Abstract:
An ink supply is contained in a manner that combines foam and free ink storage to provide high volumetric efficiency, back pressure regulation to protect against ink leakage, and a generally lower cost, easy-to-manufacture assembly. Ink leakage protection is present despite exposure of the supply to substantial variations in temperature and ambient air pressure. The container is divided, and part of the container includes porous material for storing ink. Capillary pressures of the material and of a bubble generator in the free-ink part of the container are selected to control the sequence with which ink is removed from the container parts.

Description:
TECHNICAL FIELD 
     This invention relates to an ink supply container that combines foam based and free storage of ink in a manner that provides high volumetric efficiency and protection against ink leakage. 
     BACKGROUND AND SUMMARY OF THE INVENTION 
     Replaceable ink supplies are important components of ink-jet type printers. The ink supply provides ink to a printhead that is carried in what may be called the pen of the printer. The printhead typically includes a plurality of orifices, each orifice having an associated chamber. Ink is channeled to the chamber from the ink supply. During operation of the printhead, ink droplets are fired from the chambers, through the orifices, to a printing medium such as paper. 
     In thermal-type printheads, the ink droplets are fired as a result of rapidly heating the ink in the chamber by an amount sufficient to vaporize a portion of that ink. The resultant, rapid expansion of the vapor bubble in the chamber forces out of the chamber a correspondingly sized droplet. 
     Some printer designs separate the ink supply from the printhead, which printhead is normally mounted to a carriage to reciprocate along the width of paper that is advanced through the printer. The supply resides in the printer, and an elongated tube or other means is used for interconnecting the supply to the printhead. 
     Although the printhead is a reliable and efficient means for firing ink droplets, it carries no mechanism for preventing the free flow of ink through the orifices when the printhead is not operating. As a result, ink supplied to the printhead is usually provided under a slight under pressure or back pressure. The back pressure is large enough to prevent the free flow of ink from the pen, but not so large as to prevent an activated printhead from expelling ink. This range of back pressures will be referred to as the printhead&#39;s operating range. As used here, a positive back pressure refers to a pressure within the printhead (or ink supply) that is less than ambient pressure. Thus, an increase in back pressure means an increase in the difference between ambient pressure and the relatively lower back pressure. 
     The back pressure at the printhead must be maintained within a fairly narrow operating range (to prevent leakage without causing the printhead to fail) despite severe changes in the ambient temperature and pressure that may occur, for example, when a printer is subject to altitude changes during shipping, etc. A large ambient pressure drop, for example, could overcome the back pressure in the printhead and cause ink to leak or “drool” from it. Thus, printers are provided with mechanisms that compensate for such changes. One class of mechanisms, which may be referred to as accumulators, are designed to expand and contract or otherwise compensate for pressure or volume changes inside the pen that are attributable to ambient pressure changes. 
     Accumulator mechanisms are sometimes supplemented with “bubble generators.” A bubble generator is an orifice or tubular member formed in the ink supply reservoir to allow, under certain conditions, fluid communication between the interior of the reservoir and the ambient atmosphere. The opening of the bubble generator is sized to have capillarity or capillary pressure sufficient to retain a quantity of ink in the opening as a liquid seal. The geometry of that opening is such that when the back pressure approaches the limit of the operating range of the printhead, the back pressure overcomes the capillary pressure of the bubble generator and the liquid seal is broken. Ambient air then “bubbles” into the reservoir to reduce the back pressure to an acceptable level. Ideally, when the back pressure is so reduced, ink from the reservoir reenters the orifice to reestablish the liquid seal. 
     Open-cell foam has been used as a storage medium in ink supplies. The capillary pressure of the foam provides a simple mechanism for providing back pressure for the supply. In this regard, capillary pressure is the pressure applied by a capillary member, such as the connected cells in reticulated foam, to a liquid that it contacts, such as ink. For example, foam material having a capillary pressure of 7 centimeters water column would store the ink in its cells until a suction greater than that pressure is applied to it. 
     The volumetric efficiency of an ink supply generally means the amount of ink deliverable from the supply reservoir divided by the reservoir volume. When the entire ink supply is stored in foam the volumetric efficiency of the supply suffers because of the presence of the foam material throughout the supply. The solid parts of the foam material fills volume that may otherwise be used to store ink. High volumetric efficiency is desirable for enabling as much ink as possible to be delivered to the printhead (hence to the paper) for a given size ink supply. 
     Another disadvantage with all-foam type supplies is that the level of available ink in the supply may not be as readily detectable as would be the case if the supply consisted of free ink having a discernable level. 
     Ink supplies that are contained free (that is, without the use of porous, absorbent material, such as the foam mentioned above) offer high volumetric efficiency along with a free-ink surface for detecting the ink level. Mechanisms necessarily associated with such supplies for regulating back pressure, however, tend to be complex and relatively difficult to manufacture. 
     The present invention is directed to an ink supply contained in a manner that combines foam and free ink storage to provide high volumetric efficiency, back pressure regulation to protect against ink leakage, and a generally lower cost, easy-to-manufacture assembly. 
     In a preferred embodiment of the invention, the container comprises a reservoir that is divided into two parts. One part stores free ink, and another part holds porous, absorbent “accumulator” material that stores ink and has a capillary pressure sufficient to provide back pressure in the supply. 
     Porous wicking material is in the reservoir, arranged to be in fluid communication with the free ink and with the ink in the accumulator material. The wicking material delivers both the free ink and ink in the accumulator material to an outlet in the reservoir. 
     A bubble generator is provided to connect the free-ink part of the reservoir to the ambient atmosphere. The bubble generator is designed to have a capillary pressure that is significantly higher than the capillary pressure of the accumulator material. As such, ink removed from the supply is first drawn from the accumulator material, which thus allows that drained material to thereafter act as an accumulator in the event of severe pressure or temperature changes as mentioned above. 
     After ink is removed from the accumulator material, ink is drawn via the wicking material from the free-ink supply. The bubble generator permits ingress of air as this ink is removed, thereby ensuring that the back pressure in the supply does not rise so high as to cause the printhead to fail. 
     The volumetric efficiency of the present supply is enhanced in a number of ways. For example, the significantly greater capillary pressure of the bubble generator as compared to that of the accumulator material ensures that nearly all of the ink in the accumulator material will be removed before the free-ink supply is depleted. Thus, the porous material of the accumulator will hold very little “stranded” ink when the supply is otherwise fully depleted. 
     In one embodiment of the invention, the porous material used for the accumulator material is selected to be very wettable (i.e., a zero or near-zero contact angle between the ink and surface of the material). This facilitates movement of ink from the accumulator to further minimize the amount of stranded ink. 
     The porous wicking material is selected to have a capillary pressure that is higher than that of both the accumulator material and the bubble generator. As such, the wicking material remains saturated with ink at least until all of the available accumulator ink and free ink is removed from the supply. Consequently, the wicking material provides a reliable mechanism for ensuring delivery of ink out of the supply. 
     The supply of the present invention is adaptable to be remote from the printhead, and configured with an outlet that receives a fluid interconnect mechanism for conducting ink from the supply to the printhead. A negative (suction) pressure is applied via the interconnect to remove the ink. In one embodiment, the capillary pressure of the wicking material is selected to be great enough so that ink is retained within the wicking material after removal of the accumulator ink and free ink. This design ensures that the wicking material will remain at least partly saturated, which is necessary in some instances to ensure fluidic coupling with the interconnect. This design may be useful, for example, when the fluid interconnect is periodically made and broken throughout the useable life of ink supply. 
     In another embodiment, the capillary pressure of the wicking material, while greater than that of both the bubble generator and the accumulator material, is established to be low enough to permit the suction of the interconnect to drain ink from the wicking material. Such a design may be useful, for example, in instances where the fluid interconnect is made to a full supply and not broken until the entire supply is depleted. It will be appreciated that this approach enhances the volumetric efficiency of the supply by increasing the amount of deliverable ink (that is, to include what is stored in the wicking material) for a given size of supply container. 
     As another aspect of this invention, there is provided a method for optimizing the design of the accumulator part of the ink supply so that the accumulator goals (providing back pressure regulation to avoid ink drool despite extreme changes in ambient pressure) are met with the smallest amount of accumulator material required. Minimizing the amount of accumulator material thereby minimizes the amount of ink that may be stranded in the accumulator material, which in turn increases volumetric efficiency. 
     Inasmuch as a part of the present supply contains free ink, the level of ink remaining in the supply is available for detection. 
    
    
     Other advantages and features of the present invention will become clear upon study of the following portion of this specification and the drawings. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a cross sectional diagram of an ink supply in accordance with the present invention. 
     FIG. 2 is a diagram similar to FIG. 1 for depicting certain dimensions of the supply to illustrate the aspect of this invention concerning optimizing the size of the accumulator material. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     With reference to FIG. 1, an ink supply in accordance with the present invention includes a container  20  that may be formed of any suitable lightweight material, such as plastic, that does not react with the ink it contains. FIG. 1 shows one preferred embodiment of the container  20  in cross section, with the removed portion being substantially the mirror image of the portion that is shown. It is contemplated that the principles of the present invention described below may be applied to a wide range of container configurations. 
     The container  20  includes a divider  22  that may be in the form of an integrally formed plastic wall extending downwardly from the top  24  and between the two container sidewalls, only one of which sidewalls  26  appears in FIG.  1 . 
     The divider  22  separates the container into two reservoir volumes: a free-ink volume  30 , and a capillary volume  32 . Each of these volumes stores ink, as will be explained. 
     The divider  22  is spaced from the container bottom wall  28  so that a gap  36  is present between the bottom wall  28  and the lowermost edge of the divider  22 . The gap  36  provides for fluid communication between the free-ink volume  30  and the capillary volume  32 . 
     The container  20  also includes a vent  38 , which vents the upper portion of the capillary volume  32  to ambient. Preferably, the vent  38  is sized so that it also serves as a diffusion barrier to limit mass diffusion of ink through the vent. One of ordinary skill could select a combination of vent diameter and length (The top  24  could be thickened in the vent area for providing adequate length.) so that, in accord with Fick&#39;s first law of diffusion, a working vent is in place to provide an acceptably low rate of mass diffusion for the capillary volume  32 . 
     The free-ink volume  30  is also in fluid communication with ambient atmosphere outside the container  20 . In this regard, a bubble generator  40  is incorporated into the divider  22 . As mentioned above, a bubble generator  40  permits, under certain circumstances, entry of ambient air bubbles into the ink-filled, free-ink volume  30 . The opening of the bubble generator  40  is sized to have capillary pressure sufficient to retain a quantity of ink in the lower end  42  of the opening as a liquid seal. When the back pressure inside the free-ink volume  30  approaches an upper limit, which corresponds to the limit of the operating range of the printhead, the back pressure overcomes the capillary pressure of the bubble generator and the liquid seal is broken. Ambient air then bubbles into the reservoir to reduce the back pressure to an acceptable level. Ideally, when the back pressure is so reduced, ink from the volume  30  reenters the orifice to reestablish the liquid seal. 
     As noted earlier, back pressure is referred to in a positive sense and quantified, for example, in terms of water column height. Capillary pressure, such as that of the bubble generator  40  is similarly measured. The significance of the relative values of the back pressure operating range and the capillary pressures of the various components of the present invention is described more fully below. It is noteworthy here, however, that any of a variety of bubble generator configurations may be employed with the present container configuration. For example, the bubble generator may be a separate, tubular member extending from near the bottom of the free-ink volume to terminate at any exterior wall of the container. Also, the above-mentioned vent  38  may be incorporated with or joined to the outer end of the bubble generator. 
     The capillary volume  32  includes wicking material  50 , which is placed into the bottom of that volume in contact with both sidewalls and with the front wall  52  of the container  20 . Part of the wicking material  50  is fit under the divider  22  to occlude the gap  36  so that ink passing through the gap must pass into the wicking material  50 . 
     In a preferred embodiment, the wicking material comprises a porous material that can be generally characterized as “foam,” but can be selected from any of a number of suitable materials such as bonded or bundled nylon or polyester fibers, continuous-cell polyurethane foam, glass beads, fibers or plates, or sintered plastic. The capillary pressure and wetting characteristics of the wicking material are important, as discussed below. 
     Ink in the free-ink volume  30  flows into the wicking material through the gap  36  and out of the wicking material through an outlet  54  formed in the front wall  52  of the container. In this regard, a preferred embodiment of the supply container  20  is adapted to be remote or separate from the ink-jet pen that carries the printhead. Thus, an interconnect mechanism  60  is used to connect the supply container with a remote pen (not shown). 
     The interconnect  60  is a tubular member having an outside diameter matching the diameter of the outlet  54 . The end of the interconnect  60  is inserted into the outlet into firm contact with the wicking material  50 . A flange  62  may be carried on the interconnect  60  to ensure a firm, leak-proof connection. Any of a variety of means (such as O-rings) may be employed for sealing this connection. Until this connection is made, the outlet  54  may be sealed with, for example, disposable tape, which is removed just before the interconnect  60  is inserted into the outlet  54 . 
     The ink is drawn from the wicking material by suction applied to the interconnect  60 . This is schematically shown as a pump  64  coupled to the interconnect. In this regard, the printhead may be considered a pump whereby the ejection of ink droplets therefrom, along with the capillarity channels etc leading to the chambers provide the suction for drawing ink through the interconnect. 
     Porous, absorbent accumulator material  70  is placed atop the wicking material  50  in the capillary volume. The accumulator material  70 , as was the wicking material, is generally characterized as “foam,” but can be selected from any of a number of suitable materials such as bonded or bundled nylon or polyester fibers, continuous-cell polyurethane foam, glass beads, fibers or plates, or sintered plastic. The capillary pressure and wetting characteristics of the accumulator material are discussed below. 
     The accumulator material  70  fits snugly in the capillary volume  32 , in direct contact with the wicking material  50 , with the front and sidewalls of the container, and with the divider  22 . No air space is provided between the accumulator material  70  and the wicking material  50 . A small air space is provided between the top of the accumulator material  70  and the top  24  of the container. 
     The container  20  is filled with ink by any of a variety of means. For example, the free-ink volume  30  may be filled with ink that is directed through a port through the top  24 , which port is thereafter sealed. A filled, free-ink volume has an ink level  72  very near the top  24  of the container, so that there is substantially no air gap in the top of the filled volume  30 . For illustrative purposes, FIG. 1 shows the ink level  72  at a position where it would be if some of the ink had been depleted from the supply via use of the pen. Thus, a volume of trapped air  35  also appears in FIG.  1 . 
     In a preferred embodiment, at least part of the container wall is transparent in the vicinity of the free-ink volume so that the level of ink  72  in that volume can be detected visually or by optical mechanisms. 
     The accumulator material  70  and wicking material  50  may be saturated with ink delivered by needles that protrude from a pressurized bulk source of ink and that pierce the material through the vent  38  and outlet  54 , respectively. After filling, the outlet is sealed, as noted earlier. 
     An initial back pressure is established in the filled supply. To this end, a small amount of ink may be drawn from the saturated accumulator material  70  so that the capillarity of the part saturated material provides the initial back pressure. Additionally, slight suction may be applied to the free-ink volume as that filled volume is sealed. 
     Once the interconnect  60  is coupled to the outlet  54  and suction applied thereto, ink is drawn from the volumes of the supply in a particular order in accordance with the underlying invention. More particularly, the capillary pressure of the wicking material  50  is selected or established to be greater than the capillary pressure of the bubble generator  40  and of the accumulator material  70 . As a result, the wicking material remains saturated with ink until ink is respectively drawn from the accumulator material  70  and the free-ink volume  30 . The wicking material  50 , in its saturated state, thus serves as a low resistance conduit of ink to the interconnect  60 . 
     The capillary pressure of the accumulator material  70  is selected or otherwise established to be significantly lower than the capillary pressure of the bubble generator  40 . In a preferred embodiment, the capillary pressure of the bubble generator  40  may be over 50 percent greater than that of the accumulator material. For instance, in one preferred embodiment the capillary pressure of the accumulator material is 7.5 cm water column, and the bubble generator capillary pressure is 12.5 cm water column. As a result, suction applied to the saturated wicking material  50  first draws the ink stored in the accumulator material  70  (under a relatively low capillary pressure) before ink is drawn from the free-ink volume (which can not be drawn until the relatively high capillary pressure of the bubble generator is overcome). 
     It will be appreciated that the capillary pressure of the accumulator material  70  (as well as that of the wicking material  50 ) may be established in a number of ways. For example, the aforementioned foam can be compressed in the capillary volume  32  to reduce the effective pore size of the material and thereby increase the associated capillary pressure. 
     One advantage of drawing the ink from the accumulator material first is that the portion of the accumulator material that is drained of ink is thereafter available to serve as an accumulator in the event that the partly empty container is exposed to extremes in ambient temperature and/or pressure. This will be discussed more below. 
     In order to enhance the volumetric efficiency of the supply, the accumulator material  70  is selected so that little ink will be stranded in it when the material is drained. To this end, the material is selected to have a “wetting” or “wettable” characteristic (as opposed to a non-wetting characteristic). That is, the angle between the liquid surface of the ink stored in the accumulator material and the solid surface of that material is zero, or very near zero. A wettable surface offers less resistance to ink flow than a non-wettable surface. As a result, the accumulator material  70  tends to have little stranded ink once the supply is empty. 
     Under some conventions, a contact angle of less than 90 degrees defines a wettable surface. While a contact angle of 90 degrees or less will suffice in the case at hand, it is preferred that the accumulator material have a contact angle as close to zero as practical. 
     Under normal operation of the supply (that is, ink drawn from the supply while ambient pressure and temperature have little variation), ink flows out of the free-ink volume  30  once the accumulator material  70  is drained. In this regard, the free-ink will not flow until the back pressure in that volume  30  is overcome. The capillary pressure of the bubble generator  40  regulates the back pressure in the free-ink volume by allowing ambient air bubble entry once that back pressure builds to a level slightly greater than the capillary pressure of the bubble generator. As noted above, however, the relatively lower capillary pressure in the accumulator material  70  makes flow from that material  70  the path of least resistance from a hydraulic standpoint. 
     It is noteworthy here that the highest capillary pressure in this system is that provided by the wicking material  50 . Thus, once the accumulator material  70  is drained, the lowest capillary pressure is that of the bubble generator  40 , so that ink is next drawn from the free-ink volume  30  until it is emptied. 
     In a preferred embodiment, the bubble generator  40  is configured so that its opening  42  is very near the bottom wall  28  of the container. To this end, a tubular member, separate from the divider  22  may be employed (as mentioned above) or a downward extension of the divider may be formed in the vicinity of the bubble generator so that the bubble generator reaches the bottom of the container without otherwise diminishing the gap  36 . 
     If design constraints require the bubble generator to be located above the bottom wall  28  of the container, the bubble generator could be designed to trap a small amount of ink in its end  42  to serve as a liquid seal to maintain back pressure even after the level of the free ink moves below that end  42 . The use of a trap may be omitted in instances where the bubble generator is above the bottom wall  28 . Thus, back pressure in the volume  30  will be lost once the ink level there moves to below the opening  42 . In this embodiment, the size of the accumulator material may be supplemented to absorb the volume of ink that remains in the free-ink  30  once back pressure is lost. That volume may then be drained from the accumulator material during printing. 
     It may be desirable, as noted above, to drain the wicking material  50  of ink once the free-ink volume is empty. To this end, the capillary pressure of the wicking material is established to be higher than the bubble generator capillary pressure but less than the suction applied by the interconnect  60 . Thus, the wicking material is drained after the free-ink volume  30 . 
     In the embodiment where the wicking material  50  is to be drained, it is preferred to select that material to have a very small, near zero contact angle as described above with respect to the accumulator material  70 . 
     Inasmuch as the accumulator material  70  is drained before the filled, free-ink volume  30  begins to drain, there is practically no trapped air  35  (trapped air being air introduced into the free-ink volume via the bubble generator  40 ) above the free-ink level  72  until the accumulator material  70  is drained. After such draining, the ink level  72  drops and the volume of trapped air  35  grows as ink is drawn from the free-ink volume. 
     Turning now to the accumulator aspects of the present system, a part-empty container (that is, part of the free-ink volume is empty and the accumulator material is drained) may be subjected to an ambient pressure drop (or temperature increase). In the absence of a means to compensate for such a change, a pressure gradient relative to the trapped air volume  35  would be present and the back pressure would drop to a level where ink would be free to drool from the connected printhead. 
     With the present arrangement of supply components, however, the ink volume that moves into the wicking material  50  as a consequence of the expanding trapped air volume  35  (expanding, that is, as a result of the pressure gradient just mentioned) is drawn into the accumulator material  70  to reduce pressure in the free ink volume and thereby maintain an adequate back pressure in the supply to prevent drooling from the printhead. Should the ambient pressure then return to normal (eliminating the pressure gradient), the resultant contraction of the trapped air volume  35  and attendant increase in back pressure inside the supply will draw ink back from the accumulator material  70  (again, via the saturated wicking material  50 ) until the system returns to its state of equilibrium. 
     It may be that a part empty container will be subjected to an ambient pressure increase instead of a decrease (or temperature decrease) that, in the absence of a means to compensate for such a change, would cause a pressure gradient relative to the trapped air volume  35  (the air volume would contract) such that the back pressure would rise to a level such that ink would cause the printhead to fail since it would overcome the suction of the interconnect  60 . 
     With the present arrangement of supply components, however, the back pressure increase attributable to the air volume contraction would draw air bubbles through the bubble generator  40 . In this regard, the accumulator material  70  is empty, as explained above, and the wicking material  50  has a higher capillary pressure than the bubble generator  40 , so ink would not be drawn from that saturated material  50 . Thus, the air bubbling through the bubble generator immediately reduces the back pressure rise caused by the contracting trapped air volume, thus keeping the back pressure within the operating range. 
     Should the ambient pressure thereafter return to normal (eliminating the pressure gradient), the resultant expansion of the trapped air volume  35  and attendant decrease in back pressure will allow the empty accumulator material  70  to absorb the volume of ink displaced by such expansion. Because the accumulator material has the lowest relative capillary pressure of the supply components, subsequent printing of ink will thereafter drain the accumulator material before drawing on the free-ink volume  30 . 
     In view of the foregoing, it will be appreciated that the supply system just described is robust and able to respond to an indefinite number of temperature- and pressure-change cycles irrespective of the amount of ink remaining in the supply. 
     The accumulator material  70  must be large enough to absorb all of the ink transferred to it from the free-ink volume  30  as a result of the pressure gradients and associated trapped air volume changes just described. Moreover, volumetric efficiency demands that the size of the accumulator material be optimized; that is, sized just large enough, and no larger, to compensate for a prescribed temperature or pressure change applied to a free-ink volume irrespective of the amount of ink remaining in that volume. In accordance with another aspect of the present invention, this description now turns to a technique for so optimizing that size. 
     The technique can be most readily understood with reference to FIG. 1 and 2, which diagrams represent a simple configuration of the supply container  20 . 
     Assuming for ease of explanation that the height h, depth D, and width t of the container  20  are constrained by other design considerations, the amount of accumulator material  70  needed to optimally handle the pressure gradients just described can be expressed as the variable df, which will be referred to as foam depth. 
     The largest (albeit not necessarily optimal) amount of accumulator material  70  required would be the volume necessary for absorbing substantially all of the free-ink volume should a small volume of trapped air  35  expand enough to force all of the ink from the free-ink volume. For a selected range of temperature and pressure changes, such as a designer might anticipate, and the external geometry given above, the depth of foam df can be calculated as a function of the ink height in the free-ink volume. An analysis follows, using an anticipated temperature range of 35° C. and an ambient pressure range of 255 cm of water column, which approximately corresponds to an 8000-foot elevation change. 
     
       
         
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Given: 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 γ = 
                 specific weight of ink; 
                 PI = 
                 Capillary pressure of 
               
               
                   
                   
                   
                 accumulator material 
               
               
                 h = 
                 height of the supply 
                 nl = 
                 Volumetric efficiency of 
               
               
                   
                 container 20; 
                   
                 accumulator material 
               
               
                   
                   
                   
                 (porosity × ink extraction 
               
               
                   
                   
                   
                 efficiency) 
               
               
                 D = 
                 overall depth of the 
                 T1 = 
                 initial temperature of 
               
               
                   
                 container; 
                   
                 anticipated range 
               
               
                 t = 
                 thickness of the 
                 T2 = 
                 final temperature of 
               
               
                   
                 container; 
                   
                 anticipated range 
               
               
                 hb = 
                 height of bubble 
                 Pa1 = 
                 initial ambient pressure of 
               
               
                   
                 generator above bottom 
                   
                 anticipated range 
               
               
                   
                 wall 
               
               
                 hw = 
                 height of wicking 
                 Pa2 = 
                 final ambient pressure of 
               
               
                   
                 materail above bottom 
                   
                 anticipated range 
               
               
                   
                 wall 
               
               
                 Pb = 
                 capillary pressure of 
                 Vc 
                 container volume of t D h 
               
               
                   
                 bubble generator 
               
               
                   
               
             
          
         
       
     
     One can determine the foam depth df as a function of the ink height yi1 in the free-ink volume at T1 and Pa1, along with the optimal maximum foam depth dfmax, delivered ink volume Vd (which is the volume of ink delivered from the supply), stranded ink volume Vs (which is the volume of ink stranded in the ink supply when the supply is otherwise empty), and delivered ink efficiency nd (which is the ratio of delivered ink to the container volume or Vd/Vc). 
     Before proceeding it is noted that certain assumptions are made in the analysis. They are (1) dissolved gases in the ink are neglected, (2) the air in the trapped air volume  35  is treated as an ideal gas mixture of air and water vapor, (3) the partial pressure of the vapor is equal to the saturation pressure of the vapor at the temperature of interest and can be obtained from the steam tables, and (4) condensate in the ink volume is neglected. 
     At T1 and at T2 the total moles of the trapped air equals the moles of air plus the moles of vapor, or: 
     
       
         ng1=nga1+ngv1 and 
       
     
     
       
         ng2=nga2+ngv2. 
       
     
     Assuming no air transfer (nga1=nga2) and subtracting yields 
     
       
         ng1−ng2=ngv1−ngv2  (Eq. 1) 
       
     
     Thus, the total change in moles equals the change in vapor moles due to evaporation/condensation. For an ideal gas mixture: 
     
       
         Pg Vg=ng R T 
       
     
     Solving the preceding equation for ng1, ng2, ngv1, and ngv2, and substituting in equation 1 yields: 
     
       
         ng1=(Pg1 Vg1)/R T1 
       
     
     
       
         ng2=(Pg2 Vg2)/R T2 
       
     
     
       
         ngv1=(Pgv1 Vg1)/R T1 and 
       
     
     
       
         ngv2=(Pgv2 Vg2)/R T2 
       
     
     where Pgv is the partial pressure of the vapor, considering the Dalton Model for gaseous mixtures. 
     Assuming, as noted, that the partial pressure of the vapor is equal to the saturation pressure Psat of the vapor at the temperature of interest and is obtained from the steam tables, one finds from Equation 1: 
     
       
         [(Pg1 Vg1)/T1]−[(Pg2 Vg2)/T2]=[(Psat1 Vg1)/T1]−[(Psat2 Vg2)/T2] 
       
     
     thus 
     
       
         [(Pg1−Psat1)Vg1]/T1=[(Pg2−Psat2)Vg2]/T2  (Eq. 2) 
       
     
     Since the total volume of ink does not change: 
     
       
         Vi1+Vf1=Vi2+Vf2  (Eq. 3) 
       
     
     where Vi is the volume of free ink and Vf is the volume of ink in the “foam” wicking and accumulator material. With the foregoing equations at hand, the general solution is considered under the following parameters: 
     At T1: 
     yi=yi1 
     yfl=hw (height of ink in “foam”−accumulator material empty, bubble generator active) 
     Pg1=Pa1−Pb−γ(yi1−hb) 
     Vg1=t(D−df)(h−yi1) 
     Vi1=t(D−df)yi1 
     Vf1=nl t df(yfl−hw)=0 (accumulator material initially empty) 
     Psat1 (from steam tables) 
     Pa1 (from standard atmosphere at sea level) 
     The foregoing expression, Pg1=Pa1−Pb−γ(yi1−hb), represents the pressure of the trapped air in the volume  30 . That pressure is the ambient pressure, reduced by the bubble generator capillary pressure and by the hydrostatic head pressure (the hydrostatic pressure being measured from above the height hb of the bubble generator). 
     At T2: 
     yi=yi2 
     yf2=h(assume the accumulator material is full for maximum utilization) 
     Pg2=Po−γ yi2 where 
     Po=Pa2−Pl+γ yf2 (pressure at the bottom of the container) 
     Vg2=t(D−df)(h−yi2) 
     Vi2=t(D−df)yi2 
     Vf2=nl t df(yf2−hw) 
     Psat2 (from steam tables) 
     Pa2 (from standard atmosphere at 2438 meters) 
     Recognizing that the maximum amount of accumulator material required, which corresponds to the maximum volume of free ink displaced into the foam, occurs for the condition in which the initial free ink level corresponds to a trapped air volume that will expand just enough to completely displace the free ink into the accumulator material, then the final free ink level height after such expansion can be written as: 
     
       
         yi2=0  (Eq. 4) 
       
     
     Using the previously listed parameters and substituting Equation 4 into Equation 2 yields: 
     
       
         [Pa1−Pb−γ(yi1−hb)−Psat1](h−yi1)/T1=(Pl−P1+γ h−Psat2)h/T2  (Eq. 5) 
       
     
     The solution for df is from Equation 3, substituting the appropriate parameters to yield: 
     
       
         t(D−df)yi1+0=t(D−df)yi2+nl t df(h−hw) 
       
     
     or 
     
       
         D(yi1−yi2)=df [(yi1−yi2)+nl(h−hw)], which yields to: 
       
     
     
       
         df=[D(yi1−yi2)]/[yi1−yi2+nl(h−hw)]  (Eq. 6) 
       
     
     Substituting Equation 4 into Equation 6 yields: 
     
       
         df=[D yi1]/[yi1+nl(h−hw)]  (Eq. 7) 
       
     
     Equations 5 and 7 may be solved for yi1 and df. To solve Equation 5 for yi1, it is rearranged into a polynomial in yi1: 
     
       
         [Pa1−Pb−γ(yi1−hb)−Psat1](h−yi1)=T1/T2(h)(Pa2−Pl+γ h−Psat2) 
       
     
     One can set the right half of the preceding equation equal to R, which reflects a constant dependent upon the chosen design conditions, geometries, and ink properties. Expanding that equation and collecting terms yields: 
     
       
         γ yi1 2 +(−γ h−Pa1+Pb−γ hb+Psat1)yi1+(Pa1 h−Pb h+γ h hb−Psat1 h−R)=0  (Eq. 8) 
       
     
     Equation 8 is a second degree polynomial in yi1, its coefficients depend only on constants and it may be solved using the quadratic formula: 
     Let: 
     a=γ; 
     b=−γ h−Pa1+Pb−γ hb+Psat1; 
     c=Pa1 h−Pb h+γ h hb−Psat1 h−R 
     Then: 
     
       
         yi1=[−b+/−(b 2 −4ac) ½ ]/2a  (Eq. 9) 
       
     
     The solution for yi1 is valid over the domain: 0&lt;yi1&lt;h and is typically the lesser root of Equation 9. 
     To determine the required amount of accumulator material, one substitutes the solution for yi1 into Equation 7, which is a function only of constants and yi1. 
     Using the foregoing, the required amount of accumulator material dfmax was determined to be 26.8 mm for the following container sizes and conditions: 
     
       
         
               
               
               
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 γ = 
                 10297 N/m 3   
                 Pl = 
                 7.62 cm 
               
               
                   
                 h = 
                 52.75 mm 
                 nl = 
                 0.64 
               
               
                   
                 D = 
                 65.0 mm 
                 T1 = 
                 0° C. 
               
               
                   
                 t = 
                 7.0 mm 
                 T2 = 
                 35° C. 
               
               
                   
                 hb = 
                 7.0 mm 
                 Pa1 = 
                 1033 cm H20 
               
               
                   
                 hw = 
                 9.0 mm 
                 Pa2 = 
                 778 cm H20 
               
               
                   
                 Pb = 
                 12.7 cm ink 
                 Psat1 = 
                 0.6113 kPa 
               
               
                   
                 yi1 = 
                 19.6 mm 
                 Psat2 = 
                 5.6280 kPa 
               
               
                   
                   
               
             
          
         
       
     
     Once dfmax is determined, one can determine the delivered ink volume Vd (which is the volume of ink delivered from the supply), the stranded ink volume Vs (which is the volume of ink stranded in the ink supply when the supply is otherwise empty), and the delivered ink efficiency nd (which is the ratio of delivered ink to the container volume or Vd/Vc) 
     Before these determinations, however, additional accumulator material characteristics are developed; namely: 
     n pa =porosity of accumulator material; ratio of void volume to bulk volume of the material; (for example, 0.85) 
     n pw =porosity of wicking material; ratio of void volume to bulk volume of the material; (for example, 0.85) 
     n ea =extraction efficiency of accumulator material; fraction of total ink in the material that can by extracted by the printhead; (for example, 0.75) 
     n ew =extraction efficiency of wicking material; fraction of total ink in the material that can by extracted by the printhead; (for example, 0.75) 
     n a =overall efficiency of the accumulator material; the product of n pa  and n ea    
     n w =overall efficiency of the wicking material; the product of n pw  and n ew    
     The greatest delivered ink efficiency n d  is obtained when the ink supply is designed so that both the accumulator material  70  and the wicking material  50  may be drained by the interconnect. Accordingly, the delivered ink volume for such a situation is the sum of the ink delivered from the free-ink volume, the accumulator material, and the wicking material, or: 
     
       
         Vd=t(D−dfmax)h+n a  t dfmax(h−hw)+n w  t dfmax hw  (Eq. 10) 
       
     
     The stranded ink volume is dependent on the extraction efficiency and porosity of both the accumulator material and the wicking material, or: 
     
       
         Vs=[(1−n ea )n pa  dfmax t (h−hw)]+[(1−n ew )n pw  dfmax t hw]  (Eq. 11) 
       
     
     As noted above, the delivered ink efficiency is: 
     
       
         n d =Vd/Vc  (Eq. 12) 
       
     
     Using the foregoing equations and exemplary values, the ink supply provided a delivered ink efficiency of greater than 85% which is an impressively high efficiency, especially when one considers that the supply, which combines free ink and foam-stored ink, also provides back pressure control, thus obviating the need other such regulating means. 
     Thus, while the present invention has been described in terms of a preferred embodiment, it will be appreciated by one of ordinary skill that the spirit and scope of the invention is not limited to those embodiments, but extend to the various modifications and equivalents as defined in the appended claims.