film and in the definition of the saturation pressure.
Figure 6: Thin film of thickness δ feeded on the left by the meniscus and evaporation
threw a layer of thickness Lf
5.2
Equations
Using those hypothesis, the following equations are solved :
Evaporation of the film threw the limit layer
ṁ = −D
ρt − ρb
Lf
(23)
Film cooling due to evaporation
ṁ = −
λl Tw − Tt
∆Hvap
δ
(24)
1 d 3 dpd δ
3νl dx
dx
(25)
Mass balance in the film [5]
ṁ = −
9
Diffusion in the capillary bulk
ṁ = −
R ∂ 2 ρb
L ∂x2
(26)
Saturation pressure including disjoining pressure effect [13]
−Mv
Mv psat
ρt =
exp
<Tt
5.3
A
δ3
+ Kexp
ρl RTt
l0 −δ
l0
(27)
Validation
The set of equation is solved using the MATHEMATICA software, results are compared to experimental data obtained with the Mach-Zendher interferometer. For
the resolution of the equations, conditions at the left boundary are needed for δ and
it’s derivate. Experimental values are used.
Figure 7 shows the comparison between experimental measurement and the third
model for the film thickness. For the section presented on figure 7, the gas phase
is saturated in vapor. Only at the extreme right does the vapor pressure begin to
fall. This means that evaporation don’t seem to occur at the meniscus or in this
part of the film but further from the meniscus. In the presented area, for water,
the vapor pressure equal the saturation corresponding to the wall temperature. The
heat transfer resistance from the area around the capillary to the inner wall is lower
than the heat transfer resistance to the meniscus. It leads to a higher temperature
at the inner wall that at the meniscus. This means that the vapor partial pressure
next to the meniscus is not given by the temperature predicted for the meniscus but
by a higher temperature. This can explain the discrepancies observed on figure 5
for the second model.
The model gives enough information about this situation but should not be
used for more deduction. The fact that vapor is saturated in this region implies
that the evaporation rate expression wasn’t really validated. Moreover, an extended
predictive use of the model would need to know theorically the boundary conditions
to apply.
6
Conclusion
In this paper, three models were developed to study evaporation in a circular capillary. The first one lead to the definition of a criterion to predict the apparition of
convective mouvement.
The second model focus on thermal effect, particulary the cooling of the meniscus
due to evaporation. This model give mitigated results that can be explained by a
third modeling approach.
This last one focus on the effect of thin film absorbed to the capillary inner wall.
The model shows that the cooling of the meniscus is damped by the existence of
the thin liquid film. The evaporation of the film saturate in vapor the gas phase,
10
Figure 7: Comparison between third model (continuous line) and experimental film
thickness (dots)
canceling the evaporation from the meniscus. From a global point of view, everything
happens like if evaporation was taking at a non-cooled meniscus.
7
Acknowledgement
Frédéric Debaste acknowledge financial support from the Fond pour la Formation à
la Recherche dans l’Industrie et dans l’Agriculture (Belgium)
References
[1] S. Whitaker. A theory of drying in porous media. Advances in heat transer,
13:119–203, 1977.
[2] M. Prat. Recent advances in pore-scale models for drying of porous media.
Chemical engineering journal, 86:153–164, 2002.
[3] M. Prat. Discrete models of liquid-vapour phase change phenomena in porous
media. Revue générale de Thermique, 37:954–961, 1998.
[4] B. Camassel, N. Sghaier, M. Prat, and S. Ben Nasrallah. Evaporation in a
capillary tube of square cross-section : application to ion transport. Chemical
engineering science, 60:815–826, 2005.
11
[5] A. Oron, S.H. Davis, and S.G. Bankoff. Long-scale evolution of thin liquid films.
Reviews of modern physics, 69(3):931–980, Juillet 1997.
[6] E. Hecht. Optics. Addison-Wesley Publishing Company, second edition, 1987.
[7] P. Colinet B. Haut. Surface-tension-driven instabilities of a pure liquid
layer evaporating into an inert gas. Journal of colloid and interface science,
285(1):296–305, Mai 2005.
[8] R. B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport phenomena. John
Wiley, 2ème edition, 2002.
[9] J. Bear and J.M. Buchlin. Modelling and applications of transport phenomena
in porous media, volume 5 of Theory and applications of transport in porous
media. Kluwer, 1991.
[10] J.M. Coulson and J.F. Richardson. Fluid flow, Heat transfer and mass transfer,
volume 1 of Chemical engineering. Butterworth Heinemann, 1999.
[11] P.G. de Gennes. Wetting : statics and dynamics. Review of modern physics,
53(3):827–863, Juillet 1985.
[12] U. Thiele, K. Neuffer, Y. Pomeau, and M.G. Velarde. On the importance of
nucleation solutions for the rupture of thin liquid films. Colloids and surfaces
A., 206:135–155, 2002.
[13] V.P. Carey. Heat and fluid flow in microscale and nanoscale structures, volume 13 of Development in heat transfer, chapter DSMC modeling of nearinterface transport in liquid-vapor phase-change processes with multiple microscale effects, pages 303–347. WIT Press, 2002.
12
A
Table of notations
Letter
A
Bs
Cp
D
k
K
l0
L
Lf
M
ṁ
−
→
n
p
<
Re
S
v
Meaning
Dispersion constant
Bodenstein Number
Massic specific heat
Diffusion coefficient
Thermal diffusivity
Constant in expression (22)
Constant in expression (22)
Axial length of the gazeous phase in the capillary
Length of the diffusion limit layer
Molar mass
Evaporation rate
Unit vector normal to the meniscus
Pressure
Perfect gas constant
Reynolds number
Terme source de chaleur
Velocity vector
Greek letters
α
∆Hvap
δ
µ
ν
ρ
Subscript
ad
d
g
l
sat
v
w
0
Meaning
Mass fraction of vapour
Vaporisation latent heat
Film thickness
Dynamic viscosity
Cinematic viscosity
Density
Unit
J
J/kg/ ◦ K
m2 /s
m2 /s
Pa
m
m
m
kg/mol
kg/s
Pa
J/mol/K
W
m/s
Unit
J/kg
m
kg/m/s
m2 /s
kg/m3
Meaning
Dimensionless
Grandeur se rapportant à la pression de disjonction
Gaseous phase
Liquid phase
Saturation value
Vapor component
At the capillary wall
At the capillary apex
13