Patent Publication Number: US-10788347-B2

Title: Method for estimating physical characteristics of two materials

Description:
RIGHTS OF THE GOVERNMENT 
     The invention described herein may be manufactured and used by or for the Government of the United States for all governmental purposes without the payment of any royalty. 
    
    
     FIELD OF INVENTION 
     The present invention relates to the field of monitoring material within a sensor and altering the conditions of the material based on the monitoring and particularly relates to estimating characteristics of two materials flowing within a tube based on a tomogram of the materials produced by electrical capacitance tomography and altering the flow of the materials based on the estimated characteristic. 
     BACKGROUND OF THE INVENTION 
     Flowing mixtures, such as flowing mixtures of phases of a material are monitored by a variety of techniques including electrical capacitance tomography, which produces a two-dimensional tomogram representing the permittivity of the material being sensed. This data is interpreted in different ways. For example, it may be used to calculate the liquid fraction in the material. 
     However, other characteristics of the material may be more important or have a different useful significance than a simple liquid fraction. For example, in a flow of multiphase material, such as a flow of liquid and gas, the surface areas of various interfaces provide important information. In a liquid/gas flow, the surface area of the interface between the liquid and the gas, the surface area between the liquid and the wall of a tube, and the surface area between the gas and the wall of the tube can be important. For example, in a heat exchange application, the surface area between the gas and wall would be inversely proportional to the heat flow into or out of the tube, and thus such surface area would be an important parameter to monitor in order to understand the heat transfer performance and may be used to modify or alter the flow to improve heat transfer performance. The surface area between a gas and liquid phase in a tube would also be an indicator of a pressure drop in the flow within the tube and the rate of change in such surface area would be a predictor of future waves within the flow. Thus, the gas/liquid surface area of a flow could be monitored, and alarms or trigger values could be set so as to change the flow to correct unwanted pressure drop and to avoid unwanted waves in the flow. However, in many circumstances, it is extremely difficult or impossible to monitor the aforementioned characteristic of flowing materials. What is needed in the art, therefore, is a process or method that solves or avoids the difficulty of directly monitoring the aforementioned characteristics of flow. 
     BRIEF SUMMARY OF THE INVENTION 
     Embodiments of the invention address the need in the art by providing a method which may assist in solving or avoiding the difficulties of directly monitoring the aforementioned characteristics of flow and instead may provide an accurate estimate of those characteristics by monitoring or calculating hypothetical physical characteristics of the materials. These hypothetical characteristics do not truly exist, but may be used to estimate physical characteristics that do exists. Also, the hypothetical physical characteristics may be used to control the materials to improve performance or avoid catastrophic failures. In summary, the hypothetical surfaces areas estimated by the technique described herein are a useful measure of current conditions and a predictor of future conditions in a flow of materials and thus the techniques constitute a valuable tool for monitoring flow. The present invention provides a way to estimate such surface areas quickly and reliably; and other parameters (such as heat transfer characteristics and pressure drops) may be estimated from the hypothetical characteristics. 
     In accordance with one aspect of the invention, tomography data corresponding to multiple materials (including multiple phases of the same material) is analyzed by a new method to derive information about the materials in a sensor volume in a different way. In one embodiment, a mixture of a first material and a second material is disposed in a three-dimensional sensor volume which is defined by a circumferential sensor wall and the length dimensions of a sensed volume of a sensor volume. A matrix of parallel voxels is defined within the sensor volume with each voxel having x, y and z dimensions. At least one parameter of the material is measured within the sensor volume, and a tomogram is produced. The tomogram is a two-dimensional matrix defined within a perimeter, and the matrix contains multiple values with one value being associated with each voxel. Each value represents the amount of the first material in the associated voxel. Multiple hypothetical points within the sensor volume are calculated based on the multiple values of the tomogram. The z coordinate of each point is calculated from at least one value of the tomogram, and the x and y coordinates of each point are based on the x and y coordinates of at least one associated voxel. The points are then used to calculate at least one hypothetical physical characteristic of the material within the sensor volume. For example, a hypothetical physical characteristic could be an hypothetical surface representing the interface between the first and second materials, or it could be an hypothetical surface representing the interface between the first material and the sensor wall. To be clear, the first material may be a first phase of a material (e.g., a liquid) and the second material may be a second phase of the same material (e.g., a gas). 
     These hypothetical surfaces do not exist but the areas of these surfaces are accurate estimates of surface areas that do exist. For example, the surface area of the hypothetical surface between the first and second materials is a good estimate of the surfaces areas of the interface surfaces that do exists between the first and second materials. Likewise, the area of the hypothetical surface of the wall exposed to the first material is a good estimate of the surface area of the wall that is actually exposed to the first material in the sensor. 
     In one embodiment the hypothetical characteristics may be used to control the flow of materials in a tube. For example, the hypothetical characteristic may be constantly repetitively calculated and compared to a predefined limit. If the limit is exceeded corrective action may be taken. For example, if the area of the hypothetical surface of the wall that is exposed to the first material exceeds a limit, a control signal may be generated and transmitted to a valve causing it to open and provide an increased flow of the second material in the lowing materials thereby driving down the relative amount of the first material in the sensor, which would decrease the area of the wall that is exposed to the first material. 
     In one alternative, the multiple points are used to calculate the surface area of the interface between the first and second materials. In such case, the points are assumed to be points on a hypothetical interface between the first and second materials. A surface is mapped through the points, and the area of the surface is calculated. That calculated area is the estimated surface area of the interface between the first and second materials. A smooth curved surface, or multiple smooth curved surfaces, may be mapped through the points when utilizing this technique, and the curved surface or surfaces will provide accurate results when calculating the hypothetical surface area of the mapped surface. However, a faster calculation may be achieved by mapping a multifaceted surface through the points. In this technique, a triangular planar surface is mapped between combinations of three adjacent points to produce multiple triangular surface areas. None of the triangular surface areas overlap another surface area. The area of each triangular surface area is calculated, and the sum of those surface areas constitutes the surface area of the multifaceted planar surface that was mapped through the points. The overall surface area thus calculated is the estimated surface area of the interface between the first and second materials. Even though area is usually two dimensional in nature, it may be regarded as a form of a three-dimensional data in this case since it is a measure of the area of a three-dimensional shape, namely the hypothetical mapped surface. 
     In a variation of the above method, the sensor volume that is on each side of the mapped surface may be calculated. The volume on one side of the mapped surface is the estimated volume of the first material, and the volume on the other side of the mapped surface is the estimated volume of the second material. Either technique described above, or a different technique, may be employed to define the mapped surface and thereby define the two volumes. Once the two volumes are determined, other characteristics of the volumes may be calculated such as the mass centroid of each volume. Using the mass density of the two materials, the centroids of each material and the overall centroid of the material within the sensor volume may also be calculated. Finally, in a similar manner the cross-sectional area of each material may be determined where the cross-section is taken in a direction perpendicular to the linear direction of the sensor, which is also perpendicular to the flow of material within the sensor. 
     In another alternative, the multiple points may be used to calculate the surface area of the sensor wall that is contacted by either the first or the second material. In this technique, the points may be calculated as discussed above, and only the outermost points are used to calculate the surface area of the sensor wall that is in contact with one of the materials. The outermost points will form a closed loop disposed adjacent to the wall of the sensor. The surface area of the wall on one side of the closed loop is the surface area of the first material, and the surface area of the wall on the other side of the closed loop is the surface area of the second material. In this alternative, the outermost points may be calculated in different ways. For example, the outermost points could be determined by simply using the outermost values of the tomogram to identify the z dimension of the outermost points, and the x and y coordinates of the points may be determined by reference to one or more voxels. Alternatively, the outermost points could be determined by identifying four adjacent tomogram values arranged in a square pattern, and calculating a single point based on the four adjacent tomogram values. The z dimension of the point would be based upon the values of the four adjacent tomogram values. The x and y dimensions of the point would be based upon the x and y positions of the four adjacent tomogram values or their associated voxels. 
     While the techniques of the present invention may be used in a variety of settings, a particularly useful application is for analyzing an electrical capacitance tomogram corresponding to a flowing mixture of multiple phase materials, such as a liquid and gas. In such case, a plurality of capacitive sensors are disposed in a side-by-side relationship around and adjacent to the circumferential sensor wall. Typically, the sensor wall is cylindrical. Each sensor has a width and a length, and the length of each sensor is disposed parallel to the flow direction of the flowing mixture within the sensor. The capacitance of the material in the sensor volume is measured, and the tomogram is produced in the form of a two-dimensional matrix as described above. Each value in the matrix corresponds to the electrical permittivity of the material within an associated voxel. The values in the tomogram are then used to calculate multiple points within the sensor volume with the z coordinate of each point being calculated from at least one value of the tomogram. The x and y coordinates of each point correspond to at least one associated voxel of at least one value. Physical characteristics, such as the surface area of the interface between the gas and liquid, are calculated based upon the multiple points and based upon the assumption that the multiple points lie on a hypothetical interface between the first and second phases of the material within the sensor volume. The surface areas and interfaces discussed herein that are used to produce the estimates are hypothetical surfaces. It is highly likely that the hypothetical surfaces do not exist in the form in which they are visualized, but actual surfaces in a flowing mixture do exist that closely correspond in surface area to the hypothetical surfaces. Thus, the areas of the actual are efficiently estimated using the areas of the hypothetical surfaces. 
     The methods described herein are based on a tomogram corresponding to the flow within a tube. It will be understood that multiple tomograms may be taken over time and the methods may be used to analyze each tomogram and produce an estimated area of an interface for each tomogram, and then the estimated areas may be averaged or filtered to produce smoothed area calculations over time. Also, using the multiple tomograms, an estimated area may be determined for each tomogram and the multiple estimated areas may be analyzed to determine a rate of change of the estimated area over time. In the case of a flowing multiphase material, the rate of change in the estimated area of the interface between the two phases may be used as a predictor of future waves or oscillations in the flowing fluid. In some circumstances, the presence of waves or oscillations in a multiphase flow can cause catastrophic failures and thus such oscillations are to be avoided. Thus, a computer or controller may be programmed to rapidly calculate the area of the hypothetical interface between the two phases in a flowing multiphase material. The computer then calculates the rate of change in the calculated estimated areas and compares it to a predefined limit. If the limit is exceeded, the computer then issues a command to correct the flow. For example, a command may be issued to a valve causing it to open and insert additional materials into the flowing multiphase materials and thereby corrected the dangerous condition. In extreme cases, the computer may issue a shutdown command that is transmitted to a valve or a pump which will cause the flow to immediately stop. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention may best be understood by reference to various embodiments and variations of the invention, examples of which are described below in conjunction with the drawings in which: 
         FIG. 1  is an illustration of various different types of multiphase flow in a tube; 
         FIG. 2 a    is a somewhat diagrammatic isometric view of a tubular sensor for measuring capacitance of a flowing material and producing a tomogram of the material; 
         FIG. 2 b    is a somewhat diagrammatic view of the capacitive sensor shown in  FIG. 2 a   , but the sensor in  FIG. 2 b    is unrolled and shown in a planar configuration for illustrating its construction; 
         FIG. 3  is a tomogram produced by the sensor shown in  FIG. 2   a;    
         FIG. 4 a    is an illustration of the tube containing a multiphase flow with a sensor disposed to sense the multiphase flow; 
         FIG. 4 b    is an isometric drawing of the voxel shown in  FIG. 4   a;    
         FIGS. 5 a -5 d    are isometric views of voxels with each Fig. showing a different type of multiphase flow of material within the voxel; 
         FIG. 6 a    is a tomogram produced by the sensor of  FIG. 2 a    and corresponding to the illustration of  FIG. 6   b;    
         FIG. 6 b    is an isometric illustration of multiple points in a tube that are mapped to a hypothetical surface corresponding to the interface between a a liquid phase and a gas phase flowing in the tube; 
         FIG. 7  is a graphical illustration of a tomogram showing the pixels and the elements of the tomogram along with illustrations of points that are derived from the tomogram; 
         FIG. 8  is an illustration of a non-uniform mesh defined by a plurality of points in three-dimensional space there calculated from a tomogram; 
         FIG. 9  is a graphical illustration of elements and master elements in a mesh that is calculated from and defined by a plurality of points in three-dimensional space that are calculated from a tomogram; 
         FIG. 10  is a graphical illustration of how each four-sided element of a mesh is broken into two triangular planar surfaces which are then used to calculate the surface area of a mapped surface; 
         FIG. 11 a    is a graphical illustration of the outermost boundary pixels of the tomogram; 
         FIG. 11 b    is an isometric illustration of the hypothetical boundary line to surface areas on the wall of the sensor; 
         FIG. 12  is a graphical illustration of representative consecutive boundary pixels and the vectors used to find the angle between the pixels; 
         FIG. 13  is an isometric illustration of a test surface use for estimating the error in using a trapezoidal calculation to determine the surface area of the sensor wall and contact with a particular material; 
         FIG. 14  is a graphical isometric illustration of an element or voxel for purposes of describing how the volume of an element is calculated; 
         FIG. 15  is a schematic diagram of a voxel showing two different phases within the voxel; 
         FIGS. 16 a  and 16 b    are illustrations of flow showing three-dimensional and two-dimensional views of the cross-sectional areas of each material phase in the flow; 
         FIG. 17  is an illustration of a sensor detecting the interface between to materials, salt and air, in the sensor; 
         FIG. 18  is a three dimensional illustration of 4 different tomograms representing 4 different measurements of 4 different levels of salt within a sensor; 
         FIG. 19  is a graph showing the correspondence of the theoretical solid fractions and the measured solid fractions of salt in the sensor of  FIG. 17 , and 
         FIG. 20  illustrates an apparatus for implementing the methods discussed herein. 
     
    
    
     DETAILED DESCRIPTION 
     Referring now to the drawings in which like reference characters refer to like or corresponding parts or elements throughout the several views, a technique is disclosed to estimate interfacial areas between two materials, such as a liquid and a gas. A technique is also disclosed for estimating interfacial areas between a material and a wall, such as between a liquid and wall and between a gas and a wall in two phase flow. The technique uses a standard 2D sensor in a fashion to infer 3D information about the liquid/vapor profile when the sensor length is much longer than the diameter. It will also allow the cross-sectional flow areas for the gas and liquid to be estimated as a function of the axial dimension of the sensor. The centroid of the mass in the sensor element can also be determined. Tomograms of the flow inside a sensor may, in some embodiments, be created by commercially available electric capacitance tomography (ECT) systems. The methods disclosed herein provide a quantitative interpretation of the tomogram providing estimates of 3D physical area information. 
     Two phase flows consisting of a liquid and gas phase are common in many applications such as air conditioning, chemical and petroleum industries. Predicting pressure drop and heat transfer rates in these flows typically depends on knowledge of the void fraction which can be defined on an area or volumetric basis. In many cases the flow may be pulsating and chaotic which leads to difficulty in characterizing the interfacial areas between the tube wall and each phase and the interfacial area between each phase. The impact of the interfacial areas, A lw , A gw , A lg  can be seen in the transient one dimensional momentum equations for separated flow given as: [1] (The bracketed numbers refer to references listed at the end of this Detailed Description.) 
                         ρ   l     ⁡     (         ∂       V   _     l         ∂   t       +       V   l     ⁢       ∂       V   _     l         ∂   z           )       =         γ   l     ⁢     cos   ⁡     (   θ   )         -         τ   lw     ⁢     A   lw             π   ⁢           ⁢     D   2       4     ⁢   L       +         τ   lg     ⁢     A   lg             π   ⁢           ⁢     D   2       4     ⁢   L       -       ∂   P       ∂   z           ⁢     
     ⁢     liquid   ⁢           ⁢   phase             (     1   ⁢   a     )                     ρ   g     ⁡     (         ∂       V   _     g         ∂   t       +       V   g     ⁢       ∂       g   _     l         ∂   z           )       =         γ   g     ⁢     cos   ⁡     (   θ   )         -         τ   gw     ⁢     A   gw             π   ⁢           ⁢     D   2       4     ⁢   L       +         τ   lg     ⁢     A   lg             π   ⁢           ⁢     D   2       4     ⁢   L       -       ∂   P       ∂   z           ⁢     
     ⁢     gas   ⁢           ⁢   phase             (     1   ⁢   b     )               
(The numbers enclosed within parentheses at the end of equations are equation numbers.) As may be observed in the equations, the transient liquid one dimensional momentum decreases in response to an increase in the area of the liquid/wall interface and increases in response to an increase in the area of the liquid/gas interface. Also, the transient gas one dimensional momentum decreases in response to increases in both the area of the gas/wall interface and the area of the liquid/gas interface. Thus, to understand and predict these momentums, it is important to know or estimate the aforementioned interfacial areas, A lw , A gw , A lg , and such information can be used separately or in conjunction with void fraction information to better predict physical characteristics of the flowing material, such as pressure drop and heat transfer rates.
 
     In this discussion, the following nomenclature is used: 
     Nomenclature 
     a=x 2 −x 1  the width of a pixel (m) 
     A i  area of the ith pixel (m 2 ) 
     A gw  interfacial area between gas and wall (m 2 ) 
     A l  cross sectional flow area of the liquid (m 2 ) 
     A lg  interfacial area between liquid and gas (m 2 ) 
     A lw  interfacial area between liquid and wall (m 2 ) 
     A g  cross sectional flow area of the gas (m 2 ) 
     Ā,  B  vectors 
     b=y 2 −y 1  the height of a pixel (m) 
     C i  i th  row in the connectivity matrix gives pixel number and the corner node numbers 
     D diameter (m) 
     ECT Electric capacitance tomography 
     H(x) Heaviside step function given in eq. 27 
     J i  i th  Jacobian given by eq. 7 
     L sensor length (m) 
     L liq  length of a voxel that is occupied by liquid (m) 
     m i  mass of liquid and gas in the i th  voxel (kg) 
     n 1   i , n 2   i , n 3   i , n 4   i  corner node numbers for the i th  element 
     N frame  number of frames used in a temporal average 
     NP number of active pixels used in a tomogram (e.g., 812 in the illustrated embodiment) 
     NBP number of boundary elements (e.g., 88 in the illustrated embodiment) 
     P pressure (Pa) 
     R radius of tube (3.5 mm) 
     S i  area of the i th  element on the liquid/gas interface 
     t time (s) 
       V  mean velocity (m/s) 
     x i  x coordinate of the ith node (m) 
     x ci , y ci , z ci  coordinates of the centroid of the mass in the i th  voxel 
       x ,  y ,  z  coordinates of the centroid of the mass in the sensor volume 
     z axial coordinate (m) 
       z   i  average value of the i th  element of area of wall wetted by liquid or the average of the corner nodes for a liquid/vapor surface element 
       z   gi ,  z   li  centroidal axial location for the gas and liquid respectively in the i th  voxel 
     Subscripts 
     g gas 
     l liquid 
     w wall 
     Greek Variables 
     γ specific weight (N/m 3 ) 
     Δx, Δy lengths of the sides of the square elements (m) 
     Δθ i  angle subtended by the ith area element of wall wetted by liquid (radians) 
     ε relative permittivity 
     ε *  normalized relative permittivity ratio 
       ε   *  spatial average of ε *  over a tomogram 
     &lt; ε   * (t)&gt; temporal average of  ε   *    
     θ angle (radians) 
     η vertical direction in master element 
     ξ horizontal direction in master element 
     ρ density (kg/m 3 ) 
     τ shear stress (Pa) 
     ϕ, ϕ max  angle and maximum value of the angle used in  FIG. 13  (radians) 
     φ ij  φ j −φ i ;φ=x, y, z notation used in eq. 14a. 
     ψ i  i th  shape function used in the interpolation functions given by eq. 8 
     Many techniques have been used to estimate the void fraction including optical, gamma ray attenuation, and techniques based on either electric resistance or capacitance. The present approach uses an electric capacitance tomographic (ECT) technique to estimate the liquid/vapor interface in flows that may have different physical characteristics as shown in  FIG. 1 . The flow may include bubbles of gas (phase  1 ) contained within a flow of liquid (phase  2 ) as shown by flow  20  of  FIG. 1 . Alternatively, flow  22  illustrates plugs of gas in a liquid; flow  24  shows slugs; flow  26  shows an intermittent flow of gas; flow  28  shows a stratified wavy flow of gas; and flow  30  shows an annular flow of gas in a liquid. The method described herein estimates the area interfaces of the phases regardless of the type of flow. 
     Some embodiments of the invention may use an ECT sensor  32  from Industrial Tomographic Systems [2].  FIG. 2 a    is a schematic of the sensor  32 , which consists of eight electrodes  34  on a flexible printed circuit board  36 . In  FIG. 2 a    the sensor  32  is shown in an isometric view as it appears in service wrapped around a tube  38 . The measurement volume  40  (See  FIG. 4 a   ) is nominally the volume inside the electrodes  34  but is found to be slightly larger due to fringing effects at the ends of the electrodes  34 . In  FIG. 2 b    the circuit board  36  is shown unwrapped with the length of the electrodes nominally 51 mm. The tube diameter for the data used in this application is 7 mm so the length to diameter ratio is approximately 7.3. The ECT sensor is measuring capacitance and producing values corresponding to electrical permittivity of the material in the sensor volume  40 , and a tomogram is produced in which each value (pixel) in the tomogram is a measurement of electrical capacitance corresponding to the electrical permittivity of a discrete volume (a voxel) within the sensor volume. The measured values are of course absolute value measurements of electrical permittivity that may be calibrated to accurately correspond to the electrical permittivity of the material within the voxels. Accurate absolute permittivity values may be used in the present invention but are not necessary. Instead, for example, normalized permittivity values may also be used. If gas and liquid phases of a material are present in the sensor, all permittivity values may be normalized against the permittivity of the gas. The gas permittivity and the liquid permittivity may each be determined empirically by filling the sensor with the gas and liquid separately and measuring permittivity. There is no need to calibrate those measurements to make them accurate in an absolute sense. Instead, the measurements are normalized against the measured gas permittivity. A normalized liquid permittivity is the liquid permittivity less the gas permittivity. A normalized measured permittivity when both gas and liquid are present is the measured permittivity minus the gas permittivity, and a normalized permittivity ratio is the normalized measured permittivity divided by the normalized liquid permittivity. The method uses the values of the normalized permittivity ratio, ε * , defined in eq. 2 at each pixel to develop the liquid/vapor interface. When a voxel in the sensor is full of gas, the normalized permittivity ratio is zero (0), and when voxel in the sensor is full of liquid, the normalized permittivity ratio is one (1). When the voxel is filled with half liquid and half gas, the normalized permittivity ratio is one half (0.5). 
     
       
         
           
             
               
                 
                   
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     ε g  is the relative permittivity of the gas phase 
     ε l  is the relative permittivity of the liquid phase 
     ε is the relative permittivity measured at a given pixel in the tomogram. 
     Thus a value of ε * =0 corresponds to gas and ε * =1 corresponds to a liquid. 
     A representative tomogram  42  of a two phase mixture of the refrigerant R134a is shown in  FIG. 3 . Tomogram  42  is hatched to represent the relative permittivity values in each pixel of tomogram  42 . For example, the bottom pixels are hatched with solid vertical lines to indicate a relative permittivity of 1, meaning the lower portion of the sensor value was completely filled with liquid. The hatching and the corresponding relative permittivity are shown in the legend  43 . The relative permittivity decreases in the higher areas of the tomogram, and the upper area (hatched with vertical dashes) has the lowest permittivity which is represented in this illustration as a 0.1 relative permittivity. In the past use of this tomogram, an average of the pixel values has been used as an estimate of the liquid fraction at a cross section as given in eq. 3. 
                         ɛ   _     *     ⁡     (   t   )       =         ∑     i   =   1     NP     ⁢     ɛ     *     ,   i           NP             (   3   )               
NP=812 is the number of active pixels used in the tomogram
 
The spatial value is a function of time and the temporal average given as
 
     
       
         
           
             
               
                 
                   
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     Here N frame  is the number of frames or tomograms that are used in the time average. The void fraction is then estimated as 1−&lt; ε   * (t)&gt; where the number of frames is large enough to ensure a stationary average. 
     However, this information can be used in a new method to estimate the liquid profile and thus the interfacial areas as well as the centroids of the gas and liquid regions. It also can be used to calculate the estimated volumetric void fraction of the mixture in the sensor volume. 
     Liquid/Vapor Interfacial Area 
     To estimate the liquid profile in the sensor volume, the new method assumes that the pixel value represents the volume fraction of liquid in a rectangular voxel bounded by the pixel area times the length of the sensor as shown in  FIGS. 4 a  and 4 b   , where  FIG. 4 a    diagrammatically illustrates the sensor volume  40  within a tube  38 . In  FIG. 4 a   , three illustrative voxels  52 ,  54  and  56  are shown extending for the length  50  of the sensor volume  40 . A gas slug  60  is shown flowing in the liquid  58  and the gas slug extends beyond both ends of the sensor volume  40 . 
       FIG. 4 b    shows a hypothetical voxel  62  that may exist in a particular sensor volume  40  at a hypothetical sample time. In voxel  62 , the liquid  64  and the gas  66  are separated by a single vertical interface. It is recognized that the voxels within the sensor volume  40  could have other liquid distributions than that shown in  FIG. 4 b    as illustrated in  FIGS. 5 a - d    that diagrammatically illustrate for other voxels  68 ,  70 ,  72  and  74 . In voxel  68 , a drop of liquid  64  is disposed between two regions of gas  66 , and in voxel  70  a drop of gas  66  is disposed between two regions of liquid  64 . Voxel  72  represents a flow of liquid  64  along the bottom of the voxel  72  with a flow of gas  66  just above the liquid  64 , and voxel  74  illustrates a flow in which one inclined interface is disposed between the gas  66  and the liquid  64 . The distributions shown as  5   a  and  5   b  could occur if there are small drops or bubbles so the method will not be as accurate if the features are less than the sensor length. The distribution shown as  5   c  could occur in very smooth stratified flow but is unlikely to be seen in practice. The distribution shown as  5   d  could easily occur. This, however, will be practically the same as the uniform case shown in  FIG. 4 b    because there are 812 pixels in the current sensor with a voxel length to height or width ratio of 232. The tapered section of the liquid profile in a typical flow will be very short. Thus, despite the many different theoretical profiles that might exist in various flows, for purposes of estimating interface areas as described below, the voxels may be deemed to have interfaces as shown in either  FIG. 4 b    or  FIG. 5 d   , and the interface areas may be accurately estimated using such hypothetical interfaces. 
     To estimate surface areas, the values from a tomogram, such as tomogram  80  shown in  FIG. 6 a   , are used to calculate points on a hypothetical interface between two phases of material in the sensor volume  40 . Once the points are determined, a three dimensional curved surface is mapped through the points and the surface area of that three dimensional curved surface is calculated, which is the estimated surface area between the two phases. 
     This procedure is discussed in more detail below in connection with  FIG. 6 a    which illustrates a tomogram  80  generated by a sensor  32  that is sensing a gas and liquid flow, and  FIG. 6 b   , which is a somewhat schematic illustration showing the points  82  calculated from the tomogram  80  using the method described above. The liquid vapor interface  84  within a sensor volume  40  for a horizontal tube  38  is shown in  FIG. 6 b    based on the tomogram shown in  FIG. 6 a   . The points in  FIG. 6 b    have hashing symbols corresponding to the hash shading of the pixels in  FIG. 6 a   . The volume below the points  82  is the liquid phase and the gas is represented by the volume above the points  82 . In  FIG. 6 b    the length has been normalized by the diameter. This is a qualitative indication so an approach to getting a quantitative value for the interfacial areas will be presented. 
     The following discussion for an exemplary embodiment of the invention will use a square mesh, although the described methods may be applied to any non-uniform mesh (such as shown in  FIG. 8 ) as well.  FIG. 7  illustrates the pixel  90  locations based on a square mesh, where each square in the mesh is a pixel  90 . It has a total of 32×32=1024 pixels but only 812 which are totally inside the tube  38  are used. A new mesh is created from the centroids  92  of each pixel  90 , and each square of the new mesh is an element  94 . 
     In summary, the tomogram represents a first square mesh  89  with each square  90  in the mesh representing a pixel. The first step of the disclosed method is to create a second square mesh where the centroids of the first square mesh form the corners of each square element  94  in the second mesh. The second mesh is not fully shown in  FIG. 7  to prevent clutter in the illustration, but six of the square elements  94  of the second mesh are shown in  FIG. 7 , and a magnified view of one square element  94  is shown in the upper right corner of  FIG. 7 . In the discussion above the squares in the first mesh are referred to as pixels. The squares in the second mesh are called elements, and the corners of the elements are called nodes. This terminology is employed to help distinguish the first mesh from the second mesh. After the second mesh of elements  94  is created as described above, the second mesh of elements  94  is used to calculate the points  82  as shown in  FIG. 6 b    as described below in detail. 
     A connectivity matrix is created which lists the elements  94  and the four corner node numbers going clockwise around the element  94 , C i =(i, n 1   i , n 2   i , n 3   i , n 4   i ), where i is the element number and n 1   i , n 2   i , n 3   i , n 4   i  are the node numbers for the element. 
     A matrix corresponding to the x,y,z values of the nodes is also created, where the nodes are the corners of the elements  94 , which are also the centroids of the pixels  90 . The z value is the measured permittivity ratio from the tomogram times the length  50  of the sensor. The surface elements  94  are thus created and are then mapped to a master element  100  as shown in  FIG. 9 . The surface elements  94  are mapped to the master element  100  using a linear transformation given by: 
                     ξ   =         2   ⁢     (     x   -     x   1       )       -   a     a       ;     η   =         2   ⁢     (     y   -     y   1       )       -   b     b               (       5   ⁢   a     ,   b     )               
where a=x 2 −x 1 ; b=y 2 −y 1  are the lengths of the sides of the pixels  90 . The area of the i th  element is then given as: [4]
 
 S   i =∫ −1   1 ∫ −1   1 √{square root over ( J   1   2   +J   2   2   +J   3   2   dξd η)}  (6)
 
J i  are the Jacobians and depend on the mapping function and the element geometry. The Jacobians are given by:
 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       1 
                     
                     = 
                     
                        
                       
                         
                           
                             
                               
                                 ∂ 
                                 y 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 y 
                               
                               
                                 ∂ 
                                 η 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 z 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 z 
                               
                               
                                 ∂ 
                                 η 
                               
                             
                           
                         
                       
                        
                     
                   
                   ; 
                   
                     
                       J 
                       2 
                     
                     = 
                     
                        
                       
                         
                           
                             
                               
                                 ∂ 
                                 z 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 z 
                               
                               
                                 ∂ 
                                 η 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 x 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 x 
                               
                               
                                 ∂ 
                                 η 
                               
                             
                           
                         
                       
                        
                     
                   
                   ; 
                   
                     
                       J 
                       3 
                     
                     = 
                     
                        
                       
                         
                           
                             
                               
                                 ∂ 
                                 x 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 x 
                               
                               
                                 ∂ 
                                 η 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 y 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 y 
                               
                               
                                 ∂ 
                                 η 
                               
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     For the chosen geometry and nodes, first order interpolation functions are used to describe the coordinates in terms of the transformed variables and the nodal coordinates. Higher order functions can be described but additional nodes would be needed for each element. Thus, 
                       x   =       ∑     i   =   1     4     ⁢       x   i     ⁢       ψ   i     ⁡     (     ξ   ,   η     )             ;     y   =       ∑     i   =   1     4     ⁢       y   i     ⁢       ψ   i     ⁡     (     ξ   ,   η     )             ;     z   =       ∑     i   =   1     4     ⁢       z   i     ⁢       ψ   i     ⁡     (     ξ   ,   η     )             ;     ⁢     
     ⁢           ψ   1     ⁡     (     ξ   ,   η     )       =       1   4     ⁢     (     1   -   ξ     )     ⁢     (     1   +   η     )         ;         ψ   2     ⁡     (     ξ   ,   η     )       =       1   4     ⁢     (     1   +   ξ     )     ⁢     (     1   +   η     )           ⁢     
     ⁢           ψ   3     ⁡     (     ξ   ,   η     )       =       1   4     ⁢     (     1   +   ξ     )     ⁢     (     1   -   η     )         ;         ψ   4     ⁡     (     ξ   ,   η     )       =       1   4     ⁢     (     1   -   ξ     )     ⁢     (     1   -   η     )                   (   8   )               
The x i , y i , z i  values are the coordinates for the i th  node.
 
     The Jacobians can now be found from eq. 7. This will hold for the uniform mesh shown in  FIG. 7  or a non-uniform mesh such as that shown in  FIG. 8 . All that is needed are the nodal locations. 
     As illustrated in  FIG. 9  (a graphical illustration of elements and a master element), for the uniform mesh with constant Δx and Δy the Jacobians are: 
     
       
         
           
             
               
                 
                   
                     J 
                     1 
                   
                   = 
                   
                     
                       
                         
                           - 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         y 
                       
                       8 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 z 
                                 2 
                               
                               - 
                               
                                 z 
                                 1 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               η 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 z 
                                 3 
                               
                               - 
                               
                                 z 
                                 4 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               - 
                               η 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     J 
                     2 
                   
                   = 
                   
                     
                       
                         
                           - 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       8 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 z 
                                 2 
                               
                               - 
                               
                                 z 
                                 3 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               ξ 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 z 
                                 1 
                               
                               - 
                               
                                 z 
                                 4 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               - 
                               ξ 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     J 
                     3 
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                       * 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                     
                     4 
                   
                 
               
               
                 
                   ( 
                   
                     9 
                     ⁢ 
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
     This can be shown to be equivalent to using an expression from Larson et al page  1032 . [5] The expression for the surface is given as z=f(x,y) over a square region R given by x 1 ≤x≤x 2 ; y 1 ≤y≤y 2 . The area is given as: 
     
       
         
           
             
               
                 
                   
                     S 
                     i 
                   
                   = 
                   
                     
                       ∫ 
                       
                         y 
                         1 
                       
                       
                         y 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           x 
                           1 
                         
                         
                           x 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     ∂ 
                                     f 
                                   
                                   
                                     ∂ 
                                     x 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     ∂ 
                                     f 
                                   
                                   
                                     ∂ 
                                     y 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         ⁢ 
                         dxdy 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Here the same mapping to a master element  100  and linear interpolation functions are used and the interface written as z(ξ,η). The partial derivatives are evaluated using the chain rule and the area becomes: 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         1 
                       
                       1 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           - 
                           1 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     a 
                                   
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       4 
                                     
                                     ⁢ 
                                     
                                       
                                         z 
                                         i 
                                       
                                       ⁢ 
                                       
                                         
                                           ∂ 
                                           
                                             
                                               ψ 
                                               i 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 ξ 
                                                 , 
                                                 η 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     b 
                                   
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       4 
                                     
                                     ⁢ 
                                     
                                       
                                         z 
                                         i 
                                       
                                       ⁢ 
                                       
                                         
                                           ∂ 
                                           
                                             
                                               ψ 
                                               i 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 ξ 
                                                 , 
                                                 η 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           ∂ 
                                           η 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             ab 
                             4 
                           
                           ) 
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ξ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         η 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Comparing eq. 11 with eq. 6 after eq. 7 and 8 have been substituted it can be seen to be the same for the special case of square elements. The area of a given element  94  is found using Gauss-Legendre quadrature. [6] 
     
       
         
           
             
               
                 
                   
                     S 
                     i 
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           - 
                           1 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             1 
                           
                           1 
                         
                         ⁢ 
                         
                           
                             
                               
                                 J 
                                 1 
                                 2 
                               
                               + 
                               
                                 J 
                                 2 
                                 2 
                               
                               + 
                               
                                 J 
                                 3 
                                 2 
                               
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ξ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           η 
                         
                       
                     
                     ≈ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ξ 
                                   i 
                                 
                                 , 
                                 
                                   η 
                                   j 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             w 
                             i 
                           
                           ⁢ 
                           
                             w 
                             j 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         
                           ξ 
                           i 
                         
                         , 
                         
                           η 
                           j 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               J 
                               1 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ξ 
                                   i 
                                 
                                 , 
                                 
                                   η 
                                   j 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               J 
                               2 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ξ 
                                   i 
                                 
                                 , 
                                 
                                   η 
                                   j 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               J 
                               3 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ξ 
                                   i 
                                 
                                 , 
                                 
                                   η 
                                   j 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                         w 
                         i 
                       
                     
                     = 
                     
                       
                         w 
                         j 
                       
                       = 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   
                     13 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         ξ 
                         i 
                       
                       , 
                       
                         η 
                         j 
                       
                     
                     ) 
                   
                   = 
                   
                     .577350269189626 
                     ⁢ 
                     
                         
                     
                     * 
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   - 
                                   1 
                                 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 1 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 1 
                                 , 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   - 
                                   1 
                                 
                                 , 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     13 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     The integration reduces to the sum of 4 integrand evaluations at the locations given in equation 13b. For the uniform mesh, only differences in z at the nodes need to be calculated for each element. The other terms are the same for all elements and only need to be calculated once. The total area is the sum of the areas of the elements  94 . 
     Alternative Method to Finding the Area 
     The above described method of mapping a surface to the points  82  of  FIG. 6 b    uses a precise curved three-dimensional surface and produces highly accurate results in terms of mapping a curved surface and calculating the area. However, since the objective is to estimate the surface area of the gas/liquid interface, a less precise form of mapping is normally sufficient. One alternative method to find the surface area is to subdivide all the square elements  94  into two triangles as shown in  FIG. 10 . Each triangle lies in its own plane so the cross product of vectors along two sides can be used to determine the area of each triangle. This creates two plane facets instead of the potentially curved surface that would be obtained with the previous shape functions. Here the surface area of the interface  84  of  FIG. 6 b    is given as: 
                     S   i     =       1   2     ⁢           (         y   42     ⁢     z   13       -       z   42     ⁢     y   13         )     2     +       (         z   42     ⁢     x   13       -       x   42     ⁢     z   13         )     2     +       (         x   42     ⁢     y   13       -       y   42     ⁢     x   13         )     2                   (   14   )                       ⁢           φ   ij     =       φ   j     -     φ   i         ;     φ   =   x       ,   y   ,   z             (     14   ⁢   a     )               
The differences in the x and y directions are constant for the uniform mesh and would only need to be evaluated once. For a non-uniform mesh they would need to be calculated for each element.
 
Liquid Wetted Area
 
     Another surface area that may be calculated using the pixel data from the tomogram  80  is the surface area of the interface between the liquid (or the gas) and the wall of the tube  38  within the sensor  32  ( FIG. 2 a   ). The area of the tube  38  within the sensor volume  40  that is wetted by liquid can be estimated by using the values (normalized permittivity ratios) of the boundary pixels  110  shown in the tomogram  112  of  FIG. 11 a   . First a hypothetical wall boundary between the gas and the liquid is generated using only the values of the boundary pixels, and thus the hypothetical boundary would be a curved boundary line on the surface of the tube  38 . 
       FIG. 11 b    shows a vertical sensor volume  114  within a tube  38  with a hypothetical wall boundary between liquid and gas represented by the ellipse  116 , which is a simplified example for illustration purposes. The perimeter  122  of the ellipse would constitute the boundary between the liquid and the gas along the wall of the tube  38 . The liquid is below the ellipse  116  and the gas is above the ellipse  116 , and the surface area  118  of the wall of the tube  38  below the ellipse would be the surface area of the liquid/wall interface, and the surface area  120  above the interface represents the gas/wall interface. 
     The liquid/wall interface surface area  118  is found by integration around the tube perimeter and is demonstrated using the trapezoidal rule although other methods could be used. 
                     A   wl     =         ∫   0     2   ⁢   π       ⁢   dA     =         ∫   0     2   ⁢   π       ⁢     zRd   ⁢           ⁢   θ       ≈       ∑     i   =   1     NBP     ⁢         z   _     i     ⁢   R   ⁢           ⁢     Δθ   i                     (   15   )               
NBP is the number of boundary pixels  110 . From the uniform
 
               z   i     =       L   ⁢           ⁢     ɛ     *     ,   i         ⁢           ⁢   and   ⁢           ⁢       z   _     i       =       L   2     ⁢       (       ɛ     *     ,   i         +     ɛ     *     ,     i   +   1             )     .               
For the last segment the first boundary pixel is used for z i+1 . This completes the circle around the tube.
 
     With the uniform mesh, the arc lengths corresponding to the segments defined by the boundary pixels  110  aren&#39;t the same. In some meshes Δθ i  may be a constant but the following approach will find the appropriate value using the boundary pixel  110  locations. The pixels  110  are first sorted in a clockwise or counter clockwise fashion. The location of the centroid (e.g. centroids  122   a  and  122   b ) of each pixel represents a vector (e.g., vectors  124   a  and  124   b ) extending radially from the center of the tube  38  to the pixel as shown in  FIG. 12 . By taking the dot product of the vectors associated with adjacent pixels  110 , the cosine of the angle between the vectors can be found as: 
                     AB   ⁢           ⁢     cos   ⁡     (   θ   )         =           A   _     ·     B   _       -&gt;     θ   i       =       acos   (         A   _     ·     B   _       AB     )     =     acos   (           x   i     ⁢     x     i   +   1         +       y   i     ⁢     y     i   +   1                 (       x   i   2     +     y   i   2       )     ⁢     (       x     i   +   1     2     +     y     i   +   1     2       )           )                 (   16   )               
and the area of the wall of tube  38  wetted by liquid is approximated as:
 
                     A   wl     ≈       RL   2     ⁢       ∑     i   =   1     NBP     ⁢       (       ɛ     *     ,   i         +     ɛ     *     ,     i   +   1             )     ⁢     Δθ   i                   (   17   )               
The area of the tube in contact with vapor (gas) is just the total tube area minus the liquid wetted area or
 
                       A   wg     ≈     [       π   ⁢           ⁢   DL     -       RL   2     ⁢       ∑     i   =   1     NBP     ⁢       (       ɛ     *     ,   i         +     ɛ     *     ,     i   +   1             )     ⁢     Δθ   i             ]       =     DL   ⁡     [     π   -       1   4     ⁢       ∑     i   =   1     NBP     ⁢       (       ɛ     *     ,   i         +     ɛ     *     ,     i   +   1             )     ⁢     Δθ   i             ]               (   18   )               
In practice, A wg  would calculated as A wg =πDL−A wl .
 
     The error in using this technique has two components. The first is associated with using the simple trapezoidal rule for the integration as opposed to a more complex formula. The second is approximating the values of ε *,i  at the tube wall as the values given by the centroids of the boundary pixels  110 . The error associated with using the trapezoid rule can be estimated by looking at a case where the liquid/vapor interface forms a plane  126  that intersects the sensor volume  128  as shown in  FIG. 13 . In this case the error between the exact solution and the trapezoidal rule was less than 0.35% for all values of ϕ/ϕ max ≤1, which represents all possible angles of the interface  126 . This is much smaller than the expected error in the values for ε *  (normalized relative permittivity ratio). For purposes of estimation, this error is acceptable. 
     Volumetric Void Fraction 
     It is recognized that the information taken from a tomogram such as tomogram  42  of  FIG. 3  is volumetric in nature due to the length of the electrodes  34  ( FIG. 2 b   ). The percentage of the measurement volume  40  occupied by liquid can also be estimated if there are no bubbles in the liquid or drops in the gas. The volume beneath the liquid/gas interface surface z=f(x,y) is given as
 
 V=∫∫   A   1   f ( x,y ) dxdy=∫∫   A   1   f ( x,y ) J   3   dξdη   (19)
 
The volume under a surface f(x.y) is illustrated in  FIG. 14 .
 
Using the same bilinear mapping for the elements  94  as that used to find the liquid/vapor interfacial area the volume under an element  94  is given as:
 
                     V   i     =         ∫     -   1     1     ⁢       ∫     -   1     1     ⁢       J   3     ⁢       ∑     i   =   1     4     ⁢       z   i     ⁢       ψ   i     ⁡     (     ξ   ,   η     )       ⁢   d   ⁢           ⁢   ξ   ⁢           ⁢   d   ⁢           ⁢   η             =         Δ   ⁢           ⁢   x   *   Δ   ⁢           ⁢   y     16     ⁢       ∫     -   1     1     ⁢       ∫     -   1     1     ⁢       [           z   1     ⁡     (     1   -   ξ     )       ⁢     (     1   +   η     )       +         z   2     ⁡     (     1   +   ξ     )       ⁢     (     1   +   η     )       +         z   3     ⁡     (     1   +   ξ     )       ⁢     (     1   -   η     )       +         z   4     ⁡     (     1   -   ξ     )       ⁢     (     1   -   η     )         ]     ⁢   d   ⁢           ⁢   ξ   ⁢           ⁢   d   ⁢           ⁢   η                     (   20   )               
As with the area calculation, this can be written for a non-uniform mesh with higher order interpolation functions as well. Evaluating eq. 20 gives
 
     
       
         
           
             
               
                 
                   
                     V 
                     i 
                   
                   = 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           x 
                           * 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           y 
                         
                         4 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             z 
                             1 
                           
                           + 
                           
                             z 
                             2 
                           
                           + 
                           
                             z 
                             3 
                           
                           + 
                           
                             z 
                             4 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                       * 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                       * 
                       
                         z 
                         _ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     This is just the average value,  z , at the nodes (n 1 , n 2 , n 3 , n 4 ) times the area of the pixel  90 . This is exact under the assumption of the bilinear variation of z over the element. This variation can be shown to be equivalent to z=a 0 +a 1 x+a 2 y+a 3 xy. The total volume under the liquid/vapor interface is the sum of the volumes under each element. The liquid volume fraction is the liquid volume divided by the sensor volume and the volumetric void fraction is one minus the volumetric liquid fraction. In general, the volumetric void fraction is not the same as  ε   * . 
     Centroid of the Fluid Mixture in the Sensor Volume 
     The centroid of the fluid mixture can also be found from the tomogram data. Referring to  FIG. 7 , x and y components of each centroid  92  (x ci , y ci ) and the area of each pixel  90  are known. The mass of each voxel  62  is m i =ρ i A i L. The mean density is given as
 
ρ i =ρ g +ε *,i (ρ l −ρ g )  (22)
 
The coordinates for the centroid of the mass in the sensor volume is given by
 
                     x   _     =             ∑     i   =   1     NP     ⁢       m   i     ⁢     x   ci             ∑     i   =   1     NP     ⁢     m   i         ⁢           ⁢     y   _       =             ∑     i   =   1     NP     ⁢       m   i     ⁢     y   ci             ∑     i   =   1     NP     ⁢     m   i         ⁢           ⁢     z   _       =         ∑     i   =   1     NP     ⁢       m   i     ⁢     z   ci             ∑     i   =   1     NP     ⁢     m   i                     (       23   ⁢   a     ,   b   ,   c     )               
To find  z , z ci  must be calculated. This can be found by referring to a schematic of a voxel as given in  FIG. 15 .
 
                     z   ci     =             m   li     ⁢       z   _     li       +       m   gi     ⁢       z   _     gi           m   i       =           ρ   l     ⁢     ɛ     *     ,   i         ⁢       z   _     li       +         ρ   g     ⁡     (     1   -     ɛ     *     ,   i           )       ⁢       z   _     gi           ρ   i                 (   24   )                   z   _     li     =           L   ⁢           ⁢     ɛ     *     ,   i           2     ⁢           ⁢       z   _     gi       =         L   ⁢           ⁢     ɛ     *     ,   i           +       (     1   -     ɛ     *     ,   i           )     ⁢     L   2         =       L   2     ⁢     (     1   +     ɛ     *     ,   i           )                   (       25   ⁢   a     ,   b     )               
Substituting into 23c gives
 
                     z   _     =       L   2     ⁢         ∑     i   =   1     NP     ⁢     [       ρ   g     +         (     ɛ     *     ,   i         )     2     ⁢     (       ρ   l     -     ρ   g       )         ]           ∑     i   =   1     NP     ⁢     ρ   i                   (   26   )               
Cross Sectional Flow Areas
 
     The cross-sectional flow area  150  for the gas, A g , and the cross sectional flow area  152  for the liquid, A l , phases are shown in  FIGS. 16 a  and 16 b   , and can also be estimated from the pixel values of the tomogram  80 . The liquid cross sectional area in the sensor  32  can also be approximated if the uniform cross section distribution is used. It is simply the fraction of pixels with lengths calculated from the permittivity ratios greater than a particular value of z. i.e. 
                       A   l     ⁡     (   z   )       =           π   ⁢           ⁢     D   2         4   *   NP       *       ∑     i   =   1     NP     ⁢       H   ⁡     (       ɛ     *     ,   i         -     z   L       )       ⁢           ⁢   where   ⁢           ⁢     H   ⁡     (   x   )             =     {             1   ⁢           ⁢   if   ⁢           ⁢   x     ≥   0                 0   ⁢           ⁢   if   ⁢           ⁢   x     &lt;   0                       (   27   )                   A   g     ⁡     (   z   )       =         π   ⁢           ⁢     D   2       4     -       A   l     ⁡     (   z   )                 (   28   )               
Eq. 27 assumes the area of each pixel is the same. If they are different the expression can be expressed as:
 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       l 
                     
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       NP 
                     
                     ⁢ 
                     
                       
                         A 
                         i 
                       
                       ⁢ 
                       
                         H 
                         ⁡ 
                         
                           ( 
                           
                             
                               ɛ 
                               
                                 * 
                                 
                                   , 
                                   i 
                                 
                               
                             
                             - 
                             
                               z 
                               L 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     Here A i  is the area of the i th  pixel and would be calculated once for a given mesh using Gauss-Legendre quadrature. These areas can be can be retained in this discrete form or curve fit to provide a smoother approximation to the area. This also allows an approximation of the gradients in the axial direction of the flow areas either in a finite or continuous fashion. The cross sectional area of a particular material (e.g. liquid) in the sensor (such as sensor  32 ) will vary depending on the z position within the sensor. One method of estimating an average cross sectional area of a material in the flow would be to determine the cross sectional area at multiple z positions within a sensor. For example, the cross sectional area of the liquid could be calculated at every 0.1 inch along the length of the sensor and then those areas could be averaged to determine an average cross sectional area. 
     The cross sectional area of a material at one z position in the sensor may also provide useful information. For example, the cross sectional area of a liquid in a gas/liquid flow may be calculated for the center of the sensor and such area may be calculated repeatedly over time. Each of the calculated areas may be compared to the others. If the cross sectional area of the liquid is fluctuating or oscillating over time, the frequency and amplitude of the oscillations in the cross sectional area would be a measure of the frequency and magnitude of waves in the flowing mixture within the sensor. 
     Test Results 
     The concept was tested using a vertical tube  160  with salt  162  and air  164  as two phase media in a vertical orientation as shown in  FIG. 17 . This test is using static materials in a sensor volume as opposed to flowing materials to simplify the test and to enable easy verification of the calculated information as compared to the actual information. The interface between the salt and the air can be visually or physically determined and compared to the interface that is calculated based on a tomogram. In this test, the actual position of the interface between salt and air was determined visually for four different levels of salt, and the visual measurements of the interface were very nearly the same as the interface calculated from a tomogram using the techniques described above. In this test, the interface  166  is horizontal and approximately flat, and the interface lies within the sensing volume of electrodes  168 . The theoretical volume fraction of salt based on electrode length is 
               ɛ   *     =         z   -     z   1       L     .             FIG. 18  shows the tomograms for four different levels of salt within the sensing volume of electrodes  168 .  FIG. 19  shows how the sensor model based on the concept z=Lε *  compares with the experimental results. As can be seen, the sensor is sensitive to the presence of material approximately 5 mm or 0.714 diameters outside the electrodes. This is not unexpected as the electric field extends past the ends of the electrodes. The effect can be captured by using an apparent length of 60 mm for the electrodes and provides support that the proposed methods will be accurate for liquid vapor interfaces as long as the features are longer than the length of the electrodes. Thus the technique should provide a useful metric for the interfacial areas: A lg , A lw , A gw .
 
Exemplary Apparatus
 
       FIG. 20  illustrates an exemplary apparatus  170  for implementing the methods described above. A conduit  171  provides a supply of flowing multiple materials, such as water and steam, to a pump  172  which pressurizes the mixture. The output of pump  172  is supplied through a conduit  174  to an input of a mixing valve  176 , with the flow of the mixture being indicated by arrow  184 . The mixing valve  176  also receives an input of at least one flowing material from conduit  199 , which for example may be water. The mixing valve  176  independently controls the flow from conduits  174  and  199  mixes the materials from conduits  174  and  199  and outputs the mixed materials through conduit  178  to the sensor  180 . The flow into and out of the mixing valve  176  is indicated by the arrows  184 ,  199  and  186 . The sensor  180  surrounds the conduit  178  and the flowing mixture continues uninterrupted through the sensor  180  and within the conduit  178  as indicated by flow arrow  188 . The sensor  180  corresponds to the sensor  32  discussed above. 
     The sensor  180  provides an output to a controller  200  which includes data processors and communication devices for implementing the methods. The controller  200  powers the sensor  180  and receives communications from the sensor  180  through lines  202 , which are communication lines and power lines. The controller  200  is also connected to power and control pump  192 , valve  198 , mixing valve  176  and pump  172  through the lines  202 . The pump  192  is connected to a supply conduit  190  and, in this example, is supplied with water. The output of pump  192  flows through conduit  194  to valve  198  as indicated by the flow arrow  196 , and the valve  198  controls the flow through conduit  199  to the mixing valve  176 . 
     The sensor  180  measures capacitance and those capacitance measurements are provided to the controller  200  which calculates a tomogram as discussed above with respect to sensor  32 . The controller  200  repetitively samples the sensor  180  and repetitively produces tomograms at a rate that is sufficient for a particular application, which will vary widely. In this application, the controller  200  is configured to produce tomograms at a rate of one sample per second. The controller  200  is also configured to calculate one or more of the hypothetical physical characteristics discussed above in less than one half a second. So, for example, the controller  200  may calculate a hypothetical surface area of the interface between the water and steam within the sensor  180 . In addition, the sensor  180  may calculate additional hypothetical physical characteristics, such as the hypothetical area of a wall of the sensor  180  in contact with steam. Then, the controller  200  compares the hypothetical physical characteristics against predefined limits and transmits control commands when the hypothetical physical characteristics meet or exceed the predefined limits. So, for example if the hypothetical surface area between the wall of sensor  180  and steam exceeds its predefined limit, the controller  200  issues control commands to the pump  192  and the valve  198  causing a desired amount of flow through the conduit  199  and water is introduced through the mixing valve  176  into the conduit  178 . The supply of water through the mixer  176  will decrease the amount of steam in the flowing mixture and will decrease the surface area of the sensor wall that is contacted by steam. 
     The controller  200  may also be calculating the hypothetical surface area between the water and gas within the sensor  180 . Also, it may be saving each such calculation and calculating a rate of change in the hypothetical surface area between the water and gas. When this rate of change exceeds a predefined limit, that circumstance in this particular embodiment can be predicting the formation of oscillations within the flowing mixture in the conduit  178 . In this particular embodiment, such waves would constitute a dangerous or catastrophic event. Thus, the controller  200  in response to such condition issues commands to stop the pump  172  and  192 . In addition, it will command the valve  198  and the mixing valve  176  to stop all flow through the conduit  178 , and the apparatus  170  is shut down. Alternatively, when the rate of change exceeds a predefined limit, the controller  200  may be programmed to take corrective action. For example, the pump  192 , valve  198  and mixing valve  176  may receive commands to introduce more water into the flow within conduit  178 . By increasing the water, hopefully, the rate of change in the surface area between the water and steam will reverse or stabilize. The controller  200  will continue to monitor such rate of change and will allow the embodiment to continue functioning so long as the rate of change remains below the predefined limit. It will be understood that all of the various hypothetical physical characteristics discussed herein may be calculated by the controller  200  and compared against one or more predefined limits, and in each case corrective actions may be executed when any of the hypothetical physical characteristics exceed their limits, and one of those corrective actions could be a complete shutdown of the apparatus  170 . 
     Having described several embodiments and variations of the invention in the above Detailed Description, it will be understood that the invention is capable of numerous modifications, rearrangements and substitutions of parts without departing from the spirit of the invention as defined in the Claims. 
     REFERENCES 
     
         
         [1] Wallis, G., “One-Dimensional Two Phase Flow, 1969, McGraw-Hill Inc. 
         [2] Industrial Tomography Systems plc, Sunlight House, 85 Quay Street, Manchester, M3 3JZ, UK 
         [3] Kreitzer, P., Hanchak, M. and Byrd, L., “Horizontal Two Phase Flow Regime Identification: Comparison of Pressure Signature, ECT and High Speed Visualization”, presented at 2012 ASME IMECE, Houston, Tex. 
         [4] Taylor, A. E., Mann, W. R., “Advanced Calculus”, 2 nd  ed., 1972, Xerox College Publishing 
         [5] Larson, R. E., Hostetler, R. P., Edwards, B. H., “Calculus with Analytical Geometry” 6 th  ed., 1998, Houghton Mifflin Co. 
         [6] Carnahan, B., Luther, H. A., Wilkes, J. O., “Applied Numerical Methods”, 1969, J. Wiley &amp; Sons, Inc.