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: navneetkumar@iisc.ac.in (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-

v

ion 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 .

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