U.S. patent application number 16/670608 was filed with the patent office on 2020-05-14 for methods of determining the properties of a fluid body.
The applicant listed for this patent is The Provost, Fellows, Foundation Scholars, & the Other Members of Board, of the College of the Holy. Invention is credited to Anthony James Robinson, Samuel Siedel.
Application Number | 20200150015 16/670608 |
Document ID | / |
Family ID | 51743300 |
Filed Date | 2020-05-14 |
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United States Patent
Application |
20200150015 |
Kind Code |
A1 |
Siedel; Samuel ; et
al. |
May 14, 2020 |
METHODS OF DETERMINING THE PROPERTIES OF A FLUID BODY
Abstract
A method is disclosed of determining the properties of a fluid
body in the form of a surface-attached droplet/bubble. A data set
is stored describing a plurality of droplets/bubbles of different
shapes, in which each shape is captured as a combination of two or
more linear dimensional measurements. For each shape the data set
includes one or more parameters describing the relationship between
the physical properties of a pair of fluids capable of forming that
shape as a surface-attached droplet/bubble disposed in a
surrounding fluid medium. A fluid body is provided in the form of a
surface-attached droplet/bubble and a plurality of linear
dimensional measurements are taken and are provided as an input to
a processing apparatus. The processing apparatus determines from
the data set the one or more parameters associated with the shape
described by said linear dimensional measurements. In particular
the surface tension of a fluid can be found in this way based on
simple dimensional measurements.
Inventors: |
Siedel; Samuel; (Laytown Co.
Meath, IE) ; Robinson; Anthony James; (Co. Meath,
IE) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
The Provost, Fellows, Foundation Scholars, & the Other Members
of Board, of the College of the Holy |
Dublin |
|
IE |
|
|
Family ID: |
51743300 |
Appl. No.: |
16/670608 |
Filed: |
October 31, 2019 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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15519127 |
Apr 13, 2017 |
10502672 |
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PCT/EP2015/073806 |
Oct 14, 2015 |
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16670608 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01N 2013/0241 20130101;
G01N 2013/0283 20130101; G01N 13/02 20130101 |
International
Class: |
G01N 13/02 20060101
G01N013/02 |
Foreign Application Data
Date |
Code |
Application Number |
Oct 14, 2014 |
EP |
14188853.7 |
Claims
1. A method of determining the properties of a fluid body in the
form of a surface attached droplet/bubble, comprising the steps of:
(a) storing, in a memory accessible by a processing apparatus, a
set of data describing a plurality of droplets/bubbles of different
shapes, wherein each shape is captured in said data set as a
combination of two or more linear dimensional measurements, and
wherein for each shape the data set includes one or more parameters
describing the relationship between the physical properties of a
pair of fluids capable of forming said shape when a first of said
fluids is a surface-attached droplet/bubble disposed in a
surrounding medium of the second of said fluids; (b) providing a
fluid body as a surface-attached droplet/bubble of a first fluid in
a surrounding medium of a different second fluid; (c) measuring a
plurality of linear dimensional measurements of said fluid body;
(d) providing said measurements as an input to a processing
apparatus; and (e) said processing apparatus determining from said
data set said one or more parameters associated with the shape
described by said linear dimensional measurements.
2. The method of claim 1, wherein one of the first and second
fluids is known and the other is unknown, and wherein the
properties of the known fluid permit the derivation from said one
or more parameters of corresponding properties of the unknown
fluid.
3. The method of claim 1, wherein said one or more parameters
comprise a parameter which is a function of an accelerating field,
a surface tension of one fluid at the interface with the other
fluid, and the respective fluid densities.
4. The method of claim 3 wherein the accelerating field is the
local gravitational field as characterised by the acceleration due
to gravity, g.
5. The method of claim 1, wherein the two or more linear
dimensional measurements are normalised measurements.
6. The method of claim 5, wherein the two or more linear
measurements are normalised against a further linear measurement of
the bubble/drop.
7. The method of claim 6, wherein the further linear measurement is
a base diameter or base radius of the bubble drop at a surface to
which it is attached.
8. The method of claim 1, wherein the two or more linear
dimensional measurements of the data set comprise any two of the
following measurements normalised against the remaining
measurement: height normal to attachment surface, maximum width
parallel to attachment surface, and base diameter/radius at
attachment surface.
9. The method of claim 1, wherein the two or more linear dimensions
of the data set are expressed as a combination of dimensions such
as an area or a volume.
10. The method of claim 1, wherein the data set is limited based on
one or more of the following assumptions used to create the data
set: (a) a value for one or more properties of the first fluid; (b)
a value for one or more properties of the second fluid; (c) a value
for one or more properties of the interface between first and
second fluids (d) a value for an acceleration such as gravitational
acceleration, g.
11. The method of claim 1, wherein said set of data comprises a
plurality of parameter sets, each parameter set describing a unique
solution to an equation modelling the shape of a droplet/bubble,
and each parameter set including said combination of two or more
linear dimensional measurements and said one or more parameters
describing the relationship between the physical properties of a
pair of fluids capable of providing said solution.
12. A method of obtaining the interfacial surface tension between a
liquid in a gas, comprising the steps of: performing the method of
claim 1 using a bubble of said gas in said liquid or a droplet of
said liquid in said gas in step (b), wherein the resultant shape of
the bubble/droplet is encompassed within the data set in step (a),
wherein the one or more parameters describing the relationship
between the physical properties of a pair of fluids include at
least one parameter based on interfacial surface tension, said at
least one parameter being determined in step (e).
13. A computer program product comprising instructions encoded on a
data carrier which, when executed in a computing system, are
effective to: (a) receive as an input a plurality of linear
dimensional measurements of a droplet/bubble; (b) access a memory
storing a set of data describing a plurality of droplets/bubbles of
different shapes, wherein each shape is captured in said data set
as a combination of two or more linear dimensional measurements,
and wherein for each shape the data set includes one or more
parameters describing the relationship between the physical
properties of a pair of fluids capable of forming said shape when a
first of said fluids is a surface-attached droplet/bubble disposed
in a surrounding medium of the second of said fluids; (c)
determining from said data set said one or more parameters
associated with the shape described by said linear dimensional
measurements received as an input; and (d) providing as an output
said one or more parameters.
14. An apparatus for determining the properties of a fluid body in
the form of a surface-attached droplet/bubble, comprising: (a) a
memory storing a set of data describing a plurality of
droplets/bubbles of different shapes, wherein each shape is
captured in said data set as a combination of two or more linear
dimensional measurements, and wherein for each shape the data set
includes one or more parameters describing the relationship between
the physical properties of a pair of fluids capable of forming said
shape when a first of said fluids is a surface-attached
droplet/bubble disposed in a surrounding medium of the second of
said fluids; (b) a processor programmed to receive as an input, a
plurality of linear dimensional measurements of a a fluid body as a
surface-attached droplet/bubble of a first fluid in a surrounding
medium of a different second fluid providing said measurements as
an input to a processing apparatus; and (c) a program causing said
processor to determine from said data set said one or more
parameters associated with the shape described by said linear
dimensional measurements.
15. An apparatus as claimed in claim 14, further comprising a
measurement system for making said plurality of linear dimensional
measurements, and an output there from to said processor.
Description
TECHNICAL FIELD
[0001] This invention related to the determination of physical or
thermodynamic properties of fluids.
[0002] It has particular application to determining properties
which are measurable in respect of a first fluid when disposed
within a surrounding fluid in the form of a droplet or bubble.
Examples of such properties include surface tension, Bond number
(and similar dimensionless characteristic numbers such as the
Eotvos, Goucher, and Deryagin numbers), and density difference
between phases, as wells as characteristics of the droplet or
bubble including its volume, height, maximum width, base width,
height of centre of gravity, apex curvature radius, interface area,
base contact angle and capillary length. Since certain of these
properties result from the effect of external fields and forces on
a fluid (e.g. gravitational and other accelerating fields,
electromagnetic fields etc., the method has application also in
measuring such fields and forces.
BACKGROUND ART
[0003] The measurement of the properties of a fluid, such as its
surface tension in a surrounding liquid or gas, is needed in many
applications and across many industries including the food,
textile, chemical, oil, pharmaceutical, biological and electronic
industries, and in research institutions working in these fields.
Similar to the surface tension measurement, other geometric
properties of a bubble or drop can be measured using the same
methodology, such as the bubble/drop volume, interface area, apex
curvature radius, or height of centre of gravity. The angle at the
base of the bubble/drop (known as contact angle) can also be
measured similarly. This latter measurement is particularly useful
in measuring the wettability property between a solid surface and a
fluid; something that is vital in many engineering
applications.
[0004] A common method of estimating surface tension involves
accurately measuring and analysing the shape of a drop (or bubble)
of the fluid. This shape is defined mathematically by a complex
differential equation that involves the surface tension property as
well as the respective fluid densities and gravity, which can be
measured in other ways or looked up from a table of known values.
By regression fitting a numerically integrated mathematical
solution of the equation to the experimentally measured shape of
the drop, the value of the surface tension is calculated.
[0005] FIG. 1 shows an experimental set-up for measuring the
surface tension of an unknown transparent liquid 10. A gas bubble
12 is formed from an upward-facing horizontal orifice of a needle
14 of a syringe 16 into the unknown liquid 10. The bubble 12 is
slowly and steadily injected, or simply steadily attached to the
rim of the orifice, so that its shape corresponds to a static
bubble shape. A camera 18 is used to capture the image of the
bubble shape, illuminated by a backlight 20.
[0006] FIG. 2 shows an image captured from such a system. Image
processing software can extract from this image a curve which
approximates the profile of the bubble-liquid interface, subject to
the constraints of the camera resolution and difference between the
actual profile when viewed in a true mathematical elevation versus
that captured by a camera capturing the bubble from a viewing angle
that cannot quite see both opposite edges of the profile
[0007] From a physical point of view, the shape of such an
axisymmetric bubble is a result of the hydrostatic pressure
gradients in both the liquid and gas phases and of the capillary
equilibrium at the liquid-gas interface. The mathematical equation
of the bubble profile can be written as follows (see F. J. Lesage,
J. S. Cotton and A. J. Robinson, Analysis of quasi-static vapour
bubble shape during growth and departure, Physics of Fluids, Vol.
25, p. 067103, 2013):
( .rho. l - .rho. g ) g .sigma. z = 2 R 0 - C ( z ) Eq . 1
##EQU00001##
[0008] In equation 1, .rho..sub.i, .rho..sub.g and .sigma. are
properties of the fluid, respectively the liquid and gas densities
and the interfacial surface tension; g is the gravitational
acceleration; R.sub.0 is the radius of curvature at the apex of the
bubble; z is the vertical coordinate, downward from the apex of the
bubble, and C is the curvature of the interface, which depends on
the vertical coordinate (thus expressed as C(z)). The profile of
the bubble is fully defined by this equation, and is cut by a
horizontal plane which is the horizontal surface on which the
bubble is attached.
[0009] More generally, the terms .rho..sub.i, .rho..sub.g can be
replaced by the densities of any two fluids and are not necessarily
those of a liquid and a gas. Equation 1 can therefore be understood
as covering the more general case of two fluids, not necessarily a
liquid and a gas.
[0010] The interfacial surface tension, a, is a property of the two
fluids at the interface and therefore will be different for e.g. a
water droplet in a heavy oil medium than it will for water in a
gaseous medium. While interfacial surface tension is strictly
speaking thus a property of the liquid and the gas, all gases
effectively behave the same way, and so in the specific case of a
liquid/gas interaction the surface tension is generally considered
as a property specific to the liquid.
[0011] This equation is based on the assumption that the bubble or
droplet is not freely floating but rather is attached to a surface.
(Note that droplets and bubbles are, for these purposes, different
examples of the same phenomenon, namely a discrete body of a first
fluid disposed in a second fluid.) The droplet/bubble can be
gravitationally accelerated, due to weight and buoyancy forces
arising from the density difference with the surrounding fluid,
either towards the surface in question, in which case it is a
"sessile" droplet or bubble; or away from the surface, in which
case it is a "pendant" droplet or bubble, remaining attached by
surface tension forces that oppose the gravitationally-induced
buoyancy or weight.
[0012] The bubble shown in FIG. 1 is pendant, and is equivalent to
the classic image of a droplet of water hanging from a tap (or
faucet) prior to it breaking off when it grows beyond the size
limit allowing it to remain attached under surface tension. An
example of a sessile droplet is a bead of water lying on a polished
surface, while sessile bubbles would include bubbles of air trapped
under a submerged glass surface. Note that while these simple
examples all presume a liquid-gas or gas-liquid system of two
fluids, the concepts apply equally to any two immiscible fluids of
different densities. The bodies of a first fluid disposed in a
second fluid will be referred to herein as droplets/bubbles,
droplets, or bubbles, depending on context, it being appreciated
that the terms can be used interchangeably from the point of view
of the physics involved.
[0013] Equation 1 therefore provides a relation that links
different properties of the two fluids and the gravitational
acceleration to the geometry of the droplet/bubble. Thus, if
sufficient independent parameters are known or measured, the other
parameters can be deduced using this equation.
[0014] The usual way of calculating the surface tension of a fluid
is as follows. The gravitational acceleration and the fluid
densities are considered as known (or measured with a different
method). Then, different geometrical profiles corresponding to
equation 1 are generated by numerical integration of equation 1.
Each numerical integration provides a solution which describes the
entire surface of the bubble; from the base to the tip. A
regression algorithm is utilized to obtain the solution that best
fits the experimental profile, which is obtained by image
processing. The best fitting profile provides the value of the
surface tension property.
[0015] Similarly, the volume of the drop or bubble, or the
height/position of its centre of gravity can be calculated when a
best fit geometric profile is obtained describing the curvature of
the profile.
[0016] Such methods have the drawback that they require significant
image processing capability and mathematical computational power,
as well as a good deal of post-processing time to obtain a good fit
between the mathematical model and the experimental image.
DISCLOSURE OF THE INVENTION
[0017] There is provided a method of determining the properties of
a fluid body in the form of a surface-attached droplet/bubble,
comprising the steps of: [0018] (a) storing, in a memory accessible
by a processing apparatus, a set of data describing a plurality of
droplets/bubbles of different shapes, wherein each shape is
captured in said data set as a combination of two or more linear
dimensional measurements, and wherein for each shape the data set
includes one or more parameters describing the relationship between
the physical properties of a pair of fluids capable of forming said
shape when a first of said fluids is a surface-attached
droplet/bubble disposed in a surrounding medium of the second of
said fluids; [0019] (b) providing a fluid body as a
surface-attached droplet/bubble of a first fluid in a surrounding
medium of a different second fluid; [0020] (c) measuring a
plurality of linear dimensional measurements of said fluid body;
[0021] (d) providing said measurements as an input to the
processing apparatus; and [0022] (e) said processing apparatus
determining from said data set said one or more parameters
associated with the shape described by said linear dimensional
measurements.
[0023] The number of linear dimensions is specified as two or more.
A third linear dimension may be required, depending on the dynamics
of the system. For example, a bubble/drop issuing from an orifice
of fixed dimension, which does not spread beyond the orifice, may
have an implicit base diameter which can be built into the model or
data set. Similarly a highly wetting fluid on a surface may result
in a triple contact line of fixed dimension, reducing the number of
required linear dimensions to specify a particular bubble/drop
volume and shape to two.
[0024] It has been found that a set of data can be generated which
characterises a family or universe of droplet/bubble shapes, where
each such shape is defined using a combination of at least two
linear measurements. Those linear measurements uniquely specify a
shape from among the range of possible shapes, and the shape in
turn is (based on a consideration of equation 1) attributable to
the properties of the two fluids involved and the gravitational
force. Accordingly, the experimental measurement of the required
two linear dimensions of a particular bubble allows the shape, and
the physical parameters associated with that shape, to be
pinpointed in the data set.
[0025] This means that the determination of a physical property
such as surface tension, can be obtained by modelling a universe of
droplets/bubbles, where each is characterised by a pair of linear
measurements, and then when it is desired to find the surface
tension of a fluid one can simply make a couple of accurate linear
measurements and look up the surface tension.
[0026] There are significant advantages with this method.
Measurement of the linear dimensions of a droplet/bubble can be
performed accurately without sophisticated image processing
software. This makes this method suitable for implementation in a
device which may not have sophisticated imaging or processing
power, such as with the camera on a smartphone. Because the method
involves a couple of linear measurements and a look-up operation,
it can be performed in real time, unlike a curve-fitting exercise
as in known methods. Linear dimensions can be measured with a
higher degree of accuracy compared with regression fitting
solutions of a numerical integration to a complex shape.
Consequently, a higher accuracy may be reached using this method to
measure surface tension.
[0027] It is not limited to surface tension. Once a
droplet/bubble's shape has been modelled, it is possible to store,
for that bubble, not only surface tension but also several other
parameters which are specified once the shape is determined, e.g.
volume, height of centre of gravity, contact angle, interface area,
or capillary length. Accordingly the measurement of a pair of
linear dimensions of a bubble and the provision of a data set in
which bubbles are categorised and identified according to those
measurements, allows the rapid and accurate determination of a
number of complex physical parameters which were previously
obtainable only with difficulty.
[0028] Preferably, one of the first and second fluids is known and
the other is unknown, and the properties of the known fluid permit
the derivation from said one or more parameters of corresponding
properties of the unknown fluid.
[0029] This will often be the case in an experimental or real-world
industrial set-up. The unknown fluid will be provided as a
droplet/bubble in a fluid whose properties are completely known,
such as water or air; or alternatively, the unknown fluid can be
studied by introducing into it a bubble or droplet of e.g. air,
water or a water-immiscible liquid such as toluene.
[0030] Preferably, said one or more parameters comprise a parameter
which is a function of an accelerating field, a surface tension of
one fluid at the interface with the other fluid, and the respective
fluid densities.
[0031] Preferably, the accelerating field is the local
gravitational field as characterised by the acceleration due to
gravity, g. It can be envisaged however that applications will
arise in which there is no local gravitational field, or in which
the gravitational acceleration is supplemented by another external
accelerating field. Such accelerations will influence the shape of
any droplet or bubble. Similarly other external fields and forces
can be taken into account including electromagnetic forces acting
on the fluid.
[0032] Preferably, the two or more linear dimensional measurements
are normalised measurements.
[0033] Further, preferably, the two or more linear measurements are
normalised against a further linear measurement of the
bubble/drop.
[0034] This is particularly useful as a method of standardising the
data set. It also allows a dimension which is kept constant in the
system to be used as a normalising dimension.
[0035] Preferably, the further linear measurement is a base
diameter or base radius of the bubble drop at a surface to which it
is attached.
[0036] Preferably, the two or more linear dimensional measurements
of the data set comprise any two of the following measurements
normalised against the remaining measurement: height normal to
attachment surface, maximum width parallel to attachment surface,
and base diameter (or radius) at attachment surface.
[0037] Optionally, the two or more linear dimensions of the data
set may be expressed as a combination of dimensions such as an area
or a volume.
[0038] Preferably, the data set is limited based on one or more of
the following assumptions used to create the data set: [0039] a
value for one or more properties of the first fluid; [0040] a value
for one or more properties of the second fluid; [0041] a value for
one or more properties of the interface between first and second
fluids [0042] a value for an acceleration such as gravitational
acceleration, g.
[0043] In this way, the data set can be simpler, i.e. the number of
droplet/bubble shapes can be greatly compressed if assumptions are
made, such as (most commonly) the value of the gravitational
acceleration, but also possibly the data set may be tailored to
identifying e.g. the surface tension of an unknown liquid by
looking at the shape of an air bubble within that liquid, in which
case the density of the air is known. Other simplifying assumptions
can be made in other scenarios to reduce the complexity of the data
set. Since the surface tension may be unique to the liquid, this
provides a potential method of identifying a liquid.
[0044] Preferably, said set of data comprises a plurality of
parameter sets, each parameter set describing a unique solution to
an equation modelling the shape of a droplet/bubble, and each
parameter set including said combination of two or more linear
dimensional measurements and said one or more parameters describing
the relationship between the physical properties of a pair of
fluids capable of providing said solution.
[0045] There is also provided a method of obtaining the interfacial
surface tension between a liquid in a gas, comprising the steps of:
[0046] performing any of the methods as set out in the above
statements of invention using a bubble of said gas in said liquid
or a droplet of said liquid in said gas in step (b), [0047] wherein
the resultant shape of the bubble/droplet is encompassed within the
data set in step (a), [0048] wherein the one or more parameters
describing the relationship between the physical properties of a
pair of fluids include at least one parameter based on interfacial
surface tension, said at least one parameter being determined in
step (e).
[0049] There is also provided a computer program product comprising
a set of instructions, which are effective to cause a processor to:
[0050] (a) receive as an input a plurality of linear dimensional
measurements of a droplet/bubble; [0051] (b) access a memory
storing a set of data describing a plurality of droplets/bubbles of
different shapes, wherein each shape is captured in said data set
as a combination of two or more linear dimensional measurements,
and wherein for each shape the data set includes one or more
parameters describing the relationship between the physical
properties of a pair of fluids capable of forming said shape when a
first of said fluids is a surface-attached droplet/bubble disposed
in a surrounding medium of the second of said fluids; [0052] (c)
determining from said data set said one or more parameters
associated with the shape described by said linear dimensional
measurements received as an input; and [0053] (d) providing as an
output said one or more parameters.
[0054] There is also provided an apparatus for determining the
properties of a fluid body in the form of a surface-attached
droplet/bubble, comprising: [0055] (a) a memory storing a set of
data describing a plurality of droplets/bubbles of different
shapes, wherein each shape is captured in said data set as a
combination of two or more linear dimensional measurements, and
wherein for each shape the data set includes one or more parameters
describing the relationship between the physical properties of a
pair of fluids capable of forming said shape when a first of said
fluids is a surface-attached droplet/bubble disposed in a
surrounding medium of the second of said fluids; [0056] (b) a
processor programmed to receive as an input, a plurality of linear
dimensional measurements of a a fluid body as a surface-attached
droplet/bubble of a first fluid in a surrounding medium of a
different second fluid providing said measurements as an input to a
processing apparatus; and [0057] (c) a program causing said
processor to determine from said data set said one or more
parameters associated with the shape described by said linear
dimensional measurements.
[0058] Preferably, the apparatus further comprises a measurement
system for making said plurality of linear dimensional
measurements, and an output therefrom to said processor
[0059] The measurement system may comprise an optical sensor and
imaging software calibrated to determine linear measurements of key
parameters of a droplet/bubble whose image is captured by the
optical sensor.
[0060] More preferably, the measurement system comprises a laser
source and a detector for determining interception of a laser beam
by an edge of a droplet.
[0061] Preferably, the apparatus further comprises means for
introducing said droplet/bubble into said surrounding medium.
BRIEF DESCRIPTION OF THE DRAWINGS
[0062] FIG. 1 is a diagram of an experimental set-up for measuring
the surface tension of an unknown transparent liquid;
[0063] FIG. 2 is an image of a bubble of gas within a liquid,
captured from the set-up of FIG. 1;
[0064] FIG. 3 shows the measurement of dimensions on the image of
FIG. 2;
[0065] FIG. 4 is a diagram of a capillary profile mapped against
height (y-axis) and width (x-axis) scales;
[0066] FIG. 5 is a capillary profile showing bubble families
existing at different heights;
[0067] FIG. 6 is a graphical table showing six different
droplet/bubble shapes;
[0068] FIG. 7 is a graphical representation of a data set in which
a number of parameters are plotted against Bond number (x-axis) and
normalized volume (y-axis);
[0069] FIG. 8 is a graphical representation of a subset of the
information in FIG. 7, isolating the isolines showing the
normalized width of bubbles, and also showing the families of
bubble shapes on this representation;
[0070] FIG. 9 is a graphical representation of a subset of the
information in FIG. 7, isolating the isolines showing the
normalized height of bubbles;
[0071] FIG. 10 is a reproduction of FIG. 7 overlaid with an
exploded view of a small region of the graph;
[0072] FIG. 11 is a simplified representation of the exploded
region of FIG. 10, considered in terms of the two parameters of
normalized width and height;
[0073] FIG. 12 is a graphical representation of a subset of the
information in FIG. 7, isolating the isolines showing the apparent
contact angle;
[0074] FIG. 13 is a graphical representation of a subset of the
information in FIG. 7, isolating the isolines showing the
normalized height of the centre of gravity;
[0075] FIG. 14 is a graphical representation of a subset of the
information in FIG. 7, isolating the isolines showing the radius of
curvature at the apex;
[0076] FIG. 15 is a graphical representation of a subset of the
information in FIG. 7, isolating the isolines showing the total
interface area;
[0077] FIG. 16 is a plot of Bond number isolines against normalised
width on the x-axis, and against the ratio of a bubble's height to
its width on the y-axis;
[0078] FIG. 17 is a graphical model of a bubble, obtained as a
solution to the Laplace equation;
[0079] FIG. 18 is a block diagram of a computing device;
[0080] FIG. 19 is a flowchart of program code;
[0081] FIG. 20 is a block diagram of a distributed architecture;
and
[0082] FIG. 21 is a diagram illustrating an image processing
technique for measuring bubble or drop volume.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
[0083] FIG. 3 shows the measurement of three linear dimensions of
the bubble of FIG. 2, namely the bubble's maximum width w, its
height h, and its base diameter, d. The base diameter is determined
in this case by the orifice of the needle from which the air bubble
has issued, and the width and height are functions not only of the
volume of air that has been injected through the needle but also of
the gravitational force g, the surface tension of the liquid into
which the bubble is injected .sigma., and the respective densities
of the two fluids. Accordingly, the same volume of a first fluid
provided as a droplet or bubble in a surrounding second fluid can
take up different shapes.
[0084] The shape of a bubble changes as the bubble grows. Equation
1 defines, for each pair of liquids in a given gravitational field
(thus with, .rho..sub.i, .rho..sub.g, g and .sigma. fixed) a unique
curvature or capillary profile C(z) where the actual shape of the
bubble or drop is determined by the height at which this profile is
cut by the surface to which it is attached.
[0085] To illustrate this point, FIG. 4 shows a capillary profile
20 mapped against height (y-axis) and width (x-axis) scales of
arbitrary units (in this figure, the dimensions are normalized by
the radius of curvature at the apex). The profile can be cut by a
surface 22 anywhere along its height h, giving rise to a
corresponding bubble shape 24 which is defined by the profile and
the height.
[0086] FIG. 5 shows that the same profile 20 can be cut by up to
three different surfaces 26, 28, 30 with the same radius (or base
diameter) giving rise to different bubble shapes. It is useful to
consider three different families of bubble shapes, where the order
of the family describes the number of changes of the slope of the
bubble profile. From the apex to the base, the radius is always
increasing for Family 1, the radius then decreases for Family 2 and
a neck is formed for Family 3.
[0087] The curvature of the profile in FIG. 5 is merely one example
of how a bubble or droplet may be shaped. Depending on the
competing influences of gravity and surface tension, the shape may
be quite different. A useful quantity to consider as a descriptor
of the shape is the Bond number, Bo. Bond number is a dimensionless
property of a fluid, defined as the square of the inverse of the
capillary length, normalized by the square of the characteristic
length. It captures, for a droplet/bubble, the density difference
between the fluids, the gravitational acceleration, and the surface
tension. Bond number can be considered as a measure of the
importance of surface tension forces compared to body forces. A
high Bond number indicates that the system is relatively unaffected
by surface tension effects; a low number indicates that surface
tension dominates. Intermediate numbers indicate a non-trivial
balance between the two effects.
[0088] FIG. 6 is a graphical table showing six different
droplets/bubbles. There are two bubbles in each of the families 1,
2 and 3 described above. One bubble in each family (top row) is a
shape resulting from a low Bond number while the other (bottom row)
is a high Bond number shape. From this it can be seen that Bond
number has a profound impact on droplet/bubble shape; or to be more
accurate, the physical parameters of the liquids which underlie the
Bond number have such an impact, which is handily captured by the
Bond number as a single dimensionless parameter which is
characteristic of the droplet/bubble shape and of the underlying
parameters of the liquid.
[0089] In the present innovative method, a large number of
solutions to equation 1 are numerically computed a priori, and the
solutions are stored in a database. These solutions preferably
cover the whole spectrum of bubble shapes that may be encountered.
The solutions are numerically treated, in order to extract key
geometrical features. In particular, key lengths such as bubble
base radius, bubble height and bubble maximum width are calculated.
Within certain conditions, a given set of these three linear
lengths is sufficient to identify a unique bubble shape
solution.
[0090] Using the bubble of the case of study, the three lengths can
be measured very easily and nearly instantaneously. Importantly, it
can be done very accurately as it does not rely on the use of
sophisticated image processing, mathematical development or
regression analysis (see FIG. 3).
[0091] In this case, the mentioned equation is solved for the case
of a pendant drop or bubble i.e. one which is suspended from an
orifice. The key information about the bubble can be interpolated
from the database using these three simple length measurements, two
if the base dimension is physically fixed. In order to illustrate
the database, its information is represented in FIG. 7
diagrammatically. FIG. 7 plots a number of parameters, listed in
the legend box at upper right, against Bond number (x-axis) and
normalized volume (y-axis). The volume is normalized against the
base radius of the bubble, while the normalized bubble maximum
width and height are normalized against the base radius of the
bubble.
[0092] The representation of the database of solutions in FIG. 7
contains a wealth of information, which is more readily understood
when represented in colour but the constraints of using a
black-and-white version, as in FIG. 7, render the information less
easy to readily distinguish. Accordingly, FIGS. 8 and 9 show the
normalised bubble width and normalized bubble height, respectively,
isolated from all of the other parameters other than the x- and
y-axes.
[0093] In FIG. 8, the normalized width (relative to bubble base
radius) is plotted as a series of isolines. The graph can be
considered a two dimensional map of bubble shapes, where each
bubble, existing as a unique combination of a normalized volume and
a Bond number, occupies a unique location on the map. The heavy
outer boundary line 100, 102, and 104 delineates the overall area
within which bubbles can stably exist. Internal boundary lines 106,
108 and 110 divide this overall area into three sub-areas,
delineating between bubbles of Family 1, Family 2 and Family 3 as
previously described. Family 1 bubbles exist in the large lower
area 112 defined by boundaries 104, 108 and 110. Family 2 bubbles
exist in the large upper area 114 defined by boundaries 100, 106
and 110. Family 3 bubbles exist in the small area 116 defined by
boundaries 102, 106 and 108.
[0094] It can be seen that the isolines of bubble width are close
to horizontal over much of their ranges, meaning that a given
measured width will be associated with the same normalized volume
for a fairly wide range of Bond numbers. However, for liquids with
higher Bond numbers there is a tendency for the same width to be
associated with bubbles of greater volume.
[0095] FIG. 9 shows the normalized bubble height, plotted in the
same space. At lower Bond numbers the height is constant for a
given volume, but as Bond number increases, a given height tends to
be associated with a bubble of smaller volume. This can be
understood, in the context of FIGS. 8 and 9 considered together, as
indicating that higher Bond numbers are associated with bubbles
that tend to be longer and narrower for a given volume.
[0096] FIG. 10 shows the complete graph of FIG. 7 with an exploded
region 32 in which the experimentally measured bubble is located.
FIG. 11 is a simplified representation of that exploded region,
considered in terms of the two parameters of normalized width and
height.
[0097] It can be seen from FIG. 11 that the upward (left-to-right)
tendency of the width isolines and the downward tendency of the
height isolines in this region allows a combination of a
measurement pair to be accurately localized.
[0098] For the experimental bubble the following linear
measurements were made for width, height, and base diameter (see
FIG. 3):
w=3.02 mm; h=3.78 mm; d=1.22 mm
[0099] The normalized dimensions vs. base radius are thus:
w*=2w/d=4.95; h*=2h/d=6.20
[0100] These two normalized parameters are sufficient to locate the
bubble at a specific x-y location on the graph, as illustrated in
FIG. 11. That location is at an x-axis position having a Bond
number of 0.052. The Bond number is defined by the following
equation:
B o = d 2 ( .rho. l - .rho. g ) g 4 .sigma. ##EQU00002##
[0101] Where d is the bubble base diameter, .rho..sub.i and
.rho..sub.g are respectively the liquid and gas (air) densities, g
is the gravitational acceleration and a is the surface tension. In
the experimental case, g is taken to be 9.81 ms.sup.2, and the
density of air, .rho..sub.g, is assumed to be 1 kgm.sup.-3. The
liquid density, .rho..sub.i, can be found by looking up the value
for a known liquid, or by weighing a known volume. In this case it
is found to be 1000 kgm.sup.-3.
[0102] The surface tension of the liquid can then be determined
from these values in a straightforward manner:
.sigma. = d 2 ( .rho. l - .rho. g ) g 4 B o ##EQU00003## .sigma. =
70.3 mN / m ##EQU00003.2##
[0103] Thus, using the graph (or the underlying database) and the
three simple linear measurements, together with well-known or
easily found values for the densities of fluids and gravitational
acceleration, the surface tension can be found.
[0104] It will be understood that the representation in the graphs
of FIGS. 7-11 provide for a visual understanding of the underlying
data. As described, the data values can be read directly off a
graph, but one will generally obtain more accurate results by
accessing the solutions in the database directly.
[0105] It will also be understood that blank areas as seen in FIG.
11 do not imply an absence of data--where a sufficient number of
solutions to the underlying equation have been derived, solutions
will exist in those regions lying between the isolines selected for
illustration in the graph. Thus, FIG. 11 is merely an exploded
region of a graph with isolines plotted at specific round numbers.
The underlying database has solutions in the blank regions and has
normalized heights and widths for those solutions which do not get
represented on the graph (as well as all of the other derived
values like contact angle, etc.). In the event that the measured
values fall between values in the underlying database, conventional
interpolation techniques will allow the user to readily derive the
parameters associated with the interpolated solution.
[0106] Locating the bubble shape at a specific x-y location on the
graph allows not only the Bond number to be derived but also each
of the other plotted values which have been pre-calculated for each
solution of the equation in the database. Those parameters are each
plotted on a separate graph in FIGS. 12-15.
[0107] FIG. 12 shows the apparent contact angle, .alpha., which is
plotted as a series of isolines or contour lines of constant
contact angle in the two-dimensional landscape of normalized volume
V* vs Bond number Bo. It can be seen that the contact angles
exhibit much greater resolution in some regions of the graph than
was seen for the height and width values. So for a normalized
volume V* of 10.sup.-4, a bubble with a contact angle of 10.degree.
will imply a Bond number of about 7.times.10.sup.-5, whereas a
contact angle of 15.degree. would imply a Bond number of slightly
more than 1.times.10.sup.-4' for the same normalized volume of
10.sup.-4. Again, solutions also exist in the rather large region
between 10.degree. and 15.degree. and while the graph appears
blank, the database is fully populated with solutions for contact
angles of 11.degree., 12.degree., 13.degree. and 14.degree. and
also for intermediate, non-integer values.
[0108] FIGS. 13, 14 and 15 show the remaining parameters of the
FIG. 7 graph, isolated from the other parameters and plotted in the
same space, namely the normalized height of the centre of gravity
(FIG. 13), the radius of curvature at the apex (FIG. 14) and the
total interface area (FIG. 15). For each of these parameters, the
location of a bubble shape at a specific x-y position in the 2D
space of normalized volume vs. Bond number, allows the skilled
person to read off or interpolate the associated value for the
parameter in question.
[0109] Thus one may solve a very large number of solutions of the
governing equation, and store relevant geometric features of these
solutions in a database. Then, keeping the example of the surface
tension measurement, it is only necessary to measure simple
dimensions of the drop (or bubble) such as its height, maximum
width and base width, in order to interpolate from the data in the
database the value of its surface tension. This innovative method
avoids the necessity to fit a mathematical solution to the entire
drop contour by solving all possible solutions of the equation a
priori. It results in a simpler, faster measurement, with the
possibly of much improved accuracy.
[0110] The same method can be used in order to measure the bubble
(or drop) volume, the height of its centre of gravity, its
interface area, the radius of curvature at its apex or any
geometrical feature, including the apparent contact angle at the
base of the bubble or drop. This is particularly useful to
characterize the wettability property of a fluid on a given
surface. In the case of contact angle, it is more appropriate to
use a sessile bubble (or drop) on (or below) a horizontal flat
surface. The governing equation that would describe such a
bubble/drop would still be equation 1, but with a negative value
for the gravitational acceleration.
[0111] The database and graphs of FIGS. 7-15 are complex because
they capture a great deal of information for each bubble shape
solution. It is possible to maintain instead a more simplified
database, or to represent the database as a simpler data set, if
one wishes to perform only a specific look-up operation.
[0112] For example, FIG. 16 is a graph tailored to the first
operation discussed earlier--taking a bubble's height, width and
base diameter, and finding the Bond number which is associated with
this combination. FIG. 16 plots the normalised width w* on the
x-axis, and on the y-axis, plots the ratio of the bubble's height
to its width. These two axes thus capture the three linear
measurements for a given experimental bubble. The two-dimensional
space thus created is populated with Bond number logarithmic
isolines.
[0113] Thus, by measuring the values for w, h and d as previously
described, and by calculating the quantities h/w and 2w/d, a simple
lookup will give the isoline value that overlies the identified
point. In the case of the experimental bubble discussed earlier,
the y-axis value is 1.252 and the x-axis value is again 4.95. The
Isoline value at this point is -1.28, which is log.sub.10(0.052).
This simple lookup allows a very rapid calculation of Bond number
from three measurements that can be made easily and with great
accuracy.
[0114] The accuracy of the graphical method was compared against
the leading existing techniques and against theory as follows. A
gas bubble is created in water from a 1.22 mm diameter orifice
located in a horizontal, upward-facing surface immersed in
quiescent water in terrestrial gravity conditions. (This is
basically the bubble pictured in FIGS. 2 and 3.)
[0115] The volume of gas introduced into the bubble was measured at
17.78 mm.sup.3, and the capillary length of water has a known value
of 2.7 mm. Consequently, the bubble base radius is b=0.61 mm, the
Bond number value is Bo=0.052 and the normalized volume is
V.sub.b*=78.3.
[0116] In order to find the "true" volume for the purposes of
comparing different approaches, the following technique was used.
The volume was calculated by imaging the bubble with a high
resolution photo, using the pixel locations on the interface and
then revolving to create layers of 3D disks (with bevelled edges)
and then adding the subsequent volumes to get the total bubble
volume (V.sub.b=17.78 mm.sup.3). The normalized volume is
V.sub.b*=V.sub.b/b.sup.3. FIG. 21 illustrates this technique,
showing two edge pixels located at positions (x.sub.i, y.sub.i) and
(x.sub.i+1, y.sub.i+1) defining a quadrilateral area (shaded) which
when rotated 360 degrees about the y-axis will result in a disc
volume with a bevelled outer edge. Summing the volumes of all such
discs with a suitably small incremental height between successive
edge pixels (this increment is obviously exaggerated in FIG. 21 for
clarity), gives a measurement of the bubble volume.
[0117] As a first set of data for comparison, the dimensions of the
bubble (normalized height, normalized width, normalized height of
centre of gravity, and apparent contact angle) can be measured from
the photographic image of the bubble.
[0118] As a second set of data, a normalized solution to the
Laplace equation is calculated based on the known parameters to
obtain a model of the resultant bubble, which is illustrated in
FIG. 17. From that model or simulation, numerical values can be
obtained for the same four dimensions (normalized height,
normalized width, normalized height of centre of gravity, and
apparent contact angle) as well as the normalized total interface
area, A*, and the radius of curvature at the apex, R.sub.0*.
[0119] As a third set of data, we have the data from the database
underlying the graph of FIG. 7. Using the normalized volume and the
known Bond number, the x-y position of the bubble can be localised
on the graph (see FIG. 10) and the value of each of the plotted
parameters at that x-y position can be read from the graph, to give
a set of values for the same parameters as with the second set of
data.
TABLE-US-00001 TABLE 1 Comparison of the estimation of geometrical
properties by experimental (photographic) determination, contour
simulation and use of the graph of FIG. 7 Experimental Simulated
model Read from graph h* 6.20 6.20 6.2 w* 4.98 4.96 4.95 h.sub.cg*
3.29 3.23 3.25 .alpha. 64.degree. 65.degree. 65.degree. A* -- 87.2
85 R.sub.0* -- 2.35 2.3
[0120] Table 1 sets out the comparative numerical values. It can be
seen that a very good agreement is found between the three
different methods as all values present less than 3%
discrepancy.
[0121] The method can be implemented in computer code running on a
computing device. An example of such a device is illustrated in
block diagram form in FIG. 18. The device 40 of FIG. 18 can be any
suitable computing device but preferably is a smartphone, tablet,
or similar handheld device (including a dedicated scientific
instrument). A camera 42 is provided to capture an image of a
bubble 44 under the control of a suitable input interface 46. A
display 48 allows the user to see the captured image which is
stored typically in a working memory 50 temporarily, and in a
non-volatile storage unit 52 permanently. It will be understood
that devices having this functionality to capture images of a
reasonably high resolution are ubiquitous.
[0122] The functionality need not all be carried on-board a single
device. The image of a bubble or droplet can equally be captured by
an external imaging system such as a digital camera or CCD disposed
as part of a measurement apparatus, with a connection to the
computing device.
[0123] The storage means 52 also has a database 54 within which
data, representing multiple solutions to the equation describing
the shape of a bubble or droplet, are stored. Such data typically
is organised in a manner that allows a look-up of a solution based
on a limited number of dimensional input parameters, such as a
combination of two or three linear measurements of a bubble or
droplet (or a composite parameter like a measurement of one
dimension of the bubble, normalized with respect to another
measurement). The data stored for each solution can be one or more
complex parameters of the shape of bubble described by the
solution, including without limitation the Bond number (or a
similar characteristic of the system), surface tension, contact
angle, volume, area, height of centre of gravity, radius of
curvature, or parameters of the curvature itself. Thus, the
database allows the look-up of complex parameters from simple
linear measurements.
[0124] Also stored within the storage unit 52 is program code 56
which specifies the steps to be taken in extracting the features
from an image, and deriving from the database the desired complex
parameters. A network interface, such as is known in conventional
mobile phones and tablets, allows communication from and to the
device.
[0125] The program code is preferably in the form of an "app" or
downloadable program, in the case of the device 40 being a mobile
phone or tablet.
[0126] FIG. 19 is a flowchart of the program code. In step 60 an
image is captured by the camera in response to a user input. Thus,
a user will typically launch the program, which will in turn
provide a camera view on-screen. The user will position the camera
to obtain a good bubble or droplet image (i.e. one which fills the
screen as far as practical and has clearly defined edges), then
select shutter operation. The image from the camera sensor is
typically pre-processed and stored in memory as well as being
copied onto the permanent storage (such as an internal or external
memory card).
[0127] Image processing in the software then measures the
dimensions of the image, step 62, by performing a conventional edge
detection to determine the edges of the droplet or bubble,
straightening the image according to a reference axis (such as the
axis of symmetry of the bubble or a reference edge visible in the
image), and then measuring the length of predetermined dimensions,
such as bubble height, maximum width, and width at the connection
point to a surface (or base diameter). The measured dimensions can
be determined in terms of numbers of pixels across each dimension,
or converted to actual estimated lengths e.g. in mm, according to
depth cues or assumed distance from the bubble.
[0128] Alternative, and indeed improved, methods can equally be
used to capture linear dimensions, and the technique encompasses
such methods. A limitation in the use of a camera or CCD is that
accuracy is restrained to the pixel resolution of the image.
[0129] In one improvement, the camera is omitted and instead
dimensions are captured using laser interferometry. For example, if
the drop height, width and base diameter are measured by displacing
a laser beam, and measuring when the beam is intercepted by the
drop, then two advantages arise for an enhanced accuracy: [0130]
The parallax error inherent in camera imaging with a single lens
position is suppressed [0131] The accuracy of the linear
measurement can be perhaps 1000 times than the best achievable
pixel resolution.
[0132] In step 64, the dimensions are normalized by the processor.
In the preferred method, the base diameter or radius is used to
normalize the width and height. However, this need not be the case,
and any of these dimensions can be normalized against any of the
others. For example, in the FIG. 16 example, the normalized
dimensions were width relative to base radius, and height relative
to width.
[0133] In step 66, the program accesses the database 54 to look up
solution values which correspond to the dimensions as measured and
normalized. Depending on the granularity of the database, the
resolution of the image, and the degree of rounding when
normalizing, the exact values used to look up the database index
may or may not be present. Accordingly, in decision 68, a
determination is made if the values are present for a direct look
up.
[0134] If so, step 70, then the pre-calculated values are
determined for the solution that corresponds to the normalized
linear dimensions. If not, a suitable interpolation routine is
employed in step 72 to obtain values that are deemed to be a best
match.
[0135] The retrieved or interpolated parameters can include items
such as the Bond number, from which the surface tension can be
calculated when the densities of the liquid and gas are known (and
assuming standard gravity or some adjusted accelerating field).
Post-processing steps may therefore allow the user to input
additional values and obtain derived properties like surface
tension, wettability, or any other property that can be derived or
calculated from the values retrieved from the database.
[0136] FIG. 20 illustrates a distributed architecture, in which the
device 40 is connected via a local area network or wide area
network like the Internet 80 to a server computer 82. The device 40
in this case is modified relative to the device shown in FIG. 18 as
follows: rather than carrying the full set of program instructions
and the database 54 in its storage 52, it carries a set of program
instructions for a more limited subset of the functions as will be
described below, and the database is not carried internally within
device 40, but rather is stored as a database 92 within storage 90
accessible by the server.
[0137] The server 82 has a network interface 84, a processor 86,
working memory 88, and the aforementioned non-volatile storage
medium 90. It also carries program instructions 94 relating to its
part of the functionality of the method in this client-server
relationship.
[0138] In one implementation, the program instructions at the
device are limited to capturing the image and sending the image to
the server computer over the network, with the server computer's
program causing the server to analyse the image, obtain the
dimensional values, look up the database 92, and return as an
output whatever values are sought by the device 40 for display to
the user.
[0139] In another implementation, the device carries out the image
processing and the extraction of the dimensions, and simply
transmits these to the server for look-up of the stored complex
parameters of an appropriate solution in the database. The skilled
person will appreciate that the distributed nature of computing
solutions allows many other variations in the configuration of a
suitable computer system, enabling measurements of a droplet to be
converted to useful complex parameters of the droplet, based on the
common factor of having a pre-calculated family of solutions
describing different candidate droplets to which the dimensional
measurements can be fitted to select an appropriate solution.
* * * * *