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首页> 外文期刊>International Journal of Heat and Mass Transfer >Shape factor and shape optimization for a periodic array of isothermal pipes
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Shape factor and shape optimization for a periodic array of isothermal pipes

机译：等温管道周期阵列的形状因子和形状优化

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We address the problem of two-dimensional heat conduction in a solid slab embedded with a periodic array of isothermal pipes of general cross-section. The objective of this work is two-fold: to develop a numerical procedure through which we can obtain the shape factor associated with a given configuration and, to develop a numerical shape optimization algorithm through which we can compute shapes that extremize the transport rate. The shape factor is obtained by first transforming the periodic array of pipes into a periodic array of strips using the generalized Schwarz-Christoffel transformation and, subsequently, by developing an integral equation of the first kind for the temperature gradient using the boundary element method. The integral equation is solved both numerically and analytically/asymptotically with excellent agreement between the results. The shape optimization problems, which are formulated with respect to the parameters of the generalized Schwarz-Christoffel transformation, are solved numerically to compute the shape that maximizes the cross-sectional area and the shape that minimizes the perimeter of the cross-section, given the shape factor and the distance between two consecutive pipes. It is inferred that the problems are adjoint to the transport rate minimization and transport rate maximization problems, respectively. The optimal shapes are computed numerically and validated with available analytical and numerical results for a single pipe. Furthermore, motivated by the analytical result, we propose a parametric set of equations that describe well the optimal shapes. The versatility of the Laplace equation suggests that similar formulations have applications in continuum mechanics, electricity, hydraulics and drug reduction.
机译：我们解决了在固体平板中二维热传导的问题，该平板中嵌入了具有大截面的等温管的周期性阵列。这项工作的目的有两个：开发一个数值程序，通过该程序我们可以获取与给定配置相关的形状因子；开发一个数值形状优化算法，通过该算法我们可以计算出最大化运输速率的形状。首先通过使用广义Schwarz-Christoffel变换将管道的周期性阵列转换为带状的周期性阵列，然后通过使用边界元方法针对温度梯度开发第一类积分方程，来获得形状因子。积分方程在数值上和解析/渐近上都得到了求解，结果之间具有极好的一致性。形状优化问题是针对广义Schwarz-Christoffel变换的参数制定的，可以通过数值方式求解，以计算使横截面面积最大的形状和使横截面周长最小的形状。形状因数以及两个连续管道之间的距离。可以推断，这些问题分别与最小化传输速率和最大传输速率相关。对最佳形状进行数值计算，并用单个管道的可用分析和数值结果进行验证。此外，根据分析结果，我们提出了一组参数方程，可以很好地描述最佳形状。拉普拉斯方程的通用性表明相似的公式在连续力学，电力，液压和减少毒品方面具有应用。

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