# A Novel Continuous Drop-Wise Condensation Technology for Improved Heat and
Mass Transfer Efficiencies

Ali Alshehri,1,2∗ Jonathan P. Rothstein,3 H. Pirouz Kavehpour1

1Mechanical and Aerospace Engineering Department,
University of California, Los Angeles, CA 90095, USA,
2Mechanical Engineering Department,
King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi
Arabia,
3Department of Mechanical and Industrial Engineering,
University of Massachusetts Amherst, Amherst, Massachusetts 01003-2210, USA.

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
## Abstract

> Drop-wise condensation (DWC) has been the focus of scientific research in
> vapor condensation technologies since the 20th century. Improvement of
> condensation rate in DWC is limited by the maximum droplet a condensation
> surface could sustain. Furthermore, The presence of non-condensable gases
> (NCG) reduces the condensation rate significantly. Here, we present
> continuous drop-wise condensation (CDC) to overcome the need of hydrophobic
> surfaces while yet maintaining micron-sized droplets. By shifting focus from
> surface treatment to the force required to sweep off a droplet, we were able
> to utilize stagnation pressure of jet impingement to tune the shed droplet
> size. The results show that droplet size being shed can be tuned effectively
> by tuning the jet parameters. our experimental observations showed that the
> effect of NCG is greatly alleviated by utilizing our technique. An
> improvement by at least six folds in mass transfer compactness factor
> compared to state-of-the-art dehumidification technology was possible.

## Introduction

Condensation is a perplexing problem to fully uncover, yet, its applications
play crucial roles in industrial development (?, ?, ?, ?, ?). In order to
promote more efficient applications, improving the condensation process has
been the focus of many scientific research. Various condensation heat transfer
augmentation techniques have been utilized, these techniques can be classified
to; Passive (?, ?, ?, ?, ?, ?, ?); active (?, ?); and compound techniques.
Unlike passive, active techniques require external forces to be applied either
on the heat transfer surface or working fluid. Unsurprisingly, passive
techniques have caught major attention of the scientific community because of
their easier implementation in existing industrial applications. Contrarily,
active techniques require additional equipment to exert the required forces to
improve the heat and mass transfer efficiencies. This, in turn, means higher
capital expenditures and operating expenses. Observing the state-of-the-art
literature, the most trending technology is altering surface wettability (?).
However, it is still under development due to the chemical and physical
durability issues of coating techniques (?, ?, ?, ?, ?).

Major research efforts have been focused on developing durable and cost
effective coating techniques to promote drop-wise condensation (DWC) and
mitigate the formation condensate films on surfaces, i.e. film-wise
condensation (FWC). DWC has shown to possess at least an order of magnitude
improved heat transfer coefficient compared to FWC (?). This improvement is
highly dependant on the frequency of droplet shedding on surfaces. Droplets
shedding has been achieved primarily by gravity assistance (?, ?, ?), droplet
jumping (?, ?, ?, ?), drag force (?, ?, ?, ?, ?), or by capillary driven
movement (?, ?). It has been widely accepted that droplets of diameters below
20 micron contribute about 80% of the total heat transfer to the surface (?).
This implies that removing droplets of higher diameters is preferred,
otherwise, their higher thermal resistance and coverage area will impede
further condensation. Combining superhydrophobic surfaces and a shedding
mechanism might seem to be the obvious solution for achieving efficient DWC.
However, superhydrophobic surfaces are characterized by their low nucleation
site density for condensation and higher droplet thermal resistance, therefore
presenting a conflicting purpose (?). Therefore, there is a high demand to
sustain efficient DWC with innovative techniques that go beyond surface
modifications.

An extremely important concern most passive augmentation techniques has not
resolved is the degradation of heat/mass transfer coefficients caused by the
existence of non-condensable gases (NCG) (?, ?). Experimentally, degassing
prior to running condensers has been successful in alleviating the effect of
NCG (?, ?). Despite the experimental convenience of such method, it is a
highly impractical solution in large scale condensers. NCG can find their way
into condensers via leak points or as chemical reaction products of vapor
interacting with the equipment material (?). On another front, the emerging
humidification-dehumidification (HDH) desalination technology relies heavily
on NCG as carrier gases. The premise of this technology is the low energy
required to humidify air compared to other thermal desalination counterparts
(?, ?, ?). Even though it is evident that the dehumidifier in HDH technology
is highly inefficient, the heat transfer deficit has been compensated by three
alternatives. They are; (1) extended contact area (?, ?, ?, ?) (2) direct
contact between humid air and cooling medium (?, ?, ?, ?, ?, ?, ?); and (3)
different NCG carrier (?, ?). Even though the former two solutions are
promising, the latter seems to address the problem at its core, i.e. the
effect of vapor diffusion coefficient thus condensation rate. Therefore, there
is a pressing demand on working out a solution to enhance condensation with
the presence of NCG.

To overcome the problem of sustaining efficient condensation without requiring
unstable and expensive surface modifications, an active method needs to be
designed. The active augmentation method needs to mitigate the negative effect
of NCG while maintaining practical applicability. Several active methods have
been tried, such as fluid/surface vibration (?, ?, ?, ?), electrohydrodynamic
effects (?, ?, ?, ?, ?), and rotating surfaces (?, ?, ?), to name a few. The
general goal of the different active methods is to prevent the condensate from
growing by actively sweeping it off the surface. While this has shown to be
effective, surface wettability is still important to generate DWC rather than
FWC. In addition, the effect of NCG is still not resolved with the
aforementioned methods. Here, we investigate utilizing jet impingement as an
active method for providing DWC on surfaces with varying surface wettability.
The jet impingement method not only helps with shedding droplets on wettable
surfaces but also helps with mitigating the effect of NCG. The utilization of
jet impingement in heat and mass transfer applications has been studied in
heating/cooling for single phase flow (?, ?), drying application (?), nucleate
boiling (?), and spray cooling (?). Recently, on-demand impingement of pure
steam jet has shown to alleviate the effect of NCG in accidental leakage (?,
?). We recently showed that utilizing jet impingement in condensing water
vapor in a humid environment results in a breath figure spot (?). This spot
defines the boundary over which effective condensation takes place.

To sustain efficient DWC without requiring unstable and expensive surface
modifications, we present a novel ‘continuous drop-wise condensation’ (CDC) as
a method to tune the maximum droplet size on modified and unmodified
condensation surfaces. We also present CDC as a method to improve condensation
with NCG by means of thinning the diffusion boundary layer and therefore
reducing the resistance to diffusion. Impinging a modified or unmodified
surface with a jet of humid air or pure vapor results not only in a higher
heat and mass transfer coefficients but provides an excellent droplet shedding
mechanism (Fig.1). Controlling the diameter of droplet shedding is made
possible by tuning the jet parameters, e.g. exiting diameter, velocity, and
standoff distance. To provide evidence of the proposed mechanism, several
experiments were conducted under various jet parameters as well as different
surfaces with a wide range of advancing contact angles, i.e. $\theta_{A}$ =
70o \- 160o. In addition, we utilize our experimental observation to show that
CDC provides over 6-fold improvement in compactness factor compared to state-
of-the-art dehumidifiers. Furthermore, using an analytical model, we show that
CDC provides enhancement in heat flux of over 300% compared to gravity-
assisted shedding mechanisms. This is made possible by the improved mechanism
of tuning the maximum droplet size compared to state-of-the-art techniques.
Finally, We provide a theoretical framework for understanding droplet dynamics
by comparing the different forces acting on a droplet during jet impingement.

Figure 1:  Utilizing Jet impingement as a means of continuous drop-wise
condensation (CDC). A. A schematic of CDC illustrating the condensation
mechanism utilizing jet impingement as a means of enhanced condensation rate
and droplet shedding. pure vapor or Humidified-air jet exits a tube of
diameter ($D$) at a standoff distance ($H$) with a mean velocity ($v_{j}$). B.
A still Microscopic image of CDC under a selected experimental condition ($D$
= 0.047 inches, $H$ = 0.32 inches and $Re_{j}$ = 3600). Complete description
of experimental set up is presented in methods section and supplementary
material Figure S1

## Results

### Shedding of droplets under jet impingement

The dynamics of jet impingement on a surface is characterized by a stagnation
region that spans almost two nozzle diameters (?, ?). Beyond this radial
location, a wall jet forms that behaves similar to a Blasius boundary layer.
Here we show that the force within the stagnation region provides an excellent
shedding capability. To study this, we visualize under a microscope the growth
and onset of shedding of droplets under different jet Reynolds numbers
$Re_{j}=4Q/\pi\nu D$, where $Q$ is the jet flow rate, $\nu$ is the kinematic
viscosity of humid air, and $D$ is the tube exit diameter (Fig.2). To provide
consistent comparisons, the surfaces were cleaned prior to each experimental
run as outlined in the methods section. Additionally, the advancing and
receding contact angles were measured before and after each experimental run
with no significant changes due to mobile droplet shearing effects. Here, we
utilized a hydrophobic Si wafer ($\theta_{A}$ = 107o and $\theta_{R}$ = 103o)
as the condensation surface (see methods section and Table 1). The jet flow
rate was first set to the desired value of jet mean velocity, after which the
surface temperature was brought down to the desired temperature ($T_{s}$ =
15${}^{o}C$). A high speed camera (Photron, FASTCAM Nova) attached to an
optical microscope (Nikon, AZ100) was utilized to obtain videos and images of
the condensation process (Video 3 and Fig.2).

Figure 2:  Effect of jet Reynolds number on the size of shed droplets. Images
show results of condensation experiments performed at an ambient temperature
of 21oC and surface temperature of 15oC where the relative humidity of ambient
and jet were 60% and 95%, respectively. The tested surface was a hydrophobic
Si wafer with $\theta_{A}$ = 107o and $\theta_{R}$ = 103o. Two different close
up view are shown for condensation with jet Reynolds numbers of A. $Re_{j}=$
1200, B. $Re_{j}=$ 2400, C. $Re_{j}=$ 3600, D. $Re_{j}=$ 4500, and E.
$Re_{j}=$ 6000\.

In Fig.2, we show the quasi-steady droplet distribution on the condensation
surface under different jet Reynolds numbers. In these experiments, the
relative humidity of ambient air was 60% and the jet was 95%. The tested
surface was a hydrophobic Si wafer with an advancing contact angle of
$\theta_{A}$ = 107o and a receding contact angle of $\theta_{R}$ = 103o. In
Fig.2 and the corresponding Video 3 found in the supplemental material, the
droplets can be observed to go through three main growth stages with time:
Stage I - slow growth of stationary drops beneath the impinging jet; Stage II
- fast growth as mobile droplets move radially outward merging with other
droplets; and Stage III - slow growth of droplets that have come to rest far
from the center of the impinging jet. In Stage I, single droplets residing on
nucleation sites present on the surface initially start growing by direct
condensation on their exposed surfaces. As single droplets reach a critical
size ($R_{c}\sim 1/\sqrt{4N_{s}}$, where $N_{s}$ is the nucleation site
density), they start coalescing with neighboring droplets. The growth
mechanism in Stage I can be described by a combination of direct condensation
and coalescence with neighboring droplets. As droplets grow to another
critical size, determined by the jet shedding capability, they start departing
their equilibrium location on the surface. The onset of droplet departure is
observed to be located near the stagnation region. This implies that droplets
in this region possess higher growth rates and experience higher drag forces
compared to droplet located further away. It is also observed that the
critical droplet size at the onset of departure is reduced with increasing the
jet Reynolds number. As droplets depart their first equilibrium location, they
start moving radially outward coalescing with droplets in their path (Stage
II). The action of movement and coalescence acts as an effective sweeping
mechanism for improved DWC. The growth mechanism of a mobile droplet is
determined only by coalescence and negligible direct condensation. After
droplets move to locations further from the center of the impinging jet, the
retention force due to surface tension overcomes the drag force by the gas
flow around them and they decelerate and stop. Stationary far-field droplets
can be observed clearly in Fig.2A but are out of frame in Fig.2B-C. Stationary
droplets continue to grow by smaller mobile droplets that are generated from
the stagnation region following their mother droplet path (stage III).

It is also worth noting that as droplets move along their radial path, they
leave dry traces which appear as white traces under the microscope. The
process of droplet growth then repeats on the traces until another droplet
coming from the stagnation region sweeps it away. Under ideal circumstances,
the size of the stationary droplets withing these traces is limited by the
size of the mobilized droplets. Hence, a mobilized droplet can be thought of
as the maximum size a droplet can grow. In some circumstances, surface
imperfections or dust particles can interfere with the mobile droplet sweeping
action but could be mitigated by cleaning the surface thoroughly.

In Fig.2A-E, a magnified view of the droplets formed beneath the impinging
jets are shown to illustrate the effect that jet velocity and jet Reynolds
number have on droplet mobility. At the smallest Reynolds number presented in
Figure 2A, $Re_{j}=1200$, droplets were not observed to shed even as they grew
quite large. In Fig.2B, at a $Re_{j}=2400$, droplets with radii greater than
$R_{max}$ = 33 $\mu$m were observed to shed and move radially outward
coalescing with smaller drops and growing as they moved. With increasing
Reynolds number, a further reduction in shedding drop size was observed. For
the highest flow rate tested, $Re_{j}=6000$, droplets with radii greater than
$R_{max}$ = 13 $\mu$m were observed to shed. These drop sizes are
significantly smaller than the case of gravity-assisted droplet shedding or
the shearing effect of boundary layer flows where only droplets with radii
above 250 microns shed from a hydrophobic surface (?). Our results clearly
demonstrate that the maximum condensed droplet size can be efficiently tuned
by controlling the impinging jet velocity and Reynolds number.

Surface wettability is an important factor in determining the shedding
capability of a surface regardless of the active mechanism generating the
shedding forces. In order to test the effect of wettability on CDC, the
results of a series of experiments are presented in Fig.3 for five different
surfaces with a wide range contact angles at a fixed jet Reynolds number of
$Re_{j}$ = 3600. In these experiments, ambient temperature was 21oC and the
surface temperature 15oC while the relative humidity of ambient air and the
jet were 60% and 95%, respectively. The advancing, $\theta_{A}$, and receding
contact angle, $\theta_{R}$, for each surface are presented in Table 1 along
with the contact angle hysteresis, $\theta_{A}-\theta_{R}$. The condensation
process and shedding capability is visualized in Fig.3 with videos available
as supplementary material (Videos 1-5). Two hydrophilic surfaces, one smooth
and one microstructured, with different contact angles are presented in Fig.3A
and 3B. On the smooth hydrophilic surface, Fig.3A, droplets with radii greater
than 20 $\mu$m were observed to shed. The microstructured hydrophilic surfaces
has roughly twice the contact angle hysteresis of the smooth hydrophilic
surface, $\theta_{A}-\theta_{R}=20^{o}$ vs $13^{o}$. As a result, the mobility
of the droplets is hindered by the increased interfacial pinning force on the
droplets caused by the presence of the microstructures and an increase in the
radius of the shedding drops was observed to a value of 36 $\mu$m. On the
other hand, minimizing the contact angle hysteresis, as is done for both the
hydrophobic and the nanostructured superhydrophobic surface shown in Fig.3C
and 3E, dramatically reduces the minimum droplet shedding radius by reducing
the interfacial pinning force. For example, droplet radius of the drops
shedding from the hydrophobic surface in Fig.3C was 13 $\mu$m. For the
nanostructured superhydrophobic surface shown in Fig.3E, the surface looks
clear under the microscope with no evidence of the pathlines of shedding
droplets clearly visible in Fig.3A-C. This is probably due to the low
condensation rate due in large part to the low density of nucleation sites on
these nanostructured superhydrophobic surfaces, but it could also be the
result of droplet jumping from the surface as they coalesce and interfacial
energy is recovered in the form of kinetic energy. Some evidence for the
presence of droplet jumping can be seen in the videos provided in the
supplementary materials. Droplet jumping has been shown to improve the heat
transfer to a surface during condensation (?) and will be discussed in more
detail later. Finally, we analyze the results of the microstructured
superhydrophobic surface in Fig.3D. Interestingly, even though it had the
largest advancing contact angle, the microstructured superhydrophobic surface
also had the highest contact angle hysteresis. The CDC experiments for this
were characterized by a significant pinning of droplets and a very large
variability and uncertainty in the size of the shedding droplets. As a result,
the discussion of the drop dynamics that follows will be focused primarily on
Surfaces 1, 2 and 3 for which repeatable data could be obtained.

Table 1: Advancing and Receding contact angles of the different surfaces used.
Surfaces have different wettability and contact angle hysteresis.
Name: Description | $\theta_{A}$ | $\theta_{R}$ | $\theta_{A}-\theta_{R}$ |  $cos(\theta_{A})-cos(\theta_{R}$)
---|---|---|---|---
Surface 1: Hydrophilic | $85\pm 2^{o}$ | $72\pm 2^{o}$ | $13\pm 3^{o}$ | $0.22\pm 0.05$
Surface 2: Hydrophilic Microstructured | $70\pm 2^{o}$ | $50\pm 2^{o}$ | $20\pm 3^{o}$ | $0.30\pm 0.04$
Surface 3: Hydrophobic | $107\pm 2^{o}$ | $103\pm 2^{o}$ | $4\pm 3^{o}$ | $0.03\pm 0.05$
Surface 4: Superhydrophobic Microstructured | $160\pm 2^{o}$ | $127\pm 2^{o}$ | $33\pm 3^{o}$ | $0.34\pm 0.03$
Surface 5: Superhydrophobic Nanostructured | $157\pm 2^{o}$ | $154\pm 2^{o}$ | $3\pm 3^{o}$ | $0.022\pm 0.02$
Figure 3:  Effect of surface wettability on the condensation process. The
condensation experiments were performed with an ambient air temperature of
21oC and surface temperature of 15oC, while the relative humidity of ambient
and jet were 60% and 95%, respectively. Two different close up views are shown
for condensation with varying surface wettability and contact angle hysteresis
at a fixed jet Reynolds number of $Re_{j}$ = 3600. The advancing and receding
contact angles is displayed beside images of each surface.

### Dehumidification with CDC for improved heat and mass transfer rates

In this section, we analyze the heat transfer and condensation rates for the
CDC and compare their performance with the current state of the art
dehumidifiers. To compare the heat flux or condensation rate across many
different dehumidifier designs, we compare their compactness factors. The
compactness factor indicates the heat or mass transfer rate per unit driving
potential (temperature or vapor mass fraction) per unit volume and is defined
as follows (?).

$C_{h}=\frac{hA}{V}$ (1) $C_{m}=\frac{h_{m}A}{V}$ (2)

where $C_{h}$ is the compactness factor of heat transfer unit, $C_{m}$ is the
compactness factor of mass transfer unit, $h$ is the heat transfer
coefficient, $h_{m}$ is the mass transfer coefficient, $A$ is the surface area
over which measurement takes place, and $V$ is the volume of the
dehumidification system. In order to experimentally evaluate the mass transfer
coefficient, we utilized an optical method of observing the growth of
condensate droplets near the impingement region. The mass transfer coefficient
can be written as follows.

$h_{m}=\frac{\rho_{l}}{(\omega_{\infty}-\omega_{s})A}\frac{dV_{d}}{dt}$ (3)

where $\rho_{l}$ is the liquid density, $\omega_{\infty}$ is the vapor mass
fraction at ambient conditions, $\omega_{s}$ is the vapor mass fraction
evaluated at the surface temperature and $dV_{d}/dt$ is the condensate
volumetric growth per unit time. The volumetric growth rate can be calculated
directly from the video images. To calculate the droplet volume, the droplets
are assumed to take the form of a spherical cap because their radii are
smaller than the capillary length. For a droplet that is a spherical cap with
an optically observed radius of $R$ on the hydrophobic surface, the volume of
a droplet can be calculated from

$V_{d}=\frac{\pi}{3}(2+cos\theta_{A})(1-cos\theta_{A})^{2}R^{3}$ (4)

The evolution of droplet volume with time was calculated within the
impingement region (a surface with a diameter of 1 mm). Additionally, the
number and volume of droplets shedding and leaving the impingement region was
tracked with time. Because droplets departing the impingement region collect
more liquid as the travel radially outward, this procedure provides the lower
limit of the condensation rates and the mass transfer coefficient (Fig.4A).

Figure 4:  Heat and mass transfer rate improvement of CDC. A. Experimental
evaluation of mass transfer coefficient, $h_{m}$ of CDC as a function of
impinging jet Reynolds number. The experiments were performed at an ambient
temperature of 21oC and surface temperature of 15oC where the relative
humidity of ambient and jet were 60% and 95%, respectively. The tested surface
was a hydrophobic Si wafer with $\theta_{A}$ = 107o and $\theta_{R}$ = 103o.
B. Comparison of heat and mass transfer compactness factors for different
state-of-the-art dehumidifiers; plate-and-tube dehumidifier (?), bubble column
dehumidifier (?), flat plate dehumidifier (?), moving liquid beads
dehumidifier (?). The average value of our current experiments is shown.

In Fig.4, results for the mass transfer coefficient and the mass transfer
compactness factor are presented for different jet Reynolds numbers. In these
experiments, the relative humidity of ambient air was 60% and the jet was 95%.
The tested surface was a hydrophobic Si wafer with an advancing contact angle
of $\theta_{A}$ = 107o and a receding contact angle of $\theta_{R}$ = 103o.
The mass transfer coefficient ranged from $0.2<h_{m}<0.6$ kg/m2-s while the
compactness factors obtained were in the range of $30<C_{m}<300$ kg/m3-s. The
largest values of compactness factor were found for the lowest Reynolds
numbers tested. In Fig.4B, a comparison of the compactness factor for
different dehumidifiers along with the current CDC method is presented for
condensation on hydrophobic surfaces. Four state-of-the-art dehumidifiers are
shown namely (1) plate-and-tube dehumidifier (?); (2) bubble column
dehumidifier (?), (3) flat plate dehumidifier (?), (4) moving liquid beads
dehumidifier (?). As seen in Fig.4B, even when compared against the lowest
value obtained utilizing CDC, the compactness factor of CDC exceeds the
highest state-of-the-art dehumidifier by almost 6 times. Thus, CDC provides an
extremely compact dehumidifier as well as a very compact heat transfer
technique. We expect that even higher values of compactness factor are
possible through optimization of geometric parameters like nozzle standoff
distance and impinging jet diameter.

### Pure steam condensers with CDC for improved heat and mass transfer rates

We now turn to the case of utilizing our method to improve the heat and
condensation rates for the case of pure vapor/steam. To characterize the
condensation process due to the CDC improved droplet shedding, we visualized
the condensation process on different jet Reynolds numbers (Fig.2) and
different surface wettability (Fig.3). As observed, when droplets grow to
their maximum droplet size near the stagnation region, they are shed by the
jet impingement action. In contrast, in regular gravity-assisted DWC, the
shedding mechanism is mainly due to the weight of the droplet. This requires
droplets to grow by direct condensation and coalescence with neighboring
droplets until reaching the capillary length (mm range) beyond which their
weight overcomes the surface tension force. For pure vapor condensation, the
high thermal resistance of large droplets reduces the heat transfer
significantly (Fig.S3). In addition, allowing a surface to sustain high
droplet size before shedding results in a decrease in the small droplet
density (Fig.5A). It is well established that droplets with radii below 20
$\mu m$ contribute to almost 80% of the overall heat transfer to the surface
(?). Therefore, CDC is an efficient means of reducing the maximum droplet size
and increasing the population density of droplets below 20 $\mu m$ (Fig.5A).

To evaluate the heat transfer flux, we utilize the analytical model developed
by Rose et al. (?, ?). The overall heat flux to a surface exposed to
condensation in an ambience of pure vapor is given as

$q^{\prime\prime}=\int_{r_{min}}^{r_{e}}q_{d}(r,\theta)n(r,\theta)dr+\int_{r_{e}}^{r_{max}}q_{d}(r,\theta)N(r,\theta)dr$
(5)

where $q_{d}(r,\theta)$ is the heat transfer through a single droplet with
Radius $r$ and contact angle $\theta$, $n(r,\theta)$ and $N(r,\theta)$ are the
population density of of small and large droplets, respectively. Droplet below
the critical droplet radius ($r_{e}=1\sqrt{4N_{s}}$) grow by direct
condensation (small droplets) while droplet above the critical radius grow by
direct condensation an coalescence with neighboring droplets (large droplets).
The heat transfer through a single droplet can be written as

$q_{d}(r,\theta)={\pi
r^{2}(T_{sat}-T_{s}-\frac{2T_{sat}\gamma}{\rho_{l}h_{fg}r})}(\frac{1}{2h_{i}(1-\cos{\theta})}+\frac{r\theta}{4k_{l}\sin{\theta}}+\frac{\delta_{s}}{k_{s}\sin^{2}{\theta}})^{-1}$
(6)

Detailed analysis is given in Supplementary material (section S-4). It is
worth noting that the previous equation was derived for the case of pure
vapor. The main variable CDC introduces is the modification of maximum droplet
radius value which consequently changes the heat transfer characteristics as
shown in Fig.5B. Because for this case the jet advective transport does not
introduce a thermal resistance, it acts only as an improved shedding mechanism
and no further modification is required to Eq.6. In Fig.5B, the heat flux to a
surface is improved significantly by lowering the maximum droplet radius.
Improvement as high as almost 150% in heat flux (or condensation rate) can be
obtained by utilizing a hydrophilic surface with maximum droplet radius of 20
$\mu m$. If one compares utilizing the hydrophilic surface over the
superhydrophobic surface, a maximum improvement of heat flux above 375% can be
achieved (see supplementary material section S.4). This shows that heat
transfer and consequently condensation rate can be improved by tuning the
maximum droplet size which can be achieved easily with CDC.

Figure 5: Heat transfer characterization of CDC. A. The population density of
droplets ($rN$) of radii below 20$\mu$m for different maximum droplet radius
($R_{max}$) being shed by the jet impingement action. The ordinate is defined
as the number of droplets per unit surface area. B. Heat flux
($q^{\prime\prime}$) to a surface exposed to pure vapor analytically evaluated
at different maximum droplet radius ($R_{max}$). The heat flux and maximum
droplet radius are normalized with values evaluated in case of gravity-
assisted droplet shedding (see supplementary material section S-4).

### Jet-droplet dynamics

Figure 6: A typical time evolution of droplet motion and growth. A. Time
evolution of a droplet departing its first equilibrium location and moving
radially outward. B. Typical transience plot illustrating both the location of
the droplet ($s$) and its diameter ($D$) as it coalesces with other droplets
in its path. The droplet goes through three different periods; (I) a waiting
period before droplet departure, (II) an accelerating droplet period, and
(III) a decelerating droplet period. This plot is generated for a selected
case of $Re_{j}=3600$ and hydrophobic surface ($\theta_{A}$ = 107o and
$\theta_{R}$ = 103o).

The dynamics of droplet shedding and motion is governed by an interplay
between droplet inertia, droplet retention forces due to surface tension,
viscous friction within the drop, and drag force due to flow of air around the
drop. The effect of each of these was investigated by tracking the motion and
size of individual droplets at the different stages of growth and motion as
they as they moved across different substrates under different impinging jet
conditions. In Fig.6A, a sample droplet is traced with time along the
hydrophobic surface ($\theta_{A}$ = 107o and $\theta_{R}$ = 103o) for one
representative case at a jet Reynolds number of $Re_{j}=3600$. The diameter
and location of the droplet is plotted as functions of time in Fig.6B. Three
different periods can be clearly distinguished from the data: Period I - a
waiting period before droplet departure; Period II - a period of radial
acceleration of the droplet; and Period III - a period of droplet deceleration
until the droplet comes to rest. The onset of droplet motion occurs after a
waiting period during which the droplet grows through condensation to a
critical size. At this critical size, the aerodynamic drag force acting on the
droplet becomes larger than the interfacial retention forces acting along the
contact line between the droplet and the surface. After the droplet departs
its initial location (onset of period II), it coalesces with droplets in its
path resulting in a fast growth in droplet size and an acceleration across the
surface because with increasing size the aerodynamic drag force grows faster
with droplet radius than then the interfacial retention force ($R^{2}$ vs.
$R$). During period II, the droplet roughly triples meaning the volume has
increased by a factor close to thirty. During period III, the droplet
decelerates and the rate of diameter growth slows as fewer coalescence event
occur. This deceleration occurs because the strength of the shear flow near
the wall decreases as the drop moves radially outward from the center of the
impinging jet ($1/s$). Qualitatively similar results were observed for all
surfaces tested provided the jet Reynolds number was larger than the critical
Reynolds number to initiate droplet motion.

Figure 7: Dynamics of droplet-jet interaction. A. Droplet location from the
center of the impingement region ($s$) as a function of time ($t-t_{o}$),
where $t_{o}$ is the time of onset of motion. The plot shows the effect of
different jet Reynolds numbers ($Re_{j}$) for the hydrophobic surface
($\theta_{A}$ = 107o and $\theta_{R}$ = 103o). B. The diameter of droplets at
the onset of droplet departure ($D_{o}$) is depicted for three different
surfaces at different jet Reynolds numbers ($Re_{j}$). C. For a comparison
purpose, ratio of droplet diameter at a reference location ($s_{r}=1$ mm) from
the center of the impingement region to that at the onset of motion
($D/D_{o}$) is plotted against the jet Reynolds numbers ($Re_{j}$) for three
different surfaces. D. a reference apparent speed of droplets
($v_{r}=s_{r}/(t_{r}-t_{o})$) at a reference location ($s_{r}=1$ mm) is
plotted against the jet Reynolds numbers ($Re_{j}$) for three different
surfaces.

In Fig.7A, the droplet location ($s$) is shown as a function of elapsed time
($t-t_{o}$) for four different jet Reynolds numbers ($Re_{j}$) for the
hydrophobic surface, where $t_{o}$ is the time of onset of motion. The
different curves show similar trends depicted in Fig.6B and the existence of
three different motion periods. Note that the transition from period II to
period III is delayed by lowering the jet Reynolds number. Additionally, the
droplet begins to decelerate at a location further from the center of the
impinging jet for larger jet Reynolds numbers. Ideally, we would like to
collapse all of these curves onto a single master curve. The obvious choice is
non-dimesionalize the time on the x-axis with the jet velocity divided by the
jet diameter, $(t-t_{0})V_{j}/D_{j}$. Unfortunately, this simple non-
dimensionalization does not collapse the data. Nor will the data collapse if
the critical diameter for droplet motion, $D_{0}$, is used in place of the jet
diameter, $(t-t_{0})V_{j}/D_{0}$. The dynamics of drop motion are clearly
quite complicated in this process.

In Fig.7B, the diameter of droplets is presented at the onset of departure
over a range of jet Reynolds numbers for three different surfaces:
hydrophobic, hydrophilic and microstructured hydrophilic. For the same jet
Reynolds number, the hydrophobic surface, which has the largest advancing
contact line and smallest contact angle hysteresis, consistently shows the
smallest critical droplet diameter for the onset of droplet motion. With
decreasing advancing contact angle and contact angle hysteresis, the critical
droplet diameter increases for a given jet Reynolds number with the
microstructured hydrophilic surface consistently showing the largest values of
critical diameter needed to induce droplet motion. Importantly, the critical
droplet diameter can also be tuned by controlling the jet parameters, namely
the jet velocity or Reynolds number. As can be seen in Fig.7B, increasing jet
Reynolds number results in a significant decrease in critical droplet
diameter. At large values of jet Reynolds number, a distinct scaling can be
observed that suggests, $D_{0}\sim Re_{j}^{-2}$. To investigate this further,
forces acting on the droplet at force equilibrium are analyzed just before
droplet departure.

The dominating forces are surface tension force along the droplet’s contact
line and aerodynamic drag forces on the droplet surface. Viscous friction is
assumed to be negligible because the droplet is static at this period. The
surface tension force along the contact line can be written as (?)

$F_{s}=\Gamma_{L}D_{o}\gamma(\cos\theta_{R}-\cos\theta_{A})$ (7)

where $\Gamma_{L}$ is a contact line shape factor and $\gamma$ is the
interfacial surface tension. The value of the contact line shape factor
depends on the droplet geometry and symmetry. In general, its value has been
experimentally found to lie within $1/2$ and $\pi/2$. (?, ?) It has been
observed that the state of motion of a droplet affects the surface tension
force acting on a droplet, e.g. static and dynamic behavior (?). Additionally,
external vertical forces on the droplet can play a role in pinning the droplet
(?). Rather than resorting to a different mathematical description as
suggested by Tadmor et al. (?), Eq.7 is used and the mentioned effects are
absorbed in the order one contact line shape factor, $\Gamma_{L}$.

At equilibrium, the contact line force is balanced by the aerodynamics force
generated by impinging jet that is trying to dislodge the droplet. The effects
of velocity variation across the boundary layer can be ignored as the droplets
are significantly larger than the boundary layer thickness (see supplementary
material section S-5) (?, ?, ?, ?). Thus, the drag force, which is
proportional to the stagnation pressure on the droplet, can be written as

$F_{d}=\frac{1}{8}\rho_{g}\Gamma_{A}C_{d}u_{o}^{2}D_{o}^{2}$ (8)

where $C_{d}$ is the drag coefficient of the droplet, $\rho_{g}$ is the
surrounding gas density, $\Gamma_{A}$ is a shape factor of the projected area
of the droplet in the flow direction (see supplementary material section S-5),
and $u_{o}$ is the effective air velocity around the droplet. Note that
$u_{o}$ scales with the jet mean velocity, $u_{o}\sim v_{j}$, and decays with
distance from the location of jet impingement. Balancing the forces acting on
a droplet at the onset of motion, yields the following relation.

$D_{o}=\frac{\gamma(\cos\theta_{R}-\cos\theta_{A})\Gamma_{L}}{\frac{1}{2}\rho_{g}C_{d}\Gamma_{A}}\frac{1}{v_{j}^{2}}\sim\frac{1}{Re_{j}^{2}}$
(9)

Note that the scaling of critical diameter with jet Reynolds number predicted
by Eq. 9 fits the experimental data in Fig.7B quite well for the high jet
Reynolds number cases of all three surfaces tested. However, a deviation is
observed at the lowest jet Reynolds number studied. This is likely
attributable to the complexity of evaluating the geometrical shape factors and
the drag coefficient which are not constant as assumed her, but depend on the
air velocity and diameter of the drop (?).

Following the onset of droplet motion, the droplets grow through a combination
of continued condensation, which is slow, and coalescence with smaller
stationary droplets in their path, which is fast. At least initially,
increasing droplet size is observed to result in an acceleration of the drop
and an increased droplet velocity. As shown by Eq. 7, the interfacial
retention force increases with $F_{s}\sim D$, while the aerodynamic drag force
in Eq. 8 scales increases with $F_{d}\sim D^{2}$. Hence, as the droplets
coalesce and grow beyond the critical diameter for droplet motion where these
two forces are in balance, a force imbalance favoring aerodynamic drag over
retention force exists and the droplets accelerate. In order to quantify the
rate of droplet diameter growth during this acceleration period, the droplet
diameter normalized by the critical diameter, $D_{r}/D_{o}$, was measured at a
reference location ($s_{r}=1$ mm) and is plotted against the jet Reynolds
number for each of the three surfaces. A schematic diagram is shown in in
Fig.7D as a reference. The reference location lies within the accelerating
period (Period II) for all the cases presented. In Fig.7C, the diameter of
each droplets was found to increases significantly by the time it reached a
position 1mm from the center of the impinging jet. The hydrophilic structured
surface, which had the largest contact angle hysteresis shows the smallest
increase in droplet diameter with an average of just $D_{r}/D_{o}=1.35$. This
diameter increase suggests a volume increase of roughly 2.5x meaning that on
average 2.5 coalescence events with similar sized drops have occured during
the first 1mm of travel. Within uncertainty, no significant difference between
the hydrophilic and the hydrophobic cases could be observed. On average, the
droplets on these surface grew faster than the microstructured hydrophilic
surface with an increase of in droplet diameter of roughly $D_{r}/D_{o}=1.7$
and a corresponding volume change of 5x. It is interesting to note that even
though the size of droplets is different under varying jet Reynolds numbers,
the ratio of increase is constant for the same surface wettability independent
of jet Reynolds number. This suggests a self-similarity property of the
condensation process.

In Fig.7D, the velocity of the droplets, $v_{r}$ is presented as a function of
jet Reynolds number. Here, again the velocity is measured at a location within
the accelerating region ($s_{r}=1$ mm). The droplet velocity can be seem to
vary monotonically with the jet Reynolds number. Moreover, the droplet
apparent speeds are similar for the three surfaces within the experimental
uncertainty. A simplistic equation of motion (EOM) is given in supplementary
material section S.6. In the acceleration period, the drag force is assumed to
be much greater than the retention forces. This in turn means that the droplet
motion in Period II should be independent of the surface wettability which
agrees with our experimental observation in Fig 7D.

## Discussion

A novel continuous drop-wise condensation that is efficient and compact was
achieved by utilizing jet impingement of water vapor on a cooled surface. The
fluid dynamics of an impinged jet showed an excellent shedding capability to
overcome the limitations of the state-of-the-art techniques.

In summary, we have demonstrated the capability of our CDC design in improving
the condensation process substantially compared to state-of-the-art condensers
and humidification technologies. We performed condensation experiments on
modified and unmodified Silicon substrates on a broad range of contact angles.
The drop size being shed was controlled by tuning the jet parameters namely
the jet velocity in this work. We showed that micron-sized droplets could be
shed effectively even on hydrophilic surfaces.

By comparing the condensation rate per unit volume of state-of-the-art
dehumidification technologies, we showed that our design is at least six fold
higher. This significant increase is attributed to the thinning of the
diffusion layer which is known to impede vapor condensation. We also
illustrated that by controlling the maximum droplet size being shed,
improvements as high as 375% in heat flux was possible for steam condensers in
the absence of NCG.

Finally, we discussed the droplet dynamics and growth under the jet
impingement action. By comparing the different forces acting on a droplet, we
were able to predict the size of droplet being shed under varying jet Reynolds
number and surface wettability. Furthermore, the models presented in this work
are the starting point for further optimization of the design to obtain more
compact dehumidification using CDC technique.

### Further discussion

The droplets generated withing the impingement area are of sizes less than the
capillary length of water (or bond number $Bo<<0.1$). Therefore, gravitational
force effect is negligible which in turn means the current analysis is
independent of surface orientation. Gravitational force will be effective on
the stationary droplets which reside in the region corresponding to the
decelerating period (Period III, Fig.6B). The effect of gravitational force as
well as other dominating forces, such as viscous dissipation are subjects of
future research.

It was noticed that in surfaces that are characterized by high contact angle
hystereses droplets tend to stop at random locations corresponding to the
decelerating period ( Period III in Fig.6B). This indicates that the droplet
shedding advantage of jet impingement is lost away from the impingement
region. We believe that this does not undermine the effectiveness of CDC as
most of vapor condensation takes place where shedding is significant. To
overcome the accumulation of condensate in that region, different engineering
solutions could be applied. Placing the condensation surface vertically helps
with shedding larger droplets. It is experimentally observed that stationary
droplets in regions corresponding to period III grow quickly by droplet
feeding coming from the impingement region. Other methods, such as surface
texturing or capillary wicking could be utilized to effectively drain
excessive condensate.

Surfaces with low contact angle hystereses tend to show better uniformity of
droplet shedding and stoppage location. The shedding of droplets is further
improved by droplet jumping in the case of superhydrophobic surface
($\theta_{A}$ = 157o and $\theta_{R}$ = 154o). As noticed in Fig.3E and video
S.6, droplets in the micro-scale jump off the surface due to the release of
energy upon coalescence. This phenomenon has been shown earlier to have
potential in improving DWC (?) and energy harvesting (?). A great advantage
offered by CDC is that jumping droplets are further shed by flow generated
from jet impingement. Hence, mitigate the return of micro-droplets on the
surface and therefore prevent flooding of the surface.

Supplementary Material are presented in the attached file.

## Materials and Methods

### Surface preparation and characterization

Commercially available Silicon wafers (Techgophers) were used as the base
condensation surfaces. In this work, we utilized five surface modifications
namely (1) Hydrophilic Si surface, (2) hydrophilic micro-structured Si
surface, and (3) hydrophobic silanized Si surface (4) superhydrophobic micro-
structured Si surface, and (5) superhydrophobic spray-coated Si surface.
Silicon wafers of similar thicknesses were used in this work to provide
similar thermal resistances over the different tests. Checking the temperature
distribution on the Si surfaces shows that their thermal resistance is quite
negligible compared to the vapor-NCG side dominating thermal resistance.
Before each experiment, the surfaces were cleaned with acetone (J.T.Baker),
isopropanol (J.T.Baker), ethanol (J.T.Baker) and DI water and dried with
filtered nitrogen stream. Experiments were done immediately after the cleaning
process so that the effect of organic compounds found in room environment is
negligible.

After Si wafers were cleaned different methods were utilized to change their
wettability. The untreated Si wafer provides the base surface which happens to
be slightly hydrophilic ($\theta_{A}$ = 85o and $\theta_{R}$ = 72o). For
altering the surface wettability, extra steps were performed other than the
cleaning process. Micro-posts are fabricated on top of the silicon wafer with
a diameter of 34 $\mu$m, height of 34 $\mu$m, and pitch of 50 $\mu$m. Because
water condensation exists between the posts, a Wenzel state is noticed which
renders the silicon wafer more hydrophilic ($\theta_{A}$ = 70o and
$\theta_{R}$ = 50o). To render the substrate hydrophobic ($\theta_{A}$ = 107o
and $\theta_{R}$ = 103o), the surface was silanized with a thin layer of
polydimethylsiloxane (PDMS) using a conventional dip coating method. The film
thickness is negligible compared to the thickness of the silicon wafer and
therefore does not impede the heat transfer rate. Inverted micro-posts are
fabricated on top of the silicon wafer with a diameter of 40 $\mu$m, height 40
$\mu$m, and pitch of 50 $\mu$m. To obtain superhydrophibicity, a combination
of silanization process with the roughened surface yielded a superhydrophobic
surface with a high contact angle hysteresis ($\theta_{A}$ = 160o and
$\theta_{R}$ = 127o). For the last surface, we utilized an aerosol spray
coating method to coat the silicon wafer with WX2100 (purchased from cytonix)
in which Fluorothane is the active ingredient. The resultant surface is
superhydrophobic with negligible contact angle hysteresis ($\theta_{A}$ = 157o
and $\theta_{R}$ = 154o). The contact angle is independent of the coating
thickness. Therefore, we coated the surface with about 50 $\mu$m without
significant impedance of heat transfer.

The surface wettability was characterized before and after each experimental
run to assess the homogeneity of their wettability as well as the consistency
of contact angle measurements. Droplet shape analyzer (KR$\ddot{\rm{U}}$SS,
DSA100) was used for contact angle measurements. The static advancing and
receding contact angles were measured using the protocol outlined in this
paper (?). Measurements were repeated on different spots of the Si wafers to
ensure homogeneity and consistency. Droplets with diameters less than the
capillary length were tested to ensure negligible effects of gravity. Static
advancing and receding contact angles are summarized in Table 1.

### Condensation experiments

In Fig. S1, we show a schematic of the experimental setup which consists of a
bubble humidifier, a flow system, and a cooled surface. Dry filtered air was
bubbled into a pool of room-temperature DI water through several spargers
(Ferroday). The spargers generate micron-sized bubbles which due to their high
contact area with water get humidified to above 95%. The humid air generated
exists at a room temperature ($T_{\infty}$ = 21oC $\pm$ 1 oC) and ambient
humidity of (60% $\pm$1%). Humidity of ambient air and humidified jet was
measured using a Hygrometer (VWR). The humidified air was led through a tube
(Mc-MASTER-CARR) of inner diameter ($D$ = 0.047 in) to impinge normally on the
cooled surface. For experimental convenience, the tube was bend to 90o while
allowing enough length ($L$ = 0.84 in) before the exist section ensuring fully
developed flow beyond the secondary flow region. The flow rate of humidified
air was controlled by flow-adjustment valve and measured using a rotameter
(OMEGA, model no. FMA-A2323). Volumetric flow rates tested range from 1 LPM to
5 LPM. The corresponding jet mean velocities range from 15 m/s to 75 m/s. The
humid air jet exits the tube at a standoff distance ($H$ = 0.32 in).

The condensation surface was the different Silicon wafers described earlier.
The surfaces were placed on an Aluminum substrate with a thermally conductive
paste in between. The Aluminum substrate was placed on the cold side of a
Peltier plate with a thermally conductive paste. A simple peltier plate with a
temperature controller unit was used to maintain a constant surface
temperature ($T_{s}$ = 15oC $\pm$ 1 oC). An Infra-red (IR) camera (FLIR,
A6753sc), and two flush-mounted k-type thermocouples (OMEGA, HH378) were used
to observe the condensation substrate temperature as well as the condensate
droplets. The substrate temperatures measured by the three sensors were in
agreement within 1 oC. This ruled out any possible temperature variation on
the surface and ensured that the thermal resistance of vapor-gas side was
dominant.

Systematic experiments were performed by first adjusting the flow to the
desired jet Reynolds number $Re_{j}=4Q/\pi\nu D$, where $\nu$ is the kinematic
viscosity of humid air. Then, the surface temperature was set to the desired
temperature. The condensation process was allowed to reach a quasi-steady
state by waiting for about 15 minutes before taking experimental measurements.
To visualize the condensation process an optical microscope (Nikon, AZ100)
with a high-speed camera (Photron, FASTCAM Nova) were used.

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Acknowledgements:

A. Alshehri would like to express his sincere gratitude to King Fahd
University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia.
Funding: This research was supported by the National Science Foundation under
Grant No. CBET-2032533.
Competing Interests The authors declare that they have no competing financial
interests.
Data and materials availability: Additional data and materials are available
online.

Figure S1:  Continuous Drop-wise Condensation experimental setup. A. A
schematic of CDC illustrating the general setup for condensation experiments.
Dry air is bubbled into a room-temperature pool of DI water through several
spargers (one is shown for illustration). The different parameters are
discussed in the method section. B. A side view of the condensation surface
assembly.

##  S-1 Heat Transfer of water Vapor Condensation with Humid Air Jet
Impingement

The previous section utilized a theoretical approach to understand and
quantify the heat flux to a condensation surface in the case of pure vapor
condensation, i.e. no air involved. Hence, this section is dedicated to model
the problem of air existence. It has been shown that the existence of minute
amounts of non-condensable gases (NCG), such as air reduces the condensation
rate, and thus the heat flux, tremendously. This is due to the fact that at
steady state conditions, a diffusion layer builds on top of the condensate
formed on the surface. The condensation of vapor becomes mainly dominated by
the mass diffusion through this layer. The thermal resistance of this layer
dominates the heat transfer to the surface, therefore, Eq. 24 is not valid to
describe the system. The heat transfer from the surrounding through the
droplet to the condensation surface can be estimated simply as.

$q^{\prime\prime}(r)=\dot{m}^{\prime\prime}h_{fg}+h(T_{\infty}-T_{s})$ (10)

where the mass flux to the surface can be written as

$\dot{m}^{\prime\prime}=\frac{\phi Sh\rho
D_{ij}}{D}\bigg{(}\frac{\omega_{v,\infty}-\omega_{v,o}}{1-\omega_{v,o}}\bigg{)}$
(11)

where the suction effect factor is written as

$\phi=\frac{1-\omega_{v,o}}{\omega_{v,o}-\omega_{v,\infty}}ln\bigg{(}\frac{1-\omega_{v,\infty}}{1-\omega_{v,o}}\bigg{)}$
(12)

The heat transfer coefficient can be written as

$h=\frac{Nuk}{D}$ (13)

Sherwood and Nusselt numbers for jet impingement can be obtained by utilizing
heat/mass transfer analogy along with the correlation usually used for single
round jets (?, ?).

$\frac{Sh}{Sc^{0.42}}=\frac{Nu}{Pr^{0.42}}=G(A_{r},\frac{H}{D})[2Re^{1/2}(1+0.005Re^{0.55})^{1/2}]$
(14)

where

$G=\frac{1-1.1D/r}{1+0.1(H/D-6)D/r}\frac{D}{r}$ (15)

In Fig.4B, we show the resulting heat and mass fluxes for the case of a
saturated humid air jet impinging on a cold surface at varying jet parameters.
The inputted parameters in the correlation are; saturated environment at
$T_{\infty}$ = 22oC and $T_{s}$=15oC at $P$=1 atm. The nozzle diameter is 1 mm
located at varying distance. The impingement area is is confined withing 2.5
mm.

##  S-2 Calculation of compactness factor of different state-of-the-art
condensers:

For a condenser existing in humid air environment, one can write the heat
transfer to the surface as follows

$q=hA(T_{\infty}-T_{s})$ (16)

where $h$ is the heat transfer coefficient, $A$ is the exposed surface area,
$T_{\infty}$ and $T_{s}$ are the ambient and surface temperatures,
respectively. Similarly, the mass transfer (condensation rate) can be written
as follows

$\dot{m}=h_{m}A(\omega_{\infty}-\omega_{s})$ (17)

where $h_{m}$ is the mass transfer coefficient, $\omega_{\infty}$ and
$\omega_{s}$ are the ambient and surface vapor mass fraction, respectively.
The relation between $h$ and $h_{m}$ can be inferred from heat/mass transfer
analogy as

$\frac{h}{h_{m}}=\frac{Nu}{Sh}\frac{k}{\rho
D}=\frac{CRe^{n}Pr^{m}k}{CRe^{n}Sc^{m}\rho
D}\approx\bigg{(}\frac{Pr}{Sc}\bigg{)}^{1-m}C_{pa}=Le^{1-m}C_{pa}$ (18)

The value of the exponent ($m$) depends on the correlation being used. For
instance, for a laminar boundary layer over a flat plate, a value of (1/3) is
used (?). In our case, the correlation given by Eq.14 is characterized by an
exponent of (0.42).

For comparison of the current method of condensation with other
condensers/dehumidifiers, we think CDC provides an extremely compact
condenser. Therefore, we compare the different state-of-the-art condensers by
a compactness factor which is given by the following relation

$C_{h}=\frac{hA}{V}$ (19) $C_{m}=\frac{h_{m}A}{V}$ (20)

where $C_{h}$ and $C_{m}$ are the compactness factor of heat and mass transfer
exchangers, respectively. The higher the value indicates a higher transfer
rate per unit driving potential (temperature or vapor mass fraction) per unit
volume. In Fig.4A, we show a comparison of the compactness factor for
different dehumidifiers along with the current method. We reconstruct the
results published by Sadeghpour et al. (?).

##  S-3 Heat Transfer of Pure Vapor with Jet Impingement

In this section, an estimate of the heat transfer of DWC of pure vapor is
presented. We utilize the theoretical model developed originally by Rose and
coworkers (?). The process of DWC can be summarized by the following sequence
of events; initial nucleation of vapor clusters, droplet growth by
condensation on the interface, droplet growth by coalescence, droplet fall
off, and finally re-nucleation of droplets.

The first process of DWC is heterogeneous condensation over active nucleation
sites on the condensation substrate. The active nucleation site density
($N_{s}$) depends on surface topography, Fluid’s thermophysical properties,
and subcooling degree (?, ?). The value of active nucleation site density is
in the range of 109-1015 m-2 (?). For the purpose of obtaining an approximate
comparison between jet- and gravity-assisted shedding mechanisms, we choose a
value for $N_{s}=10^{12}$ m-2. The smallest stable droplet formed by
condensation in the nucleation site is given as (?, ?)

$r_{min}=\frac{2T_{sat}\gamma}{\rho_{l}h_{fg}(T_{sat}-T_{s})}$ (21)

where $T_{sat}$, $\gamma$, $\rho_{l}$, $h_{fg}$, $T_{s}$ as saturation
temperature, surface tension, liquid density, latent heat of vaporization, and
surface tension, respectively. It has been shown experimentally that droplets
in the range $[r_{min},r_{e}]$ grow by direct condensation only, where the
effective radius results from geometrical argument as $r_{e}=1/\sqrt{4N_{s}}$.
Droplets with radii higher than the effective radius grow by direct
condensation on their surfaces as well as by coalescence with neighboring
droplets. DWC on vertical surface, i.e. typical configuration, are
characterized by the existence of a maximum droplet radius of which droplets
start sliding on the surface and consequently sweeping smaller droplets in
their path. A balance between the weight of the droplet and the surface
retention force results in the following relation of maximum radius for
gravity-assisted shedding (?)

$r_{max,g}=\sqrt{\frac{6\gamma\sin{\theta}(\cos{\theta_{r}}-\cos{\theta_{a}})}{\pi\rho_{l}g(2-3\cos{\theta}+\cos^{3}{\theta})}}$
(22)

where $\theta$, $\theta_{r}$, $\theta_{a}$ are the static, receding and
advancing contact angles respectively and $g$ is the gravitational
acceleration. To estimate the overall heat transfer rate to the condensation
surface, several researchers have utilized the theory developed by Rose and
co-workers (?). The model is centered around correlating the heat transfer
across a single droplet to the overall heat transfer across the entire
droplets on a surface. The following formula is usually considered for the
overall heat transfer rate

$q^{\prime\prime}=\int_{r_{min}}^{r_{e}}q_{d}(r,\theta)n(r,\theta)dr+\int_{r_{e}}^{r_{max}}q_{d}(r,\theta)N(r,\theta)dr$
(23)

where $q_{d}(r,\theta)$, $n(r,\theta)$ and $N(r,\theta)$ are the heat transfer
through a single droplet of radius ($r$), the small droplet size distribution
in the range $[r_{min},r_{e}]$, and large droplet size distribution in the
range $[r_{e},r_{max}]$, respectively. The detailed derivation of the
individual parameters has been discussed in several papers beginning with the
work of Rose (?).

The heat transfer across a single droplet of radius ($r$) can be represented
as a combination of Laplace pressure effect due to curvature, liquid-vapor
interfacial thermal resistance (Knudsen layer), conduction through the droplet
body, and conduction through the condensation surface. heat transfer across a
single droplet can be written as (?)

$q_{d}(r,\theta)={\pi
r^{2}(T_{sat}-T_{s}-\frac{2T_{sat}\gamma}{\rho_{l}h_{fg}r})}(\frac{1}{2h_{i}(1-\cos{\theta})}+\frac{r\theta}{4k_{l}\sin{\theta}}+\frac{\delta_{s}}{k_{s}\sin^{2}{\theta}})^{-1}$
(24)

where $k_{l}$ and $k_{s}$ are the thermal conductivity of the liquid and the
condensation surface, respectively and $\delta_{s}$ is the thickness of the
condensation surface. The liquid-vapor interfacial heat transfer coefficient
is given as (?)

$h_{i}=\frac{2\sigma_{c}}{2-\sigma_{c}}\sqrt{\frac{M}{2\pi
RT_{s}}}\frac{h_{fg}^{2}\rho_{v}}{T_{s}}$ (25)

where $sigma_{c}$, $M$, $R$ and $\rho_{v}$ are the condensation coefficient,
molecular weight of water, gas constant, and water vapor density,
respectively. The size distribution of large droplets was derived
experimentally and mathematically as (?, ?)

$N(r,\theta)=\frac{1}{3\pi
r_{max}r^{2}}\bigg{(}\frac{r}{r_{max}}\bigg{)}^{-2/3}$ (26)

Lastly, the population balance theory was used to derive the small droplet
size distribution (?). The form is given as

$n(r,\theta)=\frac{1}{3\pi
r_{max}r_{e}^{3}}\bigg{(}\frac{r_{e}}{r_{max}}\bigg{)}^{-2/3}\frac{r(r_{e}-r_{min})}{r-r_{min}}\frac{A_{2}r+A_{3}}{A_{2}r_{e}+A_{3}}\exp{(B_{1}+B_{2})}$
(27)

where the constants are given as

$\displaystyle A_{1}$ $\displaystyle=\frac{(T_{sat}-T_{s})}{2\rho_{l}h_{fg}},$
(28a) $\displaystyle A_{2}$
$\displaystyle=\frac{\theta(1-\cos{\theta})}{4k_{l}\sin{\theta}},$ (28b)
$\displaystyle A_{3}$
$\displaystyle=\frac{1}{2h_{i}}+\frac{\delta_{s}(1-\cos{\theta})}{k_{s}\sin^{2}{\theta}}$
(28c) $\displaystyle A_{4}$ $\displaystyle=\frac{A_{2}}{\tau
A_{1}}\bigg{[}\frac{r_{e}^{2}-r^{2}}{2}+r_{min}(r_{e}-r)+r_{min}^{2}\ln{\bigg{(}\frac{r_{e}-r_{min}}{r-r_{min}}\bigg{)}}\bigg{]}$
(28d) $\displaystyle A_{5}$ $\displaystyle=\frac{A_{3}}{\tau
A_{1}}\bigg{[}r_{e}-r+r_{min}\ln{\bigg{(}\frac{r_{e}-r_{min}}{r-r_{min}}\bigg{)}}\bigg{]}$
(28e) $\displaystyle\tau$
$\displaystyle=\frac{3r_{e}^{2}(A_{2}r_{e}+A_{3})^{2}}{A_{1}[8A_{3}r_{e}-14A_{2}r_{e}r_{min}+11A_{2}r_{e}^{2}-11A_{3}r_{min}]}$
(28f)

Equations 21 through 28 along with known thermophysical properties provide a
complete set of equations to obtain the overall heat transfer rate to a
surface. For instance, Fig.S3 shows the heat flux variation on a surface with
different wettability, i.e. contact angles, under gravity-assisted droplet
shedding. In the figure, we also show the maximum radius of droplets being
shed by the assistance of body weight. For a surface with static contact angle
of 90o, we notice that a heat flux of 30 kW/m2 could be transferred to the
surface with a maximum droplet radius of 390 microns. With higher contact
angle surfaces, droplets of lower radius could be shed. However, due to the
increased conduction resistance, heat flux drops beyond a contact angle of
145o.

In Fig.S4, we show for a constant static contact angle, the effect of varying
the maximum droplet radius. The cross symbols in the figure represent the case
of gravity-assisted droplet shedding as their respective static contact
angles. For instance, the heat flux to a surface could be enhanced by 150% if
a coating with static contact angle of 160o is used with a maximum droplet
radius of around 20 micron compared with the hydrophilic surface and gravity-
assisted case.

We believe that with CDC, the mechanism of droplet growth would be similar to
that of regular drop-wise condensation with the maximum droplet radius
determined by the jet impingement shedding action. Therefore, in Fig.4D we
normalize the heat flux and maximum droplet radius with with their respective
values due to gravity-assisted drop-wise condensation.

Figure S2:  Heat flux to a vertical surface exposed to pure vapor analytically
evaluated at different static contact angles with gravity-assisted shedding.
The parameters inputted into the model are; $T_{sat}$=22oC, $T_{s}$=15oC,
$\delta_{s}$ = 254$\mu$m, $k_{s}$=100W/m2K, $N_{s}$=1012 sites/m2,
$\sigma_{c}$=1, and $\theta_{A}-\theta_{R}$=5o.  Figure S3:  Heat flux to a
vertical surface exposed to pure vapor analytically evaluated at maximum
droplet Radii for the different tested surfaces. The cross symbol represents
the value obtained with gravity-assisted shedding. The parameters inputted
into the model are similar to that in Fig.S5 for meaningful comparison.

##  S-4  Discussion of drag force quantification

As observed in (Fig.1 and Fig.2 of main manuscript), the initial droplet
shedding occurs withing the stagnation region, i.e. almost within a tube
radius from the stagnation point. It is well established that for the case of
laminar non-mixing gas jet impingement, the similarity solution of Navier-
Stokes equations gives a constant momentum boundary layer thickness in the
impingement region. The equation of the boundary layer thickness is usually
written as $\delta_{o}\sim\sqrt{\nu D/u_{j}}$, where $\nu$ is the kinematic
viscosity of the jet, $D$ is the diameter of the tube exit, and $u_{j}$ is the
speed of the jet at the standoff distance (?, ?). In this work, the jet
Reynolds number was in the range of 1000-6000, therefore, the momentum
boundary layer thickness should be below the range of 15 - 35 $\mu m$,
respectively. Even though these estimations are for laminar jets ($Re<1000$),
turbulent mixing, like in our experiments, results in a lower boundary layer
thickness. Our observations of droplets size show that the boundary layer
thickness is smaller than the smallest droplet being shed. Therefore, we
expect that the flow field withing the boundary layer is unimportant.

In general, the drag force on the droplet due to the jet flow can be given by
equation 7. The projected area shape factor $\Gamma_{A}$ can be obtained
geometrically as

$\Gamma_{A}=\theta-\frac{1}{2}\sin 2\theta$ (29)

The drag coefficient based on the jet speed and droplet diameter is in the
range of 1-0.6 for Reynolds numbers of 100-500, respectively. These values
were obtained from spherical relations for the lack of better quantification
in the literature, however it is a common practice. The effective velocity
term is taken to equal the jet mean velocity value multiplied by a
proportionality constant, i.e. $u_{o}=av_{j}=Q/A_{tube}$.

##  S-5  Droplet equation of motion: simplistic approach

In this section, we present a simplistic equation of motion that represents a
one-dimensional force balance on a single droplet. In Fig.S5A, we show the
simplified physical model of a droplet with diameter ($D$) and contact angle
($\theta$) located at a distance ($x_{o}$) from the center of impingement
region. The jet issues from a tube that is located at a standoff distance
($H$) with a mean velocity ($v_{j}$). The only forces responsible for droplet
movement are drag force ($F_{d}$), surface tension force ($F_{s}$), and
viscous friction force ($F_{v}$) (Fig.S5B). Newtons seconds law is applied to
the droplet as follows.

$\rho_{l}\Gamma_{v}\frac{d(D^{3}v)}{dt}=\frac{1}{8}C_{d}\rho_{g}\Gamma_{A}(u_{o}-v)^{2}D^{2}-\Gamma_{L}D\gamma(\cos\theta_{R}-\cos\theta_{A})-\eta\Gamma_{base}R^{2}\frac{\partial
v}{\partial y}\bigg{|}_{base}$ (30)

where $\rho_{l}$ is the density of condensate liquid, $v$ is the local
velocity of the droplet, $\Gamma_{v}$ is the volumetric shape factor, $\eta$
is the dynamic viscosity, $\Gamma_{base}$ is the shape factor of the base area
of the drop. At the onset of droplet motion, only the drag and surface tension
forces are present. Equation 8 in the main manuscript is the resultant of the
force balance. In the accelerating droplet region (Period II), the drag force
becomes significantly higher than the retention forces. Hence, Eq.30 is
reduced to the following.

$\rho_{l}\Gamma_{v}\frac{d(D^{3}v)}{dt}=\frac{1}{8}C_{d}\rho_{g}\Gamma_{A}(u_{o})^{2}D^{2}$
(31)

This equation suggests a negligible effect of surface wettability on droplet
motion in regions corresponding to Period II. This is clear from the droplet
apparent velocity shown in Fig.6D. In the decelerating period (Period III),
Eq.30 should be fully used. However, because of the complexity of determining
the coefficients in the equation, presenting an outline of the equation is
sufficient.

Figure S4:  Physical model for writing the equation of motion of a single
droplet in contact with flow of an axisymmetric jet. A. one-dimensional
schematics of the pertaining parameters. B. Forces acting on a single droplet
under a generalized case of a moving droplet.

##  S-6 Videos

#### Video S1

10-second Condensation process on surface 1 (hydrophilic Silicon surface) at
different jet Reynolds numbers. The experimental conditions are as indicated
in the method section while surface characteristics are as in Table S1.

#### Video S2

10-second Condensation process on surface 2 (hydrophilic silicon surface with
high contact angle hysteresis) at different jet Reynolds numbers. The
experimental conditions are as indicated in the method section while surface
characteristics are as in Table S1.

#### Video S3

10-second Condensation process on surface 3 (hydrophobic Silicon surface) at
different jet Reynolds numbers. The experimental conditions are as indicated
in the method section while surface characteristics are as in Table S1.

#### Video S4

10-second Condensation process on surface 4 (superhydrophobic silicon surface
with high contact angle hysteresis) at different jet Reynolds numbers. The
experimental conditions are as indicated in the method section while surface
characteristics are as in Table S1.

#### Video S5

10-second Condensation process on surface 5 (superhydrophobic silicon surface
with negligible contact angle hysteresis) at different jet Reynolds numbers.
The experimental conditions are as indicated in the method section while
surface characteristics are as in Table S1.

#### Video S6

Illustration of droplet jumping on surface 5 (superhydrophobic silicon surface
with negligible contact angle hysteresis).