Computational Design of Actively-Cooled Microvascular Composites for High Temperature Applications Soheil Soghrati⇤, Ahmad R. Najafi, Kevin H. Hughes, Piyush R. Thakre,C. A. Duarte, Nancy R. Sottos, Scott R. White, and Philippe H. Geubelle Beckman Institute of Advanced Science and Engineering University of Illinois at Urbana-Champaign, Urbana, IL 61801 A novel interface-enriched generalized finite element method1, 2 is employed to perform the computational design of actively-cooled microvascular woven composites to be used as part of a multi-layered, multi-material skin for hypersonic vehicles. The numerical scheme is able to capture accurately and efficiently the gradient discontinuity along the fluid/solid interface and create a virtual model of the composite’s microstructure without using meshes that conform to the geometry of the problem. Determining the optimal configuration of the embedded microchannels that most efficiently contributes to active cooling of the material can be then accomplished with a single non-conforming mesh, which significantly reduces the computational cost of the analysis.
I.
Introduction
Several problems in materials science and engineering such as those arising in heterogeneous and composite materials involve C 0 -continuous fields, where the gradient discontinuity arises along the material interface. An example is the conjugate heat transfer problems encountered in actively-cooled microvascular materials containing an embedded network of microchannels.3–5 Recent advances in manufacturing techniques have allowed embedding microchannels with diameters ranging from 200 to 500 µm in 3D woven composites by weaving and then evacuating sacrificial PLA fibers in these materials.6 Figure 1 illustrates the structure of the sinusoidal-shaped embedded microchannels with a 500 µm diameter in a 3D woven glass/epoxy composite specimen created by the sacrificial fiber technique. These microchannels are then employed for active cooling of the domain by circulating a coolant inside the material and removing the heat from the domain. The design of such materials requires an accurate and efficient evaluation of the temperature field inside the domain to determine the optimum configuration of the embedded microchannels with respect to both the thermal and structural properties of the material. However, using the standard FEM for this class of problems necessitates the creation of many finite element meshes that conform to the microstructure of the material, which is a laborious and expensive task due to the complex microstructure of these systems. The difficulties with creating conforming meshes for the standard FEM motivates the use of more advanced numerical techniques to evaluate accurately the thermal impact of the embedded microchannels on the composite part with non-matching meshes. One of the most popular techniques in this line of work is the Generalized Finite Element Method (GFEM), for which discontinuities in the gradient field are captured through enriching the nodes of elements intersecting with the corresponding phase interfaces.7, 8 In the present study, we employ a recently introduced interface-enriched GFEM (IGFEM) formulation,1, 2 in which the enrichments are not attached to the nodes of the elements traversed by the interface, but with the points created by the intersection of the phase interface with the element edges. The unique feature of ⇤ PhD Candidate, Department of Civil and Environmental Engineering. 233 Talbot Lab, 104 S. Wright Street, Urbana, IL 61801, USA. Tel.: +1-217-819-8815, Email:
[email protected].
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Figure 1. Micro-CT image of sinusoidal-shaped microchannels embedded in a 3D woven glass fiber/epoxy matrix composite created with the sacrificial fiber manufacturing technique.
the IGFEM is the evaluation of the enrichment functions in this method, which are simply obtained from a linear combination of the standard Lagrangian shape functions of the integration elements. This formulation has several implementation advantages over conventional GFEM such as removing the difficulties associated with applying Dirichlet boundary conditions, the smaller number of enrichment degrees of freedom, and the evaluation of enrichment functions for special cases where multiple interfaces cut the edges of an element.
II.
3D IGFEM formulation
The formulation of IGFEM for 2D problems and the derivation of the enrichment functions with a detailed convergence study for solving thermal problems. i.e., the convection-di↵usion equation, is presented in [1]. In this section, we briefly explain the extension of the IGFEM to 3D problems, discretized with tetrahedral elements, and show the application of this method for evaluating the thermal response of an actively-cooled n m microvascular material. Denoting by {Ni (x) : x ! R}i=1 and { i (x) : x ! R}i=1 the set of n Lagrangian shape functions and m enrichment functions, respectively, the approximation of the solution field with the IGFEM is given by n m X X uh (x) = Ni (x) u¯i + ˆi , (1) i (x) u i=1
i=1
where u¯i and u ˆ are the standard and generalized degrees of freedom, respectively. The first term in Eqn. (1) is the standard FEM portion of the approximation, while the second term contributes to capturing the gradient discontinuity in elements cut by the phase interface by using appropriate enrichment functions. Unlike conventional GFEM, the generalized degrees of freedom in IGFEM are attached to the interface nodes, i.e., nodes created by intersecting the phase interface with the edges of the nonconforming elements. The enrichment functions associated with each interface node is evaluated by a linear combination of Lagrangian shape functions of the sub-elements created for the accurate quadrature, such that they are able to model the discontinuity in the gradient field. Figure 2 illustrates two case scenarios of a material phase interface that cuts the edges of a four-node tetrahedron element and the resulting integration elements created there. The enrichment functions corresponding to Figures 2(a) and 2(b) are respectively expressed as follows: (a) (b)
1 1
(1)
(2)
= N1 + N1 , (1) (2) = N1 + N4 ,
2 2
(1)
(2)
= N2 + N4 , (1) (2) = N2 + N5 ,
3 3
(1)
(2)
= N4 + N2 , (1) (2) = N3 + N6 ,
4
(1)
= N5
(2)
+ N5
(2)
It must be noted that the enrichment functions given in 2 may need to be multiplied by a scaling factor for situations where the interface node is too close to one of the nodes of the original mesh and sharp gradients appear in the sti↵ness matrix.2
III.
IGFEM applications
The application of IGFEM for evaluating the temperature field in an actively-cooled microvascular material is presented in Figure 3, where heat flux q = 4000 W/m2 is applied along the bottom surface of the 2 of 6 American Institute of Aeronautics and Astronautics
4
3 1
3
2
2 4
6
1
5
=
= 5 4
2 3
2
1
(2)
6
4 1
3
(1)
(1)
1
2
4
5
6 3
(2)
1 3
2
(a)
(b)
Figure 2. Evaluation of the enrichment functions in the IGFEM: two di↵erent scenarios of creating the integration subelements (which are a combination of tetrahedron and wedge elements) and the numbering of these elements used for constructing the enrichment functions.
10 ⇥ 2.7 ⇥ 5.2 mm epoxy domain ( = 0.19 W/mK), while other surfaces are subject to air with conductivity coefficient h = 7 W/mK and ambient temperature u1 = 20 C. The configuration of the embedded channels in the epoxy matrix with diameter D = 500 µm is depicted in Figure 3(b). A sinusoidal-shaped curve with amplitude A = 2 mm and wavelength = 5 mm describes the centerline of the microchannels, where water with w = 0.6 W/mK, ⇢ = 1000 kg/m3 and cp = 4185.5 J/kgK is used as the coolant. The middle channel carries a flow rate of Q = 2 ml/min with entrance temperature Te = 20 C, while the two other microchannels carry Q = 1.5 ml/min with Te = 15 C in the opposite direction. To evaluate the temperature field in the domain, convection-di↵usion equations are solved with the assumption of a fully-developed velocity profile in the microchannels.1 The solution field obtained with IGFEM solver is illustrated in Figures 3(c) and 3(d), where a structured nonconforming mesh is implemented to discretize the domain. Figure 3(c) shows the temperature field over three of the outer surfaces of the domain, for which the thermal impact of the coolant circulating in the three microchannels is clearly observable. The temperature field inside the domain, along a a vertical plane cut at y = 1.35 mm is depicted in Figure 3(d), which illustrates the discontinuity in the gradient field along the surface of the microchannel due to the thermal conductivities mismatch between the fluid/solid phases. The IGFEM can be then used as a great tool to determine the optimal configuration of the microchannels without the need of creating new conforming FEM meshes during the design process.9 For example, consider the microvascular composite system with several sinusoidal-shaped embedded microchannels as shown in Figure 4(a). The adjacent microchannels in this figure have similar configurations and thus we can further confine the computational domain by isolating a single microchannel in a sub-domain with periodic boundary conditions as illustrated in Figure 4(b). This figure also depicts the domain’s dimensions, applied thermal loads, and boundary conditions for this problem, where the temperature along the top surface is fixed at T¯ = 20 C. The coolant used for active cooling is water with thermal properties similar to those of the previous example. Homogenized thermal properties xx = 0.47, yy = 0.45, and zz = 0.40 are adopted for the microvascular composite. The optimal microchannel’s configuration is then determined such that it maintains an allowable temperature of Tall = 20 C in the microvascular material. To facilitate the design process, we define the required thermal efficiency in the actively-cooled composite as ⌘T = 1
Tmax Tref
Tall , Tall
(3)
where Tmax is the maximum temperature of the actively-cooled system and Tref is the maximum temperature in the absence of the coolant flow. The design parameters shown in Figure 4(b) consist in the amplitude A and the wavelength of the 3 of 6 American Institute of Aeronautics and Astronautics
2.7 m 10 m
mm
Te
in 0 C l/m =2 2m = Q
5.2 mm
xy Epo /mK W .19 0 = 2
y
z
4 q=
W 000
/m
in l/m .5 m C 1 Q = = 15 Te
x
(a)
y
z
x
(b)
u(thermal C) 40.9 34.4 27.9 21.4 y
z
y x
14.8
(c)
z
x
(d)
Figure 3. (a) Geometry and boundary conditions for an actively-cooled epoxy fin and (b) the embedded sinusoidal-shaped microchannels; (c) Temperature field on the surfaces and (b) inside the domain, cut along the plane y = 1.35 mm. Figures adopted from [2].
(a)
(b)
Figure 4. (a) Microvascular composite specimen with sinusoidal-shaped embedded microchannels; (b) Computational domain, where the material associated with a single microchannel is adopted with periodic boundary conditions.
microchannels, space between microchannels’ centerline w, and flow rate Q. We can then determine the allowable heat flux for each microchannel configuration that maintains the expected thermal efficiency in the actively-cooled system. Figure 5(a) illustrates the temperature field corresponding to di↵erent microchannels’ configurations, all solved by IGFEM using the same non-conforming mesh. Figures 5(b) and 5(c) show parts of the results that can be implemented to for designing the embedded microchannels. As shown in Figure 5(b), the thermal efficiency of a particular microchannel’s configuration is independent of the value of heat flux and thus the maximum allowable heat flux for each configuration can be determined according to the minimum required efficiency (illustrated with the red line). In other words, the intersection of the thermal
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efficiency associated with di↵erent microchannel’s configuration and the minimum required efficiency curve determins the allowable heat flux. Such design curves can be repeated for other values of the flow rate to determine the best thermal efficiency for di↵erent flow rates, as shown in Figure 5(c).
(a)
(b)
(c)
Figure 5. (a) Temperature field obtained with IGFEM solver in the actively-cooled composite for di↵erent microchannes’ configuration (temperatures in C); (b) Thermal efficiency corresponding to four di↵erent microchannels; (c) maximum value of the thermal efficiency in the system for di↵erent values of flow rate.
We have repeated similar analysis using the computational domain depicted in Figure 4(b) but replaced the boundary conditions along the top edge with convective boundary conditions (h = 28 W/mK and u1 = 20 C). This analysis shows that regardless of the applied thermal loads and flow rate, straight microchannels are always the best configuration for this new boundary conditions because they not only yield the lowest temperature in the composite, but also minimize the pressure drop and void volume fraction. The results presented in Figure 6(a) shows the thermal efficiency of the actively-cooled composite for di↵erent flow rates. Compared to similar results with fixed temperature along the top edge, illustrated in Figure 5(c), the current results demonstrate a higher thermal efficiency for similar values of the flow rate. Figure 6(b) shows variation of maximum temperature in the system and the average temperature of the coolant at the outlet versus variations of the flow rate.
IV.
Conclusions
The application of the IGFEM for evaluating the thermal response of actively-cooled microvascular materials has been presented. The gradient discontinuity present along the fluid/solid phase interfaces can be accurately captured by the IGFEM without using conforming meshes for the discretization of the problem domain. Thus, the IGFEM can be employed as a great tool for determining the optimal configuration of the embedded channels in microvascular materials including actively-cooled woven composites, without the need of creating conforming meshes for each channel configuration.
Acknowledgments This work has been supported by the Air Force Office of Scientific Research Multidisciplinary University Research Initiative (Grant No, FA9550-09-1-0686).
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(a)
(b)
Figure 6. (a) Maximum value of thermal efficiency vs. flow rate in the straight microchannel; (c) maximum temperature and average outflow temperature vs. flow rate.
References 1 Soghrati, S., Arag´ on, A. M., Duarte, C. A., and Geubelle, P. H., “An interface-enriched generalized finite element method for problems with discontinuous gradient fields,” International Journal for Numerical Methods in Engineering, Vol. 89, No. 8, 2012, pp. 991–1008. 2 Soghrati, S. and Geubelle, P. H., “A 3D Interface-Enriched Generalized Finite Element Method for Weakly Discontinuous Problems with Complex Internal Geometries,” Computer Methods in Applied Mechanics and Engineering, Vol. 217-220, 2012, pp. 46–57. 3 Toohey, K. S., Sottos, N. R., Lewis, J. A., Moore, J. S., and White, S. R., “Self-healing materials with microvascular networks,” Nature Materials, Vol. 6, 2007, pp. 581–585. 4 Arag´ on, A. M., Wayer, J. K., Geubelle, P. H., Goldberg, D. E., and White, S. R., “Design of microvascular flow networks using multi-objective genetic algorithms,” Computer Methods in Applied Mechanics and Engineering, Vol. 197, 2008, pp. 4399– 4410. 5 Olugebefola, S. C., Arag´ on, A. M., Hansen, C. J., Hamilton, A. R., Kozola, B. D., Wu, W., Geubelle, P. H., Lewis, J. A., Sottos, N. R., and White, S. R., “Polymer microvascular network composite,” Journal of Composite Materials, Vol. 44, No. 22, 2010. 6 Esser-Kahn, A., Thakre, P., Dong, H., Patrick, J., Sottos, N., Moore, J., and White, S., “Three Dimensional Microvascular Fiber-Reinforced Composites,” Advanced Materials, 2011. 7 Belytschko, T., Gracie, R., and Ventura, G., “A review of extended/generalized finite element methods for material modeling,” Modeling and Simulation in Materials Science and Engineering, Vol. 17, No. 4, 2009, pp. 043001. 8 Arag´ on, A. M., Duarte, C. A., and Geubelle, P. H., “Generalized finite element enrichment functions Generalized finite element enrichment functions for discontinuous gradient fields,” International Journal for Numerical Methods in Engineering, Vol. 82, 2010, pp. 242–268. 9 Soghrati, S., Thakre, P. R., White, S. R., Sottos, N. R., and Geubelle, P. H., “Computational modeling and design of actively cooled microvascular materials,” Submitted.
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