JP5734024B2 - Analysis method and program - Google Patents

Description

  The present invention relates to an analysis method and a program.

  A developing device in a color image forming apparatus that forms a full-color or multi-color image by an electrophotographic method often uses a two-component developer in which a nonmagnetic toner and a magnetic carrier are mixed from the viewpoint of color developability and mixing properties. The developing device generally used is composed of a developing chamber and an agitating chamber partitioned by a partition, and contains a two-component developer containing a nonmagnetic toner and a magnetic carrier. A developing screw and a stirring screw are installed in the developing chamber and the stirring chamber, respectively, and the developer is circulated by the conveying force of each screw.

  In this method, since the toner gradually decreases with the development process, a toner concentration control device that replenishes the developing device with an amount of toner corresponding to the reduction amount is provided, and the toner concentration of the developer is reduced. I try to keep it constant. The replenished toner is charged by being mixed with the carrier in the developing device. In addition, the toner density (that is, the ratio of the toner mass to the total mass of the carrier and the toner) representing the mixing ratio of the two-component developer in the developing unit and the spatial distribution of the toner charge amount can improve the image quality and improve the image density. It is a very important factor in stabilizing.

  On the other hand, the introduction of simulation into design activities has been promoted due to the recent demand for reduction in development speed and cost reduction, and the development and application of simulation technology are being actively carried out in electrophotography.

  In the above-described simulation technique of toner density and toner charge amount, as shown in Patent Document 1, a method of calculating toner density distribution by treating mixing and stirring of toner as fluid diffusion has been proposed. As for the calculation method of the toner charge amount associated with toner replenishment, there is a method of calculating a plurality of advection diffusion equations corresponding to each replenishment toner, and further calculating the charge amount associated with the elapsed time from the time when the replenishment toner is supplied. Proposed.

JP-A-2005-070282

  In an actual developing device, it is known that the toner charge amount imparting ability varies depending on the position of the developer device, for example, the rising speed of the toner charge amount differs between the developing chamber and the stirring chamber. It is also known that the toner charge amount varies depending on the toner density value. However, since only the replenishment elapsed time can be considered in the method of Patent Document 1, it is impossible to calculate the difference in the toner charge amount due to the difference in the position of the developer and the toner density.

  Further, in the method of Patent Document 1, in order to consider toner replenishment, it is necessary to solve the same number of advection diffusion equations as the number of times of replenishment, and there is a problem that calculation time becomes long.

  Accordingly, the object of the present invention is to consider the difference in toner charge amount depending on the toner concentration, and further, the difference in toner charge amount imparting ability depending on the position of the developing device. It is to provide a method for calculating the time change.

In order to solve the problem, the present invention adopts the following configuration. That is, a method for analyzing an advection diffusion model of toner density and toner charge amount in an electrophotographic developer, in which the information processing device performs development using the developer with respect to a condition relating to the physical quantity of the developer a setting step of setting a condition related to vessels, the information processing apparatus, a toner concentration calculation step 1 to calculate the convection-diffusion equation which represents the time variation of the spatial distribution of the toner density, the information processing apparatus, the space of the toner charge amount a toner charge amount calculation step 2 for computing the convection-diffusion equation which represents the time variation of the distribution, the information processing apparatus, calculates a charge amount variation expression representing the time variation in the toner charge amount at each position of the developing device, toner And a charge amount change calculating step 3 for updating the charge amount.

  According to the present invention, the time variation of the spatial distribution of the toner concentration and the toner charge amount in consideration of the difference in the toner charge amount depending on the toner concentration and the difference in the toner charge amount imparting ability depending on the position of the developing device. Can provide a way to calculate

1 is a configuration diagram of a toner density distribution and toner charge amount distribution calculation apparatus according to Embodiment 1. FIG. 3 is a flowchart illustrating a preferred example of a calculation flow according to the first embodiment. 6 illustrates an example of calculation results of toner density distribution and toner charge amount distribution according to the first exemplary embodiment. FIG. 9 is a configuration diagram of a toner density distribution and toner charge amount distribution calculation apparatus according to a second embodiment. 10 is a flowchart illustrating a preferred example of a calculation flow according to the second embodiment. 7 illustrates an example of calculation results of toner density distribution and toner charge amount distribution according to the second exemplary embodiment. FIG. 3 is a diagram illustrating an example of a developing device configuration according to Embodiments 1 and 2.

<Example 1>
Hereinafter, embodiments for carrying out the present invention will be described in detail with reference to the drawings as necessary. In the first embodiment, the case where the developer circulates between the developing chamber and the stirring chamber constituting the two-component developing device will be described as a representative example.
FIG. 7 shows a schematic diagram of the configuration of the target two-component developing device. Such a two-component developing device is used in an electrophotographic two-component developing process. In the drawing, reference numeral 701 denotes a stirring chamber, 702 denotes a screw contained in the stirring chamber, 703 denotes a developing chamber, and 704 denotes a screw contained in the developing chamber. As the two screws rotate, the toner and the carrier are mixed, contacted and charged, and conveyed in the direction of the rotation axis, and circulated in the direction indicated by the arrows in the figure. Reference numeral 705 denotes a toner replenishment position where the toner is replenished so as to compensate for toner consumption accompanying development.

  The essence of the present invention pays attention to the fact that the developer flow is dominant in the two-component developer, and uses the two equations of the toner concentration advection diffusion equation and the toner charge amount advection diffusion equation to determine the space between the two. It represents the distribution change. Here, in the advection diffusion equation of toner density, it is assumed that the toner advects while diffusing with the developer. In the advection-diffusion equation of toner charge amount, it is assumed that the charge held by the toner is advected while diffusing according to the toner flow. Furthermore, the temporal change in the spatial distribution of the toner concentration and the toner charge amount in the developing device is obtained by calculating in combination with a model expression representing the rise of the toner charge amount due to mixing with the developer.

  The details of these models are described in more detail below.

Equation 1 below is an advection diffusion equation of toner concentration. Here, T represents a toner concentration which is a toner mass ratio with respect to a developer mass, m represents a developer mass density distribution, v represents a developer flow rate, and D represents a diffusion coefficient.

Since the product of the toner density T and the developer mass density distribution m represents the toner mass density distribution,
Expression 1 is an expression that models a phenomenon in which the toner mass density distribution advects and diffuses according to the advancing speed and diffusion coefficient of the developer.

Equation 2 below is an advection diffusion equation of toner charge amount. Here, Q represents the toner charge amount (so-called Q / M) obtained by dividing the charge amount of the toner by the toner mass, and the others are the same as those in Equation 1.

  Since the product of the toner charge amount Q, the toner concentration T, and the developer mass density distribution m represents the spatial distribution of the toner charge amount, Equation 2 models the phenomenon of toner charge advection and diffusion as the toner advection and diffusion. It has become a formula.

The charge amount of toner is known to change with time as the toner and developer are mixed. In general, an exponential rising equation represented by the following equation 3 is often used. Here, the saturation charge amount Q _sat and the time constant τ are values representing the ability to impart a charge amount determined by the configuration of the developing device, the material characteristics of the developer, and the like. Further, t represents the rise time of the toner charge amount. When t is 0, the charge amount is 0.

Further, by differentiating Equation 3 with respect to time, the toner charge amount change ΔQ during Δt can be obtained as shown in Equation 4 below. The charge amount change calculation can be performed using this charge amount change equation.

  In the present invention, the phenomenon in which the toner advects and diffuses together with the developer and the phenomenon in which the charge amount increases when the toner is mixed with the developer are solved by coupling Equations 1, 2 and 4 together. In consideration of this, it is possible to calculate the temporal change in the spatial distribution of the toner density and the toner charge amount.

  FIG. 1 shows the configuration of an apparatus representing a preferred example according to the present invention. The apparatus includes a CPU 100, a RAM 101, a display device 102 such as a display, an input unit 103 such as a keyboard and a mouse, an external storage device 104 such as a hard disk, and a bus 105.

  The CPU 100 is an information processing apparatus, and achieves the object of the present invention by reading a program or data stored in the RAM 101 functioning as a memory and executing information processing. The CPU 100 outputs the information processing result to the display device 102 so that it can be used. Further, in the CPU 100, reference numeral 100a is a control unit that controls the entire program. Reference numeral 100b denotes a calculation condition setting unit that sets calculation conditions. Reference numeral 100c denotes a toner concentration calculation unit that solves the advection diffusion equation for the toner concentration and calculates the toner concentration. In this block, toner density calculation step 1 is executed. Reference numeral 100d denotes a toner charge amount calculation unit, which calculates the toner charge amount by solving an advection diffusion equation for the toner charge amount. Further, the equation representing the time change of the toner charge amount at each position is solved to obtain the time change amount of the toner charge amount. In this block, toner charge amount calculation step 2 and charge amount change calculation step 3 are executed. The RAM 101 stores toner charge amount change data 101e between the program 101a, calculation condition data 101b, toner density data 101c, toner charge amount data 101d, and Δt.

  Here, the contents of each data will be described. As the calculation condition data, there are values representing the developing device such as the length of the flow path of the developing device, the flow velocity distribution and diffusion coefficient of the developer, and the developer mass density distribution. In addition, there are initial conditions regarding the physical quantity of toner such as initial toner density distribution and initial toner charge amount distribution. There are also values related to calculation conditions such as the number of calculation grids and time steps. The toner density data is a toner density distribution calculated for each time step, and is a value that is sequentially updated. The toner charge amount data is a toner charge amount distribution calculated at each time step, and is a value that is sequentially updated. The toner charge amount change data during Δt is the toner charge amount change amount during Δt seconds.

Hereinafter, an example of a method for calculating the toner concentration and the toner charge amount of the two-component developer will be described. Specifically, it is assumed that the developer circulates between the developing chamber and the stirring chamber constituting the two-component developing device as a one-dimensional flow. For simplicity of explanation, it is assumed that the flow rate, the diffusion coefficient, and the mass density distribution of the developer are all the same in the developing chamber and the stirring chamber, and only the time constant τ in Equations 3 to 4 is different. Specifically, the time constant of the developing chamber is smaller than that of the stirring chamber.
Also, the temporal change in the spatial distribution of the toner density and toner charge amount in the developer unit, assuming that the toner concentration and toner charge amount in one part of the two-component developer unit are different from the values in other regions. Ask for. At this time, it is assumed that toner is not consumed due to toner replenishment or development.

  FIG. 2 is a flowchart showing the flow of the calculation method of the present invention. The flow of the toner concentration distribution calculation method for the developer will be described with reference to FIG.

  (1) In the calculation condition setting unit 100b, values such as a developer flow velocity distribution and a diffusion coefficient, a developer representing values such as a developer mass density distribution, calculation conditions such as a time step, a calculation end time, and a calculation grid are set. Stored in the calculation condition data 101b. Also, initial conditions such as initial toner density and initial charge amount are set and stored in toner density data 101c and toner charge amount data 101d (step S201).

  (2) Next, the toner concentration calculation unit 100c calculates the toner concentration distribution at time t by solving the advection diffusion equation of Equation 1 using the calculation condition data 101b and the toner concentration data 101c, and the toner concentration data 101c. (Step S202). Note that the CIP method is used for the calculation.

  (3) Next, the toner charge amount calculation unit 100d uses the calculation condition data 101b, the toner density data 101c, and the toner charge amount data 101d to solve the advection diffusion equation of Equation 2. Thus, the toner charge amount distribution at time t is calculated and stored in the toner charge amount data 101d (step S203). Note that the CIP method is used for the calculation.

(4) Next, using the calculation condition data 101b and the toner charge amount data 101d, a change amount ΔQ of the toner charge amount for Δt seconds in each calculation grid is calculated using Equation 4, and the toner charge amount between Δt is calculated. The change data 101e is stored (step S204).

  (5) Next, the sum of the toner charge amount change data 101e between the toner charge amount data 101d and Δt is obtained and stored in the toner charge amount data 101d (step S205).

  (6) It is determined whether the calculation time is the calculation end time set in (1). If true, the time is updated, and the process returns to (2) (steps S206 = Yes, S207). If it is false, the calculation is terminated.

  FIG. 3 shows an example of calculation results of the toner density distribution and toner charge amount distribution calculated in the implementation. Two cases differing only in the time constant (τ in Equation 4) of the toner charge amount in the stirring chamber are calculated, and Case 2 corresponds to a case where the time constant is larger than Case 1. Other calculation conditions are the same for Case 1 and Case 2. That is, in a part of the stirring chamber, the toner concentration is 12 wt%, and the charge amount at the same position is 21 μC / g. In other regions, the results are calculated assuming that the toner concentration is 8 wt% and the toner charge amount is uniform at 35 μC / g as an initial condition.

  3A to 3E show changes in toner density distribution over time, the vertical axis shows the toner density, and the horizontal axis shows the position in the transport direction in the developing device. (F) to (i) represent changes in the toner charge amount distribution with time, the vertical axis represents the toner charge amount, and the horizontal axis represents the position of the developing device. In all the graphs, the left half of the horizontal axis represents the stirring chamber and the right half represents the developing chamber. The developer is conveyed from the left to the right, and the agent at the right end of the developing chamber circulates to the left end of the stirring chamber. Moreover, the graph is shown for every time and shows the time transition from 0 to 4 sec.

  As can be seen from the figure, the toner density varies in the same way for both Case 1 and Case 2, and it is conveyed while diffusing as time passes, approaching a flat distribution. On the other hand, as for the toner charge amount, it can be seen that Case 1 with a smaller time constant in the stirring chamber has a larger charge amount than Case 2 when compared at the same time. Thus, it can be seen that the difference can be calculated in the case where the toner charge amount imparting ability varies depending on the position in the transport direction in the developing device.

  As described above, in the present invention, analysis is performed using the advection diffusion model of toner density and toner charge amount and the charge amount model by toner mixing. Accordingly, it is possible to calculate the temporal change in the spatial distribution of the toner density and the toner charge amount in consideration of the difference in the toner charge amount imparting ability depending on the position of the developing device.

In this embodiment, the case where the flow is regarded as a one-dimensional flow in the circulation direction has been described. However, the present invention can also be applied to a case where the flow path branches or merges like a vertical stirring type developer. is there. Further, by applying a two-dimensional or three-dimensional advection diffusion equation to Equations 1 and 2, not only a one-dimensional flow in the circulation direction but also a flow in the cross-sectional direction with respect to the circulation direction can be considered.

That is, when applied to Equation 2, the advection diffusion equation representing the temporal change in the spatial distribution of the toner charge amount is represented by Equation 5 below. Here, the toner charge amount Q, the toner mass ratio T to the developer mass, the developer mass density distribution m, the developer flow velocity v, and the diffusion coefficient D are used.

When applied to Equation 1, the advection diffusion equation representing the temporal change in the spatial distribution of the mixing ratio of toner and carrier (toner concentration) is expressed by Equation 6 below. Here, the toner mass ratio T with respect to the developer mass, the developer mass density distribution m, the developer flow velocity v, and the diffusion coefficient D are used.

  Also, when the developer flow rate, diffusion coefficient, developer mass density distribution varies depending on the position in the developing device, and when not only the time constant of the charge amount but also the saturated charge amount, Equations 1 to 4 By setting each value to be used for each position, it can be easily considered. Thus, the present invention can be easily applied to developing devices having various configurations.

  Further, in this embodiment, the flow rate and diffusion coefficient of the developer and the mass density distribution of the developer are known, but it is also possible to calculate by combining these expressions representing changes with time. As a result, the toner density and the charge amount can be calculated using the present invention even when the developer mass density distribution or the like changes with time in accordance with the developer discharge mechanism.

  In addition, although the description is made using the expression representing the exponential charging characteristics shown in Expression 3 to Expression 4, the present invention is not limited thereto. Actually, it is known that the toner density and the toner charge amount correlate with each other. However, by using a model equation of the toner charge amount in consideration of the dependency on the toner concentration, by using another model equation related to the charge amount, Calculations that take into account many factors are possible.

  In addition, in the present Example, although demonstrated as calculating Formula 1, Formula 2, and Formula 4 in order, it is computable even if it replaces order. For example, when Equation 2 is calculated before Equation 1 in the flow, the toner charge amount is calculated using the toner density obtained by Equation 1 immediately before (in the loop Δt seconds ago).

  Moreover, you may use the method of calculating simultaneously Formula 1 and Formula 2 as simultaneous equations. Specifically, it is possible to obtain the toner concentration and the toner charge amount at each time by solving simultaneous equations in which the toner charge amount and the toner concentration in Equations 1 and 2 are unknown by a method such as Newton's method. . Furthermore, the calculation may be performed using a numerical calculation method in which three equations of Equation 1, Equation 2, and Equation 4 are simultaneously calculated.

In this embodiment, the CIP method is used to solve the advection diffusion equation. However, there are many difference schemes for numerical calculation, such as upwind difference, Leap-Frog scheme, Lax-Wendoroff scheme, flux limiting method, etc. Other difference schemes may be used.

  Further, when the toner charge amount depends on the toner concentration, it can be dealt with by taking the influence into consideration. Specifically, a method of calculating Equation 4 using a value of the time constant τ depending on the toner density or the value of the saturated charge amount Qsat can be considered.

<Example 2>
In Example 1, it is assumed that there is no toner consumption due to toner replenishment and development. In the second embodiment, a method that takes into account toner consumption due to toner replenishment and development will be described. The essence of the second embodiment is to provide a process for correcting the toner density and the toner charge amount at the replenishment position and the consumption position according to the replenishment condition and the consumption condition.

  A method that takes into account toner consumption due to toner replenishment and development will be described below with reference to the form of the first embodiment.

  FIG. 4 shows a configuration of an apparatus representing a preferred example according to the present invention. In the figure, reference numerals 100 to 105 and 100a to 100d are the same as those in FIG. FIG. 401 is a replenishment / consumption calculation unit that corrects the toner density and the toner charge amount at the replenishment position and the consumption position according to the replenishment and consumption conditions. The calculation condition data includes a toner supply condition and a toner consumption condition accompanying development in addition to the data of the first embodiment.

  Hereinafter, an example of a method for calculating the toner concentration and the toner charge amount of the two-component developer will be described. Specifically, description will be made assuming that toner replenishment is considered in addition to the same conditions as in the first embodiment.

  FIG. 5 is a flowchart showing the flow of the calculation method of the present invention. The flow of the toner concentration distribution calculation method for the developer will be described with reference to FIG. Since S202 to S207 are the same as those in the first embodiment, only the differences will be described here.

  (1) In addition to the calculation condition setting of the first embodiment, the calculation condition setting unit 100b sets a replenishment position, a replenishment amount, a replenishment timing, and the like, which are toner replenishment conditions, and stores them in the calculation condition data 101b (Step 1). S201).

  (2) The replenishment / consumption calculation unit 401 determines whether or not to replenish toner at the current time based on the calculation condition data 101b (step S501). If true, the calculation condition data 101b, toner density data 101c, and toner charge amount data 101d are used to correct the toner density and toner charge amount of the calculation grid corresponding to the replenishment position. Then, the corrected value is stored in the toner density data 101c and the toner charge amount data 101d (step S502). Specifically, the toner concentration after replenishment is calculated based on the toner concentration at the replenishment position and the amount of toner replenished. For the toner charge amount, the toner charge amount after replenishment is calculated from the weight ratio between the toner charge amount at the replenishment position and the initial charge amount of the replenishment toner.

  FIG. 6 shows the calculation result when the toner having the charge amount of 0 is replenished 2.5 seconds after the calculation is started in addition to the calculation conditions of FIG. As can be seen from the figure, the time transitions of the toner concentration and the toner charge amount due to toner replenishment can be calculated.

As described above, in the present invention, even when toner replenishment is considered, analysis is performed using the advection diffusion model of toner density and toner charge amount and the charge amount model by toner mixing. This makes it possible to calculate the temporal change in the spatial distribution of the toner concentration and the toner charge amount, taking into account the difference in the toner charge amount imparting ability depending on the position of the developing device and the difference in the toner charge amount depending on the toner concentration. Toner replenishment correction is possible.

  In this embodiment, only toner replenishment is considered, but toner consumption can be easily considered by using the same method. That is, the toner consumption conditions, such as the consumption position, consumption amount, and consumption timing, are set, and the toner density and toner charge amount are corrected. This makes it possible to perform toner consumption correction that predicts changes in toner density and toner charge amount associated with image conditions.

  100a: control unit, 100b: calculation condition setting unit, 100c: toner density calculation unit, 100d: toner charge amount calculation unit, 101: RAM, 101a: program, 101b: calculation condition data, 101c: toner density data, 101d: toner Charge amount data, 101e: Toner charge amount change data between Δt

Claims (11)

An analysis method of an advection diffusion model of toner density and toner charge amount in an electrophotographic developer,
A setting step in which the information processing apparatus sets conditions relating to the physical quantity of the developer, and conditions relating to a developing device that performs development using the developer;
A toner concentration calculating step 1 in which the information processing apparatus calculates an advection diffusion equation representing a temporal change in a spatial distribution of toner concentration;
A toner charge amount calculation step 2 in which the information processing apparatus calculates an advection diffusion equation representing a temporal change in a spatial distribution of the toner charge amount;
A charge amount change calculation step 3 in which the information processing device calculates a charge amount change expression representing a change in toner charge amount with time at each position of the developing device, and updates the toner charge amount;
The analysis method characterized by having. The analysis method according to claim 1, wherein the toner density calculation step 1, the toner charge amount calculation step 2, and the charge amount change calculation step 3 are simultaneously calculated. The condition relating to the physical amount of the developer set in the setting step includes an initial toner density distribution and an initial toner charge amount distribution, and the condition relating to the developing device includes a flow path of the developer. The analysis method according to claim 1 or 2 . In the setting step, the information processing apparatus further sets a supply condition including a toner supply position in the developing unit,
The information processing apparatus, based on said supply condition, according to any one of claims 1 to 3, further comprising a toner replenishment correction step of correcting the toner density and the toner charge amount at the supplying position analysis method. The analysis method according to claim 4 , wherein the replenishment condition includes a toner replenishment amount and a replenishment timing. In the setting step, the information processing apparatus further sets a consumption condition including a toner consumption position in the developing device,
The information processing apparatus, based on the consumption condition, according to any one of claims 1 to 5, further comprising a toner consumption correction step of correcting the toner density and the toner charge amount at the consumption location analysis method. The analysis method according to claim 6 , wherein the consumption condition includes a toner consumption amount and a consumption timing. The advection diffusion equation representing the time change of the toner charge amount spatial distribution is expressed as follows: toner charge amount Q, toner mass ratio T to developer mass, developer mass density distribution m, developer flow velocity v, and diffusion coefficient D. the method of analysis according to any one of claims 1 to 7, characterized by being represented by the following formula.

The advection diffusion equation representing the temporal change in the spatial distribution of the toner concentration is expressed by the following equation when the toner mass ratio T with respect to the developer mass, the developer mass density distribution m, the developer flow velocity v, and the diffusion coefficient D are expressed. The analysis method according to any one of claims 1 to 8 , wherein:

The charge quantity change equation, the toner charge amount Q, the saturated charge amount QSAT, the time constant tau, when the toner charge amount changes ΔQ between Delta] t, according to claim 1 to 9, characterized by being represented by the following formula The analysis method according to any one of the above.

The program which makes the said information processing apparatus perform each step of the analysis method of any one of Claims 1 thru | or 10 .

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