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TRANSCRIPT
International Journal of Multiphase Flow 109 (2018) 114–122
Contents lists available at ScienceDirect
International Journal of Multiphase Flow
journal homepage: www.elsevier.com/locate/ijmulflow
Experimental investigation of effect of orientation and surface
roughness on drying of porous media consisting of rods
Navneet Kumar a , ∗, Jaywant H Arakeri a
a Department of Mechanical Engineering, Indian Institute of Science, Bangalore - 560012, India
a r t i c l e i n f o
Article history:
Received 30 December 2017
Revised 26 May 2018
Accepted 16 July 2018
Available online 30 July 2018
Keywords:
Porous Media
Evaporation
Near-zero radii contacts
Smooth and rough rods
Horizontal rods
Vertical rods
IR heating
a b s t r a c t
Studies of drying from a conventional porous medium (CPM), consisting of spheres, have shown the exis-
tence of three periods. In the first period drying rate is high and essentially depends on the atmospheric
demand while in the last stage mass transfer happens from within the porous medium and is solely gov-
erned by the internal properties. In a previous study (Kumar and Arakeri) it was shown, using a cluster
of closely packed smooth rods stacked vertically, that stage 1 was sustained till nearly the whole liquid
gets evaporated. Near-zero radii contacts, between the rods, offering infinite height rise was shown to be
the reason behind such high sustained evaporation rates and elongated duration of stage 1. Here we show
that the orientation of the rods dramatically changes the evaporation process from this vertically oriented
rod-based porous medium (VRBPM). Unlike the vertical case, in the horizontal case (HRBPM) smooth rods
do not support stage 1. In this case transition to stage 2 was due to the liquid-vapour meniscus reced-
ing continuously in the network from the start of the experiment. Rough rods pinned the liquid along
its rough surfaces and stage 1 is sustained although its duration is much smaller compared to the verti-
cal case. Maintained hydraulic connections against gravity along the roughness of the rods thus provides
a better insight towards understanding stage 1 in a conventional porous medium consisting of compli-
cated pore geometry and their connectivity. In all the experiments, infra-red heating at about 10 0 0 W/m
2
causes evaporation from an initially saturated RBPM kept in an acrylic box.
© 2018 Elsevier Ltd. All rights reserved.
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1. Introduction
Drying of a porous medium finds its application in diverse
fields, such as soils and heat pipes, and thus has been studied
extensively in field and in laboratory experiments. Although in
many porous media, the particles are in general irregular and non-
uniform most of the laboratory studies have been from porous me-
dia made up of spheres. These spheres include both mono-disperse
particles consisting of glass beads ( Shaw, 1987, Kumar and Arak-
eri, Yiotis et al., 2012 ), metal balls ( Kumar and Arakeri ) etc. and
poly-disperse ( Lehmann et al., 2008 ) particles such as quartz sand.
We term this type of porous medium consisting of 3D network of
pores as conventional porous medium (CPM). A simpler void ge-
ometry can be formed if rods with circular cross-section instead
of spheres are used. We term these systems as RBPM (rods based
porous medium) where two basic configurations of rods are: verti-
cal (VRBPM) and horizontal (HRBPM). Vertical orientation has been
recently studied ( Kumar and Arakeri ) and a few surprising features
were shown compared to CPM.
∗ Corresponding author.
E-mail address: [email protected] (N. Kumar).
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https://doi.org/10.1016/j.ijmultiphaseflow.2018.07.009
0301-9322/© 2018 Elsevier Ltd. All rights reserved.
Previous studies ( Kumar and Arakeri, Yiotis et al., 2012,
ehmann et al., 2008, Sherwood, 1929, Idso et al., 1974 ) have
hown three distinct stages of drying from a CPM; initial saturation
lso affects the existence of these stages ( Hide, 1954; Jackson et al.,
973 ). In the 1st stage, high rates of evaporation are observed;
his stage is a constant rate period (CRP) as the evaporation rate
s nearly constant ( Lehmann et al., 2008; Sherwood, 1929 ). Water
resent near the open end of a porous medium leads to such a
igh evaporation rate. Near-surface water can be sustained by liq-
id film(s) in a porous medium; this film is the source of hydraulic
onnection between the near-surface water and the bulk (drying
ront) ( Yiotis et al., 2012 ). Nuclear-based ( Pel et al., 1993; Ketelaars
t al., 1995; Valckenborg et al., 2001; Shokri et al., 2008 ) water
ontent mapping has clearly shown the presence of these liquid
lms. In cases where films are responsible for stage 1, its break-
ge (depinning) ( Kumar and Arakeri ) then leads to drastic fall in
he evaporation rate and is termed transition regime or falling rate
eriod (FRP). In the last stage i.e. stage 2, the liquid-vapour (L-V)
eniscus recedes ( Sherwood, 1929 ) from the top surface with time
nd is termed receding front period (RFP). The time of onset of FRP
trongly depends on the average pore size ( Kumar and Arakeri, Yi-
tis et al., 2012, Lehmann et al., 2008 ). Transition to RFP or FRP can
N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122 115
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Fig. 1. Schematic of the experimental setup. Rods are shown stacked horizontally.
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lso occur without the film break-up. For larger particles in a CPM
apillary film effect is negligible ( Kumar and Arakeri ) and in this
ase water moves (recedes) smoothly with time within the porous
edium; with no significant formation of islands (wet) in the net-
ork apart from some of the water getting trapped in the contacts
etween spheres.
Another relevant type of study is evaporation of a liquid from
nside of polygonal capillary. Formation of vapour fingers in the
entre of the capillary with imbibing liquid along the corners of
polygonal capillary is well known ( Singhal and Somerton, 1970;
ansohoff and Radke, 1988; Dong and Chatzis, 1995; Lenormand
t al., 1983; Lenormand and Zarcone, 1984; Lenormand, 1990 ) and
uring drying a combination of these networks ( Yiotis et al., 2003 )
as shown to sustain stage 1 numerically. These features are very
ifferent than a circular cross section capillary and during drying
xperiments capillary films, formation was nicely shown ( Chauvet
t al., 2009; Chauvet et al., 2010 ) along the corners of polygonal
apillaries. In the vertical case ( Chauvet et al., 2009; Chauvet et al.,
010 ) transition to 2nd stage was observed due to depinning of the
iquid to a lower location away from the open end of the square
apillary. In the horizontal case ( Keita et al., 2016 ) the transition
id not occur due to depinning but rather due to change in the
orner menisci geometry. These pinned liquid films were able to
ustain high mass transfer, similar to stage 1 in CPM, compared to
ircular capillaries which was first observed ( Eijkel et al., 2005 ) in
orizontal nanochannels. An extreme case of a polygonal capillary
as recently studied ( Kumar and Arakeri ) where rods were stacked
ertically (against gravity) in a closed packed structure. Note that
unit cell in such a study consists of three rods in mutual contact.
he unit pore in RBPM is the straight region formed by three circu-
ar arcs formed by three rods in contact with each other. The void
n this case thus has three corners with “near-zero radii (Kumar and
rakeri) ” which can also be thought of polygonal capillary with
ero internal angles. Limitless capillary rise ( H.M. Princen, 1969;
.M. Princen, 1969 ) can happen along these corners and forced de-
inning length was reported during the drying ( Kumar and Arak-
ri ).
The various aspects of drying of a CPM during its drying are
eviewed in a recent article ( Or et al., 2013 ). Aim of the current
tudy, though, is to investigate the effect of orientation in drying
f a RBPM where corners dominate the whole process. We focus
ostly on the characteristics of horizontal orientation and its dif-
erences with respect to the vertical orientation. Analytical results
or capillary rise between circular rods are given in ( Princen, 1970 ).
sing the experimental observations during the drying process in
RBPM we show that this configuration can be used effectively
n reducing evaporation and can also be treated as a base case of
ransition to stage 2 without any significant film effect. Further we
how the effect of rods surface roughness on drying from the two
onfigurations.
. Experimental methodology
Experiments with horizontally stacked rods were conducted
ith an acrylic box having a height of 40 mm and 76 mm x
0 mm as the cross-sectional dimensions. Experiments with verti-
ally stacked rods were conducted with a different acrylic box hav-
ng a height of 78 mm and 51 mm x 41 mm as the cross-sectional
imensions. Circular cross-section rods (2 mm & 3 mm diameter
lass and 0.7 mm diameter pencil rods) were stacked in the acrylic
ox; the length of the rods was 75 mm. Deionized water was used
s the evaporating liquid which wets both the types of rods used
n the present study. The pencil rods were of HB type (super poly-
er lead) as stated by the manufacturer (Faber-Castell).
Fig. 1 shows the experimental setup and remain similar to
he previous studies ( Kumar and Arakeri, Kumar and Arakeri ).
vaporation from the porous medium was enhanced by infrared
IR) heating from the top; the heater (20 cm x 20 cm) was placed
ertically above the drying sample at a distance of ∼18.5 cm. The
uality (wavelengths primarily between 3 μm and 5 μm ) of the IR
adiation ensures that nearly all the heat is intercepted by the
ample within a small distance ( ∼0.1 mm) near the sample top;
ransmitted radiation can therefore be ignored. Mass loss was mea-
ured using a precision weighing balance (Sartorius GPA5202). An
R camera (Fluke Ti400) was placed at a vertical inclination of
25 ° to the sample and measures the top surface temperature
f the porous medium. A T-type thermocouple was used to mea-
ure the temperature of the bottom (outer) surface of the acrylic
ox which lies in between the container and the bottom insula-
ion. Ambient (temperature and relative humidity) conditions were
easured every 15 s and were monitored using a data logger. Re-
ently ( Aminzadeh and Or, 2014; Haghighi and Or, 2015; Amin-
adeh et al., 2017; Aminzadeh and Or, 2017 ) IR camera has been
sed to great effect in obtaining the surface energy budget of
orous media. To minimize heat loss, the acrylic box was insulated
ith Styrofoam ( ∼50 mm thick) on the four sides and the bottom.
he top surface was also insulated except the drying area which
as kept open. Note that all the insulated surfaces were further
overed with thin reflecting (shiny) aluminium foil; this avoids un-
ecessary heating of the drying sample via other routes.
Packing procedure is important and a unique protocol was fol-
owed; we briefly discuss it. The box was initially filled with wa-
er till its brim and a single layer of rods were placed at the con-
ainer bottom. Rods were then added layer by layer to obtain the
RBPM; the container was shaken frequently for better packing.
uch porous media are seen in Fig. 2 (a) and (b) for 0.7 mm diam-
ter pencil rods and 3.0 mm diameter glass rods respectively; rods
re exposed length-wise. Excess water was then removed to obtain
he saturated mass of water in HRBPM. Finally in all the experi-
ents a thin (2–3 mm) layer of water was added on top of initially
aturated HRBPM, which enabled us to obtain, as reference, the
vaporation rate from a bare water surface. For comparison ver-
ically stacked rod-based porous medium (VRBPM) consisting of
.7 mm diameter pencil rods are seen in Fig. 2 (c) and (d); rods
iameter portion is exposed to the ambient. Table 1 lists the de-
ails of different experiments including both HRBPM and VRBPM.
n important parameter of a porous medium is its porosity, the
otal void volume divided by the volume of the porous medium.
ased on the measured mass of water at (near) saturation (2nd
116 N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122
Fig. 2. Pictures showing the top view of HRBPM with (a) 0.7 mm diameter pencil rods and (b) 3.0 mm diameter glass rods. Top (c) and side (d) view of VRBPM consisting of
0.7 mm diameter pencil rods. A zoomed view (e) of the top surface (c) showing a nice packed arrangement of the pencil rods with a few defects. A large defect (imperfection)
is seen at the right edge of (e). Side (non-evaporating face) views of HRBPM consisting of 0.7 mm pencil rods (f), 2.0 mm (g) and 3.0 mm (h) glass rods. The imperfections
are seen clearly for the larger diameter rods.
Table 1
List of values used in the study of HRBPM containing rods; φ represents the volume porosity.
Also listed are experimental values for VRBPM cases.
Material (mm), Dia. (mm), Orientation Saturated mass (g) Exp. φ (%) Theor. φ (%)
Glass Rods (GRH) 2.0, Horizontal 30.36 19.97 15.33
Glass rods (GRH) 3.0, Horizontal 28.80 17.18 15.17
Pencil Rods(PRH) 0.7, Horizontal 28.75 21.13 12.42
Glass rods (GRV) 2.0 , Vertical 29.41 18.88 15.26
Glass rods (GRV) 3, Vertical 28.29 18.05 16.16
Pencil rods (PRV) 0.7, Vertical 18.56 12.63 11.82
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column), we calculated the porosity values for all the cases (3rd
column) listed in Table 1 . Note that the theoretical porosity value
for an infinite array of closely packed circular rods with equal di-
ameter is 9.31%. In the presence of boundaries, the porosity in-
creases. Assuming a 2D hexagonal (having the largest coordination
number) we theoretically estimated the porosity (4th column of
Table 1 ) for the different cases accounting for the boundary ef-
fect of the planar walls. The difference between the theoretical and
experimental values in porosity is mainly due to presence of ran-
dom defects away from the wall and boundary regions where a
close packed structure is impossible. The increase in the experi-
mental porosity is decided by three factors: container dimensions,
rod diameter, and rods’ outer surface roughness. The container
imension may or may not be an integral multiple of the rod di-
meter; this decides the number of rods in the first layer. Both,
ontainer and rods are therefore important in deciding the (extent
f) imperfections during packing. Experimentally, the standard de-
iation in the rods’ diameter also increases the defects. It is ev-
dent that the rods’ roughness (of their outer surface) also influ-
nces the overall porosity. A slightly higher experimental poros-
ty value for the pencil rods is due to its lower density compared
o the glass rods. Note that, in comparison, the porosity value for
acked spheres is much higher and ranges between 31% and 45%.
Fig. 2 (e) shows a small section, 12.5 mm x 8.5 mm, of the
ig. 2 (c) for the VRBPM case. The number of rods in this zoomed
iew is ∼250 and the corresponding area occupied by the fluid
N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122 117
Fig. 3. Experimental data for the two orientations of RBPM consisting of smooth 2 mm diameter glass rods showing (a) evaporation rate versus time and (b) evaporation
rate versus saturation.
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s calculated to be ∼9.5%; a value very close to the no bound-
ry case. There are defects (such as the one at the right edge of
ig. 2 (e)) which are present in such systems and cannot be avoided
nless a well-designed boundary is intentionally manufactured. In
ase of HRBPM, the imperfections are not clearly visible in the
op (exposed) view rather we look for defects in the side (non-
vaporating face) view. Fig. 2 (f)–(h) show the side views of HRBPM
onsisting of 0.7 mm pencil, 2.0 mm, and 3.0 mm glass rods respec-
ively. Note that these images were captured at the end of the ex-
eriment; some water in the form of isolated water droplets can
e seen in these images near the bottom of the container. Note
hat the defects are clearly visible in the case of glass rods; more
learly for 3.0 mm rods case.
. Results and discussion
The raw mass loss data measured using the scale is converted
o get the evaporation rate, expressed in the units of ‘mm/d’ . We
rst discuss the evaporation characteristics of a HRBPM and com-
are it with that of a VRBPM, both consisting of 2 mm diameter
mooth glass rods. Note that the exposed (to the ambient) regions
re the diameter and length sides in VRBPM and HRBPM respec-
ively. For both the experiments excess water ( ∼ 5 g or 2.4 mm for
ertical glass rods (GRV) case and ∼ 7 g or 1.8 mm for horizontal
lass rods case (GRH) case) was put on top of the saturated porous
edia.
Variation with time of the evaporation rate is seen in Fig. 3 (a).
n the presence of IR heating the evaporation rate is seen increas-
ng. It reaches a maximum ( ∼ 20 mm/d) value soon. Excess wa-
er is evaporated in nearly 2.44 h and 2.95 h for GRH and GRV
ases respectively. Note that the initial distance between the sur-
ace and heater was 18.5 cm and when excess water evaporates
ompletely the heat flux reaching the rods top surface reduces
lightly. Once the rods are exposed higher evaporation rate (stage
) is sustained for GRV case as seen in Fig. 3 (a); near-zero radii
f contacts between the rods act as capillary films which supply
ater from deeper regions to the exposed top surface. However in
RH case the rate of evaporation reduces considerably, compared
o GRV case, and stage 1 lasts only till the top layer of rods are wet.
ear-zero radii contacts (in GRH case) transport water horizontally
and not vertically as in GRV case) in GRH case and thus hydraulic
onnectivity in the vertical direction is not maintained around the
ods outer surfaces which results in unavailability of water near
he top surface; stage 1 is therefore absent. Transition to stage 2
s sudden in GRV case while a gradual transition is seen for GRH
ase. In GRV, transition is due to depinning of water films while
uch depinning has no meaning in GRH case. We see that a simple
rientation (rods stacked horizontally) inhibits the hydraulic con-
ectivity to such an extent that films connecting the top surface
nd the bulk liquid below can no longer form resulting in a clas-
ic transition to stage 2 almost immediately. In CPM, composed of
pherical particles as in soils, the longer duration of first stage is
ue to films connecting the top surface to the bulk liquid being
aintained, which when broken, leads to transition from stage 1.
Instead of versus time, evaporation rate is plotted versus satura-
ion (S), ratio of instantaneous mass of water and initial saturated
ass of water, in Fig. 3 (b). The figure should be viewed from right
o left as indicated. At S = 1 evaporation rate is nearly 20 mm/d for
oth cases. Stage 1 is sustained for GRV case till about S = 0.10
hile in GRH case stage 1 is very short. Consider the top layer
f rods, between S = 1 to 0.8, in GRH case. The contacts (of rods
ith other rods) exposed to the ambient are wet while the ambi-
nt air enter from the sides (between the rods and side walls) and
nvades the void (between rods on top and just below it). Since the
lm exposed to the ambient is connected through the bulk menis-
us (between three horizontal rods) stage 1 exists. Once the top
–2 layers of rods is completely occupied by the gas phase and
ecome dry stage 1 ends, as there is no communication between
he voids formed by the rods in upper layers and in lower regions.
learly, due to lack of any significant film formation, against grav-
ty, in GRH case the transition to stage 2 is gradual and water, al-
ost everywhere, recedes uniformly within the network.
It is worthwhile to look at the temperature evolution of the
orous medium top surface in both the cases. It has been shown
arlier ( Kumar and Arakeri ) that in presence of external heat load
R camera clearly captures the wet and dry regions on (or near)
he surface for a CPM. As long as stage 1 is sustained the top sur-
ace temperature is low and during stage 2 it increases to a much
igher value ( Kumar and Arakeri ). IR images for the two present
ases (GRH & GRV) are seen in Fig. 4 ; three different instants are
hosen for analysis. Ascending order of temperature is indicated
y colours as: purple < blue < cyan < green < yellow < orange < red.
ote that the colour bar is dynamic i.e. they represent different
emperature ranges at different times. For both the cases at near
aturation (S ∼ 1) most of the surface is occupied by the colder re-
ions as seen in Fig. 4 (a) and (d) for GRH and GRV respectively;
ut slightly higher temperature is seen for GRH case owing to its
ower evaporation rate (17 mm/d) compared to 20 mm/d for GRV
ase. At a later time, t ∼ 6 h, evaporation rates are nearly 9 mm/d
nd 18 mm/d for the GRH and GRV cases respectively, and due to
he lower evaporation rate the GRH surface temperature is higher
y about 10 °C, ( Fig. 4 (b)) compared to GRV case ( Fig. 4 (d)). Low
118 N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122
Fig. 4. IR images of the top surface of the two porous media: (a–c) horizontal and (d–f) vertical case respectively. Variation, along a horizontal line (see (f)), of temperature
is plotted is shown in (g). Temperature data to be seen from left to right.
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temperature regions is limited to a few rods at this instant for GRH
case ( Fig. 4 (b)) indicating that no significant film reaching the top
surface is formed. At t ∼ 20 h, evaporation rates are nearly 2 mm/d
and 14 mm/d for the GRH and GRV cases. Note that at his instant
S = 0.46 and 0.15 respectively for the two cases. As expected the
surface temperature is much higher in case of GRH ( Fig. 4 (e)). IR
images thus clearly reflect the significant differences in the evapo-
ration rates when the rods’ orientation is changed.
Next we look at the surface temperature variation along a white
line marked “L0” in Fig. 4 (f) at different times; temperature along
this line is plotted versus the pixel number in Fig. 4 (g) for all the
six IR images. In the absence of stage 1, temperature of GRH case
is seen increasing rapidly compared to the GRV which has stage 1,
higher evaporation rates, for much longer duration. For example,
at t ∼ 20 h, surface temperatures are nearly 41 °C and 59 °C for
GRV and GRH cases respectively. Waviness profile in case of GRH
is likely due to the curved surfaces of the rods. Note that the IR
images were taken by blocking the IR radiation from the heater
for a few seconds. This assured that reflected IR radiation from the
porous medium surface did not enter the IR camera and gave the
true temperature value.
We now discuss the effect of rods surface roughness in deter-
mining the evaporation characteristics. Surface of the glass (2 mm
diameter) and pencil (0.7 mm diameter) rods are seen in Fig. 5 (a)
nd (b) respectively. The pencil rod surface has roughness of the
rder of 10 μm while the glass rod is very smooth. We show that
or the horizontal orientation, hydraulic connectivity can be main-
ained along these rough grooves around the rod leading to ap-
earance of stage 1, similar to what is obtained in CPM but which
s not present in case of smooth rods HRBPM.
Evaporation rates are plotted against saturation for HRBPM
ases, pencil (PRH) and glass rods (GRH) in Fig. 6 (a). For com-
arison the curve for vertically oriented pencil rods (PRV) is also
hown. We see that unlike the smooth rods (GRH) stage 1 is sus-
ained in PRH till a saturation of about 50%. The evaporation rates
n stage 1 are higher than in GRH case but lower than in PRV case.
f we assume that the hydraulic connectivity is maintained in PRH
ase then water between the rods will essentially be present as
hin films throughout the rod length. This configuration thereby
ranslates to evaporation from a 2D periodic array of line sources,
imilar to a previous ( Suzuki and Maeda, 1968 ) numerical study,
eparated by a distance equal to rod diameter. Note that evapo-
ation in PRV (or GRV) cases will be 3D as it happens from point
ources ( Schlünder, 1988 ) rather than line sources. Higher evapora-
ion rate in stage 1 in case of PRV is due to 3D nature of equicon-
entration (vapour shells ( Stefan, 1881; Brown and Escombe,
900; Bange, 1953 )) lines supporting slightly higher evaporation
ates compared to the PRH case. The value of saturation where
N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122 119
Fig. 5. Pictures showing 2 mm diameter smooth glass (a) and 0.7 mm diameter rough pencil (b) rods respectively.
Fig. 6. Variation of evaporation rates (a) and average surface temperature (b) with saturation. Also shown is the curve for pencil rods stacked vertically. In all the cases heat
flux received by the top surface was about 10 0 0 W/m
2 .
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ransition to stage 2 happens in PRH is not quite clear. Experiments
ave been done with one type and value of roughness. A system-
tic study can be done how different roughness values will affect
he duration of stage 1. However one thing is clear that in hori-
ontal orientation with smooth rods, films reaching the top sur-
ace are absent and thus stage 1 is almost non-existent, whereas
ougher rods do support film formation. In a CPM, stage 1 is sus-
ained by films forming through the 3-D pore geometry reaching
he top surface.
We now discuss the surface temperature evolution and its re-
ation to evaporation rate using surface heat budget. The surface
emperature is obtained by the IR images by taking the average
hroughout the exposed surface. For averaging, a large ‘window’
approximately 200 × 200 pixels) was chosen in the IR images; the
R camera has a sensor array of 320 × 240 pixels. We show that
he average surface temperature is an indirect measure of evapora-
ion rate. Fig. 6 (b) shows the variation of the average surface tem-
erature versus the saturation. For PRH case two zones are seen.
one ‘1 ′ represents stage 1 where temperature increases slowly
nd zone ‘2 ′ represents the transition regime. In stage 1 the evap-
ration rate (see Fig. 6 (a)) and thus latent heat loss reduces gradu-
lly, and the surface temperature rises resulting in increasing con-
ective and radiative heat losses making up for the reducing latent
eat loss. Note that for the case of PRV a similar trend (‘4 ′ ) is seen
n stage 1 although this trend will last till nearly S = 0.1. For the
RH case stage 1 does not exist and a rapid increase in tempera-
ure with decreasing S is seen in this case (marked ‘3 ′ in Fig. 6 (b)).
ote that the trend of ‘3 ′ in GRH case is very similar to ‘2 ′ , transi-
vion regime of PRH case. In stage 2 the surface attains much higher
alues for all the cases as seen in Fig. 6 (b).
odelling evaporation based on surface temperature
After the top layers become dry in case of smooth rods we have
een that the evaporation rate reduces due to lack of any signifi-
ant film formation leading to very short stage 1. The liquid-vapour
L-V) menisci formed within the porous medium may assumed
o be horizontal. The menisci receive heat required for evapora-
ion through conduction via the dry rods from the top. Based on
he surface temperature of the porous medium we can predict the
vaporation rate in this case. In the steady state the incoming heat
ill be lost completely and thus we can write the surface energy
alance for the top surface as:
total = I con v + I rad + I re f + I sur f
cond (1)
here I total is the total incident IR heat flux (in W/m
2 ) on the
op surface which gets distributed into convective heat loss ( I conv ),
adiative heat loss ( I rad ), reflected heat ( I ref ), and heat conducted
rom the surface into the porous medium ( I surf cond ). The measured
urface and the ambient temperatures are used to get:
con v = h ( T s − T a ) ; I rad = σ [ ∈ s T 4
s ∈ a (1 − F sh ) T 4
a ] (2)
here T s and T a are the average surface and ambient temperatures
espectively, σ is the Stefan-Boltzmann constant, ∈ s and ∈ a are the
missivity of the rods and the ambient respectively, and f sh is the
iew factor between the rods top surface and the heater. h is the
120 N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122
Table 2
Range of parameters for estimating the convective heat loss from
the porous medium top surface to the ambient.
T s ( °C) T a ( °C) Ra Nu h
41 – 61 26.5 – 27.5 1600 – 4000 3.8 – 4.7 8.8 – 11
Fig. 7. Estimated evaporation rates from the model ( • ) at different saturation values
according to Eqs. (8) and (9) . Also shown are the experimentally (o) found evapo-
ration rates.
Fig. 8. [Colour online] Variation of evaporation rate versus the average depth of
water in different types of porous media. Data for vertical rods and spheres are
also shown for comparison.
e
h
p
m
s
t
s
c
e
e
t
p
o
m
h
s
t
t
convective heat transfer coefficient which can be estimated using
the Nu-Ra correlations. Ra is defined as:
Ra =
g [
( T s −T a ) 0 . 5 ( T s + T a )
]L ∗3
να; L ∗ =
Area
P erimeter (3)
Here g is the gravitational acceleration, L ∗ is a characteristic length
(defined as the ratio of area to the perimeter of the exposed hor-
izontal surface), ν and α are the kinematic and thermal diffusiv-
ities of air respectively. Note that density difference, required for
the natural convection, is mainly due to the temperature difference
while the contribution of water vapours is relatively small and thus
is neglected. For the present geometry and the incident heat flux
we estimated that Ra < 10 4 and thus we can use the following re-
lation to get Nu :
Nu = 0 . 59 Ra 0 . 25 =
hL ∗
k (4)[ Lloyd and Moran, 1974 ]
Where k is the thermal conductivity of air. Eq. (4) is used to esti-
mate h and I conv in Eq. (2) ; Table 2 shows the range of parameters
used. For the calculation of Ra the change in values of the fluid
(air) properties are assumed not to change with the average fluid
temperature, which is reasonable since the temperature range is
not high.
The conducted heat from the surface into the porous medium
can be further written as:
I sur f
cond = I bottom
cond + I leak + I sens + I lat (5)
Where I bottom
cond is the heat conducted from the l -V menisci to the
porous medium within, I leak is the heat leak through the insula-
tion, I sens is used up in heating the porous medium sensibly (heat
stored), and I lat is the latent heat loss term. We estimated that I leak
and I sens is nearly 10% and 5% of I total respectively. Further consider-
ing an emissivity of 0.92 for the glass rods we get the reflectivity
as 8% of I total ; transmissivity is assumed to be 0 for IR radiations.
Eq. (5) combined with Eq. (1) can thus be rewritten as:
I lat = I total − ( I con v + I rad ) −(I re f + I leak + I sens + I bottom
cond
)(6)
Nearly 10% of the incident heat flux is conducted to the bottom
and we can further assume
I bottom
cond = 0 . 1 I total (7)
Using estimates for above mentioned terms and Eq. (7), Eq. (6) be-
comes
I lat = 0 . 67 I total − ( I con v + I rad ) (8)
The evaporation rate ( E ) is related to I lat by
E ( mm/d ) = 86 . 4 I lat /λ (9)
Where λ is the latent heat of vaporization (2260 J/g). In a pre-
vious study ( Kumar and Arakeri ) we estimated I total to be about
10 0 0 W/m
2 . We use the same value of incident heat flux since the
distance between the rod’s top surface and the IR heater remained
same as before ( Kumar and Arakeri ).
The evaporation rates are estimated from Eqs. (8) and (9) at
the instances where IR images were captured. At these values of
S the estimated evaporation rates from the model are compared
with the experimental values as seen in Fig. 7 . Model predicted
slightly higher values of E than the experimental counterpart but
the trend is captured quite well. If we consider the lower bound in
stimating I total and take the upper bounds in estimating the other
eat losses, rates from the model becomes nearly equal to the ex-
erimental values. Towards S = 0, the model results are expected to
atch the experimental values closely. We don’t have any data to
upport this argument; towards lower S values, the time required
o evaporate increased non-linearly. A complete model requires the
olution of the 1D heat conduction equation at the l -V menisci in-
orporating the thin film evaporation near the solid boundary. It
ssentially is an extension of the Stefan’s problem with an added
ffect of temperature gradients within the dry and wet regions of
he porous medium. A complete evaporation model is thus com-
lex. However, the presented model captures the essential features
f evaporation from stack of horizontal rods.
Present experiments are compared with spheres-based porous
edia (CPM). Interest lies in the similarities or differences ex-
ibited by the rods with its orientations when compared to the
pheres. Saturation values are not good as a comparison tool since
hey depend on the height of the porous column. Better is to use
he average depth of water in the porous media measured from
N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122 121
t
H
R
v
l
v
e
c
f
t
t
1
s
o
t
c
4
s
w
s
p
w
w
t
t
d
r
m
m
t
T
v
t
a
a
r
o
m
d
i
i
t
p
o
i
u
o
d
t
A
C
W
s
R
S
K
Y
L
K
S
I
H
J
P
K
V
S
S
R
D
L
L
L
Y
C
C
K
E
P
P
O
P
A
H
A
A
S
S
S
op (H water ) which is calculated as:
water =
Total evaporated mass
Column height ∗φ (10)
ate of evaporation for different types of porous media are plotted
ersus H water in Fig. 8 . Irrespective of rods diameter (solid dashed
ines) high evaporation rate is maintained till H water ∼ 70 mm for
ertical cases; transition to stage 2 depends weakly on rod diam-
ter. In CPM, however, transition strongly depends on the parti-
le diameter. At the end of stage 1, H water ∼ 30 mm and 12 mm
or 2.5–3.0 mm and 0.70–0.85 mm diameter glass beads respec-
ively. Horizontally stacked rough (pencil) rods sustained stage 1
ill H water ∼ 19 mm. Decrease in the evaporation rate during stage
is steeper than the vertical cases while transition to stage 2 is
till rapid. For smooth rods in horizontal orientation transition is
bserved from the very beginning followed soon by stage 2. The
ransition in this case is gradual, similar to that in spheres (CPM)
ases.
. Conclusion
Drying characteristics of porous medium consisting of rods,
tacked horizontally (HRBPM), have been studied and compared
ith that of vertically stacked rods (VRBPM) and CPM consisting of
pheres. In the vertical orientation with diameter side exposed, the
orous medium sustained high evaporation rates. Water depth till
hich this high evaporation rate is sustained was unusually high
hen compared to similarly sized spheres. This is possible due to
he near-zero radii contacts transporting water all the way to the
op (against gravity) where evaporation occurs.
The near-zero radii contacts in HRBPM transports water pre-
ominantly in the horizontal direction, but not in the vertical di-
ection against gravity. Due to this reason, in case of the porous
edium consisting of smooth rods, stage 1 lasts only till the top
ost voids are wet. Stage 1 is thus very short and the evapora-
ion regime directly entered into transition followed by stage 2.
he gradually reducing evaporation rate in smooth rods case was
ery similar to the one found in the spheres cases. Surprisingly if
he rods are rough stage 1 was clearly sustained even if the rods
re horizontally stacked and preferential hydraulic connectivity is
long the rod lengths.
Rough rods produced enough hydraulic connectivity, around the
ods, against the gravity in order to sustain stage 1. This property
f rough rods is very similar to the concept of capillary film for-
ation in case of spheres where the film path is highly tortuous
ue to curvilinear paths around the spheres. Roughness of the rods
n vertical cases was found not to alter the evaporative character-
stics. However it is surprising that changing the same configura-
ion from vertical to horizontal brings such a major change in the
roperties of the porous medium. A systematic study of the effect
f changing roughness values in same diameter rods would be an
nteresting addition to this work.
A simple energy balance for the porous medium top surface
sing the surface temperature may be used to predict the evap-
ration rate. A complete model would require solution of the con-
uction equations through the porous medium in conjunction with
hose for phase change and vapour flow.
cknowledgments
We gratefully acknowledge funding by Robert Bosch Centre for
yber Physical System under the grant RBCCPS/ME/JHA/PC-0013 .
e also thank financial assistance provided by Indian Space Re-
earch Organization under the grant ISTC/MME/JHA/374 .
eferences
haw, T.M. , 1987. Drying as an immiscible displacement process with fluid counter-
flow. Phys. Rev. Lett. 1671 59.15 .
umar, N., and Arakeri, J.H., 2018. Evaporation from confined porous media due tocontrolled IR heating from above, Transp. Porous Med. https://doi.org/10.1007/
s11242- 018- 1120- 4 . iotis, A.G. , Salin, D. , Tajer, E.S. , Yortsos, Y.C. , 2012. Drying in porous media with
gravity-stabilized fronts: experimental results. Phys. Rev. E 86.2, 026310 . ehmann, P. , Assouline, S. , Or, D. , 2008. Characteristic lengths affecting evaporative
drying of porous media. Phys. Rev. E 77.5, 056309 .
umar, N., and Arakeri, J.H., 2018. Sustained high evaporation rates from porousmedia consisting of packed circular rods, Int. J. Therm. Sci. https://doi.org/10.
1016/j.ijthermalsci.2018.07.035 . herwood, T.K. , 1929. The drying of solids—II. Ind. Eng. Chem. 21.10, 976–980 .
dso, S.B. , Reginato, R.J. , Jackson, R.D. , Kimball, B.A. , Nakayama, F.S. , 1974. The threestages of drying of a field soil. Soil Sci. Soc. Am. J. 38.5, 831–837 .
ide, J.C. , 1954. Observations on factors influencing the evaporation of soil moisture.Soil Sci. Soc. Am. J. 18.3, 234–239 .
ackson, R.D. , Kimball, B.A. , Reginato, R.J. , Nakayama, F.S. , 1973. Diurnal soil-water
evaporation: time-depth-flux patterns. Soil Sci. Soc. Am. J. 37.4, 505–509 . el, L. , Ketelaars, A .A .J. , Adan, O.C.G. , Van Well, A .A . , 1993. Determination of mois-
ture diffusivity in porous media using scanning neutron radiography. Int. J. HeatMass Transf. 36 (5), 1261–1267 .
etelaars, A .A .J. , Pel, L. , Coumans, W.J. , Kerkhof, P.J.A.M. , 1995. Drying kinetics: acomparison of diffusion coefficients from moisture concentration profiles and
drying curves. Chem. Eng. Sci. 50 (7), 1187–1191 .
alckenborg, R.M.E. , Pel, L. , Hazrati, K. , Kopinga, K. , Marchand, J. , 2001. Pore waterdistribution in mortar during drying as determined by NMR. Mater. Struct. 34
(10), 599–604 . hokri, N. , Lehmann, P. , Vontobel, P. , Or, D. , 2008. Drying front and water content
dynamics during evaporation from sand delineated by neutron radiography. Wa-ter Resour. Res. 44 (6) .
inghal, A.K. , Somerton, W.H. , 1970. Two-phase flow through a non-circular capillary
at low Reynolds numbers. J. Can. Pet. Technol. 9, 03 . ansohoff, T.C. , Radke, C.J. , 1988. Laminar flow of a wetting liquid along the corners
of a predominantly gas-occupied noncircular pore. J. Colloid Interface Sci. 121.2,392 .
ong, M. , Chatzis, I. , 1995. The imbibition and flow of a wetting liquid along thecorners of a square capillary tube. J. Colloid Interface Sci. 172.2, 278 .
enormand, R. , Zarcone, C. , Sarr, A. , 1983. Mechanisms of the displacement of one
fluid by another in a network of capillary ducts. J. Fluid Mech. 135, 337 . enormand, R. , Zarcone, C. , 1984. Role of roughness and edges during imbibition in
square capillaries, SPE annual technical conference and exhibition. Soc. Pet. Eng .enormand, R. , 1990. Liquids in porous media. J. Phys. Condens. Matter SA79 .
iotis, A.G. , Boudouvis, A.G. , Stubos, A.K. , Tsimpanogiannis, I.N. , Yortsos, Y.C. , 2003.Effect of liquid films on the isothermal drying of porous media. Phys. Rev. E 68,
037303 .
hauvet, F. , Duru, P. , Geoffroy, S. , Prat, M. , 2009. M, Three periods of drying of asingle square capillary tube. Phys. Rev. Lett. 103.12, 124502 .
hauvet, F. , Duru, P. , Prat, M. , 2010. Depinning of evaporating liquid films in squarecapillary tubes: Influence of corners’ roundedness. Phys. Fluids 22.11, 112113 .
eita, E. , Koehler, S.A. , Faure, P. , Weitz, D.A. , Coussot, P. , 2016. Drying kinetics drivenby the shape of the air/water interface in a capillary channel. Eur. Phys. J. Soft
Matter 39 (2) .
ijkel, J.C.T. , Dan, B. , Reemeijer, H.W. , Hermes, D.C. , Bomer, J.G. , van den Berg, A. ,2005. Strongly accelerated and humidity-independent drying of nanochannels
induced by sharp corners. Phys. Rev. Lett. 95, 256107 . rincen, H.M. , 1969a. Capillary phenomena in assemblies of parallel cylinders: I.
Capillary rise between two cylinders. J. Colloid Interface Sci. 30.1, 69 . rincen, H.M. , 1969b. Capillary phenomena in assemblies of parallel cylinders: II.
Capillary rise in systems with more than two cylinders. J. Colloid Interface Sci.30.3, 359 .
r, D. , Lehmann, P. , Shahraeeni, E. , Shokri, N. , 2013. Advances in soil evaporation
physics — a review. Vadose Zone J. 12, 4 . rincen, H.M. , 1970. Capillary phenomena in assemblies of parallel cylinders: III. Liq-
uid columns between horizontal parallel cylinders. J. Colloid Interface Sci. 34.2,171 .
minzadeh, M. , Or, D. , 2014. Energy partitioning dynamics of drying terrestrial sur-faces. J. Hydrol. 519, 1257–1270 .
aghighi, E. , Or, D. , 2015. Thermal signatures of turbulent airflows interacting with
evaporating thin porous surfaces. Int. J. Heat Mass Transf. 87, 429–446 . minzadeh, M. , Breitenstein, D. , Or, D. , 2017. Characteristics of Turbulent Airflow De-
duced from Rapid Surface Thermal Fluctuations: An Infrared Surface Anemome-ter. Boundary Layer Meteorol. 1–16 .
minzadeh, M. , Or, D. , 2017. Pore-scale study of thermal fields during evaporationfrom drying porous surfaces. Int. J. Heat Mass Transf. 104, 1189–1201 .
uzuki, M. , Maeda, S. , 1968. On the mechanism of drying of granular beds. J. Chem.
Eng. Jpn. 1 (1), 26–31 . chlünder, E.U. , 1988. On the mechanism of the constant drying rate period and its
relevance to diffusion controlled catalytic gas phase reactions. Chem. Eng. Sci.43 (10), 26 85–26 88 .
tefan, M.J. , 1881. Uber die Verdampfnng aus einem kreisf6rmig oder elliptisch be-grenzten Becken. - Sitzungsber. Kais. Akad. Wiss. Wien. Mathem. - naturwiss.
943 CI. II, LXXXIII .
122 N. Kumar, J.H. Arakeri / International Journal of Multiphase Flow 109 (2018) 114–122
B
Brown, H.T. , Escombe, F. , 1900. Static diffusion of gases and liquids in relation to theassimilation of carbon and translocation in plants. In: Proceedings of the Royal
Society of London, 67, pp. 124–128 .
ange, G.G.J. , 1953. On the quantitative explanation of stomatal transpiration. ActaBotanica Neerlandica 2 (3), 255–297 .
Lloyd, J.R. , Moran, W.R. , 1974. Natural convection adjacent to horizontal surface ofvarious plan forms. J. Heat Transf. 96 no. 4, 4 43–4 47 .