Foam: 

High Frequency Fill Level Inspection

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complex system of polydisperse gas bubbles, separated by draining films 

Knowingly, soft-drinks obtained by adding CO2 to sweetened water are markedly foaming.   But, what is foam ?    An extremely complex system of polydisperse gas bubbles, separated by draining films.   The total amount of liquid lies in two separate forms: liquid and draining films like in the figure below, at left side.   Until very recently, the main studies of foam were empiric and based on the studies of the physicist Joseph Plateau.   Bubbles, something we all played with when children, in the reality are one of the most difficult object of Mathematical Physics.   Foam dynamics, however visible at naked eye, is a process involving simultaneously the modification occurring in a multitude of objects.   Each object with a size different than others.   The mathematical model where they all have identical diameter and volume is just a useful banalization.   And it is because of this intrinsic complexity that foam had, until recently, mainly been studied by the mere observation of its effects, say in an empiric way. 




Bubbles’ equilibrium is based on the minimization of surface area of liquid films:

  • three lamellae of foam meet at an edge, implying three coplanar angles of 120°;
  • four edges meet at a point, determining angles ~109°.

where the foam bubbles’ shapes are a consequence of the fact that bubbles’ own dynamics follows these Laws.  The special case of a foam composed of same volume bubbles, is  what we observe after each one of them has assumed the geometric shape of a regular pentagonal dodecahedron.  


  When all bubbles in a foam have same volume, they all assume the geometrical shape of the regular pentagonal dodecahedron (image credit VG-3/CC 3.0)



In the foam bed, thin liquid films separate the pentagonal dodecahedral bubbles from each other. Each bubble has twelve liquid films that it shares with twelve surrounding bubbles.   The edge at which these films meet is called a "Plateau border".    Each bubble has thirty Plateau Borders as its edges; three bubbles share each Plateau border.    Mind the familiar beer aspect when it’s in a glass.    They are the Plateau Borders those responsible for the visible thinning of foam films and foam drainage.    Bottling Companies and Filling specialists know how relevant is to prevent the oxidation resulting after a direct contact between the beverage and the oxygen in the air.  


  Nanometric details of the Plateau Border visible in a picture obtained by a Cryo-Scanning Electron Microscope (image credit N. Duerr-Auster/ETH/2008)




Foam shape


Prevention made necessary because of the joint effect of gravity force and Plateu physical laws, starting the:

  • thinning process;
  • drainage process.     

A deep exam of these processes is out of the scopes of these pages.   However, the Law of capillarity of Laplace’s implies a pressure gradient.  Pressure gradient directed from the center of the liquid film, towards the center of the Plateau border.   As an example, originating the familiar drainage of beer from films into Plateau Borders, we all have seen.   


  Ongoing thinning and drainage in a foam



Following the local geodesics, are the Plateau Borders those providing the path for the liquid drainage. 



    Laplace’s Law of capillarity implies a pressure gradient. Pressure gradient directed from the center of the liquid film, toward the center of the Plateau border.  As an example, originating the familiar drainage of beer from films into Plateau Borders. The Plateau Borders whom, following the local geodesics, provide the path for the liquid drainage



Electronic Inspection of a poliphase medium


We have examined here the simplest case, no foam at all: we’ll see in the following how-to treat the case with foam.   The figure at right side shows the special foam curve, programmed along an inspector commissioning phase, to reach minimum false rejects and maximum fill level inspection performances, when checking for under and over filled bottles.   The underfilling boundary appears light-blue coloured and over filling boundary depicted with blue coloured.   Each white colour dot in between light and dark blue splines represents, for each one bottle, the couple of independent measurements (x, y) where:    

  

               (x, y)   =   (Foam, Liquid) 


 Foam-phase + liquid phases, after a sample of one thousand accumulated measurements. Each  white colour dot represents a the couple of measurements (Foam; Liquid) referred to different statuses of the water



In the graphic above, the ordinates associated to each white dot in the vertical axis, are related to those in the horizontal axe only because referred to one and the same bottle.    But, they are not function of the abscissas associated to the same point in the horizontal axe.   Also, they are the results of measurements fulfilling the condition of independence.   For example, meaning that a change to the conditions of the Foam-phase measurement does not cause effects on the result of the measurement of the Liquid-phase for the same bottle.   Graphics has been rendered monometric to easy its comprehension.   Foam compensation module let the Fill Level Inspection have two separate hardware channels along which two distinct informations are separately processed.   


   The necks full of foam in a European famous Brewery, 10 metres after the starwheel release point of a Filler Machine.  Beer remains densely foamy along ~ 8 to 15 m after bottles’ are released by the Filler.  Where to inspect the fill level ?   A basic empirical rule of design prescribes at least 1 meter of distance by the release (tangent) point per each 10000 containers-per-hour of production speed.  To have the Full Bottle Inspector closer means to artificially inhibit its Quality Control function




“Observable” is the polyphase Beverage in the Neck

beer-immediately-after med

To understand the rationale lying back of the solution hinted by the 2-dimensional graphics before, we invite the Readers to consider that each bottle is filled by a beverage which is a poliphase mix.   Mix created in the Filler Machine and reaching the Electronic Inspector, where each one bottle shows distinct effects for its own set of amounts of molecules in the phases:

  1. liquid-phase, 
  2. foam-phase, 
  3. solid-phase.

Partition true after two remarks:

  • Physics does not conceive any “foam phase”.  Foam is considered an extremely complex system of polydisperse gas bubbles, separated by draining films.  A complex kind of gaseous-phase;
  • “solid phase”, however in negligible amounts, really exists in the bottles, when considering that many Breweries in the World limit the dagnine effects of the excessive foaming, filling beer infeeding the Filler Machine at temperatures as low as the range (4 - 7)ºC suggest.   A few water molecules in solid-phase (crystals) shall be really mixed to the majority of liquid-phase molecular bonds, and the minority of foam-phase (gaseous-phase). 

 The presence of a foamy neck of the bottle implies the necessity for a special version of the HF fill level inspection, thinked on the base of a foam compensation module.  Two separate hardware channels along which two distinct informations have to be separately processed.  What, for liquids like still and carbonated water is unnencessary, results vital for carbonated soft-drinks with added sugar









































These different phases hint to the different interactions with the interactant, the 21 MHz electromagnetic wave.   Different interactions when having a neck with different balances of gas (CO2) and matter (water) in liquid- and foam-phases.    Difference since nearly two centuries synthesized in Maxwell's recognition that the speed v of an electromagnetic wave in the medium (in our application, the composite medium filled bottle neck plus air, interposed between neck and radiator), is different than the speed of light c in the vacuum: 


  phase velocity of light in the medium   ≠  speed of light in the vacuum 

                                    

                                      v    c  ≡  ε0  μ)-1/2

where:

  • ε0    is the value of the dielectric permittivity of the vacuum,
  • μ0    is the value of the magnetic permeability of the vacuum.

As an example, introducing water in liquid-phase in the neck of a bottle at a reference temperature of 20 ºC, they are observed the effects of a: 

  • huge increase (80.1 times) of the dielectric permittivity ε with respect to the value present in a vacuum ε(no water), 
  • nearly neglibile effect for the magnetic permeability  μ = μ0 μrel   

Now, consider that: 

  1. due to the negligible interaction between a PET or glass dielectric and an electromagnetic wave at 21 MHz, the observable B is the Beverage in the neck of the bottle in all its phases, and the air interposed between neck and radiator;
  2. liquid-phase and foam-phase are two ways in which a single observable B, the Beverage, may present itself.  Each one characterised by different physical properties (i.e., angle existing between electric potential and electric current vectors, amplitude of the electric potential, impedance, etc.); 
  3. the container capacity defines a superior limit to the observable, a limit to the sum of its liquid-phase and foam-phase;  
  4. the measurements of the liquid-phase and foam-phase are independent and referred to orthogonal properties.  Orthogonality displayed as a 90º angle existing between their axes (liquid-phase axe oriented as the foam-phase axe, plus 90º); 
  5. considering negligible the humidity of the air, the vectorial superposition of the liquid-phase and of the foam-phase (or their linear combination), is the observable B;
  6. the white dots grouped below at left side, represent bottles whose observable represent common values of the amplitude of the (liquid-phase, foam-phase) vectorial superposition;
  7. the space far from the area where the white dots appear grouped, represents anomalous values for the amplitude of the vector superposition.  A total difference with respect to the vector average amplitude, function of any or more of many independent variables like the:
    1. amount of molecules in liquid-phase;
    2. amount of molecules in foam-phase;
    3. ambient temperature in the medium between the bottle neck and the radiator;
    4. beverage temperature;
    5. efficiency of the earthing system, closing the RF measurement circuit by mean of the Conveyor;
    6. relaxation time, allowing time to let gas bubbles be drained along the Plateau Borders, time dependent on the bottles’ linear speed; 
    7. ….. 

          with their permutations going in the thousands.  

Vectorial Superposition Foam Liquid Environmental factors. “Observable” is the polyphase Beverage in the Neck

beer-immediately-after med
To understand the rationale lying back of the solution hinted by the 2-dimensional graphics before, we invite the Readers to consider that each bottle is filled by a beverage which is a poliphase mix.   Mix created in the Filler Machine and reaching the Electronic Inspector, where each one bottle shows distinct effects for its own set of amounts of molecules in the phases:

liquid-phase, 
foam-phase, 
solid-phase.
Partition true after two relevant remarks:

Physics does not conceive any “foam phase”.  Foam is considered an extremely complex system of polydisperse gas bubbles, separated by draining films.  A complex kind of gaseous-phase;
“solid phase”, however in negligible amounts, really exists in the bottles, when considering that many Breweries in the World limit the dagnine effects of the excessive foaming, filling beer infeeding the Filler Machine at temperatures as low as the range (4 - 7)ºC suggest.   A few water molecules in solid-phase (crystals) shall be really mixed to the majority of liquid-phase molecular bonds, and the minority of foam-phase (gaseous-phase). 
 The presence of a foamy neck of the bottle implies the necessity for a special version of the HF fill level inspection, thinked on the base of a foam compensation module.  Two separate hardware channels along which two distinct informations have to be separately processed.  What, for liquids like still and carbonated water is unnencessary, results vital for carbonated soft-drinks with added sugar















































































These different phases hint to the different interactions with the interactant, the 21 MHz electromagnetic wave.   Different interactions when having a neck with different balances of gas (CO2) and matter (water) in liquid- and foam-phases.    Difference since nearly two centuries synthesized in Maxwell's recognition that the speed v of an electromagnetic wave in the medium (in our application, the composite medium filled bottle neck plus air, interposed between neck and radiator), is different than the speed of light c in the vacuum: 

phase velocity of light in the medium   ≠  speed of light in the vacuum 

                                    

                                      v  ≠  c  ≡  ( ε0  μ0 )-1/2

where:

ε0    is the value of the dielectric permittivity of the vacuum,
μ0    is the value of the magnetic permeability of the vacuum.
As an example, introducing water in liquid-phase in the neck of a bottle at a reference temperature of 20 ºC, they are observed the effects of a: 

huge increase (80.1 times) of the dielectric permittivity ε with respect to the value present in a vacuum ε0 (no water), 
nearly neglibile effect for the magnetic permeability μ = μ0 μrel   
Now, consider that: 

due to the negligible interaction between a PET or glass dielectric and an electromagnetic wave at 21 MHz, the observable B is the Beverage in the neck of the bottle in all its phases, and the air interposed between neck and radiator;
liquid-phase and foam-phase are two ways in which a single observable B, the Beverage, may present itself.  Each one characterised by different physical properties (i.e., angle existing between electric potential and electric current vectors, amplitude of the electric potential, impedance, etc.); 
the container capacity defines a superior limit to the observable, a limit to the sum of its liquid-phase and foam-phase;  
the measurements of the liquid-phase and foam-phase are independent and referred to orthogonal properties.  Orthogonality displayed as a 90º angle existing between their axes (liquid-phase axe oriented as the foam-phase axe, plus 90º); 
considering negligible the humidity of the air, the vectorial superposition of the liquid-phase and of the foam-phase (or their linear combination), is the observable B;
the white dots grouped below at left side, represent bottles whose observable represent common values of the amplitude of the (liquid-phase, foam-phase) vectorial superposition;
the space far from the area where the white dots appear grouped, represents anomalous values for the amplitude of the vector superposition.  A total difference with respect to the vector average amplitude, function of any or more of many independent variables like the:
amount of molecules in liquid-phase;
amount of molecules in foam-phase;
ambient temperature in the medium between the bottle neck and the radiator;
beverage temperature;
efficiency of the earthing system, closing the RF measurement circuit by mean of the Conveyor;
relaxation time, allowing time to let gas bubbles be drained along the Plateau Borders, time dependent on the bottles’ linear speed; 
….. 
          with their permutations going in the thousands.   The gaseous-phase (foam-phase) and liquid-phase of water in a closed bottle can be considered unrelated physical properties.   The vectors' orthogonality reflects the fact their measurements' independence.    All High Frequency Fill Level Inspections treating foaming beverages are sensible to a long row of other variables, in the graphic above simplified for reasons of clarity and synthesized as: “environment”.   Here, as environment is meant the vector superposition of the projections in that direction of many other terms.  Terms which, each their own  boundaries, are variable independent by the bottle's liquid content.    Examples the environmental temperature, environmental humidity, container speed during filling, filling pressure, off-axis in millimetres of the High Frequency Fill Level Inspection bridge with respect to the bottle, etc.   

Then, the High Frequency bridge's final measurement, namely the vector impedance Z, can be represented as a function of the vectorial superposition of foam-phase, liquid-phase and many others without relation with the bottle’s content in millilitres, but intervening in the measurement.    As a consequence, the module of the vector impedance Z results fluctuating, originating the inspection’s notorious False Rejects.   Comparing this graphics with an analogue prepared by the data arising after one consecutive year of operation of an X-ray or gamma-ray fill level inspection, presents a bold reality to the Bottler.  The HF Fill Level Inspection ranks as the worst technology which could have been thinked to care Quality and Production interests.   Hundredths of times more expensive than a LASER or Infra Red Fill Level Inspection, and exposed to superior false rejects (losses).   False rejects which could be prevented, without downgrading the inspection Quality, only paying a well definite price nearly no Bottler accepts to pay (nor was informed he’d have had to pay).   Namely, sensitivity thresholds readjustments of the under- and over-filling threshold polygonal made by Electronic Maintenance Technicians 3-4 times along the 24 hours, all days along 10-15 years.

  The amounts of water in gaseous-phase (foam-phase) and liquid-phase in a closed bottle can be considered unrelated physical properties. Vectors' orthogonality reflects their measurements' independence.  All High Frequency Fill Level Inspections treating foaming beverages are sensible to a long row of other variables, in the graphic above simplified for reasons of clarity and synthesized as: “environment”.   Here, as environment is meant the vector superposition of the projections along that direction of many other terms.   Environmental temperature, environmental humidity, container speed, filling pressure, off-axis in millimetres of the HF Fill Level Inspection bridge with respect to the bottle, etc.   The High Frequency bridge's final measurement, the vector impedance Z, can be represented as a function of the vectorial superposition of foam-phase, liquid-phase and many others without relation with the bottle’s content in millilitres.  As a consequence, the module of the vector Z results fluctuating, originating the inspection’s notorious False Rejects




















































Orthogonal Vectors and Scalars 

The gaseous-phase (foam-phase) and liquid-phase of water in a closed bottle can be considered unrelated physical properties.   The vectors' orthogonality reflects the fact their measurements' independence.    All High Frequency Fill Level Inspections treating foaming beverages are sensible to a long row of other variables, in the graphic above simplified for reasons of clarity and synthesized as: “environment”.   Here, as environment is meant the vector superposition of the projections in that direction of many other terms.  Terms which, each their own  boundaries, are variable independent by the bottle's liquid content.    

Some examples being the environmental temperature, environmental humidity, container speed during filling, filling pressure, relevant mechanical factors like the off-axis in millimetres of the High Frequency Fill Level Inspection bridge with respect to the bottle, etc.   On the first of the variables listed before, the environmental temperature, we’ll recall here subjects treated with more details elsewhere in these pages.   To have a complete idea of the reasons why the temperature of the beverage, a scalar amount, plays a role so important during fill level inspections, it results necessary to extend the notation for the dielectric permittivity ε out of the familiar field RC of Real numbers to the field of Complex numbers.   The Complex vector dielectric permittivity, for an em wave ω (wave number ω = 2 π ν ):   

                                   

                                     ε (ω)  =  ε′ (ω)  +  i ε′′ (ω)


has a real and an imaginary part, both representing components of the frequency response of the medium, differentiated following their phases:

                                                 ε′     in-phase          

                                                 ε′′    out-of-phase

and where:

                    ε′    →    refraction of the electromagnetic wave;

                    ε′′   →    absorption of the electronic, vibrational and rotational transitions;


This last imaginary component  i ε′′ (ω)  is  related  to  the  dissipation  of  energy  within  the medium.   Energy dissipation is related to the average molecular and bonds’ motion, say to the beverage water temperature.   As an effect, the dielectric permittivity ε will change continously, as temperature decrease. 

In conclusion, the High Frequency bridge's final measurement, namely the vector impedance Z, is vectorial superposition of foam-phase, liquid-phase plus many other variables unrelated with the bottle’s content in millilitres but intervening in the measurement.    As a consequence, the module of the vector impedance Z results fluctuating, originating the inspection’s notorious False Rejects.   Comparing this graphics with an analogue prepared by the data arising after one consecutive year of operation of an X-ray or gamma-ray fill level inspection, presents a bold reality to the Bottler.    The HF Fill Level Inspection ranks as the worst existing technology when studying how-to care Quality and Production interests.   Hundredths of times more expensive than a LASER or an Infra Red Fill Level Inspection, and exposed to worse false rejects (losses).   False rejects which could be prevented, without downgrading the inspection Quality, only paying a well definite price nearly no Bottler accepts to pay (nor was informed he’d have had to pay).   Namely, sensitivity thresholds readjustments of the under- and over-filling threshold polygonal made by Electronic Maintenance Technicians 3-4 times along the 24 hours, all days along 10-15 years.  



Sample Space Partitioning

We listed before at the § 7.1., 7.2., 7.3.,…, 7.7., some of the multiple causes which let the amplitude of the vectorial superposition: 


  The amounts of gaseous-phase (foam-phase) and liquid-phase of water in a closed bottle can be considered unrelated physical properties. The vectors' orthogonality reflects the fact their measurements' independence







                                       foam  +  liquid


of the measurements of the impedance Z be a constantly fluctuating spread random variable.   If a statistical amount of bottles, all of them filled “exactly the same content” (same liquid-phase, foam-phase, at the same pressure, speed, temperature, etc.) should be consecutively inspected, the expected result is still a widely dispersed function as that visible below at right side.    Spreading due to multiple factors acting alone, or in combination with other terms whose permutations goes in the hundredths.   The evaluation of the liquid- and foam-phases is accomplished in a graphic where the measured properties are attributed to mutually orthogonal axes.  Orthogonal because, as an example, a small change in one of the variables determining the liquid-phase measurement does not determines any related sensible change in the measurement of the foam-phase of that individual bottle.   The graphic space is partitioned like visible in the figure on side.   Here shown both the over- and under-filling threshold limits.   In the example, the: 

  • dark-blue overfilling threshold limit reduced to a single upper value because referred to a Bottling Company whose Quality Control Dept.’s policies do not consider the over-filled bottles to be interpreted as Positives (defects);
  • light-blue underfilling threshold limit is the white coloured open polygonal.  Its contour is modelled around the expected spread distribution of the couples of measurements.  When the balance of the liquid- and foam-phases results in a certain bottle below the ordinate locally imposed by the contour, the container is considered underfilled and designated to the rejection.

Partitioning parameterised during Commissioning phase, thus fine-adjusting the sample space of the High Frequency Fill Level Inspection with Foam Compensation.   Purpose of the sample space partitioning reflects that two of the many variables intervening in the process, are strictly related.   Other variables, on the opposite, have the capability to emulate a defective bottle.   Then, if the possibility of partitioning should not exist, these events should in the end be unavoidably translated in False Positives, or False Rejects’ losses on production.




The many alternative meanings of “Fill Level-defective” 

when inspecting foaming Beverages

 


The many alternative meanings of “Fill Level-defective” 

The anomalous amplitude of the vector superposition has several keys of interpretation.  It originates by any or more of the following Root Causes:

underfilling, caused by a leaking closure which allowed the beverage to get out before fill level inspection;
underfilling, caused by an emergency stop of the Filler and Closer Machines, after the bottles were filled and before to be closed, which let the foaming beverage get out of the open bottle;
underfilling, in presence of a correct closing, due to an insufficient vectorial superposition of the beverage in its liquid- and foam-phases;
underfilling, associated to a PET bottle anomalously deformed by CO2 pressure, resulting an actual False Positive; 
underfilling, associated to a bottle inspected by mean of an inspection bridge temporarily damped by a precedent open and foaming bottle, resulting an actual False Positive; 
underfilling, after an extended stop of the Filler Machine tank and then also of the beverage's flow which let the Filler tank and piping start to slowly escalate their Temperature toward conditions of thermal equilibrium with the Environment.  In these conditions, at the restart of the Production the observable B, the Beverage, shall result hot.   So hot that the second-order effects of the known relation existing between the measured neck's impedance and the beverage Temperature, are amplified til inducing actual False Positives;
overfilling, in presence of a correct closing, say an excessive vectorial superposition of the beverage in its liquid- and foam-phases;

  Massive rejects are expected by the Fill Level Inspections treating foaming beverages, after an extended stop of the Filler Machine tank and then also of the beverage's flow cooling the Filler tank and piping.  During the stop they start to slowly escalate their Temperature toward conditions of thermal equilibrium with the Environment.  In these conditions, at the restart of the Production the observable B, the Beverage, shall result hot.  “Hot” is in the reality a laymen way to express the sensible increase in the dimension of the state space representing the expected outcomes of the Fill Level Inspection of that observable.  So hot that the second-order effects of the known relation existing between the measured neck's impedance and the beverage Temperature, results amplified til inducing actual False Positives




















The anomalous amplitude of the vector superposition has several keys of interpretation.  It originates by any or more of the following Root Causes:

  • underfilling, caused by a leaking closure which allowed the beverage to get out before fill level inspection;
  • underfilling, caused by an emergency stop of the Filler and Closer Machines, after the bottles were filled and before to be closed, which let the foaming beverage get out of the open bottle;
  • underfilling, in presence of a correct closing, due to an insufficient vectorial superposition of the beverage in its liquid- and foam-phases;
  • underfilling, associated to a PET bottle anomalously deformed by CO2 pressure, resulting an actual False Positive; 
  • underfilling, associated to a bottle inspected by mean of an inspection bridge temporarily damped by a precedent open and foaming bottle, resulting an actual False Positive; 
  • underfilling, after an extended stop of the Filler Machine tank and then also of the beverage's flow cooling Filler tank and piping.  In these cases, they start to slowly escalate their Temperature toward conditions of thermal equilibrium with the Environment.  In these conditions, at the restart of the Production the observable B, the Beverage, shall result hot.   So hot that the second-order effects of the known relation existing between the measured neck's impedance and the beverage Temperature, are amplified til inducing actual False Positives;
  • overfilling, in presence of a correct closing, say an excessive vectorial superposition of the beverage in its liquid- and foam-phases;
  • …...

Finally, the Foam Compensation hardware, firmware and software for liquids like still and carbonated water are unnencessary, but vital for carbonated soft-drinks with added sugar.  






This website has no affiliation with, endorsement, sponsorship, or support of Heuft Systemtechnik GmbH, MingJia Packaging Inspection Tech Co., Pressco Technology Inc., miho Inspektionsysteme GmbH, Krones AG, KHS GmbH, Bbull Technology, Industrial Dynamics Co., FT System srl, Cognex Co., ICS Inex Inspection Systems, Mettler-Toledo Inc., Logics & Controls srl, Symplex Vision Systems GmbH, Teledyne Dalsa Inc., Microscan Systems Inc., Andor Technology plc, Newton Research Labs Inc., Basler AG, Datalogic SpA, Sidel AG, Matrox Electronics Systems Ltd. 



                                                                                                                                                                                                                                                                                                                                                                                                                                                         
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