不同的材料和操作条件的范围内的模具挤出电线包覆的设.doc

不同的材料和操作条件的范围内的模具挤出电线包覆的设计与实验验证的一种优化方法1. N. Lebaal 1，*，2. 2 S.强力，3. 3 FM施密特，4. D.Schläfli 4高分子工程与科学第52卷， 第12期， 2675页至2687页，2012年12月抽象这篇文章的目的是要确定的导线涂层衣架熔体分配器几何形状，以确保均匀的出口速度分布，将最适应较宽的材料范围内和多个操作条件（即，模具壁的温度和流速的变化）。计算方法采用有限元（FE）分析，以评估性能的模具设计，包括了克里金插值和序列二次规划算法更新模具的几何非线性约束优化算法。两个优化问题的解决，最好的办法是考虑到生产的最佳分销商。Taguchi方法是用来研究的效果的操作条件下，即，熔融和模壁的温度，流速和材料变化，上的速度分布的最佳模。在所选择的例子中，通过考虑由刀具几何形状的几何限制的导线涂覆模具的几何形状进行了优化。最后获得最佳的模具与实验数据的比较，有限元分析和优化结果进行了验证。下面所描述的实验的目的是调查的材料变化的效果。高分子。ENG。SCI，2012年。©2012年塑料工程师协会简介衣架熔体分配器（图1）是常用的在电线包覆过程。它的任务是在导 体周围均匀的熔体分配。平衡流量通过一个模具，实现了整个模具出口的速度分布均匀的分布是挤压模的设计的最困难的任务之一。图1。衣架熔体经销商。对于的聚合物挤出行业中，最具挑战性和挑战性的工作是探讨如何减少甚至消除模具校正。在一般情况下，增加芯片土地的查询结果，在显着的流动阻力，其效果是改善最终的熔体分配的长度。然而，这增加土地的长度可能迅速导致模头的压降的过度增加。甲夹紧酒吧更新也可以优化 1 ，得到均匀的模具出口处的速度。但是，使用此夹紧栏也导致模头的压降，这可能导致的模体偏转的增加。因此，信道的几何形状（歧管）的衣架型模头应在这样一种方式，在模具出口的速度分布均匀而不过分提高模头的压降得到优化。的聚合物挤压模具的设计是复杂的，树脂的粘度和剪切速率之间的非线性关系。到模具中，使以得到均匀的速度，流过的分布是一个函数的总吞吐量，因此，该树脂的剪切稀化的功能和散热。计算机模拟的挤出过程中，必须考虑到该聚合物的非线性材料行为，并准确地预测在模具内的压力和温度分布。挤压模具的性能取决于，除其他外，在流路的设计和操作条件下，通过在挤出过程中 2，3 。这可能会导致材料具有非常不同的流变学特性的设计材料相比，性能降低到不可接受的水平的问题。Chen等人的。 4 表明，使用田口方法的操作条件下，材料的变化和模具的几何形状，在模具出口的速度分布上有很大的影响。王等人 5 研究歧管角的效果和歧管的横截面的轮廓上的流量分布的衣架型模头，利用三维有限元（FE）与等温流的假设和幂律流体的软件。实验的设计也被用来由尤尼斯等人研究的效果在聚合物挤出工艺参数。 6 。他们使用的统计的方法，使用一个阶乘实验设计流变机制提供的描述，通过数学的相互作用，和研究中，聚合物熔体流动指数和挤压温度对晶体的形状和尺寸 7  的效果。卡内罗等。 8 研究了不同挤压条件下的矩形聚丙烯配置文件的效果。田口实验设计用于确定最相关的处理变量。他们的结论是确定的挤压型材的机械性能是最显着的处理变量的挤出温度。挤压的铝挤压型材的流动平衡和温度演变过程参数的效果已经由Bastani等研究。 9 。作者通过选择工艺参数的适当组合一个二维模型中的出口速度和出口温度的径向变化最小，并得出结论，最小化的出口温度和速度，可以导致在温度和速度的均匀性下降的交叉部所产生的部分。在不同的聚合物的流变学的多样性也需要个别优化每种聚合物的模具。聚合物和模具通道几何的组合通常需要额外的设备，如夹紧列 10 。在这种情况下，可以使用的试验和错误的方法，以获得均匀的模具出口处的速度。的聚合物的流变学这种复杂性进一步提高模具的优化问题的难度。如果聚合物流变学还没有考虑准确地优化模具的同时，预测的速度，压力和温度场预计将有较大的误差，这可能导致在非最佳的模具设计。然而，在理论上是可能的设计的模具中的流动分布是独立的流动性能，特别是，独立的剪切稀化的程度。冬季和Fritz  11 ，提出了一个理论，衣架模具的设计，圆形或矩形截面分水器。对于一个给定的纵横比（高度/宽度）的歧管，该理论预测材料独立的流动分布。然而，Lebaal等。 12 显示使用三维仿真软件，和实验验证，在实践中可能不是最优的，通过该方法得到的分布。Smith等人 13 优化的扁平模头设计，操作以及在多点温度。作者表明，出口速度分布的影响，通过熔融温度。事实上，幂律流变模型参数的材料根据熔体温度变化。为了简化优化方法，使用的润滑近似模型等温幂律流体的流动。所使用的优化算法的大部分需要大量的模拟结果，这一事实增加了计算时间。这意味着，对于复杂的几何形状，挤压模具的分析所需的计算资源和时间是相当的。为了防止或至少减少这种缺点，Shahreza等。 14 提出了一个有趣的优化过程来实现均匀的出口流动的熔融聚合物的更新模，与在模头出口的横截面的各种厚度。模具出口速度是根据三维流动模拟的结果。的设计灵敏度分析，使用直接的鉴别方法，可以很容易地纳入一个FE的代码，计算目标函数的梯度。对于为此目的Sienz等。 15 提出了一种程序，使用优化异型材挤出模具的设计灵敏度分析。诺夫雷加等。 16 提出的异型材挤出模具，模具设计的基础上的有限体积方法和优化算法（SIMPLEX和启发式方法）的代码，以优化的流道。提出了两种优化策略的长度控制的基础上，第一个和第二个的厚度是根据。作者得出这样的结论：在挤压模具的长度控制的基础上进行了优化的厚度的基础上进行了优化的那些相比，具有更高的灵敏度的处理条件。模具设计在聚合物挤出过程中，穆等人最近提出的元模型优化策略。 17 提出了基于BP神经网络的优化策略，以及非支配排序遗传算法（NSGA-II），以优化挤压模具。NSGA-II进行评估所建立的神经网络模型，其目标函数的全局优化设计变量的搜索。用有限元模拟耦合模式国境的优化算法的软件，以确保最终产品的尺寸精度。为此目的采取相对的速度差和溶胀比的目标函数。这种优化工具（模式FRONTIER）是有趣的和易于使用的其他聚合物加工，如注塑机的性能优化 18 。在这项工作中，一个强大和有效的优化方法已发展为线涂装工艺，测试使用不同的策略。该方法包括耦合与几何体和网格生成器和3D计算的软件（Rem3D®）基于有限元方法，来模拟非等温的聚合物流的优化例程。根据出口流分布的均匀性作为目标函数采取的流量平衡原理建立的优化模型，在模具中的最大压力，得到的约束函数，和模头的结构参数的设计变量。能够预测，在可接受的计算时间，速度，压力，剪切场和温度场分布的有限元模拟。结果，通过目标和约束函数的计算。序列二次规划（SQP）算法来解决非线性约束的优化问题，优化设计变量的搜索。上述优化的方法也应用于钢丝衣架型模头的几何形状，范围广泛的材料和多个操作条件下，实现了良好的性能，以达到最佳的设计。实验结果表明，它是可行的，合理的。建模与仿真挤压过程中进行使用3D计算软件的功能实体REM3D®。从Navier-Stokes方程的不可压缩方程的流动方程的推导。不可压缩粘性流动的混合有限元方法。流求解器使用四面体单元与线性连续插值的压力和速度的速度和气泡富集。质量，动量和能量守恒方程，按照材料的行为，从速度，压力和温度场的确定。· （1）使用行为法得到的粘度对剪切速率和温度的关系。根据冬季和弗里茨 11 ，Schläfli 19 ，和Smith  13 ，出口速度分布的真正分销商依赖的粘度应变率曲线的斜率（幂指数）。这使得敏感的材料和流量变化的出口的速度分布。为了分析的效果的材料变化的分布的结果，选择了两种不同的聚合物（图2）。一种低密度聚乙烯（LDPE）引用LDPE 22D780，使用，因为它的流变行为。值得注意的是，牛顿之间的过渡区域的宽度（恒定的粘度）和幂律（线性）区域是重要的。引用的Lupolen 1812D，第二个材料被选中。在这种情况下，记录日志的粘度曲线是线性的（几乎没有牛顿或恒定粘度部）的粘度的温度依赖性是比较小的。图2。粘度的LDPE（22D780的Lupolen 1812D）“卡罗阿累尼乌斯法律。”Carreau-Yasuda/Arrhenius粘度模型是用来描述依赖的粘度（）的温度和剪切速率（）：· （2）同· （3）其中， 0，  ， T 文献，一个，和米为材料参数。从数据基地REM3D®商业软件（MatDB®）的两种聚合物（表1），得到的流变性质。两个其它的热塑性材料为实验选择，线性低密度聚乙烯（LLDPE“LLN 1004 YB”）和聚（氯乙烯）（聚（氯乙烯），PVC“FKS 910”）。物料 0  帕斯卡·秒米米s PaTref K KLDPE 22D78083140.15922406247311703Lupolen 1812D434340.347105554736156LDPE 22D780的Lupolen 1812D的流变参数见表1。 By symmetry, only one half die is modeled for a flow of 120 kg/h. This corresponds to a volume flow of 34,400 mm3/s. The entrance melt temperature (Tm) and wall die temperature (Tf) are Tm = 180°C and Tf = 185°C, respectively.OPTIMIZATION STRATEGYThis section describes the coat hanger melt distributor design problem. First, the design variables and the parameterization of the die manifold is explained and then the objective and constrained functions used in the optimization problem are defined. Finally, the optimization procedure is illustrated.The optimization method used in this work is based on the Kriging interpolation and SQP algorithm. The Kriging consists in the construction of an approximate expression of objective and constrained functions using evaluation points starting from a composite design of the experiment 20. Then, the approximated problem is solved using the SQP algorithm to obtain the optimal solution.Die Design VariablesFor a given die diameter (2R), a slit height (h), and an initial manifold of constant width (W) (Fig. 3), the manifold thickness H() and the contour lines yc() can be calculated by the mean of the analytical model presented by Winter and Fritz 10 as follows:· (4)· (5)Lebaal et al. 12 already showed the limitations of this analytical model. However the authors 12 note, that, for a geometry obtained using this model, the material has a weak influence on the exit velocity distribution.Figure 3. Coat-hanger distribution system. (a) and optimization variables W(y), H() (a) andW(y), H(y) (b).Within this work, we want to obtain some die geometries that will be machined afterward. Indeed, they are very often subject to geometrical requirements related to the manufacturing process. Within this framework, during the optimization procedure, several geometrical constraints dependent on the manufacturing process and to the tool geometry are applied.In our case, these geometrical requirements imposed by the machine tools are: the tool cutting edge radius (RF) and diameter (D). The manifold will be milled by a tool of diameter 8 mm. This implies that the minimal manifold width Wmin should not be lower than 8 mm. The second requirement is the tool cutting edge radius RF = 3 mm, which will be taken into account during the milling of the part geometry.Also, other geometrical limitations related to the tooling, which must be adapted to the optimal die. To achieve this goal, several geometrical constrained must be imposed (Fig. 4). The width of entry Wentry must be equal to 20 mm; the maximum length (y) of the manifold should not exceed 85 mm. The overall length of the die is 95 mm. The overall length of the flow before the flux separator is of 112.5 mm. To obtain a length of the manifold which does not exceed the imposed length of 85 mm, the manifold contour lines is calculated for a constant width of W = 10 mm.Figure 4. Sketch of extrusion tool (a) and coat-hanger distribution system (b).For a diameter of 55 mm, a slit height of 3 mm and an initial manifold of constant width, the contour lines yc() and the thickness variation H() of the manifold are calculated starting from Eqs. 4 and 5.During the optimization procedure, the external contour lines of the die (determined by the initial parameters) remain constant. Consequently, two variables will be optimized to ensure better exit velocity distribution: manifold thickness and manifold width variation (Fig. 3).Two cases are proposed to optimize the wire coat hanger melt distributor. In the first (case 1) the manifold thickness is varying linearly along the die circumferenceH():· (6)The constants c0, c1 are determined by the following boundary conditions:· (7)In the second (case 2), H varies linearly along the length of the die (H(y) as follows:· (8)with: h being the slit die and Hk the manifold thickness at the die entrance. This second variable can vary during the optimization procedure as follows: 5  Hk  15 mm.For the two cases, during the optimization procedure, the manifold width (variable W) varies linearly according to the die length (y). The entrance manifold width must be equal to Wentry = 20 mm and at the exit it should not be lower than the tool machining diameter. The latter parameter can vary during the optimization procedure and is limited by 8  Wk  20 mm.· (9)where P = 1 y is the polynomial basis function, and  are the unknown coefficients that are determined by the boundary conditions:· (10)One important need is to have a design process which is less dependent on personal experience. To automate the optimization procedure and to save time, a die design code has been developed in MATLAB®. This code carries out the automatic search for the flow channel geometry and allowing the CAD to be processed and the die geometry to be changed automatically. From Eqs. 4 and 5, the manifold contour line is obtained. Then, with the optimization variables, the manifold thickness and width variations independently of the external contour line are obtained. From the manifold contour line, width and thickness, a three-dimensional mesh of the coat hanger melt distributor is generated.Objective and Constrained FunctionsSince the primary function of the wire coat-hanger melt distributor is to produce a uniform flow distribution across the die, this also means to achieve the minimum velocity dispersion (E(x). The objective function is a positive exit flow uniformity index that becomes zero for perfect uniformity. Other considerations include the limitation of pressure to the one obtained by the initial geometry; this condition is translated by a constrained function (g(x).The optimization problem is defined as follows:· (11)where (J(x) being the normalized objective function, is function of the vector of design variables (x) and is obtained with the help of the velocity dispersion (E(x), defined as follows:· (12)and E0 and P0 are respectively the velocity dispersion (dimensionless velocity uniformity index) and the pressure in the initial die, which is given by the initial optimization parameters (Table 2), N the total number of nodes at the die exit in the middle plane, vi the velocity at an exit node, and v the average exit velocity defined as:· (13)The constrained function (g(x) is selected in a way to be negative if the pressure is lower than the pressure obtained by the initial die design (the pressure must be lower than the initial pressure).Optimization resultsInitialCase 1 W, H(x)Case 2 W, H(y)CPU time 18h4018h16Iterations033Objective function f10.1340.14Improvement of the velocity distribution %-8786Constraint function P/P010.920.97Global relative deviation E %19.772.652.77Global relative deviation of the average velocities E115.2514.6813.2Variable W mm208.038Variable H mm710.367.23Table 2. Summary of the optimization resultsOptimization ProcedureTo find the global optimum parameters with the lowest cost and a good accuracy, the Kriging interpolation, described in the next section, is adopted and coupled with SQP algorithm. The Kriging interpolation consists in the construction of an approximate expression of the objective and constraint functions (Eq. 11), starting from a limited number of evaluations of the real function. In this method, the approximation is computed by using the 15 evaluation points obtained by composite design of experiments.The SQP algorithm is used to obtain the optimal approximated solution which respects the imposed nonlinear constraints. Since the successive evaluations of the approximated functions does not take much computing time, once the approximated objective and constraint functions are built, and to avoid falling into a local optimum, an automatic procedure is used which allows to resolving the optimization problem using SQP algorithms, starting from each point of the experimental design. Then, the best approximated solution among those obtained by the various optimizations is taken into account.After that, successive local approximations are built, in the vicinity of the optima by taking into account the weight function of Gaussian type, the aim of the weight function is to slightly change the interpolations and makes the approximations more accurate locally, around the best optimum. The iterative procedure stops when the successive optimum of the approximated function are superposed with a tolerance = 106. Finally, another evaluation is carried out to obtain the real response in the optimization iteration.An adaptive strategy of the search space is applied to allow the location of the global optimum. During the progression of the procedure, the region of interest moves and zooms by reducing the search space by 1/3 on each optimum. In addition, an enrichment of the interpolation is made by recovering responses already calculated, and which are located in the new search space. The iterative procedure is stops when the successive points are superposed with a tolerance = 103.Kriging InterpolationThe Kriging interpolation is used in many works 21, 22, to approximate a complexes function effectively. This method is applied in this work to approximate the objective and constraint functions in an explicit form, according to the optimization variables. The approximated relationship of the objective and constraint function can be expressed as follows:· (14)with, p(x) = p1 (x), . . . , pm (x)T, where m denotes the number of the basis function in regression model, a = a1, . . . , amT is the coefficient vector the x is the design variables,  (x) is the unknown objective or constraint interpolate function, and Z(x) is the random fluctuation. The term pT (x)a in Eq. 14 indicates a global model of the design space, which is similar to the polynomial model in a moving least squares approximation. The second part in Eq. 14 is a correction of the global model. It is used to model the deviation from pT (x)a so that the whole model interpolates response data from the function.The output responses from the function are given as:· (15)From these outputs the unknown parameters a can be estimated:· (16)where P is a vector including the value of p(x) evaluated at each of the design variables and R is the correlation matrix, which is composed of the correlation function evaluated at each possible combination of the points of design:· (17)· (18)A Gaussian type weight function with a circular support is adopted for the Kriging interpolation expressed as follows:· (19)where  is the distance from a discrete node xi to a sampling point x in the domain of support with radius rw, and c is the dilation parameter.  is used in computation.The second part in Eq. 14 is in fact an interpolation of the residuals of the regression model pT(x)a. Thus, all response data will be exactly predicted; is given as:· (20)where rT is defined as follow:·