heat transfer and dynamics of impinging droplets on hot horizontal surfaces

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Heat Transfer and Dynamics of ImpingingDroplets on Hot Horizontal SurfacesRanjeet P. Utikar a , Sanjesh K. Pathak b , Anurag Mehra c & Vivek V.Ranade ba Department of Chemical Engineering , Curtin University ofTechnology , GPO Box U1987, Perth, WA, 6845, Australiab Industrial Flow Modelling Group , National Chemical Laboratory , Pune,411008, Indiac Department of Chemical Engineering , Indian Institute of Technology ,Bombay, Powai, 400 076, IndiaPublished online: 16 Dec 2011.

To cite this article: Ranjeet P. Utikar , Sanjesh K. Pathak , Anurag Mehra & Vivek V. Ranade (2010) HeatTransfer and Dynamics of Impinging Droplets on Hot Horizontal Surfaces, Indian Chemical Engineer, 52:4,281-303, DOI: 10.1080/00194506.2010.547769

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Heat Transfer and Dynamics of Impinging

Droplets on Hot Horizontal Surfaces

Ranjeet P. Utikar1*, Sanjesh K. Pathak2, Anurag Mehra3 &

Vivek V. Ranade2

1Department of Chemical Engineering, Curtin University of Technology, GPO Box U1987,

Perth, WA 6845, Australia2Industrial Flow Modelling Group, National Chemical Laboratory, Pune 411008, India3Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai 400

076, India

Abstract: In fluidised bed olefin polymerisation reactors, a liquid monomer is added for

enhancing heat removal (super-condensed mode). This broadens the operating window and can

substantially increase the capacity of a given reactor hardware. Design and location of liquid

injection nozzles play a key role in dictating the performance of condensed mode operations.

An understanding of the fundamental characteristics of droplet impingement onto solid particles

is critical for proper design. In this paper, we study the interactions between liquid droplets and

hot solid surfaces across various boiling regimes. High-speed digital imaging is used to capture

the droplet vaporisation process. Water droplets impacting on a solid wall with different Weber

numbers are investigated. The effect of various parameters such as surface and initial droplet

diameter, liquid velocity and surface tension is studied. Experimental data of dynamic drop

impact on a flat hot surface are reported. A phenomenological model is developed to describe

the droplet vaporisation phenomena in single phase and nucleate boiling regimes. The results

from the phenomenological model are compared with the experimental data presented here and

the experimental results reported in the literature. The study provides useful clues for

understanding the real case when solid particles are much larger than the injected liquid

droplets.

Keywords: Droplet, Boiling, Liquid injection, Polyolefin, Modelling, Heat transfer, Surface

interaction, Dynamics.

Paper received: 28/05/2009; Revised paper accepted: 31/03/2010

*Author for Correspondence. E-mail: [email protected]

INDIAN CHEMICAL ENGINEER – 2010 Indian Institute of Chemical Engineers

Vol. 52 No. 4 December 2010, pp. 281�303

Print ISSN: 0019-4506, Online ISSN: 0975-007X, http://dx.doi.org/10.1080/00194506.2010.547769S

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http://dx.doi.org/10.1080/00194506.2010.547769

IntroductionIn fluidised bed olefin polymerisation reactors, a liquid monomer is added for enhancing heat

removal (super-condensed mode). The injected liquid evaporates by coming in contact with hot

solids. This broadens the operating window and can substantially increase (by 50�100%) the

capacity of a given reactor hardware [1]. At macroscopic level, liquid injection into the fluidised

bed has been studied by Fan et al. [2], Leclere et al. [3] and Werther and Bruhns [4]. The design

and the location of liquid injection nozzles play a key role in dictating the performance of

condensed mode operations. If the evaporation of injected liquid is not fast enough, it may result

in defluidisation of the bed. An understanding of the fundamental characteristics of droplet

impingement onto solid particles is critical for proper design. Study of droplet impingement will

help in gaining insight into the problem and allow us to design better injection nozzles.

The actual physics of contact between hot solids and volatile liquid drops is quite complex.

Various mechanisms of heat transfer exist depending on the operating parameters. When a liquid jet

is injected in gaseous environment, it successively fragments into ligaments that break down into

drops [5]. However, when a liquid jet is injected in fluidised bed, different behaviours can be seen

depending on the relative size of solid particles and liquid drops. Mechanisms have been proposed

to describe these different behaviours. Figure 1 shows two such mechanisms proposed by Bruhns

and Werther [6] (Fig. 1a) and Nayak et al. [7] (Fig. 1b). If the evaporation times are large, the

droplet forms agglomerate. The agglomerate is broken due to evaporation from the surface finally

(in case of non-porous particles) or due to pore diffusion and evaporation from surface (in case of

porous particles). When particles are larger than liquid droplets and if temperature gradients are

large, droplet levitation can occur due to vapour generated during evaporation. Additionally, many

competing phenomena (like heat transfer, drop spreading, phase change) exist depending on the

operating parameters. Depending on the temperature difference, heat transfer may occur in different

regimes (single phase, nucleate boiling, transition boiling, film boiling). If the temperature gradient

between the surface and droplet is high, the droplet levitates on the surface, a phenomena known as

Leidenfrost effect. In case of polyolefin fluidisation, the problem is complicated further as we have

a wide particle size distribution and non-smooth particle surfaces. The mechanism of heat transfer is

not clear. Very limited information is available on these aspects in the literature. As a first step, an

idealised system would prove to be useful in studying relative importance of various parameters as

it offers greater control on isolating and understanding the effect of a single parameter. In case of

polypropylene polymerisation, the particles are much larger than the liquid droplets. We can

idealise the problem by considering drop impact on flat solid surfaces. Therefore, we focus our

attention on a single droplet impacting a heated, dry, flat surface. For simplicity, we study the

impact of water droplets on solid surfaces. The considered problem will provide useful clues for

understanding the real case when solid particles are much larger than the injected liquid droplets (as

in the case in olefin polymerisation reactors).

The interaction between liquid droplets and hot solid surfaces across various boiling regimes

is studied. High-speed digital imaging is used to capture the droplet vaporisation process. Water

droplets impacting on the solid wall with different Weber numbers are investigated. The effect

of various parameters such as surface and initial droplet diameter, liquid velocity and surface

tension is studied. Experimental data of dynamic drop impact on flat hot surface are reported

over a wide range of operating conditions. A phenomenological model is developed to describe

INDIAN CHEMICAL ENGINEER Vol. 52 No. 4 December 2010

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the droplet vaporisation phenomena in single phase and nucleate boiling regimes. The results from

the phenomenological model are compared with the experimental data presented here and the

experimental results reported in the literature.

Literature reviewSeveral related studies involving evaporation of small, gently deposited drops have been reported

(see, for example, Yarin [8] and references cited therein). Wachters et al. [9] performed experiments

involving impinging water droplets. In their study, approximately 2 mm droplet impacted a hot

(up to 3708C) polished gold surface. Xiong and Yuen [10] studied the impact of liquid droplets

with diameters less than 1 mm, including water and several hydrocarbon fuels. The droplets

impacted on a stainless steel plate heated up to the Leidenfrost temperature. Chandra and Avedisian

[11] studied the evolution of droplet shapes when an n-heptane droplet struck a heated surface.

They showed that the wetted area and the spreading rate of the droplet are independent of surface

temperature during the early period of impact. Qiao and Chandra [12] measured the influence of a

surfactant on the evaporation rates of liquid droplets. The presence of a surfactant greatly reduced

Fig. 1. Mechanism of liquid injection into a fluidised bed as proposed by (a) Bruhns and Werther [6] and (b)

Nayak et al. [7].


Droplet impingement on hot horizontal surfaces 283

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the droplet evaporation times and increased the surface heat transfer rate. Chandra et al. [13] studied

the effect of varying initial liquid-solid contact angle on the evaporation of single droplets of water

deposited on a stainless steel surface. The surface was maintained at temperatures up to the

saturation temperature. Their results showed that during evaporation the droplet diameter remained

relatively constant as the contact angle continually decreased. Once a critical angle was reached, the

contact angle became fixed while the diameter decreased. Bernandin et al. [14] studied the impact of

water droplets on heated surfaces with different roughness for a range of Weber numbers from 20 to

220 and surface temperatures from 1008C to 2808C. Crafton and Black [15] studied evaporation of

gently deposited n-heptane and water droplets on several metallic surfaces that were initially

maintained below the saturation temperature of the liquid. They used droplets with post-impact

diameters of approximately 1 mm for water and 5 mm for n-heptane.

Two approaches are followed in modelling the drop impact phenomena on a solid surface. In

the first approach, the droplet geometry is assumed to be of a predefined shape. Heat and mass

balance equations are then solved to predict geometry and temperature variations. Liao [16] follows

this approach to modelling single and multiple droplets. Ruiz [17] assumed spherical droplets and

solved the governing equations using the control volume method on a curvilinear grid to study the

effect of droplet geometry evolution on the Nusselt number (Nu), heat transfer coefficient and heat

transfer rates. This approach is limited to only single phase and conjugate boiling regimes;

nevertheless, these models are a valuable tool as design models. As the temperature is increased, the

droplet shape no longer follows a specific shape. The second approach involves actual prediction of

droplet geometry.

In recent years, computational fluid dynamics (CFD) models have been proposed to study the

interactions between liquid droplets and solid surfaces. Harvie and Fletcher [18, 19] presented a

volume of fluid (VOF)-based model to simulate the behaviour of an axisymmetric volatile liquid

droplet on a hot solid surface in a film boiling regime. The VOF algorithm was used to model the

gross deformation of the droplet. A separate one-dimensional (1D) algorithm was used to model

the flow within the viscous vapour layer that exists between the droplet and the solid surface.

Gunjal et al. [20] presented a CFD model, based on the VOF approach to simulate drop dynamics of

cold drops colliding on flat surfaces over a range of Reynolds numbers (550�2500) and Weber

numbers (2�20). Ge and Fan [21] developed a 3D model to study the fundamental nature of the heat

transfer phenomenon of a sub-cooled droplet upon impact with a superheated flat surface. The

numerical technique adopted in this model involves a Eulerian, fixed grid and finite-volume

algorithm coupled with the level-set methods that tracks the deformation of the droplet surface.

The heat transfer properties in each phase are solved with a micro-scale vapour flow model applied

to determine the vapour pressure force during the contact process between the droplet and

the superheated surface. The model is applied to simulate the impact phenomenon of water and

n-heptane on a heated surface with different properties. The impact of the n-heptane droplet induces

a much lower temperature drop at the solid surface compared with that of the water droplet. The

decrease in the solid surface temperature occurs mainly during the droplet-spreading stage, and

it recovers at the recoiling stage due to the presence of the fully established vapour layer.

Nikolopoulos et al. [22] numerically studied the flow development during normal impingement of

droplets onto a hot wall by using a finite volume Navier-Stokes equation flow solver incorporating

the VOF methodology. Use of an evaporation model predicting the vapour produced during impact,


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together with the numerical solution of two additional transport equations for the temperature and

vapour concentration fields, has allowed estimations of the coupled hydrodynamic and thermo-

dynamic processes. The numerical model utilises an adaptive local grid refinement technique at the

liquid-gas interface which has allowed prediction of the flow development taking place during

droplet impingement on a heated surface with temperature below or above the Leidenfrost point.

The CFD models are capable of revealing important characteristics of the droplet-particle

collision such as the droplet-particle contact area, the vapour layer thickness, and the heat flux

distribution, which may not be easily obtained from experiments. However, these CFD models are

still in infancy and need to be validated thoroughly before using them for design purpose.

Comprehensive experimental data that span the entire boiling curve are required to validate these

models. Experimental data that reveal instantaneous changes in droplet geometry, contact diameter,

contact angle and evaporation times, especially on small drops, will be useful in carrying out such

validation process. The data reported in this paper will certainly be useful for this purpose.

Experimental setup and procedureTo study the droplet vaporisation process, simultaneous measurements of droplet geometry and

surface temperature were taken. High speed imaging was used to capture the droplet geometry.

A flush mounted thermocouple was used to measure the temperature. The experimental setup is

shown schematically in Fig. 2.

Experimental setup

The setup consisted of a droplet generator, CCD camera, hot solid surface, light source, temperature

measurement device and heating devices. Details of the equipment are given below.

A syringe with volume 10 ml, least count 0.1 ml and needle inner diameter 0.3 mm was used to

generate droplets having a wetted diameter ranging from 1 to 5 mm. The syringe was located at the

centre of the surface. A mechanical arrangement was made to adjust the horizontal and vertical

Fig. 2. Experimental setup for studying droplet vaporization process.



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positions of the syringe precisely. Droplets were generated manually. A high-speed CCD camera

(Redlake MotionPre, frame rate 2000 fps) was used for image capturing. The camera was located at

15 cm from the droplet generator and a zoom lens (18-180/2.5) was used for recording the images.

The camera was focused on an area of about 15 mm�15 mm and images of resolution 55 pixels/

mm were acquired in a movie form. Image processing software Image-Pro plus (from Media

Cybernetics, USA) was used for processing of the imaging data. A bright white light was placed in

front of the camera after the droplet and a light diffuser was placed in front of the light source

so as to eliminate harsh shadows. To avoid radiant heating a screen was used and the light source

was kept at large distance from the droplet. Two solid surfaces, copper and iron, were chosen for

the study because it was possible to cover the relevant range of surface temperatures: copper

(30�1008C) and iron (30�3008C). Polished copper surface having dimension 29 mm�29 mm�5 mm and iron surface having dimension 200 mm�150 mm�25 mm were used for

experimentation. Prior to each experiment the surfaces were cleaned with isopropyl alcohol and

then dried with a soft non-scratching cotton cloth. Both the plates were insulated at the bottom to

avoid heat transfer in the downward direction. A flush mounted thermocouple (T type, least count

0.18C) was placed at the centre of the surface. The droplet was placed at the centre using a syringe.

Electrical heating was used to provide constant heat flux, and to maintain uniform temperature at

the surface.

Experimental procedure

Experiments were carried out in closed surroundings to limit ambient air movement. Ambient

temperature and humidity were measured during each experiment. For one set of conditions several

experiments were performed to ensure that the measured values were within 5%. Readings were

taken when the surrounding was at steady state. Experiments were carried out over different boiling

regimes for a wide range of surface temperature and droplet Weber numbers.

When a sessile liquid droplet interacts with a hot surface, the rate of heat transfer is governed

by fluid and solid thermal properties as well as surface roughness and temperature. The difference

between the surface temperature and drop temperature (wall superheat temperature) is generally

noted as the single most important parameter in boiling and is used to classify the four distinct heat

transfer regimes as shown in Fig. 3. These regimes are (I) film evaporation or single phase boiling,

(II) nucleate boiling, (III) transition boiling and (IV) film boiling. Figure 3 shows the boiling curve

(Fig. 3a) that relates wall heat flux to wall superheat temperature, the evaporation rate curve (Fig.

3b) and actual photographs of droplet in all four boiling regimes (Fig. 3c). The droplet behaviour

and the corresponding heat transfer are characteristically different in each of the four regimes.

Single phase boiling

In the film evaporation or single phase regime, heat from the surface is conducted through the liquid

film and is dissipated by evaporation at the liquid-gas interface. The regime is characterised by long

evaporation times. A single phase boiling regime can exist several degrees above the liquid

saturation temperature, depending on the surface roughness and liquid properties (up to 1058C for

water as shown in Fig. 3c�I).


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Nucleate boiling

The nucleate boiling regime (Fig. 3c-II) begins at a surface temperature referred to as the boiling

incipience point, which is above the liquid saturation temperature. As the surface temperature

increases above the incipience point, vapour bubble production starts due to nucleation and the

corresponding heat flux increases dramatically. The upper limit of the nucleate boiling regime

is known as the critical heat flux (CHF). It corresponds to the maximum heat flux and minimum

drop lifetime. In both regimes I and II, the droplet wets the surface and spreads into a thin liquid

film.

Fig. 3. Boiling regimes for sessile liquid droplets: (a) pool boiling curve; (b) droplet evaporation curve;

(c) experimental photographs.



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Transition boiling

In the transition regime (Fig. 3c-III), the droplet starts separating from the surface due to the vapour

layer generated due to rapid evaporation. The droplet therefore experiences only partial surface

wetting. As partial surface wetting develops, liquid cohesion forces begin to overcome surface

adhesion forces. This causes the droplet to bead up, thus reducing evaporation rates and increasing

droplet lifetime.

Film boiling

As the superheated wall temperature increases further, any liquid that comes into contact with the

solid surface evaporates almost instantaneously creating a vapour layer around the droplet. At the

upper end of the transition boiling regime, the vapour layer grows substantially to prevent any

significant contact between the droplet and surface and the droplet levitates. At this upper limit,

referred to as the Leidenfrost Point (LFP), the droplet evaporation time reaches a maximum. At

surface temperatures above the LFP, the droplet remains separated from the surface by a thin

insulating vapour layer, as shown in Fig. 3c-IV, through which heat is conducted from the surface.

In film boiling, the droplet lifetime decreases with an increase in surface temperature as radiation

heat transfer becomes significant in addition to conduction.

All four regimes were captured by a CCD camera. A frame rate of 50 fps was used for the

single phase and nucleate boiling regimes, whereas the transient and film boiling regimes were

captured at a frame rate of 200 to 2000 fps depending on the impact velocity of the droplet.

Recorded images were processed with the image analysis software, Image-Pro plus (Media

Cybernetics). For calibration, a test material of known dimension was recorded during each set of

experiments. Brightness and contrast of the images were adjusted so that a clear three-phase

interface position could be measured.

The temperature of the surface corresponding to the four regimes is given in Table 1. Droplets

with diameter of 2.6590.1 mm (for water) were used with constant wall temperature condition in

all the four regimes. Two sets of observations with different temperatures were made in the

transition regime and film boiling regime for better understanding of the phenomenon.

For each experiment, the entire evaporation process was recorded on video. Transient

variations in droplet geometry, distance of centre of gravity of the droplet from the surface and

droplet behaviour were measured by analysing every individual frame during each experiment.

Measurements verified that for single phase and nucleate boiling regimes, the free surface had a

Table 1. Surface temperature maintained to capture boiling regime

Serial no. Regime Set Surface temperature (8C)

I Single phase 1 103

II Nucleate boiling 1 110

III Transition boiling 1 140

Transition boiling 2 160

IV Film boiling 1 220

Film boiling 2 280


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spherical shape throughout the lifetime of the water droplet [13]. Instantaneous values of the contact

angle between the evaporating droplets and the heater surface, evaporation rate and spatially

averaged heat flux were then calculated from the diameter and height measurements. The

evaporation time was measured up to a surface temperature of 2008C. Beyond this temperature, the

droplet levitation is pronounced and measurement of evaporation time was not possible due to

droplet escape from the surface. For measurement of evaporation time, images were captured until

complete evaporation occurred, so that time of evaporation could be accurately determined.

The dynamics of droplet impact (contraction, expansion, etc.) does not play a significant role

in the single phase and nucleate boiling regimes. The time scale of evaporation is much larger than

the droplet impact time scales. Therefore, the effect of droplet dynamics (Weber number, We) was

studied only in the transient and film boiling regimes. Experiments were done over a range of

Weber numbers (3 to 230) by changing the velocity of the droplet while other parameters remained

constant. Impact velocity was changed by changing the height of the needle. Water droplets having

an impact diameter of 2.6590.1 mm and an impact velocity of 0.3 to 2.5 m/s were used to study the

evaporation behaviour.

Mathematical modelIt is difficult to develop a mathematical model that can represent the interactions of liquid droplets

with hot solid surfaces across the entire boiling curve. As pointed out earlier, the CFD models are

still in infancy and require further development before these can be applied successfully. The value

of phenomenological models as a first step towards more complex models has been proven by

earlier researchers [16, 17] in this situation. There are several benefits of this type of model. It can

be used to study several parameters such as initial surface temperature, liquid subcooling, varying

properties of solids and liquid materials in spray cooling (from liquid injection nozzles). A droplet

evaporation model can also be used to determine optimum droplet parameters, evaporation time,

heat transfer rate and distributions as well as effect of droplet geometry on heat transfer rates. A first

principles model was therefore developed to study the evaporation of a liquid droplet on a hot

horizontal surface. Model equations were written to describe simultaneous heat and mass transfer of

liquid droplet with geometrical variations. This section details the governing equations. The model

schematic is shown in Fig. 4. The following assumptions were made while developing the model:

1. Drop-wise evaporation in a constant environment was considered.

2. Droplet was assumed to be spherical in shape up to complete evaporation.

3. During the first stage of evaporation, base area of droplet is constant and contact angle

decreases until the receding value was reached. After this point contact angle remained

constant and base area of droplet decreases.

Evolution of Droplet Geometry

Based on experimental observations, small water droplets were assumed to evaporate with a shape

that approximates a segment of sphere or a spherical cap as shown in Fig. 4a. The volume of a

spherical cap was calculated as follows:

V ¼ p

6h 3r2 þ h2� �

(1)



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where V, h, r and d are the volume, height, wetted radius and wetted diameter of the droplet,

respectively. The rate of change of the droplet volume was determined based on the rate of change

of mass, on droplet surface:

dV

dt¼

dmdt

� �q

(2)

The rate of change of mass (dm/dt) was determined by the moisture balance of the droplet (Eq. (9)).

Based on the calculated volume, the geometrical properties of the spherical cap can be

determined. While the wetted radius remains constant, the height of the droplet was determined by

the following equation:

h ¼ 3V

pþ r6 þ 9V 2

p2

� �1=224

35

1=3

�r2 3V

pþ r6 þ 9V 2

p2

� �1=224

35�1=3

(3)

And the contact angle was calculated as:

h ¼ 2 tan�1 2h=d

(4)

Fig. 4. Model schematic: (a) droplet geometry; (b) direction of convective heat transfer in droplet; (c) energy

balance for droplet.


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When the contact angle reached the receding value, the geometry was calculated based on a

spherical cap with constant contact angle. At the receding contact angle, the droplet height and

contact diameter were determined using the radius of curvature, which was calculated as:

Rc ¼3V

pð2� 3 cos hþ cos3 hÞ

" #1=3

(5)

The height and contact diameter at constant receding contact angle were then calculated using the

following equations:

h ¼ Rcð1� cos hÞ (6)

d ¼ 2Rc sin h (7)

Balance Equations

Balance equations for heat and mass were written to calculate the droplet temperature and the mass

of the droplet.

Energy Balance for Droplet

Energy balance of droplet includes sensible heat exchange by convection with ambient air, surface

and latent heat exchange due to evaporation or condensation. Convective heat transfer from solid-

liquid and vapor-liquid interfaces and latent heat of vaporisation were considered. Various energy

fluxes are shown in Fig. 4.

d

dtqdVdCpdTd

¼ hdc AdcðTs � TdÞ þ hds Ads Ta � Tdð Þ þ KwAds xamb � xdð Þk (8)

where Ts, Ta, Td, Vd, rd, Cpd and l are surface temperature, air temperature, droplet temperature,

droplet volume, droplet density, droplet specific heat and droplet latent heat, respectively.

Moisture Balance

With changing temperature and time the mass of droplet will change. Moisture balance was written

to calculate the amount of liquid on surface. This information was in turn used to calculate the

geometry of the droplet.

d

dtmwð Þ ¼ Kw Ads xamb � xdð Þ (9)

The droplet surface area (Ads) and droplet contact area (Adc) vary with the amount of evaporation.

These areas were calculated using Eqs (10) and (11), respectively.

Adc ¼p

4d2 (10)

Ads ¼p

83d2 þ 4h2� �

(11)



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vamb and vd are specific humidity of ambient and at droplet surface, these were calculated using the

correlations given by Crafton and Black [15] as:

x ¼ 0:622pv

p� pv

(12)

pv ¼ psat: / (13)

psat: ¼ 2337 exp 67891

293:15� 1

Td

!� 5:03 ln

Td

293:15

� � !(14)

where pv is the vapour pressure, psat is the saturation pressure, p is the ambient pressure and f is the

equilibrium mole fraction.

The mass transfer coefficient (Kw) was calculated from

Kw ¼hds

Cpa Leð Þ2=3

(15)

hds, hdc, Cpa and Le are the surface to droplet (water) heat transfer coefficient, and droplet to air (air)

convective heat transfer coefficient, specific heat capacity of air and effective length, respectively.

The natural convection heat transfer coefficient was calculated based on the Nusselt number

(Nu):

Nu ¼ hdsd

kair

(16)

where d is the characteristic length of the geometry as shown in Fig. 4c. The natural convection heat

transfer rate between the droplet surface and the surrounding air is very small because the surface

area is small and the temperature difference between the droplet surface and the surrounding air is

also small. Given this condition, Nu can be approximated by its conduction limit value of 2.0 [17].

The natural convection heat transfer coefficient for surface to water (hdc) was the adjustable

parameter in the model. The coefficient was set to 1200 W/m2 K for 608C and to 1700 W/m2 K for

808C. The liquid properties are listed in Table 2. The liquid-solid contact angle of 75928 was used.

The governing differential equations were solved using backward difference formula method.

Freely available numerical library sundials (http://www.llnl.gov/CASC/sundials/) were used for this

purpose.

Results and discussion

Droplet Behaviour in Different Boiling Regimes

Fig. 5 displays a series of representative images for vaporisation of a water droplet

(diameter �2.6690.1 mm) having initial velocity of 0.16 m/s (We �0.94). At time t �0 the

droplet is spherical in shape. The images were taken at three temperatures which correspond to

nucleate regime (1168C), transient regime (1618C) and film boiling regime (2208C). In the single

phase regime (not shown) and the nucleate boiling regime at lower Weber numbers, the droplet

forms a spherical cap when it impacts the surface. It then spreads and recoils to attain a spherical


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http://www.llnl.gov/CASC/sundials/

shape. The time required to attain equilibrium is a few milliseconds (compared to vaporisation times

of a few seconds). Thus the impact dynamics play a less significant role during vaporisation process.

At higher Weber numbers, however, the vaporisation is faster as the droplet spreads making a large

area available for heat transfer. As the temperature of the surface increases, the time scales of droplet

dynamics and vaporisation become comparable and vaporisation starts during the spreading and

recoiling process itself (transient boiling).

As the temperature is increased further into the film boiling regimes, surface wetting decreases.

The droplet starts to rebound. A vapour film is created around the droplet due to high evaporation

rates. This causes the levitation of the droplet and the Leidenfrost effect can be observed. In the

Leidenfrost or film boiling regime, the droplet evaporation time increases as the time of contact

between the drop and the surface is much smaller compared to the other heat transfer regimes.

The droplet vaporisation process at different Weber numbers is shown in Fig. 6. For the same

droplet size (2.7 mm) and surface temperature (2508C) the droplet vaporisation process is captured.

The droplet impact velocity is used to achieve Weber numbers from 5 to 214. At lower Weber

numbers, the droplet vaporisation takes place as the drop spreads and recoils. At higher Weber

numbers, splashing of droplet occurs, producing smaller droplets which evaporate individually.

The captured image sequences of vaporisation at different Weber numbers and in different

boiling regimes were analysed to get dynamic information on droplet geometry. A typical analysis

sequence is shown in Fig. 7. The droplet vaporisation process for a 2.65 mm water droplet placed

on a horizontal surface maintained at 2358C was recorded. For each sequence the droplet dynamics

during the collision were described quantitatively by two parameters: the height of the centre of

gravity (CG) and the projected area of the droplet. The projected area was measured from a 2D

image captured by the CCD camera. The centre of gravity and the height of CG were measured

between base of the surface and mass centre of the droplet, by image processing.

The dynamic variation in the distance of CG of the drop from the surface is shown in Fig. 7.

Representative images and the variation in the projected surface area are shown at top of the figure.

The projected area of the droplet was normalised with respect to initial droplet area and plotted

against time. The droplet levitation starts within 25 ms of the impact. The droplet rebounds four

Table 2. Properties used for simulation

Temperature

(8C)

Density

(kg/m3)

Surface

tension (N/m)

Thermal

conductivity

(W/m K)

Specific heat

(J/kg K)

Latent heat

(J/kg)

Prandtl

number

27 997.10 0.0717 0.613 4179 2 437 500 5.83

32 995.02 0.0709 0.620 4178 2 425 600 5.20

37 993.00 0.0700 0.628 4178 2 413 700 4.62

42 991.10 0.0692 0.634 4179 2 401 800 4.16

52 987.17 0.0675 0.645 4182 2 378 000 3.42

62 982.30 0.0658 0.655 4186 2 354 200 2.88

72 976.56 0.0641 0.665 4191 2 329 300 2.45

82 970.87 0.0623 0.671 4199 2 304 300 2.14

92 963.39 0.0605 0.677 4209 2 278 300 1.91

100 957.85 0.0589 0.680 4217 2 256 900 1.76



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times before breaking into two smaller droplets. The graph shows the variation of the distance of

CG for the larger of the two droplets generated after the break-up process. The smaller droplet

rebounds to a much higher level compared to the bigger drop. If the break-up process generates very

small droplets, these droplets behave as ‘‘satellite drops’’. The projected area continuously

decreases as the drop sheds more and more vapour. As expected, there is a sudden decrease in the

projected area after droplet break-up occurs. With such high frequency images it is possible to get

detailed transient variation in important geometrical parameters of the droplet. Such data are highly

valuable for validating results from CFD models.

The effect of Weber number on evaporation time is shown in Fig. 8. The data shown are for the

impact of water droplet with diameter 2.65 mm at different surface temperatures (140�2008C). At

lower surface temperatures (higher evaporation times), the Weber number does not have any

Fig. 5. Boiling regimes for a sessile water drop.


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significant effect on the evaporation time. As noted earlier, the time scale of droplet dynamics is of

the order of 0.25 ms, thus time required for the droplet to reach equilibrium shape is much smaller

than the evaporation time. Therefore, the evaporation time remains almost constant. For higher

surface temperatures, however, the evaporation time decreases with the Weber number up to a

Fig. 6. Sequence of behaviour of water droplet in film boiling regime with different Weber numbers

(Ts�258C).



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certain limit and then remains constant. As droplet Weber number increases, spreading of droplet

increases, and hence the heat transfer area increases and the evaporation time decreases. At higher

Weber numbers more than one competing phenomena, like spreading, recoiling, bouncing and

corona formation, are pronounced. These complex interactions govern the heat transfer

characteristics. CFD models will be useful in understanding such interactions and establishing

proper relationships between various design parameters.

Analysis of Single Phase Boiling Regime

During the single phase boiling regime, the droplet remains spherical throughout the evaporation

process. Figure 9 depicts the evaporation of water droplet in single phase boiling regime. The solid

surface was maintained at 808C. Figure 9a shows the images of water droplet at successive time

intervals. It was observed that during the initial period of droplet vaporisation, the wetted perimeter

remained constant while contact angle decreased. As the contact angle reaches a critical value, the

contact angle remained the same and the wetted perimeter decreased rapidly, thus validating our

assumption (3) for the first principles model. It was observed that 3D effects became apparent

towards the end of the evaporation process. Figure 9b shows a comparison of evaporation times

between experimental measurements and model predictions at different droplet volumes. The

experimental observations varied95%, due to variation in droplet contact angle and environmental

Fig. 7. Dynamic behaviour of water droplet in film boiling regime (Ts�2358C).


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conditions. Good agreement is seen between measured and predicted values for larger droplet sizes.

For smaller droplet sizes, the model predicted longer evaporation times because, at these small

droplet sizes, 3D effects were much stronger compared to the larger ones.

To study single phase boiling regime, experiments were designed in such a fashion that they

complement available experimental data in the literature. The heat transfer characteristics of water

droplet are shown in Fig. 10. Over the limited range of droplet sizes that were placed on the heated

surface, the evaporation rate increased nearly linearly with an increase in the wetted diameter of the

droplet. The calculated evaporation rates were used to estimate the average heat flux taken by

the droplet from the surface, by assuming that heat is removed solely due to the phase change of the

droplet.

heat flux ¼ _mL

A0

(17)

The area used to calculate the heat flux was equal to the original surface area covered by the droplet

(that is, a circle with diameter do) after spreading. The average heat flux decreases steadily with the

droplet diameter. At larger droplet diameters, lower heat flux was observed. This is because, for

larger droplets, the sensible energy required to heat the droplet from the ambient temperature to the

temperature of the heated surface is comparable to the heat required to vaporise the droplet. These

observations are consistent with the earlier study of Crafton and Black [15]. The comparison of

measured evaporation rates and model predictions for all the available data on single phase boiling

is shown in Fig. 11. The model accurately predicts the evaporation rates for a wide range of

evaporation rates.

Droplet Geometry Evolution

Typical trends in the variation of droplet wetted diameter, height, contact angle and volume are

shown in Fig. 12. Experimental results are compared with model prediction for three different

Fig. 8. Effect of Weber number on evaporation time in transient and film boiling regimes.



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droplet volumes (3, 7 and 10 ml). From the experimental data, contact angle and volume of the

droplet were calculated using measured values of droplet wetted diameter and height. The volume

calculations were verified by measuring dimensions of droplet of known volume and back

calculating the volume, whereas initial contact angle and droplet wetted diameter were input to the

model for calculating droplet height and volume.

The normalised height of water droplet as a function of time is shown in Fig. 12a. Initial droplet

height for 3, 7 and 10 ml was 0.98, 1.3 and 1.46 mm, respectively. The height is normalised with

respect to the maximum recorded droplet height during the evaporation. The height of the water

droplets decreases nearly uniformly during the axisymmetric evaporation process. Normalised

wetted diameter of water droplet evaporating on a hot horizontal surface is shown in Fig. 12b. The

diameter is normalised with respect to the maximum recorded wetted diameter during the

evaporation. The wetted diameter remains almost constant for most of the evaporation process. Near

the end of evaporation, the droplet diameter exponentially decreases. The drop in diameter is more

pronounced for smaller droplets. This may be due to increased capillary forces for the smaller

droplet that would cause it to experience a greater tendency to shrink during the evaporation

process. Due to the assumption in developing the model equations, the model predicts constant

droplet diameter till a critical contact angle is reached and then the diameter decreases linearly.

Figure 12c shows comparison between measured and predicted droplet volume. Good

agreement was seen between measured and predicted values. The volume was normalised with

Fig. 9. Evaporation of water droplet in single phase boiling regime. (a) Sequence of water droplet evaporates

at 808C (total evaporation time �56.12 s). (b) Comparison of evaporation time as a function of droplet

volume.


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respect to the maximum calculated volume during the evaporation, which was nearly equal to initial

droplet volume. The volume of droplet decreases during the evaporation process. It was identified

that for the same initial volume, if the droplet had a larger spread (large wetted diameter or lower

contact angle) it evaporated quickly compare to small spread of droplet. Figure 12d shows the

measured and predicted variation of contact angle as a function of time. The initial contact angle

was 75948. As the droplet evaporates, the contact angle decreases until it reaches the value of the

receding contact angle. The value of receding contact angle was measured to be 15928. The

contact angle remains constant after the receding contact angle value is reached while the contact

diameter decreases. The contact angle decreases faster for smaller droplets indicating an increased

evaporation rate.

Fig. 10. Heat transfer characteristics of water droplet at 808C. (a) Evaporation rate as a function of water

droplet wetted diameter. (b) Heat flux as a function of water droplet wetted diameter.



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Fig. 11. Comparison of measured and predicted evaporation rates.

Fig. 12. Transient variation in droplet geometry parameters during droplet vaporisation at 808C. (a)

Normalised height. (b) Normalised wetted diameter. (c) Normalised volume. (d) Contact angle.


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Using the mathematical model, the effect of key design variables can be easily studied. The

effect of liquid-solid contact angle on the evaporation time and wetted diameter is shown in Fig. 13.

The liquid-solid contact angle depends mainly on surface tension of the liquid. The contact angle

was varied between 208 and 908 for a constant volume (4.5 ml) droplet. It should be noted that

addition of a small concentration of surfactant can change the surface tension without greatly

affecting the thermo-physical properties of the liquid. For example, by adding 0.08 g and 0.8 g of

sodium dodecyl sulfate in 800 g of water, the contact angle on a stainless steel surface decreases

from 908 to 558 and 208, respectively [13]. With decrease in contact angle, spread of droplet on the

surface increases. This increases the heat transfer area and consequently reduces the evaporation

time. Both of these effects enhance the rate of cooling at the cost of heat flux, as the heat flux

decreases due to increase in droplet wetted area.

ConclusionThe heat transfer and interaction between small liquid droplets and hot horizontal surfaces were

studied over different boiling regimes. Detailed observations of the interaction were made using

high speed imaging. A first principles model was developed to describe the evaporation process in

the single phase boiling regime. The model was validated using experimental and literature data.

Some key findings of this work are listed below.

1. For single phase and nucleate boiling regimes, the droplet impingement dynamics has little

role in the evaporation process. In the higher boiling regimes, the droplet impingement

dynamics time scales are similar to evaporation time scales and the evaporation process is

affected by the droplet impingement dynamics.

2. During the evaporation in single phase and nucleate boiling regimes, the contact angle

decreases continuously until it reaches the receding value (15928 in case of water) and then it

Fig. 13. Effect of contact angle on wetted diameter and evaporation time.



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remains constant. This means that the wetted diameter remains almost constant throughout the

vaporisation process, only near the end of droplet lifetime it shows an appreciable decrease.

3. One can control the vaporisation time by controlling the contact angle. For example, in case of

water over iron surface, decreasing the contact angle from 908 to 208 reduced the evaporation

time by 58% and increased the evaporation rate by 139%. In reality, this can be achieved by

adding small amounts of surfactants into the liquid.

4. Weber number significantly affects the evaporation time of water droplet after 1208C; a

decrease in droplet spreading time and a more violent break-up were observed with increasing

Weber number.

5. In film boiling regime, a droplet floats on its own vapour. At the same time, spreading,

recoiling and rebounding occur. Here, fluid dynamics play a very important role in the

transport processes during the evaporation process. Detailed CFD models that account all

these complexities will be helpful in understanding these.

The generated data in all the boiling regimes would be useful for comparison and validation of CFD

models. Despite of its limitations, the developed model can serve as a useful tool in designing and

optimizing nozzles used in the condensed mode operations.

References

1. Sinclair, K., 1995, SPE, Polyolefins IXth International Conference, Texas, USA.

2. Fan, L.S., Lau, R., Zhu, C., Vuong, K., Warsito, W., Wang, X. and Liu, G., ‘‘Evaporative Liquid Jets in

Gas Liquid Solid Flow System’’, Chem. Eng. Sci., 56, pp. 5871�5891 (2001).

3. Leclere, K., Briens, C., Gauthier, T., Bayle, J., Guigon, P. and Bergougnou, M., ‘‘Experimental

Measurement of Droplet Vaporization Kinetics in a Fluidized Bed’’, Chem. Eng. Process., 43, pp. 693�699 (2004).

4. Werther, J. and Bruhns, S., ‘‘3-D Modeling of Liquid Injection into Fluidized Beds’’, Int. J. Chem.Reactor Eng., 2, A31. Available at: http://www.bepress.com/ijcre/vol2/A31 (2004).

5. Lefebvre, A., Atomization and Sprays, CRC Press, Boca Raton USA (1989).

6. Bruhns, S. and Werther, J., ‘‘An Investigation of the Mechanism of Liquid Injection into Fluidized Beds’’,

AIChE J., 51, pp. 766�775 (2005).

7. Nayak, S.V., Joshi, S.L. and Ranade, V.V., ‘‘Modeling of Vaporization and Cracking of Liquid Oil

Injected in a Gas-Solid Riser’’, Chem. Eng. Sci., 60, pp. 6049�6066 (2005).

8. Yarin, A.L., ‘‘Drop Impact Dynamics: Splashing, Spreading, Receding, Bouncing’’, Annu. Rev. FluidMech., 38, pp. 159�192 (2006).

9. Wachters, L.H.J., Bonne, H. and Van Nouhuis, H.J., ‘‘The Heat Transfer from a Hot Horizontal Plate to

Sessile Water Drops in the Spheroidal State’’, Chem. Eng. Sci., 21, pp. 1047�1056 (1966).

10. Xiong, T.Y. and Yuen, M.C., ‘‘Evaporation of a Liquid Droplet on a Hot Plate’’, Int. J. Heat MassTransfer, 34, pp. 1881�1894 (1991).

11. Chandra, S. and Avedisian, C.T., ‘‘Observation of Droplet Impingement on a Ceramic Porous Surface’’,

Int. J. Heat Mass Transfer, 35, pp. 2377�2388 (1992).

12. Qiao, Y.M. and Chandra, S., ‘‘Evaporative Cooling Enhancement by Addition of a Surfactant to Water

Drops on a Hot Surface’’, ASME National Heat Transfer Conference, 2, Portland, Oregon, 6�8 August

(1995), pp. 63�71 (1995).

13. Chandra, S., di Marzo, M., Qiao, Y.M. and Tartarini, P., ‘‘Effect of Liquid Solid Contact Angle on

Droplet Evaporation’’, Fire Safety J., 27, pp. 141�158 (1996).


302 UTIKAR ET AL

Dow

nloa

ded

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er 2

014

http://www.bepress.com/ijcre/vol2/A31

14. Bernandin, J.D., Stebbins, C.J. and Mudawar, I., ‘‘Effect of Surface Roughness on Water Droplet Impact

History and Heat Transfer Regimes’’, Int. J. Heat Mass Transfer, 40, pp. 73�88 (1997).

15. Crafton, E.F. and Black, W.Z., ‘‘Heat Transfer and Evaporation Rates of Small Liquid Droplets on Heated

Horizontal Surface’’, Int. J. Heat Mass Transfer, 27, pp. 1187�1200 (2004).

16. Liao, Y., 1992, ‘‘Dropwise Evaporative Cooling of Solid Surfaces’’, PhD thesis, University of Maryland

(1992).

17. Ruiz, O.E., (2000), ‘‘Numerical Analysis of the Dropwise Evaporation Process’’, PhD dissertation,

Georgia Institute of Technology, Atlanta, GA (2000).

18. Harvie, D.J.E. and Fletcher, D.F., ‘‘A Hydrodynamic and Thermodynamic Simulation of Droplet Impacts

on Hot Surfaces, Part I: Theoretical Model’’, Int. J. Heat Mass Transfer, 44, pp. 2633�2642 (2001).

19. Harvie, D.J.E. and Fletcher, D.F., ‘‘A Hydrodynamic and Thermodynamic Simulation of Droplet Impacts

on Hot Surfaces, Part II: Validation and Applications’’, Int. J. Heat Mass Transfer, 44, pp. 2643�2659

(2000).

20. Gunjal, P.R., Ranade, V.V. and Chaudhari, R.V., ‘‘Experimental and Computational Study of Liquid

Drop over Flat and Spherical Surfaces’’, Catalysis Today, 79�80, pp. 267�273 (2004).

21. Ge, Y and Fan, L.S., ‘‘3-D Modeling of the Dynamics and Heat Transfer Characteristics of Subcooled

Droplet Impact on a Surface with Film Boiling’’, Int. J. Heat Mass Transfer, 49, pp. 4231�4249 (2006).

22. Nikolopoulos, N., Theodorakakos, A. and Bergeles, G., ‘‘A Numerical Investigation of the Evaporation

Process of a Liquid Droplet Impinging onto a Hot Substrate’’, Int. J. Heat Mass Transfer, 50, pp. 303�319 (2007).



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heat transfer and dynamics of impinging droplets on hot horizontal surfaces

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