Patent Publication Number: US-2007109334-A1

Title: Method of ink evaporation prediction for an ink reservoir

Description:
BACKGROUND OF THE INVENTION  
      1. Field of the Invention.  
      The present invention relates to determining an amount of ink depleted from an ink reservoir, and, more particularly, to a method of ink evaporation prediction for an ink reservoir.  
      2. Description of the Related Art.  
      Ink jet disposable printhead cartridges include an ink reservoir that contains ink that is used to print on a print medium, such as paper. Typically, the ink level indicators on the printer in the Windows driver can keep track of the ink level based on counting the ink drops jetted on the print medium. In addition, the drops jetted during a printhead maintenance operation can be tracked as well. However, ink volume losses can occur in ways that cannot be tracked by only counting jetted ink dots. As used herein, the terms “ink dots” and “ink drops” are synonymous.  
      For example, it has been recognized that a significant loss of ink volume in a printhead cartridge can occur through evaporation. The evaporation occurs, for example, through the vent in the cartridge lid, through the nozzle openings in the printhead nozzle plate (even when capped), through the plastic cartridge body and through the cap seals. The loss rate depends, for example, on temperature and humidity, as well as the construction of the lid vent, cartridge material, etc.  
      What is needed in the art is a method of ink evaporation prediction for an ink reservoir.  
     SUMMARY OF THE INVENTION  
      The present invention provides a method of ink evaporation prediction for an ink reservoir.  
      The invention, in one form thereof, is directed to a method that establishes an empirical evaporation curve representing evaporation characteristics for an ink reservoir type, the ink reservoir belonging to the ink reservoir type; and establishes an evaporation prediction curve for the ink reservoir that approximates the empirical evaporation curve.  
      In another form thereof, the invention is directed to a method of ink evaporation prediction for an ink reservoir having ink evaporation characteristics represented by an empirical evaporation curve determined for an ink reservoir type, the ink reservoir belonging to the ink reservoir type, the method associating a respective rate of evaporation to each of a plurality of time segments associated with the empirical evaporation curve, the respective rate of evaporation being based on a respective approximation algorithm associated with each of the plurality of time segments.  
      In still another form thereof, the invention is directed to a printhead comprising memory. The memory stores parameters associated with an evaporation prediction curve for an ink reservoir that approximates an empirical evaporation curve. A printer in which the printhead is installed executes instructions to: determine an evaporation amount based on the evaporation prediction curve for the ink reservoir; and use the evaporation amount to compensate for an evaporation loss for the ink reservoir by adjusting a cumulative actual ink drop count to form an evaporation compensated drop count.  
      An advantage of certain embodiments of the present invention is that the method of ink evaporation prediction for an ink reservoir, such as for example, an ink reservoir associated with an ink jet printhead cartridge, tracks an empirically modeled evaporation profile established for a particular ink reservoir type to which the ink reservoir belongs, thereby permitting evaporation compensation from a time of initial ink reservoir fill to the time of complete exhaustion of the usable ink in the ink reservoir. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      The above-mentioned and other features and advantages of this invention, and the manner of attaining them, will become more apparent and the invention will be better understood by reference to the following description of an embodiment of the invention taken in conjunction with the accompanying drawings, wherein:  
       FIG. 1  is an imaging system embodying the present invention.  
       FIG. 2  depicts a plurality of evaporation prediction curves established in accordance with an embodiment of the present invention and based on a plurality of combinations of parameters that may be stored in a memory associated with a particular ink reservoir.  
       FIG. 3  depicts an empirical evaporation curve representing evaporation characteristics associated with a particular type of ink reservoir, and an exemplary evaporation prediction curve established in accordance with an embodiment of the present invention.  
       FIG. 4  is a general flowchart of a method that estimates an amount of ink contained in an ink reservoir.  
       FIG. 5  is a flowchart of a method that may be utilized in implementing an evaporation amount determination act of the method of  FIG. 4 . 
    
    
      Corresponding reference characters indicate corresponding parts throughout the several views. The exemplifications set out herein illustrate embodiments of the invention, and such exemplifications are not to be construed as limiting the scope of the invention in any manner.  
     DETAILED DESCRIPTION OF THE INVENTION  
      Referring now to the drawings, and particularly to  FIG. 1 , there is shown an imaging system  6  embodying the present invention. Imaging system  6  may include a host  8 , or alternatively, imaging system  6  may be a standalone system.  
      Imaging system  6  includes an imaging apparatus  10 , which may be in the form of an ink jet printer, as shown. Thus, for example, imaging apparatus  10  may be a conventional ink jet printer, or may form the print engine for a multi-function apparatus, such as for example, a standalone unit that has faxing and copying capability, in addition to printing  
      Host  8 , which may be optional, may be communicatively coupled to imaging apparatus  10  via a communications link  11 . Communications link  11  may be, for example, a direct electrical connection, a wireless connection, or a network connection.  
      In an embodiment including host  8 , host  8  may be, for example, a personal computer including a display device, an input device (e.g., keyboard), a processor, input/output (I/O) interfaces, memory, such as RAM, ROM, NVRAM, and a mass data storage device, such as a hard drive, CD-ROM and/or DVD units. During operation, host  8  may include in its memory a software program including program instructions that function as a printer driver for imaging apparatus  10 . The printer driver is in communication with imaging apparatus  10  via communications link  11 . The printer driver, for example, may include a halftoning unit and a data formatter that places print data and print commands in a format that can be recognized by imaging apparatus  10 . In a network environment, communications between host  8  and imaging apparatus  10  may be facilitated via a standard communication protocol, such as the Network Printer Alliance Protocol (NPAP). The NPAP includes a multitude of predefined Network Printer Alliance (NPA) cominands, and facilitates the generation of new NPA commands.  
      In the embodiment of  FIG. 1 , imaging apparatus  10  includes a printhead carrier system  12 , a feed roller unit  14 , a sheet picking unit  16 , a controller  18 , a mid-frame  20  and a media source  21 .  
      Media source  21  is configured to receive a plurality of print media sheets from which an individual sheet of print media  22  is picked by sheet picking unit  16  and transported to feed roller unit  14 , which in turn further transports print media sheet  22  during a printing operation. The sheet of print media  22  may be, for example, plain paper, coated paper, photo paper and transparency media.  
      Printhead carrier system  12  includes a printhead carrier  24  for carrying a color printhead  26  and/or a monochrome printhead  28 . A color ink reservoir  30  is provided in fluid communication with color printhead  26 , and a monochrome ink reservoir  32  is provided in fluid communication with monochrome printhead  28 . Those skilled in the art will recognize that color printhead  26  and color ink reservoir  30  may be formed as individual discrete units, or may be combined as an integral unitary printhead cartridge. Likewise, monochrome printhead  28  and monochrome ink reservoir  32  may be formed as individual discrete units, or may be combined as an integral unitary printhead cartridge.  
      The amount of available ink in an ink reservoir, such as for example, color ink reservoir  30  or monochrome ink reservoir  32 , when initially filled with ink, and prior to any evaporation, is referred to as the total yield, T 0 Yield, of the ink reservoir. T 0 Yield may be represented, for example, by an ink drop count, which in turn may be correlated to an approximate page count, if desired. An amount of ink depleted from the ink reservoir may be determined, for example, by counting the number of ink drops expelled from the ink reservoir by the associated printhead, and by compensating for ink evaporation losses, regardless of whether any ink was expelled from the ink reservoir during a printing or maintenance operation.  
      Printhead carrier  24  is guided by a pair of guide members  34 , which may be, for example, in the form of guide rods, guide channels, or a combination thereof. The axes  34   a  of guide members  34  define a bi-directional scanning path for printhead carrier  24 , and thus, for convenience the bi-directional scanning path may be referred to as bi-directional scanning path  34   a . Printhead carrier  24  is connected to a carrier transport belt  36  that is driven by a carrier motor  40  via carrier pulley  42 . Carrier motor  40  has a rotating carrier motor shaft  44  that is attached to carrier pulley  42 . At the directive of controller  18 , printhead carrier  24  is transported in a reciprocating manner along guide members  34 . Carrier motor  40  may be, for example, a direct current (DC) motor or a stepper motor.  
      The reciprocation of printhead carrier  24  transports ink jet printheads  26 ,  28  across the sheet of print media  22 , such as paper, along bi-directional scanning path  34   a  to define a print zone  50  of imaging apparatus  10 . The reciprocation of printhead carrier  24  occurs in a main scan direction  52  that is parallel with bi-directional scanning path  34   a , and is also commonly referred to as the horizontal direction. During each scan of printhead carrier  24  during printing, the sheet of print media  22  is held stationary by feed roller unit  14 .  
      Mid-frame  20  provides support for the sheet of print media  22  when the sheet of print media  22  is in print zone  50 , and in part, defines a portion of a print media path  54  of imaging apparatus  10 .  
      Feed roller unit  14  includes an index roller  56  and corresponding index pinch rollers (not shown). Index roller  56  is driven by a drive unit  60 . The index pinch rollers apply a biasing force to hold the sheet of print media  22  in contact with respective driven index roller  56 . Drive unit  60  includes a drive source, such as a stepper motor, and an associated drive mechanism, such as a gear train or belt/pulley arrangement. Feed roller unit  14  feeds the sheet of print media  22  in a sheet feed direction  62 , designated as an X in a circle to indicate that the sheet feed direction is out of the plane of  FIG. 1  toward the reader.  
      Controller  18  includes a microprocessor having an associated random access memory (RAM) and read only memory (ROM). Controller  18  executes program instructions to effect the printing of an image on the sheet of print media  22 , and executes further instructions to communicate with and monitor the operations of printheads  26 ,  28 . Controller  18  is electrically connected and communicatively coupled to printheads  26 ,  28  via a communications link  64 , such as for example a printhead interface cable. Controller  18  is electrically connected and communicatively coupled to carrier motor  40  via a communications link  66 , such as for example an interface cable. Controller  18  is electrically connected and communicatively coupled to drive unit  60  via a communications link  68 , such as for example an interface cable. Controller  18  is electrically connected and communicatively coupled to sheet picking unit  16  via a communications link  70 , such as for example an interface cable.  
      As an example, one of color printhead  26  and color ink reservoir  30  may have attached thereto a memory  72  for storing information relating to color printhead  26  and/or color ink reservoir  30 , such as for example, an identification number, a value representing an amount of usage of color printhead  26  and/or color ink reservoir  30 , and one or more values representing time. Memory  72  may be, for example, a one time programmable memory. For example, memory  72  may be formed integral with other electrical components on the silicon of color printhead  26 . Color printhead  26  may be configured to eject a single color of ink, or may be configured to eject multiple colors of ink, such as for example, two or more combinations of various colors of ink, e.g., black, cyan, magenta, yellow, diluted colors, orange, green and any other colors known in the art. Color ink reservoir  30  may be configured to carry a single color of ink, or may be configured to carry multiple colors of ink, such as for example, two or more combinations of various colors of ink, e.g., black, cyan, magenta, yellow, diluted colors, orange, green and any other colors known in the art. Controller  18  communicates with memory  72  via communications link  64 .  
      Also, one of monochrome printhead  28  and monochrome ink reservoir  32  may have attached thereto a memory  74  for storing information relating to monochrome printhead  28  and/or monochrome ink reservoir  32 , such as for example, a supply item identification number, a value representing an amount of usage of monochrome printhead  28  and/or monochrome ink reservoir  32 , and one or more values representing time. Memory  74  may be, for example, a one time programmable memory. For example, memory  74  may be formed integral with other electrical components on the silicon of monochrome printhead  28 . Controller  18  communicates with memory  72  via communications link  64 .  
       FIGS. 2 and 3  are graphical depictions of evaporation curves established and/or used in accordance with embodiments of the present invention.  
       FIG. 2  shows a plurality of evaporation prediction curves  75  generated in accordance with embodiments of the present invention. The evaporation prediction curves  75  are based on a plurality of combinations of parameters, such as time parameters, that may be stored in a memory, such as memory  72  or memory  74 , associated with a particular ink reservoir, such as one of ink reservoirs  30 ,  32  that in some embodiments may be integral with printheads  26 ,  28 , respectively. The evaporation prediction curves  75  assume no ejection of ink from the ink reservoir.  
      In the exemplary curves of  FIG. 2 , various scenarios for evaporation losses are plotted in association with predetermine times, e.g., T 0 , T 1 , T 2  and T 3 . Time T 0  may be, for example, a time of initial fill of the ink reservoir. Time T 1  may be an amount of time, e.g., in months, measured from initial time T 0 , to when each of the exemplary evaporation prediction curves  75  shown in  FIG. 2  is at a first percentage of total yield (T 0  Yield), e.g., 85 percent. Time T 2  may be an amount of time, e.g., in months, measured from time T 1 , to when each of the exemplary evaporation prediction curves  75  shown in  FIG. 2  is at a second percentage of total yield (T 0  Yield), e.g., 67 percent; and time T 3  may be an amount of time, e.g., in months, measured from time T 2 , that it takes for the evaporation curve to go to zero percent of total yield (T 0  Yield).  
       FIG. 3  shows an exemplary evaporation prediction curve  78  (represented by a solid line) established in accordance with the present invention. Evaporation prediction curve  78  is established for an ink reservoir so that it approximately tracks an empirical evaporation curve  76  (represented by a dashed line) associated with an ink reservoir type, wherein the ink reservoir being considered is of that ink reservoir type. As an example, times T 1 , T 2  and T 3  may be represented in memory  72  corresponding to color ink reservoir  30 , or memory  74  of corresponding monochrome ink reservoir  32 , by three binary bits in memory, e.g., 12 months =101 b, 6 months =010 b, 4 months =001 b, and 2 months =000 b. The approximation of empirical evaporation curve  76  is achieved by dividing the associated empirical evaporation curve  76  into consecutive time segments, e.g., T 0  to T 1 , T 1  to T 2 +T 1 , and T 2 +T 1  to T 3  +T 2  +T 1 , and then associating a rate of evaporation to each of the segments. Thus, for example, the time segments may extend from an initial time T 0 , prior to any evaporation loss, to a final time (e.g., T 3  +T 2  +T 1 ) when the evaporation loss would deplete a usable supply of ink in the ink reservoir. The rate of evaporation for each of the time segments may be represented, for example, by a respective algorithm, such as for example, linear equations, as more fully described below.  
      Memory  72  associated with color printhead  26  and/or color ink reservoir  30  may include, for example. thirty-two or more bits reserved for an identification number for color printhead  26  and/or color ink reservoir  30 , which may be set by the manufacturer or generated randomly upon installation in imaging apparatus  10 ; eight or more bits may be used as a usage gauge to maintain a record of usage of color printhead  26  and/or color ink reservoir  30 , with each bit representing a level of depletion of ink from color ink reservoir  30 ; and four or more sets of time bits, represented for example as T 0   c , T 1   c , T 2   c  and T 3   c , each including three or more time tracking bits, that may be used to represent time. The letter “c” is used for convenience to designate that the time is associated with a color ink reservoir, and corresponds to times T 0 , T 1 , T 2  and T 3  shown in  FIGS. 2 and 3 .  
      By attaching memory  72  to color printhead  26  and/or color ink reservoir  30 , in essence, information stored in memory  72  associated with color printhead  26  and/or color ink reservoir  30  travels, respectively, with color printhead  26  and/or color ink reservoir  30  from one imaging apparatus to another. Alternatively, time information, such as one or more of times T 0   c , T 1   c , T 2   c  and T 3   c , may be stored in host  8  or imaging apparatus  10   
      Memory  74  of monochrome printhead  28  and/or monochrome ink reservoir  32  may include for example, thirty-two or more bits reserved for an identification number for monochrome printhead  28  and/or monochrome ink reservoir  32 , which may be set by the manufacturer or generated randomly upon installation in imaging apparatus  10 ; eight or more bits may be used as a usage gauge to maintain a record of usage of monochrome printhead  28  and/or monochrome ink reservoir  32  with each bit representing a level of depletion of ink from monochrome ink reservoir  32 ; and four or more sets of time bits, represented by T 0   m , T 1   m , T 2   m  and T 3   m , each including three or more time tracking bits, that may be used to represent time. The letter “m” is used for convenience to designate that the time is associated with a monochrome ink reservoir, and corresponds to times T 0 , T 1 , T 2  and T 3  shown in  FIGS. 2 and 3 .  
      By attaching memory  74  to monochrome printhead  28  and/or monochrome ink reservoir  32 , in essence, information stored in memory  74  associated with monochrome printhead  28  and/or monochrome ink reservoir  32  travels, respectively, with monochrome printhead  28  and/or monochrome ink reservoir  32  from one imaging apparatus to another. Alternatively, time information, such as one or more of times T 0   m , T 1   m , T 2   m  and T 3   m , may be stored in host  8  or imaging apparatus  10 .  
       FIG. 4  is a general flowchart of a method that estimates an amount of ink contained in an ink reservoir. It is to be understood that the discussion that follows applies to either of color printhead  26  and/or color ink reservoir  30 , or monochrome printhead  28  and/or monochrome ink reservoir  32 , as discrete components or when integrated into a unitary printhead cartridge. However, for convenience and ease of understanding, the description of the invention that follows will be directed to an example using monochrome printhead  28  and/or monochrome ink reservoir  32 . Further, the previously identified time designations for the monochrome implementations, i.e., T 0   m , T 1   m , T 2   m , T 3   m , simply will be referred to using the general time designations T 0 , T 1 , T 2 , and T 3 .  
      At step S 100 , time is tracked since the initial fill, or refilling, of ink reservoir  32 , or the installation of ink reservoir  32  in imaging apparatus  10 . This may be performed by controller  18  and/or host  8  by determining an initial time T 0  for ink reservoir  32 , tracking a total accumulated time period Tt since the initial time T 0 , and comparing the total accumulated time period Tt to a time threshold, such as for example, time T 1 . In one embodiment, for example, time T 1  may be at least three months.  
      To obtain the total time the printhead associated with ink reservoir  32  has been in operation, several implementations are possible. One would be write an initial value Tt into memory  74 , and increment value Tt over time.  
      Another possibility would be to write the host date into memory  74  at the time of installation of printhead  28  and/or ink reservoir  32 . For example, in one embodiment that utilizes host  8  to calculate time, host  8  may send an NPA Ext Inkjet Cartridge Information command to controller  18  of imaging apparatus  10  that contains the host&#39;s date and the identification (ID) of the host. The host date may be, for example, a 16-bit value defined as the number of days since Jan. 1, 2001. The NPA command can be sent prior to every print job, following an NPA Start Job command. Firmware in controller  18  of imaging apparatus  10  uses the date in the current NPA command to calculate the difference in time (delta) since the last NPA command. The total accumulated time Tt since printhead installation may be stored in the memory, such as memory  74 , associated with the ink reservoir in a time parameter T 4 , which is written by the firmware. Total accumulated time Tt may be represented, for example, by a six bit binary array, with each bit of T 4  representing, for example, one months or 30 days. Therefore, when the total accumulated time increases by 30 days, another fuse will be blown in T 4 .  
      Alternatively, host  8  could send the date and the host ID to imaging apparatus  10  in the print job start header information, rather than use an NPA command. If imaging apparatus  10  records a time from the print header of a print job that is less than a previous recorded time, imaging apparatus  10  will reset the current time only if the Host ID for the current job is the same as the Host ID for the previous job.  
      As a further alternatives, if a real time clock (RTC) is used, the install date loaded into memory, such as memory  74 , would yield the total time Tt since installation. For more robustness, two dates could be loaded into memory  74 : 1) the install date and 2) the date when ink reservoir  32  went empty. The subtraction of the two dates would document the length of time printhead  28  and/or ink reservoir  32  was in operation based on relative dates in case the RTC time is significantly different than world time.  
      The firmware in imaging apparatus  10  may, for example, keep a record of the last used monochrome, color dye, and color pigmented ink reservoirs and/or printheads. The record may include the total dot counts, and the total accumulated time since installation. For examples, if a monochrome printhead cartridge is replaced with a color pigmented printhead cartridge, the dot count and the accumulated time for the monochrome printhead cartridge may be stored in the memory. Thus, when the monochrome printhead cartridge is returned to replace the color pigmented printhead cartridge, the monochrome printhead cartridge may be treated just as if it had not been removed.  
      If a printhead and/ or ink reservoir is installed with a blank identification (ID), then imaging apparatus  10  recognizes the printhead and/or ink reservoir as being new and will read the parameters, e.g., T 0 Yield, T 0 , T 1 , T 2 , and T 3  from the memory associated with the printhead and/or ink reservoir. These parameters may be stored in the memory associated with the ink reservoirs, for example, during a manufacturing operation. The total dot count and the total accumulated time Tt locations in memory  74  will be set to zero.  
      Further, if a printhead and/or ink reservoir is newly installed with a non-blank ID, but has not been recorded by the firmware of controller  18 , then the firmware may use the total dot count stored in the memory associated with the newly installed printhead and/or ink reservoir. Any remainder dot counts in memory of the last printhead and/or ink reservoir installed of that type may also be added to the total dot counts of the newly installed printhead. However, the total accumulated time will be set to the value in T 4  of memory  74 .  
      At step  102 , a cumulative actual ink drop count of ink drops expelled from ink reservoir  32  is determined. Each drop, or dot, jetted from printhead  28  is counted by controller  18 , or alternatively host  8  as ink is used from ink reservoir  32  The ink usage may be tracked by setting a bit in the ink usage gauge array of memory  74  when the accumulated count counted by controller  18 , or alternatively host  8  reaches the next usage gauge threshold boundary. For example, usage threshold boundaries may be established in the ink usage array of memory  74  to represent 1,000,000 dots each, and an additional usage bit is set as each threshold boundary is reached. Thus, the cumulative actual ink drop count of ink drops may be maintained in memory  74 , or may be maintained in controller  18 , or alternatively host  8 , by retrieving ink usage information from memory  74 .  
      At step S 104 , an evaporation amount associated with the ink reservoir, such as ink reservoir  32 , is determined in accordance with an embodiment of the present invention, and a compensated drop count is established. The details of determining the evaporation amount in step S 104  will be provided following this discussion of the general method. In summary, however, the evaporation amount may be represented by evaporation prediction curve  78  of  FIG. 3 . Referring to  FIG. 3 , before time threshold T 1 , a first rate of evaporation is used. Upon reaching time T 1 , another rate of evaporation is used. Upon reaching accumulated time T 1 +T 2 , still another rate of evaporation is used. For example, upon reaching time threshold T 1 , i.e., if the total accumulated time period Tt is equal to or greater than time threshold T 1 , then a second rate of evaporation is used to compensate for an evaporation loss for ink reservoir  32  by adjusting the cumulative actual ink drop count to form an evaporation compensated drop count.  
      More particularly, for example, the rate of evaporation is used to calculate the amount of ink loss from ink reservoir  32  due to ink evaporation. The ink loss due to the evaporation amount is converted to an equivalent ink drop count, wherein the sum of the cumulative actual ink drop count is added to the equivalent ink drop count to form the evaporation compensated drop count. When the evaporation compensated drop count reaches the next usage threshold boundary, the next bit in the usage gauge in memory  74  associated with ink reservoir  32  will be set.  
      At step S 106 , by knowing the evaporation compensated drop count, e.g., the sum of the cumulative actual ink drop count and the evaporation equivalent ink drop count, as well as the initial drop count (estimated) at initial time T 0 , i.e., when ink reservoir  32  is full, then an amount of remaining ink available from ink reservoir  32  can be readily determined by subtracting the evaporation compensated drop count from the initial drop count.  
       FIG. 5  is a flowchart of a method that may be utilized in implementing the act of determining the evaporation amount in step S 104  of  FIG. 4 .  
      At step S 104 - 1 , an empirical evaporation curve is established for an ink reservoir type. Referring to  FIG. 3 , empirical data is collected by making evaporation measurements relating to a particular ink reservoir type to establish empirical evaporation curve  76  for the ink reservoir type. The ink reservoir type may be identified, for example, based on the ink type (e.g., color, monochromatic, pigment, dye, dilute, etc.). fluid capacity, and configuration, For example, color ink reservoir  30  may be associated with one ink reservoir type, whereas monochrome ink reservoir  32  may be associated with another ink reservoir type. The empirical evaporation curve  76  for the ink reservoir type may be maintained at the manufacturing site, or alternatively, may be stored in the memory to be associated with an ink reservoir belonging to that ink reservoir type. For example, an empirical evaporation curve for a particular monochrome ink reservoir type may be stored in inemory  74  associated with monochrome ink reservoir  32 , and may be stored in the form of a look-up table.  
      At step S 104 - 2 , an evaporation prediction curve  78  is established for the ink reservoir, such as for example monochrome ink reservoir  32 , that approximates, e.g., approximately tracks, empirical evaporation curve  76 . The act of approximating empirical evaporation curve  76  can be performed by changing a slope of the evaporation prediction curve at predetermined points in time, e.g., T 1 , T 2 +T 1 , and T 3 +T 2 +T 1 , as shown in  FIG. 3 , to approximate a slope of the empirical evaporation curve  76 . Time values for T 0 , T 1 , T 2  and T 3  may be stored in the memory, e.g., memory  74 , associated with the ink reservoir, e.g., monochrome ink reservoir  32 . Thus, as shown in the example of  FIG. 3 , the rate of change in the slope of evaporation prediction curve  78 , i.e., the rate of evaporation, changes as time increases. More particularly, the slope i.e., rate of evaporation, of the evaporation prediction curve at time T 0  in  FIG. 3  is selected to correspond generally to the slope of a corresponding portion of a empirical evaporation curve  76 , e.g., from time T 0  to time T 1 . The slope, i.e., rate of evaporation, of the evaporation prediction curve  78  at time T 1  in  FIG. 3  is selected to correspond generally to the slope of a corresponding portion of empirical evaporation curve  76 , e.g., from time T 1  to time T 2 +T 1 . The slope, i.e., rate of evaporation, of the evaporation prediction curve  78  at time T 2 +T 1  in  FIG. 3  is selected to correspond generally to the slope of a corresponding portion of empirical evaporation curve  76 , e.g., from time T 2 +T 1  to time T 3 +T 2 +T 1 .  
      Thus, by utilizing multiple rates of evaporation in establishing evaporation prediction curve  78 , evaporation prediction curve  78  more closely tracks the profile, e.g., slope, of the corresponding portion of empirical evaporation curve  76  than would have been the case if a single straight line approximation of evaporation was used.  
      In the example shown in  FIG. 3 , at time T 1 , the amount of ink was determined to be about 85 percent of the initial claimed yield T 0 Yield designated by evaporation prediction curve  78  at time T 0 . At time T 2 +T 1 , the amount of ink was determined to be about 67 percent of the initial claimed yield T 0 Yield designated by ink evaporation prediction curve  78  at time T 0 . At time T 3 +T 2 +T 1 , evaporation prediction curve  78  will go to zero.  
      In specific example that follows, the firmware in controller  18  will use the date information to calculate the change in time, e.g., delta time, since the last print job. The firmware will begin determining, e.g., accumulating, an amount of evaporated ink using the equations:  
       rate   =     -       T   ⁢           ⁢   0   ⁢   Yield   *   0.15       T   ⁢           ⁢   1             
       Yield   =       rate   *     Time   Current       +     T   ⁢           ⁢   0   ⁢   Yield           
 
 wherein: 
 
      rate is the rate of evaporation;  
      T 0 Yield is the total yield of the ink reservoir e.g., ink reservoir  32 , at time T 0 ;  
      T 1  is a first length of time measured from the time T 0 ;  
      Time current  is the total accumulated time Tt; and  
      Yield is the ink evaporation amount, i.e., loss, of the ink reservoir.  
      When the delta time reaches time T 1 , the firmware will begin determining, e.g., accumulating, an amount of evaporated ink using the equations:  
       rate   =     -       T   ⁢           ⁢   0   ⁢   Yield   *   0.18       T   ⁢           ⁢   2             
       Yield   =       rate   *     Time   Current       +       T   ⁢           ⁢   0   ⁢   Yield   *     (       (       T   ⁢           ⁢   1   *   0.67     -     (       T   ⁢           ⁢   2     +     T   ⁢           ⁢   1       )       )     *   0.85     )         T   ⁢           ⁢   2             
 
 wherein: 
 
      rate is the rate of evaporations;  
      T 0 Yield is the total yield of the ink reservoir, e.g., ink reservoir  32 , at time T 0 ;  
      T 1  is a first length of time measured from the time T 0 ;  
      T 2  is a second length of time measured from time T 1 ;  
      T 2 +T 1  is the sum of times T 1  and T 2  (see, for example,  FIG. 3 );  
      Time current  is the total accumulated time Tt; and  
      Yield is the ink evaporation amount of the ink reservoir.  
      When the delta time reaches time T 2 +T 1 , the firmware will begin determining, e.g., accumulating, an amount of evaporated ink using the equations:  
       rate   =     -       T   ⁢           ⁢   0   ⁢   Yield   *   0.67       T   ⁢           ⁢   3             
       Yield   =       rate   *     Time   Current       +       T   ⁢           ⁢   0   ⁢   Yield   *     (       (       T   ⁢           ⁢   3     +     T   ⁢           ⁢   2     +     T   ⁢           ⁢   1       )     *   0.67     )         T   ⁢           ⁢   3             
 
 wherein: 
 
      rate is the rate of evaporation;  
      T 0 Yield is the total yield of the ink reservoir, e.g., ink reservoir  32 , at time T 0 ;  
      T 1  is a first length of time measured from the time T 0 ;  
      T 2  is a second length of time measured from time T 1 ;  
      T 3  is a third length of time measured from time T 2 ;  
      T 3  +T 2 +T 1  is the sum of times T 1 , T 2  and T 3  (see, for example,  FIG. 3 );  
      Time current  is the total accumulated time Tt; and  
      Yield is the ink evaporation amount of the ink reservoir.  
      In embodiments utilizing host  8  in case the host computer&#39;s time becomes incorrect, the maximum delta in the rate of evaporation may be based on a maximum delta time e.g., a delta time of two weeks. For example, if the rate of evaporation is 200 pages/month and the delta time calculated is 3 months, then the evaporation may be limited to 100 pages. However, the time may be set based on the time read from the print header even if the delta in time is greater than two weeks.  
      While this invention has been described with respect to embodiments of the invention, the present invention may be further modified within the spirit and scope of this disclosure. This application is therefore intended to cover any variations, uses, or adaptations of the invention using its general principles. Further, this application is intended to cover such departures from the present disclosure as come within known or customary practice in the art to which this invention pertains and which fall within the limits of the appended claims.