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i
Sessile Drop Evaporation in Relation to Leidenfrost Phenomenon
A thesis
Submitted to
Department of Mechanical Engineering
Bangladesh University of Engineering and Technology
by
Md. Zahabul Islam (Student No: 0510030)
Shuvra Banik (Student No: 0510094)
Under the supervision of
Dr. Aloke Kumar Mozumder
In partial fulfillment of the
Requirement for the Degree
of
Bachelor of Science in Mechanical Engineering.
February, 2011
ii
DECLARATION
This to certify that the presented paper is the outcome of the accomplishment of the project and
thesis on “Sessile Drop Evaporation in Relation to Leidenfrost Phenomenon” carried out by
the students of Mechanical Engineering Department, BUET, Dhaka under the supervision of Dr.
Aloke Kumar Mozumder, Associate Professor, Mechanical Engineering Department, BUET,
Dhaka and it has not been submitted anywhere for any award of degree or diploma, nor it has
been published in any technical journal.
Name Student Number Signature
1. Md. Zahabul Islam 0510030
2. Shuvra Banik 0510094
SUPERVISOR
-----------------------------------------------
Dr. Aloke Kumar Mozumder
Associate Professor
Dept. of Mechanical Engineering
BUET, Dhaka-1000.
iii
ACKNOWLEDGEMENTS
The authors are grateful to Dr. Aloke Kumar Mozumder, Associate Professor, Department of
Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka for his
guidance, close supervision, inspiration and constructive suggestions to attain the desired
experimental goal.
The authors also acknowledge their gratitude to the Department of Mechanical Engineering,
Bangladesh University of Engineering and Technology (BUET) for providing necessary
financial aid and other facilities to conduct the research. The authors also express their
thankfulness to the personnel of different shops and laboratories for their help during the
fabrication of the experimental setup.
iv
CONTENTS
Declaration ii
Acknowledgements iii
List of Tables viii
List of Figures ix
Abstract xii
Nomenclature xiv
1 Introduction 1-3
2 Literature Review 4-11
2.1 Experiment Conducted by Leidenfrost 4
2.2 Stability of Leidenfrost phenomenon 5
2.3 Momentum, Heat and Mass transfer processes 6
2.4 Application 6
2.5 Boiling and Leidenfrost Effect 7
3 Development of Model 12-23
4 Experimental setup and procedure 24-30
4.1 Introduction 24
4.2 Schematic diagram of experimental setup 24
4.2.1 Metal Blocks 25
4.2.2 Stand 26
4.2.3 Heater 27
4.2.4 Thermocouple 27
4.2.5 Variac 28
4.2.6 Syringe 28
4.3 Working Fluids 29
v
4.3.1 Water 29
4.3.2 NaCl solution 29
4.3.3 Methanol 29
4.3.4 Ethanol 29
4.4 Experimental Procedure 29
5 Results and Discussions 31-82
5.1 Analysis of Theoretical model 31
5.2 Analysis of Experimental data 38
5.2.1 Liquid Variation 38
5.2.1.1 Effect of latent heat of vaporization 38
5.2.1.2 Effect of specific heat, thermal
conductivity and density of liquid
39
5.2.1.3 Effect of boiling temperature of liquid 39
5.2.2 Diameter Variation 39
5.2.3 Material Variation 46
5.3 Experimental Boiling Curve 52
5.3.1 Experimental Boiling curve of water 53
5.3.2 Experimental Boiling curve of methanol 55
5.3.3 Experimental Boiling curve of ethanol 57
5.4 Inverse boiling curve 59
5.5 Engineering Correlation of experimental data 61
5.5.1 Experimental Correlation for Aluminum 62
5.5.2 Experimental Correlation for Brass 63
5.5.3 Experimental Correlation for Copper 64
5.6 Comparison of Theoretical and Experimental result 66
5.6.1 Comparison curve for distilled water on different
metal surfaces
66
5.6.2 Comparison curve for methanol on different metal
surfaces
71
5.6.3 Comparison curve for ethanol on different metal
surfaces
75
vi
6 Conclusions 80-82
6.1 Conclusions 80
6.2 Further Work 81
6.3 Recommendations 81
REFERENCE 83-84
APPENDICES 85-107
A Saturation Properties of Liquid 86
A.1 Saturation Properties of Methanol 86
A.2 Saturation Properties of Ethanol 87
A.3 Saturation Properties of Water 88
B Experimental Data 89
B.1 Evaporation time of Distilled water on Aluminum
Surface
89
B.2 Evaporation time of NaCl Solution on Aluminum
Surface
90
B.3 Evaporation time of Ethanol on Aluminum Surface 90
B.4 Evaporation time of Methanol on Aluminum
Surface
91
B.5 Evaporation time of Distilled water on Brass
Surface
91
B.6 Evaporation time of NaCl Solution on Brass
Surface
92
B.7 Evaporation time of Ethanol on Brass Surface 92
B.8 Evaporation time of Methanol on Brass Surface 93
B.9 Evaporation time of Distilled water on Copper
Surface
93
B.10 Evaporation time of NaCl Solution on Copper
Surface
94
B.11 Evaporation time of Ethanol on Copper Surface 94
B.12 Evaporation time of Methanol on Copper Surface 95
B.13 Evaporation time of Distilled water on Mild Steel
Surface
95
B.14 Evaporation time of NaCl Solution on Mild Steel
Surface
96
B.15 Evaporation time of Ethanol on Mild Steel Surface 96
vii
B.16 Evaporation time of Methanol on Mild Steel
Surface
97
C Summary of Theoretical and Experimental Result
98
C.1 Comparison for small diameter liquid droplet 98
C.2 Comparison for large diameter liquid droplet 99
C.3 Comparison of Leidenfrost point temperature for
for different liquid droplets (small and large
diameter)
different liquid droplets
100
D Program Code 101
D.1 Estimation of theoretical time 101
D.2 Engineering Correlation of experimental data 103
viii
LIST OF TABLES
B1 Evaporation time of Distilled water on Aluminum Surface 89
B2 Evaporation time of NaCl Solution on Aluminum Surface 90
B3 Evaporation time of Ethanol on Aluminum Surface 90
B4 Evaporation time of Methanol on Aluminum Surface 91
B5 Evaporation time of Distilled water on Brass Surface 91
B6 Evaporation time of NaCl Solution on Brass Surface 92
B7 Evaporation time of Ethanol on Brass Surface 92
B8 Evaporation time of Methanol on Brass Surface 93
B9 Evaporation time of Distilled water on Copper Surface 93
B10 Evaporation time of NaCl Solution on Copper Surface 94
B11 Evaporation time of Ethanol on Copper Surface 94
B12 Evaporation time of Methanol on Copper Surface 95
B13 Evaporation time of Distilled water on Mild Steel Surface 95
B14 Evaporation time of NaCl Solution on Mild Steel Surface 96
B15 Evaporation time of Ethanol on Mild Steel Surface 96
B16 Evaporation time of Methanol on Mild Steel Surface 97
ix
LIST OF FIGURES
2.1 A Leidenfrost drop in cross section 4
2.2 (a) A bubble forms in the crevice of a scratch along the bottom of a
pan of water. (b–f ) The bubble grows, pinches off, and then ascends
through the water
8
2.3 Boiling curve for water 9
2.4 Drop lifetimes on a hot plate 11
3.1 Geometry of sessile droplet for the model 12
3.2 Determination of radiation shape factor at side surface of liquid
droplet
19
4.1 Schematic diagram of the experimental setup 24
4.2 Metal Block 26
4.3 Stand 26
4.4 Heater 27
4.5 K type thermocouple meter. 27
4.6 Variac 28
4.7 Syringes used in the experiment 28
5.1 Experimental evaporation time and model predicted Leidenfrost time
for small diameter Methanol on Copper surface
32
5.2 Experimental evaporation time and model predicted Leidenfrost time
for small diameter Methanol on Aluminum surface
32
5.3 Experimental evaporation time and model predicted Leidenfrost time
for large diameter Methanol on Aluminum surface
33
5.4 Experimental evaporation time and model predicted Leidenfrost time
for small diameter Ethanol on Brass surface
35
5.5 Experimental evaporation time and model predicted Leidenfrost time
for large diameter Ethanol on Brass surface
36
5.6 Experimental evaporation time and model predicted large diameter 36
x
Leidenfrost time for Ethanol on Mild steel surface
5.7 Comparison curve of Droplet Evaporation Time of Distill Water,
NaCl solution, Methanol, Ethanol on Aluminum surface
40
5.8 Comparison curve of Droplet Evaporation Time of Distill Water,
NaCl solution, Methanol, Ethanol on Brass surface
42
5.9 Comparison curve of Droplet Evaporation Time of Distill Water,
NaCl solution, Methanol, Ethanol on Copper surface
44
5.10 Comparison curve of Droplet Evaporation Time of Distill Water,
NaCl solution, Methanol, Ethanol on Mild steel surface
45
5.11 Comparison of Droplet Evaporation Time of Distilled water on four
different surfaces
47
5.12 Comparison of Droplet Evaporation Time of NaCl solution on four
different metal surfaces
49
5.13 Comparison of Droplet Evaporation Time of Methanol on four
different surfaces
50
5.14 Comparison of Droplet Evaporation Time of Ethanol on four different
surfaces
51
5.15 Boiling curve of Water on different material surfaces 54
5.16 Boiling curve of Methanol on different material surfaces 56
5.17 Boiling curve of Ethanol on different material surfaces 58
5.18 Schematic of a typical Inverse boiling curve and boiling curve of a
Liquid
59
5.19 Empirical correlation of total evaporation time for Aluminum surface 62
5.20 Empirical correlation of total evaporation time for Brass surface 63
5.21 Empirical correlation of total evaporation time for Copper surface 64
5.22 Empirical correlation of total evaporation time for Mild steel surface 65
5.23 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of water on Aluminum surface
67
5.24 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of water on Brass surface
68
5.25 Comparison graph for total evaporation time (τ) of Water vs surface 69
xi
temperature (𝑇𝑠) on Copper surface
5.26 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of water on Mild steel surface
70
5.27 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of methanol on Aluminum surface
71
5.28 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of methanol on Brass surface
72
5.29 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of Methanol on Copper surface
73
5.30 Comparison graph of total evaporation time (τ) vs. surface
temperature (𝑇𝑠) of methanol on Mild steel surface
74
5.31 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of ethanol on Aluminum surface
76
5.32 Comparison graph of total evaporation time (τ) vs surface
temperature (𝑇𝑠) of ethanol on Brass surface
77
5.33 Comparison graph of total evaporation time (τ) vs surface temperature
(𝑇𝑠) of ethanol on Copper surface
78
5.34 Comparison graph of total evaporation time (τ) vs surface
temperature (𝑇𝑠) of ethanol on Mild steel surface
79
xii
ABSTRACT
A model for the prediction of evaporation time in film boiling region of a sessile drop of liquid
on a hot metallic surface has been developed in the present study. The droplet is assumed to have
a stable vapor film formed beneath the droplet and is supported by the excess vapor pressure of
the film. Heat is assumed to be transferred to the liquid droplet by conduction and radiation
through vapor film. The model develops an iterative process for the refinement of calculated
mass transfer rate from the bottom surface of the droplet. Sessile drop of four different liquids
such as distilled water, saturated NaCl solution, methanol and ethanol having two different
diameters (2.5 and 2.75mm) were used to conduct an experiment for a wide range of solid
surface temperatures (60-400 oC) which verifies the proposed model at Leidenfrost point. Four
solid surfaces copper, aluminum, brass and mild steel were used to conduct the experiment. The
Leidenfrost time (complete evaporation time corresponding to Leidenfrost temperature)
predicted from the proposed model has been compared with the experimental value; the
predicted time is 50-80 % of the experimental one in most of the cases. In the present study heat
flux has been determined. Using these heat fluxes, boiling curves have been generated for each
liquid on different material surfaces. By using experimental data correlation constants have been
developed for four different metal surfaces (aluminum, copper, mild steel and brass) considering
four liquids on each metal surfaces. A comparison has been made above Leidenfrost temperature
between experimental data and correlated data using these correlation constants; correlated time
is 80-95% of the experimental one in most of the cases.
xiii
NOMENCLATURE:
A Projected base area of droplet (m2)
Al Bottom surface area of liquid (m2)
Ap Test surface area (m2)
As Side surface area of liquid (m2)
C1
Correlation constant from experimental data [ - ]
C2 Correlation constant from experimental data [ - ]
cp Specific heat of vapor (J. Kg−1. K−1)
C Concentration of liquid (Kg/m3)
D Diffusivity of liquid (m2/s)
Fpl Non dimensional radiation shape factor
between test surface and liquid
[ - ]
g Gravitational acceleration (m/s2)
hg Latent enthalpy of vaporization of liquid droplet (J/Kg)
h* Reduced latent heat of vaporization (J/Kg)
kv Thermal conductivity of vapor (W. m−1. K−1)
m b Evaporation rate of mass from bottom surface
of the liquid
(Kg/s)
m s Evaporation rate of mass from side surface of
the liquid
(Kg/s)
M Molecular mass of liquid (Kg/mol)
p Pressure at the bottom of the droplet (Pa)
xiv
po Atmospheric pressure (Pa)
ps Saturation pressure of liquid (Pa)
q Time averaged heat flux (W/m2)
Q Cb (p→l) Heat flux from test surface to bottom surface of
liquid by conduction
(W/m2)
Q Rb (p→l) Heat flux from test surface to bottom surface of
liquid by radiation
(W/m2)
Q Rs (p→l) Heat flux from test surface to side surface of
liquid by radiation
(W/m2)
Q b Total heat flux from test surface to bottom
surface of liquid
(W/m2)
Q s Total heat flux from test surface to side surface
of liquid
(W/m2)
ro Initial radius of liquid droplet (m)
ro′ Radius of spherical liquid droplet considering
spherical coordinate
(m)
rp Radius of test surface (m)
r Radius of liquid droplet at any time (m)
R Universal gas constant (J. mol−1. K−1)
T Variable temperature (⁰C)
TL Leidenfrost point Temperature (⁰C)
Tp Test surface temperature (⁰C)
Ts Saturation temperature of liquid (⁰C)
xv
∆T = ( Tp − Ts) (⁰C)
u Average radial velocity of vapor (m/s)
U Radial velocity of vapor (m/s)
v Vertical velocity of vapor (m/s)
V Volume at any time (m3)
Vo Initial volume of the liquid droplet (m3)
Z Vertical coordinate (m)
Greek symbols
δ Vapor film thickness at the bottom (𝑚)
𝜏 Total evaporation time of liquid droplet (𝑠)
𝜌𝑙 Density of liquid (𝐾𝑔/𝑚3)
𝜌𝑣 Density of vapor (𝐾𝑔/𝑚3)
𝜇𝑣 Viscosity of vapor (𝑁. 𝑠/𝑚−2)
𝜀𝑙 Non dimensional emissivity of liquid
𝜀𝑝 Non dimensional emissivity of test surface
𝜎 Stefan-Boltzmann constant (𝑊.𝑚−2.𝐾−4)
𝜆′ =ℎ𝑔+0.5𝐶𝑝*∆T (J/Kg)
xvi
Subscripts
B Bottom surface of liquid droplet
C Conduction heat transfer
L Liquid phase
P Plate or test surface
R Radiation heat transfer
S Saturation phase of liquid droplet
V Vapor phase
1
CHAPTER
INTRODUCTION 1
When a sessile drop of liquid comes into contact with a very hot metal surface, nucleate boiling
is not usually found because a thin vapor film at the contact surface of the droplet and hot metal
surface is formed instantly. The higher the metal surface temperature, the thicker is the vapor
layer and at a certain temperature a stable vapor film is formed. This stable film impedes the
conductive heat transfer through it as its thermal conductivity is much lower compared to that of
the liquid. So the evaporation time reaches to a maximum. This phenomenon was first
investigated by Johann Gottlob Leidenfrost [1] in 1756 and is named the Leidenfrost
phenomenon in honor of him. The temperature at that point is known as Leidenfrost temperature
and the corresponding evaporation time is known as Leidenfrost time. After Leidenfrost
temperature, the heat transfer rate to the liquid droplet again increases with the increase in
temperature because radiation heat transfer starts to dominate which in turn decrease the
evaporation time considerably. This point is practically significant because it states that a higher
metal surface temperature doesn‟t guarantee the higher amount of heat transfer to the liquid
which can be utilized in the design of boiler, cooling tower and many other heat addition and
removal equipments.
Since Leidenfrost published the phenomenon, a numerous number of research papers and
technical notes have been published. Gottfried et al. [2] investigated the behavior and
evaporation rate of small droplet of liquid on a hot flat surface experimentally with no evident of
bouncing, spitting or hissing. They developed an analytical model to predict evaporation rate of
droplets. In their model they assumed the liquid droplet to be separated from the heating surface
by a vapor film which also provided the excess vapor pressure to support the liquid mass. Starov
and Sefiane [3] proposed a theoretical description of evaporation of sessile drop. They described
linear dependency of evaporation rate on droplet‟s base radius. Their analysis showed that most
of the evaporation was concentrated near the contact line.
2
Baumeister et al. [4] analyzed water droplets of volume 0.05 to 1 cc. In this volume range, they
suggested an analytical model based on flat disk geometry. The droplet was assumed to be in a
steady state condition and using momentum, energy, continuity equation along with boundary
conditions, they derived a formula to find the droplet evaporation time. Chatzikyriakou et al. [5]
used CFD simulations of both sessile droplets resting upon a vapor cushion and droplets
bouncing off a hot solid surface. Experimentally they found that the droplet can rebound from
the vapor layer. A good agreement with the experimental observations was found with their
simulation. Wachters et al. [6] developed an theoretical model for heat transfer from hot
horizontal plate to sessile water drops in spheroidal state. They neglected radiation heat transfer
in their model and only put concentration on conduction heat transfer to the bottom surface of
liquid through the vapor film. Xie and Zhou [7] conducted a theoretical analysis for liquid
droplet impinging on a solid wall near Leidenfrost point. They divided the evaporation process
into two stages recoil stage and spherical stage and built heat transfer models on both stages
respectively. The maximum contact radius was also calculated by a theoretical model. Crafton
and Black [8] observed and quantified the evaporation rates of small liquid droplets of Water and
n-Heptane on Aluminum and Copper surfaces. From the measured quantities, they calculated
contact angle and evaporation rate. They used these results to predict the heat transfer on surface
and compared it with experimental results.
Nguyen and Avedisian [9] presented a numerical solution for the problem of film evaporation of
a liquid droplet on a horizontal surface. They assumed the horizontal surface having a constant
surface temperature which was considered to be isolated from the ambience. There are many
other scientists and researchers who conducted experiment on this phenomenon [10-12]. Yao and
Cai [13] studied the dynamics of water drops impacting at small angles on hot surfaces. The
Experiments were conducted using a monosize droplet stream and a rotating disk. When the
impact angle was decreased, the Leidenfrost temperature was found to be reduced. Correlations
were established for the description of this behavior. Nagai and Nishio [14] studied Leidenfrost
temperature on a very smooth surface. The Leidenfrost temperature was measured on single-
crystal and metal plates. The maximum surface roughness of the former was 0.03 μm, and that of
the latter was 1.25 μm. Results of the experiment showed that the Leidenfrost temperatures on
these two surfaces did not differ from each other as long as the surfaces were the same in
wettability and thermal conductivity (or thermal diffusivity). Based on the theory of Baumeister
3
et al. [4], I. Michiyoshi, K. Makino [16] determine time averaged heat flux. They studied the heat
transfer characteristics of evaporation of a single droplet of pure water placed on smooth surfaces
of copper, brass, carbon steel and stainless steel at temperature ranging 80⁰C to 450⁰C .They
analyzed heat transfer characteristics by correlating heat flux with ∆T=( 𝑇𝑝 − 𝑇𝑠).
In the present study, an analytical model will be proposed for the prediction of sessile drop
evaporation time at Leidenfrost temperature on hot solid surface. The model will be verified with
some experimental data. In the proposed model, conduction and radiation heat transfer along
with mass diffusion have been successfully included. It was roughly observed in the experiment
that for small droplet diameter, the liquid droplet usually flattens on the hot metal surface having
a thin vapor cushion beneath it. In the proposed model, it is considered that the liquid droplet on
the heated surface to have an almost cylindrical shape with a very little height. The vapor layer
thickness is considered to be uniform during the entire vaporization process and the vertical
velocity of the vapor leaving the bottom surface of the droplet has been considered to be
uniform. Heat is assumed to be transferred at the bottom surface of the liquid by conduction and
radiation. The side surface is assumed to get the heat energy by radiation only. Mass diffusion is
also considered from side surface of droplet in the analysis.
A correlation was developed in reference [18] considering only the heat is transferred from the
hot surface to the droplet by conduction through the vapor film and by radiation. Mass of the
droplet is removed by evaporation and diffusion. The droplet evaporation time of a specific
liquid on different plates depends upon the thermo-physical properties of the corresponding
plate. Evaporation times of different liquids on different solid surfaces are compared in this
dissertation by graphical presentation. The experimental data has been correlated in terms of
dimensionless groups resulting from the analytical model by Lee. In the present study heat
transfer characteristics of evaporation of small droplets of distilled water, saturated NaCl
solution, methanol and ethanol settled on aluminum, copper, mild steel and brass surface with
temperature ranging from 60 to 400⁰C will be studied. The experimental data has been plotted to
obtain evaporation time (τ) versus surface temperature (Ts) curve, which has an inverse trend of a
typical boiling curve and can be called as Inverse Boiling Curve. By determining heat flux,
boiling curve for different liquids (water, methanol and ethanol) has been generated.
4
CHAPTER
LITERATURE REVIEW 2
2.1 Experiment Conducted by Leidenfrost
Figure 2.1 A Leidenfrost drop in cross section
Leidenfrost conducted his experiments with an iron spoon that was heated red-hot in a fireplace.
After placing a drop of water into the spoon, he timed its duration by the swings of a pendulum.
He noted that the drop seemed to suck the light and heat from the spoon, leaving a spot duller
than the rest of the spoon. The first drop deposited in the spoon lasted 30 s while the next drop
lasted only 10 s. Additional drops lasted only a few seconds.
Leidenfrost misunderstood his demonstrations because he did not realize that the longer-lasting
drops were actually boiling. When the temperature of the plate is less than the Leidenfrost point,
the water spreads over the plate and rapidly conducts energy from it, resulting in complete
vaporization within seconds.
When the temperature is at or above the Leidenfrost point, the bottom surface of a drop
deposited on the plate almost immediately vaporizes. The gas pressure from this vapor layer
5
prevents the rest of the drop from touching the plate (Fig. 4). The layer thus protects and
supports the drop for the next minute or so. The layer is constantly replenished as additional
water vaporizes from the bottom surface of the drop because of energy radiated and conducted
through the layer from the plate. Although the layer is less than 0.1 mm thick near its outer
boundary and only about 0.2 mm thick at its center, it dramatically slows the vaporization of the
drop.
2.2 Stability of Leidenfrost Phenomenon
The question of the stability of the Leidenfrost phenomenon usually quickly reduces to a
discussion of the Leidenfrost point and how it was determined, since most workers are agreed
that film boiling becomes increasingly stable relative to nucleate and mixed modes at increasing
surface temperatures. However, very little agreement exists between various workers on the true
value of the Leidenfrost point for any given set of conditions.
Baumeister [4] report maintaining stable film boiling for small droplets in air down to a surface
temperature less than a degree above saturation, while Wachters [20] found a similar for water
droplets in film boiling in dry air at a surface temperature as low as 75ºC. In fact, Wachters
argues that the absolute minimum surface temperature for the Leidenfrost phenomenon is equal
to the wet-bulb temperature of the surrounding atmosphere; to quote his explanation
“When the drop bottom temperature has a value below the boiling point, the narrow layer under
the drop contains a mixture of vapor and air. In this mixture the vapor concentration is in
equilibrium with the drop bottom temperature. However, at the outer rim of the drop bottom the
dry surroundings. This is a one-way diffusion which involves a drift velocity of the gas mixture
and generates a radial pressure gradient, higher than the atmospheric pressure”.
The explanation appears plausible, but the quantitative expressions have not been worked out.
Baumeister and Wachters both emphasize the need for extremely smooth surface and
suppression of disturbances in the droplet to achieve these extremely low temperatures. The
6
droplets are initially deposited on quite hot surfaces which are then cooled to low temperatures.
Too rapid cooling of the surface leads to premature collapse, presumably because the droplet
oscillations have not been adequately damped out in this time. With care, a wire can be inserted
into the droplet to damp out the oscillations; the droplet is then partially supported by surface
tension on the wire, and the unconstrained force balance is upset. Both of these workers also
used test surfaces that were slightly concave underneath the droplet, and this undoubtedly
contributed to droplet stabilization.
2.3 Momentum, Heat and Mass transfer processes
A number of attempts have been made to analyze quantitatively the momentum, heat and mass
transfer processes during the Leidenfrost phenomenon. Several of these contain errors of
assumption or execution. All of these analyses assume that vapor is generated on the lower
surface of the drop by heat conduction through the vapor layer, that the vapor is in laminar flow
under the droplet, and that the integrated product of the local excess pressure (above
atmospheric) and the horizontal projection of the droplet lower surface is equal to the droplet
weight.
2.4 Application
Failure of tube walls of steam boiler is a common problem. In a nuclear reactor with boiling
coolant, the transition from nucleate to film boiling occurs at constant heat flux and can be
accompanied by a very large increase in wall temperature, most descriptively called burnout.
Conversely, once a reactor has had a coolant flow failure and surface has become very hot, film
boiling will occur, and one way to make a small amount of coolant contact a large amount of
surface is to spray it in as a fog. This technique, spray or fog cooling, has been tested, and it is a
variant of the convective Leidenfrost phenomenon in that major interest is attached to the impact
characteristics of the droplets on the surface.
7
Several other applications of the phenomenon are closely related in basic mechanism to those
above
The use of a water spray to cool steel billets or the rolls in rolling mill operations.
Water spray during continuous casting.
The design of quick response steam generators by spraying liquid on a hot surface.
The stable operation of a steam iron with a changing water inventory.
Film cooling of a rocket nozzle, either by breakdown of a continuous liquid film or direct
spray injection.
Cool-down of cryogenic liquid storage tanks and transfer lines during filling. An
interesting corollary problem is the possibility of minimizing cryogenic liquid loss by
deliberate production of a vapor film next to the wall by film boiling.
Use of air-dropped solutions to control forest fires.
2.5 Boiling and Leidenfrost Effect
Let us consider a pan where water to be heated from below by a flame or electric heat source. As
the water warms, air molecules are driven out of solution in the water, collecting as tiny bubbles
in crevices along the bottom of the pan (Fig. 1a). The air bubbles gradually inflate, and then they
begin to pinch off from the crevices and rise to the top surface of the water (Figs. 1b–f ). As they
leave, more air bubbles form in the crevices and pinch off, until the supply of air in the water is
depleted. The formation of air bubbles is a sign that the water is heating but has nothing to do
with boiling.
8
Figure 2.2 (a) A bubble forms in the crevice of a scratch along the bottom of a pan of water. (b–f
) The bubble grows, pinches off, and then ascends through the water
Water that is directly exposed to the atmosphere boils at what is sometimes called its normal
boiling temperature TS . For example, TS is about 100ºC when the air pressure is 1 atm. Since the
water at the bottom of your pan is not directly exposed to the atmosphere, it remains liquid even
when it superheats above TS by as much as a few degrees. During this process, the water is
constantly mixed by convection as hot water rises and cooler water descends.
9
Figure 2.3 Boiling curve for water.
If the pan‟s temperature is continuing to increase, the bottom layer of water begins to vaporize,
with water molecules gathering in small vapor bubbles in the now dry crevices, as the air bubbles
do in Fig. 1. This phase of boiling is signaled by pops, pings, and eventually buzzing. The water
almost sings its displeasure at being heated. Every time a vapor bubble expands upward into
slightly cooler water, the bubble suddenly collapses because the vapor within it condenses. Each
collapse sends out a sound wave. Once the temperature of the bulk water increases, the bubbles
may not collapse until after they pinch off from the crevices and ascend part of the way to the top
surface of the water. This phase of boiling is labeled „„isolated vapor bubbles‟‟ in the boiling
curve.
10
If the pan‟s temperature is more increased, the clamor of collapsing bubbles first grows louder
and then disappears. The noise begins to soften when the bulk liquid is sufficiently hot that the
vapor bubbles reach the top surface of the water. There they pop open with a light splash. The
water is now in full boil.
If the pan‟s temperature is further increased the vapor bubbles next become so abundant and
pinch off from their crevices so frequently that they coalesce, forming columns of vapor that
violently and chaotically churn upward, sometimes meeting previously detached „„slugs‟‟ of
vapor.
The production of vapor bubbles and columns is called nucleate boiling because the formation
and growth of the bubbles depend on crevices serving as nucleating sites (sites
of formation).
If the pan‟s temperature is raised past the stage of columns and slugs, the boiling enters a new
phase called the transition regime. Then each increase in the pan‟s temperature reduces the rate
at which energy is transferred to the water. The decrease is not paradoxical. In the transition
regime, much of the bottom of the pan is covered by a layer of vapor. Since water vapor
conducts energy about an order of magnitude more poorly than does liquid water, the transfer of
energy to the water is diminished. The hotter the pan becomes, the less direct contact the water
has with it and the worse the transfer of energy becomes.
At this stage, the whole of the bottom surface is covered with vapor. Then energy is slowly
transferred to the liquid above the vapor by radiation and gradual conduction. This
phase is called film boiling.
Jearl Walker of Cleveland State University performed an experiment for finding an elementary
relationship between lifetime of drops and pan temperature. Drops of water having uniform size
were released from a syringe to the hot plate and the survival time of the drop was measured.
The data was plotted and the graph shows a curious peak.
11
Figure 2.4 Drop lifetimes on a hot plate
When the plate temperature was between 100 and about 200ºC, each drop spread over the plate
in a thin layer and rapidly vaporized. When the plate temperature was about 200ºC, a drop
deposited on the plate beaded up and survived for over a minute. At even higher plate
temperatures, the water beads did not survive quite as long. The temperature corresponding to
the peak in a graph is generally known as the Leidenfrost point.
12
ro
Projected area = 𝜋𝑟𝑜2
Total surface area = 4𝜋𝑟𝑜
2
r
z
v
Profile of u
ro Thin vapor film
𝑄𝑅𝑠 (𝑝→𝑙)
𝑄𝐶𝑏(𝑝→𝑙) + 𝑄𝑅𝑏(𝑝→𝑙) Hot test surface
δ u
CHAPTER
DEVELOPMENT OF MODEL 3
Considering a droplet of liquid to be a sphere and of radius, 𝑟𝑜 , when dropped very gently on a
solid surface from a syringe, it takes heat from the solid surface and evaporates. For the proposed
model it is considered that the droplet becomes flattened when it falls on the hot test surface. So
the liquid droplet on the metal surface will be considered as a cylinder with a very little height. It
will be assumed that this flattened liquid will occupy a contact surface area equal to the projected
area of the initial spherical droplet. A schematic view of the geometry for the proposed model is
given in Fig. 3.1.
Fig. 3.1 Geometry of sessile droplet for the model
13
Heat transfer from a hot surface to a liquid is a very complex process and several physical
processes occur simultaneously. For simplicity of analysis, the following assumptions are made
throughout the theoretical development.
i. Heat is transferred to the bottom surface of the liquid droplet by conduction and
radiation through the vapor film.
ii. The side surface of the droplet is heated by radiation heat transfer only.
iii. The vapor flow at the bottom of the droplet is laminar and viscous.
iv. A stable vapor layer of uniform thickness is considered throughout the theory.
v. Mass diffusion from the droplet has been considered only on the side surface.
vi. As the vapor layer is very thin, it is assumed that the vertical velocity will not
influence the heat transfer to the bottom of the liquid droplet. Vertical velocity has
been considered to be constant in the analysis.
vii. Though actually the bottom surface of the liquid is not completely flat, for simplicity
of analysis it has been considered as a flat surface.
viii. As the vapor layer is considered to have an extreme radius at r = 𝑟𝑜 it is considered that
at this point the pressure will be the atmospheric pressure, 𝑝𝑜 .
Now, conduction heat transfer through bottom surface of the liquid can easily be found by
considering the steady state one dimensional Fourier‟s heat conduction equation
𝛿2𝑇
𝛿𝑧2 = 0 (3.1)
By integrating and putting boundary conditions at z=0, T=Tp, and at z=δ, T=Ts, the temperature
profile beneath the droplet is found. Substitution of this temperature profile in the differential
conduction heat transfer formula yields in conduction heat flux through the bottom surface,
Integrating, ∫𝛿(𝛿𝑇
𝛿𝑧)=∫0. 𝛿𝑧
𝛿𝑇
𝛿𝑧 =c1
∫𝛿𝑇= c1∫δz
14
T=c1z+c2 ∙∙∙∙∙∙∙∙∙∙ (3.a)
Boundary conditions, at z=0, T=Tp, and at z=δ, T=Ts
For, z=0, Tp=0+c2 i.e. c2= Tp
So, from equation (a), T=c1z+ Tp ∙∙∙∙∙∙∙∙∙∙ (3.b)
And, from equation (3.b)
For z=δ, T=c1δ+ Tp,
c1=𝑇𝑠−𝑇𝑝
𝛿
Now, from equation (b), T=𝑇𝑠−𝑇𝑝
𝛿𝑧 +Tp ∙∙∙∙∙∙∙∙∙∙ (3.c)
So, conduction heat transfer from hot metal surface to bottom surface of liquid is
Qcb(p→l)=-kv𝐴𝑙𝛿𝑇
𝛿𝑧
Qcb(p→l)=-kv𝐴𝑙 𝛿
𝛿𝑧 (
𝑇𝑠−𝑇𝑝
𝛿𝑧 +Tp) ……………[from eqn (3.c)]
Qcb(p→l)=-kv𝐴𝑙 𝑇𝑠−𝑇𝑝
𝛿
Qcb(p→l)=kv 𝐴𝑙 𝑇𝑝−𝑇𝑠
𝛿
QCb (p→l)
𝐴𝑙= kv
𝑇𝑝−𝑇𝑠
𝛿
𝑄 𝐶𝑏(𝑝→𝑙) = kv
𝑇𝑝−𝑇𝑠
𝛿 (3.2)
We will consider this initial heat flux will continue up to last for the ease of calculation.
Now, radiation heat flux from hot metal surface to bottom of liquid droplet may be expressed by
the following equation [15],
QRb(p→l)= 𝐸𝑝 −𝐸𝑏𝑙
1−εp
휀𝑝 𝐴𝑝+
1
𝐴𝑝 𝐹𝑝𝑙+
1−휀𝑙휀𝑙𝐴𝑙
𝑄𝑅𝑏(𝑝→𝑙)=𝜍(𝑇𝑝
4 −𝑇𝑠4)
1−εp
휀𝑝 𝐴𝑝+
1
𝐴𝑝 𝐹𝑝𝑙+
1−휀𝑙휀𝑙𝐴𝑙
15
𝑄𝑅𝑏(𝑝→𝑙)= 𝜍(𝑇𝑝
4 −𝑇𝑠4)𝐴𝑙
1−εp
휀𝑝
𝐴𝑙𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑙𝐴𝑝
+1−휀𝑙
휀𝑙
𝑄𝑅𝑏 (𝑝→𝑙)
𝐴𝑙=
𝜍(𝑇𝑝4 −𝑇𝑠
4)1−εp
휀𝑝
𝐴𝑙𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑙𝐴𝑝
+1−휀𝑙
휀𝑙
𝑄 𝑅𝑏 𝑝→𝑙 =
𝜍 𝑇𝑝4 –𝑇𝑠
4 1−휀𝑝
휀𝑝
𝐴𝑙𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑙𝐴𝑝
+1−휀𝑙
휀𝑙
(3.3)
We will consider this initial heat flux will continue up to last for the ease of calculation.
Radiation shape factor for bottom surface, 𝐹𝑝𝑙 can be found using formula of shape factor
between two parallel coaxial disks given by reference [15],
𝐹𝑝𝑙 = 𝑋− 𝑋2 −4
𝑅2𝑅1
2
12
2 (3.4)
Where, 𝑅1 =𝑟𝑝
𝛿 , 𝑅2 =
𝑟𝑜
𝛿 and 𝑋 =
1+(1+𝑅22)
𝑅12
As the droplet is of very little volume, these two heat fluxes 𝑄 𝐶𝑏(𝑝→𝑙) and 𝑄
𝑅𝑏 𝑝→𝑙 will be
considered to retain their value to be constant up to the complete vaporization of the droplet.
So, total heat flux at bottom,
𝑄 𝑏 = kv
𝑇𝑝−𝑇𝑠
𝛿 +
𝜍(𝑇𝑝4 −𝑇𝑠
4)1−휀𝑝
휀𝑝
𝐴𝑙𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑙𝐴𝑝
+1−휀𝑙
휀𝑙
(3.5)
From Eq. (5), it is evident that total heat flux at the bottom of the droplet is only a function of
vapor layer thickness, 𝛿.
This 𝑄 𝑏 vaporizes the liquid at bottom and superheats the vapor at an average temperature of
𝑇𝑝 +𝑇𝑠
2 . So,
𝑄𝑏 =𝑚 𝑏𝑔+𝑚 𝑏𝑐𝑝(𝑇𝑝 +𝑇𝑠
2 - 𝑇𝑠)
16
𝑄𝑏 =𝑚 𝑏𝑔+𝑚 𝑏𝑐𝑝(𝑇𝑝 −𝑇𝑠
2)
𝑄𝑏 = 𝑚𝑏
𝑡(𝑔+𝑐𝑝
𝑇𝑝 −𝑇𝑠
2 )
𝑄𝑏
𝐴𝑙 =
𝑚𝑏
𝑡𝐴𝑙(𝑔+𝑐𝑝
𝑇𝑝 −𝑇𝑠
2)
𝑄 𝑏=
𝑚𝑏𝛿
𝑡𝐴𝑙𝛿(𝑔+𝑐𝑝
𝑇𝑝 −𝑇𝑠
2)
𝑄 𝑏 =
𝑚𝑏𝛿
𝑉 𝑡(𝑔+𝑐𝑝
𝑇𝑝 −𝑇𝑠
2)
𝑄 𝑏 = 𝜌𝑣 𝑣(𝑔+𝑐𝑝
𝑇𝑝 −𝑇𝑠
2) (3.6)
Again, we will consider this initial heat flux will continue up to last for the ease of calculation.
Eqs. (5) and (6) yield,
𝑣 =
𝑘𝑣 𝑇𝑝−𝑇𝑠
𝛿 +
𝜍(𝑇𝑝4 −𝑇𝑠
4)
1−휀𝑝휀𝑝
𝐴𝑙𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑙𝐴𝑝
+1−휀𝑙
휀𝑙
𝜌𝑣 (𝑔+𝑐𝑝𝑇𝑝 −𝑇𝑠
2)
(3.7)
Now, simplified from of Navier-Stokes equation for cylindrical vapor layer at bottom, is given
by,
𝛿𝑝
𝛿𝑟 = 𝜇𝑣
𝛿2𝑢
𝛿𝑧2
By integrating twice and considering no slip condition at the bottom vapor layer (Fig. 3.1),
average velocity can be obtained as follows,
By Integrating,
∫𝛿
𝛿𝑧(
𝛿𝑢
𝛿𝑧) 𝛿𝑧=
1
𝜇𝑣
𝛿𝑝
𝛿𝑟∫ 𝛿𝑧
𝛿𝑢
𝛿𝑧=
1
𝜇𝑣
𝛿𝑝
𝛿𝑟𝑧+𝑐3
Again integrating,
17
𝑢=1
𝜇𝑣
𝛿𝑝
𝛿𝑟
𝑧2
2+𝑐3𝑧 +𝑐4 ……… (3.d)
Boundary condition at the vapor layer
Here, 𝑢 = 𝑓(𝑟, 𝑧)
At no 𝑠lip condition,
𝑢 𝑟, 0 = 0 ……… (3.e)
𝑢 𝑟, 𝛿 = 0 ……… (3.f)
For condition (e), using equation (3.d)
0=1
𝜇𝑣
𝛿𝑝
𝛿𝑟
02
2+𝑐3. 0 +𝑐4
𝑐4=0
𝑢=1
𝜇𝑣
𝛿𝑝
𝛿𝑟
𝑧2
2+𝑐3𝑧 ……… (3.g)
For condition (3.f), using equation (3.g)
0=1
𝜇𝑣
𝛿𝑝
𝛿𝑟
𝛿2
2+𝑐3𝛿
𝑐3=- 1
𝜇𝑣
𝛿𝑝
𝛿𝑟
𝛿
2
𝑐3=- 1
2𝜇𝑣
𝛿𝑝
𝛿𝑟𝛿
So from equation (3.g)
𝑢=1
𝜇𝑣
𝛿𝑝
𝛿𝑟
𝑧2
2−
1
2𝜇𝑣
𝛿𝑝
𝛿𝑟𝛿. 𝑧
𝑢=1
2𝜇𝑣
𝛿𝑝
𝛿𝑟(𝑧 − 𝛿)𝑧
Now average velocity can be obtained from the above equation as follows
𝑢 =1
𝛿∫ 𝑢 𝛿𝑧
𝛿
0
𝑢 =1
𝛿∫
1
2𝜇𝑣
𝛿𝑝
𝛿𝑟 𝑧 − 𝛿 𝑧 𝛿𝑧
𝛿
0
18
𝑢 =1
𝛿
1
2𝜇𝑣
𝛿𝑝
𝛿𝑟[𝑧3
3−
𝑧2
2𝛿]0
𝛿
𝑢 =1
𝛿
1
2𝜇𝑣
𝛿𝑝
𝛿𝑟[𝛿3
3−
𝛿3
2]
𝑢 = - 𝛿2
12𝜇𝑣
𝛿𝑝
𝛿𝑟 (3.8)
Balance of flow rate at bottom gives,
2𝜋𝑟𝛿𝑢 =𝜋𝑟2𝑣
Substitution of 𝑢 in it from Eq. (3.8) and integration using boundary conditions at r=0, pressure
is 𝑝 and at r = 𝑟𝑜 pressure is 𝑝𝑜yield in,
2𝜋𝑟𝛿(−𝛿2
12𝜇𝑣
𝛿𝑝
𝛿𝑟)=𝜋𝑟2𝑣
𝛿𝑝
𝛿𝑟 =−
6𝜇𝑣𝑟𝑣
𝛿3
∫ 𝛿𝑝 =𝑝
𝑝𝑜 −
6𝜇𝑣𝑣
𝛿3 ∫ 𝛿𝑟𝑟
𝑟𝑜
𝑝 − 𝑝𝑜 = −3𝜇𝑣𝑣
𝛿3 (𝑟2 − 𝑟𝑜2)
𝑝 − 𝑝𝑜 =3𝜇𝑣𝑣
𝛿3 (𝑟𝑜2 − 𝑟2) (3.9)
This is the pressure distribution of vapor film at the bottom.
Force balancing at bottom at initial condition gives,
𝜌𝑙𝑔𝑉𝑜=∫ (𝑝 − 𝑝𝑜)2𝜋𝑟𝛿𝑟𝑟𝑜
0
Here, 𝑉𝑜=4
3 𝜋𝑟𝑜
3=Initial droplet volume
𝜌𝑙𝑔𝑉𝑜=∫3𝜇𝑣𝑣
𝛿3 (𝑟𝑜2 − 𝑟2) 2𝜋𝑟𝛿𝑟
𝑟𝑜
0
𝜌𝑙𝑔4
3 𝜋𝑟𝑜
3 =2𝜋3𝜇𝑣𝑣
𝛿3 ∫ r(𝑟𝑜2 − 𝑟2) 𝛿𝑟
𝑟𝑜
0
19
Vapor surface
Hot test surface
Liquid surface
Imaginary surface used for
finding Radiation shape factor
for side surface, 𝐹𝑝𝑙
2
1
4
3
2′
3′
4′
3
2𝑔𝜌 𝑙𝑟𝑜3𝛿3
9𝜇𝑣𝑣 =
𝑟𝑜
4
𝛿 = [9
8
𝜇𝑣𝑟𝑜𝑣
𝜌 𝑙𝑔 ]
1
3 (3.10)
This is the final expression of vapor film thickness, δ. It can be determined by iterations using a
computer program.
Now, radiation heat flux from hot metal surface to side surface of liquid droplet can be expressed
by the following Eqn. [3.15]
𝑄 𝑅𝑠(𝑝→𝑙) =
𝜍(𝑇𝑝4 −𝑇𝑠
4)1−εp
휀𝑝
𝐴𝑠𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑠𝐴𝑝
+1−휀𝑙
휀𝑙
(3.11)
Radiation shape factor for side surface, 𝐹𝑝𝑙 can be found using formula of shape factor [3.15]
between two finite, coaxial cylinders and two parallel, coaxial disks.
Fig. 3.2 Determination of radiation shape factor at side surface of liquid droplet
Figure 3.2 provides required configuration for determination of desired shape factor. For lower
half of the image, considering symmetry and reciprocal relation, the following relation can be
obtained,
𝐹12 =𝐴2
2𝐴1(1 −
𝐴4
𝐴2𝐹42) (3.12)
20
Here,
𝐹42 =1
𝑋−
1
𝜋𝑋 𝑐𝑜𝑠−1(
𝐵
𝐴 −
1
2𝑌 [𝐴2 + 4𝐴 − 4𝑋2 + 4]
1
2 𝑐𝑜𝑠−1 𝐵
𝐴𝑋 + 𝐵 𝑠𝑖𝑛−1(
1
𝑋) −
𝜋𝐴
2 }
(3.13)
Where, =𝑟𝑝
𝑟𝑜 , 𝑌 =
𝛿
𝑟𝑜, 𝐴 = 𝑋2 + 𝑌2 − 1 and 𝐵 = 𝑌2 − 𝑋2 + 1 and
𝐴2 = 2𝜋𝑟𝑜𝛿, 𝐴1 = 𝜋(𝑟𝑝2 − 𝑟𝑜
2)
Again for complete image [Fig. 3.2]
𝐹1→22′=
𝐴22 ′2𝐴1
(1−𝐴44 ′𝐴22 ′
𝐹44 ′→22 ′) (3.14)
𝐹1→22′ can be found by using the Eq. (3.13)
Where, =𝑟𝑝
𝑟𝑜 ,𝑌 =
𝛿+4
3𝑟𝑜
𝑟𝑜, 𝐴 = 𝑋2 + 𝑌2 − 1 , 𝐵 = 𝑌2 − 𝑋2 + 1 and 𝐴44 = 2𝜋𝑟𝑝(𝛿 +
4
3𝑟𝑜),
𝐴22′ = 2𝜋𝑟𝑜(𝛿 +4
3𝑟𝑜)
Here, 4
3𝑟𝑜 is the height of the liquid droplet on the test surface as the volume of the initial
spherical liquid droplet is equal to the volume of the cylindrical shaped droplet on the test
surface.
And, finally,
𝐹𝑝𝑙 = 𝐹1→22′ − 𝐹12 (3.15)
Where 𝐹1→22′ and 𝐹12 can be found using Eqs. (3.14) and (3.12) respectively.
So, total radiation heat transfer from hot metal surface to side surface of liquid droplet,
21
𝑄 𝑠 = 𝑄
𝑅𝑠(𝑝→𝑙)
𝑄 𝑠 =
𝜍(𝑇𝑝4 −𝑇𝑠
4)1−εp
휀𝑝
𝐴𝑠𝐴𝑝
+1
𝐹𝑝𝑙
𝐴𝑠𝐴𝑝
+1−휀𝑙
휀𝑙
(3.16)
For mass diffusion from side surface of liquid droplet steady state mass diffusion equation for
cylindrical coordinate could be used. For simplicity of calculation spherical coordinate has been
used with an equivalent cylindrical surface area.
Now, for spherical coordinate, steady state mass transfer equation
𝛿
𝛿𝑟 𝑟2 𝛿𝐶
𝛿𝑟 = 0 (3.17)
Having boundary conditions At 𝑟 = 𝑟𝑜′ , 𝐶 =
𝑝𝑠𝑀
𝑅𝑇𝑠 and at 𝑟 = ∞, 𝐶 = 0
By integrating,
𝑟2 𝛿𝐶
𝛿𝑟=𝑐5
𝛿𝐶
𝛿𝑟=
𝑐5
𝑟2
Again integrating,
𝐶 = −𝑐5
𝑟+ 𝑐6 ……… (3.h)
From equation (3.21) for the second boundary condition,
0 = −𝐶5
∞+ 𝐶6
𝐶6 = 0
Putting the value of 𝐶6 in equation (3.h)
𝐶 = −𝐶5
𝑟 ……… (3.i)
22
Now, from equation (3.22) at the first boundary condition,
𝑝𝑠𝑀
𝑅𝑇𝑠 =−
𝐶5
𝑟𝑜′
𝐶5 = −𝑝𝑠𝑀𝑟𝑜
′
𝑅𝑇𝑠
So, from equation (3.i), we get
𝐶 =𝑝𝑠𝑀𝑟𝑜
′
𝑅𝑇𝑠𝑟
So, rate of mass transfer from the side surface of the liquid to atmosphere is
𝑚 𝑠 = −𝐷𝐴𝑠 𝑑𝐶
𝑑𝑟 𝑟 = 𝑟𝑜
′
𝑚 𝑠 = −𝐷𝐴𝑠 𝑑𝑑𝑟
(𝑝𝑠𝑀𝑟𝑜
′
𝑅𝑇𝑠𝑟 ) 𝑟 = 𝑟𝑜
′
𝑚 𝑠 = 𝐷𝐴𝑠 𝑝𝑠𝑀
𝑅𝑇𝑠
1
𝑟𝑜′
𝑚 𝑠 =𝐷𝐴𝑠𝑝𝑠𝑀
𝑅𝑇𝑠𝑟𝑜′ (3.18)
Now we will develop a relationship between cylindrical coordinate and spherical coordinate as
follows
2𝜋𝑟𝑜 ∗4
3𝑟𝑜 = 4𝜋𝑟𝑜
′2
𝑟𝑜′ =
√2𝑟𝑜
√3
So, from eqn (3.18)
𝑚 𝑠 =√3𝐷𝐴𝑠𝑝𝑠𝑀
√2𝑅𝑇𝑠𝑟𝑜 (3.19)
For a refined value of 𝑚 𝑏 , total heat balance should be considered as,
𝑄 𝑏𝐴𝑏 + 𝑄
𝑠𝐴𝑠 = 𝑚 𝑏(𝑔 +𝑐𝑝
2 (𝑇𝑝 − 𝑇𝑠)) +𝑚 𝑠𝑔
𝑚 𝑏 =𝑄 𝑏𝐴𝑏 +𝑄 𝑠𝐴𝑠−𝑚 𝑠𝑔
𝑔+𝑐𝑝
2 (𝑇𝑝−𝑇𝑠)
(3.20)
23
Again balancing the total evaporation rate,
−𝜌𝑙𝑑𝑉
𝑑𝑡= 𝑚 𝑏 + 𝑚 𝑠
Now integrating,
∫ 𝑑𝑉0
𝑉𝑜= −
𝑚 𝑏 +𝑚 𝑠
𝜌 𝑙∫ 𝑑𝑡
𝜏
0
Which gives,
𝜏 =𝜌 𝑙
𝑚 𝑏 +𝑚 𝑠𝑉𝑜 (3.21)
Writing a computer program, total vaporization time, 𝜏 can be found from Eq. (3.21). Value of
𝑚 𝑏 and 𝑚 𝑠 can be found using Eqs. (3.20) and (3.19) respectively.
24
CHAPTER
EXPERIMENTAL SETUP AND 4 PROCEDURE
4.1 Introduction
The sessile drop apparatus was used to study the evaporation characteristics of droplet on a
heated surface. In particular, the liquid-solid interface temperature corresponding to the
Leidenfrost Temperature was determined from droplet evaporation curve for different materials
of different liquid.
4.2 Schematic diagram of experimental setup
Fig. 4.1 Schematic diagram of the experimental setup
Insulator
Thermocouple
Variac
AC supply
Test surface
Stand
Cartridge
heater
Syringe
25
The experimental setup is shown in Figure 4.1 which consists of the following components
• Metal blocks
• Stand
• Heater
• Variac
• Thermocouple
• Dropper
The working liquids that are used in the in the experiment are as follows
• Water
• NaCl solution
• Methanol
• Ethanol
4.2.1 Metal Blocks
Four different metal blocks are used to find out the Leidenfrost Temperature and variation of
drop evaporation time. They are as follows
• Mild steel
• Copper
• Aluminum
• Brass
26
(a) Top view (b) Side view
Figure 4.2 Metal Block
Each block has same dimension. The height and diameter of the blocks were 3 in and 3.5 in
respectively. A thermocouple was installed 2 mm below the test surface. Two cartridge heaters
were used. The first one is located 1 inch beneath and second one was 4 inch beneath the test
surface of each block. The test surfaces of the blocks are polished by emery paper (1200 grade
(000) and 1600 grade (0000)).
4.2.2 Stand
A supporting structure shown in Figure 4.2 is used in experiment to hold the metal blocks above
the earth surface. The dimensions of the stand are 1 ft × 1 ft × 3 ft. Stand hold the block above 2
ft from the earth surface. Four glass plates of 1 ft × 1 ft are attached with the stand to protect
from the air flow over the test surface.
Figure 4.3 Stand
27
4.2.3 Heater
Heater is used for heating the metal blocks. 500 Watt cartridge (Figure 4.4) heater is used. Heater
is placed 1 in beneath the test surface by drilling the block.
Figure 4.4 Heater
4.2.4 Thermocouple
K-type thermocouple is used (Figure 4.5) to determine the center temperature of the testing
surface. A thermocouple was installed 5 mm in below the test surface. But we should installed
the thermocouple at the center of the test surface where liquid droplet falls. We installed the
thermocouple at the center of the test surface, heat may conduct with the thermocouple wire and
evaporation time should vary and we will not get correct evaporation temperature.
Figure 4.5 K type thermocouple meter.
28
4.2.5 Variac
The heat supplied to the metal block through joule heating. A cartridge heater is fitted in the
blocks. Regulated electrical energy is supplied to the heater during the experiment. The
resistance element that output is controlled by variac connected to the 220 Volt laboratory
power.
Figure 4.6 Variac
4.2.6 Syringe
A couple of syringes were used to drop gently the liquid droplets on the test surface. Two
different types of needles were used with the syringe to produce two different droplet diameters;
2.50 mm and 2.75 mm. The syringe was held perpendicular to the horizontal test surface and
droplets were released from about two inches from the surface.
Figure 4.7 Syringes used in the experiment
29
4.3 Working Fluids
The working fluids used in this experiment are Water(H2O), NaCl solution(H2O+NaCl),
Methanol(CH3OH) and Ethanol(C2H5OH). Fluid is heated up on the test surface, boils and
evaporate. and we measure the drop evaporation time. Some important properties of working
fluids are mentioned below.
4.3.1 Water
Water is available in nature. But natural water is not pure. Many salts are dissolved in natural water.
Water used in experiment is distilled water from BUET boiler lab. It boils at 373.15 K at 101.325 kPa.
4.3.2 NaCl Solution
We use NaCl solution in experiment. We add (2± 0.01) gm NaCl salt in 100 ml distilled Water.
4.3.3 Methanol
Methanol is a colorless, flammable liquid. Pure methanol melts at 175.2 K, boils at 327.85 K and
molecular weight is 32. The commercial use of methanol has sometimes been prohibited. Large amount
of it are used in the synthesis of formaldehyde. Methanol is often called wood alcohol because it was once
produced mainly as a byproduct of destructive distillation of wood. Methanol is also used as a solvent for
varnishes and lacquers as antifreeze and as gasoline extender in the production of gasohol.
4.3.4 Ethanol
Ethanol can be produced by formation of carbohydrates, which occur naturally and abundantly in some
plants like sugarcane and from starchy materials like potato and corn. It boils at 351.3 K. Ethanol and
methanol both also used as fuels in SI engines.
4.4 Experimental Procedure
As seen from the Fig. 4.1, the test surface was heated from the bottom by using two cartridge
heaters. The power supply to the block was regulated using the variac to reach desired surface
temperature of the test surface. When the temperature reached at a predetermined value, a droplet
of working liquid was dropped gently to the center of the heating surface with a syringe; complete
evaporation time was measured using a stopwatch. The droplet temperature was equal to the
30
room temperature (30⁰C±5%) when it was dropped. The surface temperature was sensed by the
thermocouple, whose bead was located 3 mm beneath the center of the test surface and the digital
temperature reading was taken from the meter. Few numbers of observed phenomena during the
droplet evaporation was captured using a video camera. The droplet‟s initial diameter was
calculated from the total measured volume of 30 droplets at room temperature considering each
droplet to be a little sphere. To reduce error, this was done three times and the average diameter
was taken.
When the plate temperature reached at steady state the syringe was filled with liquid and
mounted. Bottom end of the syringe was pressed slowly and a droplet formed on the tip of the
needle of the syringe until the droplet weight becomes sufficient to detach from the tip. The
stopwatch was used to record the time of evaporation of droplets and its accuracy was 0.01sec. To
minimize the measured time error, three evaporation times were recorded for each temperature
and then averaged together. The experiment conducted for the test surface temperature with an
increment of 10⁰C up to ten surface temperature of 100⁰C and later the increment was changed to
25 ºC up to the test surface temperature 400⁰C.
31
CHAPTER
RESULTS AND DISCUSSIONS 5
5.1 Analysis of Theoretical model
Experimental droplet vaporization time (taken for complete evaporation) has been investigated
as a function of test surface temperature for four test metal surfaces, four different liquid and two
different droplet diameters (2.5mm and 2.75 mm). A numerous number of graphs have been
found within a temperature range from 60 to 400 oC. Resulting graphs are found to have shapes
just opposite of a typical boiling curve, as expected which is defined as the „Inverse Boiling
Curve‟ in the present study. It is because in typical boiling curve, heat flux is plotted as a
function of temperature difference. And in this experiment, evaporation time has been plotted as
a function of test surface temperature. The droplet getting higher heat flux will evaporate
quickly, and so the time and heat flux relationship is just opposite and it has become an evident
in the experimental graphs. Leidenfrost temperatures are easily determined from the graphs
where maximum vaporization time is required in the film boiling region. Based on the prescribed
model, a computer program has been generated. Theoretical Leidenfrost time has been estimated
from this program. For comparison purposes some experimental and theoretical (at Leidenfrost
point) times have been presented in graphs from Figs. 5.1-5.6. The lower temperature than the
Leidenfrost point in the boiling curve has not been included in the present model. This is because
the theory has been developed considering a stable vapor film beneath the liquid droplet. But,
before the Leidenfrost temperature, the film developed is not in a stable condition and gradually
increases in thickness of the vapor film with the increase in time up to Leidenfrost point.
32
Fig. 5.1 Experimental evaporation time and model predicted Leidenfrost time for small diameter
Methanol on Copper surface
Fig. 5.2 Experimental evaporation time and model predicted Leidenfrost time for small diameter
Methanol on Aluminum surface
0
5
10
15
20
25
30
35
40
0 100 200 300 400
Vap
oriz
tion
tim
e (s
ec)
Temperature (⁰C)
Methanol (Small diameter) on Copper surface
Theoritical Experimental
MethanolCopper2.50 mm
0
5
10
15
20
25
0 100 200 300 400 500
Vap
oriz
tion
tim
e (s
ec)
Temperature (⁰C)
Methanol (Small diameter) on Aluminum surface
Theoritical Experimental
MethanolAluminum2.50 mm
33
Fig. 5.3 Experimental evaporation time and model predicted Leidenfrost time for large diameter
Methanol on Aluminum surface
Figure 5.1 shows experimental vaporization time of Methanol for do = 2.5 mm on copper surface
along with theoretical approximation of Leidenfrost time. The resulting graph with experimental
data has a shape just opposite of a typical boiling curve and the reason has been described earlier.
For surface temperature around 100 oC, the evaporation time is very small might be because of
nucleate boiling. With the increase of temperature, the vapor layer develops and the heat flux to
the droplet falls eventually. As a result, total vaporization time increases. It expresses the partial
film boiling (transition boiling from nucleate to stable film boiling) region (From temperature
about 100 to 175 oC). A stable vapor layer forms at around 175
oC and it is the Leidenfrost
temperature where the vaporization time is the maximum. After 175 oC, the radiation heat
transfer becomes dominating mode which increases the heat flux and decreases the total
vaporization time. So the region in the graph from 175 to about 400 oC is film boiling region.
The Leidenfrost time from experiment as shown in the Fig. 5.1 is around 33 sec whereas, this
value is about 17 sec as predicted from the analytical solution [Eqn. 21]. The theoretical value is
about 50% of the experimental one.
0
5
10
15
20
25
30
0 100 200 300 400
Vap
oriz
tion
tim
e (s
ec)
Temperature (⁰C)
Methanol (Large diameter) on Aluminum surface
Theoritical Experimental
MethanolAluminum2.75 mm
34
Figure 5.2 shows experimental vaporization time of Methanol for do = 2.5mm on Aluminum
surface along with the theoretical approximation of Leidenfrost time. The shape of the graph is
opposite to the shape of boiling curve as described earlier. Here the nucleate boiling region is
again at the vicinity of 100 oC. The region between 100 to 225
oC is partial film boiling region.
The Leidenfrost temperature here is around 225 oC. The graph between 225 to 400
oC shows the
film boiling region where radiation heat transfer is dominating. The Leidenfrost time from
experiment as shown in the Fig. 5.2 is around 22 sec whereas, this value is about 17 sec as
predicted from the analytical solution [Eqn. 21]. The theoretical value is about 80% of the
experimental one.
Figure 5.3 shows experimental vaporization time of Methanol for do = 2.75mm on Aluminum
surface along with theoretical approximation of Leidenfrost time. As shown in the Fig. 5.3, the
Leidenfrost temperature is here also 225oC. This reveals that the size of sessile drop does not
have any influence on Leidenfrost temperature (both are around 225 oC as shown in the Figs. 5.2
and 5.3) though it effects the Leidenfrost time (for do = 2.5mm, Leidenfrost time is about 22 sec
as shown in the Fig. 5.2 and for do = 2.75 mm Leidenfrost time is around 26 sec as shown in the
Fig. 5.3. The Leidenfrost time for larger droplet is more than that of smaller one. It is expected
because a larger droplet will require more heat and time to fully evaporate. The temperature
range for nucleate boiling, partial film boiling and film boiling for large diameter Methanol
droplet on Aluminum surface are almost same as those of small diameter Methanol droplet on
Aluminum surface as shown in Fig. 5.2. The Leidenfrost time as measured in the experiment (as
shown in the Fig. 5.3) is approximately 25 sec and on the other hand this value is around 20 sec
as predicted from the model [Eqn. 21]. The predicted value from the proposed model is around
80% of the experimental measured value.
If it is compared Figs. 5.1-5.3; Fig. 5.2 and 5.3 show a fair agreement between the Leidenfrost
time predicted from the proposed model and the experimental data. Figure 5.1 shows that the
agreement between the experimental Leidenfrost time and the theoretical Leidenfrost time is not
as good as compared to the other two as shown in the Figs. 5.2 and 5.3. It can be explained by
thermal conductivity of the test surface. Thermal conductivity of Copper is almost double than
that of Aluminum. A metal having higher thermal conductivity has higher capability to supply
intense heat to the liquid droplet on its surface. As a result a thicker vapor blanket will generate
35
instantly which will impede the conduction heat transfer to the liquid and in turn a higher
Leidenfrost time should be found. It is evident from Fig. 5.1. The Leidenfrost time for Methanol
on Copper surface (as shown in the Fig. 5.1) is more than the Leidenfrost time for the same
liquid and same droplet diameter on Aluminum surface (Fig. 5.2). For simplicity, thermal
conductivity of test metal surface has not been incorporated in present theoretical model. As a
result the experimental Leidenfrost time is higher than the theoretical prediction of the
Leidenfrost time on the copper surface. In future study, the solid surface material‟s conductivity
will be tried to include which will hope to more close agreement of the experimental data with
the model.
Fig. 5.4 Experimental evaporation time and model predicted Leidenfrost time for small diameter
Ethanol on Brass surface
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400
Vap
oriz
tion
tim
e (s
ec)
Temperature (⁰C)
Ethanol (Small diameter) on Brass surface
Theoritical Experimental
EthanolBrass2.50 mm
36
Fig. 5.5 Experimental evaporation time and model predicted Leidenfrost time for large diameter
Ethanol on Brass surface
Fig. 5.6 Experimental evaporation time and model predicted large diameter Leidenfrost time for
Ethanol on Mild steel surface
0
10
20
30
40
50
0 100 200 300 400
Vapo
riztio
n tim
e (s
ec)
Temperature (⁰C)
Ethanol (Large diameter) on Brass surface
Theoritical Experimental
EthanolBrass2.75 mm
0
5
10
15
20
25
30
35
0 100 200 300 400
Vap
oriz
tion
tim
e (s
ec)
Temperature (⁰C)
Ethanol (Large diameter) on Mild steel surface
Theoritical Experimental
EthanolMild stel2.75 mm
37
Figure 5.4 shows experimental vaporization time of Ethanol for do = 2.5 mm on Brass surface
along with theoretical approximation of Leidenfrost time. Leidenfrost temperature here is around
150 oC and the Leidenfrost time is about 43 sec for the experimental case and around 17 sec
predicted from the model. Figure 5.5 represents the experimental time of Ethanol for do = 2.75
mm on Brass surface along with the theoretical approximation of Leidenfrost time, the
Leidenfrost temperature here is around 150 oC. Here the experimental Leidenfrost time value is
more or less 46 sec and the predicted value is about 20 sec. Again it is an evident from these two
figures (Figs. 5.4-5.5) that droplet size does not have any influence on the Leidenfrost
temperature. Experimental Leidenfrost time as shown in the Fig. 5.4 is less than that of as shown
in the Fig. 5.5. This again depicts that the larger the droplet size, the larger the Leidenfrost time.
Figure 5.6 expresses comparison between experimental vaporization times of Ethanol with do =
2.75mm on Mild steel surface along with the theoretical approximation of Leidenfrost time. Here
the Leidenfrost temperature is about 175oC, the experimental Leidenfrost time is about 30 sec
and the predicted value is around 20 sec. For all the three conditions (Fig. 5.4-5.6), the deviations
of the model predicted time values from the experimental times vary from 40 to 65 %. Here,
among the three conditions (Figs. 5.4-5.6), the agreement for the Leidenfrost time between the
model predicted and the experimental is better for the case of mild steel as it has the lower
thermal conductivity.
Sessile drop evaporation on a hot metal surface is a very complex phenomenon. The droplet on
the hot metal surface goes through dancing and jumping or in another way touching and
detaching of the droplet on the metallic surface. When the vapor beneath the droplet makes
floating the droplet (due to reaction force of the vapor) on hot metallic surface, there forms a
vapor layer which results in drop of heat transfer. In this way, when the heat transfer reduces, the
vapor pressure no longer remains able to sustain the weight of the droplet on the solid surface.
Once the droplet touches the hot solid surface, a large amount of heat transfer occurs and the
vapor pressure again floats the droplet. So the heat transfer to the droplet occurs in an
interrupted way. This continues until the weight of the droplet is reduced due to evaporation and
it is then balanced by the vapor pressure beneath the droplet (vapor pressure is almost not
reducing practically because it may be assumed that the solid surface temperature remains more
or less constant during the evaporation of the droplet). It is to be mentioned here that the heat
capacity of the experimental solid block is infinite compared to liquid droplet. At Leidenfrost
38
temperature a stable vapor film is formed below the droplet. Based on this stable film layer,
present model has been developed. For simplicity of the model, the dancing and bouncing
phenomena have been discarded. The effect of radiation heat transfer has been successfully
inserted into the theory and has also been taken in account at the final calculation and the
theoretical approximation of Leidenfrost time has been found fair enough compared to the
experimental Leidenfrost time.
5.2 Analysis of Experimental data
The experimental total vaporization time results are shown in (Figure 5.7 to Figure 5.14). The
mean point are plotted and the range of experimental results. The temperature which gives
maximum evaporation time is presumed to be the minimum heat flux at which stable film boiling
can exist and is termed as Leidenfrost temperature. The Leidenfrost temperature is not a strong
function of size, as has been noted over a much wider size range.
In the present study, complete evaporation time of a sessile droplet of liquid as a function of test
surface temperature of four different materials for four different liquids with two different
droplet diameters are analyzed. A numerous number of graphs having different combinations are
obtained here due to involvement of various experimental parameters. The representative
characteristics among all the experimental conditions will be described here.
5.2.1 Liquid variation
In this experiment we have plotted evaporation time of different liquids on a specific metal
surface (Figure 5.7 to Figure 5.10). Different liquids on a specific metal surface took different
time to evaporate. The factors that affect the evaporation time of a liquid droplet are as follows
5.2.1.1 Effect of latent heat of vaporization
A liquid having a higher latent heat of vaporization should take more time to evaporate. This
phenomenon is verified in our experiment (Figure 5.7 to Figure 5.10). Water has the maximum
heat of vaporization compared to methanol and ethanol so it takes the highest time to evaporate
among the all liquids for different metal surfaces (Aluminum, Brass, Copper and Mild steel).
39
5.2.1.2 Effect of specific heat, thermal conductivity and density of liquid
Evaporation time and Leidenfrost point temperature of the liquid depends on the specific heat,
thermal conductivity and density of the liquid as shown in the model of Henry [17]. Higher the
specific heat, thermal conductivity and density of the liquid Leidenfrost temperature will also be
higher as we observe in the experiment (Figure 5.9). Droplet evaporation time is maximum for
copper and minimum for mild steel (Figure 5.8 and Figure 5.9).
5.2.1.3 Effect of boiling temperature of liquid
Evaporation time also depends on boiling temperature of the liquid. The liquid which has higher
boiling point will take more time to evaporate. In this experiment we observe this phenomenon
as water has highest boiling point (100⁰C) comparing to methanol (64.7⁰C) and ethanol (78.3⁰C)
so water takes highest time to evaporate (Figure 5.11 and Figure 5.13).
5.2.2 Diameter variation
For a specific liquid, larger diameter droplet should take more time to evaporate. This
phenomenon is verified in our experiment (Figure 5.7 to Figure 5.14). As diameter increases
volume and mass of the liquid also increases due to this total amount of heat required by larger
diameter liquid droplet to evaporate is higher than the smaller diameter liquid droplet. Although
the evaporation time for larger diameter liquid droplet is higher than smaller diameter liquid
droplet but Leidenfrost point temperature remain same for both diameters as Leidenfrost point
temperature is independent of diameter. This phenomenon has been also proved by experimental
data (as shown in Figure 5.7 to Figure 5.10).
40
(a)
(b)
Fig. 5.7 Comparison curve of Droplet Evaporation Time of Distill Water, NaCl solution, Methanol, Ethanol on Aluminum surface
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(°C)
Diameter: 2.50 mmAluminium:Dist water:Sd
Aluminium:NaCl sol:Sd
Aluminium:Ethanol:Sd
Aluminium:Methanol:Sd
0
20
40
60
80
100
120
140
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(°C)
Diameter: 2.75 mmAluminium:Dist water:Ld
Aluminium:NaCl sol:Ld
Aluminium:Ethanol:Ld
Aluminium:Methanol:Ld
41
Droplet evaporation time curve of water on Aluminum surface shows that the both droplet takes
maximum time to evaporate completely at a temperature of 200 ⁰C and corresponding time for
smaller diameter is around 55sec. The complete evaporation time for larger diameter droplet is
around 65 sec.
Droplet evaporation time curve of NaCl solution on Aluminum shows that the both droplet takes
maximum time to evaporate completely at a temperature of 225 ⁰C and corresponding time for
smaller diameter droplet is around 45sec. The complete evaporation time for larger diameter
droplet is around 55 sec.
Droplet evaporation time curve of methanol on Aluminum shows that the both droplet takes
maximum time to evaporate completely at a temperature of 225 ⁰C and corresponding time for
smaller diameter droplet is around 20sec. The complete evaporation time for larger diameter
droplet is around 25 sec.
Droplet evaporation time curve of ethanol on Aluminum shows that the both droplet takes
maximum time to evaporate completely at a temperature of 200 ⁰C and corresponding time for
smaller diameter droplet is around 25sec. The complete evaporation time for larger diameter
droplet is around 30 sec.
42
(a)
(b)
Fig. 5.8 Comparison curve of Droplet Evaporation Time of Distill Water, NaCl solution, Methanol, Ethanol on Brass surface
0
20
40
60
80
100
120
140
0 100 200 300 400
Tim
e(s)
Temperature(°C)
Diameter: 2.50 mm Brass:Distill water:Sd
Brass:NaCl solution:Sd
Brass:Ethanol:Sd
Brass:Methanol:Sd
0
20
40
60
80
100
120
140
0 100 200 300 400
Tim
e(s)
Temperature(°C)
Diameter: 2.75 mm Brass:Distill water:Ld
Brass:NaCl solution:Ld
Brass:Ethanol:Ld
Brass:Methanol:Ld
43
Droplet evaporation time curve of water on Brass surface shows that the both droplet takes
maximum time to evaporate completely at a temperature of 175 ⁰C and corresponding time for
smaller diameter is around 110sec. The complete evaporation time for larger diameter droplet is
around 120 sec.
Droplet evaporation time curve of NaCl solution on Brass shows that the both droplet takes
maximum time to evaporate completely at a temperature of 200 ⁰C and corresponding time for
smaller diameter droplet is around 60sec. The complete evaporation time for larger diameter
droplet is around 70 sec.
Droplet evaporation time curve of methanol on Brass shows that the both droplet takes maximum
time to evaporate completely at a temperature of 150 ⁰C and corresponding time for smaller
diameter droplet is around 40sec. The complete evaporation time for larger diameter droplet is
around 45 sec.
Droplet evaporation time curve of ethanol on Brass shows that the both droplet takes maximum
time to evaporate completely at a temperature of 150 ⁰C and corresponding time for smaller
diameter droplet is around 43sec. The complete evaporation time for larger diameter droplet is
around 47sec.
(a)
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature (⁰C)
Diameter: 2.50 mm Copper:dist water sd
Copper:NaCl sol sd
Copper:Ethanol:sd
Copper:Methanol:sd
44
(b)
Fig. 5.9 Comparison curve of Droplet Evaporation Time of Distill Water, NaCl solution, Methanol, Ethanol on Copper surface
Droplet evaporation time curve of water on Copper surface shows that the both droplet takes
maximum time to evaporate completely at a temperature of 200 ⁰C and corresponding time for
smaller diameter is around 70sec. The complete evaporation time for larger diameter droplet is
around 80 sec.
Droplet evaporation time curve of NaCl solution on Copper shows that the both droplet takes
maximum time to evaporate completely at a temperature of 225 ⁰C and corresponding time for
smaller diameter droplet is around 60sec. The complete evaporation time for larger diameter
droplet is around 65 sec.
Droplet evaporation time curve of methanol on Copper shows that the both droplet takes
maximum time to evaporate completely at a temperature of 175 ⁰C and corresponding time for
smaller diameter droplet is around 34sec. The complete evaporation time for larger diameter
droplet is around 40 sec.
Droplet evaporation time curve of ethanol on Copper shows that the both droplet takes maximum
time to evaporate completely at a temperature of 175 ⁰C and corresponding time for smaller
0
20
40
60
80
100
120
0 100 200 300 400
Tim
e (s
)
Temperature( C)
Diameter: 2.75 mm Copper:Dist Water:Ld
Copper:NaCl sol:ld
Copper:Ethanol:Ld
Copper:Methanol:Ld
45
diameter droplet is around 35sec. The complete evaporation time for larger diameter droplet is
around 40 sec.
(a)
(b)
Fig. 5.10 Comparison curve of Droplet Evaporation Time of Distill Water, NaCl solution, Methanol, Ethanol on Mild steel surface
0
10
20
30
40
50
60
70
80
0 100 200 300 400
Tim
e(s)
Temperature(°C)
Diameter: 2.50 mmMild steel:Distill water:Sd
Mild steel:NaCl solution:Sd
Mild steel:Ethanol:Sd
Mild steel:Methanol:Sd
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400
Tim
e(s)
Temperature(°C)
Diameter: 2.75 mmMild steel:Distill water:Ld
Mild steel:NaCl solution:Ld
Mild steel:Ethanol:Ld
Mild steel:Methanol:Ld
46
Droplet evaporation time curve of water on Mild steel surface shows that the both droplet takes
maximum time to evaporate completely at a temperature of 250 ⁰C and corresponding time for
smaller diameter is around 50sec. The complete evaporation time for larger diameter droplet is
around 55 sec.
Droplet evaporation time curve of NaCl solution on Mild steel shows that the both droplet takes
maximum time to evaporate completely at a temperature of 275 ⁰C and corresponding time for
smaller diameter droplet is around 40sec. The complete evaporation time for larger diameter
droplet is around 45 sec.
Droplet evaporation time curve of methanol on Mild steel shows that the both droplet takes
maximum time to evaporate completely at a temperature of 200 ⁰C and corresponding time for
smaller diameter droplet is around 22sec. The complete evaporation time for larger diameter
droplet is around 24 sec.
Droplet evaporation time curve of ethanol on Mild steel shows that the both droplet takes
maximum time to evaporate completely at a temperature of 175 ⁰C and corresponding time for
smaller diameter droplet is around 25 sec. The complete evaporation time for larger diameter
droplet is around 30 sec.
5.2.3 Material variation
We kept the liquid fixed and changed the metal surfaces while plotting Fig. 5.11 to Fig. 5.14. For
each liquid, we have plotted the total droplet vaporization time as a function of surface
temperature of four different metal surfaces. We find a general trend that, vaporization time
required for a specific liquid at a specific temperature is different for different metal surfaces. It
happens due to different values of specific heat, thermal conductivity and density of the metal.
For all cases, according to the model of Henry [17] if specific heat, thermal conductivity and
density of metal is high Leidenfrost point temperature will be low.
From Fig. 5.9 and Fig. 5.10 we observe that Leidenfrost time for methanol on copper surface is
150°C and on mild steel surface is 200°C (as copper has the highest density and thermal
47
conductivity while mild steel has the lowest among these four metals). In this experiment, brass
takes the maximum time, then copper, aluminum and mild steel respectively. The effect of
diameter of liquid droplet is also verified from experimental results and graphs.
(a)
(b)
Fig. 5.11 Comparison of Droplet Evaporation Time of Distilled water on four different surfaces
0
20
40
60
80
100
120
140
0 50 100 150 200 250 300 350 400
Tim
e(s
)
temperature(°C)
Diameter: 2.50mm Distilled water:copper:sd
Dist water:Mild steel:sd
Dis water:Aluminum:sd
Dist water:Brass:sd
0
20
40
60
80
100
120
140
0 50 100 150 200 250 300 350 400
Tim
e(s
)
Temperature(℃)
Diameter: 2.75mm Dist water:copper:LD
Distwater:Mild steel:Ld
Dist water:Aluminum:Ld
Dist water:Brass:Ld
48
For both water droplets (small and large diameter) Figure 5.11(a-b) shows that, water droplet
evaporation time is maximum for Brass surface. Droplet evaporation time decreases in the order
of copper, aluminum and Mild steel. According to the model of Henry [17] if specific heat,
thermal conductivity and density of metal is high Leidenfrost point temperature will be low.
From Fig. 5.11 we observe that Leidenfrost temperature for distilled water on copper surface is
150°C and mild steel surface is 200°C (as copper has the highest density and thermal
conductivity while mild steel has the lowest among these four metals).
(a)
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(℃)
Diameter: 2.50mmNaCl sol:copper:sd
NaCl sol:Mild steel:sd
NaCl sol:Aluminum:sd
NaCl sol:Brass:Sd
49
(b)
Fig. 5.12 Comparison of Droplet Evaporation Time of NaCl solution on four different metal surfaces
For both NaCl solution droplets (small and large diameter) Figure 5.12(a-b) shows that, NaCl
solution droplet evaporation time is maximum for Brass surface. Droplet evaporation time
decreases in the order of copper, aluminum and Mild steel. From Figure 5.12(a-b) we observe
that Leidenfrost temperature for NaCl solution on copper surface is 200°C and mild steel surface
is 275°C (as copper has the highest density and thermal conductivity while mild steel has the
lowest among these four metals).
0
20
40
60
80
100
120
140
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(℃)
Diameter: 2.75mmNaCl sol:copper:Ld
NaCl:Mild steel:Ld
NaCl sol:Aluminum:Ld
NaCl sol:Brass:Ld
50
(a)
(b)
Fig. 5.13 Comparison of Droplet Evaporation Time of Methanol on four different surfaces
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(℃)
Diameter: 2.50mm
Methanol:Copper:sd
Methanol:Mild steel:sd
Methanol:Aluminum:sd
Methanol:Brass:sd
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(℃)
Diameter: 2.75mmMethanol:Copper:Ld
Methanol:Mild steel:Ld
Methanol:Aluminum:Ld
Methanol:Brass:Ld
51
For both Methanol droplets (small and large diameter) Figure 5.13(a-b) shows that, Methanol
droplet evaporation time is maximum for Brass surface. Droplet evaporation time decreases in
the order of copper, aluminum and Mild steel. From Figure 5.13(a-b) we observe that
Leidenfrost temperature for methanol on copper surface is 175°C and mild steel surface is 200°C
(as copper has the highest density and thermal conductivity while mild steel has the lowest
among these four metals).
(a)
(b)
Fig. 5.14 Comparison of Droplet Evaporation Time of Ethanol on four different surfaces
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(℃)
Diameter: 2.50 mmEthanol:copper:Sd
Ethanol:Mild steel:sd
Ethanol:Aluminum:sd
Ethanol:Brass:sd
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(℃)
Diameter: 2.75mmEthanol:copper:Ld
Ethanol:Mild steel:Ld
Ethanol:Aluminum:Ld
Ethanol:Brass:Ld
52
For both Ethanol droplets (small and large diameter) Figure 5.14(a-b) shows that, Ethanol
droplet evaporation time is maximum for Brass surface. Droplet evaporation time decreases for
copper, aluminum and Mild steel respectively. From Figure 5.14(a-b) we observe that both for
copper and mild steel surfaces Leidenfrost temperature of ethanol is identical and its value is
175⁰C.
Leidenfrost temperature values were obtain for water, NaCl solution, methanol and ethanol on
aluminum, copper, brass and mild steel surface. The Leidenfrost temperature is nearly identical
for aluminum, brass and mild steel surfaces but is slightly higher for the copper surface
[Bernardin and Mudawar, 1999]. The higher Leidenfrost value of copper surface is speculated to
be the result of surface roughening which accompanied large amounts of surface oxidation
during heating. The higher drop evaporation value of copper surface is speculated to be the result
of higher conductivity than other metal. Higher conductivity means higher heat transfer through
the metal. It means, when liquid touch the metal, large amount vapor will produce due to higher
heat transfer rate and surface is completely covered by a vapor blanket and then heat transfer
from the surface to the liquid occurs by conduction through vapor. Droplet was supported by the
vapor film slowly boil away.
5.3 Experimental boiling curve
Boiling curve of water, methanol and ethanol on four metal surfaces (aluminum, brass, copper,
mild Steel) as a function of test surface temperature are presented in Figure 5.15 to Figure 5.17.
Michiyoshi, Makino [16] have simplified the time averaged heat flux based on the theory of
Baumeister et. al [4]. Combining heat balance and time averaged heat transfer coefficient, time
averaged heat flux can be determined. According to Baumeister et al. [4] heat transfer coefficient
is
α= 1.1×1.5(𝐾𝑣
3𝜌𝑙𝜌𝑣 ∗𝑔
µ𝑣𝑉13∆𝑇
)1
4 (5.22)
Here, h*= 𝑔 [1 + (7
20)𝐶𝑝(
∆𝑇
𝑔)]−3
53
From the heat balance the following equation can be obtained,
𝜌𝑙 𝑔 𝑑𝑉
𝑑𝑡= α (V) A (V) ∆T (5.23)
The resulting equation for the time averaged heat flux is
q= 1.5
τ×
ρl hg
1.813× 3 × Vo
1
3 (5.24)
By inserting complete droplet evaporation time in Eqn (5.24), heat flux is determined and by
plotting this heat flux with respect to surface temperature boiling curve is obtained.
5.3.1 Experimental Boiling curve of water
By using Eqn (5.24), heat flux during sessile drop evaporation has been estimated as shown in the
Figure 5.15(a-b) (for all the experimental conditions, the working pressure remain constant as of
atmospheric pressure. The experiments were started from 60⁰C of test surface for all the cases and the
experiment were conducted up to 400⁰C. Depends on the wall superheat (=surface temperature˗ liquid
saturation temperature) different mode of heat transfer could be obtained.
(a)
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
0 50 100 150 200 250 300 350 400 450
Hea
t fl
ux
(W/m
2)
Surface Temperature (⁰C)
Boiling curve of distilled water(diameter: 2.50mm)
ALUMINUM
BRASS
COPPER
MILD STEEL
54
(b)
Fig. 5.15 Boiling curve of Water on different material surfaces
Immediate after the 60⁰C of test surface temperature and before 100⁰C, convection boiling can
be predicted as the dominating mode of heat transfer for water evaporation as shown in Figure
5.15(a-b). Above 100⁰C of test surface temperature, nucleate boiling becomes the dominating
mode and consequently sharp increase of heat flux is observed for all the materials (aluminum,
brass, copper and mild steel). At around 105-130⁰C (5-30⁰C of wall superheat for water) the heat
flux reaches the maximum value (it is defined as critical heat flux).
In the transition boiling region all curves deviates from each other. This can be explained by
considering thermal diffusivity. We consider the two cases mild steel and copper as the former
has lowest thermal diffusivity and the later one has the maximum thermal diffusivity.
The film boiling region starts at the Leidenfrost point temperature; at this point heat flux is
minimum. Above this temperature heat flux increases as radiation heat transfer gradually
dominates, this is true for all liquids for different material surface. If the thermal diffusivity of
1.E+04
1.E+05
1.E+06
1.E+07
0 100 200 300 400
Hea
t flu
x (W
/m2)
Temperature(⁰C)
Boiling curve of water (diameter: 2.75 mm)
Aluminum
Brass
Copper
Mild Steel
55
the material is high Leidenfrost point temperature TL will start at low ∆T. Leidenfrost point
temperature depends on material. The higher the thermal diffusivity of material lower the
Leidenfrost temperature which has already established in our experimental graph. We have
noticed in our experimental graph that the Leidenfrost point temperature of higher thermal
diffusivity material (copper, aluminum) is lower and ranging between 200-225⁰C, and for lower
thermal diffusivity material (mild steel) is higher (>225⁰C).
5.3.2 Experimental Boiling curve of methanol
(a)
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
0 50 100 150 200 250 300 350 400 450
Hea
t fl
ux
(W/m
2)
Surface Temperature (⁰C)
Boiling curve of methanol(diameter: 2.50mm)
ALUMINUM
BRASS
COPPER
MILD STEEL
56
(b)
Fig.5.16 Boiling curve of Methanol on different material surfaces
In the nucleate boiling region Figure 5.16(a-b) we find that the curve linearly increases for all the
material and they almost merges (not perfectly) to each other. The maximum heat flux is
obtained at a temperature above 25-40⁰C above the saturation temperature. Material with higher
thermal diffusivity has critical heat flux (CHF) at lower surface temperature (for copper ~90⁰C),
similarly material with lower thermal diffusivity has CHF at higher surface temperature (for
Mild steel ~100⁰C). CHF is maximum for mild steel surface and the numerical value is
approximately 15.26 MW/𝑚2.
In the transition boiling region all curves deviates from each other. This can be explained by
considering thermal diffusivity. We consider the two cases mild steel and copper as the former
has lowest thermal diffusivity and the later one has the maximum thermal diffusivity. The curve
for mild steel deviates at higher ∆T ranges than for copper plate.
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
0 100 200 300 400
Hea
t flu
x(W
/m2
)
Temperature(⁰C)
Boiling Curve of Methanol(diameter: 2.75 mm)
Aluminum
Brass
Copper
Mild Steel
57
The film boiling region starts at the point where heat flux is minimum is known as Leidenfrost
point temperature. Above this temperature heat flux also increases as radiation heat flux
gradually dominates, this is true for all liquids for different material. The higher the thermal
diffusivity the lower ∆T at TL . We know that Leidenfrost point temperature depends on material
surface. The higher the thermal diffusivity of material lower the Leidenfrost temperature which
has been also established in our experimental graph. We noticed in our experimental graph that
Leidenfrost point temperature of higher thermal diffusivity material (such as copper) is lower
and ranging between 175-200⁰C, and for lower thermal diffusivity material (such as mild steel)
is higher (200-225⁰C).
5.3.3 Experimental Boiling curve of ethanol
(a)
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
0 50 100 150 200 250 300 350 400 450
Hea
t fl
ux
(W/m
2)
Surface Temperature (⁰C)
Boiling curve of ethanol(diameter : 2.50 mm)ALUMINUM
BRASS
COPPER
MILD STEEL
58
(b)
Fig. 5.17 Boiling curve of Ethanol on different material surfaces
Starting at the nucleate boiling region we find that the curve Figure 5.17(a-b) linearly increases
for all the material and they almost merges to each other. The maximum heat flux (CHF) is
obtained at a temperature above 20-30⁰C above the saturation temperature. CHF is maximum for
mild steel surface and the numerical value is approximately 10.55 MW/𝑚2.
In the transition boiling region all curves deviates from each other as happen in the previous
cases. This can be explained by considering thermal diffusivity. Considering two cases mild steel
and copper as the former has lowest thermal diffusivity and the later one has the maximum
thermal diffusivity. The curve for mild steel deviates at higher ∆T ranges than for copper plate.
The film boiling region starts at the Leidenfrost point temperature; at this point heat flux is
minimum. Above this temperature heat flux also increases as radiation heat flux gradually
dominates, this is true for all liquids for different material surface. The higher the thermal
diffusivity the lower ∆T at TL .Leidenfrost point temperature depends on material. The higher the
thermal diffusivity of material lowers the Leidenfrost temperature which has been also
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
0 100 200 300 400
Hea
tflu
x(W
/m2
)
Temperature(⁰C)
Boiling Curve of Ethanol(diameter: 2.75 mm)
Aluminum
Brass
Copper
Mild Steel
59
established in our experimental graph. We noticed in our graph that Leidenfrost point
temperature of higher thermal diffusivity material is lower and ranging between175-200⁰C, and
for lower thermal diffusivity material is higher (200-225⁰C).
5.4 Inverse boiling curve
In this experiment we have plotted evaporation time versus metal surface temperature thus we
have obtained curves (Figure 5.7 to Figure 5.10) which is inverse of boiling curve may be called
as inverse boiling curve. A typical inverse boiling curve Figure 5.18 is classified into four
regimes film evaporation, nucleate boiling, transition boiling and film boiling.
Fig. 5.18 Schematic of a typical Inverse boiling curve and boiling curve of a Liquid
The region a-b is known as film evaporation region which is below the saturation temperature of
liquid. As point „b’ is the saturation temperature, below this temperature there is no boiling,
Tim
e / H
eat
flux
Surface temperature( C)
Typical inverse boiling and boiling curve
Inverse boiling
curve
Boiling curvea
b
c
d
60
liquid droplet evaporates at atmospheric pressure. Gradually less time required for boiling when
the temperature approaches to the saturation point. This is due to the increase in heat flux as
temperature increases. The trend of the graph in this region is downward as shown in Figure 18.
The region b-c is known as nucleate boiling region. This region starts from saturation point up to
the point at which heat flux is maximum. As the surface temperature is increasing above
saturation point heat flux also increases due to the combined effect of liquid entrainment and
evaporation. In this region evaporation time gradually decreases and reaches a minimum value at
CHF (critical heat flux). This region is the most desirable region in boiling phenomenon.
The region c-d is transition boiling region also known as unstable film boiling. In this region heat
flux decreases for further increase in temperature due to the large fraction of heater surface is
covered by vapor film. As heat flux in this region in decreasing order so evaporation time must
be in increasing order which has been already established by experimental data.
The region above d is known as film boiling region. The point „d‟ is known as Leidenfrost point
at which heat flux is minimum. At this point metal surface is completely covered by vapor film.
As heat flux at this point is minimum so evaporation time at this point is maximum which is
undesirable. So during boiling this point must be avoided. Above point d heat flux is increasing
as the surface temperature is increasing. The heat transfer rate increases with increasing excess
temperature above saturation point as a result of heat transfer from the heated surface to the
liquid through the vapor film by radiation heat transfer which become significant at higher
temperature.
In our experimental curve (which is similar to Figure 5.18), we notice that from saturation
temperature up to temperature at which heat flux is maximum, the required vaporization
decreases. This happens due to the less vapor bubble or absence of vapor film under the droplet.
So, the total heat is transferred to the droplet from the hot metal surface by conduction. As the
temperature increases, a vapor layer is formed beneath the droplet and increases in thickness and
consequently the vaporization time also increases. This happens up to the Leidenfrost point. At
Leidenfrost point, the film thickness reaches to its maximum value and this lessens the
conductive heat transfer rate to its minimum value. So the time required for vaporization reaches
to its maximum value. After this point radiation heat transfer starts to dominate which gradually
decrease the vaporization time as heat flux increases.
61
5.5 Engineering Correlation of experimental data
The theoretical development in reference [2] is purely analytical. It does not require any
experimental data (except physical properties) in the prediction for droplet evaporation time. Due
to the complicated iterative computations to obtain the correct values for droplet evaporation
time, it is necessary in engineering calculations relatively simple equation which would imply
the correct functional dependence upon variables and allow a prediction for droplet evaporation
time without any time consuming iteration. In order to obtain such an empirical correlation of the
experimental data, a functional equation between the dependent variables and the independent
variables must be obtained.
From the theoretical development, we find in reference [2] that heat is transferred from the plate
to the droplet by conduction and radiation, neither one of which may be neglected in general.
The evaporation rate per unit area for a spherical droplet is on the order of L 𝑟𝑜/𝜏 and this
quantity is equal to the sum of the heat transferred by conduction and radiation divided by .
Functional arguments are developed in detail in reference [18], but the resulting equation is
𝜌𝑙𝑟𝑜
𝜏 = 1C [
𝑘∆𝑇𝑟𝑜𝑔𝜌𝑣(𝜌𝑙−𝜌𝑣)
µ𝑣𝜆 ′]
1
2 + 2C [𝜍𝜖𝑝 𝑇𝑝
4−𝑇𝑠4
𝜆 ′] ……………………….. (25)
Where, 1C and 2C are constants to be evaluated from the experimental data.
The first and second part of the above equation represents the conduction and radiation heat
transfer respectively.
62
5.5.1 Experimental Correlation for Aluminum
(a)
(b)
Fig. 5.19 Empirical correlation of total evaporation time for Aluminum surface
0 0.02 0.04 0.06 0.08 0.1 0.120
0.02
0.04
0.06
0.08
0.1
0.12
Obseved value
Calc
ula
ted v
alu
e
Correlation curve for Aluminum(Diameter 2.50 mm)
water
methanol
ethanol
equation
0 0.02 0.04 0.06 0.08 0.1 0.120
0.02
0.04
0.06
0.08
0.1
0.12
Obseved value
Calc
ula
ted v
alu
e
Correlation curve for Aluminum(diameter: 2.75 mm)
water
methanol
ethanol
equation
63
In both cases, twenty three data points representing the full range of experimental condition were
selected for aluminum and used to calculate C1 and C2 by least squares fitting. The resulting
correlation for four liquids on aluminum is-
𝜌 𝑙𝑟𝑜
𝜏 = 0.0206[
𝑘∆𝑇𝑟𝑜𝑔𝜌𝑣(𝜌 𝑙−𝜌𝑣)
µ𝑣𝜆 ′]
1
2 +1215 [𝜍𝜖𝑝 𝑇𝑝
4−𝑇𝑠4
𝜆 ′] ………………… (26)
5.5.2 Experimental Correlation for Brass
(a)
(b)
Fig. 5.20 Empirical correlation of total evaporation time for Brass surface
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Obseved value
Cal
cula
ted
valu
e
Correlation curve for Brass(Diameter 2.50 mm)
water
methanol
ethanol
equation
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Obseved value
Cal
cula
ted
valu
e
Correlation curve for Brass(Diameter 2.75 mm)
water
methanol
ethanol
equation
64
In both cases, twenty seven data points representing the full range of experimental condition
were selected for brass and used to calculate C1 and C2 by least squares fitting (Fig. 5.13). The
resulting correlation for four liquids on brass is-
𝜌 𝑙𝑟𝑜
𝜏 = 0.0169[
𝑘∆𝑇𝑟𝑜𝑔𝜌𝑣(𝜌 𝑙−𝜌𝑣)
µ𝑣𝜆 ′]
1
2 + 500[𝜍𝜖𝑝 𝑇𝑝
4−𝑇𝑠4
𝜆 ′] ………………… (27)
5.5.2 Experimental Correlation for Copper
(a)
(b)
Fig. 5.21 Empirical correlation of total evaporation time for Copper surface
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Obseved value
Cal
cula
ted
valu
e
Correlation curve for Copper( Diameter 2.50 mm)
water
methanol
ethanol
equation
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Obseved value
Cal
cula
ted
valu
e
Correlation curve for Copper(Diameter: 2.75 mm)
water
methanol
ethanol
equation
65
Twenty five data points representing the full range of experimental condition were selected for
copper and used to calculate C1 and C2 by least squares fitting (Fig. 5.12). The resulting
correlation for four liquids on copper is-
𝜌 𝑙𝑟𝑜
𝜏 = 0.0160[
𝑘∆𝑇𝑟𝑜𝑔𝜌𝑣(𝜌 𝑙−𝜌𝑣)
µ𝑣𝜆 ′]
1
2 + 1170[𝜍𝜖𝑝 𝑇𝑝
4−𝑇𝑠4
𝜆 ′] ………………… (28)
5.5.3 Experimental Correlation for Mild Steel
(a)
(b)
Fig. 5.22 Empirical correlation of total evaporation time for Mild steel surface
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Obseved value
Calc
ula
ted v
alu
e
Correlation curve for Mild Steel(Diameter 2.50 mm)
water
methanol
ethanol
equation
0 0.02 0.04 0.06 0.08 0.1 0.120
0.02
0.04
0.06
0.08
0.1
0.12
Obseved value
Cal
cula
ted
valu
e
Correlation curve for Mild Steel(diameter: 2.75 mm)
water
methanol
ethanol
equation
66
In both cases, twenty three data points representing the full range of experimental condition were
selected for mild steel and used to calculate C1 and C2 by least squares fitting. The resulting
correlation for four liquids on mild steel is-
𝜌 𝑙𝑟𝑜
𝜏 = 0.0158 [
𝑘∆𝑇𝑟𝑜𝑔𝜌𝑣(𝜌 𝑙−𝜌𝑣)
µ𝑣𝜆 ′]
1
2 + 1440[𝜍𝜖𝑝 𝑇𝑝
4−𝑇𝑠4
𝜆 ′] ………………… (29)
5.6 Comparison of Theoretical and Experimental Result
Finally a comparison has been made for surface temperature above the Leidenfrost point
between data points obtained from correlation and data points obtained from experiment. This
comparison has been shown by plotting curve (Fig. 5.14 and Fig. 5.15). Correlated Equations
(Eqn 26 to Eqn 29) give very accurate time ranging maximum error limit 20 percents, in most of
the cases less than 10 percents.
5.6.1 Comparison curve for distilled water on different metal surfaces
(a)
0
10
20
30
40
50
60
70
80
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.50 mmCorrelation: Al-dist water
Experimental: Al-dist water
67
(b)
Fig. 5.23 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of water
on Aluminum surface
(a)
0
10
20
30
40
50
60
70
80
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.75 mmCorrelation: Al-dist water
Experimental: Al-dist water
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.50 mmCorrelation: Brass -Dist waterExperimental: Brass-Dist water
68
(b)
Fig. 5.24 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of water
on Brass surface
(a)
0
10
20
30
40
50
60
70
80
90
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.75 mmCorrelation: Brass -Dist waterExperimental: Brass-Dist water
0
10
20
30
40
50
60
70
80
90
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.50 mmCorrelation: Copper -Dist waterExperimental: Copper-Dist water
69
(b)
Fig. 5.25 Comparison graph for total evaporation time (τ) with surface temperature (𝑇𝑠) of Water
on Copper surface
(a)
0102030405060708090
100
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.75 mmCorrelation: Copper -Dist waterExperimental: Copper-Dist water
0
10
20
30
40
50
60
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.50 mmCorrelation: Steel -Dist water
Experimental: Steel-Dist water
70
(b)
Fig. 5.26 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of water
on Mild steel surface
Comparison curves above the Leidenfrost point for distilled water on four different material
surfaces (Aluminum, Brass, Copper and Mild steel) have shown in Figure 5.23 to Figure 5.26.
Above the Leidenfrost point experimental data and correlated data are well matched as shown in
Figure 5.23 to Figure 5.26. For Aluminum and Brass, variation between experimental data and
correlated data are 10% in both cases above the transition region. For Copper, variation between
experimental data and correlated data are 15% in both cases above the transition region. But for
Mild steel variation is higher than other three metal surfaces and it is within 20%.
0
10
20
30
40
50
60
70
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.75 mmCorrelation: Steel -Dist water
Experimental: Steel-Dist water
71
5.6.2 Comparison curve for methanol on different metal surfaces
(a)
(b)
Fig. 5.27 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
methanol on Aluminum surface
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(°C)
Diameter: 2.50 mmCorrelation: Al-methanol
Experimental: Al-methanol
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(°C)
Diameter: 2.75 mmCorrelation: Al-methanolExperimental: Al-methanol
72
(a)
(b)
Fig. 5.28 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
methanol on Brass surface
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.50 mmCorrelation: Brass -MethanolExperimental: Brass -Methanol
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.75 mmCorrelation: Brass -Methanol
Experimental: Brass -Methanol
73
(a)
(b)
Fig. 5.29 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
Methanol on Copper surface
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.50 mmCorrelation: Copper -Methanol
Experimental: Copper -Methanol
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.75 mm
Correlation: Copper -MethanolExperimental: Copper -Methanol
74
(a)
(b)
Fig. 5.30 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of methanol on
Mild steel surface
0
5
10
15
20
25
30
35
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.50 mmCorrelation: Steel - Methanol
Experimental: Steel -Methanol
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.75 mmCorrelation: Steel - Methanol
Experimental: Steel -Methanol
75
Comparison curves above the Leidenfrost point for methanol on four different material surfaces
(Aluminum, Brass, Copper and Mild steel) have shown in Figure 5.27 to Figure 5.30. Above the
Leidenfrost point experimental data and correlated data are well matched as shown in Figure
5.27 to Figure 5.30. For Aluminum and Brass, variation between experimental data and
correlated data are within10% in both cases above the transition region which is similar to
distilled water. For Copper and Mild steel, variation between experimental data and correlated
data are 15% in both cases above the transition region. Above Leidenfrost point as temperature
increases accuracy for both Copper and Mild steel increase.
5.6.3 Comparison curve for ethanol on different metal surfaces
(a)
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(°C)
Diameter: 2.50 mmCorrelation: Al-ethanol
Experimental: Al-Ethanol
76
(b)
Fig. 5.31 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
ethanol on Aluminum surface
(a)
0
10
20
30
40
50
60
70
80
0 50 100 150 200 250 300 350 400 450
Tim
e(s)
Temperature(°C)
Diameter: 2.75 mmCorrelation: Al-ethanolExperimental: Al-Ethanol
0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.50 mmCorrelation: Brass -EthanolExperimental: Brass -Ethanol
77
(b)
Fig. 5.32 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
ethanol on Brass surface.
(a)
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.75 mmCorrelation: Brass -Ethanol
Experimental: Brass -Ethanol
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.50 mmCorrelation: Copper -EthanolExperimental: Copper -Ethanol
78
(b)
Fig. 5.33 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
ethanol on Copper surface.
(a)
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(ᵒC)
Diameter: 2.75 mmCorrelation: Copper -EthanolExperimental: Copper -Ethanol
0
5
10
15
20
25
30
35
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.50 mmCorrelation: Steel -Ethanol
Experimental: Steel -Ethanol
79
(b)
Fig. 5.34 Comparison graph of total evaporation time (τ) with surface temperature (𝑇𝑠) of
ethanol on Mild steel surface.
Comparison curves above the Leidenfrost point for ethanol on four different material surfaces
(Aluminum, Brass, Copper and Mild steel) have shown in Figure 5.31 to Figure 5.34. Above the
Leidenfrost point experimental data and correlated data are well matched as shown in Figure
5.31 to Figure 5.34. For Aluminum, Brass and Copper, variation between experimental data and
correlated data are within 5% in all cases above the transition region. For Mild steel, variation
between experimental data and correlated data are within 10% above the transition region.
Above Leidenfrost point as temperature increases accuracy for all metal surfaces increases.
0
5
10
15
20
25
30
35
0 50 100 150 200 250 300 350 400
Tim
e(s)
Temperature(⁰C)
Diameter: 2.75 mmCorrelation: Steel -Ethanol
Experimental: Steel -Ethanol
80
CHAPTER
CONCLUSIONS 6
6.1 Conclusions
Leidenfrost phenomenon is a very complex matter due to involvement of numerous numbers of
physical parameters. This makes the analysis of the most intrinsic details difficult. Here a model
has been proposed and verified with experimental results. From the experimental results, several
key conclusions concerning the influential parameters (Leidenfrost Temperature) can be drawn.
The fundamental understandings are summarized below:
1. The major contribution to the heat transfer is the conductive and radiative mode.
Conduction heat transfer mode is dominant below the Leidenfrost point and radiation
heat transfer becomes dominant above the Leidenfrost point.
2. Radiation heat transfer has been successfully included in the theoretical model and using
an iterative computer program, radiation effects on the droplet evaporation time has been
incorporated.
3. A stable layer of vapor has been considered beneath the droplet and the weight of the
droplet is balanced with the pressure beneath the droplet.
4. Size of droplet has been found to have no influence on the Leidenfrost temperature.
5. Data are well correlated, allowing prediction of total vaporization time to within ±20
percent. The comparison has been made between actual graph obtain in the experiments
and those obtained by correlation.
6. The heat flux is inversely proportional to the evaporation time. Heat flux increases after
the Leidenfrost point so evaporation time decreases.
81
7. The experimental curve (evaporation time versus temperature) curve is opposite to
boiling curve as heat flux is inversely related to the evaporation time.
8. Droplet size has been found to influence the Leidenfrost time. The larger the droplet size;
the higher is the total evaporation time.
6.2 Further Work
In this science era, faster growing technology promotes mankind to augment their standard of
living. The prerequisite for this technological development is the research and development in
the sector of science and technology. Human beings are eagerly waiting for the latest invention
which leads to the consequence that research and development are an endless job. In this context,
the research work delineated in this dissertation may serve as a primary foundation for some of
the phenomena which will lead to some future study :
1. The Leidenfrost phenomenon for two-component solutions.
2. The Leidenfrost phenomenon for cryogenic fluids.
3. The Leidenfrost phenomenon for various liquids on composite material.
4. The role played by heating surface conditions at the Leidenfrost point.
5.3 Recommendations
1. Instead of the four fluids used in the experiment other fluids can be chosen having
larger specific heat.
2. Detailed theoretical and experimental studies are needed to accurately predict the
behavior of the Leidenfrost Temperature.
82
3. During the evaporation period droplet always vibrated a little at their fundamental
frequency. In order to increase the accuracy this oscillation phenomenon should take in to
account.
4. Instead of 25˚C, temperature should be increased by 5˚C during the experiment for
accurate prediction of droplet evaporation time.
5. If the surface roughness of the metal is changed, detail behavior of the Leidenfrost
Temperature will be obtained.
83
REFERENCES
[1] J. G. Leidenfrost, De Aquae Communis Nonnullis Qualitatibus Tractatus, (A Tract About
Some Qualities of Common Water), Duisburg on Rhine (1756)
[2] B.S. Gottfried, C.J. Lee and K.J. Bell, The leidenfrost phenomenon: film boiling of liquid
droplets on a flat plate, International Journal of Heat and Mass Transfer, Volume 9, Issue 11,
November 1966, pp. 1167-1188
[3] Victor Starov and Khellil Sefiane, On evaporation rate and interfacial temperature of volatile
sessile drops, Colloids and Surfaces A: Physicochemical and Engineering Aspects, Volume 333,
Issues 1-3, 5 February 2009, pp.170-174
[4] Baumeister, K. J., Hamill, T. D., Schoessow, G. J., Schwartz, F. L., Film boiling heat transfer
to water drops on a flat plate, Jan 1, 1965 ,Technical report, NASA-TM-X-52103
[5] D. Chatzikyriakou, S.P. Walker, G.F. Hewitt, C. Narayanan and D. Lakehal, Comparison of
measured and modeled droplet–hot wall interactions, Applied Thermal Engineering, Volume 29,
Issue 7, May 2009, pp. 1398-1405
[6] L. H. J. Wachters, H. Bonne and H. J. van Nouhuis, The heat transfer from a hot horizontal
plate to sessile water drops in the spheroidal state, Chemical Engineering Science, Volume 21,
Issue 10, October 1966, pp. 923-936
[7] Heng Xie, Zhiwei Zhou, A model for droplet evaporation near Leidenfrost point, Technical
Note, International Journal of Heat and Mass Transfer, Volume 50, Issues 25-26, December
2007, pp. 5328-5333
[8] Elyssa F. Crafton, W. Z. Black, “Heat transfer and evaporation rates of small liquid droplets
on heated horizontal surfaces”, International Journal of Heat and Mass Transfer, Volume 47,
Issues 6-7, March 2004, pp. 1187-1200
84
[9] T. K. Nguyen, C. T. Avedisian, Numerical solution for film evaporation of a spherical liquid
droplet on an isothermal and adiabatic surface, International Journal of Heat and Mass Transfer,
Volume 30, Issue 7, July 1987, pp. 1497-1509
[10] K.E. Nicholds, Leidenfrost, Cryogenics, Volume 10, Issue 1, February 1970, pp. 45-47
[11] George S. Emmerson, The effect of pressure and surface material on the leidenfrost point of
discrete drops of water, International Journal of Heat and Mass Transfer, Volume 18, Issue 3,
March 1975, pp. 381-386
[12] J. Kistemaker, The spheroidal state of a waterdrop: The leidenfrost phenomenon, Physica,
Volume 29, Issue 2, February 1963, pp. 96-104
[13] Shi-Chune Yao, and Kang Yuan Cai, The dynamics and leidenfrost temperature of drops
impacting on a hot surface at small angles, Experimental Thermal and Fluid Science, Volume 1,
Issue 4, October 1988, pp. 363-371
[14] Niroh Nagai, Shigefumi Nishio, Leidenfrost temperature on an extremely smooth surface,
Experimental Thermal and Fluid Science, Volume 12, Issue 3, April 1996, pp. 373-379
[15] J. P. Holman, Heat Transfer, 8th
Edition, McGraw-Hill, INC., International Edition 1997,
Chapter 8, pp. 405 -413
[16] I. Michiyoshi, K. Makino, “Heat transfer characteristics of evaporation of liquid droplet on
heated surfaces”, Int. J. Heat Mass Transfer, Vol. 21, pp 605-613
[17] Henry, R. E. [1974], “A Correlation for the Minimum Film Boiling Temperature”, Chem.
Eng. Prog. Symp. Ser 70, 81–90.
[18] C. J. Lee, “A Theoretical and. Experimental Investigation of the Leidenfrost Phenomenon
for Small. Droplets.”Ph.D. Thesis, Oklahoma State University (1965).
[19] Hosler, E. R., and Westwater, J. W., "Film Boiling on a Horizontal Plate," ARS J., April
1962, pp. 553-558.
[20] Wachters, L. H. J., H. Bonne, and H. J. van Nouhuis, Chem. Eng. Sci., 21, 923-936 (1966).
85
APPENDICES
86
APPENDIX
SATURATION POROPERTIES A
OF LIQUID
A.1 Saturation Properties of Methanol
Chemical formula CH3OH
Molecular weight 32.00
Critical temperature 513.15 K
Critical Pressure 7950 kPa
Critical density 275 kg/m3
Table A.1 Saturation properties of methanol
87
A.2 Saturation Properties of Ethanol
Chemical formula CH3CH2OH
Molecular weight 46.1
Critical temperature 516.25K
Critical Pressure 6390 kPa
Critical density 280 kg/m3
Table A.2 Saturation properties of ethanol
88
A.3 Saturation Properties of Water
Chemical formula H2O
Molecular weight 18.0156
Critical temperature 647.3 K
Critical Pressure 22,129 kPa
Critical density 351kg/m3
Table A.3 Saturation properties of water
89
APPENDIX
EXPERIMENTAL DATA B
Table B.1 Evaporation time of Distilled water on Aluminum Surface
Temperature(⁰C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 73.24 73.05 71.10 72.46 108.2 107.10 110.20 108.50
80 69.21 68.94 67.52 68.56 75.12 77.31 78.22 76.88
90 43.42 44.23 45.12 44.26 58.19 59.22 59.39 58.93
100 17.85 16.09 17.56 17.17 17.12 18.97 18.25 18.11
125 0.47 0.47 0.45 0.46 1.37 1.30 1.25 1.31
150 1.57 1.90 1.65 1.71 2.11 2.36 2.21 2.23
175 22.40 23.50 22.9 22.93 26.32 25.38 25.91 25.87
200 55.89 56.70 56.30 56.29 65.84 67.21 66.81 66.62
225 55.81 55.10 55.09 55.33 67.52 67.19 67.92 67.54
250 51.39 47.89 53.95 51.08 58.12 63.29 60.22 60.54
275 46.99 47.19 45.64 46.61 56.21 59.99 60.19 58.80
300 42.29 41.49 37.54 40.44 49.87 51.58 52.81 51.42
325 38.12 35.23 36.45 36.6 44.52 48.15 43.25 45.31
350 32.99 31.59 34.69 33.09 42.91 40.92 39.86 41.23
375 29.98 31.01 32.80 31.26 41.12 37.92 38.82 39.29
400 30.04 29.14 28.34 29.17 36.21 34.06 33.82 34.70
t1, t2, t3= Droplet evaporation time
𝑡𝑎𝑣𝑔 = Average value of t1, t2 and t3
90
Table B.2 Evaporation time of NaCl Solution on Aluminum Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 108.60 104.90 107.10 106.86 132.19 128.98 129.54 130.24
80 69.23 70.64 70.21 70.03 86.21 84.34 90.61 87.05
90 45.81 51.23 50.19 49.08 61.42 58.23 57.19 58.95
100 27.98 28.86 29.68 28.84 33.78 34.91 34.31 34.33
125 0.51 0.48 0.42 0.47 0.50 0.47 0.44 0.47
150 0.75 0.97 0.61 0.78 1.22 1.07 1.1 1.13
175 23.05 22.89 22.95 22.96 23.21 24.12 21.84 23.06
200 40.15 40.56 40.29 40.33 50.20 48.49 52.55 50.41
225 47.59 48.69 46.15 47.48 54.22 54.58 55.19 54.66
250 44.98 46.89 45.69 45.85 52.11 54.51 51.18 52.60
275 42.12 43.01 42.44 42.52 47.58 51.19 52.22 50.33
300 36.59 38.10 39.99 38.23 40.11 43.59 44.95 42.88
325 35.95 36.86 35.42 36.08 38.84 39.55 42.11 40.17
350 31.18 32.51 29.45 31.05 33.94 37.86 36.84 36.21
375 30.05 28.22 28.5 28.92 29.84 31.14 30.99 30.65
400 22.98 24.54 25.83 24.45 25.87 29.95 26.88 27.57
Table B.3 Evaporation time of Ethanol on Aluminum Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 7.37 7.83 6.92 7.37 9.39 9.83 8.83 9.35
80 4.31 4.19 4.83 4.44 5.73 5.23 4.82 5.26
90 1.65 1.70 1.43 1.59 1.92 1.53 2.21 1.89
100 0.34 0.37 0.35 0.35 0.56 0.47 0.43 0.49
125 0.84 0.85 0.82 0.84 1.37 1.65 1.51 1.51
150 2.52 2.48 2.51 2.50 2.49 2.72 2.61 2.61
175 11.65 12.47 12.15 12.09 12.78 12.31 12.56 12.55
200 24.33 25.00 25.19 24.84 28.12 29.02 28.93 28.69
225 22.39 22.52 22.19 22.37 26.45 26.86 25.23 26.18
250 22.3 18.55 19.45 20.10 22.12 23.54 23.82 23.16
275 15.65 17.10 18.12 16.96 21.01 20.86 21.54 21.14
300 13.99 15.11 14.25 14.45 17.88 15.59 18.11 17.19
325 13.15 14.21 13.24 13.53 16.11 15.21 14.55 15.29
350 13.21 9.41 12.55 11.72 14.20 12.55 13.12 13.29
375 9.55 10.52 10.11 10.06 12.31 10.80 10.50 11.20
400 7.88 8.11 9.95 8.65 9.56 10.12 8.95 9.54
91
Table B.4 Evaporation time of Methanol on Aluminum Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
60 4.82 4.31 4.90 4.68 6.39 5.98 6.21 6.19
70 2.74 2.52 2.83 2.70 4.96 4.83 4.63 4.81
80 1.66 1.54 1.31 1.50 1.85 1.97 1.73 1.85
90 1.27 1.19 0.99 1.15 1.52 1.32 1.79 1.54
100 0.25 0.40 0.31 0.32 0.25 0.25 0.38 0.29
125 1.12 1.32 1.04 1.16 1.62 1.84 1.32 1.59
150 3.21 2.98 3.45 3.21 3.56 3.86 3.99 3.80
175 10.99 10.96 10.31 10.75 13.86 14.15 14.40 14.14
200 20.12 20.43 20.84 20.46 21.02 22.12 21.54 21.56
225 21.58 21.83 21.02 21.48 25.56 25.74 24.86 25.39
250 18.54 18.12 17.89 18.18 22.23 22.56 21.86 22.22
275 15.19 16.21 16.11 15.84 19.26 20.39 18.51 19.39
300 12.81 12.54 13.856 13.07 16.59 15.23 17.46 16.43
325 10.54 11.32 12.5 11.45 13.21 15.64 14.44 14.43
350 8.51 8.84 9.86 9.07 12.54 13.66 14.11 13.44
375 9.12 7.64 8.45 8.40 11.54 10.68 11.25 11.16
400 6.86 7.20 7.66 7.24 10.86 8.65 9.95 9.82
Table B.5 Evaporation time of Distilled water on Brass Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 119.56 118.57 117.23 118.45 123.01 122.45 119.24 121.57
80 78.96 77.65 78.99 78.53 82.55 84.23 86.11 84.29
90 40.01 42.21 39.99 40.74 45.21 46.53 48.21 46.65
100 4.15 4.58 5.95 4.89 5.21 5.68 4.89 5.26
125 39.89 41.52 42.52 41.31 43.22 44.56 47.86 45.21
150 81.19 80.12 82.34 81.22 88.21 89.45 86.25 87.97
175 110.42 108.56 107.68 108.89 119.21 118.59 117.56 118.45
200 79.98 80.12 78.98 79.69 84.55 83.21 88.99 85.58
225 65.21 66.58 67.28 66.35 67.54 68.59 69.84 68.66
250 55.12 56.89 57.65 56.55 59.21 58.63 61.12 59.65
275 46.82 45.21 47.38 46.47 49.84 46.86 51.89 49.53
300 43.21 42.89 41.26 42.45 46.12 45.22 46.52 45.95
325 39.22 37.84 40.21 39.09 40.12 39.86 42.21 40.73
350 35.52 34.26 32.81 34.20 36.78 37.96 38.11 37.62
375 31.12 33.54 32.99 32.55 34.22 35.21 37.12 35.52
92
Table B.6 Evaporation time of NaCl Solution on Brass Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 110.12 108.54 107.39 108.68 113.21 112.45 112.54 112.73
80 69.11 68.42 66.89 68.14 75.23 74.22 76.21 75.22
90 30.45 32.3 31.26 31.34 38.82 39.91 40.3 39.68
100 0.52 0.62 0.49 0.54 0.72 0.78 0.88 0.79
125 5.11 5.69 4.89 5.23 6.30 6.49 7.12 6.64
150 26.34 25.42 26.89 26.22 29.2 28.62 30.40 29.41
175 45.29 44.26 43.87 44.47 48.21 44.56 49.12 47.29
200 61.10 62.89 59.46 61.15 68.12 69.52 67.86 68.50
225 55.89 54.67 55.16 55.24 60.22 61.35 62.22 61.26
250 48.22 47.69 49.59 48.50 52.26 54.34 50.84 52.48
275 45.27 46.32 44.22 45.27 48.02 47.76 48.95 48.24
300 41.23 40.25 42.26 41.25 42.22 41.15 43.11 42.16
325 38.96 36.84 39.92 38.57 40.44 41.56 37.89 39.96
350 36.53 38.82 35.12 36.82 36.92 38.12 38.82 37.95
375 30.40 33.21 32.25 31.95 35.59 34.32 34.99 34.97
Table B.7 Evaporation time of Ethanol on Brass Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 6.21 6.39 5.89 6.16 8.04 8.32 8.49 8.28
80 3.09 3.19 3.42 3.23 4.31 4.51 3.92 4.25
90 1.21 1.79 2.21 1.74 2.64 2.54 2.78 2.65
100 0.51 0.52 0.49 0.51 0.79 0.82 0.93 0.85
125 14.16 12.29 13.21 13.22 17.52 16.86 18.12 17.50
150 43.19 45.23 41.15 43.19 46.93 47.12 45.98 46.68
175 34.22 36.95 35.21 35.46 38.12 37.23 39.22 38.19
200 31.24 32.15 30.99 31.46 34.54 33.22 35.15 34.31
225 26.59 24.43 27.11 26.04 29.33 30.4 31.12 30.28
250 22.88 24.56 23.12 23.52 26.88 25.43 27.66 26.66
275 22.62 21.13 23.65 22.47 24.21 23.56 24.82 24.20
300 18.78 19.96 20.26 19.67 21.30 22.34 20.86 21.50
325 18.82 16.52 17.45 17.60 19.98 19.53 20.86 20.12
350 15.82 16.93 14.12 15.62 19.42 18.52 17.12 18.35
375 11.96 13.56 12.45 12.66 15.42 18.46 16.86 16.91
93
Table B.8 Evaporation time of Methanol on Brass Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
60 5.64 5.24 5.42 5.43 7.11 7.19 6.60 6.97
70 4.51 4.29 4.11 4.30 5.75 5.53 5.21 5.50
80 1.24 1.39 1.58 1.40 1.92 2.01 1.83 1.92
90 1.18 0.92 0.93 1.01 1.83 1.45 1.51 1.60
100 0.40 0.42 0.49 0.44 0.67 0.79 0.59 0.68
125 12.71 11.98 13.42 12.70 14.83 13.21 15.23 14.42
150 39.18 40.22 38.45 39.28 44.23 45.89 44.85 44.99
175 30.99 31.24 32.86 31.70 38.21 39.12 37.54 38.29
200 27.89 25.32 26.22 26.48 31.12 32.23 33.42 32.26
225 24.86 22.13 25.86 24.28 29.98 30.44 28.45 29.62
250 22.83 21.45 24.44 22.91 28.01 27.05 26.98 27.35
275 19.86 18.92 21.45 20.08 23.45 24.52 22.12 23.36
300 18.72 19.24 17.92 18.63 20.54 21.23 19.82 20.53
325 15.69 14.23 16.32 15.41 18.82 17.64 16.98 17.81
350 11.63 12.87 13.14 12.55 16.62 15.42 14.23 15.42
375 9.50 10.55 11.10 10.38 11.99 12.45 13.86 12.77
Table B.9 Evaporation time of Distilled water on Copper Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 98.23 96.54 98.99 97.92 110.23 108.74 106.54 108.50
80 78.25 80.23 75.44 77.97 84.56 86.23 87.23 86.01
90 54.12 56.22 59.42 56.59 60.22 62.45 63.11 61.93
100 32.13 31.12 34.12 32.46 49.23 48.15 50.23 49.20
125 40.32 39.73 41.51 40.52 55.12 56.31 54.23 55.22
150 46.95 48.33 49.13 48.14 63.22 66.18 64.37 64.59
175 61.11 63.43 65.43 63.32 70.13 69.35 68.13 69.20
200 70.13 70.79 71.93 70.95 79.87 83.12 80.90 81.29
225 59.10 61.15 58.24 59.50 65.43 67.84 70.26 67.84
250 50.92 49.85 51.41 50.73 60.35 59.87 58.85 59.69
275 47.34 45.13 44.21 45.56 57.36 54.33 55.83 55.84
300 40.13 41.13 42.27 41.18 50.41 49.81 47.90 49.37
325 38.13 37.35 36.56 37.35 43.13 42.22 40.95 42.10
350 33.14 34.91 35.16 34.40 35.60 37.83 36.92 36.78
375 28.72 29.10 27.63 28.48 30.13 31.95 29.24 30.44
94
Table B.10 Evaporation time of NaCl Solution on Copper Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 90.12 89.85 86.99 88.99 105.66 106.21 104.44 105.44
80 70.12 69.89 71.12 70.38 80.22 81.11 81.19 80.84
90 52.12 49.53 50.11 50.59 54.23 56.81 59.12 56.72
100 35.63 34.93 33.17 34.58 38.36 40.16 39.9 39.47
125 8.42 9.33 10.16 9.30 9.84 11.16 13.18 11.39
150 14.16 15.99 13.23 14.46 19.16 20.11 21.5 20.26
175 50.13 52.32 51.31 51.25 55.13 57.95 58.14 57.07
200 59.73 58.82 60.18 59.58 63.13 64.93 65.81 64.62
225 51.32 54.23 52.31 52.62 59.18 57.83 56.97 57.99
250 45.16 46.33 44.19 45.23 52.19 52.96 51.12 52.09
275 43.59 42.11 42.05 42.58 51.24 50.86 51.65 51.25
300 39.54 40.21 39.86 39.87 49.11 48.11 47.33 48.18
325 34.31 33.99 36.11 34.80 41.26 43.25 40.21 41.57
350 35.13 34.36 32.23 33.91 38.95 37.26 35.10 37.10
375 30.12 29.29 28.29 29.23 30.16 31.27 32.94 31.46
Table B.11 Evaporation time of Ethanol on Copper Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 4.51 4.19 3.99 4.23 4.74 4.93 5.11 4.93
80 4.04 3.70 3.65 3.80 4.25 4.42 4.11 4.26
90 1.73 1.42 1.39 1.51 2.31 2.59 2.05 2.32
100 1.05 0.89 0.78 0.91 1.31 1.49 1.79 1.53
125 5.11 4.95 6.36 5.47 6.12 5.95 6.35 6.14
150 14.95 15.16 13.24 14.45 17.23 19.36 18.43 18.34
175 36.93 34.82 35.18 35.64 39.32 40.53 41.95 40.60
200 34.11 33.28 31.13 32.84 37.38 36.43 35.93 36.58
225 30.17 32.16 29.32 30.55 33.26 31.19 32.92 32.46
250 26.33 24.56 25.93 25.61 28.33 27.56 26.93 27.61
275 23.14 22.23 20.94 22.10 25.44 23.24 22.11 23.60
300 19.86 17.93 18.34 18.71 20.26 21.96 19.98 20.73
325 17.3 16.82 15.91 16.68 17.31 18.46 16.13 17.30
350 13.21 13.59 14.99 13.93 14.17 15.93 16.12 15.41
375 10.13 11.13 12.92 11.39 13.16 12.16 11.49 12.27
95
Table B.12 Evaporation time of Methanol on Copper Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
60 5.47 4.83 5.29 5.19 8.35 7.91 7.61 7.96
70 4.91 3.98 4.01 4.30 5.11 5.49 4.89 5.16
80 1.61 1.79 1.93 1.78 2.32 2.14 2.93 2.46
90 0.41 0.31 0.48 0.40 0.62 0.71 0.98 0.77
100 0.54 0.47 0.63 0.55 0.74 0.89 0.63 0.75
125 1.01 1.12 0.95 1.03 1.32 1.49 1.53 1.45
150 8.45 8.93 7.96 8.45 9.55 10.11 11.21 10.29
175 31.13 33.56 34.92 33.20 35.78 39.24 40.21 38.41
200 29.85 28.76 29.31 29.31 30.15 31.26 32.46 31.29
225 26.26 27.36 24.35 25.99 27.12 28.24 25.93 27.10
250 24.31 23.23 21.15 22.90 24.39 23.26 25.53 24.39
275 20.14 19.27 19.97 19.79 21.23 20.23 23.21 21.56
300 18.16 17.26 17.99 17.80 19.39 18.98 17.93 18.77
325 16.46 15.31 14.36 15.38 17.52 16.81 18.33 17.55
350 13.18 12.96 12.32 12.82 14.34 14.54 15.1 14.66
375 10.16 9.87 11.34 10.46 11.82 10.92 12.25 11.66
Table B.13 Evaporation time of Distilled water on Mild Steel Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 70.12 69.86 68.98 69.65 74.53 76.55 73.82 74.97
80 50.22 54.11 53.21 52.51 54.53 55.61 58.79 56.31
90 20.12 23.46 22.42 22.00 26.98 25.45 26.01 26.15
100 2.23 2.19 2.35 2.26 4.88 4.79 4.99 4.89
125 1.15 1.11 1.32 1.19 2.21 2.31 2.49 2.34
150 4.14 4.35 4.49 4.33 5.15 5.32 5.31 5.26
175 8.15 8.85 7.92 8.31 9.35 9.23 9.56 9.38
200 11.13 11.63 10.56 11.11 13.27 13.22 13.59 13.36
225 14.35 15.15 14.35 14.62 17.44 17.21 17.56 17.40
250 49.89 49.32 50.01 49.74 55.17 55.39 56.32 55.63
275 44.23 45.15 44.21 44.53 49.18 50.23 50.62 50.01
300 39.24 40.23 42.21 40.56 41.25 41.33 41.93 41.50
325 34.17 33.96 34.52 34.22 36.26 36.56 35.91 36.24
350 29.33 29.63 29.12 29.36 30.16 30.58 31.23 30.65
375 22.15 22.65 21.38 22.06 25.14 26.36 25.39 25.63
96
Table B.14 Evaporation time of NaCl Solution on Mild Steel Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 63.11 64.23 62.1 63.15 65.42 66.46 66.11 65.99
80 41.23 42.22 40.19 41.21 49.12 48.25 47.12 48.16
90 16.12 14.99 15.21 15.44 22.99 23.56 23.01 23.19
100 1.66 1.69 1.81 1.72 3.88 3.79 3.98 3.88
125 0.37 0.32 0.41 0.37 0.59 0.62 0.55 0.59
150 0.79 0.89 0.65 0.78 0.99 1.25 1.05 1.10
175 1.81 1.32 1.66 1.60 2.53 2.45 2.96 2.65
200 8.52 7.95 7.88 8.12 10.11 9.86 9.72 9.90
225 12.11 13.01 12.56 12.56 13.98 13.52 14.02 13.84
250 19.25 19.23 19.54 19.34 22.21 22.58 22.69 22.49
275 40.1 40.55 41.23 40.63 45.32 45.62 45.98 45.64
300 37.83 37.62 37.12 37.52 39.27 39.37 39.03 39.22
325 32.54 32.56 32.33 32.48 36.34 36.54 36.51 36.46
350 27.15 27.89 27.48 27.51 29.95 29.33 29.3 29.53
Table B.15 Evaporation time of Ethanol on Mild Steel Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
70 2.56 2.46 2.72 2.58 3.77 3.95 3.99 3.90
80 1.45 1.61 1.31 1.46 2.36 2.29 2.59 2.41
90 0.38 0.42 0.32 0.37 0.46 0.48 0.38 0.44
100 0.37 0.34 0.31 0.34 0.41 0.45 0.49 0.45
125 0.44 0.41 0.39 0.41 0.55 0.62 0.67 0.61
150 18.47 18.52 18.69 18.56 19.9 19.36 19.81 19.69
175 26.12 25.86 26.02 26.00 29.86 30.12 29.54 29.84
200 23.54 24.11 22.86 23.50 24.12 23.89 24.51 24.17
225 18.19 18.69 18.42 18.43 18.93 18.51 17.65 18.36
250 15.76 15.45 15.78 15.66 17.37 17.52 16.89 17.26
275 13.73 13.58 13.69 13.66 15.25 15.69 15.98 15.64
300 11.58 11.23 11.53 11.45 12.43 12.89 11.99 12.44
325 10.12 10.23 10.69 10.35 11.99 11.56 11.59 11.71
350 9.73 9.36 9.56 9.55 10.25 10.21 10.39 10.28
97
Table B.16 Evaporation time of Methanol on Mild Steel Surface
Temperature(C) Small diameter Large diameter
t1 t2 t3 𝑡𝑎𝑣𝑔 t1 t2 t3 𝑡𝑎𝑣𝑔
60 2.85 2.77 2.62 2.75 4.91 5.02 7.61 5.85
70 1.16 1.10 1.29 1.18 2.52 2.43 4.89 3.28
80 0.87 0.81 0.78 0.82 1.09 1.32 2.93 1.78
90 0.27 0.21 0.32 0.27 0.31 0.49 0.98 0.59
100 0.33 0.31 0.28 0.31 0.37 0.39 0.63 0.46
125 2.52 2.62 2.39 2.51 2.20 2.12 1.75 2.02
150 13.21 12.86 12.54 12.87 14.12 14.86 14.34 14.44
175 20.12 19.87 20.45 20.15 20.98 21.01 21.14 21.04
200 22.98 22.86 23.01 22.95 23.56 23.84 23.42 23.61
225 17.13 17.32 17.83 17.43 16.15 16.01 17.21 16.46
250 13.25 13.78 13.23 13.42 15.58 15.39 15.01 15.33
275 11.13 11.24 11.85 11.41 13.27 13.87 12.87 13.34
300 9.83 9.37 9.85 9.68 10.19 10.25 10.20 10.21
325 8.12 8.31 8.29 8.24 9.63 9.99 9.12 9.58
350 7.67 7.63 8.31 7.87 8.11 8.01 8.36 8.16
98
APPENDIX
SUMMARY OF THEORETICAL
AND EXPERIMENTAL RESULT C
C.1 Comparison of Theoretical and Experimental Result for small diameter liquid droplet
Leidenfrost time(sec)
Liquids Experimental
Al Brass Cu MS
Water 56.29 108.89 70.95 49.74
NaCl solution 47.48 61.15 59.58 40.63
Methanol 21.48 39.28 33.20 22.95
Ethanol 24.84 43.19 35.64 26.00
Leidenfrost time(sec)
Liquids Theoretical (from correlation)
Al Brass Cu MS
Water 67.95 96.99 83.50 44.45
NaCl solution - - - -
Methanol 24.42 45.08 40.07 29.42
Ethanol 24.22 39.64 34.70 32.22
99
C.2 Comparison of Theoretical and Experimental Result for large diameter liquid droplet
Leidenfrost time(sec)
Liquids Experimental
Al Brass Cu MS
Water 67.54 118.45 81.29 55.63
NaCl solution 54.66 68.50 64.62 45.64
Methanol 25.39 44.99 38.41 23.61
Ethanol 28.69 46.68 40.60 29.84
Leidenfrost time(sec)
Liquids Theoretical (from correlation)
Al Brass Cu MS
Water 72.99 94.43 93.10 43.61
NaCl solution - - - -
Methanol 26.86 43.28 44.07 25.26
Ethanol 26.05 37.99 38.08 25.47
100
C.3 Comparison of Leidenfrost point temperature for different liquid droplets (small and
large diameter)
Leidenfrost temperature(˚C)
Liquids Experimental
Al Brass Cu MS
Water 200 175 200 250
NaCl solution 225 200 200 275
Methanol 225 150 175 200
Ethanol 200 150 175 175
101
APPENDIX
PROGRAM CODE D
C.1 Estimation of theoretical time
clear all;
close all;
clc
%Constants
sigma=5.669e-8;
g=9.81;
pi=3.1416;
Ru=8.3145;
%Temperature and pressure
Tp_dummy=(150+273.15); %Metal surface temperature
for i=19
Tp(i)=Tp_dummy+25;%Tp(1)=125 deg centigrade,....Tp(12)=400 deg centigrade.
Tp_dummy=Tp(i);
end
for i=19
%Rest temperature and pressure
Ts=(78.3+273.15); %Liquid saturation temperature
Tv=(Tp(i)+Ts)/2; %Average vapor temperature
Ps=101.3e3; %Partial pressure of diffusing vapor
M=46e-3; %Molecular weight of water
%Radius
ro=1.23e-3; %Initial droplet radius
rp=.0889; %Metal surface radius
%Area
Ap=pi*rp^2;%Metal surface area
Alb=pi*ro^2;%Flat liquid droplet bottom surface area
Ap_ring=pi*(rp^2-ro^2);%Ring shaped metal surface area
Als=2.67*pi*ro*ro;%Cylindrical liquid droplet side surface area
%Volume
102
Vo=(4*pi*ro^3)/3;
%Emissivity
ep=.26; %Metal surface emissivity (of copper)
el=0.07;
%Properties at Ts (of water)
rhol=757;
kv_sat=19.9e-3;
rhov_sat=1.435;
hg=963e3;
cpv_sat=1.83e3;
muv_sat=10.4e-6;
D=.119e-4;
%Properties at Tv (of water)
kv=kv_sat*(Tv/2);
rhov=rhov_sat*(2/Tv);
cpv=cpv_sat*(Tv/2);
muv=muv_sat*(Tv/2);
%Vapor layer thickness
delta=1e-3; %let
%Iteration for delta
while 1
%Shape factor from metal surface to flat bottom of liquid droplet
Fbpl=Fb(ro,rp,delta);
%Shape factor from metal surface to cylindrical side surface of liquid droplet
Fspl=Fs(ro,rp,delta);
Qcbflux(i)=kv*((Tp(i)-Ts)/delta);%Conduction heat flux from metal surface to flat bottom of
liquid droplet
Qrbflux(i)=(sigma*(Tp(i)^4-Ts^4))/(((1-ep)/ep)*(Alb/Ap)+Alb/(Ap*Fbpl)+(1-
el)/el);%Raidation heat flux from metal surface to flat bottom of liquid droplet
Qbflux(i)=Qcbflux(i)+Qrbflux(i);%Total heat flux at the flat bottom of liquid droplet
v(i)=(Qbflux(i))/(rhov*(hg+.5*cpv*(Tp(i)-Ts)));%Vertical velocity at flat bottom of liquid
droplet
delta_new=((9*muv*v(i)*ro)/(8*rhol*g))^(1/3);
if abs(delta_new-delta)<1e-6
delta=delta_new;
break;
end
delta=delta_new;
103
end
delta_matrix(i)=delta;
Qcsflux=(kv*(Tp(i)-Ts))/delta;%(May be discarded)Conduction heat flux from metal surface to
cylindrical side of liquid droplet
Qrsflux(i)=(sigma*(Tp(i)^4-Ts^4))/(((1-ep)/ep)*(Als/Ap_ring)+Als/(Ap_ring*Fspl)+(1-
el)/el);%Radiation heat flux from metal surface to cylindrical side of liquid droplet
mpr_s(i)=(sqrt(2)*D*Als*Ps*M)/(Ru*Ts*ro);%Total mass flow rate from cylindrical droplet
side
mpr_b(i)=(Qbflux(i)*Alb+Qsflux(i)*Als-mpr_s(i)*hg)/(hg+.5*cpv*(Tp(i)-Ts));%Total mass
flow rate from flat droplet bottom
mpr(i)=mpr_b(i)+mpr_s(i);%Total mass flow rate from droplet
t(i)=(rhol*Vo)/mpr(i); %Total vaporization time
end
disp('Theoritical result for distilled water on copper surface');
final_value_table=[t'];
disp(' Tp t');
disp(final_value_table);
plot((Tp-273.15),t); %Tp has taken in degree by -273.15
xlabel('Temperature of metal surface(Degree centigrade)');
ylabel('Vaporization time, t');
C.2 Engineering Correlation of experimental data
% Code for correlation constants
clear all;
close all;
clc
y=[0.0209,0.0213,0.0231,0.0253,]; %observed value
x1=[0.7634,0.8489,0.9251,0.9940,]; %observed value,,
x2=1e-4.*[0.0180,0.0293,0.0448,0.0654,]; %observed value
p=(sum(x1.*x1));
q=(sum(x1.*x2));
r=(sum(x2.*x2));
A=[p q;q r];
C=[sum(x1.*y);sum(x2.*y)];
B=inv(A)*C; %constants
104
for i=1length(y);
Y(i)=B(1)*x1(i)+B(2)*x2(i);
end
% Code for correlation graph
y1=y(1,18);y2=y(1,915);y3=y(1,1623);
Y1=Y(1,18);Y2=Y(1,915);Y3=Y(1,1623);
sumY=sum(Y);
sumy=sum(y);
m=sumY/sumy;
x11=1length(Y);
x11=x11*max(y)/length(Y);
y11=x11*m;
y12=x11*1;
figure (2);
axis tight
plot(y1,Y1,'ro')
hold all
plot(y2,Y2,'bs')
hold all
plot(y3,Y3,'mv')
hold all
plot(x11,y12,'g')
xlabel('Obseved value');
ylabel('Calculated value');
legend('water','methanol','ethanol','equation')
% Code for plotting heat flux graph
clear all;
close all;
clc
r=1.23e-3;
v=(4*pi*r^3)/3;
Ta=[70,80,90,100,125,150,175,200,225,250,275,300,325,350,375];
Tb=[70,80,90,100,125,150,175,200,225,250,275,300,325,350,375];
Tc=[70,80,90,100,125,150,175,200,225,250,275,300,325,350,375];
Ts=[70,80,90,100,125,150,175,200,225,250,275,300,325,350,375];
ta=[72.46333,68.55667,44.25667,17.16667,]; %Observed time
105
tb=[118.4533,78.53333,40.73667,4.893333,]; %Observed time
tc=[97.92,77.97333,56.58667,32.45667,]; %Observed time
ts=[69.65333,52.51333,22,2.256667,]; %Observed time
pl=1000;
hg=2256.7e3;
disp('HEAT FLUX OF WATER ON ALUMINUM')
qa=(4.921332e-3*pl*hg)./ta;
disp(qa)
plot(Ta,qa,'ko')
hold all;
xlabel('surface Temperature(degree C)');
ylabel('heat flux(Watt per meter square)');
disp('HEAT FLUX OF WATER BRASS')
qb=(4.921332e-3*pl*hg)./tb;
disp(qb)
plot(Tb,qb,'k*')
hold all;
xlabel('surface Temperature(degree C)');
ylabel('heat flux(Watt per meter square)');
disp('HEAT FLUX OF WATER ON COPPER')
qc=(4.921332e-3*pl*hg)./tc;
disp(qc)
plot(Tc,qc,'kv')
hold all;
xlabel('surface Temperature(degree C)');
ylabel('heat flux(Watt per meter square)');
disp('HEAT FLUX OF WATER ON MILD STEEL')
qs=(4.921332e-3*pl*hg)./ts;
disp(qs)
plot(Ts,qs,'k+')
hold all;
xlabel('surface Temperature(degree C)');
ylabel('heat flux(Watt per meter square)');
legend('Heat flux on AL','Heat flux on BR','Heat flux on Cu','Heat flux on MS');
plot(Ta,qa,'k',Tb,qb,'k',Tc,qc,'k',Ts,qs,'k')
106
%Code for comparison of theoretical (correlated) and experimental graph
clear all;
close all;
clc
r=1.23e-3;
ea=.05;eb=.09;ec=.045;es=.1;
pl=958.3 ;
pv=.597;
kv=25e-3;
Ts=100;
u=12.55e-6;
s=5.669e-8;
h=2256.7e3;
cp=2.03e3;
Tp=[150,175,200,225,250,275,300,325,350,375,400];
dT=Tp-Ts;
h1=h+0.5*cp.*dT;
disp('water-al')
x1a=sqrt((kv*g*pv*(pl-pv).*dT.*r)./(u*h1));
x2a=((s*ea*(Tp.^4-Ts^4))./h1);
xa=x1a.*0.0197+x2a.*1280.5;
ta=(pl.*r)./xa;
disp('correlated value of ta ')
disp(ta)
disp('water-br')
x1b=sqrt((kv*g*pv*(pl-pv).*dT.*r)./(u*h1));
x2b=((s*eb*(Tp.^4-Ts^4))./h1);
xb=x1b.*0.0169+x2b.*501.9980;
tb=(pl.*r)./xb;
disp('correlated value of tb ')
disp(tb)
disp('water-cu')
x1c=sqrt((kv*g*pv*(pl-pv).*dT.*r)./(u*h1));
x2c=((s*ec*(Tp.^4-Ts^4))./h1);
xc=x1c.*0.0160+x2c.*1171.9;
tc=(pl.*r)./xc;
disp('correlated value of tc')
disp(tc)
disp('water-st')
x1s=sqrt((kv*g*pv*(pl-pv).*dT.*r)./(u*h1));
x2s=((s*es*(Tp.^4-Ts^4))./h1);
107
xs=x1s.*0.0143+x2s.*1483.4;
ts=(pl.*r)./xs;
disp('correlated value of ts')
disp(ts)