Abstract:
A system and method for using an element made of porous ceramic materials such as zeolite to constrain the flow of gas molecules to the free molecular or transitional flow regime. A preferred embodiment of the gas pump may include the zeolite element, a heater, a cooler, passive thermal elements, and encapsulation. The zeolite element may be further comprised of multiple types of porous matrix sub-elements, which may be coated with other materials and may be connected in series or in parallel. The gas pump may further include sensors and a control mechanism that is responsive to the output of the sensors. The control mechanism may further provide the ability to turn on and off certain heaters in order to reverse the flow in the gas pump. In one embodiment, the pump may operate by utilizing waste heat from an external system to induce transpiration driven flow across the zeolite. In another embodiment, the pump may selectively drive and direct gas molecules depending on the molecular size and the interaction between the gas molecule and the zeolite element.

Description:
PRIORITY CLAIM 
       [0001]    This application claims priority to U.S. Provisional Patent Application No. 61/020,126 entitled “THE USE OF A ZEOLITE MATERIAL WITHIN THE FLOW CHANNEL OF A GAS PUMP BASED ON THERMAL TRANSPIRATION”, which was filed on Jan. 9, 2008 by Yogesh B. Gianchandani, the contents of which are expressly incorporated by reference herein. 
     
    
     BACKGROUND 
       [0002]    Pumps are devices used to move fluids, such as gases or liquids. Displacement of fluid is achieved by physical or mechanical means. Pumps may be used to evacuate gas from a confined space, thereby creating a vacuum. Conversely, pumps may also be used to draw in gas from one environment to another. In another example, pumps may be used to pressurize a sealed volume or to generate a pressure gradient along a restricted flow path. 
         [0003]    Most pumps are not suitable for miniaturization as they possess mechanical parts or require a low backing pressure that makes it necessary to use a backing pump. Miniaturized pumps, such as micropumps and mesoscale pumps, can suffer from poor performance and reliability, or introduce undesired vibrations into a system. 
         [0004]    Thermal transpiration pumps work by maintaining a temperature difference across an orifice under rarefied conditions. However, there is room for improvement in throughput, range of pressure under operating conditions, operating voltage, energy efficiency, and other aspects affecting cost, manufacturability and performance. 
         [0005]    The foregoing examples of the related art and limitations related therewith are intended to be illustrative and not exclusive. Other limitations of the related art will become apparent upon a reading of the specification and a study of the drawings. 
       SUMMARY 
       [0006]    The following examples and aspects thereof are described and illustrated in conjunction with systems, tools, and methods that are meant to be exemplary and illustrative, not limiting in scope. In various examples, one or more of the above-described problems have been reduced or eliminated, while other examples are directed to other improvements. 
         [0007]    A technique provides a system and method for constraining gas molecules to the free molecular or transitional flow regime using nanoporous ceramic materials in gas pumps based on the principle of thermal transpiration. 
         [0008]    A system based on the technique may comprise a single nanoporous ceramic element or may comprise multiple layers of one or more types of nanoporous ceramic materials. A temperature difference may be achieved across the nanoporous ceramic element by the use of one or more heaters, thereby creating a flow of gas molecules through the nanoporous ceramic element. 
         [0009]    A method based on the technique may provide differential molecular pumping speeds for different gas molecules of varying sizes. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]      FIG. 1  depicts an exploded view of a thermal transpiration driven gas pump with a nanoporous ceramic element. 
           [0011]      FIG. 2  depicts an alternative embodiment of a thermal transpiration driven gas pump using nanoporous ceramic elements. 
           [0012]      FIG. 3  depicts an example of a nanoporous ceramic element including multiple layers of one or more types of ceramic materials. 
           [0013]      FIG. 4  depicts an alternative embodiment for the encapsulation shown in  FIG. 1 . 
           [0014]      FIG. 5  depicts an example of a thermal transpiration driven gas pump that provides different flow rates for different gas molecules. 
           [0015]      FIG. 6  depicts an example of an arrangement comprising various types of ceramic elements arranged in series or parallel along a flow path. 
           [0016]      FIGS. 7A and 7B  depict an example of a sequence of steps required to estimate some of the potential performance parameters for a transpiration driven Knudsen pump. 
           [0017]      FIG. 8  depicts the modeled pressure in the hot chamber. 
           [0018]      FIG. 9  depicts the idealized theoretical mass flow rate of air across a zeolite element subject to a given temperature drop across its thickness. 
       
    
    
     DETAILED DESCRIPTION 
       [0019]    In the following description, several specific details are presented to provide a thorough understanding. One skilled in the relevant art will recognize, however, that the concepts and techniques disclosed herein can be practiced without one or more of the specific details, or in combination with other components, etc. In other instances, well-known implementations or operations are not shown or described in detail to avoid obscuring aspects of various examples disclosed herein. 
         [0020]    A technique provides gas pumping by thermal transpiration using nanoporous ceramic materials to constrain the gas molecules to free molecular or transitional flow regime at pressures up to around atmospheric pressure. A method and system based on the technique may provide differential pumping rates for different gas molecules. The degree of differential pumping is determined primarily by the size of the gas molecules and their rates of interaction with the matrix of the nanoporous ceramic element. 
         [0021]    In a non-limiting example, the nanoporous ceramic element may be zeolite. Zeolites are hydrated alumino-silicate minerals with an “open” structure with a large surface area to volume ratio. They are characterized by an interconnected network of nanopores, which are typically in the range of 0.3 nm to 10 nm. Zeolites can be naturally occurring or may be synthesized. 
         [0022]    The Knudsen number (Kn), which is used as a parameter to characterize various gas flow regimes, is defined as the ratio of the mean free path of gas molecules (i.e. the average distance traveled by a molecule between two successive collisions) to the hydraulic diameter of the channel (i.e. the equivalent diameter to circular ducts). These flow regimes, which include free molecular, transitional, slip and viscous, correspond to Kn&gt;10, 0.1&lt;Kn&lt;10, 0.01&lt;Kn&lt;0.1 and Kn&lt;0.01, respectively. For the free molecular or transitional flow conditions to be satisfied at pressures near atmospheric pressure, the gas flow channels must have a hydraulic diameter (d h ) on the order of 100 nm or less. 
         [0023]    A thermal transpiration driven vacuum pump, also known as Knudsen pump, works by the principle of thermal transpiration as manifest in the equilibrium pressures of two chambers that are maintained at different temperatures, while connected by a channel that permits gas flow in the free molecular or transitional flow regimes, but not in the viscous regime. By equating the molecular flux between these chambers, it can be shown that the idealized ratio of the pressures is related to the ratio of their absolute temperatures by: 
         [0000]    
       
         
           
             
               
                 P 
                 2 
               
               
                 P 
                 1 
               
             
             = 
             
               
                 ( 
                 
                   
                     T 
                     2 
                   
                   
                     T 
                     1 
                   
                 
                 ) 
               
               
                 1 
                 2 
               
             
           
         
       
     
         [0024]    A Knudsen pump has high structural efficiency because of the lack of moving parts. Thermal transpiration, the mechanism for a Knudsen pump, has its observable effects on the gas molecules flowing across the channels with Knudsen number (Kn) greater than 0.1. 
         [0025]      FIG. 1  depicts a diagram 100 of an exploded view of a thermal transpiration driven gas pump with a nanoporous ceramic element.  FIG. 1  includes a first part of an encapsulation  101 , a second part of an encapsulation  105 , heaters  102 , passive thermal elements  103 , nanoporous ceramic element  104 , sensors  106 , feedback control  107 , coolers  108 , provisions for sensors  109 , and ports  110 . 
         [0026]    In the example of  FIG. 1 , the nanoporous ceramic element  104  may be disposed within an encapsulation. In a non-limiting example, the encapsulation may include a first encapsulation  101  and a second encapsulation  105 , which are configured to provide a seal around the nanoporous ceramic element  104  (with the exception of the inlet/outlet ports  110 ). The encapsulation may be bonded to the nanoporous ceramic element  104 , thereby restricting gas molecules passing through the device to flow through the nanoporous ceramic element  104 . Encapsulations  101  and  105  may be made of a thermally insulating material, such as polyvinyl chloride (PVC), to minimize the parasitic losses of heat from the device. 
         [0027]    In the example of  FIG. 1 , the heaters  102  may be resistive heaters. The heaters can be operated in such a way as to create a temperature difference between two sides of the nanoporous ceramic element  104 . A single heater may also be employed instead of two heaters as illustrated in  FIG. 1 . Alternatively, other mechanisms may be employed to provide the temperature difference, such as cooling the gas on one side of the nanoporous ceramic element  104  (for example, using coolers  108 ), using heat from a source outside of the device (such as scavenging waste heat from an independent system), or any other means of cooling or heating. The temperature difference may be created using at least one of the coolers  108  with at least one of the heaters  102  in conjunction or combination. 
         [0028]    Coolers  108  may be finned conductors providing passive cooling or heat sinks with liquid pumped through for active cooling. Heaters  102  and coolers  108  may be selectively turned on to control the temperature difference across the nanoporous ceramic element  104 , and to control the gas flow rate and/or direction of flow. 
         [0029]    In the example of  FIG. 1 , passive thermal elements  103  are disposed on either side of the nanoporous ceramic element  104  within the encapsulation  101  and  105 . The passive thermal elements  103  may be made of a material with high thermal conductivity, such as, in a non-limiting example, aluminum or silicon, and may have an array of holes through which a gas can flow. The size of the holes should be such that gas molecules within the passive thermal elements  103  are in the viscous flow regime. The high thermal conductivity of the passive thermal elements  103  and their proximity to heaters  102  means that the thermal elements  103  will reach a temperature close to that of the heaters  102 . In another embodiment, a heater may be directly fabricated onto the passive thermal element  103 , or the passive thermal element  103  may act as a heater and/or cooler itself. 
         [0030]    The nanoporous ceramic element  104  has a plurality of interconnected molecular sized pores throughout the volume. In a non-limiting example, the nanoporous ceramic element  104  may consist of zeolite or a combination of zeolite and other materials. The zeolite may be naturally occurring or synthesized. 
         [0031]    Sensors  106  may be disposed within provisions  109  to measure temperature, pressure, and/or flow rate across the nanoporous ceramic element  104 . The pressure, temperature and flow rate data may be analyzed and used by the feedback control  107  to reversibly control the temperature difference and hence the gas flow rate across the nanoporous ceramic element  104 . 
         [0032]    In operation, a temperature difference may be maintained between two sides of a nanoporous ceramic element  104 . The size of the pores of the ceramic element  104  constrains a gas to the free molecular or transitional flow regime within the matrix of the ceramic element  104 , even if the gas is at atmospheric pressure. The temperature difference generates a flow across the nanoporous ceramic element  104  due to thermal transpiration. Heat transfer between the hot side and the cold side of the nanoporous ceramic element  104  is reduced due to the low thermal conductivity of the ceramic element  104 , thus allowing for greater and more efficient temperature differences. Gas flowing through the device will enter the device through one of the ports  110 . The passive thermal element  103  allows the gas to achieve a desired temperature before the gas reaches the nanoporous ceramic element  104 . 
         [0033]      FIG. 2  depicts an alternative embodiment of a thermal transpiration driven gas pump using nanoporous ceramic elements.  FIG. 2  includes encapsulation  202 , first nanoporous ceramic element  204 , second nanoporous ceramic element  206 , first passive thermal element  208 , second passive thermal element  210 , third passive thermal element  212 , fourth passive thermal element  214 , heater  216 , inlet ports  218 , and outlet port  220 . 
         [0034]    The elements are similar to those as described with reference to  FIG. 1 . In the example of  FIG. 2 , the first nanoporous ceramic element  204  is disposed between the first passive thermal element  208  and the second passive thermal element  210 . The second nanoporous ceramic element  204  is disposed between the third passive thermal element  212  and the fourth passive thermal element  214 . Heater  216  is in thermal contact with both the second passive thermal element  210  and the third passive thermal element  212 . These elements are sealed within encapsulation  202 . The nanoporous ceramic elements  204  and  206  and heaters provide a molecular (or transitional) flow regime and temperature gradient, respectively, such that a gas flow is created between the inlet ports  218  and the outlet port  220  due to thermal transpiration. 
         [0035]      FIG. 3  depicts a diagram  300  of a nanoporous ceramic element including multiple layers of one or more types of ceramic materials.  FIG. 3  includes first nanoporous ceramic layer  301 , second nanoporous ceramic layer  302 , third nanoporous ceramic layer  303 , fourth nanoporous ceramic layer  304 . 
         [0036]    In the example of  FIG. 3 , the nanoporous ceramic element includes multiply stacked layers of one or more types of nanoporous ceramic materials. Stacking layers of nanoporous ceramic materials may act in favor of thermal efficiency of the device by disrupting the path of phonons moving across the thickness of the nanoporous ceramic element. In another embodiment, passive thermal elements, heaters, and/or coolers may be disposed between the stacked layers. 
         [0037]      FIG. 4  depicts an alternative embodiment for the encapsulation shown in  FIG. 1 . The encapsulation  400  is hollowed to accommodate a thermally conductive base  405 , which provides greater uniformity in temperature across the facet of the ceramic element  104 . It may also serve as a heat sink that maintains the cold end of the ceramic element  104  close to room temperature.  FIG. 4  includes port provisions  401  and  406 , sensor provision  402 , and thermally conductive base  405 . 
         [0038]    In the example of  FIG. 4 , port provisions  401  and  406  may be used for inlet or outlet of gas flow. Sensor provisions  402  may accommodate various sensing elements to measure, for example, the gas flow rate through the nanoporous ceramic element, the temperature, or other variables. 
         [0039]    The thermally conductive base  405  may be used to create a temperature gradient across the nanoporous ceramic element  104 . In a non-limiting example, the thermally conductive base  405  may absorb all the necessary heat from an outside source and may therefore not require a heater as described in  FIG. 1 . In one embodiment, thermally conductive base  405  may be connected to a cooler  108 . In another embodiment, the thermally conductive base  405  may be used in combination or conjunction with a heater and/or cooler, as described with reference to  FIG. 1 . Thermally conductive base  405  may be made of copper, and may be used for thermal coupling of the transpiration driven gas pump with heat from an external system. 
         [0040]      FIG. 5  depicts a diagram  500  of a thermal transpiration driven gas pump that provides different flow rates for different gas molecules.  FIG. 5  includes nanoporous ceramic element  501 , seal  502 , encapsulations  503  and  505 , sensors  504 , passive thermal elements  506 , heaters  507 , sensor provisions  508 , port provisions  509 , and feedback control system  5   10 . 
         [0041]    The transpiration driven flow speeds may depend on the mass of the gas molecules and their rates of interaction with the matrix of the nanoporous ceramic element  501 . This may lead to different flow characteristics for different gases. The interaction between the gas molecules and the ceramic element  501  may further be controlled by coating the surface of the matrix of the ceramic element  501 . The coating may comprise of one or more types of layers of polymer that may be treated chemically. 
         [0042]    In the example of  FIG. 5 , encapsulations  503  and  505 , sensors  504 , passive thermal elements  506 , heaters  507 , sensor provisions  508 , port provisions  509 , and feedback control system  510  are similar to those as described in reference to  FIG. 1 . 
         [0043]    In the example of  FIG. 5 , the nanoporous ceramic element  501  is configured to provide a flow path that is long compared to the mean free path of the gas molecules. The nanoporous ceramic element  501  may be shaped in lithographically fabricated flow channels and may be sealed, as indicated by seal  502 , to prevent the gas molecules from escaping through the edges of the nanoporous ceramic element  501 . 
         [0044]    The lithographically fabricated flow channels may include a micromachined recess on the surface of a glass wafer. Ends of the nanoporous ceramic element  501  may have encapsulations  503  and  505 , which have provisions for inlet/outlet  509 . The device encapsulations  500  may further comprise passive thermal elements  506  and heaters  507  required to reversibly control the differential pumping of the gas. Encapsulations  503  and  505  may have provisions  508  for sensors  504  that can sample temperature, pressure and flow rate of the gas sample entering and leaving the nanoporous ceramic element  501 . The pressure, temperature and flow rate data may be used to provide feedback to the control system  510 , which regulates the gas flow rate across the nanoporous ceramic element  501 . 
         [0045]      FIG. 6  depicts an example  600  of an arrangement comprising various types of ceramic elements arranged in series or parallel along a flow path.  FIG. 6  includes nanoporous ceramic sub-elements  602 - 610 . 
         [0046]    In the example of  FIG. 6 , the nanoporous ceramic element, as described with reference to  FIGS. 1 and 5 , is divided into sub-elements  602 - 610 , which may be of varying sizes, shapes and materials. Sub-elements  602 - 610  may or may not have independent heaters associated with them. The sub-elements  602 - 610  may be arranged in series along the flow path such that the gas molecules must sequentially pass through each one, or they may be arranged in parallel, such that each gas molecule may pass through only one. This arrangement may further provide a means for physically separating the flow path of certain types of molecules. 
         [0047]      FIGS. 7A and 7B  (herein referred to as  FIG. 7  collectively) depict an example of a flowchart for estimating performance parameters for a transpiration driven pump. These parameters may include the percent porosity of the nanoporous ceramic element, effective leakage aperture of a defect, correction for thermal contact resistance, correction for the delay in heating of the air trapped in the hot chamber and so on. 
         [0048]    In the example of  FIG. 7 , the flowchart starts at module  702  with choosing a time step (Δt) and calculating interpolated temperature in the hot chamber (Th_int) and in the cold chamber (Tc_int). 
         [0049]    In the example of  FIG. 7 , the flowchart continues to module  704  with estimating the initial number of molecules entrapped in the hot chamber. The initial number of molecules relates to the dead volume (V) of the entrapped gas, its temperature (T) and pressure (P) by the correlation 
         [0000]    
       
         
           
             
               PV 
               
                 
                   k 
                   B 
                 
                  
                 T 
               
             
             , 
           
         
       
     
         [0000]    where k B  is the Boltzmann constant. 
         [0050]    In the example of  FIG. 7 , the flowchart continues to module  706  with selecting the percent porosity (Por) of the nanoporous ceramic element, selecting the effective aperture diameter for gas leakage through macrocracks for the duration the heater is on (D_ap_on), and selecting the effective aperture diameter for gas leakage through macrocracks for the duration the heater is off (D_ap_off). Por D_ap_on and D_ap_off may be selected such that it minimizes the least squared error between the modeled pressure in the hot chamber (Ph_mod) and the interpolated value (Ph_int) of the experimentally measured pressure (Ph_exp) in the hot chamber. Ph_int may be a cubic interpolation of Ph_exp of the form e.t 3 +f.t 2 +g.t+h=Ph_int, where the coefficients e,f, g and h may depend on Ph_exp. 
         [0051]    In the example of  FIG. 7 , the flowchart continues to module  708  with calculating the final pressure for the current time step. The final pressure may depend on the temperature rise over the duration Δt. 
         [0052]    In the example of  FIG. 7 , the flowchart continues to module  710  with calculating the average temperature and pressure over the time step. The average temperature and pressure may be assumed to be the average temperature and pressure over current time period for the purpose of subsequent calculation over this time step. 
         [0053]    In the example of  FIG. 7 , the flowchart continues to module  712  with calculating the number of molecules (N_pos) leaking out of the hot chamber through the aperture by virtue of Poiseuille&#39;s law over the time Δt, and calculating the number of molecules (N_tt) pumped into the hot chamber due to thermal transpiration flow across the nanopores of the ceramic element over the time Δt. This accounts for the transpiration flow due to temperature gradient and back flow due to the pressure gradient. The calculation of N_pos and N_tt may use average temperature and pressure over the current time step. 
         [0054]    In the example of  FIG. 7 , the flowchart continues to module  714  with estimating the final number of molecules in the hot chamber at the end of Δt. The final number of molecules after time step Δt may be given by the algebraic sum of N_pos, N_tt and the initial number of molecules in the hot chamber. 
         [0055]    In the example of  FIG. 7 , the flowchart continues to module  716  with calculating the modeled pressure in the hot chamber (Ph_mod). P_mod at a particular time-step may depend on the number of molecules remaining the chamber, temperature and pressure. 
         [0056]    In the example of  FIG. 7 , the flowchart continues to module  718  with determining: 
         [0000]    
       
         
           
             
               ɛ 
               = 
               
                 
                   
                     [ 
                     
                       
                         1 
                         n 
                       
                        
                       Σ 
                        
                       
                         
                            
                           
                             Ph_int 
                             - 
                             Ph_mod 
                           
                            
                         
                         2 
                       
                     
                     ] 
                   
                   
                     1 
                     2 
                   
                 
                 ≤ 
                 
                   err 
                    
                   
                       
                   
                    
                   1 
                 
               
             
             , 
           
         
       
     
         [0000]    where ε is the root mean square deviation of Ph_mod with respect to Ph_int, n is the total number of interpolation points, and err1 is the tolerance limit on the root mean square deviation. 
         [0057]    If the decision at module  718  is yes, then the flowchart continues to module  720  with choosing the rate of increase of temperature difference (RITD_on) between Tc_mod and Tc_exp for the duration when heater is on, choosing the rate of decrease of temperature difference (RDTD_off) between Tc_mod and Tc_exp for the duration when heater is off, and calculating Tc_mod. Due to thermal contact resistance Tc_mod is expected be higher than Tc_exp at all times. RITD_on and RDTD_off represent the loss in the performance due to the thermal contact resistance. 
         [0058]    In the example of  FIG. 7 , the flowchart continues to module  722  with calculating the modeled pressure in the hot chamber (Ph_mod). Ph_mod at this step accounts for the loss in performance due to the thermal contact resistance. 
         [0059]    In the example of  FIG. 7 , the flowchart continues to module  724  with determining: 
         [0000]    
       
         
           
             
               ɛ 
               = 
               
                 
                   
                     [ 
                     
                       
                         1 
                         n 
                       
                        
                       Σ 
                        
                       
                         
                            
                           
                             Ph_int 
                             - 
                             Ph_mod 
                           
                            
                         
                         2 
                       
                     
                     ] 
                   
                   
                     1 
                     2 
                   
                 
                 ≤ 
                 
                   err 
                    
                   
                       
                   
                    
                   2 
                 
               
             
             , 
           
         
       
     
         [0000]    where ε is the root mean square difference between Ph_mod and Ph_int, and err2 is the tolerance limit on the root mean square deviation. 
         [0060]    If the decision at module  724  is yes, then the flowchart continues to module  726  with choosing the factor (TCF_on) by which the time constant of heating of air is higher than Th_exp for the duration when heater is on, choosing the factor (TCF_off) by which the time constant of heating of air is higher than Th_exp for the duration when heater is off, and calculating the modeled temperature of air in the hot chamber (Th_air). TCF_on and TCF_off account for the delay in heating and cooling of air molecules, entrapped in the hot chamber, with respect to the heater itself. 
         [0061]    In the example of  FIG. 7 , the flowchart continues to module  728  with calculating the modeled pressure in the hot chamber (Ph_mod). Ph_mod at this step accounts for the delay in the heating of the air in the hot chamber. 
         [0062]    In the example of  FIG. 7 , the flowchart continues to module  730  with determining: 
         [0000]    
       
         
           
             
               ɛ 
               = 
               
                 
                   
                     [ 
                     
                       
                         1 
                         n 
                       
                        
                       Σ 
                        
                       
                         
                            
                           
                             Ph_int 
                             - 
                             Ph_mod 
                           
                            
                         
                         2 
                       
                     
                     ] 
                   
                   
                     1 
                     2 
                   
                 
                 ≤ 
                 
                   err 
                    
                   
                       
                   
                    
                   3 
                 
               
             
             , 
           
         
       
     
         [0000]    where ε is the root mean square difference between Ph_mod and Ph_int, and err3 is the tolerance limit on the root mean square deviation. These deviations are representative numbers for variation of between Ph_mod as compared to Ph_int in these steps. 
         [0063]    If the decision at module  730  is yes, then the flowchart terminates. If the decision at module  718 ,  724 , or  730  is no, then the flowchart continues to module  706 . 
         [0064]      FIG. 8  depicts the modeled pressure in the hot chamber (Ph_mod) as determined by a method as described with reference to  FIG. 7 . Ph_mod takes into account some of the performance parameters, such as defects in the ceramic matrix, effect of delay in the heating of the air entrapped in hot chamber (Th_air), elevated temperature at the cold end of the ceramic element due to the thermal contact resistance (Tc_mod) and so on. 
         [0065]      FIG. 9  depicts the idealized theoretical mass flow rate of air across a zeolite element (48 mm in diameter and 2.3 mm thick) subject to a given temperature drop across its thickness. The predictions are based on a semi-analytical model for gas flow in the free molecular and transitional flow regimes. 
         [0066]    According to a known model, the average mass flow rate across a narrow channel, by the virtue of thermal transpiration, is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       M 
                       . 
                     
                     avg 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             Q 
                             T 
                           
                            
                           
                             
                               
                                 T 
                                 h 
                               
                               - 
                               
                                 T 
                                 c 
                               
                             
                             
                               T 
                               avg 
                             
                           
                         
                         - 
                         
                           
                             Q 
                             P 
                           
                            
                           
                             
                               
                                 P 
                                 h 
                               
                               - 
                               
                                 P 
                                 c 
                               
                             
                             
                               P 
                               avg 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         π 
                          
                         
                             
                         
                          
                         
                           a 
                           3 
                         
                          
                         
                           P 
                           avg 
                         
                       
                       l 
                     
                      
                     
                       
                         ( 
                         
                           m 
                           
                             2 
                              
                             
                               k 
                               B 
                             
                              
                             
                               T 
                               avg 
                             
                           
                         
                         ) 
                       
                       
                         1 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T h  and P h  are the temperature and pressure on the hot end of the nanoporous channel, T c  and P c  are the temperature and pressure on the cold end of the nanoporous channel, T avg  and P avg  are the average temperature and pressure in the nanoporous channel, m is mass of a gas molecule, k B  is the Boltzmann constant, a is the hydraulic radius of the narrow tube, and l is the length of the nanoporous channel. Q P  and Q T  are the pressure and temperature coefficients that depend on rarefaction parameter δ avg  given by 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                     avg 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             π 
                             3 
                           
                           2 
                         
                         ) 
                       
                       
                         1 
                         2 
                       
                     
                      
                     
                       
                         
                           aD 
                           2 
                         
                          
                         
                           P 
                           avg 
                         
                       
                       
                         
                           k 
                           B 
                         
                          
                         
                           T 
                           avg 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where D is the collision diameter of the gas molecules under consideration. 
         [0067]    The analytical model described above, coupled with various performance parameters, may be used to describe a representative simulation model for thermal transpiration pumping through the nanoporous ceramic element. 
         [0068]    The simulation model also serves as a platform for benchmarking various material properties and design features that may affect the performance of a transpiration driven gas pump. These include, for example:
       The percentage porosity of the ceramic element Por and the effective diameter of the leak aperture D_ap_on or D_ap_off are two of the most important parameters that may affect the steady state pressure attained by the device.   Loss in performance due to the thermal contact resistance may play a major role in the deterioration of transpiration based gas pumping in continuous operation.   The time constants of heating and cooling of the air entrapped in the hot chamber of the device may cause an initial pressure spike that occurs before the pressure down to a steady state value.       
 
         [0072]    A single stage transpiration driven gas pump, with 48 mm diameter and 2.3 mm thick zeolite element, subjected to a temperature gradient of 15.7 K/mm may produce a flow rate of approximately 0.1-10 ml/min against a back pressure of about 50 Pa offered by a typical measurement set-up. The matrix of the zeolite element, which is assumed to have pore diameter 0.45 nm and porosity (Por) of 34%, may have structural defects or leakage through the seals that would be accounted for by the effective leakage aperture (D_ap_on and D_ap_off). 
         [0073]    While operating with sealed outlet, a typical variation of pressure in the hot chamber (Ph_mod) may appear as in  FIG. 8 . This transient pressure profile, which is primarily dependent on thermal transpiration flow across the zeolite element, corresponds to the variation of temperature in the hot and the cold chambers. The temperature in the cold chamber is assumed to regulate the temperature at the cold end of the zeolite (Tc_mod). This temperature rise over time is due to the thermal contact resistance at the interface of various thermal elements. The temperature at the hot end of the zeolite is assumed to be regulated by the bulk air temperature (Th_air) entrapped in the hot chamber. The matrix of the zeolite element is assumed to have pore diameter 0.45 nm and porosity (Por) of 34%. Further, the zeolite matrix is assumed to have effective leak aperture diameters (D_ap_on and D_ap_off) of about 20 μm, which may be due to structural defects in the matrix of the zeolite element or due to the leakage through the seals. 
         [0074]    During the intial phases of the device operation, thermal expansion of the gas entrapped in the hot chamber may be more prominent, which would result in a sharp rise in the pressure in the hot chamber ( FIG. 8 ). The pressure rise due to the thermal expansion of gas would be subsequently neutralized by the Poiseuille flow that may be responsible for the backflow of gas molecules from hot chamber to the cold chamber. Finally, while operating in steady state, thermal transpiration would be the dominant phenomenon and it would result in a higher steady state pressure. As soon as the heater is turned off the transpiration driven flow would cease and hence the Poiseuille flow may play a dominant role in equilibrating the pressure between the hot chamber and the ambient. 
         [0075]    The pressure profile (Ph_mod), as predicted by the simulation model (based on the algorithm presented in  FIG. 8 ), takes into account the design and material choices and assumptions listed above, and may be representative of a typical experimentally observed pressure (Ph_exp), such that the root mean square deviation (err1, err2 and err3) between the two is on the order of 1 kPa. The root mean square deviations err1, err2 and err3 serve as the convergence criteria for various simulation steps. 
         [0076]    A semi-analytical model for the gas flow in free molecular and transitional flow regime may be used to estimate the idealized pumping efficiency of the transpiration driven gas pump.  FIG. 9  suggests that under idealized conditions a 2.3 mm thick zeolite element with 48 mm diameter may generate a flow rate of about 0.1 sccm for a temperature drop of about 38 K. The idealized model assumes: (a) perfect structure of zeolite, which has no macro cracks, (b) perfect thermal contact at all interfaces, (c) uniform in-plane temperature, (d) negligible flow resistance offered by all other elements, except the zeolite element. 
         [0077]    The model may be further used to estimate the idealized differential pumping capabilities of a Knudsen pump. The model predicts that for a temperature gradient of about 15.7 K/mm across the zeolite element, the hydrogen gas molecules, which are two and a half times smaller than nitrogen molecules, are pumped about four times faster. Moreover, Poiseuille flow may also provide a mechanism for differential pumping within the zeolite element. Under idealized conditions, for pressure driven flow of 21 kPa/mm across the zeolite element, with zero temperature gradient, hydrogen molecules are expected to move four times faster than nitrogen molecules.