Abstract:
A system and method of balancing a hydrogen feed for a fuel cell to optimize flow of hydrogen through the fuel cell, wherein a pressure drop through parallel feed channels and active area channels of the fuel cell is balanced.

Description:
FIELD OF THE INVENTION 
     The invention relates to a fuel cell and more particularly to a system and method of balancing a hydrogen feed for the fuel cell to optimize flow of hydrogen through the fuel cell. 
     BACKGROUND OF THE INVENTION 
     Fuel cell systems are increasingly being used as a power source in a wide variety of applications. Fuel cell systems have been proposed for use in power consumers such as vehicles as a replacement for internal combustion engines, for example. Such a system is disclosed in commonly owned U.S. patent application Ser. No. 10/418,536, hereby incorporated herein by reference in its entirety. Fuel cells may also be used as stationary electric power plants in buildings and residences, as portable power in video cameras, computers, and the like. Typically, the fuel cells generate electricity used to charge batteries or to provide power for an electric motor. 
     Fuel cells are electrochemical devices which directly combine a fuel such as hydrogen and an oxidant such as oxygen to produce electricity. The oxygen is typically supplied by an air stream. The hydrogen and oxygen combine to result in the formation of water. Other fuels can be used such as natural gas, methanol, gasoline, and coal-derived synthetic fuels, for example. 
     The basic process employed by a fuel cell is efficient, substantially pollution-free, quiet, free from moving parts (other than an air compressor, cooling fans, pumps and actuators), and may be constructed to leave only heat and water as by-products. The term “fuel cell” is typically used to refer to either a single cell or a plurality of cells depending upon the context in which it is used. The plurality of cells is typically bundled together and arranged to form a stack with the plurality of cells commonly arranged in electrical series. Since single fuel cells can be assembled into stacks of varying sizes, systems can be designed to produce a desired energy output level providing flexibility of design for different applications. 
     Different fuel cell types can be provided such as phosphoric acid, alkaline, molten carbonate, solid oxide, and proton exchange membrane (PEM), for example. The basic components of a PEM-type fuel cell are two electrodes separated by a polymer membrane electrolyte. Each electrode is coated on one side with a thin catalyst layer. The electrodes, catalyst, and membrane together form a membrane electrode assembly (MEA). 
     In a typical PEM-type fuel cell, the MEA is sandwiched between “anode” and “cathode” diffusion mediums (hereinafter “DM&#39;s”) or diffusion layers that are formed from a resilient, conductive, and gas permeable material such as carbon fabric or paper. The DM&#39;s serve as the primary current collectors for the anode and cathode as well as provide mechanical support for the MEA. The DM&#39;s and MEA are pressed between a pair of electronically conductive plates which serve as secondary current collectors for collecting the current from the primary current collectors. The plates conduct current between adjacent cells internally of the stack in the case of bipolar plates and conduct current externally of the stack (in the case of monopolar plates at the end of the stack). 
     The secondary current collector plates each contain at least one active region that distributes the gaseous reactants over the major faces of the anode and cathode. These active regions, also known as flow fields, typically include a plurality of lands which engage the primary current collector and define a plurality of grooves or flow channels therebetween. The channels supply the hydrogen and the oxygen to the electrodes on either side of the PEM. In particular, the hydrogen flows through the channels to the anode where the catalyst promotes separation into protons and electrons. On the opposite side of the PEM, the oxygen flows through the channels to the cathode where the oxygen attracts the hydrogen protons through the PEM. The electrons are captured as useful energy through an external circuit and are combined with the protons and oxygen to produce water vapor at the cathode side. 
     When laying out a pattern for the flow field, it is desirable to have all of the flow channels the same length to balance the flow amongst the channels for uniform distribution of reactant flow, as a first approximation. In some flow fields, however, it is desirable to branch channels or to connect channels in an active area to fewer feed channels to allow smaller headers for smaller overall stack size. In other flow field designs, the feed channels may be positioned outside of the active area of the flow field since the DM has been removed from these regions to allow nesting of the plate halves for a smaller overall stack size. In this case, the feed channels may or may not be branched. 
     In the above configurations, uniform channel lengths do not provide uniform reactant distribution for the anode when hydrogen is used. For hydrogen anode flow, there is a significant change in volume from inlet to outlet as the hydrogen gas is consumed within the active area of the fuel cell. The flow will be unevenly distributed with more flow on one side of the flow field. The volume flow is greater in the inlet branched or non-active channels, so more pressure drop occurs in these channels per length than for the outlet branched or non-active channels. Therefore, for channels on a side which have a longer inlet feed channel length, the flow will be reduced compared to channels on the opposite side which have longer outlet channel lengths. 
     It would be desirable to produce a plate for a fuel cell wherein a hydrogen feed is balanced to optimize flow of hydrogen through the fuel cell. 
     SUMMARY OF THE INVENTION 
     Consistent and consonant with the present invention, a plate for a fuel cell wherein a hydrogen feed is balanced to optimize flow of hydrogen through the fuel cell, has surprisingly been discovered. 
     In one embodiment, the plate comprises a plate having a flow field formed therein, the flow field defined by a plurality of channels formed on an outer surface thereof, the channels of the flow field adapted to provide communication between a source of gas and an exhaust header; and an active region of the flow field including at least a portion of the channels, the active region including a diffusion medium adjacent thereto, at least a portion of the gas consumed in the active region to cause a difference in volumetric flow of the gas upstream of the active region and downstream of the active region, wherein at least a portion of the channels includes a branched section forming a plurality of branched channels to facilitate a balancing of a flow of the gas therethrough. 
     In another embodiment, the plate comprises a plate having a first side and a second side; and a flow field formed in a first side of the plate, the flow field further comprising an inlet feed region; an outlet feed region; a plurality of inlet flow channels formed in the inlet feed region on an outer surface of the plate in communication with a source of gas; a plurality of outlet flow channels formed in the outlet feed region on an outer surface of the plate in communication with an exhaust header; a plurality of intermediate flow channels formed in the flow field providing communication between the inlet flow channels and the outlet flow channels; and an active region including at least a portion of the intermediate flow channels, the active region including a diffusion medium adjacent thereto, at least a portion of the gas consumed in the active region to cause a difference in flow of the gas through the inlet flow channels and the outlet flow channels, wherein at least one of the inlet flow channels communicates with at least two of the intermediate flow channels and at least one of the outlet flow channels communicates with at least two of the intermediate flow channels to facilitate a balancing of a flow of the gas through the flow field. 
     The invention also provides methods of balancing fuel flow through a flow field in a fuel cell. 
     In one embodiment, the method of balancing fuel flow through a flow field in a fuel cell comprises the steps of providing a plate having the flow field formed therein, the flow field defined by a plurality of channels formed on an outer surface of the plate; providing a source of gas, wherein the channels provide communication between a source of gas and an exhaust header; providing a diffusion medium adjacent at least a portion of the channels to form an active region of the flow field, wherein at least a portion of the gas is consumed in the active region to cause a difference in volumetric flow of the gas upstream of the active region and downstream of the active region; providing a branched section in at least a portion of the channels, the branched section including a plurality of branched channels; and positioning the branched channels in desired ones of the channels to facilitate a balancing of a flow of the gas through the flow field. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       The above, as well as other advantages of the present invention, will become readily apparent to those skilled in the art from the following detailed description of a preferred embodiment when considered in the light of the accompanying drawings in which: 
         FIG. 1  is an exploded perspective view of a fuel cell stack; 
         FIG. 2  is a schematic view of a flow field of an anode plate including branched flow according to the prior art; 
         FIG. 3  is a graph showing an anode stoichiometry profile for the flow field of  FIG. 2 ; 
         FIG. 4  is a schematic view of a flow field of an anode plate without branched flow according to the prior art; 
         FIG. 5  is a schematic view of a flow field of an anode plate including branched flow according to an embodiment of the invention; 
         FIG. 6  is a schematic view of a flow field of an anode plate including branched flow according to another embodiment of the invention; and 
         FIG. 7  is a schematic view of a flow field of an anode plate including branched flow according to another embodiment of the invention. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The following detailed description and appended drawings describe and illustrate various exemplary embodiments of the invention. The description and drawings serve to enable one skilled in the art to make and use the invention, and are not intended to limit the scope of the invention in any manner. In respect of the methods disclosed, the steps presented are exemplary in nature, and thus, the order of the steps is not necessary or critical. 
       FIG. 1  shows a two-cell bipolar PEM fuel cell stack  10 . Although a bipolar PEM fuel cell stack is shown, it is understood that other fuel cell types and configurations can be used without departing from the scope and spirit of the invention. It is also understood that fuel cell stacks having more cells and plates can be and typically are used. 
     The fuel cell stack  10  includes a first membrane-electrode-assembly (MEA)  12  and a second membrane-electrode assembly  14 . An electrically conductive, liquid-cooled, bipolar plate  16  is disposed between the first MEA  12  and the second MEA  14 . The first MEA  12 , the second MEA  14 , and the bipolar plate  16  are stacked together between clamping plates  18 ,  20  and monopolar end plates  22 ,  24 . The clamping plates  18 ,  20  are electrically insulated from the monopolar end plates  22 ,  24 . 
     A working face of each of the monopolar end plates  22 ,  24 , as well as both working faces of the bipolar plate  16  include a plurality of grooves or channels  26 ,  28 ,  30 ,  32  formed therein. The channels  26 ,  28 ,  30 ,  32  define a so-called “flow field” for distributing a fuel and an oxidant gas over the faces of the MEA&#39;s  12 ,  14 . In the embodiment described herein, the fuel is hydrogen and the oxidant is oxygen, although it is understood that other fuels and oxidants can be used as desired. 
     Nonconductive gaskets  34 ,  36 ,  38 ,  40  are respectively disposed between the monopolar end plate  22  and the first MEA  12 , the first MEA  12  and the bipolar plate  16 , the bipolar plate  16  and the second MEA  14 , and the second MEA  14  and the monopolar end plate  24 . The gaskets  34 ,  36 ,  38 ,  40  provide a seal and electrically insulate the monopolar end plate  22  and the first MEA  12 , the first MEA  12  and the bipolar plate  16 , the bipolar plate  16  and the second MEA  14 , and the second MEA  14  and the monopolar end plate  24 . 
     Gas-permeable diffusion media  42 ,  44 ,  46 ,  48  abut respective electrode faces of the first MEA  12  and the second MEA  14 . The diffusion media  42 ,  44 ,  46 ,  48  are respectively disposed between the monopolar end plate  22  and the first MEA  12 , the first MEA  12  and the bipolar plate  16 , the bipolar plate  16  and the second MEA  14 , and the second MEA  14  and the monopolar end plate  24 . 
     The bipolar plate  16  is typically formed from an anode plate (not shown) and a cathode plate (not shown). The anode plate and the cathode plate are bonded together to form a coolant chamber therebetween. The channel  28  is formed in the anode plate and channel  30  is formed in the cathode plate to form the respective flow fields. 
       FIG. 2  shows a flow field  60  of an anode plate (not shown) according to the prior art. The flow field  60  includes an inlet feed region  62  and an outlet feed region  64 . A plurality of inlet flow channels  66  is formed in the inlet feed region  62  and a plurality of outlet flow channels  68  is formed in the outlet feed region  64 . The inlet flow channels  66  are in communication with an anode inlet header (not shown), and the outlet flow channels  68  are in communication with an anode exhaust header (not shown). Each of the inlet flow channels  66  and the outlet flow channels  68  are in communication with a pair of intermediate branched flow channels  70 . The flow channels  66 ,  68 ,  70  are adapted to provide a flow path from a source of an anode gas or fuel (not shown) to the exhaust header as indicated by the arrows I, E. An active region S is represented by the shaded area of the flow field  60 . 
     In operation, the fuel is caused to flow into the flow field  60  through the inlet flow channels  66  from the source of fuel. When the fuel reaches the branched channels  70  downstream of the inlet feed region  62 , the fuel flowing through the inlet flow channels  66  is divided into two branched channels  70 . The fuel continues to flow through the branched channels  70  to the outlet flow channels  68  where the fuel is combined from two branched channels  70  into one of the outlet flow channels  68 . 
     As is well known in the art, the fuel flows through the channels  66 ,  70 ,  68  in the active region S of the flow field  60  and is consumed during the chemical reaction resulting in the formation of water and electricity. Due to this consumption of fuel, the volume of fuel flowing from the anode exhaust header is less than the volume of fuel flowing to the anode inlet header. Therefore, a difference in the volumetric flow exists between the anode inlet header and the anode exhaust header. Since the inlet flow channels  66  differ in length from the outlet flow channels  68  communicating with the same branched channels  70 , the difference between the volumetric flow at the anode inlet header and the anode exhaust header results in a higher pressure drop in the inlet flow channels  66  and a lower pressure drop in the outlet flow channels  68 . This results in an uneven distribution of fuel between each of the individual flow channels  66 ,  70 ,  68  across the flow field  60 . Consequently, some areas of the flow field  60  will have higher flows of fuel than other areas of the flow field  60 . This uneven distribution of fuel in the flow field  60  is undesirable as anode stoichiometry distribution is affected.  FIG. 3  is a graph  72  showing an anode stoichiometry profile for the flow field  60  of  FIG. 1 . 
       FIG. 4  illustrates a flow field  80  of an anode plate (not shown) according to the prior art. The flow field  80  includes an inlet feed region  82  and an outlet feed region  84 . A plurality of inlet flow channels  86  is formed in the inlet feed region  82  and a plurality of outlet flow channels  88  is formed in the outlet feed region  84 . The inlet flow channels  86  are in communication with an anode inlet header (not shown), and the outlet flow channels  88  are in communication with an anode exhaust header (not shown). Each of the inlet flow channels  86  and the outlet flow channels  88  are in communication with an intermediate flow channel  90 . The flow channels  86 ,  88 ,  90  are adapted to provide a flow path from a source of an anode gas or fuel (not shown) to the exhaust header as indicated by the arrows I, E. An active region S is represented by the shaded area of the flow field  80 . The inlet feed region  82  and the outlet feed region  84  are located outside of the active region S and are non-active. Typically, where the inlet feed region  82  and the outlet feed region  84  are inactive, the diffusion media has not been added to these regions to permit nesting of plate halves to reduce the overall stack size of the fuel cell. 
     In operation, the fuel is caused to flow into the flow field  80  through the inlet flow channels  86  from the source of fuel. The fuel then flows through the intermediate channels  90  and through the outlet flow channels  88 . 
     As is well known in the art, the fuel flows through the channels  90  in the active region S of the flow field  80  and is consumed. As described above for  FIG. 2 , the volume of fuel flowing from the anode exhaust header is less than the volume of fuel flowing to the anode inlet header, and a difference in the volumetric flow exists between the anode inlet header and the anode exhaust header. Additionally, both the lengths of each of the inlet flow channels  86  and the lengths of each of the outlet flow channels  88  are different. Due to the difference in length of the inlet flow channels  86  and the outlet flow channels  88 , as well as the difference between the volumetric flow at the anode inlet header and the anode exhaust header, a difference in pressure drop between the inlet flow channels  86  and the outlet flow channels  88  exists. Thus, an uneven distribution of fuel between each of the individual flow channels  86 ,  90 ,  88  exists across the flow field  80 . As a result, some areas of the flow field  80  will have higher flows of fuel than other areas of the flow field  80 . 
       FIG. 5  illustrates a flow field  100  of an anode plate (not shown) according to an embodiment of the invention. The flow field  100  includes an inlet feed region  102  and an outlet feed region  104 . A plurality of inlet flow channels  106  is formed in the inlet feed region  102  and a plurality of outlet flow channels  108  is formed in the outlet feed region  104 . The inlet flow channels  106  are in communication with an anode inlet header (not shown), and the outlet flow channels  108  are in communication with an anode exhaust header (not shown). 
     Each of the inlet flow channels  106  and the outlet flow channels  108  are in communication with intermediate flow channels  110 . In the embodiment shown, the inlet flow channels  106  and the outlet flow channels  108  communicate with a plurality of branched intermediate flow channels  110  between two and eight. It is understood that the inlet flow channels  106  and the outlet flow channels  108  can communicate with more or fewer intermediate flow channels  110  as desired. The flow channels  106 ,  108 ,  110  are adapted to provide a flow path from a source of an anode gas or fuel (not shown) to the exhaust header as indicated by the arrows I, E. An active region S is represented by the shaded area of the flow field  100 , and encompasses the inlet region  106  and the outlet region  108 . The active region S includes a diffusion medium adjacent those regions. It is understood that the inlet region  106  and the outlet region  108  can be located outside of the active region S, as will be described herein for other embodiments of the invention. 
     In operation, the fuel is caused to flow into the flow field  100  through the inlet flow channels  106  from the source of fuel. The fuel then flows through the intermediate channels  110  and through the outlet flow channels  108 . As the fuel flows through the channels  106 ,  110 ,  108  in the active region S, the fuel is consumed. In order to compensate for the differences of the volume of fuel flowing through the anode inlet header and the anode exhaust header due to the consumption of the fuel, the number of intermediate flow channels  110  in communication with each of the inlet flow channels  106  and each of the outlet flow channels  108  is controlled. 
     In the inlet feed region  102 , the volumetric flow is high. Thus, if the inlet flow channel  106  is relatively short, a larger number of intermediate flow channels  110  are in communication therewith. If the inlet flow channel  106  is relatively long, a smaller number of intermediate flow channels  110  are in communication therewith. In the outlet feed region  104 , the volumetric flow is low. Thus, if the outlet flow channel  108  is relatively short, a larger number of intermediate flow channels  110  are in communication therewith. If the outlet flow channel  108  is relatively long, a smaller number of intermediate flow channels  110  are in communication therewith. This balances the pressure drop, and therefore the flow, across each of the inlet flow channels  106 , the intermediate flow channels  110 , and the outlet flow channels  108 , thereby optimizing an anode stoichiometry distribution for the fuel cell. 
     In order to determine the correct balance, Equation 1 can be used: 
     
       
         
           
             
               
                 
                   dP 
                   = 
                   
                     
                       C 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       μ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Lvol 
                     
                     
                       2 
                       ⁢ 
                       
                         D 
                         h 
                         2 
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     C is a constant, μ is viscosity, L is a length of the channel, vol is the volume flow, D h  is the channel hydraulic diameter, and A is the channel cross-sectional area. This equation can also be used to get the same pressure drop between feed channels connected to different integer numbers of non-branched or active area channels. As an example, where the inlet flow channels  106  or the outlet flow channels  108  are of different lengths, the longer channels require proportionately less flow. Therefore, the volume flow required per channel according to Equation 1 would be adjusted inversely proportional to the length of the channels. Accordingly, fewer channels can be fed per feed channel for the longer channels. It is understood that the length of a feed channel could be increased by providing a non-direct path to achieve a desired feed channel length. It is also understood that although the configuration of the anode-inlet feed channels and the anode outlet feed channels can also be applied to the cathode feed channels, it is typically not necessary. The volume of the air used for the cathode reactant gas does not change as much as the hydrogen fuel because even though the oxygen is consumed from the air, most of the air used is non-reacting nitrogen. 
       FIG. 6  illustrates a flow field  120  of an anode plate (not shown) according to another embodiment of the invention. The flow field  120  includes an inlet feed region  122  and an outlet feed region  124 . A plurality of inlet flow channels  126  is formed in the inlet feed region  122  and a plurality of outlet flow channels  128  is formed in the outlet feed region  124 . The inlet flow channels  126  are in communication with an anode inlet header (not shown), and the outlet flow channels  128  are in communication with an anode exhaust header (not shown). 
     Each of the inlet flow channels  126  and the outlet flow channels  128  are in communication with intermediate flow channels  130 . In the embodiment shown, the inlet flow channels  126  and the outlet flow channels  128  communicate with a plurality of branched intermediate flow channels  130  between two and eight. It is understood that the inlet flow channels  126  and the outlet flow channels  128  can communicate with more or fewer intermediate flow channels  130  as desired. The flow channels  126 ,  128 ,  130  are adapted to provide a flow path from a source of an anode gas or fuel (not shown) to the exhaust header, as indicated by the arrows I, E. An active region S is represented by the shaded area of the flow field  120 . The inlet feed region  122  and the outlet feed region  124  are located outside of the active region S, and are non-active. The active region S includes a diffusion medium adjacent thereto. Typically, where the inlet feed region  122  and the outlet feed region  124  are inactive, the diffusion media has not been added to these regions to permit nesting of plate halves to reduce the overall stack size of the fuel cell. 
     In operation, the fuel is caused to flow into the flow field  120  through the inlet flow channels  126  from the source of fuel. The fuel then flows through the intermediate channels  130  and the outlet flow channels  128 . As the fuel flows through the intermediate flow channels  130  in the active region S, the fuel is consumed. In order to compensate for the differences of the volume of fuel flowing through the anode inlet header and the anode exhaust header due to the consumption of the fuel, the number of intermediate flow channels  130  in communication with each of the inlet flow channels  126  and each of the outlet flow channels  128  is controlled. 
     In the inlet feed region  122 , the volumetric flow is high. Thus, if the inlet flow channel  126  is relatively short, a larger number of intermediate flow channels  130  are in communication therewith. If the inlet flow channel  126  is relatively long, a smaller number of intermediate flow channels  130  are in communication therewith. In the outlet feed region  124 , the volumetric flow is low. Thus, if the outlet flow channel  128  is relatively short, a larger number of intermediate flow channels  130  are in communication therewith. If the outlet flow channel  128  is relatively long, a smaller number of intermediate flow channels  130  are in communication therewith. This balances the pressure drop, and therefore the flow, across each of the inlet flow channels  126 , the intermediate flow channels  130 , and the outlet flow channels  128 , thereby optimizing an anode stoichiometry distribution for the fuel cell. 
     In  FIG. 7 , a flow field  140  of an anode plate (not shown) is illustrated according to another embodiment of the invention. The flow field  140  includes an inlet feed region  142  and an outlet feed region  144 . A plurality of inlet flow channels  146  is formed in the inlet feed region  142  and a plurality of outlet flow channels  148  is formed in the outlet feed region  144 . The inlet flow channels  146  are in communication with an anode inlet header (not shown), and the outlet flow channels  148  are in communication with an anode exhaust header (not shown). 
     Each of the inlet flow channels  146  and the outlet flow channels  148  are in communication with intermediate flow channels  150 . As shown, the inlet flow channels  146  and the outlet flow channels  148  communicate with a plurality of branched intermediate flow channels  150  between two and nine. It is understood that the inlet flow channels  146  and the outlet flow channels  148  can communicate with more or fewer intermediate flow channels  150  as desired. The flow channels  146 ,  148 ,  150  are adapted to provide a flow path from a source of an anode gas or fuel (not shown) to the exhaust header, as indicated by the arrows I, E. An active region S is represented by the shaded area of the flow field  140 , and encompasses the inlet region  146  and the outlet region  148 . The active region S includes a diffusion medium adjacent those regions. It is understood that the inlet region  146  and the outlet region  148  can be located outside of the active region S, as previously described herein. 
     In operation, the fuel is caused to flow into the flow field  140  through the inlet flow channels  146  from the source of fuel. The fuel then flows through the intermediate channels  150  and the outlet flow channels  148 . As the fuel flows through the channels  146 ,  150 ,  148  in the active region S, the fuel is consumed. In order to compensate for the differences of the volume of fuel flowing through the anode inlet header and the anode exhaust header due to the consumption of the fuel, the number of intermediate flow channels  150  in communication with each of the inlet flow channels  146  and each of the outlet flow channels  148  is controlled. 
     In the inlet feed region  142 , the volumetric flow is high. Thus, if the inlet flow channel  146  is relatively short, a larger number of intermediate flow channels  150  are in communication therewith. If the inlet flow channel  146  is relatively long, a smaller number of intermediate flow channels  150  are in communication therewith. In the outlet feed region  144 , the volumetric flow is low. Thus, if the outlet flow channel  148  is relatively short, a larger number of intermediate flow channels  150  are in communication therewith. If the outlet flow channel  148  is relatively long, a smaller number of intermediate flow channels  150  are in communication therewith. This balances the pressure drop, and therefore the flow, across each of the inlet flow channels  146 , the intermediate flow channels  150 , and the outlet flow channels  148 , thereby optimizing an anode stoichiometry distribution for the fuel cell. For the inlet feed region  142  and the outlet feed region  144  orientation shown in  FIG. 7 , the intermediate flow channels  150  are of different lengths. The additional flow required for the longer intermediate flow channels  150  is taken into consideration by using Equation 1 to determine the desired pressure drop for each feed channel to balance the flow. 
     It is understood that the size (hydraulic diameter or area) of feed channels can be adjusted according to Equation 1 for each of the embodiments described herein to achieve the desired balance between feed channels. For example, in the embodiment shown in  FIG. 7 , a group of flow channels  146 ,  148  may connect to the same integer number of intermediate flow channels  150  as the flow channels are of different lengths. The size of longer flow channels  146 ,  148  of this group is increased compared to the shorter flow channels  146 ,  148  of this group. Also, as the flow channels  146 ,  148  connect to an integer number of intermediate channels, some adjustment of the flow channel  146 ,  148  sizes can be used to adjust to the same pressure drop as other flow channels  146 ,  148 . 
     From the foregoing description, one ordinarily skilled in the art can easily ascertain the essential characteristics of this invention and, without departing from the spirit and scope thereof, can make various changes and modifications to the invention to adapt it to various usages and conditions.