Air bubble growth prediction method, air bubble growth prediction device and air bubble growth prediction program of foam resin molding, and recording medium

Abstract

To execute a simulation in consideration of coalescence of air bubbles in a growth process of a foam resin, and improve prediction accuracy.SOLUTION: An air bubble growth prediction method of a foam resin molding includes: a proportional coefficient determination step S11 of determining a proportional coefficient about the following expression in which a moving distance when two air bubbles are contacted in a liquid phase, then are repelled and separated from each other is represented by radii of air bubbles, a distance between centers of the air bubbles and a proportional coefficient; and an analysis step S2 of executing a simulation in a condition that a large number of air bubbles exist in a liquid phase with a predetermined volume, and calculating an air bubble state of a foam resin molding, wherein the analysis step includes a simulation condition to determine whether or not the air bubbles coalesce, on the basis of the surface tension of the air bubbles and the viscosity of the liquid phase, when the growing air bubbles are contacted, and calculate a moving distance when the air bubbles do not coalesce, irrespective of the setting of the distance between the centers of the air bubbles to be less than the sum of each air bubble radius, and separate the repelled air bubbles on the basis of the calculation result.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present disclosure relates to a method for predicting bubble growth in a foamed resin molded product, a bubble growth prediction device, a bubble growth prediction program, and a recording medium. [Background technology]

[0002] Molded foam plastic products, which are widely used in automotive exterior and interior materials, electrical appliances, various containers, etc., have sound-absorbing and heat-insulating properties. The cell diameter and cell density of such foam plastic molded products are affected by molding conditions such as temperature and pressure, which determine the performance of the molded product.

[0003] In conventional development of foamed plastic molded products, it is common to make prototypes under different conditions such as temperature and pressure, and analyze the foam structure and properties of the molded product to approach the desired performance. However, this method requires a great deal of time and cost to arrive at molding conditions that can achieve the desired performance.

[0004] In recent years, therefore, attempts have been made to simulate bubble growth during the molding process of foamed resins and predict in advance the molding conditions necessary to obtain foamed resin molded products that exhibit the desired performance. For example, Patent Document 1 discloses a method for simulating the foaming state in a die during extrusion foaming, in which the actual bubble nucleation rate, which is the value at which bubbles essentially begin to be generated, is applied to a bubble nucleation rate equation to estimate the foaming initiation pressure. It also discloses a method for using the results of this simulation to simultaneously consider bubble nucleation and bubble growth.

[0005] Known bubble nucleation models and bubble growth models are used to simulate such foamed resins (e.g., Non-Patent Documents 1-3). Non-Patent Document 1 proposes a nucleation rate equation as a bubble nucleation model. Non-Patent Document 2 proposes a bubble growth model consisting of three basic equations: a motion equation for the bubble interface, a mass balance equation for the gas inside the bubble, and an equation for gas diffusion near the bubble interface. Non-Patent Document 3 also describes an extension of known bubble nucleation and bubble growth models that takes into account the overlap of concentration boundary layers that occur between approaching bubbles. [Prior art documents] [Patent documents]

[0006] [Patent Document 1] Japanese Patent Application Laid-Open No. 2000-135731 [Non-patent literature]

[0007] [Non-Patent Document 1] M. Blander, JL Katz, The thermodynamics of cluster formation in nucleation theory, J. Statistical Physics 4, 1972 [Non-patent document 2] Shafi, MA, Joshi, K., Flumerfelt, RW, bubble size distributions in freely expanded polymer foams, Chemical Engineering Science 52 ,1997 [Non-patent document 3] Sun Ying, Experimental Study and Simulation of Physical Foaming Process of Polymer Using High-Pressure Gas, Doctoral Dissertation, Graduate School of Engineering, Hiroshima University, 2015 (https: / / ir.lib.hiroshima-u.ac.jp / 00040233) Summary of the Invention [Problem to be solved by the invention]

[0008] Among the structural factors of foamed plastic molded products, the ones that particularly affect their sound-absorbing properties are the bubble diameter and the size of the passages (throat diameter) created by the coalescence of the bubbles. Finding molding conditions that precisely control these factors makes it possible, for example, to give foamed plastic molded products sound-absorbing properties that absorb a specific range of frequencies. There is a significant difference in sound-absorbing properties between a state in which bubbles are in contact without coalescence and a state in which bubbles are coalesced. Therefore, in order to more precisely predict the bubble state of foamed plastic molded products and find molding conditions that accurately achieve the desired performance, it is necessary to include bubble coalescence in the simulation conditions.

[0009] However, although Patent Document 1 discloses a simulation that takes into account bubble nucleation and bubble growth, it does not mention the coalescence of bubbles that come into contact with each other. In such conventional simulations, all bubbles are assumed to be independent, and bubble growth is assumed to stop when all the gas required for bubble generation is consumed, and the coalescence of bubbles is not taken into account. Non-Patent Document 3 also describes the consideration of overlapping concentration boundary layers that occur when bubbles come into close contact with each other, but does not mention the coalescence of bubbles.

[0010] As described above, in the conventional technology, the merging of bubbles is not taken into account in the simulation of the foamed resin molding process, and therefore the structure produced by the simulation may differ from the structure formed by the actual molding process, which may result in differences in performance.

[0011] Therefore, the objective of this disclosure is to improve prediction accuracy in a bubble growth prediction method, bubble growth prediction device, bubble growth prediction program, and recording medium for foamed resin molded products by fully taking into account the merging of bubbles during the growth process of foamed resin. [Means for solving the problem]

[0012] In order to solve the above problems, a method for predicting bubble growth in a foamed resin molded product disclosed herein includes the steps of: A method for predicting the bubble state of a foamed resin molded product obtained by foaming a foamed resin through computer simulation, comprising: a proportionality coefficient determination step for determining the proportionality coefficient of the following equations (1) and (2), which express the distance traveled when two bubbles come into contact with each other in a liquid phase and then repel each other and move away from each other, based on an experiment or simulation using two bubbles in a liquid phase, based on the bubble radius, the distance between the bubble centers, and a proportionality coefficient; an analysis step of executing a simulation of foam resin molding assuming that a large number of bubbles are generated and grow in a liquid phase in a predetermined volume, and calculating the bubble state of the foam resin molded product; The analyzing step When the grown bubbles come into contact with each other, it is determined whether the bubbles will coalesce or not based on the surface tension of the bubbles and the viscosity of the liquid phase; and The simulation conditions include the following: when the center-to-center distance between bubbles becomes smaller than the sum of the bubble radii but the bubbles do not coalesce, the movement distance is calculated based on the following formulas (1) and (2) using the proportional coefficients, and the bubbles that have repelled each other based on the calculation results are separated from each other (where, in formulas (1) and (2), r1 is the radius of bubble 1, r2 is the radius of bubble 2, L 12 is the distance between the centers of bubbles 1 and 2, d is the distance traveled, and k is the proportionality coefficient.)

[0013]

number

[0014] This configuration incorporates the possibility that growing bubbles may coalesce into a single bubble when they come into contact with each other into the simulation conditions, enabling more precise prediction of the bubble state of a foamed resin product and the identification of molding conditions that achieve the desired performance with high accuracy. Furthermore, by taking into account that bubbles that come into contact do not coalesce but instead repel each other and move away, and calculating the distance traveled by the repelled bubbles, more accurate predictions are possible. The distance traveled by the repelled bubbles can be calculated using the above formula, but the proportionality coefficient used in the formula is determined in advance and then input into the foamed resin molding simulation. When simulating the generation and growth of a large number of bubbles in a given volume in a foamed resin molding scenario, including calculations that involve fluid analysis of the repelled bubble movement, it is expected to take a long time to obtain results. However, by specifying a proportionality coefficient and performing a simulation as in this configuration, it is possible to obtain results in a shorter time than with the previously described method.

[0015] In one embodiment, a threshold determination step, which includes, before the analyzing step, executing a simulation assuming that two bubbles are present in a liquid phase and that the two bubbles approach and coalesce, calculating a data group of the surface tension and liquid viscosity of the bubbles when the two bubbles coalesce, and outputting the data group as thresholds of the surface tension and liquid viscosity for determining whether the two bubbles will coalesce or not; and determining a contact angle, which is an angle formed by tangents to the bubbles at the contact point when the two bubbles come into contact, for determining a contact angle threshold for determining whether the two bubbles will coalesce or not; The analysis step reads the threshold values ​​of the surface tension and liquid phase viscosity and the threshold value of the contact angle, and includes in its simulation conditions that the bubbles coalesce when the surface tension and liquid phase viscosity of the grown bubbles exceed the threshold values ​​of the surface tension and liquid phase viscosity when the contact angle between the bubbles exceeds the threshold value of the contact angle.

[0016] According to this configuration, when simulating the generation and growth of a large number of bubbles in a predetermined volume assuming foam resin molding, if calculations are performed including a fluid analysis to determine whether the bubbles will coalesce, it is expected that an extremely long time will be required to obtain results. However, if the simulation is performed as in this configuration, by providing threshold values ​​for the surface tension and liquid-phase viscosity at the time of bubble coalescence, it is possible to significantly shorten the time required to obtain results compared to the above-mentioned method.

[0017] The data group is a table of surface tension and liquid viscosity of bubbles when two bubbles coalesce, and the analysis step preferably reads the table data before executing a simulation.

[0018] According to this configuration, it is possible to obtain simulation results in a shorter time than when performing a simulation that takes into account the effects of surface tension and liquid viscosity for all of the numerous bubbles in foam resin molding.

[0019] In one embodiment, the analyzing step calculates the bubble generation rate based on the following formula (3), and calculates the bubble growth rate based on the following formulas (4) to (6). Note that the bubble generation rate calculated based on the above formula (1) is the number of bubbles generated per unit time or per unit volume.

[0020]

number

[0021] (In equation (3), γ is the surface tension at the bubble / liquid interface, m is the mass per gas molecule, k B is the Boltzmann constant, T is the temperature, and P G is the internal pressure of the bubble, P L is the environmental pressure, N A is the Avogadro constant, and c is the gas concentration in the liquid phase.

[0022]

number

[0023] (In equations (4) to (6), η is the viscosity of the liquid phase, R is the bubble radius, r is the radial position within the bubble, R' is the gas constant, D is the diffusion coefficient, γ is the surface tension at the bubble / liquid interface, t is time, and P G is the internal pressure of the bubble, P L is the ambient pressure, and c is the gas concentration in the liquid phase. According to this configuration, it is possible to apply a known calculation model to the generation and growth of bubbles and then perform a simulation that takes into account the coalescence of bubbles, thereby enabling highly accurate predictions to be made in a short amount of time.

[0024] One aspect of the bubble growth prediction device for a foamed resin molded product disclosed herein is: An apparatus for predicting the bubble state of a foamed resin molded product formed by foaming a foamed resin through computer simulation, At least one calculation unit is provided, The calculation unit The distance traveled by two bubbles in a liquid phase when they come into contact with each other and then repel each other is expressed in terms of the radius of the bubbles, the distance between the centers of the bubbles, and a proportionality coefficient, and the proportionality coefficient is determined based on an experiment or simulation using two bubbles in a liquid phase and is input into the following equations (1) and (2): In calculating the bubble state of a foamed resin molded product by performing a foamed resin molding simulation assuming that a large number of bubbles are generated and grow in a liquid phase in a predetermined volume, When the grown bubbles come into contact with each other, it is determined whether the bubbles will coalesce or not based on the surface tension of the bubbles and the viscosity of the liquid phase; and When the center-to-center distance between bubbles becomes smaller than the sum of their radii but the bubbles do not coalesce, the movement distance is calculated based on the following equations (1) and (2) using the proportional coefficients, and the bubbles that have repelled each other based on the calculation results are separated from each other, and this is included in the simulation conditions. (Note that in equations (1) and (2), r1 is the radius of bubble 1, r2 is the radius of bubble 2, and L 12 is the distance between the centers of bubbles 1 and 2, d is the distance traveled, and k is the proportionality coefficient.)

[0025]

number

[0026] This configuration takes into account the possibility that growing bubbles may coalesce into a single bubble when they come into contact with each other, and that bubbles that come into contact may repel each other and separate without coalescing, thereby enabling more precise prediction of the bubble state in a foamed resin product and identifying molding conditions that achieve the desired performance with high accuracy. A proportional coefficient is determined in advance for the above equation that calculates the distance traveled by repelled bubbles, and the proportional coefficient is input to perform a foamed resin molding simulation. When simulating the generation and growth of a large number of bubbles in a given volume, simulating foamed resin molding, the time required to obtain results can be shortened compared to when calculations are performed that also include fluid analysis of the movement of repelled bubbles.

[0027] Furthermore, one aspect of the program for predicting bubble growth in a foamed resin molded product disclosed herein is: A program for analyzing, by computer simulation, the state of bubbles in a foamed resin molded product formed by foaming a foamed resin, Step A involves inputting a proportionality coefficient determined based on an experiment or simulation using two bubbles in a liquid phase into the following equations (1) and (2), which express the distance traveled when two bubbles come into contact with each other in a liquid phase and then repel each other and move away from each other, using the bubble radius, the distance between the bubble centers, and a proportionality coefficient. and step B, which causes a computer to execute a simulation of foam resin molding assuming that a large number of bubbles are generated and grow in a liquid phase in a predetermined volume, and calculates the bubble state of a foam resin molded product; The step B is When the grown bubbles come into contact with each other, it is determined whether the bubbles will coalesce or not based on the surface tension of the bubbles and the viscosity of the liquid phase; and When the center-to-center distance between bubbles becomes smaller than the sum of their radii but the bubbles do not coalesce, the simulation conditions include calculating the travel distance based on the following formulas (1) and (2) using the proportional coefficients, and separating the bubbles that have repelled each other based on the calculation results. (In formulas (1) and (2), r1 is the radius of bubble 1, r2 is the radius of bubble 2, L 12 is the distance between the centers of bubbles 1 and 2, d is the distance traveled, and k is the proportionality coefficient.)

[0028]

number

[0029] This configuration takes into account the possibility that growing bubbles may coalesce into a single bubble when they come into contact with each other, and that bubbles that come into contact may repel each other and separate without coalescing, thereby enabling more precise prediction of the bubble state in a foamed resin product and identifying molding conditions that achieve the desired performance with high accuracy. A proportional coefficient is determined in advance for the above equation that calculates the distance traveled by repelled bubbles, and the proportional coefficient is input to perform a foamed resin molding simulation. When simulating the generation and growth of a large number of bubbles in a given volume, simulating foamed resin molding, the time required to obtain results can be shortened compared to when calculations are performed that also include fluid analysis of the movement of repelled bubbles.

[0030] One aspect of the recording medium disclosed herein is: A computer-readable recording medium storing the above-mentioned bubble growth prediction program for a foamed resin molded product. [Effects of the Invention]

[0031] As described above, according to the present disclosure, a simulation is performed taking into full consideration the coalescence of bubbles during the growth process of the foamed resin, thereby improving prediction accuracy. [Brief explanation of the drawings]

[0032] [Figure 1] 1 is a flowchart showing an outline of a method for predicting bubble growth in a foamed resin molded product. [Figure 2] This is a model diagram showing an example of two bubbles coalescing in a liquid phase. [Figure 3] FIG. 1 shows a simulation of the coalescence of two bubbles in the liquid phase. [Figure 4] FIG. 1 is a model diagram illustrating the contact angle when two bubbles come into contact in a liquid phase. [Figure 5] FIG. 1 is a block diagram illustrating an example of a bubble growth prediction device. [Figure 6] 1 is a flowchart showing an outline of a bubble growth prediction method according to a first embodiment. [Figure 7] 1 is a graph showing the relationship between the distance between bubbles and time for each viscosity of the liquid phase. [Figure 8] 1 is a graph showing the relationship between the distance between bubbles and time for each viscosity of the liquid phase. [Figure 9] 10 is table data showing an example of a data group of surface tension and liquid viscosity of bubbles when the bubbles coalesce. [Figure 10] 1 is a graph of average cell diameter showing a comparison between predicted results and experimental results for Example 1 and Comparative Example. [Figure 11] 1 is a graph of average cell diameter showing a comparison between predicted results and experimental results for Example 1 and Comparative Example. [Figure 12] 1 is a graph of average cell diameter showing a comparison between predicted results and experimental results for Example 1 and Comparative Example. [Figure 13] 1 is a graph of bubble number density showing a comparison between predicted results and experimental results for Example 1 and Comparative Example. [Figure 14] 1 is a graph of bubble number density showing a comparison between predicted results and experimental results for Example 1 and Comparative Example. [Figure 15] 1 is a graph of bubble number density showing a comparison between predicted results and experimental results for Example 1 and Comparative Example. DETAILED DESCRIPTION OF THE INVENTION

[0033] Hereinafter, embodiments of the bubble growth prediction method, bubble growth prediction device, bubble growth prediction program, and recording medium disclosed herein will be described in detail with reference to the drawings. The following description of the preferred embodiments is merely exemplary in nature and is not intended to limit the present disclosure, its applications, or its uses in any way.

[0034] <Method for predicting bubble growth in foamed resin molded products> The bubble growth prediction method of the present disclosure is a method for predicting the bubble state of a foamed resin molded product, which is formed by foaming a foamed resin due to changes in temperature and pressure, by computer simulation. The bubble growth prediction method of the present disclosure is performed, for example, using a bubble growth prediction device described below.

[0035] The foamed resin molded product to be analyzed is not limited to a specific molding method, but examples include molded products manufactured by injection molding. Specific examples include automobile parts, which have heat insulation and sound absorption properties and are used as interior and exterior components.

[0036] A foamed resin is a mixture of a resin and a foaming agent. Well-known thermoplastic resins can be used as the resin. Specific examples of the resin include polypropylene resin, polyethylene resin, polyacetal resin, polyamide resin, polyurethane resin, polystyrene resin, and nylon. These resins may be used alone or in combination of two or more. The foaming agent is not limited as long as it can be used to form a foamed resin, and examples include chemical foaming agents and physical foaming agents.

[0037] 1 is a flowchart showing a method for predicting bubble growth in a foamed resin molded product. As shown in FIG. 1, this analysis method includes a condition determination step S1 and an analysis step S2.

[0038] The condition determination step S1 is a step of determining in advance the conditions necessary for simulating the bubble state of foamed resin molding in a short time in the analysis step S2. The analysis condition determination step S1 includes a proportionality coefficient determination step, a threshold value determination step, a crystallization temperature determination step, and a volumetric shrinkage rate determination step. By including one or more of these determination steps, the analysis time and accuracy are significantly improved compared to performing a simulation in the analysis step S2 that includes those conditions.

[0039] In the analysis step S2, a simulation is performed after reading in the numerical values, thresholds, etc. obtained in the analysis condition determination step S1. In the analysis step S2, a simulation of foamed resin molding is performed assuming that a large number of bubbles are generated and grow in a foamed resin in a predetermined volume, and the bubble state of the foamed resin molded product is calculated. In the analysis step S2, the simulation is performed including a determination of whether or not the grown bubbles will coalesce when they come into contact with each other, based on the surface tension of the bubbles and the viscosity of the foamed resin.

[0040] Specific means for each step will be described below.

[0041] -Proportional coefficient determination process- When two bubbles in a liquid phase approach and come into contact with each other, they may coalesce into a single bubble, or they may repel each other and separate. Figure 2 is a model diagram showing an example of the coalescence of two bubbles in a liquid phase. In Figure 2, r1 is the radius of bubble 1, r2 is the radius of bubble 2, L12 is the center-to-center distance between bubble 1 and bubble 2, and r3 is the size of the passage (throat diameter) connecting the internal space of bubble 1 and the internal space of bubble 2 when the two bubbles coalesce.

[0042] The proportionality coefficient determination process is a process for determining the proportionality coefficient in the formula for calculating the distance traveled by bubbles when two bubbles come into contact and move repulsively without coalescing. The distance traveled by two bubbles in the liquid phase when they come into contact and then move apart repulsively is determined by the radii r1 and r2 of the bubbles and the distance L between the centers of the bubbles. 12and proportionality coefficient k, the calculation can be performed using the following equations (1) and (2). In the proportionality coefficient determination step, the proportionality coefficient in the following equations (1) and (2) is determined to find the moving distance d. The proportionality coefficient is determined by analyzing, based on experiments or simulations, the phenomenon in which two bubbles in a liquid phase approach and come into contact with each other, and then move away from each other by repelling each other without coalescing. The value of the proportionality coefficient is positive when two contacting bubbles are moved in directions away from each other.

[0043]

number

[0044] -Threshold determination process- The threshold determination step is a step of determining a threshold for determining whether two bubbles will coalesce when they come into contact, using a data set of the surface tension of the bubbles and the viscosity of the liquid phase when the two bubbles coalesce. The threshold determination step includes: executing a simulation assuming that two bubbles exist in a liquid phase and that the two bubbles approach and coalesce; outputting a data set of the surface tension of the bubbles (interfacial tension between the bubble and the resin interface) and the liquid phase viscosity when the two bubbles coalesce as a threshold for determining whether the two bubbles will coalesce; and determining a contact angle, which is the angle formed by the tangents of the bubbles at the point of contact when the two bubbles come into contact, as a contact angle, and determining a contact angle threshold for determining whether the two bubbles will coalesce.

[0045] The thresholds for the surface tension and liquid viscosity of bubbles can be determined using general fluid analysis software. Specifically, for example, COMSOL (registered trademark) Multiphysics by COMSOL AB is used to perform a simulation in which two bubbles of a predetermined size exist in a liquid phase, move toward each other, come into contact, and coalesce. FIG. 3 is a diagram showing a simulation of two bubbles coalescing in a liquid phase. Through this simulation, the surface tension and liquid viscosity of the bubbles when the bubbles coalesce are determined as a data group. From the analysis results, a data group of the surface tension and liquid viscosity of the bubbles when the two bubbles coalesce is output as a threshold for determining whether the two bubbles will coalesce. This data group is preferably in the form of table data.

[0046] FIG. 4 is a model diagram illustrating the contact angle when two bubbles 1 and 2 come into contact in a liquid phase. When the two bubbles 1 and 2 come into contact, the angle formed by the tangents L1 and L2 of the bubbles 1 and 2 at the contact point P is defined as the contact angle θ. Depending on the magnitude of the contact angle θ, the two bubbles may or may not coalesce. The threshold value of the contact angle θ for determining whether the two bubbles will coalesce can be set to, for example, 90°. It can be assumed that the two bubbles will coalesce when the contact angle θ is less than 90°, and will not coalesce when the contact angle θ is 90° or greater. The contact angle θ may also be determined by detailed analysis.

[0047] -Crystallization temperature determination process- When the liquid phase reaches the crystallization temperature, crystallizes, and hardens, the growth of the bubbles stops. The crystallization temperature determination process is a process for determining the crystallization temperature of the liquid phase. Specific methods for determining the crystallization temperature of the liquid phase include, for example, a method of measuring using a differential scanning calorimeter or the like based on JIS K7121:2012, and a method of calculating by performing a crystallization simulation. If the physical properties of the resin used as the liquid phase are known, the physical property information of the resin may be referenced in literature, etc.

[0048] -Volumetric shrinkage rate determination process- When the liquid phase reaches the crystallization temperature, crystallizes, and hardens, the volume of the hardened liquid phase shrinks compared to before hardening. The volume shrinkage rate determination step is a step of determining the volume shrinkage rate of the hardened liquid phase. The volume shrinkage rate of the hardened liquid phase can be calculated, for example, by measuring the volumes before and after hardening experimentally and using the following formula: Volumetric shrinkage rate [%] = (volume before curing - volume after curing) / volume before curing x 100 The shrinkage rate can also be determined by software based on the change in density before and after hardening.

[0049] -Analysis process- The analysis step not only simulates foam molding based on a known bubble nucleation model and bubble growth model, assuming that a large number of bubbles are generated and grow in a foamed resin in a predetermined volume, but also includes a simulation condition that determines whether the grown bubbles will coalesce when they come into contact with each other, based on the surface tension of the bubbles and the viscosity of the liquid phase. The simulation, which takes into account bubble coalescence, predicts the growth of bubbles in a foamed resin product with higher accuracy than conventional methods. The analysis results, for example, include the bubble number density and bubble diameter of the foamed resin product, and simulated images of the bubbles.

[0050] For the bubble nucleation model, a known basic theory of bubble nucleation in foaming based on classical nucleation theory can be used. For example, the following equation (3) can be used as the homogeneous nucleation rate equation for bubble generation in a homogeneous medium. In equation (3), γ is the surface tension at the bubble / liquid interface, m is the mass per gas molecule, and k B is the Boltzmann constant, T is the temperature, and P G is the internal pressure of the bubble, P L is the environmental pressure, N A is the Avogadro constant, and c is the gas concentration. The number of bubbles generated per unit time or unit volume can be calculated using the following formula (1).

[0051]

number

[0052] Various bubble growth models have been proposed depending on the assumed manufacturing method and conditions, with the most representative basic models being the Newtonian model and the non-Newtonian model. In both models, a single spherical bubble exists in an infinitely expanding solution of liquid and gas phases, and the bubble grows due to the diffusion and inflow of gas from the liquid phase. Here, we will briefly explain the Newtonian model, in which the liquid and gas phase solutions are Newtonian fluids (constant viscosity). In this model, the following assumptions are made: 1. The bubbles are perfectly spherical. 2. The system is isothermal and Henry's law holds at the gas-liquid interface. 3. The polymer phase is completely homogeneous and incompressible. 4. The gas inside the bubbles is an ideal gas.

[0053] Under the above assumptions, the Newtonian model consists of the following three basic equations (4), (5), and (6): where η is the viscosity of the liquid phase, R is the bubble radius, R' is the gas constant, D is the diffusion coefficient, γ is the surface tension at the bubble / liquid interface, t is time, and P G is the internal pressure of the bubble, P L is the environmental pressure, and c is the gas concentration in the liquid phase. Equation (4) is the equation of motion for the gas-liquid interface, and shows that the bubble diameter changes due to the imbalance between the pressure difference inside and outside the bubble and the interfacial tension, with viscosity acting as resistance to the rate of change. Equation (5) is the mass balance equation for gas inside the bubble, and shows that the amount of gas in the bubble increases as gas diffuses and flows into the bubble from the liquid phase through the bubble interface. Equation (6) is the diffusion equation at the gas-liquid interface, and shows that gas diffuses according to the gas concentration distribution around the bubble.

[0054]

number

[0055] The analysis step performs calculations not only based on this bubble generation and growth model, but also taking into account whether bubbles will coalesce when they come into contact with each other. Whether or not bubbles will coalesce when they come into contact with each other after growth can be determined by analysis based on the surface tension of the bubbles and the viscosity of the liquid phase. By reading the numerical values, thresholds, etc. obtained in the analysis condition determination step S1 and then executing the analysis step S2, calculations that take into account bubble coalescence can obtain more accurate results and analysis results in a shorter time.

[0056] In the analysis step S2, which is based on the threshold determination step as part of the analysis condition determination step S1, a simulation is performed after reading in the threshold values ​​for the data group and the contact angle threshold value. This analysis step S2 includes, as a simulation condition, a condition that bubbles coalesce when the surface tension and liquid viscosity of the bubbles exceed the surface tension and liquid viscosity threshold values ​​when the bubbles come into contact with each other, and when the contact angle between the bubbles exceeds the contact angle threshold value. If the analysis step, which performs calculations on a large number of bubbles in a predetermined volume, is performed without setting a threshold value, analysis of coalescence determination based on the surface tension and liquid viscosity of all bubbles will be performed, which may require a long time to obtain analysis results. By taking the threshold determination step into account, it is possible to not only improve the accuracy of the analysis results but also significantly shorten the analysis time.

[0057] In the analysis step S2 based on the proportional coefficient determination step as the analysis condition determination step S1, first, the center-to-center distance L between the bubbles is calculated. 12 When the sum of the radii of the bubbles (r1 + r2) is reached, it is determined whether the bubbles will coalesce based on the surface tension of the bubbles and the viscosity of the liquid phase. Then, at a certain calculation time, the center-to-center distance L12 between the bubbles becomes smaller than the sum of the radii of the bubbles (r1 + r2) and the distance (L 12When it is determined that the bubbles do not coalesce even though (r1 + r2) and the bubbles have bitten into each other, the moving distance d is calculated based on the above formulas (1) and (2) using the proportionality coefficient k determined in the proportionality coefficient determination step, and the simulation condition includes that the repelled bubbles move away from each other by the moving distance d. For example, the larger the bite when the bubbles contact each other and the smaller L 12 is, the larger the moving distance d becomes.

[0058] In the analysis step S2 based on the crystallization temperature determination step as the analysis condition determination step S1, the simulation is executed after reading the crystallization temperature of the liquid phase. This analysis step S2 includes, as simulation conditions, that a large number of bubbles are generated and grow in the liquid phase while the temperature and pressure change over time, and the growth of the bubbles stops when the liquid phase temperature reaches the crystallization temperature. In the analysis step S2 based on the crystallization temperature determination step as the analysis condition determination step S1, in particular, it is possible to improve the prediction accuracy of the bubble diameter of the foamed resin molded product. In the conventional bubble growth prediction method, the analysis is performed on the assumption that a certain amount of dissolved gas exists in the liquid phase and the growth of the bubbles stops when all of it is consumed. However, by performing the analysis so that the growth of the bubbles stops depending on the crystallization temperature, it is possible to cope with cases where the amount of dissolved gas in the liquid phase increases or decreases due to chemical reactions or the like, and it becomes possible to predict with higher accuracy.

[0059] In the analysis step S2, which is based on the volumetric shrinkage rate determination step as part of the analysis condition determination step S1, a simulation is performed after reading in the volumetric shrinkage rate of the liquid phase. This analysis step S2 includes the following simulation conditions: a large number of bubbles are generated and grow in the liquid phase as the temperature and pressure change over time; when the liquid phase reaches a predetermined temperature, the volume of the liquid phase is reduced based on the volumetric shrinkage rate; and the distance between the bubbles is shortened based on the reduced volume. When the analysis step S2 is performed based on the volumetric shrinkage rate determination step and the crystallization temperature determination step as part of the analysis condition determination step S1, the predetermined temperature can be the crystallization temperature. The predetermined temperature is not limited to the crystallization temperature; multiple temperatures for the volumetric shrinkage rate determination step may be determined, and the shrinkage of the liquid phase may be calculated in stages. In the analysis step S2, which is based on the volumetric shrinkage rate determination step as part of the analysis condition determination step S1, the accuracy of predicting the bubble number density of a foamed resin molded product can be particularly improved.

[0060] It is possible to improve prediction accuracy by performing any one of the proportionality coefficient determination process, the crystallization temperature determination process, and the volume shrinkage rate determination process before the analysis process S2 and reflecting the results in the analysis process S2, but it is also possible to significantly shorten the analysis time by taking the threshold determination process into consideration.

[0061] <Device for predicting bubble growth in foamed resin molded products> Fig. 5 shows an example of the configuration of a bubble growth prediction device 1 for a foamed resin molded product according to this embodiment (hereinafter also referred to as "prediction device 1"). The prediction device 1 is a device that predicts bubble growth in a foamed resin molded product through computer simulation using the bubble growth prediction method described above. Note that the prediction device 1 is merely one example of a bubble growth prediction device for a foamed resin molded product according to the present disclosure, and the configuration of the bubble growth prediction device is not limited to this example.

[0062] The prediction device 1 includes a computer 10, an input unit 20, and an output unit 30. The analysis device 1 is a CAE (Computer Aided Engineering) system that basically includes the computer 10. The computer 10 includes a calculation unit 11, a storage unit 12, a reading unit 13 for acquiring information stored in various recording media 100, and the like.

[0063] The calculation unit 11 is, for example, a CPU, and controls the entire computer 10 and executes various programs. Based on information stored in the memory unit 12, information input via the input unit 20, and information acquired from the recording medium 100 via the reading unit 13, the calculation unit 11 analyzes the behavior of bubbles in the liquid phase, such as bubble movement, bubble contact and coalescence, bubble generation and growth, and bubble contact and repulsion, and can predict bubble growth in a foamed resin molded product. As a result of the bubble growth prediction, for example, the bubble number density and bubble diameter of the foamed resin molded product can be calculated.

[0064] The storage unit 12 is, for example, a ROM, a RAM, a hard disk, etc., and stores the calculation formulas and algorithms used in the calculation process by the calculation unit 11. The calculation formulas are, for example, the above formulas (1) to (5), etc.

[0065] The input unit 20 is an input device for inputting and setting various analysis conditions in response to, for example, the operation of a designer. The analysis conditions include, for example, a proportionality coefficient used in a calculation formula, a threshold value, a crystallization temperature, a volumetric shrinkage rate, a type of resin, and molding conditions for foam resin molding. The input unit may be, for example, a keyboard, a mouse, a touch panel, or the like.

[0066] The output unit 30 outputs the growth prediction of the foamed resin molded product obtained by the calculation unit 11. The output unit 30 is not limited to, but may be, for example, a display device such as a display that outputs visual information to a designer. The output unit 30 enables the designer to grasp the growth state of bubbles in the internal structure of the foamed resin molded product as numerical information such as the bubble number density and bubble diameter, or visual information such as a three-dimensional simulation image.

[0067] <Program for predicting bubble growth in foamed resin molded products and its recording medium> At least some of the steps of the method for predicting bubble growth in a foamed resin product are programmed as a bubble growth prediction program for a foamed resin product. That is, the bubble growth prediction program for a foamed resin product according to this embodiment is a program that causes a computer to execute at least the following steps: (a) analysis condition determination procedure A, which determines in advance the conditions necessary for quickly simulating the bubble state of a foamed resin product in the analysis step S2; and (b) simulation of a foamed resin product assuming that a large number of bubbles are generated and grow in a foamed resin in a predetermined volume, thereby calculating the bubble state of the foamed resin product. Note that step A includes a proportionality coefficient determination procedure A1, a threshold value determination procedure A2, a crystallization temperature determination procedure A3, and a volumetric shrinkage rate determination procedure A4. Step A includes one or more of steps A1 to A4. This bubble growth prediction program is stored in, for example, the memory unit 12 and can be executed by the calculation unit 11. Furthermore, the bubble growth prediction program is not limited to being stored in the storage unit 12, but can be recorded on various well-known computer-readable recording media 100, such as optical disk media and magnetic tape media. By loading such a recording medium into the reading unit 13 and reading out the bubble growth prediction program, the program can be executed.

[0068] <Example> Next, specific examples will be described. This analysis method is a method for predicting bubble growth in foamed resin molded products using the finite element method through computer simulation, and is performed using, for example, the above-mentioned analysis device 1. In the examples, calculations were performed using polypropylene as the resin and CO2 as the blowing agent. Note that, for comparison with the comparative examples and experimental examples described below, the values ​​obtained in the experimental examples and the parameters used in the comparative examples were used. The calculation formulas required to calculate these parameters are publicly known, and since the formulas described in Non-Patent Document 3 were used, detailed explanations will be omitted. Furthermore, since the calculation formulas used in the analysis process are also those described in Non-Patent Document 3, only a brief explanation will be given and details will be omitted.

[0069] Example 1 Fig. 6 is a flowchart showing an outline of the bubble growth prediction method of Example 1. First, as the analysis condition determination step S1, a proportionality coefficient determination step S11, a threshold value determination step S12, and a volumetric shrinkage rate determination step S13 were performed. Note that the proportionality coefficient determination step S11, the threshold value determination step S12, and the volumetric shrinkage rate determination step S13 may be performed in any order. Fig. 6 is a simplified flowchart for the sake of convenience, and the above steps are not necessarily performed in the order shown in Fig. 6.

[0070] In the proportionality coefficient determination step S11, a value for the proportionality coefficient was considered that would reduce the difference from the experimental results of the experimental example described below. The proportionality coefficient was determined to be 0.001.

[0071] In the threshold determination step S12, a simulation was performed using the two-phase laminar flow interface in COMSOL® Multiphysics (manufactured by COMSOL AB) under compressible fluid and gravity-free conditions. Two bubbles of a given size were present in a liquid phase, and the bubbles moved toward each other, came into contact, and coalesced. The surface tension of the bubbles and the viscosity of the liquid phase were calculated as a data set. Figures 7 and 8 are graphs showing the relationship between the distance between two bubbles and time for each viscosity of the liquid phase. Figure 7 shows a viscosity sweep for a surface tension of 10 mN / m, and Figure 8 shows a viscosity sweep for a surface tension of 100 mN / m. In Figures 7 and 8, line L represents the boundary between whether or not two bubbles will coalesce. The two bubbles coalesce when moving toward the origin of the graph from line L, while the two bubbles do not coalesce when moving away from the origin of the graph from line L. From the analysis results, a data set of the surface tension and liquid viscosity of the bubbles when the two bubbles coalesce is output as a threshold value for determining whether the two bubbles will coalesce. This data set is preferably in the form of a table data such as that shown in Fig. 9. The contact angle threshold value was determined to be 90°.

[0072] In the volumetric shrinkage rate determination step S13, the shrinkage rate was determined on the software from the change in density of the polypropylene before and after curing. If the density and volume of the resin before curing are ρ,V, and the density and volume of the resin after curing are ρ',V', then according to the law of conservation of mass, ρV = ρ'V'. Therefore, the volumetric shrinkage rate is V' / V = ρ / ρ'. The volumetric shrinkage rate was calculated by inputting ρ, ρ', and the curing temperature into the software.

[0073] The input information was the proportionality coefficient obtained in the analysis condition determination step S1, the table data of the threshold values ​​of the bubble surface tension and liquid viscosity, the contact angle threshold value, and the volume shrinkage rate, as well as information necessary for simulating the generation and growth of bubbles in the liquid phase. The information necessary for simulating the generation and growth of bubbles in the liquid phase is, for example, parameters necessary for using the bubble generation rate equation and bubble growth rate equation, and the values ​​described in Non-Patent Document 3 (see Non-Patent Document 3, pp. 39-42, pp. 60-63) were used.

[0074] In the examples, the analysis step S2 was performed on a Linux machine and a Windows machine with Cygwin installed, with the source code written in C and the compiler gcc. The simulation used individual bubbles in a liquid phase at a given volume as calculation elements, and calculated the bubble generation and growth, as well as the contact and coalescence with other bubbles, using a time integration method with a fixed time step. The analysis step S2 used the calculation model described in Non-Patent Document 3, and the simulation flow was constructed based on the simulation flow (see Non-Patent Document 3, pp. 53-59, 63-65).

[0075] The analysis step S2 includes a bubble generation process S21, a bubble growth process S22, a contact / coalescence start process S23, a coalescence determination process S24, a coalescence process S25, a non-coalescence process S26, a liquidus temperature determination process S27, a volume shrinkage process S28, and a dissolved gas determination process S29.

[0076] Specifically, the bubble generation process S21 and the bubble growth process S22 were calculated using a known calculation method by Sun et al. (Non-Patent Document 3). The calculation method in Non-Patent Document 3 simultaneously handles the generation and growth of multiple bubbles, and takes into account the concentration boundary layer in the gas concentration distribution formed around the bubbles. During the foaming process, as bubble nucleation and growth progress, if the concentration boundary layers of adjacent bubbles overlap, the gas concentration distribution changes, which affects the bubble nucleation rate and growth rate, and the phenomenon of concentration boundary layer overlap is taken into account.

[0077] In the bubble generation process S21, the bubble generation rate is calculated by dividing it into homogeneous nucleation and heterogeneous nucleation, and the homogeneous nucleation rate J hom The heterogeneous nucleation rate J is calculated using the following equation (7) by Blander and Katz (M. Blander and J. Katz, Bubble Nucleation in Liquids. AIChE Journal, 21 (1975) 833-848). he(JS Colton and NP Suh, The nucleation of microcellular thermoplastic foam with additives: Part I: Theoretical considerations, Polymer Engineering & Science, 27 (1987), 485-492) (JS Colton and NP Suh, The nucleation of microcellular thermoplastic foam with additives: Part II: Experimental results and discussion, Polymer Engineering & Science, 27 (1987), 493-499) (JS Colton and NP Suh, Nucleation of Microcellular Foam, Theory and Practice, Polymer Engineering & Science, 27 (1987), 500-503) were used and integrated into (10). In the following equations (7) to (10), γ is the surface tension, m is the mass per gas molecule, k B is the Boltzmann constant, T is the temperature, and P G is the internal pressure of the bubble, P L is the environmental pressure, N is the number of molecules, f1 is the frequency factor, c is the gas concentration in the liquid phase, and θ is the contact angle.

[0078]

number

[0079] The number of bubbles to be generated up to the current step was calculated from the bubble nucleation rate obtained from the above calculation. The following process was repeated for ungenerated bubbles until that number was reached. The created coordinates were stored in an array variable. The bubble radius was set to the radius of the initially generated bubble. The pressure inside the bubble was calculated, and the surface area and volume were calculated. The number of moles of gas inside the bubble was calculated and subtracted from the number of moles of dissolved gas. The number of generated bubbles was set to +1.

[0080] In the bubble growth process S22, the bubble growth rate was calculated using a bubble growth model in a Newtonian fluid by Barlow and Langlois (EJ Barlow and WE Langlois, Diffusion of gas from a liquid into an expanding bubble, IBM J. 6 (1962), 329-337). The equation of motion at the gas-liquid interface was given by Equation (11), and the physical property balance equation for the gas inside the bubble was given by Equation (12). Diffusion at the gas-liquid interface was calculated based on a simple equation (13) by Payver (P. Payvar, Mass transfer-controlled bubble-growth during rapid decompression of a liquid, International J. Heat and Mass Transfer 30 (1987) 699-706). In Equations (11) to (13), η is the viscosity of the liquid phase, R is the bubble diameter, R' is the gas constant, D is the diffusion coefficient, and c ∞ ,c R are the gas concentrations at infinity and near the gas-liquid interface, respectively, and δ is the thickness of the concentration boundary layer.

[0081]

number

[0082] The thickness of the concentration boundary layer and the gas concentration at the gas-liquid interface were calculated using the above formula, as well as the amount of gas transferred from the liquid phase to the bubble. If the amount of transfer was one molecule or more, the following processing was performed: the number of moles in the bubble was updated (added), the number of dissolved gas moles transferred into the bubble was subtracted, the volume was calculated from the bubble's current true radius, the internal pressure of the bubble was calculated, the equation of motion for the bubble diameter was calculated to calculate the rate of change of the bubble radius, the true radius was updated by the rate of change of the bubble radius multiplied by the time step, the surface area of ​​the bubble was updated, the bubble volume was updated, and the difference from the original volume was tallied. In addition, the diffusion coefficient for bubble movement was calculated using the Einstein-Stokes equation, and the amount of movement per step was calculated from that diffusion coefficient. Random orientations were created in all directions in a spherical coordinate system, and the bubble was moved by converting it to an xyz coordinate system and converting it into partial coordinates.

[0083] In the contact / coalescence initiation processing step S23, processing was performed when bubbles come into contact and begin to coalesce. First, a variable was initialized to aggregate the amount of movement of each bubble in the original process. For pairs of bubbles, the distance between the bubble centers was calculated, and the periodic boundary conditions were processed and converted to real coordinates. The square of the distance and the distance were calculated, and other variables were prepared. Three forms of separation, inclusion, and contact were distinguished from the distance between the bubbles and their respective radii. In the case of contact, a contact flag was set and the contact angle was calculated.

[0084] In the coalescence determination process S24, it was determined whether or not the contacting bubbles would coalesce. The coalescence of the bubbles was determined based on thresholds for the surface tension, liquid phase viscosity, and contact angle of the bubbles. If the surface tension, liquid phase viscosity, and contact angle of the contacting bubbles all exceeded their thresholds, it was determined that the bubbles would coalesce, and the process proceeded to the coalescence process S25. If any of the surface tension, liquid phase viscosity, and contact angle of the contacting bubbles did not exceed their thresholds, it was determined that the bubbles would not coalesce, and the process proceeded to the non-coalescence process S26.

[0085] In the coalescence process S25, it is assumed that coalescence of bubbles progresses, and calculations are performed separately for the bubbles after coalescence (coalesced bubbles) and the bubbles before coalescence (coalesced bubbles).

[0086] Calculation of coalescing bubbles If the radius could not be determined immediately after coalescence, the initial bubble radius was set. The bubble volume calculated from the current bubble diameter was subtracted from the tabulated variable. The number of moles coalescing per step was calculated from the bubble coalescence rate and added to the number of moles of bubbles. The new bubble diameter was calculated by convergence calculation. The volume was calculated from the new radius and added to the tabulated variable.

[0087] Calculations for combined energetic bubbles A pair of bubbles (a, b) from which the original bubbles merged into the aforementioned target bubble was searched for. The number of moles of both bubbles was added. For the merged bubble a, the volume was calculated from the new radius and subtracted from the tabulated variable, and the number of moles was subtracted based on the number of moles of both bubbles. If the number of moles was one molecule or less, the bubble was eliminated; otherwise, the new radius and new surface area were calculated and the volume was added to the tabulated variable. For the merged bubble b, the volume was calculated from the new radius and subtracted from the tabulated variable. The number of moles was subtracted based on the molar ratio of both bubbles. If the number of moles was one molecule or less or bubble a had disappeared, the bubble was eliminated; otherwise, the new surface area was calculated and the volume was added to the tabulated variable.

[0088] When the radii of both bubbles that merged disappeared, the process of terminating the merger was performed.

[0089] In the anti-coalescence process S26, the volume of each bubble collapsed due to contact was calculated. The temperature gradient at the midpoint between the two bubbles was calculated, and if a temperature gradient existed, the temperature, viscosity, and surface tension were modified as follows. The film thickness between the bubbles was calculated, and the liquid volume of the film portion was calculated. The film thickness and film diameter were saved in the contact information array variable. The volume of each bubble was reduced by the amount hidden by contact. If a repulsion coefficient was set, the bubble was moved. The amount of movement was calculated as the overlap distance multiplied by the proportionality coefficient in equation (6) above. When the proportionality coefficient was positive, the contacting bubbles were moved in directions away from each other. The aggregate of the above contact movement amounts was reflected in the coordinates of each bubble. In addition, the apparent radius was changed to match the volume increase due to the above contact.

[0090] If the liquidus temperature determination process S27 determined that the current density was lower than the density at the time of crystallization and the current temperature was higher than the crystallization temperature, the volume shrinkage process S28 was performed. Each process in the analysis step S2 was repeated until the requirements for starting the volume shrinkage process S28 were met. In the volume shrinkage process S28, the new density was first calculated by adding the current density multiplied by the density change rate and the time step. The difference between the new volume and the current volume was added to the volume change summary variable. The current volume was calculated by dividing the initial weight by the current density. The new volume was calculated by dividing the initial weight by the new density. The calculated cell volume was corrected based on this summary of the volume change. The current volume was calculated from the cell size. Different processes were performed depending on the cell size fixation flag. 1. When the x and y axes are fixed: Calculate the shrinkage rate (add the total to the current volume and divide by the current volume) and multiply it by the z-axis size 2. When the z-axis is fixed: Calculate the shrinkage rate in the same way as above, take the square root, and multiply it by the x-axis size and y-axis size, respectively. 3. If not fixed: Calculate the shrinkage rate as above, take the square root, and multiply it by each of the x-axis, y-axis, and z-axis sizes. In the dissolved gas determination process S29, the number of moles of dissolved gas in the liquid phase is confirmed, and if the number of moles is negative, it is assumed that dissolved gas no longer exists, and the process of the analysis process S2 is terminated. The analysis process S2 was repeated until dissolved gas no longer exists.

[0091] When the analysis step S2 was completed, the cell size, concentration, number of bubbles, average bubble diameter, and bubble number density were output as the analysis results.

[0092] As described above, in the example, when performing a simulation taking into account bubble coalescence, the conditions necessary for bubble coalescence were determined in advance in the analysis condition determination step, and simulation results were obtained within one day. If a simulation were performed that also included fluid analysis taking into account bubble coalescence, it would likely take several days to obtain results, so it can be said that a dramatic reduction in calculation time has been achieved.

[0093] <Comparative Example> The results of a conventional bubble growth prediction method were used as a comparative example. This comparative example does not include an analysis condition determination step like the bubble growth prediction method of the present disclosure, and does not consider the coalescence of bubbles. For this comparative example, the calculation results described in Non-Patent Document 3 above were used (see Non-Patent Document 3, p. 70, Fig. 4-6-3). In the results of Non-Patent Document 3, the horizontal axis is expressed as the difference between saturation pressure and ambient pressure, which corresponds to the passage of time. Therefore, the horizontal axis was converted to time and superimposed on the results of Example 1 (Figs. 10 to 15).

[0094] <Experimental Example> Next, to verify the correlation between the bubble growth prediction methods of the Examples and Comparative Examples and the experimentally obtained values, a comparison was made with the experimental results. The actual measured values ​​of the average bubble diameter and bubble number density described in Non-Patent Document 3 (see Non-Patent Document 3, pp. 29-33, p. 70 Fig. 4-6-3) were used as the experimental example values ​​and were superimposed on the results of Example 1 (Figs. 10 to 15).

[0095] -Comparison results- Figures 10 to 15 are graphs comparing the results of Example 1, the Comparative Example, and the Experimental Example. Figures 10 to 12 show the average bubble diameter, and Figures 13 to 15 show the bubble number density. Figures 10 and 13 show the results at a temperature of 473 K, a saturation pressure of 10 MPa, and a depressurization rate of 0.75 MPa / s. Figures 11 and 14 show the results at a temperature of 473 K, a saturation pressure of 15 MPa, and a depressurization rate of 0.75 MPa / s. Figures 12 and 15 show the results at a temperature of 473 K, a saturation pressure of 20 MPa, and a depressurization rate of 0.75 MPa / s. The results of the Comparative Example showed a tendency for the average bubble diameter to differ significantly from the experimental value over time, while the results of Example 1 showed a good correlation with the experimental values ​​for both the average bubble diameter and the bubble number density. When the error from the experimental value was calculated, Example 1, which took into account bubble coalescence, had an error of 13%, while the Comparative Example, which did not take into account bubble coalescence, had an error of as much as 61%. It has been confirmed that the prediction method of the present disclosure, which predicts bubble growth by taking into account the coalescence of bubbles, can achieve significantly improved accuracy compared to conventional prediction methods.

[0096] <Example 2> In Example 1, the analysis was performed using a proportionality coefficient of 0.001 in the proportionality coefficient determination step S11, while in Example 2, the analysis was performed using a proportionality coefficient of 0.1. Other analysis conditions were the same as in Example 1. When the results of Example 2 were applied to Figures 10 to 15, they nearly overlapped with the results of Example 1. One reason for the lack of a significant difference in the results between Example 1 and Example 2 is that in this embodiment, a resin used for sound-absorbing materials or heat insulating materials was assumed as the foamed resin. This is thought to be because the analysis was performed on a foamed resin molded product with a high expansion ratio and a large number of bubbles, so the distance traveled by contacting bubbles due to repulsion was short, making it difficult to see the effect of determining the proportionality coefficient and taking the repulsion movement distance into account. Under analysis conditions assuming a foamed resin with a lower expansion ratio, a significant difference occurred in the calculation results between cases where the repulsion movement of contacting bubbles was considered and cases where it was not considered. This suggests that there is sufficient significance in performing the analysis step via the proportionality coefficient determination step.

[0097] <Comparative Example 2> In Example 1, a volumetric shrinkage rate determination step was included, and the inter-cell distance was adjusted taking into account the shrinkage rate of the liquid phase after curing. However, in Comparative Example 2, an analysis was performed without including the volumetric shrinkage rate determination step. Other analysis conditions were the same as in Example 1. When the results of the Comparative Example were applied to FIGS. 10 to 15, they nearly overlapped with the results of Example 1. In this embodiment, a resin used for sound-absorbing materials or heat insulating materials was assumed as the foamed resin. Because the analysis was performed on a foamed resin molded product with a high expansion ratio and a large number of cells, even when the shrinkage rate was considered, the amount of change in the liquid phase was small, making it difficult to see the effect. Under analysis conditions assuming a foamed resin with a lower expansion ratio, the volume of the liquid phase increases, and the amount of change is also large. Therefore, there is a significant difference in the calculation results between cases where the shrinkage of the liquid phase is considered and cases where it is not considered. Therefore, it is considered that there is sufficient significance in performing the analysis step via the volumetric shrinkage rate determination step. [Explanation of symbols]

[0098] 1. Bubble growth prediction device for foamed resin molded products 10. Computers 11 Arithmetic section 12 Storage section 13 Reading unit 20 Input section 30 Output section 100 Recording Media S1 Analysis condition determination process S2 analysis process S11 Proportionality coefficient determination process S12 Threshold determination process S13 Volumetric shrinkage rate determination process S21 Bubble generation treatment S22 Bubble growth treatment S23 Contact / Merge Start Processing S24 Combination determination process S25 Combination Processing S26 Non-coalescence processing S27 Liquid phase temperature determination process S28 Volumetric shrinkage treatment S29 Dissolved gas determination process

Claims

1. A method for predicting the bubble state of a foamed resin molded product obtained by foaming a foamed resin through computer simulation, comprising: a proportionality coefficient determination step for determining the proportionality coefficient of the following equations (1) and (2), which represent the distance traveled when two bubbles come into contact with each other in a liquid phase and then repel each other and move away from each other, based on an experiment or simulation using two bubbles in a liquid phase, using the following equations (1) and (2), which represent the proportionality coefficient in terms of the radius of the bubbles, the distance between the centers of the bubbles, and the proportionality coefficient; an analysis step of executing a simulation of foam resin molding assuming that a large number of bubbles are generated and grow in a liquid phase in a predetermined volume, and calculating the bubble state of the foam resin molded product; The analyzing step When the grown bubbles come into contact with each other, it is determined whether the bubbles will coalesce or not based on the surface tension of the bubbles and the viscosity of the liquid phase; and When the center-to-center distance between bubbles becomes smaller than the sum of the bubble radii but the bubbles do not coalesce, the simulation conditions include calculating the travel distance based on the following formulas (1) and (2) using the proportional coefficient, and separating the bubbles that have repelled each other based on the calculation result. A method for predicting bubble growth in foamed resin molded products. [Equation 1] (However, in formulas (1) and (2), r １ is the radius of bubble 1, r ２ is the radius of bubble 2, L １２ is the distance between the centers of bubbles 1 and 2, d is the distance traveled, and k is the proportionality coefficient.)

2. In claim 1, a threshold determination step, which includes, before the analyzing step, executing a simulation assuming that two bubbles are present in a liquid phase and that the two bubbles approach and coalesce, calculating a group of data on the surface tension and liquid viscosity of the bubbles when the two bubbles coalesce, and outputting the group of data as thresholds of the surface tension and liquid viscosity for determining whether the two bubbles will coalesce or not; and determining a contact angle, which is the angle formed by the tangents of the bubbles at the contact point when the two bubbles come into contact, for determining a contact angle threshold for determining whether the two bubbles will coalesce or not; The analyzing step includes, as a simulation condition, a condition that, after reading the threshold values ​​of the surface tension and the liquid phase viscosity and the contact angle, the bubbles coalesce when the surface tension and the liquid phase viscosity of the bubbles exceed the threshold values ​​of the surface tension and the liquid phase viscosity when the grown bubbles come into contact with each other, and the contact angle between the bubbles exceeds the threshold value of the contact angle. A method for predicting bubble growth in foamed resin molded products.

3. In claim 2, The data group is a table of surface tension and liquid viscosity of bubbles when two bubbles coalesce, and the analysis step reads the table data and then executes a simulation. A method for predicting bubble growth in foamed resin molded products.

4. In any one of claims 1 to 3, The analysis step calculates the bubble generation rate based on the following formula (3), and calculates the bubble growth rate based on the following formulas (4) to (6). [Equation 2] (In equation (3), γ is the surface tension at the bubble / liquid interface, m is the mass per gas molecule, k B is the Boltzmann constant, T is the temperature, and P Ｇ is the internal pressure of the bubble, P Ｌ is the environmental pressure, N Ａ is the Avogadro constant, and c is the gas concentration in the liquid phase.) [Equation 3] (In equations (4) to (6), η is the viscosity of the liquid phase, R is the bubble radius, r is the radial position within the bubble, R' is the gas constant, D is the diffusion coefficient, γ is the surface tension at the bubble / liquid interface, t is time, P Ｇ is the internal pressure of the bubble, P Ｌ is the ambient pressure, and c is the gas concentration in the liquid phase.

5. An apparatus for predicting the bubble state of a foamed resin molded product formed by foaming a foamed resin through computer simulation, At least one calculation unit is provided, The calculation unit The distance traveled by two bubbles in a liquid phase when they come into contact with each other and then repel each other is expressed in terms of the radius of the bubbles, the distance between the centers of the bubbles, and a proportionality coefficient, and the proportionality coefficient is determined based on an experiment or simulation using two bubbles in a liquid phase and inputted into the following equations (1) and (2): In calculating the bubble state of a foamed resin molded product by performing a foamed resin molding simulation assuming that a large number of bubbles are generated and grow in a liquid phase in a predetermined volume, When the grown bubbles come into contact with each other, it is determined whether the bubbles will coalesce or not based on the surface tension of the bubbles and the viscosity of the liquid phase; and When the center-to-center distance between bubbles becomes smaller than the sum of their radii but the bubbles do not coalesce, the device calculates the travel distance based on the following equations (1) and (2) using the proportional coefficient, and includes in the simulation conditions that the bubbles that have repelled each other based on the calculation result will move away from each other. [Equation 4] (However, in formulas (1) and (2), r １ is the radius of bubble 1, r ２ is the radius of bubble 2, L １２ is the distance between the centers of bubbles 1 and 2, d is the distance traveled, and k is the proportionality coefficient.)

6. A program for analyzing, by computer simulation, the state of bubbles in a foamed resin molded product formed by foaming a foamed resin, Step A involves inputting a proportionality coefficient determined based on an experiment or simulation using two bubbles in a liquid phase into the following equations (1) and (2), which express the distance traveled when two bubbles come into contact with each other in a liquid phase and then repel each other and move away from each other, using the bubble radius, the distance between the bubble centers, and a proportionality coefficient: and step B, which causes a computer to execute a simulation of foamed resin molding assuming that a large number of bubbles are generated and grow in a liquid phase in a predetermined volume, and calculates the bubble state of a foamed resin molded product; The procedure B is When the grown bubbles come into contact with each other, it is determined whether the bubbles will coalesce or not based on the surface tension of the bubbles and the viscosity of the liquid phase; and When the center-to-center distance between bubbles becomes smaller than the sum of their radii but the bubbles do not coalesce, the simulation conditions include calculating the travel distance based on the following formulas (1) and (2) using the proportional coefficients, and separating the bubbles that have repelled each other based on the calculation results. A program to predict bubble growth in foamed resin molded products. [Equation 5] (However, in formulas (1) and (2), r １ is the radius of bubble 1, r ２ is the radius of bubble 2, L １２ is the distance between the centers of bubbles 1 and 2, d is the distance traveled, and k is the proportionality coefficient.)

7. 7. A computer-readable recording medium having recorded thereon the program for predicting bubble growth in a foamed resin molded product according to claim 6.

Images