Abstract:
Systems and methods for solid oxide fuel cell (SOFC) surface analysis. Exemplary embodiments include systems and methods for solid oxide fuel cell (SOFC) surface analysis, including a SOFC having a ceramic surface, a scanner adjacent the ceramic surface for collecting data related to the ceramic surface, a structure for retaining the SOFC with respect to the scanner, a device for collecting a processing the ceramic surface data and a process residing on the device, the process for analyzing and presenting the ceramic surface data.

Description:
BACKGROUND 
       [0001]    The present disclosure generally relates to solid oxide fuel cells (SOFC) and more particularly to system and method for SOFC surface analysis. 
         [0002]    SOFC using ceramic membranes are used as energy conversion devices. In general, the cell is multi-layer structure fabricated by sintering or deposition method. The surface of the cell is interface for current collection, also for sealing. The flatness and smoothness of SOFC are critical parameters for the contact and seal. Cell flatness problems are related to how to improve fuel cell performance and consistency. Current methods attempt to address cell flatness issues, typically by evaluating density and quality of ceramic coating, but do not address leakage problems of the ceramic surface. Instead, current methods provide a measurement of permeability due to the combined effects of cracks, voids, and porosity. Other methods have been used to inspect the surface quality of ceramic coating, while still other methods are used to measure the gross permeability of the coating. However, none of the methods provides details about the factors affecting the permeability and simultaneously give a quantitative permeability measurement of selected regions in the coating. In addition, the ceramic coatings in SOFC (and in other applications, such as protective and thermal barrier coatings in turbines combustors and airfoils) are currently inspected with gas leak tests, which provide a gross estimate of the coating permeability without distinguishing between the various causes of the leakage, such as, micro-crack, mud cracks, seal defects, voids, and porosity. 
         [0003]    As such, there is a persistent need for systems and methods for SOFC surface analysis. 
       SUMMARY 
       [0004]    Disclosed herein is a solid oxide fuel cell (SOFC) surface analysis system, including a SOFC having a ceramic surface, a scanner adjacent the ceramic surface for collecting data related to the ceramic surface, a structure for retaining the SOFC with respect to the scanner, a device for collecting and processing the ceramic surface data and a process residing on the device, the process for analyzing and presenting the ceramic surface data. 
         [0005]    Additional embodiments include a solid oxide fuel cell (SOFC) surface analysis method, including identifying a surface of a SOFC for analysis, generating a measurement wave on the surface, receiving data from the reflected wave from the surface, processing the data to determine surface irregularities and defects and generating a graphical presentation of the surface irregularities and defects. 
         [0006]    Further disclosed herein is a system for determining irregularities and defects on a ceramic surface of a SOFC, the system including the ability to retain the SOFC, generate a measurement wave for shining on the SOFC, generate an emitted wave from the SOFC, collect emission data from the SOFC ceramic surface and process the emission data for a determination of the surface irregularities and defects of the ceramic surface. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]    The disclosure and embodiments thereof will become apparent from the following description and the appended drawings, in which the like elements are numbered alike: 
           [0008]      FIG. 1  illustrates an exemplary embodiment of a surface analysis apparatus; 
           [0009]      FIG. 2  illustrates a flow chart of an exemplary surface analysis method; 
           [0010]      FIG. 3  illustrates an exemplary data processing method; 
           [0011]      FIG. 4  illustrates an exemplary 3D rendition of edge area of original data; 
           [0012]      FIGS. 5   a  and  5   b  illustrate projections of the original surface by two different wavelets in accordance with exemplary embodiments; 
           [0013]      FIGS. 6   a  and  6   b  illustrate projections of the original surface by two different wavelets in accordance with exemplary embodiments; 
           [0014]      FIG. 7  illustrates an original x-ray photo of a section of a SOFC cell in accordance with exemplary embodiments; 
           [0015]      FIG. 8  illustrates a wavelet transformation of the photo of  FIG. 7  in accordance with exemplary embodiments; 
           [0016]      FIG. 9  illustrates an exemplary dye penetrant system; 
           [0017]      FIG. 10  illustrates a flow chart of an exemplary surface analysis method; 
           [0018]      FIG. 11  illustrates an exemplary data processing method; 
           [0019]      FIG. 12  illustrates an exemplary dye penetrant inspection movie  1200 ; 
           [0020]      FIGS. 13   a  and  13   b  respectively illustrate an intensity versus time plot of the calibrant and target of  FIG. 12 ; 
           [0021]      FIG. 14  illustrates a plot of normalized target intensity and derivatives; 
           [0022]      FIGS. 15   a - 15   c  illustrate three exemplary plots in accordance with exemplary embodiments; and 
           [0023]      FIG. 16  illustrates an exemplary plot in accordance with exemplary embodiments. 
       
    
    
     DETAILED DESCRIPTION 
       [0024]    Exemplary systems and methods described herein provide surface analysis of SOFC defects, bumps etc. In a first exemplary embodiment, a system and method for multi-resolution thin film surface analysis implementing wavelet transformation is provided. In a second exemplary embodiment, a system and method for inspection and permeability measurement using dye penetrants is provided. 
       Multi-Resolution Thin Film Surface Analysis 
       [0025]    Exemplary embodiments include systems and methods for the surface analysis of solid-oxide ceramic cells to characterize the surface flatness and smoothness, which provides input for QC and manufacturing. Surface data collection systems and methods acquire cell surface data automatically in a continuous mode, machine intelligence (wavelet transforms) software analyzes the data and software provides warnings for quality control. In other exemplary embodiments, infrared scan/photo, MRI, etc can be implemented. In addition, different data analysis algorithms, such as Fourier transforms and windowed Fourier transforms can be implemented. As such, methods for continuously 3-D measurements of ceramic surface, basis function, mesh generation, and parameters using in the wavelet transforms can be implemented to analyze SOFC ceramic surfaces 
         [0026]      FIG. 1  illustrates an exemplary embodiment of a surface analysis apparatus  100 . Apparatus  100  includes conveyor belt  105  onto which sample  110  is placed for measurements. Individual dot points  111  are representative of measurements of irregularities of sample  110 . Measurement device (generally having both a measurement wave generator and a scanner for receiving back-scattered waves)  115  is disposed at a location above conveyor belt  105  and sample  110 . A measurement area  117  is associated with a scanning area of measurement device  115 , through which sample  110  is measured. In general, conveyor belt can be displaced in a direction as indicated by arrow A. Measurement device  115  can be moved along a direction as indicated by arrows B, C. In general, direction of movement B, C of measurement device  1115  is orthogonal to movement A of conveyor belt  105 . System  100  can further include computing and data acquisition device  120 , which can be a desk top computer, lap top computer, PDA, etc. Computing and data acquisition device  120  can further be coupled to a storage medium  125  for collection and storage of data and applications such as analysis application  130 . Computing and data acquisition device  120  can further include a graphical user interface (GUI) for presentation and display of analyzed data 
         [0027]      FIG. 2  illustrates a flow chart of an exemplary surface analysis method  200 . In accordance with exemplary embodiments, method  200  generally includes two parts: the first part is a method of surface measurement; and the second part is a method of surface data analysis. For the first part, at step  205  conveyor belt  105  moves sample  110  (e.g., ceramic cell) through the measurement area  117  of measurement device  115 . Measurement device  115  (e.g., sensor, scanner, wave projector, etc.) moves perpendicularly (orthogonally) to sample  110  movement. At step  210 , measurement device  115  generated a measurement wave. The measurement wave can be ultrasound or laser, or other reflective waves, and x-ray. It is understood that in other exemplary embodiments and implementations, the measurement wave can be other wavelengths and frequencies. At step  215 , measurement data is collection. A sensor on measurement device  115  receives the reflection and sends signals indicative of distance, thickness and other parameters to computing and data acquisition device  120  at step  220 . In another exemplary implementation, measurement device can be a high resolution camera in which photos of the sample  110  can be taken at step  125 . The photos can be single shot or multiple shots with multiple-resolution. 
         [0028]    For the second part of method  200 , a gauge to measure the oscillations on the surface, collected at step  220  is processed at step  225 . The oscillations on the surface of sample  110  are irregular, and typically non-periodical. The data analysis step  225  decomposes the irregular oscillation into composite wavelets (small waves) with known period and location, as available by machine intelligence. This multiple wavelets decomposition is a gauge, which measures similarity of the oscillation on the surface to the wave. Before quality control, criteria are set at step  230 . In general, the criteria are from cell test results with different cell surface characteristics. The characteristics include, but are not limited to: ratio between wave size and wave magnitude; and intensity of wave and the distribution (locations). In an exemplary implementation, the good performance cell sets the tolerance for the two criteria. Historical data that has been collected from prior scans and samples at step  235  is compared with the presently collected data at step  240 . From a statistical point-of-view, a new ceramic surface compares to the historical data taken from previous tests. At step  245 , the sample&#39;s performance is determined. If the new surface data from sample  110  falls out of the good range, the cell is more likely to have lower performance. If the new surface data is within the tolerance, the cell passes the QC. 
         [0029]    At step  225  above, the data is processed.  FIG. 3  illustrates an exemplary data processing method  300 . In general, at step  305 , the collected oscillation data is decomposed into multiple waves, or wavelets. At step  310 , the individual wavelets are identified and isolated. At step  315 , the characteristics of the wavelets are identified and categorized, including, but not limited to, the frequency and wavelength, and amplitude of the wavelets. In general, the higher resolution wavelets are retained in order to determine the sample irregularities at step  320 . In exemplary implementations, the higher frequency wavelets are retained for this determination. At step  325 , the method  300  generates projections of the original surface. From these projections, using the higher resolution wavelets, the method  300  determines the surface irregularities of the sample  110  at step  330 . 
       EXAMPLE 1 
       [0030]    As discussed above, the method  200  allows data analysis using the wavelets. A sample is scanned using an Acu-guage laser scanner.  FIG. 4  illustrates an exemplary 3D rendition  400  of edge area of original data. As illustrated, the surface is curvy and with bumps on the surface. In general, the waves on the surface of the sample are not regular. The waves are composite of small waves with different wavelengths and magnitude. By implementing methods  200 ,  300 , the bumpy surface is decomposed into a combination of many waves in multi-resolution level, as discussed above. The higher frequency of the wavelets that are used, the finer resolution that is obtained. The original data can be decomposed by many wavelets with different wavelength.  FIGS. 5   a  and  5   b  illustrate projections  500 ,  550  of the original surface by two different wavelets.  FIG. 5   a  illustrates a projection  500  of a wavelet with wavelength p ρ =1 mm (i.e., wavelength in the ρ direction) and p θ =2.0 radians (i.e., wavelength in the θ direction). Projection  500  illustrates a larger wavelet that fits the larger curve of the original data.  FIG. 5   b  illustrates a projection  550  of a wavelet with wavelength p ρ =4 mm and p θ =0.5 radians. Projection  550  of the smaller wavelet fits the bumps of the original data better than projection  500 . Since the wavelength of the wavelets is predetermined, the wavelength represents the size of the bump that it fits. 
         [0031]      FIGS. 6   a  and  6   b  illustrate projections  600 ,  650  of the original surface by two different wavelets.  FIG. 6   a  illustrates a projection  600  of a wavelet with wavelength p ρ =1 mm and p θ =2.0 radians.  FIG. 6   b  illustrates a projection  650  of a wavelet with wavelength p ρ =4 mm and p θ =0.5 radians. The projections  600 ,  650  are 2D representations of the wavelets. With finite support, the function aids to locate bumps on the surface of the sample. The smaller wavelets are capable of catching the higher frequency characteristics. 
       EXAMPLE 2 
       [0032]    The present example illustrates cell faults detection analysis using an x-ray photo, in which a ceramic cell sample surface has been photographed by x-ray.  FIG. 7  illustrates an original x-ray photo  700  of a section of a SOFC cell in accordance with exemplary embodiments. The dark marks are speculated as heterogeneous particles or defects. Wavelet transformations in accordance with exemplary embodiments are implemented to identify boundary, size, location, etc. of the defects. The wavelets have a wavelength of 64 units. 
         [0033]      FIG. 8  illustrates a wavelet transformation  800  of the photo of  FIG. 7  in accordance with exemplary embodiments.  FIG. 8  illustrates a presentation of the transformation results in which the transform matches the photo  700  closely. 
       Inspection and Permeability Measurements Using Dye Penetrant 
       [0034]    Exemplary embodiments further include a system and method implementing fluorescent dye inspection, which is used for visual inspection of surface defects in various applications, to quantitatively measure the permeability of the ceramic coating as well as to provide a visual image of the coating defects showing their size, location, and orientation. In accordance with exemplary embodiments, systems and methods monitor the amount of dye leaking through the coating, by recoding the intensity of the light emitted by the dye, and uses methods similar to those used in transient IR measurement of thermal diffusively, to measure the permeability of the coating. 
         [0035]    The application of the dye penetrant (or other fluids) in a leak test configurations is implemented to obtain quantitative measurement of the permeability as well as a visual image of the through thickness defects that affect the permeability (such as cracks, porosity, and voids). The implementation of fluorescent dye in a leak test provides visual images of the ceramic coating defects, which provides details about the size, shape, and location of the factors affecting the coating permeability and acts as a mean to distinguish between them. Further, exemplary embodiments of the systems and methods extract features from the dye emitted light intensity profile to measure the permeability of selected regions in the coating. Therefore, the systems and methods can quantitatively measure permeability of fuel cell&#39;s ceramic coatings due to porosity (with options to measure in specific region only) and separate that from other factors contributing to permeability, such as cracks and voids in the coating. 
         [0036]    A visual image of the coating (the fluorescent dye leaking through the fuel cell to the coating surface) can be obtained, making it possible to distinguish between the various defects, as well as provides a quantitative measure of the permeability of the coating. Therefore, the systems and methods can be implemented for inspection during manufacturing of fuel cells or products with similar inspection needs. For example, products that have ceramic coatings or permeable layers subjected to cracking, such as protective and thermal barrier coatings used in turbines for improved impact and erosion resistance. 
         [0037]      FIG. 9  illustrates an exemplary dye penetrant system  900 . System  900  generally includes sample  905 , which can be a SOFC having fuel inlets/outlets  906  and a ceramic coating  907 . SOFC sample  905  can further include internal corrugated sheet  908  and filter  909 . Sample  905  is in communication with a dye penetrant supply  910  via fuel inlets/outlets  906 . Dye penetrant supply  910  can therefore provide dye to the sample  905  as needed. System  900  further includes scanner  915 , which can be a digital camera in accordance with exemplary embodiments. Scanner  915  can further include a filter  916  to allow capture of a particular band of wavelengths, generally representative of the dye color. System  900  further includes ultraviolet (UV) light source  920  for providing UV light  925  directed toward dye-penetrated ceramic surface  907 , emissions  930  from which are collected by scanner  915  (discussed further in the description below). System  900  generally further includes a light-tight box  935  into which a portion of sample  905  (i.e., the ceramic coating  907 ), scanner  915  and UV light source  920  are disposed, such that proper generation of UV light  925  and recordation of emissions  930  are not affected by ambient sources. Ceramic surface  907  can be disposed in light-tight box  935  via fissuring  940  on an upper surface of light-tight box  935 . 
         [0038]    System  900  further include computing and data acquisition device  945 , which can be a desktop computer, lap top computer, PDA etc. Computing and data acquisition device  945  can be used for acquiring the emission data as well as processing of the data. Computing and data acquisition device  945  can further be coupled to a storage medium  950  for collection and storage of data and applications such as analysis application  955 . Computing and data acquisition device  945  can further include a graphical user interface (GUI) for presentation and display of analyzed data. Analysis application can be used to acquire the data, process the images and perform the calculations for the quantitative measurement of permeability. 
         [0039]    As discussed above, dye penetrant can be used to obtain a visual inspection of defects and a quantitative measurement of permeability due to porosity simultaneously is unique. Data processing and mathematical calculations are implemented to measure the effective permeability of the sample to be tested from the dye intensity versus time profile. It is understood that in other exemplary embodiments and implementations, other devices can be analyzed in addition to SOFC, including, but not limited to protective and thermal barrier coatings in turbines combustors and airfoils, etc. 
         [0040]      FIG. 10  illustrates a flow chart of an exemplary surface analysis method  1000 . Once sample  905  has been suitably affixed to light-tight box  905  and coupled to dye penetrant supply  910 , UV light source  920  generates UV light  925  onto sample  905  at step  1005 . When UV light  925  hits sample  905 , the dye on the sample&#39;s surface  907  emits visible light  930 , which is recorded by scanner  915  (e.g., digital monochromic camera having optical filter  916  around the wavelength of the dye), at step  1010 . In one exemplary implementation, the dye and filter  916  can be green, such that emitted light  930  is also in the green wavelength area. Dye is then inserted into the sample via the fuel inlets/outlets  906  at step  1015 . At step  1020 , a sufficient time period passes to allow the dye to leak through and into sample  905 . Then the UV light is shut down at step  1025 . 
         [0041]    Data is captured by computing and data acquisition device  945  and stored in storage medium  950  and processed by application  955  at step  1030 .  FIG. 11  illustrates an exemplary data processing method  1100 . In general, at step  1105 , the data is collected as individual frame during emission from the sample  905 . At step  1110 , the frames are compiled into a single film clip, which can be calibrated to a calibrant (e.g., a tape with dye, dye in a transparent container, etc.), which is discussed further below with respect to  FIG. 12 . At step  1115  application  955  creates profiles of dye intensity versus time. At step  1120  a curve-fitting algorithm is applied to estimate the diffusion time constant as described below. 
         [0042]      FIG. 12  illustrates an exemplary frame in a dye penetrant inspection movie  1200  created at step  1110  above. Inspection movie clip frame  1200  includes calibrant  1205  as discussed above as well as a region of interest in the image  1210 . 
         [0043]    As such the methods  1000 ,  1100  discussed above obtain quantitative measurements of the permeability of the ceramic coating  907 . Analysis application  955  can include algorithms for correlating features in the intensity profile to the rate at which the dye leaks out of the sample  905 . 
         [0044]      FIGS. 13   a  and  13   b  respectively illustrate intensity versus time plots  1300 ,  1350  of the calibrant  1205  and target  1210  of  FIG. 12 . Furthermore,  FIG. 14  illustrates a plot  1400  of normalized target intensity and derivatives. In general, the time of inflection point (2 nd  derivative of intensity profile=0) or other time characteristics from the slope (1 st  derivative) where noise due to numerical differentiation is less e.g., times of maximum slope and 50% of maximum slope is calculated by application  955  and could be used to estimate the diffusion coefficient of the sample. These time values are also optionally displayed on GUI of computing and data acquisition device  945 . 
         [0045]    In accordance with exemplary embodiments, the governing equation for the 1-D dye diffusion through the sample  905  having dye concentration of C=C(x,t) is C t (x,t)=αC xx (x,t) having solution discussed below. 
         [0046]    The effective diffusion coefficient of the cell multi-layered structure is α. Certain boundary conditions can be applied in order to solve the equations. For example, initially, the dye concentration was zero everywhere, such that C(x,t&lt;t 0 )=0. At time t=t 0 , a dye concentration, C 0 , is applied and maintained at the surface of the filter  909 , where x=0. In general, no dye diffusion occurred beyond the sample coating surface  907 , where x=L for C x (L,t)=0. Therefore, the solution for the above referenced equation C t (x,t)=αC xx (x,t) is: 
         [0000]    
       
         
           
             
               
                 C 
                  
                 
                   ( 
                   
                     x 
                     , 
                     t 
                   
                   ) 
                 
               
               = 
               
                 
                   C 
                   0 
                 
                 [ 
                 
                   1 
                   - 
                   
                     
                       4 
                       π 
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 
                                   2 
                                    
                                   L 
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                                 
                                   4 
                                    
                                   
                                     L 
                                     2 
                                   
                                 
                               
                                
                               α 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                         
                         n 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               n 
               = 
               1 
             
             , 
             3 
             , 
             5 
             , 
             … 
              
             
                 
             
             , 
             ∞ 
           
         
       
     
         [0047]    Assuming that the emitted light intensity  930  is linearly proportional to the dye concentration at the surface x=L and that the maximum intensity is I 0  leads to the following expression of the intensity and its time derivatives at the surface L: 
         [0000]    
       
         
           
             
               
                 I 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   I 
                   0 
                 
                 [ 
                 
                   1 
                   - 
                   
                     
                       4 
                       π 
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                    
                                   
                                       
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                                 
                                   4 
                                    
                                   
                                     L 
                                     2 
                                   
                                 
                               
                                
                               α 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                         
                         n 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               n 
               = 
               1 
             
             , 
             3 
             , 
             5 
             , 
             … 
              
             
                 
             
             , 
             ∞ 
           
         
       
       
         
           
             
               
                 
                   I 
                   t 
                 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   I 
                   0 
                 
                  
                 
                   
                     π 
                      
                     
                         
                     
                      
                     α 
                   
                   
                     L 
                     2 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     n 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                           2 
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           
                             
                               - 
                               
                                 n 
                                 2 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                           
                             4 
                              
                             
                               L 
                               2 
                             
                           
                         
                          
                         α 
                          
                         
                             
                         
                          
                         t 
                       
                     
                   
                 
               
             
             , 
             
               n 
               = 
               1 
             
             , 
             3 
             , 
             5 
             , 
             … 
              
             
                 
             
             , 
             ∞ 
           
         
       
       
         
           
             
               
                 
                   I 
                   tt 
                 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   - 
                   
                     I 
                     0 
                   
                 
                  
                 
                   
                     
                       
                         π 
                          
                         
                             
                         
                       
                       3 
                     
                      
                     
                       α 
                       2 
                     
                   
                   
                     4 
                      
                     
                       L 
                       4 
                     
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                         n 
                          
                         
                             
                         
                       
                       3 
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                           2 
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           
                             
                               - 
                               
                                 n 
                                 2 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                           
                             4 
                              
                             
                               L 
                               2 
                             
                           
                         
                          
                         α 
                          
                         
                             
                         
                          
                         t 
                       
                     
                   
                 
               
             
             , 
             
               n 
               = 
               1 
             
             , 
             3 
             , 
             5 
             , 
             … 
              
             
                 
             
             , 
             ∞ 
           
         
       
     
         [0048]      FIG. 15   a  illustrates the plot  1500  of I/I0 versus t*α/L 2  at the boundary x=L. In addition,  FIGS. 15   b  and  15   c  respectively illustrate a plot  1550  of normalized I, I t  and I tt  versus t*α/L 2 , and a zoomed plot  1575  to show the time characteristics discussed above. As such, an experimental profile of the dye intensity and its derivatives can be obtained, referring again to  FIG. 14 . By comparing a set of two or more of the time characteristics of the theoretical intensity profile to the ones obtained experimentally, one can estimate the diffusion time constant τ=L 2 /α. 
         [0049]    Furthermore, it was shown that instead of using such time characteristics of the intensity profile, better estimations of the sample diffusion time constant could be obtained by performing a curve fit of the intensity profile obtained experimentally to that predicted theoretically.  FIG. 16  shows a plot  1600  of the experimental and theoretical intensity profiles and the value of τ obtained by performing the curve fit process. 
         [0050]    As described above, the exemplary embodiments can be in the form of computer-implemented processes and apparatuses for practicing those processes. The exemplary embodiments can also be in the form of computer program code containing instructions embodied in tangible media, such as floppy diskettes, CD ROMs, hard drives, or any other computer-readable storage medium, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the exemplary embodiments. The exemplary embodiments can also be in the form of computer program code, for example, whether stored in a storage medium, loaded into and/or executed by a computer, or transmitted over some transmission medium, loaded into and/or executed by a computer, or transmitted over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein, when the computer program code is loaded into an executed by a computer, the computer becomes an apparatus for practicing the exemplary embodiments. When implemented on a general-purpose microprocessor, the computer program code segments configure the microprocessor to create specific logic circuits. 
         [0051]    This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to make and use the invention. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.