Simulation of heat transport in textiles inspired by polar bear fur

The demand for weight-efficient thermally insulating materials is particularly acute due to changing climate conditions and persistent energy challenges.^1,2 Textiles, in particular, are widely employed for their thermally insulating properties in fields such as performance clothing,^3,4 shelters,^5–7 medical dressings,^8,9 and cold-weather outerwear.^10,11 Such textiles are generally evaluated based on their weight, cost, appearance, and thermal insulation.^2,12–15 In the drive to improve these metrics, researchers have turned to biomimicry, a field where motifs found in nature are adapted to become synthetic structures.^16–30 In the context of thermal insulation, the polar bear is a particularly inspiring natural example, as these animals maintain a body temperature of 36.9 °C with minute fluctuations of ∼0.5 °C even in environments as cold as −40 °C.^31–38 A number of factors have been implicated in this feat, including thick layers of subcutaneous fat,³² the large thickness of their fur,³³ the ability of this fur to be transparent to thermal radiation, allowing such radiation to be absorbed by the skin,^34–36 and the hollow structure of the fibers in their fur.^37,38 Among these adaptations, the hollow fibers of their fur stand out as particularly intriguing for textile design. Further evidence of this adaptation’s utility is that it is also seen in other cold-climate mammals such as reindeer, caribou, and moose.^39–42 However, the specific degree to which the hollow nature of the fur contributes to its thermal insulation has not been reported, nor have the thermal properties of textiles based on hollow fibers been analyzed in detail. Melt spinning has been used to make hollow polyester fibers,⁴³ and these have been experimentally shown to have improved thermal resistance relative to solid-core fibers in non-woven fabrics.⁴⁴ However, the main commercial avenue for hollow-core fibers has been filtration due to their high surface area-to-volume ratio.⁴⁵ Given recent innovations in advanced fiber design and extrusion techniques,^46–52 fundamental knowledge and design heuristics are needed to guide the development of advanced textiles.

Here, we analyze the thermal insulation of textiles based on fibers with hollow and solid cores to determine the role of the hollow core in their thermal properties. Initially, a parametric model is constructed to represent textiles with varying core diameters and layers. Next, a series of multiphysics simulations are performed to determine the effective medium thermal conductivity of each textile geometry. A notable result of this study is that textiles based on hollow-core fibers conduct significantly less heat than their solid-core counterparts, with fibers with a core-to-fiber diameter ratio of 0.95 exhibiting a 33% reduction in thermal transport. A more striking comparison emerged from considering multi-layered textiles. For example, multi-layer hollow-core textiles exhibited a four-fold decrease in heat flux relative to single-layer solid-core textiles with the same mass per area. These simulations highlight that hollow-core fibers are well suited for applications in which gravimetric, rather than volumetric, thermal insulation is a priority, as is the case for many clothing applications.

The unit cell had a cross-sectional area of 4D × 4D, which is as closed-packed as possible, preserving the same fiber density in both lateral dimensions. This unit cell was selected to preserve the oscillatory nature of the weft’s path while maintaining contact between the warp and weft, thus providing a realistic depiction of fibers in a textile.

The modeled textile was used as the basis for a simulation of heat transport [Figs. 1(b) and 1(c)]. The top and bottom boundaries of this region were set to constant temperatures to model thermal diffusion in a cold environment such as the natural habitat of the polar bear. In particular, the upper boundary was held at a constant temperature of 253.15 K to represent free air. The bottom boundary was held at 310.15 K to represent the physiological temperature of mammal skin. Periodic boundary conditions were applied to all other sides of the simulation region to simulate the behavior of a continuous textile. Air was used to fill the interstitial spaces around and inside the fibers. The thermal conductivities of air and the fiber were set to 0.025 and 0.24 mW/mm⋅K (value taken for cotton fiber),⁵³ respectively. We assumed that in this thin boundary layer near the textile, free convection was negligible,⁵⁴ which was confirmed using multiphysics simulations. The overall study volume was 0.512 mm³, with a length of 0.8 mm, a width of 0.8 mm, and a height of 0.8 mm. A constant fiber diameter of 0.2 mm was selected. Finite element analysis (FEA) simulations of heat transfer were carried out using COMSOL Multiphysics using the steady-state thermal transport module. The unstructured tetrahedral mesh was constructed using the built-in physics controller with a minimum element size of 8 μm and a maximum element size of 64 μm. The properties of the model and the simulation are detailed in Table I.

In order to estimate the effect of porosity on the magnitude of thermal conductivity, we simulated the heat transport across textiles composed of fibers with solid cores and textiles composed of fibers with hollow cores [Fig. 2(a)]. Specifically, we found that solid-core fibers exhibited a heat flux Q_c = 1.46 mW/mm² under the simulation conditions, while fibers with d/D = 95% exhibited Q_c = 0.98 mW/mm², showing a ∼33% reduction in heat transport. To justify the use of steady-state thermal simulations, we perform transient thermal transport analysis and find that all structures reach steady-state conditions in <16 ms [Fig. 2(b)]. To explore this process in more depth, we repeated this simulation while sweeping d from 0.02 to 0.19 mm in steps of 0.01 mm. The transition from large heat transport with solid cores to smaller heat transport with hollow cores was found to be monotonic and smooth, with the most dramatic change at d/D → 1 [Fig. 2(c)]. This result makes it clear that hollow fibers result in lower heat transfer, motivating their study for use in harsh weather environments. The reduction in heat transport with increasing porosity can be attributed to the hollow cores interrupting the flow of heat through the fibers.

Effective medium theory can be used to understand and quantify the degree to which being hollow improves fiber performance relative to other possible configurations of materials. Specifically, we may compute an effective-medium thermal conductivity k_e that is based on Q_c = k_eΔT/h, where h is the height of the simulation domain and ΔT is the temperature difference across the domain. For this analysis, we employ h = 0.6 mm, so that we only consider heat flow across the region with fibers. Based on this, we find that the solid core fiber exhibited k_e = 0.077 mW/mm⋅K, while the textile with d/D = 95% exhibited k_e = 0.030 mW/mm⋅K. Therefore, the total thermal transport can be reduced by more than a factor of two by making the fibers hollow cores.

All the observed structures exhibited E_f ∼ 50%, showing that being hollow does not make the architecture intrinsically more insulating per unit mass. In fact, lower Φ_f, which corresponds to larger hollow cores, is slightly less efficient. Instead, the main reason why these are better insulators is that more of their total volume is made up of air.

To test the hypothesis that textiles consisting of layers of hollow fibers will further reduce thermal transport, we developed computational models of textiles with two, three, and four layers of warp and weft.⁵⁹ The practicality of multi-layer woven structures is supported by prior work realizing such structures using weaving.^60,61 These computations revealed that Q_c decreased with increasing textile thickness. For instance, at d/D = 50%, we found Q_c = 1.367 mW/mm² for a single layer and Q_c = 0.859 mW/mm² for a two-layer textile. The 37% percent decrease in Q_c while the thickness doubled is a smaller decrease than one would expect if the textile was behaving as a simple effective medium. This is because the air boundary layer above the textile has a constant thickness. This result emphasizes the common principle of conductive thermal insulation, which is that air has the lowest thermal conductivity in the system, so engineering the system to have more entrapped air is a major benefit. To explore how this trend depends on d/D and the number of layers, we repeated these computations with up to four layers with d varied from 0% to 95% of D [Fig. 3(a)]. As expected, larger d or more layers decreased thermal transport.

While it is clear that hollow fibers or multiple layers improve thermal insulation, this is but one consideration when designing textiles. Another key consideration is the mass per unit area. To quantify this, we define the textile mass per unit skin area M_t. It is not a priori clear that trends in M_t will have a simple connection to trends in Q_c, as adding layers or decreasing d will affect M_t. Therefore, we compute M_t for each textile and plot Q_c vs M_t [Fig. 3(b)]. When visualized together, it is clear that adding layers vastly reduces thermal transport. Perhaps more strikingly, the hollow core fibers are sufficiently light that multiple layers can obtain lower heat transport than a single layer of solid core textile without increasing the mass. In particular, a four-layer fabric with d/D = 0.85 has the same M_t as a single layer textile based on a single fiber while being four times less thermally conductive. These results highlight the opportunity of hollow-core fibers to provide a path to thermal insulation that is highly efficient per unit mass.

In order to explore the role of fiber material, we repeated this analysis by varying fiber thermal conductivity k_f. Fiber k_f was set to tabulated values for common textile materials.⁶² The resulting thermal transport increased monotonically with k_f [Fig. 3(c)]. For fibers with very small or very large k_f, being hollow did not affect thermal transport, as in both limits, the thermal conductivity of air limited heat flow. For intermediate values of k_f, hollow fibers exhibited markedly reduced thermal transport. This finding suggests that for fibers with k_f ∼ 0.75 mW/mm⋅K, the impact of fibers being hollow is especially high in terms of managing thermal transport.

To objectively compare the thermal insulation performance of all geometries that were considered, we use a figure of merit (FOM) defined using FOM = ΔT/(Q_C⋅M_T). This reflects the thermal insulation provided per unit mass of a given textile and monotonically increases with increasing d/D [Fig. 3(d)]. Interestingly, the number of layers considered has a weak effect on this FOM, as the added layer increases thermal insulation and mass in nearly equal measure. Comparing the FOM computed for textiles based on hollow cotton fibers with tabulated experimental values for a variety of textiles,^58,63 it is clear that being hollow can increase this FOM by >10×. This is a particularly exciting prospect when considering performance textiles such as those based on solid-core Kevlar-PBI/Nomex fibers as a starting part for improvement.

In summary, this work utilized extensive finite element analysis to determine the role that hollow cores have on the thermal transport of woven textiles. First, we found that making the fibers hollow offers a modest reduction in thermal transport while slightly reducing the effective medium efficiency. In other words, the textile did not insulate better than would be expected given that more of the more thermally conductive fiber had been replaced by less thermally conductive air. This observation points to hollow cores being primarily useful as a tool for light-weighting insulating materials. The main quantitative result of this study is that transitioning to hollow-core fibers can reduce thermal transport by 33% and decrease mass density by >90% relative to solid-core fibers. We anticipate that the dual benefits of better thermal insulation and lighter mass will motivate further study into how to efficiently realize hollow-core structures for thermal insulation applications. To emphasize this point, we compute that four-layer textiles based on hollow-core fibers can reduce the overall thermal conductivity by more than a factor of four without adding to the mass per unit area. As the textile architecture studied here can be readily woven using looms and associated techniques, there are no fundamental barriers to experimentally realizing the structures discussed here on a large scale. Given the high importance of insulation normalized by mass in, for example, performance clothing or thermal insulation in aerospace, these results motivate the study of advanced fiber geometries to incorporate structures such as hollow cores.

The authors acknowledge support from the Toyota Research Institute and the Department of Mechanical Engineering at Boston University. The authors acknowledge Dr. Timothy J. Lawton for helpful discussions.

The authors have no conflicts to disclose.

A.D.A.: Investigation, software, analysis, and writing. K.A.B.: Conceptualization, writing, and supervision.

Adedire D. Adesiji: Formal analysis (equal); Investigation (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Keith A. Brown: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon request.